the mechanics of cell motility and a unifying theory for
TRANSCRIPT
The Mechanics of Cell Motility
and a Unifying Theory for
Characterizing Directed Motion
by
Alex Loosley
B. Sc, Simon Fraser University, Burnaby, BC, Canada, 2009
Sc. M., Brown University, Providence, RI, USA, 2011
A Dissertation submitted in partial fulfillment of the
requirements for the Degree of Doctor of Philosophy
in the Department of Physics at Brown University
Providence, Rhode Island
May 2015
c© Copyright 2015 by Alex Loosley
This dissertation by Alex Loosley is accepted in its present form
by the Department of Physics as satisfying the
dissertation requirement for the degree of Doctor of Philosophy.
Date
Jay X. Tang, Ph.D., Advisor (�)
Recommended to the Graduate Council
Date
Jonathan S. Reichner, Ph.D., Reader (⊥)
Date
James M. Valles Jr., Ph.D., Reader (‖)
Date
Robert A. Pelcovits, Ph.D., Reader (∴)
Approved by the Graduate Council
Date
Peter Weber, Ph.D, Dean of the Graduate School
iii
Vitæ
Alex J Loosley was born in Vancouver, British Columbia, Canada, on May 26,
1985 to Ellen B Loosley. Alex liked extreme sports while growing up, thrilling himself
by skateboarding, snowboarding, sailing, and playing ice hockey. At Templeton Sec-
ondary School, Alex accelerated through science and math courses (except biology),
actively participating in math competitions, and eventually graduating as the top
math student in his year.
During his college years at Simon Fraser University, Alex received a rigorous ed-
ucation in physics. In addition to learning from many fantastic professors (too many
to name), Alex benefited from being around many bright award winning classmates
including Cisco Gooding (NSERC CGS M and D, Quantum Gravity), Michael Mc-
Dermott, (NSERC CGS M and D, High Energy Theory), Joel Zylberberg (Fulbright
Scholar, Theoretical Neuroscience), and Karen Chan (NSERC CGS M and D, Theo-
retical Chemistry) to name a few. It was truly a special group.
Later, Alex moved to Providence, Rhode Island, to pursue a Ph.D. in Physics
under the guidance of Profs. Jay X. Tang and Jonathan Reichner. When not working
hard on his Ph.D. projects, Alex could be found refereeing ice hockey at the minor
and college levels, or racing sailboats in Newport, Rhode Island.
iv
An abbreviated C.V. and other resources can be found at:
http://www.brown.edu/research/labs/tang-biophysics/lab-members/alex-loosley
v
“It’s not poison, it’s clove.”
Xian speaking about die Clovenstoop
vi
Dedicated to My Family
(ABCDE, R)
vii
Acknowledgements
I would not be here writing this Ph.D. thesis without my research advisers Prof.
Jay Tang, Prof Jonathan Reichner, and Dr. Xian O’Brien. Jay, you have been there
for me from day 0. I will also be grateful that you brought me into the physics
department at Brown. Thanks also for many hours spent playing a key role in my
development as a scientist, from reading fellowship applications to supporting me
through a number of projects. Jonathan, I still have not finished that 4-D vector
analysis of god knows what that you asked me to do the first time you met me. You
are a fantastic translator of biology into layman’s terms, a subject which at one point
in my life appeared to be a forest too dense to navigate. To be able to speak to
somebody in personalized metaphors is a gift. Xian, I should have moved my office
to the division of surgical research near you long before I finally did to begin my fifth
year. Every young graduate student could benefit from being around good senior
graduate students and post-docs. You exemplify the role of a guide to a younger
researcher and I am a better researcher when I am around you. Thank you for always
being happy to heed my ideas and answer questions, no matter when I decided to
spontaneously pop into your office to present them. Thank you to all of my thesis
readers Profs. Jay Tang, Jonathan Reichner, James Valles Jr. and Robert Pelcovits.
I have been blessed with many good lab mates over my years as a graduate student.
My apologizes in advance if I miss anybody. In the Tang Lab and department of
physics, Angus McMullen, you have been an outstanding friend and lab mate with
viii
many qualities that I have tried emulate. I am sorry that Yinda and I almost killed
your future wife the day you met her. Dr. Jingjing Wang, thanks for the many fun
science conversations and being a great desk mate. Drs. Patrick Oakes and Hyeran
Kang, I have always looked up to you both. Thank you Ryan Handoko, Wei Li, and
Cole Morrissette for all the hard work you put into my projects. Mike Morse, Jun He,
Nelson Leung, Barbara Dailey, Sabina Griffin, and Mary Ellen Woycik, thank you. In
the Reichner Lab and in the division of surgical research, Dr. Meredith Crane (Dingo
Dingo Dingo), Dr. Angel Byrd, Courtney Johnson, Estefany Flores, Dr. Dipan
Patel, Dr. Liz Lavigne, Dr. Hilary McGruder, Patty Young, Tracy Monteforte and to
anybody else that I missed, thank you all. Kyle Glass and Prof. Sunil Shaw, thanks
for letting me mess around with your microscope and flow chamber. Thanks to Prof.
Christian Franck, Eyal Bar-Kochba, and Dr. Jennet Toyjanova. Jennet, thanks for
all the reagent care packages and helping me be a better presenter. Also, thanks to
past advisers and colleagues that helped me get to Brown, especially Prof. Eldon
Emberly.
The summer of 2012 was the best summer of my life, and completely transformed
my way of thinking science. That summer in Singapore at the Mechanobiology In-
stitute (MBI), I worked with many fantastic researchers doing nanopillar array force
spectroscopy. Feroz Musthafa, my mentor, worked with me from 9pm to 5am just so
I could stay in my home time zone. Thanks for fabricating all of my nanopillar arrays
and bringing me on many exciting motor cycle rides. Aneesh Sathe, you helped get
my experiments up and running, endowing me with the cell culture and microscopy
skills to succeed. You along with Feroz also brought me into your family introduc-
ing me to many people I will never forget: Sharvari Sathe, Schweta Pradip Jadhav,
Soumya Mohanty, Shelly Kaushik, Aarthi Ravichandran, Rohith Kutty, and Dipanjan
ix
Bhattacharya. Nils Gauthier, our experiments failed miserably except for that one
fibroblast. I wish I had more time to do science with you. Cheng-han Yu, you got me
to think of the cell membrane as the 2-D fluid that it is. Prof. Michael Sheetz, I was
always amazed that you would spontaneously take time out of your busy schedule to
see how I was doing. I now find myself hearing your voice as a read your manuscripts.
Many of your words of wisdom have become permanent pillars of my way of thinking.
Thanks also to Thomas Masters, Esther Anon, Bo Yang, Prabuddha Gupta, Alvin
Guo (hook’n me up with reagents day and night), Keiko Kawauchi (your HL-60 cell
line saved my summer research plan), Andrea and Christina (you taught me the im-
portance of coffee breaks), Joseph Tarango, Dacotah Melicher, Katelyn Goetz, Tanya
Gordonov, Hau Chan, Loretta Au and to anybody that I missed, there were so many
people that made an impact on my life that summer.
Outside of academia, I want to graciously thank my entire family. I have amazing
parents that have supported me every step of the way. Cisco Gooding, you helped
shape me as a physicist I am today. Callan MacKinlay, Racan Souiedan, Wendy
and Mitchell Parks, Angus and Jenica McMullen, Christina Miles, Marc Bertucci,
Kyle Helson, Lu Lu, Mingming Jiang, Simone Konig, I could go on. Thank you
fundamental constants of the universe for enabling this beautiful existence.
x
Preface
This dissertation covers nearly five years of cell mechanics and motility research
findings as a graduate student in the Department of Physics at Brown University.
Most of this work was done as part of a collaboration with Profs. Jay X. Tang
(Physics), Xian O’Brien (Surgical Research), and Jonathan S. Reichner (Surgical
Research). This collaboration was originally fostered by the work of previous graduate
student, Dr. Patrcik Oakes. Patrick laid the groundwork for many of the experiments
described in this dissertation and has been supportive from afar over the last five years.
Six chapters grouped into two parts are presented here to report on an analyti-
cal tool for characterizing directed motion, multiple spring models that recapitulate
the shape and movement of motile cells, and the effects of pharmacological inter-
vention on immune cell motility and mechanics. Chapter 1 features a review style
blanket overview of cell motility and mechanobiology. Part 1 incorporates Chapters
2 and 3 about analytical tools for characterizing cell motility and the mechanisms
of mechanosensitive neutrophil motility in 2D. Chapter 2 describes a novel analysis
tool for measuring the extend of honing in directed motion. This tool was derived
from random walk theory, coming together through many critical back and forth dis-
cussions between Xian, Jonathan, Jay, and myself. Both Xian and Dr. Kate Oakley
provided the experimental data that was used to test our tool. Chapter 3 shows the
application of a simplified version of this tool to probe the mechanism of substrate
stiffness mediated migration of neutrophils. Xian and Jonathan were the driving
xi
forces behind the work and much of this chapter is made up of text and figures from
our manuscript published in the Journal of Leukocyte Biology [1] (Copyright 2014 by
the Federation of American Societies for Experimental Biology). In Part 2, the scope
of this dissertation broadens with the additional consideration of cell mechanics in
describing cell motility. In Chapter 4, several toy models of cell motility using springs
and stick-slip adhesion as ingredients, are presented. Much of this work was inspired
by a Biophysics Journal Club presentation by Dr. Jingjing Wang, who did a presen-
tation about bipedal locomotion in crawling cells. This chapter contains a modified
version of our manuscript published in Physical Review Letters E [2] (Copyright 2012
by the American Physical Society, ). Chapter 5 contains much of the experimental
data I have collected over five years in collaboration with Xian, Jonathan, and Jay.
Several research assistants, namely Mr. Ryan Handoko, Ms. Wei Li, and Mr. Cole
Morrissette, helped with experiments and data analysis. Dr. Eliza Fox and Jonathan
Reichner were instrumental in setting up an IRB for the acquisition of neutrophils
from sepsis patients in the Rhode Island Hospital Intensive Care Unit. This lab work
would also not be possible without the general assistance of Dr. Angel Byrd and
(soon to be) Dr. Courtney Johnson. The original findings in this dissertation are
tied together in a final chapter containing unpublished data for future directions, and
closing remarks section.
Alex Loosley
Providence, Rhode Island
xii
Publications
• Alex J. Loosley and Jay X. Tang. “Stick-slip motion and elastic coupling
in crawling cells.” Phys. Rev. E 86: 031908 (2012)
• Xian M. O’Brien, Alex J. Loosley, Kate E. Oakley, Jay X. Tang, and
Jonathan S. Reichner. “Technical advance: introducing a novel metric, di-
rectionality time, to quantify human neutrophil chemotaxis as a function of
matrix composition and stiffness.” J. Leukocyte Biol. 95: 993-1004 (2014)
• Alex J. Loosley, Xian M. O’Brien, Jonathan S. Reichner, and Jay X. Tang.
“Describing Directionally Biased Paths by a Characteristic Directionality
Time.” submitted (2014)
xiii
Contents
Vitæ iv
Dedication vii
Acknowledgements viii
Preface xi
Publications xiii
List of Tables xx
List of Figures xxi
Chapter 1. Introduction 1
1. A General Overview of Cell Motility 2
2. Dissertation Outline 7
Part 1. Cell Kinematics 9
Chapter 2. Directionality Time: A New Theory for Characterizing Directed
Motion 10
Forward 11
1. Introduction 11
1.1. Commonly Used Analytical Tools for Characterizing Migration 11
1.2. Sampling Interval Dependent Metrics 12
xiv
1.3. Sampling Interval Independent Metrics 13
2. Results 15
2.1. Mean Squared Displacement Analysis 15
2.2. Deriving the Directionality Time Model 16
2.3. Computational Modeling to Test Robustness 22
2.4. Application to Real Data 27
2.5. Neutrophil Chemotaxis 30
3. Discussion and Future Direcitons 33
4. Supplementary Material 36
4.1. Methods 36
4.2. Appendix A: Analytical Modeling 36
4.3. Appendix B: Deviations Caused by Variances and Ergodicity 46
Chapter 3. Directionality Time Analysis Identifies Rigidity Sensing Pathway
Through αMβ2 Integrins 51
Forward 52
1. Introduction 52
1.1. Neutrophils 52
1.2. Neutrophil Mechanosensing 52
1.3. Integrins and Mechanosensing Mechanisms 53
1.4. Integrin Ligand Engagement 54
2. Materials and Methods 56
2.1. Reagents 56
2.2. Neutrophil Isolation 56
2.3. Substrate Preparation 56
xv
2.4. Neutrophil Chemotaxis 58
2.5. Microscopy 58
2.6. Cell Tracking Tools and Analytics 58
2.7. Statistics 59
2.8. Online Supplemental Material 59
3. Results and Discussion 60
3.1. Neutrophil morphology is dependent on substrate stiffness and
independent of ligand coating 61
3.2. The MSD mechanosensitivity of the neutrophil chemotaxis toward fMLP
is ligand-dependent 63
3.3. The TAD mechanosensitivity of neutrophil chemotaxis toward fMLP is
ligand-dependent 64
3.4. The tortuosity of neutrophil chemotaxis toward fMLP is ligand-dependent
and independent of substrate stiffness 65
3.5. Neutrophils migrating on Fgn-coated substrates toward fMLP show
mechanosensitive changes in td and β− 67
3.6. Neutrophils migrating on Col-coated substrates toward fMLP show a td
independent of substrate stiffness 67
3.7. Neutrophils migrating on Fn-coated substrates toward fMLP show
mechanosensitive differences in td and β− 69
3.8. The mechanosensitive component of directionality for neutrophils
migrating on Fn-coated substrates toward fMLP is dependent on
β2 integrins 70
Part 2. Integrating Cell Kinematics with Cell Mechanics 78
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Chapter 4. Cell Mechanics and Motility: Springs and Stick-Slip Adhesion 79
Forward 80
1. Introduction 80
2. Methods 82
2.1. Model overview 82
2.2. Simulation methods and criteria for characterizing dynamics 86
2.3. Choice of model parameters 88
2.4. Simulation benchmarking 89
3. Results 90
3.1. Viable spring configurations 90
3.2. Crawling dynamics depend on cell aspect ratio 95
4. Discussion 96
5. Supplementary Material 101
5.1. Analysis of fish keratocyte movies 101
5.2. Shape dynamics of fast crawling keratocytes 101
5.3. The two dimensional model 103
5.4. Comparing 2-D Elastic Configurations 107
Movie Legends (Movies Online) 112
Chapter 5. Immune System Modifier β-Glucan Regulates Motility Through the
Actin Cytoskeleton 114
Abstract and Forward 115
1. Introduction 115
1.1. Neutrophils 115
1.2. Traction Forces 116
xvii
1.3. Force Motility Relationships 116
1.4. β-Glucan 117
2. Materials and Methods 118
2.1. Reagents 118
2.2. Substrate Preparation 118
2.3. Neutrophil Preparation 119
2.4. Microscopy 120
2.5. Cell Tracking and Migration Analysis 120
2.6. Traction Force Microscopy 121
2.7. Force and Energy Measurements 121
2.8. Dipole Moment Analysis 123
2.9. Statistics 123
3. Results 123
3.1. Spread Area and Motility Biphasically Dependent on BG Concentration 123
3.2. Mechanotactic Conservation of Energy and Contractility Time 124
3.3. Optimal Motility Correlated Inversely With Mechanical Output 128
3.4. Dipole Case Study: Forces Perpendicular to the Direction of Motion 129
3.5. Estimating Active Contractile Force and Cytoskeletal Stiffness 132
4. Discussion 134
5. Supplementary Material 138
5.1. Supplementary Movie Legends 138
5.2. Supplementary Tables and Figures 140
Chapter 6. Future Directions and Closing Remarks 143
Forward 144
xviii
1. Future Project 1: Signaling During Early Stage Chemotaxis 145
1.1. Introduction 145
1.2. Computational Model 147
1.3. Results 153
1.4. Discussion 155
1.5. Pseudocode 157
2. Future Project 2: Neutrophils From Patients With Sepsis Show Novel
Stress Distributions During Chemotaxis 159
2.1. Introduction 159
2.2. Results 159
2.3. Discussion 162
3. Closing Remarks 164
Chapter 7. Extras 170
1. Cell Trajectory Simulator Class (MATLAB) 170
2. How to Save $150 On A New Mercury Lamp 179
Bibliography 180
xix
List of Tables
2.1 Mathematical notation used to derive directionality time and inter-
pret cell motility .................................................................................... 37
3.1 Mechanosensitive parameters of human neutrophil chemotaxis to-
ward fMLP ............................................................................................ 74
4.1 List of parameters and physiologically viable values for the 2D stick-
slip model .............................................................................................. 89
5.1 Table of Alternating Conditional Expectations values .......................... 140
xx
List of Figures
2.1 The effects of sampling interval on measurements and characteriza-
tion of migration trajectories................................................................. 14
2.2 Log-log EASD slopes of the 1D persistent biased random walk and
its correspondence to the directionality time model.............................. 21
2.3 The universal fit for measuring directionality time ............................... 23
2.4 Tangent-bias correlation plotted against the von Mises distribution
bias factor.............................................................................................. 24
2.5 Time scales at which βPBRW(t) converges to the directionality time
model when measurement errors are nonzero (σm > 0)......................... 25
2.6 Additional data plots investigating the robustness of the directional-
ity time model ....................................................................................... 27
2.7 Sampling interval dependent metrics applied to simulated 2D persis-
tent biased random walk trajectories .................................................... 28
2.8 Step-by-step flow-chart for processing experimental migration data
to measure directionality time............................................................... 31
2.9 Measuring directionality time from experimental data.......................... 32
2.10 Sampling interval dependent metrics applied to chemotactic poly-
morphonuclear neutrophils trajectories ................................................. 34
2.11 Explicit demonstration of the measurement error correction on ex-
perimentally measured log-log MSD slope............................................. 35
xxi
3.1 Introduction to Integrins ....................................................................... 55
3.2 Mobility and TAD of neutrophil chemotaxis toward fMLP on surfaces
of varying stiffness depend on ligand coating ........................................ 62
3.3 Tortuosity of neutrophil chemotaxis toward fMLP is ligand-dependent
but independent of substrate stiffness ................................................... 66
3.4 Cells migrating on Fgn-coated gels toward fMLP have similar MSD
and RMS speed but show a mechanosensitive change in td and β− ...... 68
3.5 Neutrophils migrating on Col IV-coated substrates toward fMLP
show td, β−, and β+ independent of substrate stiffness ......................... 69
3.6 Neutrophils migrating on Fn-coated substrates toward fMLP show
mechanosensitive differences in MSD, td, and β−. ................................. 71
3.7 The mechanosensitive component of directionality for neutrophils mi-
grating on Fn-coated substrates toward fMLP is dependent on β2
integrins................................................................................................. 73
4.1 1-D model schematic and stick-slip adhesion definitions ....................... 85
4.2 2-D model schematic ............................................................................. 87
4.3 Overview and dynamics of viable 2-D stick-slip toy models (Configs.
1-4) ........................................................................................................ 92
4.4 Characterized dynamical response of Config. 4 with respect to pa-
rameters KD, KL and g ......................................................................... 94
4.5 Phase diagrams of Config. 4 dynamics: cell shape matters ................... 97
4.6 Cell aspect ratio, speed, and amplitude of cell length oscillation distri-
butions measured from previously published movies of fish keratocytes102
xxii
4.7 Bipedal locomotion and lateral oscillations of the nucleus observed
in fish keratocytes.................................................................................. 104
4.8 Representative dynamical responses of four select configurations of
the 2-D cell crawling model ................................................................... 109
4.9 Drag force sensitivity analysis of Config. 4 ........................................... 113
5.1 Traction force microscopy cartoon with representative DIC and trac-
tion field frames..................................................................................... 121
5.2 Neutrophil motility and mechanical output depend on BG concen-
tration and substrate stiffness ............................................................... 125
5.3 Two graphs showing the Pearson product-moment correlation coeffi-
cients (PCCs) between variables describing the motility and mechan-
ical output of neutrophils ...................................................................... 129
5.4 Slower moving cells exert more energy contracting their substrate
than faster moving cells......................................................................... 130
5.5 Dipole moment analysis of a motile BG0 cell migrating on a 5 kPa
substrate................................................................................................ 133
5.6 Estimating total myosin mediated contractile force and cell stiffness ... 134
5.7 The perceived cell boundary may differ from the actual cell boundary 141
5.8 Control experiments showing that the BG effect cannot be explained
by changes in osmolarity ....................................................................... 141
5.9 Neutrophil adhesion rate, activity rate, and contractility rate .............. 142
6.1 Schematic showing the chemical signaling of chemotaxis and the
downstream cellular components that drive cell migration ................... 145
6.2 Classification of neutrophil traction stress fields during 2D chemotaxis148
xxiii
6.3 The LEGI model ................................................................................... 150
6.4 LEGI model results using Cartesian coordinate system ........................ 154
6.5 LEGI model results using polar coordinate system............................... 156
6.6 Neutrophil activation, endothelium transmigration, and chemotaxis
through the extracellular matrix ........................................................... 160
6.7 Differential Traction Force Mapping of representative neutrophils
from healthy (H) and septic (S) donors................................................. 161
6.8 Neutrophils from healthy (H) and septic (S) donors differ in adhesion
and Mean Square Displacement ............................................................ 163
6.9 Schematics summarizing all the force modes of motile and non motile
neutrophils............................................................................................. 165
7.1 The $165 kymograph............................................................................. 179
xxiv
Abstract of “The Mechanics of Cell Motility and a Unifying Theory for
Characterizing Directed Motion”
by Alex Loosley, Ph.D., Brown University, May 2015
This dissertation is split into two parts with the first focusing on characterizing cell
motility, and the second focusing on linking cell motility with cell mechanics. Cellular
motility has been routinely characterized by parameters such as migration speed,
tortuosity, and persistence time. However, many of these parameters are generally
not reproducible because their numerical values depend on technical parameters such
as the experimental sampling interval and measurement error. In Part 1, we use
random walk theory, simulations, and experimental data to address the need for
a metric that quantifies the directionality of a migration path in a manner that
decouples from technical parameters. We call this novel metric directionality time
because it can be interpreted as the minimum observation time required to determine
that the migratory motion is directed. Along with measures of migration speed and
persistence, we used the directionality time metric to determine the directedness
of chemotactic neutrophil migration paths as a function of physiologically relevant
substrate stiffnesses and compositions. We find that engagement of the β2-integrin,
CR3, is required for the substrate stiffness dependent regulation of neutrophil honing
in 2D on fibronectin.
In Part 2, we investigate the role of cell mechanics in cell motility. As cells migrate,
they exert cytoskeletal driven contractile forces on their environment. Using spring
models with stick-slip adhesion, we are able to predict how the mechanical properties
and shape of the cell affect its dynamics and traction forces. We then switch the
focus back to neutrophils and explore the relationship between neutrophil contractile
xxv
force, motility, and the biological response modifier β-glucan, a substance that has
shown promise as a clinic grade therapeutic. Using live cell imaging and traction force
microscopy to measure neutrophil traction forces as a function of substrate stiffness
and β-glucan concentration, we report that increasing the concentration of β-glucan
leads to diminished traction forces. Our findings indicate that biological response
modifiers may act through the modulation of cell mechanics and motility. Further,
we report an overall inverse correlation between migration speed and mechanical
output that is affected by β-glucan. These data should be useful for modeling cell
motility.
CHAPTER 1
Introduction
.
1
1. A General Overview of Cell Motility
Cell motility is the ability of a cell to actively locomote from one place to another. Such
locomotion can arise in the form of swimming [3, 4] or crawling [5–7] with the latter the
focus of this dissertation. Crawling motility is required for many physiological processes that
sustain life. For example, in wound healing, an influx of inflammatory cells clear harmful
pathogens and dead material [8]. Following the inflammatory response, motile fibroblasts
must arrive and proliferate at the scene of the injury where they facilitate extracellular
matrix synthesis and repair [9, 10]. Finally, epithelial cells collectively migrate in a motion
called plithotaxis to close tissue lesions [11–13].
Research aiming to describe and elucidate the mechanisms of cell motility has been
under way since the 1680s when the first study of bacterial motion was completed by van
Leeuwenhoek [14]. This was also one of the first studies of micro-organisms using a micro-
scope, which had been invented nearly 100 years prior. In the twentieth, ground breaking
optical techniques, such as differential interference contrast (DIC), confocal microscopy, and
the discovery of green fluorescent protein (GFP) [15], have paved the way for the acquisition
of massive amounts of data in real time at spatial resolutions down to 10−7 m (0.1µm), two
orders of magnitude smaller than the size of most cells. These technologies, aided by a
sharp upswing in computational power, have led to an influx of quantitative data describ-
ing cell migration and a new frontier of opportunities for interdisciplinary collaborations
between biologists and physicists who wish to understand the physics behind the question,
how do crawling cells locomote? In the chapters that follow, this question is answered by
combining the principles of classical mechanics, fluid dynamics, and random walk theory
with the underlying biomolecular mechanisms that generate and sustain active matter.
Biomolecular mechanisms are often initiated by a physical interaction, and often lead
to a subsequent physical interaction. For this reason, biomolecular mechanisms and their
corresponding physical interactions are indubitably connected. Active matter can be defined
2
as matter that spontaneously converts a fuel source (energy source) into active forces.
Without active matter, cellular processes such as morphogenesis [16], mitosis [17], and cell
motility [18], would not be possible.
At least two sets of components within the cell are necessary for active matter driven
cell crawling motility. The first is the actin cytoskeleton, a viscoelastic network that spans
the cell and is made up of actin filaments called F-actin, cross-linking components, and
molecular motors. F-actin is highly rigid at the length scale of cells, with a filament per-
sistence length of 17.7µm [19] (cell lengths ∼ 10 to 15µm). Thus, the actin cytoskeleton
acts as a strong scaffolding medium maintaining cell shape [20, 21] and providing a base
for molecular cargo transport [22]. The actin cytoskeletal network is highly dynamic and
can be disassembled into globular actin monomers called G-actin, which can be reused to
polymerize new F-actin elsewhere. As part of the motility process, actin polymerization at
the leading edge of the cell generates localized protrusions of the cellular plasma membrane,
while molecular machines such as ATP driven myosin motors cause contraction of the actin
cytoskeleton. As the actin cytoskeleton contracts, depolymerization mediated by cofilin
[23, 24] occurs near the center of the cell, and material is recycled for F-actin polymeriza-
tion and protrusion at the edges of the cell. As a result of both depolymerization at the
center and polymerization at periphery, the actin cytoskeleton flows toward the center of
the cell in what is called retrograde flow.
The second set of components required for crawling motility is adhesion between the cell
and its substrate. Without an adhesive clutch to anchor the cell, it would not be possible
for protrusion at the leading edge of the cell to propel the cell forward. For many cells, this
anchorage is achieved through transmembrane proteins called integrins [25]. Integrins form
bonds with the substrate and a host of intracellular components, ultimately coupling the
substrate to the actin cytoskeleton [26]. Retrograde flow of the actin cytoskeleton leads to
3
traction forces that are inward directed (contractile) and highly correlated across the cell
body [27–29].
In past decades, techniques have been invented to detect cellular traction forces, provid-
ing a new spotlight for illuminating features of the cytoskeleton/adhesion driven cell motility
process. In 1980, Harris et al. were the first to see traction forces by culturing fibroblasts on
very thin silicon substrates that wrinkled under cellular traction stress [30]. Later in 1994,
Lee et al. improved the spatiotemporal resolution of this procedure by implanting small
latex beads into the silicon substratum to measure traction force generated displacements
[31]. In 1999, Dembo and Wang were the first to report traction force measurements using
bead embedded polyacrylamide substrates, an improvement over silicon substrates because
of their well characterized rheological properties [32]. Dembo and Wang assumed that the
size of their substrates was effectively infinite compared to the 3T3 fibroblasts they studied
in order to apply the Boussinesq equation to map micrometer sized bead displacements to
traction stresses on the order of 0.1 to 10 nN/µm2. This study marked the beginning of
modern traction force microscopy measurements. In the last 15 years, numerous improve-
ments have been made to resolution, accuracy, and throughput of calculating traction forces.
This includes greatly reducing the computational power required to measure traction forces
by solving the inverse Boussinesq equation in Fourier space [33], improving resolution us-
ing multiple colors of fluorescent displacement markers [34], assuming finite traction stress
regions instead of point sources [35], and properly accounting for large nonlinear substrate
displacements and rotations [36]. Traction force microscopy has now also been adapted
for measurements in 3D [37–39], and has even been combined with total internal reflection
fluorescence (TIRF) microscopy to precisely co-localize cell-substrate contact regions with
measurements of traction stress. [40].
In this dissertation, traction force microscopy is used to address two questions: How
do cells coordinate actin cytoskeletal mechanics and adhesions to achieve locomotion in a
4
specific direction? And, how much force (or energy) does the cellular machinery apply to
(or use in contracting) its environment during locomotion? The latter is important because
it is part of an even bigger question currently under intense investigation: What is the
relationship between cellular traction forces and motility [41, 42]?
The ability for cells to detect and adapt their function based on the mechanical prop-
erties of their environment is another important factor that plays a significant role in cell
motility. This phenomenon, sometimes called mechanosensing, has been the topic of nu-
merous studies over the last 15 years [1, 29, 32, 43–55].
Mechanosensing is one of the central themes of this dissertation, specifically the role
of mechanosensing in neutrophils. Neutrophils play a primary role in the inflammatory
response as part of the innate host defense against pathogenic bodies. Vital to their role
is the ability to migrate from blood vessels, through the extracellular matrix, to sites of
inflammation. During the course of migration, neutrophils encounter a variety of tissues
characterized by differing physical topologies and stiffnesses ranging in Young’s moduli from
10−1 kPa for brain tissue, to more than 106 kPa for bone [29]. That cell motility depends on
the mechanical properties of a cell’s surroundings is now widely accepted [41, 48, 49]. How
this mechanosensitivity comes about is another matter. We ask the following questions:
Does mechanotactic regulation depend on tissue type? Which integrins are involved in the
mechanotactic regulation of cell motility? And, what specific aspect of motility do specific
integrins regulate? Note above that the units kPa were used to describe substrate stiffness.
This is to be consistent with the bulk of the recent literature. However, most other quantities
in this dissertation, including applied stress, have been written using the fundamental size,
time, and force scales of cell motility: µm, s, and nN, respectively (1 nN/µm2 = 1 kPa).
These scales are convenient for making quick and dirty estimates of quantities based on the
data presented in this dissertation.
5
In order to answer the second question, precise analytical tools are required to charac-
terize the trajectories of cell migration. Common has been the usage of persistent random
walk theory to fit the mean squared displacement (MSD) of non directed migration trajecto-
ries thereby characterizing them in terms of parameters for migration speed and persistence
time [56–62]. When cell migration is directed (i.e. biased towards an external cue), one
more parameter is necessary to fully characterize the motion. One parameter that has been
used is the fraction of all turning angles between ±φ, where φ is an arbitrary angle typ-
ically chosen to be ≤ 90o [1, 48, 63, 64]. Another extra parameter that has been used is
tortuosity, the end to end distance divided by the arc length of the migration trajectory
(also known as chemotactic index [1, 49, 50], directionality fraction [65], and straightness
index [60, 66]). Both of these two parameters have shortcomings. Mainly, they are sampling
interval dependent. The sampling interval used to collect data biases the measurement of
both turning angle fraction and tortuosity (more about this in Chapter 2). Some rules for
characterizing MSD scaling exponent crossover (currently used in physics [67] and contin-
uum mechanics [68]) could potentially be developed into a directed cell migration analysis
tool that is sampling interval independent. But, these theories are not designed to analyze
noisy and potentially nonergodic migratory motion. To address the need for an analytical
tool that measures the honing portion of directed motion, we used random walk theory to
derive a tool called directionality time. Armed with the proper analytical tools to character-
ize directional cell migration, we identified the integrins responsible for the mechanotactic
regulation of 2D neutrophil honing towards a source of chemoattractant.
Finally, towards the end of this dissertation, we combine the study of motility and
traction microscopy to investigate the effects of biological response modifiers on the cell
mechanics of immune cells. As a model biological response modifier, we use is β-glucan
(BG) from yeast that has been purified into a clinical grade, water soluble form [69]. BG
of this form has been acts as a biological response modifier by improving wound healing
6
in mice [70], by showing promise as an anti-cancer agent in in vivo tumor models [71–
73] and clinical trials [74], and by enhancing chemotactic honing of neutrophils on glass
[75, 76]. To promote an effect in neutrophils, BG is known to bind and allosterically
regulate the affinity of Complement Receptor 3 (CR3, also known as αMβ2, Mac-1, and
CD11b/CD18) [76–79], an integrin that plays a significant role in mechanosensitive behavior
[1, 48]. Therefore, it is rational to suggest that BG affects biological response modification
through the regulation of cell mechanics. Altogether, BG has the potential to be a positive
pharmacological intervention and we use live cell tracking and traction force microscopy to
investigate the role of cell mechanics as part of this intervention.
2. Dissertation Outline
Although the primary cell directly studied in this dissertation is the human neutrophil,
many of the analytical techniques developed herein are generally applicable to a variety of
other motile organisms, and our findings provide clues towards a broad understanding of
crawling motility in all cell types. Below is the story of cell motility beginning in Chapter
2 by first using random walk theory to develop the analytical tools necessary to analyze
motility kinematics. Next, in Chapter 3, these tools are applied to mathematically charac-
terize directed cell migration in terms of speed, persistence, and honing. This analysis is
applied to determine the specific integrins on neutrophils that mechanotactically regulate
each of these components of cell migration. In Chapter 4, we switch gears ever so slightly
to explain cell motility and traction forces using minimalist toy models using springs and
stick-slip adhesion as ingredients. Next, in Chapter 5, we bring together the studies of
traction forces, mechanosensing, and cell motility to determine which factors of mechanical
output correlate to optimal neutrophil motility. As part of this study, we show for the first
time the relationship between immune system priming with β-glucan, and traction forces,
demonstrating that the immunological response is tied to the mechanical output of immune
7
cells. The dissertation finishes with a future directions Chapter that contrasts the motil-
ity and traction forces of BG treated neutrophils with data from neutrophils treated with
lipopolysaccharide, or harvested from human patients with sepsis.
8
Part 1
Cell Kinematics
CHAPTER 2
Directionality Time: A New Theory for Characterizing
Directed Motion
10
Forward. In this chapter, we review and develop the analytical tools necessary to
characterizing directed cell migration. These tools are applied in Chapter 3 to study the
role of integrins in regulating directed cell motility as a function of substrate stiffness.
1. Introduction
Directed cell migration is the process where a single cell or a group of cells bias their
direction of locomotion by coupling to an external cue. One example of an external cue is
a chemical gradient. Migration in the direction of (or opposite to) a particular chemical
gradient is called chemotaxis. Besides chemotaxis, there are many types of directed cell
migration named according to the external cue, including gravitaxis [80], aerotaxis [81],
durotaxis [82], haptotaxis [83], and plithotaxis [13]. These processes are ubiquitous in
nature, facilitating the innate and adaptive immune systems [84, 85], sexual reproduction
[86], embryonic development [87], cancer metastasis [88, 89], and more. The efficacy to
which cells are able to carry out these functions is often tied to the characteristics of their
migration, including migration speed, persistence, and tortuosity. These characteristics
can be quantified to determine which biochemical and biomechanical factors affect cell
migration, and by how much.
1.1. Commonly Used Analytical Tools for Characterizing Migration. Mean
squared displacement (MSD) is one of the most common metrics for measuring migration
speed and distance traveled because it is easily interpretable and readily derived from math-
ematical models of motion. Numerous studies that characterize directed migration use MSD
in conjunction with at least one other metric for quantifying path persistence or tortuosity
[1, 41, 48, 49, 90]. Three examples of such metrics used are: the distribution of turning an-
gles between discrete measurements of centroid displacements (turning angle distribution,
TAD); tortuosity (also known as straightness index [60, 66], chemotactic index [49, 50], or
directionality ratio [65]) defined as the end-to-end distance traveled divided by the total
11
migration path length; and tangent-tangent correlation, which describes the correlation in
migration path orientation over a specific length or time interval.
1.2. Sampling Interval Dependent Metrics. In order to gain insight from quanti-
tative characterizations of the migration path, the numerical values of the metrics applied
need to be reproducible from one set of experiments to another. Such values should also
reflect the true kinematic properties of migration by decoupling from pseudo random kine-
matics induced by measurement error along the migration path. The shortcoming of TAD,
tortuosity, and tangent-tangent correlation is that they each implicitly depend on sampling
interval, ∆t, which is the time interval between position measurements. Sampling interval
can be chosen arbitrarily, implying that TAD, tortuosity, and tangent-tangent correlation
curves are not generally reproducible without applying equivalent sampling intervals across
all experiments. Even when sampling intervals are accounted for, a sampling interval de-
pendent metric only characterizes migration at an arbitrarily chosen time scale at which
the metric may or may not decouple from measurement error.
To visualize sampling interval dependence, consider two experimental measurements of
a migration path, one using a “long” sampling interval, ∆t = ∆t>, the other using a “short”
sampling interval, ∆t = ∆t< (Fig. 2.1 a, top and bottom, respectively). Circles are centroid
positions and the corresponding perceived migration paths (blue line segments connecting
circles) are juxtaposed against the true migration path (thick grey curve). The deviation
between centroid positions (rn = r(tn), where tn = n∆t, n = 1, 2, ..., N , ∆t = ∆t< or ∆t>)
and the true migration path represents centroid measurement error, σm, which depends
on factors such as image resolution and cell boundary detection accuracy. Angles between
successive blue line segments, φn, are turning angles (−π < φ ≤ π). Taking into account
all turning angles, normalized TADs, ρφn(φ; ∆t), are calculated for both sampling intervals
(Fig. 2.1 b). As the sampling interval increases towards the total duration of observation,
the TAD curve becomes sharply peaked at φ = 0. Conversely, as the sampling interval
12
decreases towards zero, the effects of diffusive kinematics and centroid measurement error
flatten the TAD curve. Hence, TAD depends notably on sampling interval. One measure of
persistence is the so-called turning angle persistence (TAD persistence), the fraction of all
turning angles between ±π2 (shaded area under TAD curves in Fig. 2.1 b). TAD persistence
depends on the sampling interval just as TAD does.
The dependence of tortuosity on sampling interval is apparent when considering the
limit that the sampling interval approaches the total duration of the migration path. In
this limit, total path length approaches the end-to-end length resulting in a tortuosity
of 1. When sampling interval decreases, the total path length increases due to centroid
measurement error and the underlying fractal nature of the migration path itself [91, 92].
1.3. Sampling Interval Independent Metrics. Tangent-tangent correlation, v(t+ τ) · v(t),
is the time averaged cosine of the angle between tangent vectors v(t+ τ) and v(t) that are
separated by a time interval τ . The overline denotes an average over all time t. When ap-
plied to discrete experimental data captured at a specific sampling interval, v(t) is replaced
by vn (Fig. 2.1 a). Tangent-tangent curves are sampling interval dependent because the
uncertainty of measuring tangent vector direction increases with increasing sampling inter-
val. However, there is a sampling interval independent measure called persistence time that
derives from the tangent-tangent correlation curve. Persistence time, tp, is the time scale
over which directional orientation of the migration path remains correlated. Measurements
of persistence time are sampling interval independent as long as the sampling interval is
small enough for it to be resolved. In general, persistence time would be measured as a fit
parameter in a model used to fit the tangent-tangent correlation curve over all time scales.
Persistence time and migration speed together can characterize non-directional motion,
but an additional metric is needed to characterize directional motion. We recently discussed
the need for such a metric in Ref. [1]. The purpose of this work is to derive a sampling
interval independent metric that measures directionality. One approach could entail fitting
13
a b
c
∆t = ∆t<
r1
r2
rn
^
v1
^
v2
^
vnφ1
φ2
φn
∆t = ∆t>
φ
ρφn(φ;∆t<)
−π −π2
π π02
−π −π2
π π02
ρφn(φ;∆t>)
φ
Θn+1Ln
Θn
Rn
Rn+1
êy
êx
Figure 2.1. The effects of sampling interval on measurements and char-acterization of migration trajectories. (a) A migration path sampled witha long sampling interval, ∆t = ∆t> (top, outlined yellow circles) and a shortsampling interval, ∆t = ∆t< (bottom, outlined white circles). Using the longsampling interval diagram, the observed migration path is formed by connectingmeasurements of centroid positions, r1, r2, ..., with lines (in red). Unit tangentvectors are shown as v1, v2, ... while turning angles are defined as the angle be-tween successive tangent vectors, φ1, φ2, .... Differences in centroid position fromthe true migration path (thick grey curve) represent measurement error. (b)Turning angle distributions (TAD), ρφn(φ; ∆t), based on both the long (top) andshort (bottom) sampling intervals. A measure of migration persistence knownas TAD persistence is the area under the TAD curve between ±π
2(shaded).
TAD persistence depends notably on sampling interval. Similar diagrams canbe used to show the sampling interval dependence of metrics such as tortuosityand tangent-tangent correlation. (c) A diagram visually defining terms nec-essary to analytically derive the directionality time model. A cell is depictedas a random walker that steps (R1,R2, ...,Rn, ...) located in a 2-D coordinatesystem defined by the unit vectors ex and ey. Capital letters correspond torandom variables. For each step, there is a corresponding step length Ln andpolar angle Θn. The latter is defined with respect to unit vector ex and is notto be confused with the corresponding turning angle, Φn = |Θn −Θn−1|
data such as TAD persistence or tortuosity to a model with a set of fit parameters. However,
TAD persistence and tortuosity models are difficult to calculate and interpret. Mean squared
displacement (MSD) models are easier to calculate and interpret and several models have
already been analytically derived [60, 93, 94]. Nevertheless, this approach of fitting to a
model only works if the underlying kinematics of the migrating cell are understood a priori
such that a model can be chosen. While one can attempt to fit more than one model to
14
determine which fits best, changes to MSD from one model to the next can be small with
respect to the error bars on an experimental MSD curve. Hence, there is the possibility
of a causality loop - one cannot accurately understand a set of migration paths without
knowledge of the underlying process and corresponding random walk model, but one is
unsure of the corresponding random walk model without understanding the migration paths.
To circumvent this causality loop, we take a bottom up approach to derive a novel sampling
interval independent metric called directionality time based on the slope of MSD in log-log
coordinates. We show that the directionality time concept is broadly applicable to many
types of directional motion and demonstrate its implementation on data of directionally
migrating chemotactic neutrophils.
2. Results
2.1. Mean Squared Displacement Analysis. Leading to the definition of direc-
tionality time, we begin with the assertion that ensemble averaged squared displacement
follows a power law ⟨r2(t)
⟩∼ tα (2.1)
where square brackets 〈〉 denote the ensemble average over squared displacements measured
at time t. To be precise about the type of averaging, we call this quantity the ensemble
averaged squared displacement (EASD) instead of MSD. The exponent α characterizes the
motion. A constant value of α = 1 indicates diffusive (random) motion whereas α = 2
indicates ballistic (directed) motion. Other values represent subdiffusive motion (0 < α <
1), superdiffusive motion (1 < α < 2), or no motion at all (α = 0).
When few sets of trajectories are available for the ensemble average, time averaged
squared displacement (TASD) can be calculated to reduce statistical noise. The TASD of
the ith migration path is given by
r2i (τ) = |ri(t+ τ)− ri(t)|2 (2.2)
15
where the overline denotes an average over time t. Squared displacement that is first time
averaged and then ensemble averaged,⟨r2(τ)
⟩, is hereby referred to as MSD as this is how
it is often defined in many studies [1, 48, 95].
Ergodicity, ξi, is the conversion factor that maps EASD as a function of time t, to TASD
(and MSD) as a function of time interval τ :
r2i (τ) = ξi(τ)〈r2(t = τ)〉 (2.3)
where subscript i is the migration path index. The time averaging that goes into calcu-
lating MSD smooths out the effects of variables that may be changing over time, such as
instantaneous speed. When such factors change significantly, 〈ξ〉 6= 1, and the migration
paths are said to be nonergodic. Otherwise, the migration paths are ergodic and EASD
and MSD are interchangeable (i.e. 〈r2(τ)〉 = 〈r2(t = τ)〉).
The slope of EASD plotted in log-log coordinates is an approximate measure of the
EASD exponent α (Eq. 2.1) and therefore a measure that characterizes trajectory diffusivity
and/or directedness. Using Eq. 2.1, and noting that α can change with time t, one can define
the log-log EASD slope
β(t) =dα
dtt ln t+ α. (2.4)
When EASD exponent α is constant, β = α. Otherwise, β(t) is an estimator of α(t), and
therefore an estimator of how diffusive or ballistic the motion is at a particular time, t. Log-
log MSD slope, β(τ), can also be calculated to characterize motion as a function of time
interval τ . Two questions arise: What is the mathematical form of these β curves? Can a
β curve be used to determine a sampling interval independent quantity that characterizes
directionality?
2.2. Deriving the Directionality Time Model. When observing a directionally
biased random walk, there exists a sufficiently large sampling interval such that the motion
16
will appear to be ballistic. Put in terms of log-log MSD slope, β(τ) → 2 as τ → ∞ for
a directionally biased random walk. The idea of using log-log MSD slope to measure this
particular time interval was recently hypothesized in our recent article about neutrophil
chemotaxis [1]. We suggested an empirical fit function for β(t) to estimate the location of
the directionality time transition. Here in this article, we rigorously develop the concept of
directionality time from the bottom up by analytically deriving a universal β(t) fit model
and using biased and persistent random walk modeling to characterize its robustness. Direc-
tionality time is defined as the time scale above which motion appears ballistic (directional)
and can be loosely interpreted as the time it takes for a random walker to orient towards
an external cue.
To determine the mathematical meaning of directionality time and its applicability, log-
log EASD slope, β(t), is analytically derived for three directionally biased random walk
models:
(1) Drift Diffusion (DD)
(2) 2D Stepping Biased Random Walk (2D-SBRW)
(3) 1D Persistent Biased Random Walk in Continuous Time (1D-PBRW)
To optimize the flow of this chapter, detailed step by step derivations of β(t) for each
model have been relegated to Appendix A (Sec. 4.2) at the end of this chapter, while the
calculations and significant findings are summarized here.
DD (model 1) was a suitable starting point because these processes are readily under-
stood. For DD with a diffusion constant D, and drift speed u, the log-log EASD slope is
shown in Appendix A (Sec 4.2.2) to be
βDD(t) =1 + 2t
td
1 + ttd
(2.5)
where td = 2dDu2 defines directionality time. Note that β(t) begins at β(0) = 1 and asymp-
totes towards 2. This is the signature of a directionally biased random walk. Directionality
17
time is the time at which β = 32 where the migration transitions from diffusive to direc-
tional. As D increases and/or u decreases, more observation time is required to determine
that the motion is directionally biased.
A 2D-SBRW (model 2) describes an object that steps from one discrete position, Rn,
to the next, Rn+1 (n = 0, 1, 2, ...), such that the displacements between successive steps are
biased towards a particular direction, ex (Fig. 2.1 c). Notationally, all random variables
are assigned capital letters. Using Ln and Θn to denote step lengths and polar angles
(orientation) respectively, the stepwise EASD can be shown to be
⟨R2n
⟩= n〈L2〉+ n(n− 1)〈L cos Θ〉2
(see Ref. [94] for the derivation). Note that the motion is diffusive (〈R2n〉 ∼ n) when n is
small and directional (〈R2n〉 ∼ n2) when n is large. This step by step random walk has no
persistent directionality at low n because the direction of motion changes with each step.
By defining a constant instantaneous speed v, the approximation n ≈ vt〈L〉 can be used to
derive EASD as a function of time t instead of step number n. In the time representation,
this is a model of a biased random walk (BRW) instead of an SBRW. Differentiating in
log-log coordinates gives the log-log EASD slope
βBRW(t; td) =1 + 2t
td
1 + ttd
(2.6)
which is functionally identical to βDD (Eq. 2.5), now with directionality time given by
td =〈L〉v
(〈L2〉 − 〈L cos Θ〉2
〈L cos Θ〉2
). (2.7)
The functional form of EASD slope is no different between models 1 and 2, only the math-
ematical constants that constitute directionality time have changed.
18
This generalized directionality time equation can be understood by considering the
following example. Consider the case where the probability of measuring a step length L
between l and l + dl is given by a Poissonian probability density function (PDF), ρL(l) =
1lp
e− llp , and the orientation Θ is independent from step length. Then directionality time
simplifies to
td = tp
(2− c2
c2
)(2.8)
where c ≡ 〈cos Θ〉, and tp =lpv represents the reorientation time, the average time to take
one step in a new direction. The term c is tangent-bias correlation (similar to tangent-
tangent correlation). Values of c2 range from 0, corresponding to no orientational bias
(PDF ρΘ(θ) = 12π , where −π < θ ≤ π), to 1, corresponding to maximal orientational or
anti-orientational bias (PDF ρΘ(θ) = δ(θ) or δ(θ− π), where δ is the Dirac delta function).
Directionality time depends only on the reorientation time and the extent to which the
orientation is biased when a reorientation event occurs, increasing with the former and
decreasing with the latter. In particular, the term 2−c2c2
ranges from 1 at maximal bias, to
∞ at no bias. It may appear odd that td → tp (the equivalent of one step) when the system is
perfectly directional (c2 → 0). However, this is no more than a subtlety of stepping random
walks. There is no change in position defined at t < tp because of the way continuous time
was substituted in for discrete stepping number (t ≈ ntp). Therefore, the minimum time to
determine that movement is directionally biased will always be greater than or equal to tp.
No information can be gained from a random walker that has not yet taken any steps. Since
each step is accompanied by reorientation, this model cannot be used to derive a log-log
EASD slope equation that accounts for persistence.
In order to consider the relation between directionality time and persistence, a con-
tinuous time random walk model must be derived, noted as the PBRW (model 3). This
model is derived in 1D for simplicity using the biased telegrapher equation [93, 96]. To
put the biased telegrapher equation in context, the unbiased telegrapher equation is used
19
to derive the dynamics and EASD of persistent random walks that describe the kinematics
of chemokinesis [41, 61], as well as the motion of grasshoppers and kangaroos [93]. In this
model, an object moves with constant speed v, either left (−x direction) or right (+x di-
rection) for some random run time (T (l) or T (r), respectively) before switching directions.
Bias is induced by drawing left and right run times from nonequivalent distributions and is
encapsulated by tangent-bias correlation c = 〈T (r)〉−〈T (l)〉〈T (r)〉+〈T (l)〉 . The log-log EASD slope of the
PBRW (Eq. 2.31) is more complicated than that for the BRW and DD because direction-
ality over short time scales caused by persistence induces a zero-time log-log EASD slope
βPBRW(0) = 2. As t increases and the orientational correlation of persistent motion is lost,
βPBRW(t) dips towards 1. Except when c = 0, βPBRW(t)→ 2 as t→∞, as is the signature
of directionally biased motion.
These βPBRW(t) curves are plotted Fig. 2.2 a for multiple values of c2 (solid curves).
In this plot, time is in units of λ−1+ , which is related to the average run time (persistence
time). At sufficiently large time scales
βPBRW(t > tBRW) ' βBRW(t; td) =1 + 2t
td
1 + ttd
(2.9)
where tBRW is the convergence time above which the difference between βPBRW and βBRW
is less than 5% (Fig. 2.2 b, curve). Directionality time can be measured by fitting βPBRW(t)
to the corresponding βBRW(t; td) at time scales larger than t = tBRW. The resulting mea-
surement of td from this fit is given by
td =1
λ+
2(1− 2c2)
c2. (2.10)
As with the BRW, td →∞ for a random walk that is unbiased. When the bias is sufficient
(12 ≤ c2 ≤ 1), the gap in time scales between short time scale persistent directionality and
long time scale biased directionality vanishes such that the random walk appears directional
at all time scales. By construction, we redefine negative values of directionality time to
20
0 in this domain to be consistent with the interpretation that directionality time is the
observation time necessary to determine that motion is directional (Fig. 2.2 c).
0 20 40 60 80 1000.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
time, t (λ-1)+
log−
log
EASD
slo
pe, β
(t)
approximate (BRW)
exact (PBRW)
c2 = 0.00
c2 = 0.04
c2 = 0.16c2 = 0.36
c2 = 0.64
c2 =1.00
a b
c
0 0.5 10
20
40
60
80
dire
ctio
nalit
y tim
e, t d
(λ-1
)+
tangent-bias correlation, c2
0
5
10
<1%
<5%
<10%
>10%
15
conv
erge
nce
time,
t BRW
(λ-1
)+
Figure 2.2. Log-log EASD slopes of the 1D-PBRW and its correspon-dence to the directionality time model. (a) Log-log EASD slopes of the1D-PBRW model (βPBRW(t), solid curves) asymptotically approach the direc-tionality time model (βBRW(t; td), dashed curves) given a specific value of di-rectionality time td. These slopes are plotted against time for different valuesof the squared tangent-bias correlation, c2 = 〈cos Θ〉2. In these plots, timeis nondimensionalized by the persistence time-like parameter, λ−1
+ . Correlatedorientation over short time scales gives β(0) = 2 regardless of directional bias.Except when the motion is non-directional (i.e. c2 = 0), βPBRW(t) dips from2 to 1, before asymptoting back toward 2 as t → ∞. (b) The convergencetime above which the difference between between βPBRW(t) and βBRW(t; td) isless than 10, 5, and 1% are shown in gray scales plotted against c2. The 5%convergence time, denoted tBRW (black solid curve), is used for calculations inthis work. (c) The corresponding directionality time plotted against c2. Thetransition from random motion to directional motion
(βBRW(td) = 3
2
)does not
occur in the domain, 0.5 ≤ c2 ≤ 1. In this domain, directionality time is definedto be zero because the motion appears directional at all time scales.
The result βDD(t) = βBRW(t) ' βPBRW(t) at large time scales implies that one unique fit
function can be used to measure td from any set of idealized migration path measurements.
For brevity, we refer to this fit function as the βBRW-model. The βBRW-model is independent
21
of the type of random walk process, returning values of td ranging from 0 when all time
scales appear directional, to ∞ when motion is completely unbiased.
2.3. Computational Modeling to Test Robustness. We used computer simula-
tions of 2D-PBRWs to test the robustness of the βBRW-model. A diagram of the 2D-PBRW
is shown in Fig. 2.3 a. Continuous time was simulated in time steps of δt. Run times,
Tn (times between reorientations), were drawn from Poisson distributions corresponding
to persistence times, tp, ranging from 0.4 to 3.6 s (Fig. 2.3 b). At the end of each run
time, a reorientation occurred. Polar angles, Θn, were drawn from von Mises distributions
[97] centered about θ = 0 in the ex direction. The von Mises distribution width, set by
parameter κ (Fig. 2.3 c), was selected from values between 0 (uniform distribution) and
10 (relatively narrow Gaussian-like curve). The correspondence between κ and 〈cos Θ〉 is
shown in Fig. 2.4. Positions along the 2D-PBRW were sampled periodically with sample in-
terval ∆t and Gaussian noise with width σm was added to each sampled position coordinate
so that simulated data would resemble experimental data similar to those of directionally
migrating neutrophils [1, 48].
To investigate the effects of measurement error on measures of directionality time, two
ensembles of simulated trajectories were computed, one with σm = 0 and other with σm =
1µm. These simulated trajectories were ergodic (data not shown) and EASD was equated
to MSD (β(t)↔ β(τ), t interchangeable with τ). The corresponding measurements of β(τ)
are shown Fig.2.3 d (black and red, respectively). Visually, the deviations between the two
data sets are greatest at short time scales (shaded region). Specifically, β(0) = 0 when
σm > 0, instead of β(0) = 2 when σm = 0. Effectively, measurement error hides the ability
to identify persistence.
Recall that βPBRW(t) converges to the βBRW-model above time tBRW (see Eq. 2.9 and
Fig.2.2 b). Measurement error sets an additional convergence time, tσm , above which the
βBRW-model is valid. This time is derived analytically in the Appendix B (Sec. 4.3) by
22
00
0.5
1.0
1.5
-π π
κ=0,2,10
ρΘ
(ra
d-1
)
polar angle, θ (rad)
êy
êx
0 2 4 6 8 10
0.2
0
0.4
0.6
0.8tp = 0.4,2.0,
3.6s
run time, t (s)
ρT
(s-1
)
0 50 100 150 200 2500
0.5
1.0
1.5
2.0
2.5
σm = 1µm
τmin = 50.4s
σm = 0µm
βBRW-model
single parameter
exponential !tlo
g-l
og
MS
D s
lop
e, β
(τ)
time interval, τ (s)
Θn+1L
n Ln=vTn
Θn
δtσm
a
b d
c
Figure 2.3. Measuring directionality time from simulated 2D-PBRWmigration paths that resemble experimental data. (a) A schematic of the2D-PBRW model and its corresponding random variables. A simulated randomwalker travels with constant speed v in a straight line with polar angle Θn. Aftertraveling in one direction for a run time Tn, the random walker reorients andcontinues in a new direction, Θn+1. For each random walk, positions were sam-pled in increments of sampling interval ∆t. Measurement error σm was addedto each sampled position coordinate to generate migration paths resemblingexperimental data. (b) Run times, Tn, were randomly selected from Poissondistributions with average run times of tp, known as the persistence time. (c)Polar angles, Θn, were randomly selected from the von Mises distribution withbias factor κ. The value κ= 0 corresponds to unbiased motion. Increasing κcorresponds to more directional bias. (d) Log-log MSD slopes showing β(τ) fortwo values of centroid measurement error: σm=0, 1µm (black and red, respec-tively; ×’s for τ < τmin and •’s for τ > τmin; parameters tp = 3.6 s, κ = 1.5,v= 0.3µm/s, ∆t = 1 s; ensemble sizes of n = 400). The minimum time inter-val above which the βBRW-model fit decouples from the effects of measurementerror and persistence was estimated at τmin = 50.4 s. Fitting the βBRW-modelto data above this minimum time interval (green curve) gave virtually identicaldirectionality times for both values of measurement error. Data in the τ < τmin
domain (×’s in the shaded region) were not used for fitting. A single parameterexponential fit function, 2− e−t/td (cyan curve), also fit in the τ > τmin domain,shows that a heuristic fit model does not accurately measure directionality time.
23
0 5 1 0
0
0.5
1.0
von Mises bias factor, κta
ng
en
t-b
ias
corr
ela
tio
n, <co
s(Θ
) >
Figure 2.4. Tangent-bias correlation, c = 〈cos Θ〉, plotted against thevon Mises bias factor, κ. This curve shows the correspondence betweenc used in the analytically derived biased random walk models, and κ used inthe simulated biased random walk model. This curve is calculated analytically
from the definition of the von Mises distribution. Specifically, 〈cos Θ〉 = I1(κ)I0(κ)
where In is the modified Bessel function of the first kind, order n. To sample csomewhat evenly, values κ = 0, 0.5, 1, 1.5, 2, 6, 10 were used in the simulationsof persistent biased random walks discussed in this manuscript.
adding a measurement error term to the EASD of the PBRW model and calculating the
time above which this modified PBRW model converges to within 5% of the βBRW-model:
tσm ≈ 4.5√
2dσm
vrms(∞). (2.11)
(Fig. 2.5) Here, d is the number of dimensions, and vrms(∞) = limτ→∞
vrms(τ) = |c|v is the
root mean squared (RMS) speed asymptote, a measurable quantity that is sampling interval
independent.
24
0 0.5
ε=10
ε=5ε=2ε=0
1.00
50
100
conv
erge
nce
time,
t BRW
(λ-1
)
+tσm
exact
tangent-bias correlation, <cos(Θ)>2
Figure 2.5. Time scales at which βPBRW(t) converges to the βBRW-model when measurement error is nonzero (σm > 0). Times abovewhich the difference between βPBRW(t) and the βBRW-model is less than 5% are
shown as solid curves correspond to values of the constant ε =√
2dσmλ+
v, which
is proportional to measurement error. The convergence time is estimated by theequation λ+tσm ≈ 4.5C
c, as derived in Appendix A. When implementing a βBRW-
model fit to measure directionality time, β(τ) curves are fit at time intervalsτ > tσm to decouple directionality time from measurement error.
A minimum fitting time, τmin, is defined as the larger of tBRW and tσm . In this example,
τmin = tσm ≈ (4.5)(2) (1µm)(0.6)(0.3µm/s) = 50.4 s. Data in the τ > τmin domain (Fig. 2.3 d, •’s)
were fit to the βBRW-model (green curve) illustrating that the effects of measurement error
and persistence can be decoupled from measures of directionality time by leaving out data
below the time interval τmin. Although backwards extrapolation is required to measure
directionality time in this example, backwards extrapolation is not generally required (i.e.
when td > τmin).
The possibility of using a heuristic model to measure directionality time is investigated
by fitting the data in the τ > τmin domain to a single parameter exponential function, 2−
e−t/td (Fig. 2.3 d, cyan curve). This model does not fit the data as well as the βBRW-model,
returning overestimates of directionality time. Overall, measurements of directionality time
using a heuristic model such as this generally satisfy the overall objective of measuring the
time scale at which motion transitions from random to directional. However, lost when
25
using a heuristic fit model is the consistency and interpretability of directionality time with
respect to an analytical framework that characterizes directional motion.
Simulated migration paths corresponding to a range of parameter combinations (tp, κ, σm)
were analyzed to answer the following questions related to the robustness of the βBRW-
model. For what type of motion does the βBRW-model fit well, and when does it fail? And
to what extent does directionality time decouple from measurement error by fitting at times
τ > τmin? Goodness of fit was tested using the reduced chi-squared metric, χ2ν (Fig. 2.6 a,
left column). Fits labeled “good” (unshaded regions) were those with χ2ν ≤ 1. All other fits
were labeled “problematic” (yellow shaded regions). The βBRW-model fits were good over
most of the parameter space. For non-directional motion (κ = 0), many of the βBRW-model
fits were problematic, although the model still returned characteristically large directional-
ity times. The βBRW-model fits were also problematic for directional motion with small tp
and td compared to the time scale of noise, tσm (i.e. tp and td � tσm). Measurements of
directionality time indicated that values of td increased slightly as σm increased (Fig. 2.6 a,
right column). This weak monotonic coupling between td and σm occurs because measure-
ment error decreases the value of β(τ) at all time scales, thus slightly increasing the value
of td obtained from fitting to the βBRW-model. This coupling was only significant at the
lower limits of tp (Fig. 2.6 b). Overall, these simulation results show that if motion is direc-
tional and tp and td are not masked by the time scale of noise, tσm , then the βBRW-model
fits robustly and returns measurements of directionality time that are negligibly skewed by
measurement error.
For comparison, the sampling interval dependent metrics (TAD, TAD persistence, tor-
tuosity, and tangent-tangent correlation) were calculated for the simulation trajectories
(Fig. 2.7). As expected, the numerical values associated with these metrics varied sig-
nificantly with sampling interval, coupling strongly to measurement errors at the smaller
sampling intervals.
26
0 50 100
0.5
1.0
dire
ctio
nal m
igra
tion
e�
cien
cy,
<R(t) > / v
t
directionality time, td (tp)
σ m (µ
m)
012
+λ-1 (s)
3.60.4
<R(t)> vt0x
r
y ρR
σc = 2µm
σc = 1µm
σc = 0µm
von Mises bias factor, κ0 2 6 10 0 2 6 10
directionality time, td (s)
0 25 50
robustness
problematic �tgood �t
0.41.22.02.83.6
0.41.22.02.83.6
0.41.22.02.83.6pe
rsis
tenc
e tim
e, t p
(s)
a b
Figure 2.6. Additional data plots investigating the robustness of thedirectionality time model. (a) Goodness of fit and directionality time mea-surements across a range of parameter combinations (same parameters as inFig 2.3 d but ∆t = 10 s and tp, κ, σc all vary). Squares on phase diagramsindicate parameter combinations where ensembles of random walks were sim-ulated (n = 4000 migrations per ensemble). Goodness of fit was measured by
calculating reduced chi squared, χ2ν =
∑n
(β(τn)−βBRW(τn;td))2
σ2β(τn)
, where σβ is the
standard deviation (spread) on values of β(τn) used for fitting. A fit was cat-egorized as “good” if χ2
ν ≤ 1, otherwise it was categorized as “problematic.”The directionality time model worked robustly except near κ = 0 and when tpand td � τmin. The latter is the limit where measurements of directionalitytime can no longer be resolved. (b) Directional migration efficiency is inverselyproportional to directionality time (in units of tp). Directional migration effi-ciency is defined as the ensemble averaged distance travelled 〈R(t)〉 divided bythe distance a ballistic walker would have travelled if it had the same speed, vt.This plot shows that td is a proxy for the distance a walker will travel, and thattd best decouples from measurement error as persistence time tp increases.
2.4. Application to Real Data. Under idealized circumstances, the βBRW-model
may be used to fit ensembles of data. However, as we saw with the simulated data, adding
measurement error made the data less ideal. Small ensemble sizes, nonergodicity, and
changing experimental conditions all cause experimental data to vary. To improve the
27
turning angle (rad)
−π π π−π
20
2
pro
ba
bil
ity
0
0.1
0.2
0.7
0 µm0.1µm
1s 4s 20s
sampling interval
∆tσc = ...
0 50 1000.2
0.4
0.620s
4s
1s
0.8
1.0
time interval, τ (s)
tan
ge
nt-
tan
ge
nt
corr
ela
tio
n
ba
c
1s 4s 20s0.8
0.9
1.0
pe
rsis
ten
ce
sampling interval
1s 4s 20s0.4
0.6
0.8
1.0
tort
uo
sity
sampling interval
Figure 2.7. Sampling interval dependent metrics applied to simulationdata (2D-PBRW, tp = 3.6 s, κ = 1.5, and v = 0.3µm/s). The position ofthe random walk was sampled every ∆t = 1, 4, or 20 s, with sampling errorsof σm = 0 (blue bars and thin blue curves) or 0.1µm (green bars and thickgreen curves). (a) Ensemble averaged turning angle distributions (TAD) basedon sampling intervals of 1 s (solid curves) and 4 s (short-dashed curves). TADpersistence is the fraction of all turning angles between ±π
2(inset, error bars
are standard error of the ensemble mean). When σm = 0, TAD persistence issmallest when ∆t ≈ tp. This is time scale of reorientation and consequently thesampling interval with which the motion appears most random. Small amountsof measurement error (σm = 0.1 µm) hide the persistence of motion that wouldotherwise be measured at the smallest sampling interval ∆t = 1 s. Thesedata show that TAD persistence is sampling interval dependent. (b) Ensembleaveraged tortuosity (error bars are standard error of the ensemble mean). Aswith TAD persistence, tortuosity is sampling interval dependent, increasing with∆t. (c) Tangent-tangent correlation curves. Tangents are calculated based onthe forward displacement between nearest sampled points. Therefore tangent-tangent correlation is sampling interval dependent. In particular, randomnessat short time scales are not resolved as ∆t increases. The persistence time,tp, can be back-measured from the tangent-tangent correlation curves if thereis sufficient temporal resolution (∆t ≤ tp). Each of these metrics are samplinginterval dependent and couple to measurement error at short sampling intervals.Hence, these metrics are not generally comparable from one experiment to thenext.
reliability and accuracy of the directionality time measurement, we now examine three
causes of deviation between experimental measurements of β(τ) and the βBRW-model. These
causes of deviation are: position variance, persistence, and nonergodicity. The causes of
28
position variance can be further subdivided into three categories: implicit position variance,
measurement error, and parametric variance. Methods for handling these deviations are
addressed below.
Implicit position variance is already accounted for in the βBRW-model and does not
cause deviations. Deviations caused by persistence and measurement error induced position
variance can be decoupled from the βBRW-model by fitting above a minimum fit time τmin, as
discussed above (c.f. Fig. 2.3 d). Therefore, one only needs to be concerned with deviations
caused by parametric variance and nonergodicity.
Parametric variance, denoted σ2p, is the variance in distance traveled caused by variance
of random walk parameters across the ensemble (population heterogeneity). For example,
the variance in distance traveled at long time scales due to a spread in the instantaneous
speed parameter, δv, is σ2p = c2(δv)2t2. Parametric variance systematically increases mea-
surements of EASD and is the dominant cause of deviation between β(t) and the βBRW-
model at large time scales (see Eq. 2.37). Deviations caused by parametric variance cannot
be corrected by fitting above a minimum fit time, τmin.
Ergodicity is the conversion factor that maps EASD to TASD (and MSD), given in
Eq. 2.3. When a process is ergodic, EASD and MSD are equivalent, and so are calculations
of β based on either EASD or MSD. When a process is nonergodic, for example when the
instantaneous migration speed changes significantly over time across the entire ensemble,
measurements of β(τ) based on MSD will deviate from the βBRW-model which is derived
from the EASD. Deviations caused by ergodicity occur at all time scales and, like deviations
caused by parametric variance, cannot be corrected for by fitting above a minimum fit time,
τmin.
We show in Appendix B (Sec. 4.3, Eq. 2.44) that log-log MSD slope β(τ) can be de-
composed into three terms that correct for deviations caused by position variance and
29
ergodicity:
β(τ) = βBRW(τ) + βσ(τ) + βξ(τ). (2.12)
The term βσ accounts for deviations caused by position variance (primarily parametric
variance), and βξ accounts for deviations caused by nonergodicity. The former is calcu-
lated by using experimentally observed variances in distance traveled to estimate how much
parametric variance and measurement error have an effect on MSD, and the corresponding
log-log MSD slope. The resulting position variance modification term is
βσ(τ) =d log
[〈r2(τ)〉
〈r2(τ)〉−σ2r(τ)
]d log[τ ]
(2.13)
where σ2r (τ) is the variance on the mean distance traveled at time interval τ (see Appendix B,
Sec. 4.3, for the derivation). The ergodicity term can also be calculated from experimental
data using the relation βξ(τ) = d log[ξ(τ)]d log[τ ] , but this term can be relatively noisy because
time averaging cannot be applied to reduce statistical noise. Often, migration data can be
made nearly ergodic by choosing to analyze truncated migration segments over which the
instantaneous speed remains relatively constant. In such cases, no ergodicity correction is
necessary. When the migration paths cannot be partitioned into ergodic segments, βξ(τ)
should be estimated by simulation rather than directly calculated from the experimental
data. A recipe for these simulations can be found in Appendix B (Sec. 4.3).
2.5. Neutrophil Chemotaxis. In this section, an implementation of the βBRW-model
to measure directionality time is demonstrated on migration paths of chemotactic human
polymorphonuclear neutrophils following the step-by-step procedure outlined in Fig. 2.8.
Neutrophil migration paths, r(t), and centroid measurement errors, σm, were obtained
from O’Brien et al. [1]. Two sets of data were analyzed, each containing the trajectories of
chemotactic neutrophils migrating on the 2D surfaces of polyacrylamide gels (Young’s modu-
lus: 10 kPa) towards a source of the chemoattractant formyl-methionyl-leucyl-phenylalanine.
30
The difference between the two data sets was the coating on the gel surface, either human
fibrinogen (Fgn) or human type IV collagen (Col IV). Tangent-tangent correlation measure-
ments gave a persistence time, tp, upper bound of 5 s (Fig. 2.10 c). EASD was calculated but
was noisy. Therefore, MSD was also calculated to reduce the statistical noise (Fig. 2.9 a).
Data was checked for ergodicity by measuring the ensemble averaged instantaneous speed
(EAIS) over time. EAIS was approximately constant when the migration paths were trun-
cated at 400 s and no ergodicity correction was applied to this data. If EAIS were changing
significantly in time, the ergodicity correction term βξ(τ) could have been calculated using
the recipe in Appendix B (Sec. 4.3).
track migration:
determine positions, r(t)
& measurement error, σc
calculate EASD, <r2(t)>Is the data noisy?
calculate ergodicity, ξi2(τ)
& ergodicity correction βξ(τ)
via simulations
measure persistence time,
tp, from tangent-tangent
correlation
measure rms speed
asymptote, vrms(∞)
calculate minimum
βBRW-model !tting
time scale, τmin
Fit to βBRW-model to measure
directionality time, td
equate MSD with EASD,
<r2(τ)> = <r2(t=τ)>Is the data nonergodic?
calculate MSD, <r2(τ)>
yes
noyes
no
calculate log-log MSD slope
β(τ) and correct with terms
βσ(τ) and, if applicable, βξ(τ)
Figure 2.8. Step-by-step flow-chart for processing experimental mi-gration data to measure directionality time.
Next, vrms(∞) was calculated (Fig. 2.9 a, bottom-right inset): vrms(∞) ≈ 0.15 and
0.13µm/s for Fgn and Col IV, respectively. Taken together with tp, the minimum fit time
was calculated using Eq. 2.11 for both data sets: τmin ≈ 60 s and 69 s for Fgn and Col
IV, respectively. These minimum fit times were used later to fit the variance corrected
31
a
b
101 20 30 50 200 300102
101
102
103
104
MS
D, <r
2> (µ
m2)
Fgn Col IV
50µm
0.1
0.2
0.3
100 2000 300
time interval, τ (s)
RM
S s
pe
ed
, vrm
s
(µm
/s)
time interval, τ (s)
slope = 1
slope = 2
0 50 100 150 200 250 300
0
0.5
1.0
1.5
2.0
2.5
log
-lo
g M
SD
slo
pe
, β(τ
)
time interval, τ (s)
0
20
30
10
Fgn Col IV
dir
ect
ion
ali
ty t
ime
, td (
s)
ligand
Fgn
Col IV
Figure 2.9. Measuring directionality time from experimental data. (a)The observed mean squared displacements (MSDs) of chemotactic human poly-morphonuclear neutrophils migrating on fibrinogen (Fgn, blue, circles, n=25)and collagen IV (Col IV, red, squares, n=19) coated polyacrylamide gels witha Young’s modulus of 10 kPa. The shaded regions indicate the standard errorsof the MSD. The corresponding trajectories (top-left) and RMS speed (bottom-right) are also shown. The RMS speed asymptotes, vrms(∞), for Fgn and ColIV are approximately 0.15 and 0.13µm/s, respectively. (b) Log-log MSD slopescorrected for position variance, β−βσ, are plotted against time interval τ and fitto the βBRW-model (solid curves) to obtain directionality times, td. The shadedregions indicate standard errors on log-log MSD slope at time intervals abovethe minimum fitting time. The resulting directionality times are depicted in thebar graph to the bottom-right, with error bars corresponding to 68% confidenceintervals.
32
log-log MSD slope, β − βσ (see Fig. 2.9 b), to the βBRW-model and time intervals τ > τmin
(regions with shaded error bars). The resulting directionality times were 10.7 ± 1.0 s and
21.2 ± 5.1 s for Fgn and Col IV, respectively (Fig. 2.9 b, bottom-right inset). With 19
neutrophil trajectories in the Col IV data set, compared to 25 in the Fgn data set, the
Col IV fit was noisier. For comparison, the sampling interval dependent metrics (TAD,
TAD persistence, tortuosity, and tangent-tangent correlation) were calculated for this data
(Fig. 2.10). The numerical values of these measurements vary significantly with sampling
interval and their coupling to measurement errors is unknown.
3. Discussion and Future Direcitons
We have completed a three part description of directionality time, beginning with an
analytical derivation, checked by computational simulations, and applied to noisy real world
ensembles of directed migration paths. In comparison to the sampling interval dependent
metrics (see Figs. 2.7 and 2.10), the directionality time metric is nearly independent of
parameters constrained by the experimental apparatus and/or chosen arbitrarily by humans.
Whereas speed and persistence time are sufficient for characterizing non directed migration,
one additional metric, directionality time, along with speed and persistence time suffices to
characterize directed migration.
The theoretically derivation of the directionality time model and its corrections to ac-
count for measurement error (βσ) and nonergodic motion (βξ) are a step forward for the
analysis of directed motion. Of the two corrections, however, only measurement error was
fully demonstrated here. The experimental data used in this demonstration was nonergodic,
but made to be ergodic by constraining the analysis time window to 400 s. In Fig. 2.9 b,
the experimentally measured log-log MSD slope was corrected for measurement error using
Eq. 2.13. An explicit demonstration of the difference between the non corrected (β) and
measurement error corrected (β−βσ) is shown in Fig. 2.11. Without the measurement error
correction, the corresponding log-log MSD slopes asymptoted to values of approximately
33
100 200 30000.2
0.4
0.6
0.8
1.0
time interval, τ (s)
tan
ge
nt-
tan
ge
nt
corr
ela
tio
n
60s
10s
turning angle (rad)
−π ππ−π
20
2
10s 60s0.6
1.0
0.8
turn
ing
an
gle
pe
rsis
ten
ce
∆t
pro
ba
bil
ity
60s
10s
0
0.05
0.10
0.15
0.20a b
c
ColFgn
10s 60s
sampling interval,
∆t
Fg
n
Co
l IV
tort
uo
sity
10s 60s0.4
0.6
0.8
1.0
∆t
Fg
n
Co
l IV
Figure 2.10. Sampling interval dependent metrics applied to chemo-tactic polymorphonuclear neutrophils on fibrinogen (Fgn, blue colors)and human collagen IV (Col IV, red colors) coated polyacrylamidegels with elastic modulus of 10 kPa. (a) Ensemble averaged turning an-gle distributions (TAD) are plotted based on measurements of turning anglesat time intervals of 10 s (solid curves) and 60 s (dashed curves). TAD persis-tence, calculated as the fraction of all turning angles between ±π
2, is shown in
the inset (error bars are standard error of the ensemble mean). (b) Ensembleaveraged tortuosity measured at the same sampling intervals (error bars arestandard error of the ensemble mean). (c) Tangent-tangent correlation curvesalso measured at sampling intervals ∆t = 10 and 60 s. Tangents are calculatedbased on the forward displacement between nearest sampled points. Thereforetangent-tangent correlation curves show more correlation when calculated atthe 60 s sampling interval, compared to the 10 s interval. For both Fgn andCol IV, tangent-tangent correlation curves drop towards their asymptote at apersistence time of tp < 10 s. Regardless of the sampling interval, chemotaxison Fgn is more correlated than chemotaxis on Col IV, a result that is consis-tent with our measurements of directionality time. Each of these metrics aresampling interval dependent and couple to measurement error. Hence, thesemetrics are not generally comparable from one experiment to the next.
1.7 and 1.85 (Col IV and Fgn, respectively), instead of 2 as predicted by the βBRW-model.
With the measurement error correction, these asymptotes came closer to a value of 2, at
1.8 and 1.9, respectively. Overall, the measurement error correction moved the experimen-
tally measured log-log MSD slope closer to the analytically derived βBRW-model, a partial
indication that the correction was valid.
34
The erogidic correction, βξ, needs to be analyzed in a similar manner. First, the βξ cor-
rection should be confirmed using measurement error free (clean) nonergodic data generated
by computer simulation. Second, the βξ correction should be demonstrated explicitly with
nonergodic experimental data. These two tasks have begun in earnest, with the latter being
done on a mix or ergodic and nonergodic neutrophil migration paths collected by O’Brien
et al. to determine the precise role of extracellular matrix composition on mechanosensing
and neutrophil migration.
Part of this work is discussed in the next chapter, but there is still no demonstration
of the βξ correction because the findings of the next chapter were made at a time before
we took ergodicity into consideration. Instead, in the next chapter, we use a heuristically
derived directionality time model that fits the data well and provides a good relative com-
parison of directionality times across the datasets it is applied to. The heuristic measure of
directionality time is used to determine that neutrophils regulate chemotactic honing as a
function of substrate stiffness using the β2-integrin, CR3.
0 50 1001.0
1.5
2.0
time interval, τ (s)
β−βσ, Fgnβ−βσ, Col IV β, Fgn β, Col IV
log
-lo
g M
SD
slo
pe
, β(τ
)
Figure 2.11. Explicit demonstration of the measurement error correc-tion on experimentally measured log-log MSD slope. Log-log MSD slopedata correspond to Fig. 2.9. Compared to the non corrected measurements oflog-log MSD slope (β), the measurement error corrected slopes (β − βσ) havelong time interval asymptotes closer to 2. Therefore, the measurement error cor-rected data points are better explained by the βBRW model, which asymptotesto 2.
35
4. Supplementary Material
4.1. Methods.
4.1.1. Migration Paths and Centroid Measurement Error. Differential interference con-
trast (DIC) image sequences capturing directionally migrating Polymorphonuclear Human
Neutrophils were obtained from Ref. [1] along with cell centroid positions, ri(tn), where i is
the migration path index and tn is the time measured in multiples of the sampling interval,
tn = n∆t (n = 1, 2, ...). Centroid measurement error was estimated as follows. A cell
was chosen at random and manually outlined five times. The five corresponding centroid
positions were determined using the regionprops algorithm in MATLAB (the MathWorks;
Natick, MA). The centroid measurement error of that cell, σm, was calculated as the RMS
displacement from the mean centroid position.
4.1.2. 2D PBRW Simulations and Data Fits. All simulation data were generated in
MATLAB. All data was fit using custom MATLAB software. This code is available for
download at http://www.github.com/aloosley (see Chapter 7, Sec. 1, pg. 170). Data fits
were calculated using the Levenburg-Marquardt least squared fitting algorithm [98] built
into MATLAB.
4.2. Appendix A: Analytical Modeling.
4.2.1. Appendix A Introduction and Notation. In this appendix, three random walk
models are discussed. Positions, R, are specified by a two dimensional (2D) Cartesian
coordinate system with unit vectors ex and ey (Fig. 2.1 c). The orientation of movement
is described by polar angle θ (π < θ ≤ π) defined with respect to the positive x-axis. The
random walker begins by moving with speed V1 in a particular direction Θ1 for a particular
time T1, before reorienting and moving with a new speed V2, in a new direction Θ2, for
another time period given by T2. This process continues and is altogether described by
speeds Vn, angles Θn, and step durations Tn (n = 1, 2, ...), which are random values of
speed v, polar angle θ, and time t, respectively. Each random variable is associated with a
36
notation usageδt simulation time step
∆t sampling intervalt timeτ time interval
bold variables vectorsCAPITALIZED variables random variables
ρT(t)
probability density function (PDF),(the probability of measuring random variable
T between the values t and t+ dt)
Rrandom variable for position used in analytical models(corresponding to the PDF, ρ
R(r))
Θrandom variable for 2D orientation(the polar angle of the instantaneous velocity vector)
overline average over time t〈〉 average over ensemble or analytical PDF
c = 〈cos Θ〉 tangent-bias correlation (averaged over ρΘ(θ))
r experimentally measured positionv instantaneous velocity
vrms root mean squared speedφ turning angle (not to be confused with polar angle θ)
v(t+ τ) · v(t) tangent-tangent correlationtp persistence timetd directionality time
Table 2.1. Mathematical notation used to derive directionality timeand interpret cell motility.
probability density function (PDF), ρV (v), ρΘ(θ), and ρT (t). For each random walk model
described in this appendix, we define the relevant variables, parameters, and distributions
using the mathematical notation summarized in Table 2.1, and use these to determine the
ensemble averaged squared displacement (EASD), 〈R2(t)〉, and log-log EASD slope, β(t).
From beta(t), we derive the model specific meaning of directionality time, td, the measure
that is central to this work. A universal, random walk model invariant fit function is derived
to measure td from EASD data.
4.2.2. Model 1: Drift Diffusion (DD). A walker undergoing a Diffusion in 2D (without
drift) obeys the following rules:
37
(1) Stepwise Position: The position after n steps is given by the random variable
Rn =n∑j=1
VjTj [cos(Θj)ex + sin(Θj)ey].
(2) Constant step duration: ρT (t) = δ(t− λ−1), i.e. Tj = λ−1 for all j
(3) No directional bias: ρΘ(θ) = 12π where −π < θ ≤ π
(4) Constant track speed: Vj = v
where δ(...) is the Dirac delta function, λ is the reorientation frequency, and the other terms
are defined in the Appendix A Introduction. The notion for average step duration, λ−1, is
used for consistency with the models below. The addition of drift to a diffusion process adds
an additional term to the stepwise position equation. Without loss of generality, one can
define a drift velocity along the x-axis, u = u ex. Subsequently, Rule 1 would be redefined
as
1’. Stepwise Position: The position after n steps is given by the random variable
Rn =n∑j=1
v
λ[cos(Θj)ex + sin(Θj)ey] +
nu
λex.
One example of DD would be the motion of an object trapped at an air-water interface
in the presence of a surface current (air or water). If the object were charged, drift could
also be induced by applying a uniform electric field (i.e. carrier drift) [99].
The ensemble averaged squared displacement (EASD) of a DD in d dimensions has been
shown to be ⟨R2(t)
⟩= 2dDt+ u2t2 (2.14)
where D ≡ v2
λ is the so-called diffusion constant [93]. At large times, motion is directional
(〈R2(t)〉 → u2t2 ∼ t2) with a root mean squared (RMS) speed, vrms(t) =
√〈R2(t)〉t , that
asymptotes to vrms(∞) = u. Differentiation in log-log coordinates
(d log[〈R2(t)〉]
d log[t] = t〈R2(t)〉
d〈R2(t)〉dt
)gives the log-log EASD slope
βDD(t; t0) =1 + 2 t
t0
1 + tt0
(2.15)
where time constant, t0 = 2dDu2 , incorporates both model parameters D and u. Defining
directionality time as the time when β = 32 provides a time scale where the migration
38
transitions from diffusive to directionally drifting. This definition gives td = t0 or
td =2dD
u2. (2.16)
The greater the diffusion constant and/or lower the drift velocity, the more time is required
to observe a directional bias.
4.2.3. Model 2: 2-D Stepping Biased Random Walk (2D-SBRW). A walker undergoing
a 2-D Stepping Biased Random Walk (2D-SBRW) obeys the following rules:
(1) Stepwise position: The position after n steps is given by the random variable
Rn =
n∑j=1
Lj [cos(Θj)ex + sin(Θj)ey].
(2) Random step length: Lj is drawn from a PDF ρL(l)
(3) Directional Bias: Θj is drawn from a circular distribution (i.e. von Mises,
Wrapped Normal, etc.[97]) with PDF ρΘ(θ) that is symmetric about θ = 0 (−π <
θ ≤ π).
(4) Constant track speed: Vi = v
It can be shown that the EASD after n steps of a 2D-SBRW is given by
⟨R2n
⟩= n
⟨L2⟩
+ n(n− 1) 〈L cos Θ〉2 + 〈L sin Θ〉2 (2.17)
(see Ref. [94] for a derivation). If the orientation angles {Θj} are independent of the step
lengths {Lj} and the directional bias PDF is symmetric about θ = 0 such that 〈sin Θ〉 = 0
(c.f. rule 3), then the EASD simplifies to
⟨R2n
⟩= n
⟨L2⟩
+ n(n− 1)〈L〉2c2 (2.18)
where c = 〈cos Θ〉 is called tangent-bias correlation because it correlates the random walk
orientation (tangent vectors) to the direction of bias (θ = 0). Squared tangent-bias correla-
tion, c2, ranges from 0, corresponding to no orientational bias(ρΘ(θ) = 1
2π where − π < θ ≤ π),
to 1, corresponding to maximal orientational bias (ρΘ(θ) = δ(θ) or δ(θ − π)).
39
Note that the motion is diffusive (〈R2n〉 ∼ n) when n is small and directional (〈R2
n〉 ∼ n2)
when n is large. This step by step random walk has no persistent directionality at low n
because the direction of motion changes with every step, n. In order to calculate log-log
EASD slope, β(t), an average track speed v is defined and the relation n ≈ vt〈L〉 is used to
transform 〈R2n〉 → 〈R2(t)〉. In the time representation, this is a model of a biased random
walk (BRW). Converting the step lengths into step durations (L = vT ) gives
〈R2(t)〉 ≈ v2
[(〈T 2〉〈T 〉2
− c2
)〈T 〉t+ c2t2
]. (2.19)
Like with DD, the t2 term dominates at large times and RMS speed asymptotes to
vrms(∞) = |c|v. (2.20)
Differentiating in log-log coordinates
(d log[〈R2(t)〉]
d log[t] = t〈R2(t)〉
d〈R2(t)〉dt
)gives the log-log EASD
slope
βBRW(t; t0) =1 + 2t
t0
1 + tt0
(2.21)
where t0 = 〈T 〉
(〈T2〉〈T 〉2−c2
c2
). Note that βBRW has the same functional form as βDD (Eq. 2.15).
Only the mathematical constants have changed. Analogous to the DD treatment, defining
directionality time at the transition point, β = 32 , gives td = t0, or
td = 〈T 〉
〈T 2〉〈T 〉2 − c
2
c2
. (2.22)
This generalized directionality time equation is more easily understood by considering an
example. Consider the case where the probability of measuring a step length T between t
and t+ dt is given by a Poissonian PDF, ρT (t) = 1tp
e− ttp . Then, 〈T 〉 = tp, 〈T 2〉 = 2t2p, and
directionality time simplifies to
td = tp
(2− c2
c2
). (2.23)
40
Importantly, directionality time is independent of speed v. It depends only on the average
step duration tp and the extent to which the orientation is biased when a reorientation
occurs (given by c2). Directionality time decreases when either the reorientation rate (1/tp)
or orientational bias (c2) increase. In particular, the term 2−c2c2
scales from 1 at maximal
bias, to ∞ at no bias. It may appear odd that td → tp (the average time of one step) when
the system is perfectly directional (c → 0). However, this is no more than a subtlety of
stepping random walks. Based on the way EASD was converted from the stepping number
domain to the continuous time domain, positions are only allowed to change in increments
of 〈T 〉 = tp. Therefore, the minimum time to determine movement is directionally biased
will always be greater than or equal to tp. No information can be gained from a random
walker that has not yet taken any steps. Since each step is accompanied by reorientation,
this model cannot be used to derive a log-log EASD slope equation that accounts for short
time scale directionality known as persistence. In order to consider the relation between
directionality time and persistence, we consider a continuous time random walk model in
the following section.
4.2.4. Model 3: 1-D Persistent Biased Random Walk (1D-PBRW) in Continuous Time.
A walker undergoing a 1-D Persistent Biased Random Walk (1D-PBRW) obeys the following
rules:
(1) Stepwise position: The position after n steps is given by the random variable
Xn =
n∑j=(odd)
VjT(r)j −
n∑i=(even)
VjT(l)j
, where l and r indicate movement to the
left (-x direction) and right (+ x direction), respectively.1
(2) Variable leftward and rightward step durations: The leftward and rightward
durations are drawn from Poissonian PDFs: ρT
(k)(t) = λke−λkt, where k = l, r.
1Although the stepwise position is useful for introducing the model, below we derive the positiondistribution in continuous time.
41
(3) Directional Bias: Set by the difference between the average rightward and left-
ward step durations in rule 2: c ≡ 〈cos Θ〉 = 〈T (r)〉−〈T (l)〉〈T (r)〉+〈T (l)〉 = λl−λr
λl+λr
(4) Constant track speed: Vj = v
The 1D-PBRW walker alternates between two possible directional states, traveling left (l)
or right (r). To allow for persistence, a PDF for the position of the PBRW object, ρX (x, t),
is derived in continuous time from a partial differential equation (as opposed to from the
stepwise position). It can be shown (see Refs. [93, 96]) that PDFs of a 1D-PBRW are
solutions of the biased telegrapher equation
∂2ρX∂t2
+ (λl + λr)∂ρX∂t
+ v (λl − λr)∂ρX∂x
= v2∂2ρX∂x2
(2.24)
where parameters λ−1l and λ−1
r are the average walk times in the leftward and rightward
direction (between alternations), and v is the instantaneous speed (c.f. rules 2 and 4).
Ensemble averaged displacement (EAD), 〈X(t)〉 =∫∞−∞ xρX (x, t)dx, can be calculated in
two steps. Step one, multiply both sides of Eq. 2.24 by x. Step two, integrate by parts over
the entire x-axis. Assuming the position distribution ρX (x, t) and its first spatial derivative
∂ρX
∂x tend to zero as x→ ±∞, integration by parts gives the ordinary differential equation
d2 〈X(t)〉dt2
+ λ+d 〈X(t)〉
dt= vλ− (2.25)
where parameters λ± ≡ λl±λr are introduced to simplify the equations. The frequency λ−
defines the degree of directional bias: leftward when λ− < 0 and rightward when λ− > 0.
The frequency λ+ is inversely related to the average step duration, tp = 12
(λ−1l + λ−1
r
),
known as persistence time [93].
Solving Eq. 2.25 with initial conditions 〈X(t)〉|t=0= 0 and d〈X(t)〉dt |t=0= 0 gives the EAD
〈X(t)〉 = vλ−λ+
[t− 1
λ+
(1− e−λ+t
)]. (2.26)
Note that when there is no bias (i.e. λ− = 0), EAD is identically equal to 0.
42
The EASD, 〈X2(t)〉 =∫∞−∞ x
2ρX (x, t)dx, is calculated analogously to the EAD. Mul-
tiplying both sides of the biased telegrapher equation (Eq. 2.24) by x2 and integrating
gives an ordinary differential equation containing 〈X(t)〉 and 〈X2(t)〉 terms. Substitution
of 〈X(t)〉 using Eq. 2.26 gives the ordinary differential equation
d2⟨X2(t)
⟩dt2
+ λ+d⟨X2(t)
⟩dt
= 2v2
{1 +
λ2−λ+
[t− 1
λ+
(1− e−λ+t
)]}. (2.27)
Solving for 〈X2(t)〉 using the initial conditions 〈X2(t)〉|t=0= 0 and d〈X2(t)〉dt |t=0= 0 gives the
EASD
⟨X2(t′)
⟩=
v2
λ2+
[2(3c2 − 1
) (1− e−t′
)+ 2
(1− c2
(2 + e−t
′))
t′ + c2t′2]
(2.28)
where t′ ≡ λ+t is nondimensionalized time and c = λ−λ+
is the tangent-bias correlation as
described in the previous section (c.f. rule 3). When there is bias (c2 > 0),⟨X2(t)
⟩→ c2v2t2
in the large time scale limit (t′ → ∞), corresponding to directional motion with an RMS
speed asymptote
vrms(∞) = |c|v (2.29)
just like in the BRW model (model 2). At maximal bias, c2 = 1, the walker travels in a
straight line with speed v. If there is no bias, c2 = 0, the t′2 term vanishes and the t′ term
indicative of diffusive motion dominates at large time scales (t′ →∞). In this special case,
EASD simplifies to the persistent random walk (PRW) equation commonly used to model
the movement of chemokinetic cells,
⟨X2(t)
⟩= 2v2tp
[t− tp
(1− e−t/tp
)](2.30)
[59, 61, 93], where tp ≡ λ−1+ = 1
2〈T(l)〉 = 1
2〈T(r)〉 is the so-called persistence time, half the
average time the walker moves in one direction before switching to the other.
43
The log-log EASD slope
(d log[〈X2(t)〉]
d log[t] = t〈X2(t)〉
d〈X2(t)〉dt
)is
βPBRW(t′; c) =(2− 4c2)t′ + 2c2t′2 + (4c2 − 2)t′e−t
′+ 2c2t′2e−t
′
(6c2 − 2) + (2− 4c2)t′ + c2t′2 + (2− 6c2)e−t′ − 2c2t′e−t′. (2.31)
These curves are plotted for multiple values of tangent-bias correlation c in Fig. 2.2 a
(solid curves). Of particular interest is the form of a directionality time that would be
calculated from the log-log EASD slope in Eq. 2.31. Persistence over short time scales
causes a βPBRW(0) = 2 and instead of a monotonic increase of βPBRW(t) from 1 to 2 as
with βDD(t) and βBRW(t) (Eqs. 2.15 and 2.21, respectively), βPBRW dips towards a value of
1 before eventually asymptoting back towards 2 as t→∞. At sufficiently large time scales,
the constant terms and terms with decaying exponentials in Eq. 2.31 become negligible and
βBRW(t′; t′0 > t′BRW) ≈1 + 2 t
′
t′0
1 + t′
t′0
(2.32)
where t′0 = 2(1−2c2)c2
(Fig. 2.2 a, dashed curves). This is known as the βBRW-model. The
convergence time scale above which the βPBRW(t) is approximated by the βBRW-model in
Eq. 2.32, denoted t′BRW, is defined as the time above which∣∣∣βPBRW(t′)−βBRW(t′; t′0)
βBRW(t′; t′0)
∣∣∣ < 5%
(Fig. 2.2 b, solid black curve). For the purpose of fitting the βBRW-model to PBRW data,
choosing to fit data at times above the convergence time scale is important. Fitting at times
above 10λ−1+ will produce good fits because 10λ−1
+ > tBRW for all but the most unbiased
motion, in which case directionality time has little meaning regardless.
Notably, at time scales t′ > t′BRW, βPBRW has the same functional form as the βDD
and βBRW (Eqs. 2.15 and 2.21, respectively). Analogous to both DD and the 2D-SBRW,
defining directionality time as the time when βBRW = 32 gives t′d = t′0 = 2(1−2c2)
c2. In
the interpretation that directionality time is a proxy for the observation time necessary to
distinguish directionality from randomness, we redefine directionality time to vanish in the
44
12 < c2 < 1 domain, signifying directional motion at all time scales:
td =
1λ+
2(1−2c2)c2
, 0 < c2 < 12
0 12 < c2 < 1
(2.33)
(Fig. 2.2 c). Importantly, directionality time depends only on the reorientation time scale,
λ−1+ , and reorientation bias, c2. Directionality time does not depend on speed v.
4.2.5. Model 3A: 1-D Persistent Biased Random Walk with measurement error (PBR-
WwE). The 1D-PBRW (model 3) can be modified to account for measurement errors,
σm. Assuming positions along the PBRW are sampled and each position coordinate has
a Gaussian distribution with standard deviation σm, the change to EASD is: 〈X2(t)〉 →
〈X2(t)〉+ 2σ2m, where 〈X2(t)〉 is given in Eq. 2.28. The log-log EASD slope becomes
βPBRWwE(t′; c) =(2− 4c2)t′ + 2c2t′2 + (4c2 − 2)t′e−t
′+ 2c2t′2e−t
′
(6c2 − 2) + (2− 4c2)t′ + c2t′2 + (2− 6c2)e−t′ − 2c2t′e−t′ +2σ2
mλ2+
v2︸ ︷︷ ︸≡ε2
(2.34)
where the term added with respect to Eq. 2.31 is marked by an underbrace and defined as
ε2 for brevity. As with the 1-D PBRW without measurement error, the 1-D PBRWwE also
converges to βBRW(t). The convergence times above which the difference between βPBRWwE
and βBRW is less than 5% are shown in Fig. 2.5 (solid curves) for different values of the
constant ε. These convergence times increase with increasing centroid measurement error
and/or decreasing instantaneous speed. When ε2 is sufficiently large, the convergence time is
estimated by comparing ε2 to c2t′2 (see the denominator of Eq. 2.34), giving tσm ≈ 4.5√
2σmvrms(∞) =
4.5 ελ+c
(Fig. 2.5, dashed curves), where vrms(∞) = |c|v as derived in Eq. 2.29. The factor of
4.5 was chosen so the convergence time scale corresponds to the 5% difference threshold.
A larger factor can be chosen to lower the percent difference threshold. This convergence
time scale can be used to set a minimum fit time (c.f. τmin in Fig. 2.3 d). Fitting the
45
βBRW-model at times above the minimum fit time significantly decouples directionality time
measurements from measurement error. Note that tσm is sampling interval independent.
As the dimensionality of the random walk increases, so does the effect of measurement
error. Specifically, 〈X2(t)〉 → 〈X2(t)〉 + 2dσ2m where d is the number of dimensions. Also
taking into the account the convergence time tBRW due to persistence over short time scales,
the generalized convergence time scale, known as the minimum fit time tmin, is the greater
of:
tmin = max
tσm ≈ 4.5
√2dσm
vrms(∞)
tBRW ≈ 10 1λ+≈ 10tp
(2.35)
both of which are readily measured and sampling interval independent. Fitting experimental
data to the βBRW-model at times above tmin decouples measurements of directionality time,
td, from the effects of persistence and measurement error.
4.3. Appendix B: Deviations Caused by Variances and Ergodicity. In Appen-
dix A, we showed that biased random walks could be described by the βBRW-model
βBRW(t; td) =1 + 2t
td
1 + ttd
(2.36)
which can be used to measure directionality time, td. This analytical fit model implicitly
accounts for the variance of position, R, caused by variance in the step time distribution,
ρT (t), and the orientation distribution, ρΘ(θ) (c.f. Appendix A, rules 2 and 3). However, it
does not account for other sources of variance associated with measurements of real data,
or variance of parameters such as speed, v, which can take on a different value for each
random walker in an ensemble. We will refer to these three categories of variances as:
implicit variance, measurement error, and parametric variance, respectively. The implicit
variance is given by, σ2R(t) = 〈R2(t)〉 − 〈R(t)〉2. The variance attributed to measurement
error is 2dσ2m, where d is the number of dimensions. Finally, parametric variance is denoted
σ2p(t). Adding the variances in quadrature and using the 1D-PBRW model to calculate
46
implicit variance, 〈X2(t)〉 − 〈X(t)〉2, from Eqs. 2.26 and 2.28, the composite variance of
position X is
σ2X,real(t) =
v2
λ2+
(5c2 − 2) +2v2
λ+(1− c2)t+ O
(e−λ+t
)+ O
(te−λ+t
)︸ ︷︷ ︸
implicit variance,
σ2X
+
2σ2m︸︷︷︸
measurement variance
+ c2(δv)2t2︸ ︷︷ ︸parametric variance,
σ2p
(2.37)
where O(f(t)) denotes a term of order f(t). Note here that d = 1. For demonstrative
purposes, only one type of parametric variance is written in Eq. 2.37, the speed variance (at
large time scales). There can be other parametric variance terms but in general, σ2p(t) ∼
t2. Therefore, at large time scales, parametric variance is the dominant cause of position
variance.
The EASDs derived analytically in Appendix A assumed zero measurement error and
parameter variance. When measurement error and parameter variance are nonzero as is
the case with experimental measurements, EASD is increased with respect to our analytical
models
〈R2(t)〉 = 〈R2(t)〉BRW + σ2(t) (2.38)
where σ2(t) ≡ 2dσ2m+σ2
p(t) incorporates all measurement and parameter uncertainties, and
it has been assumed that each of these uncertainties is random and does not change 〈R(t)〉.
Ensemble averaged data is often noisy. As more migration trajectories are tracked,
the standard errors of ensemble averaged statistics decrease. However, ensemble averaged
statistics such as EASD are often still too noisy to fit with many experimental designs.
Statistical noise can be reduced by first time averaging the data before calculating the
47
ensemble average. The time averaged squared displacement (TASD), for example, is given
by
R2i (τ) =
1
(tdur,i − τ)
∫ (tdur,i−τ)
0|Ri(t+ τ)−Ri(t)|2 dt (2.39)
where i is an integer path index (i.e. path 1, 2, 3,...) and tdur,i is the total duration of the
ith path. The difference between TASD and EASD is described by
R2i (τ) = ξi(τ)〈R2(t = τ)〉 (2.40)
where the factor ξi is known as ergodicity. Mean squared displacement (MSD) is calculated
by ensemble averaging TASD (averaging over all trajectories, i)
〈R2(τ)〉 = 〈ξi(τ)〉〈R2(t = τ)〉 (2.41)
For β-curves measured from MSD instead of EASD is
β(τ) =d log
[〈R2(τ)〉
]d log[τ ]
. (2.42)
Rewriting 〈R2(τ)〉 in terms ξ, 〈R2〉BRW, and σ2 using Eqs. 2.41 and 2.38 gives
β(τ) =d log
[〈ξ(τ)〉〈R2(τ)〉BRW
(1 + σ2(τ)
〈R2(τ)〉BRW
)]d log[τ ]
. (2.43)
The log-log MSD slope can thus be expanded into its final form
β(τ) = βBRW(τ) + βσ(τ) + βξ(τ) (2.44)
where βBRW(τ) is given by Eq. 2.36, βσ(τ) is given by
βσ(τ) =d log
[1 + σ2(τ)
〈R2(τ)〉BRW
]d log[τ ]
(2.45)
48
and βξ(τ) is given by
βξ(τ) =d log [〈ξ(τ)〉]d log[τ ]
. (2.46)
The variance correction βσ(τ) becomes negligible as time increases unless σ2(τ) is of
order τ2 or greater such that σ2(τ)→∞ faster than 〈R2(τ)〉BRW →∞. At large time scales,
the position variance is dominated by parametric variance and not intrinsic variance (c.f.
Eq. 2.37). Therefore, the standard deviations of the mean distance traveled can be used to
calculate σ2(τ) in the variance correction term, βσ(t). At large time scales, the variance
correction term can be completely determined from the experimental data by using Eq. 2.38
to substitute 〈R2(τ)〉BRW for 〈R2(τ)〉 − σ2(τ) into Eq. 2.45, giving
β(τ)−d log
[〈R2(τ)〉
〈R2(τ)〉−σ2(τ)
]d log[τ ]
− d log [〈ξ(τ)〉]d log[τ ]
= βBRW(τ). (2.47)
Here, all terms that can be measured experimentally have been moved to the left of
the equality, while the fit model has been moved to the right. Eq. 2.47 is only valid
at sufficiently large time intervals when the effects of persistence on β(τ) are negligible
(i.e. βPBRW(τ) ≈ βBRW(τ), c.f. Eqs. 2.31 and 2.21), and when the effect of implicit
variance is small relative to the combined measurement and parametric variances (i.e.[〈r2(τ)〉 − 〈r(τ)〉2
]�[σ2m + σ2
p(τ)], c.f. Eq. 2.37). The minimum fitting time interval
τmin defined in Eq. 2.35 can be used to satisfy these conditions.
As a final note, calculating the ergodicity correction term βξ(τ) from experimental data
is not always helpful because its uncertainty is often relatively large and ergodicity cannot
be time averaged to reduce statistical noise. Therefore, the correction term βξ(τ) should
only be calculated if the system is believed to be nonergodic (i.e. when the parameters that
describe an experimentally measured migration change significantly over the duration of
that migration). If the data is nonergodic and βξ(τ) is unacceptably noisy when calculated
from experimental data, sometimes truncating the migration paths to equalize (and/or
shorten) their durations will make the data nearly ergodic. This was the procedure used
49
in order to ignore the ergodicity correction term (βξ(τ) = 0) in the chemotactic neutrophil
example from the main text (Fig. 2.9).
If the data is still nonergodic after truncation, βξ(τ) can be simulated as follows. First,
make an initial measurement of the persistence time and directionality time from the trun-
cated migration paths. Based on these initial measurements, parameters tp and κ = κ(c2)
can be chosen for a corresponding 2D-PBRW simulation that will recapitulate the experi-
mental data. Second, calculate the ensemble averaged instantaneous speed (EAIS) of the
truncated migration paths over their entire duration 0 ≤ t ≤ tdur. When the data is non-
ergodic, there will typically be an overall acceleration or deceleration associated with the
EAIS. Third, run 2D-PBRW simulations with instantaneous speeds, V (t), randomly se-
lected from the experimentally measured EAIS distribution each time a reorientation event
occurs. Simulate enough trajectories so that a smooth ensemble averaged ergodicity curve as
a function of time interval, ξ(τ), can be calculated. Finally, calculate ergodicity correction
term βξ(τ) using Eq. 2.46.
50
CHAPTER 3
Directionality Time Analysis Identifies Rigidity Sensing
Pathway Through αMβ2 Integrins
Portions of this chapter have been published, XM O’Brien, AJ Loosley, KE Oakley,
JX Tang, and JS Reichner. J Leukocyte Biol 96, 993-1004 (2014), Copyright 2014 by the
Federation of American Societies for Experimental Biology. Adaptations of text and figures
from this publication are presented here with permission from the Federation of American
Societies for Experimental Biology, and XM O’Brien. All supplementary material can be
found online.
51
Forward. In this chapter, we return to the usage of turning angle distributions (TADs),
tortuosity, and directionality time (td) to study the role of integrins in regulating directed
neutrophil motility in 2D as a function of substrate stiffness. The fit model used to measure
directionality time in this chapter is different compared to the last chapter. Instead of the
βBRW-model derived from random walk theory (Eq. 2.9), a heuristically derived exponential
version (denoted βexp) is used instead. At the time that we analyzed the data that went
into this chapter, directionality time was only a partially developed idea for a sampling
interval independent metric slated to be a proverbial change of direction for a field that
far too often used sampling interval dependent metrics to characterize cell motility. While
the βexp-model lacks a bottom-up derivation and precise mathematical meaning of its fit
parameters, the βexp-model does generate intuitive, heuristically interpretable data that
can be compared across the data sets it is applied to. In this chapter, the directionality
time metric revealed that stiffness dependent 2D honing depended on the engagement of
the β2, CR3. To give these results more context, an extended overview on integrins has
been written into this chapter.
1. Introduction
1.1. Neutrophils. Neutrophils serve as the body’s first line of defense against invading
pathogens, transmigrating though vascular endothelium and chemotaxing through vicinal
extracellular matrix (ECM) to reach the target area of infection or injury [8, 100]. Improper
neutrophil activity has grave clinical consequences: insufficient activity results in recurrent
life-threatening infections and impaired wound-healing, whereas excessive activity leads to
an exaggerated immune response, resulting in tissue damage.
1.2. Neutrophil Mechanosensing. Mechanosensing of cells within tissues refers to
the ability of the cells to perceive differences in the mechanical properties of its environment.
52
The mechanical environment has been shown to influence gene expression, proliferation, cy-
toskeletal organization, and survival significantly in a number of cell types [45, 101–104].
Neutrophil emigration into tissue is a multistep process, culminating with entry into notably
different tissue microenvironments. In so doing, cells interpret a number of biochemical cues
that hone their directed movement and regulate their host-defence functions within injured
or infected tissues. The physical properties of the tissue microenvironment contribute sig-
nificantly to the regulation of neutrophil migration speed and generation of traction [48, 49].
Relatively few host-defense functions take place in circulating or nonadherent neutrophils
nor does the physiological context in which they occur resemble rigid surfaces, such as glass
or plastic (Young’s modulus of ∼ 106 to 107 kPa [105]). With the use of polyacrylamide
gels of physiologically relevant stiffnesses coated with the ECM protein fibronectin (Fn),
Oakes et al. [48] found that human neutrophils migrating up a concentration gradient of
the chemoattractant Formyl-Methionyl-Leucyl-Phenylalanine (fMLP) were faster but less
directed on softer gels (Young’s modulus of 10 kPa) than they were on stiffer gels (Young’s
modulus of 100 kPa).
1.3. Integrins and Mechanosensing Mechanisms. Integrins are transmembrane
adhesion-receptors that play a significant role in mechanosensing [29]. Integrins dynami-
cally form and break bonds with the ECM as well as several intermediate proteins that link
integrins to the actin cytoskeleton [46]. Depending on factors such as integrin conformation
and distribution on the cell surface, traction forces can be applied to the ECM through
integrin mediated coupling to the actin cytoskeleton. Typically, force transduction occurs
through the formation of integrins clusters of at least 3 to 5 integrins, the minimal number
required to couple to the actin cytoskeleton [106]. Traction forces are normally contractile
due to the inward direction of actin retrograde flow. The bonds and proteins that link the
ECM to integrins, and integrins to the actin cytoskeleton, form a constitutive system that
effectively behaves like a catch bond between the ECM and the actin cytoskeleton. As the
53
force on the adhesion complex increases, the effective catch bond strengthens. Furthermore,
other integrins diffusing through the plasma membrane aggregate near the adhesion com-
plex overtime, bind ECM, and further strengthen the overall link between ECM and the
cytoskeleton.
Measuring rigidity fundamentally equates to determining the stress required to strain
a substrate by a specific amount, or vice verse, determining the strain required to stress a
substrate by a specific amount. In some cases, it is the magnitude of applied strain that
appear to be conserved across changing substrate stiffness [48, 53, 107, 108], while in other
cases it is the magnitude of the applied stress that appears to be conserved [51, 109]. One
example of a mechanosensing mechanism is stress induced protein unfolding that exposes
new effector binding or phosphorylation sites [110, 111]. The downstream response to
increased effector binding and/or phosphorylation provides an indirect measure of the strain
response with respect to the applied stress, which therefore makes it an indirect measure of
substrate rigidity. The forces involved between single integrins and the ECM range from 1
to 30 pN, but as we will see in chapter 5, the size of contractile forces from mature adhesion
complexes are as large as 100 nN in neutrophils.
1.4. Integrin Ligand Engagement. Integrins are distinguishable by their α and β
subunits (Fig. 3.1) [26, 112, 113]. Fn is a canonical β1-integrin ligand, and α5β1 is a key
mediator of neutrophil migration on this substrate. β1 integrins have been shown to play
a role in the cellular mechanosensing apparatus in cells other than neutrophils [114, 115].
Fn, however, is also recognized by several other receptors, one of which is the β2 integrin,
αMβ2 (CR3). To examine the contribution of CR3 in transducing mechanotactic cues,
we analyzed the migration of human neutrophils up a concentration gradient of fMLP on
polyacrylamide gels, ranging in stiffness from 10 to 100 kPa, coated with Fn or the CR3-
restricted ligand, fibrinogen (Fgn), which is known to be associated with the ECM under
inflammatory conditions, forming the provisional fibrin matrix at virtually all sites of tissue
54
αM αX
α1α2
α5β2
β1
Figure 3.1. Schematic of integrin types. Integrins are transmembrane re-ceptors that can be distinguished by their α and β subunits. The integrinsrelevant to this study (highlighted yellow) are those that bind the ligands colla-gen, fibronectin, and fibrinogen. These include α1β1 (VLA-1) and α2β1 (VLA-2) which bind collagen, α5β1 (VLA-5) which binds fibronectin, αXβ2 (CR4)which binds fibrinogin, and lastly, αMβ2 (CR3) which binds both fibronectinand fibrinogen. This figure was adapted from Ref. [113] with information fromRef. [112].
damage [116]. We also extended our study of neutrophil migration to include gels coated
with type IV Col (Col IV), an ECM ligand recognized by β1 but not β2 integrins.
In this chapter, the role of specific groups of integrins in the mechanoregulation of
neutrophil chemotaxis is determined from observations of chemotaxis on the ECM pro-
teins, Fn, Fgn, and Col IV, which bind to specific integrin subgroups. Knowing which
integrins engaged during chemotaxis, we decompose the migration trajectories into param-
eters that characterize honing, persistence, and speed to determine that the ligand-specific
mechanosensitivities for each parameter appear to be regulated independently. The roles of
specific integrins are further elucidated by performing integrin blocking experiments. Direc-
tionality time measurements with and without β2 integrin blocking show that β2 integrins
modulate honing on a 2D surface in response to substrate stiffness.
55
2. Materials and Methods
2.1. Reagents. Anti-Fgn (85D4) and anti-Fn (FN-15) antibodies were purchased from
Sigma-Aldrich (St. Louis, MO, USA); anti-type IV Col pAb and anti-integrin β1-blocking
antibody (P5D2) were purchased from Millipore (Billerica, MA, USA); and anti-integrin β2-
blocking antibody (TS1/18) was purchased from Pierce Biotechnology (Rockford, IL, USA).
L-15 and HBSS were purchased from Invitrogen (Carlsbad, CA, USA), and polymyxin B and
sulfo-SANPAH were purchased from Pierce Biotechnology. Human Fn isolated from plasma
(>95% purity) and human type IV Col from human placenta (>95% purity) were purchased
from Sigma-Aldrich. Human Fgn isolated from plasma (>95% purity) was purchased from
Molecular Innovations (Detroit, MI, USA), and IRDye800-conjugated goat anti-mouse IgG
was obtained from Rockland Immunohistochemicals (Gilbertsville, PA, USA).
2.2. Neutrophil Isolation. Under the approval and guidelines of the Rhode Island
Hospital Institutional Review Board, neutrophils were isolated from healthy human vol-
unteers by collection into EDTA-containing Vacutainer tubes (BD Biosciences, San Jose,
CA, USA). Histopaque 1077 (Ficoll Histopaque) was used for the initial cell separation,
followed by gravity sedimentation through 3% dextran (average 400-500 kDa MW). Con-
taminating erythrocytes were removed by hypotonic lysis, yielding a neutrophil purity of
> 95%. Neutrophils were suspended in HBSS (without Ca2+ or Mg2+) on ice until use in
the experiments. All reagents contained < 0.1 pg/ml endotoxin, as determined by Limulus
amoebocyte lysate screening (BioWhittaker, Walkersville, MD, USA).
2.3. Substrate Preparation. Migration experiments were carried out in heatable
glass-bottom DeltaT dishes (Bioptechs, Butler, PA, USA) on 10 kPa, 50 kPa, or 100 kPa
gels, prepared following a method described by Pelham and Wang [101]. Briey, a solution
of acrylamide and bisacrylamide was polymerized using tetramethylethylenediamine and
ammonium persulfate. Gels were made in DeltaT dishes using AbGene frames (AbGene,
Epsom, UK) as molds. The gels were allowed to polymerize at room temperature. Once
56
polymerized, gels were soaked overnight in PBS, allowing unpolymerized acrylamide to
diffuse out. The final size of the gels was ∼1 cm× 1 cm× 300µm.
Gel stiffness was regulated by varying the percentage of bisacrylamide in relation to the
percentage of acrylamide in the initial mixture [44], and elasticity was confirmed with an
AR2000 oscillating plate rheometer (TA Instruments, New Castle, DE, USA) [117]. The
gels were coated with Fgn, Col IV, or Fn using the chemical cross-linker sulfo-SANPAH,
which was allowed to bond covalently to the acrylamide gel for ∼1 h at room temperature.
Gels were washed three times with 50 mM HEPES (pH 7) and then incubated in 200µL of
20µg/mL Fgn, Col IV, or Fn, all diluted in 50 mM HEPES (pH 7), while exposed to UV
using a Foto/Prep I transilluminator (Fotodyne, Hartland, WI, USA), equipped with 15 W,
312 nm bulbs with an energy output of ∼ 4000µW/cm2 for 12 min at room temperature,
washing three times with PBS before use.
The elasticity of the gels has been shown to be unaffected by the protein-coating pro-
cedure, and the density of protein on the surface of the gel is unaffected by the elasticity
of the gels [43, 48]. The uniformity of the protein coating on the acrylamide substrate was
confirmed by immunofluorescence. Gels were cast on activated glass slides and coated with
Fgn, Col IV, or Fn, as described above or 90:10 and 10:90 mixtures by volume of 20µg/ml
Fgn and 20µg/ml Fn. Gels were blocked with 200 µL of 2% BSA at room temperature for
30 min before being incubated with a 100µg/ml solution of mouse anti-human Fgn, Col IV,
or Fn antibody as appropriate in 2% BSA for 1 h at room temperature. The gels were then
washed three times with PBS. Next, gels incubated with 1:10,000 IRDye800-conjugated
anti-mouse IgG in 2% BSA for 1 h at room temperature. After incubation, the gels were
washed three times with PBS and scanned using the Odyssey Infrared Imaging System
(LI-COR, Lincoln, NE, USA). Each gel was divided into four, 5 mm2 regions of interest.
Fluorescence intensity from regions of interest was obtained for six gels from two to four
independent experiments/condition.
57
2.4. Neutrophil Chemotaxis. For chemotactic migration assays, 1-2×106 neutrophils
were resuspended at 37C in 2 mL L-15-glu. Neutrophils were allowed to settle for approxi-
mately 2 min. A femtotip (Eppendorf North America, New York, NY, USA) was filled with
5µL of a 1-mM concentration of bacterial fMLP and placed with the tip at the migration
surface within the field of view. When indicated, neutrophils were pretreated on ice for
30 min with 10 µg/ml β2-blocking antibody (TS1/18), β1-blocking antibody (P5D2), or
isotype control in L-15-glu, which was maintained for the duration of the experiment.
2.5. Microscopy. A Nikon TE-2000U inverted microscope (Nikon, Melville, NY,
USA) coupled to a CoolSNAP HQ CCD camera (Roper Scientific, Martinsried, Germany)
or an iXonEM + 897E back-illuminated Electron Multiplying CCD camera (Andor, Belfast,
UK), outfitted with a Bioptechs (Butler, PA, USA) stage heater and a 20x Nikon Plan
apochromat objective, was used for all experiments. DIC images were captured over 30 min
on a 10 s interval using Elements (Nikon). All data were analyzed using Excel (Microsoft,
Redmond, WA, USA), ImageJ (U.S. National Institutes of Health, Bethesda, MD, USA),
and MATLAB (MathWorks, Natick, MA, USA) computational software.
2.6. Cell Tracking Tools and Analytics. The neutrophils chosen for analysis ex-
hibited movement for at least 300 s. Under the conditions analyzed in this study, 25 to 30%
of cells migrated to the fMLP point source in a given 30-min observation period. Cells were
tracked frame-by-frame using custom MATLAB software [48]. Individual cell boundaries
were determined through thresholding of the DIC images with respect to a median image,
calculated by taking the median intensity of all images in a sequence. Each cell centroid
was determined through a center of mass calculation, based on the cell border. Finally, cell
migration trajectories, ~r(t), were assembled from the time-course of all centroid positions.
Mean squared displacement (MSD),⟨r2(τ)
⟩, was calculated as a function of time
interval, τ , as described in Chapter 2 (Eq. 2.2). From MSD, the log-log MSD slope,
58
β(τ) =d[〈r2(τ)〉]
dlog[τ ] , was determined. The directionality of the cells in this study were charac-
terized fitting β(τ) to a three parameter exponential fit function,
βexp = β∞ − (β∞ − β0) exp
(−τtd
)(3.1)
where fit parameter td characterizes the time scale at which migration transitions from
random to directed, fit parameter β0 heuristically describes the randomness of motion at
τ = 0, and β∞ heuristically describes the directedness of motion at long time intervals
(τ →∞).
To further analyze the directionality transition, the time averaged quantities, β− and β+
were calculated as the time averages of βexp at time intervals τ < td and τ > td respectively.
Root mean squared speed (RMSS) was calculated by taking the square root of MSD
at time interval, τ , and dividing by τ . TADs were calculated as the distribution of angles
between sequential displacement vectors ~r(t + ∆t) − ~r(t) binned in 10◦ increments. The
sampling interval of ∆t = 60 s was used for all TAD calculations. Positive angles indicate
counter-clockwise turns, whereas negative angles indicate clockwise turns. Tortuosity was
calculated as the fraction: end-to-end displacement of the migration path divided by to-
tal migration path length. Like TAD, tortuosity depends on sampling interval ∆t and is
routinely reported for ∆t = 10 s, as we do here.
2.7. Statistics. Data were pooled from a minimum of three independent experiments
representing three to six different donors, with an n equal to 15-40 cells for each condition.
ANOVA with Newman-Keuls post hoc analysis, as appropriate, was performed using MAT-
LAB (MathWorks) or Excel (Microsoft) running the statistiXL data package (Needlands,
Western Australia). The null hypothesis was rejected if P<0.05.
2.8. Online Supplemental Material. Supplemental Fig. 1 shows that protein coat-
ing is unaffected by gel stiffness, and mixtures of Fn and Fgn are cross-linked to the gel
59
in proportion to their percentage in the applied protein mixture. In Supplemental Fig.
2, neutrophils migrating on Col IV-coated substrates toward fMLP show no difference in
MSD, td, β−, or β+ upon blocking of β2 integrins. Supplemental Fig. 3 shows that blocking
β1 integrins do not change MSD or transition significantly from random to directed motion
in cells migrating on Fgn-coated gels toward fMLP. Video 1 shows neutrophils migrating
on Fgn-coated gels of 10 kPa and 100 kPa stiffness toward a fMLP point source; Video 2,
neutrophils migrating on Col-coated gels of 10 kPa and 100 kPa stiffness toward a fMLP
point source; Video 3, neutrophils migrating on Fn-coated gels of 10 kPa and 100 kPa stiff-
ness toward a fMLP point source; and Video 4, neutrophils migrating on Fn-coated gels
of 10 kPa+β2-integrin block and 100 kPa+β2-integrin block stiffness toward a fMLP point
source.
3. Results and Discussion
In this work, we demonstrate that the process of human neutrophil chemotaxis toward a
fMLP point source is ligand-directed and is underpinned additionally by distinct constituent
processes directing cellular speed, direction, and persistence of motion that can be mod-
ulated independently by the tissue microenvironment and integrin engagement. A novel
feature of this work is the development of biophysical tools, some introduced for the first
time here, which allow fine dissection of the mechanisms that lead to the mechanosensitive
differences in neutrophil chemotaxis.
Previous work from our lab [48] found that the mechanical properties of a Fn-coated
substrate strongly affected neutrophil morphology, with cells spreading over a larger area
and more quickly on substrates of greater stiffness. In addition, they also observed that
neutrophil function was affected on stiffer substrates, resulting in slower migration, but
with an increase in directedness, such that a greater net distance was traveled over time.
Their finding that the elastic properties of the substrate dictate the ability and efficiency of
the neutrophils to adhere and migrate, taken in the context that the mechanical properties
60
of cells and tissues can be altered dramatically in states of disease or inflammation [118],
suggests that studies of cell adhesion and migration on flexible substrates represent a more
physiologically appropriate and biologically relevant approach than similar studies on plastic
or glass. Physiologically, these findings also suggest that neutrophil function may be subject
to the relative tonicity of the tissue in which the response is taking place.
We were interested in investigating if these findings were restricted to Fn or if they
could be extended to matrices more reflective of an inflammatory site and recognized by
other integrin families. To that end, polyacrylamide gels of 10 kPa, 50 kPa, and 100
kPa stiffness were coated with Fgn, Col IV, or Fn using the photoactivatable cross-linker,
sulfo-SANPAH. In these experiments, ligand density was held constant across conditions
to allow the isolation of substrate stiffness as an independent variable. Immunological
detection was used to confirm that the protein coating was uniform and proportional and
that the density of protein was independent of substrate stiffness (Supplemental Fig. 1
found online, see [1]). Neutrophils do not migrate through a matrix-coated polyacrylamide
gel, and therefore, they provide us an experimental system that allows for the interrogation
of cellular mechanosensing, based purely on the mechanical properties of the substrate at
constant ligand density.
3.1. Neutrophil morphology is dependent on substrate stiffness and inde-
pendent of ligand coating. Consistent with cell morphology on Fn, neutrophils adhered
to coated substrates show distinct changes in morphology that are dependent on substrate
stiffness and independent of ligand coating. A dramatic increase in spreading was observed
on 100 kPa gels compared with that on 10 kPa gels, regardless of ligand Fgn, Col IV, or
Fn at 120 s after initial cell adhesion, a time when cells typically reach their stable spread
area [48] (Fig. 3.2 a). Strikingly, the spread area of cells on 100 kPa substrates was more
than double that of cells on 10 kPa substrates (Fig. 3.2 a).
61
10s
20s
30s
40s50s
60s
70s
b c
time interval, τ (s)
Fn Fgn1
0 k
Pa
10
0 k
Pa
Cola
g
f
1000
1000 0
100
10
10
1100
Slope = 1
Slope =
2
MS
D (
µm
)2
Linear Motion:
300
Random Motion:
time interval, τ (s)
*
turn
ing
an
gle
dis
trib
uti
on
angle (deg) angle (deg) angle (deg)
1000
1000 0
100
10
d
MS
D (
µm
)2
1200
1000
800
600
400
200
MSD at 150s
*
MS
D (
µm
2)
**
**
**
**
**
*
10
1
100 300
RM
SS
(µ
m/s
)
0
0.0
0.1
0.2
0.3
0.4
e
Col
10 100 10 100
Fn
10 100
Fgn
50 50 50
Slope = 1
Slope =
2
10s
20 30s
40 0s
60
70s
Figure 3.2. Mobility and TAD of neutrophil chemotaxis toward fMLPon surfaces of varying stiffness depend on ligand coating. (a) Humanneutrophils adhered to Fgn-, Col IV-, orFn-coated gels show distinct changes inmorphology that are dependent on substrate stiffness. Inset on each micrographis the average spread area of migrating neutrophils at 120 s after adhesion (20×bright-field magnification; bar=50µm). (b) For a migrating cell, trajectory plotswere generated at 10 s increments between cell centroids. (caption continued onpg. 64)
62
3.2. The MSD mechanosensitivity of the neutrophil chemotaxis toward
fMLP is ligand-dependent. Unlike the ligand-independent effect in cell morphology,
the migration dynamics on these single substrates were found to have three distinct pat-
terns of behavior in response to mechanotactic cues. Neutrophils were allowed to migrate
on Fgn-, Col IV-, or Fn-coated substrates toward a fMLP point source, and migrating
neutrophils were tracked over a 30-min period using time-lapse DIC images acquired every
10 s (Supplemental Videos 1-3 found online). With the use of these images, we were able
to track individual cells, calculate their centroid, and generate cell-migration trajectories
(Fig. 3.2 b). We used these trajectories to quantify migration dynamics.
The MSD of tracked cells is a measure of the net distance that an average cell will
travel during a particular interval of time. The dependence of the MSD on the time interval
portrays the type of underlying motion: for a simple random walk, the MSD is linear with
time, whereas for directed motion along a straight line, the MSD increases quadratically
with time. Generally, MSD is proportional to τα, where exponent α characterizes the
directionality of motion. When MSD is plotted against time interval τ in log-log coordinates,
the exponent α can be estimated by the slope of the curve, β (Fig. 3.2 c, see Chapter 2
Eq. 2.4). A value of β = 1 indicates motion that appears random. A value of β = 2 indicates
purely directed motion at that given time scale (Fig. 3.2 c). At long-time intervals, where
the displacement of the migrating cell approximates a straight line from the starting position
to the fMLP point source, the slope of the MSD curve approaches two, whereas at shorter
time intervals, which reveal the contours of the natural path wiggle of the cell migration,
the slope tends toward one. Overall, β increases from one to two as the time interval τ
increases.
(Fig. 3.2 caption continued) (c) Idealized MSD plots in log-log coordinates. The slope characterizesthe type of motion. A constant value of one indicates diffusive motion, whereas a constant valueof two indicates ballistic motion. (d) MSD plotted, based on the average migration trajectories ofhuman primary neutrophils migrating on Fgn-, Col IV-, and Fn-coated gels of 10 kPa, 50 kPa, or 100kPa stiffness toward a fMLP point source. Migration was tracked over a 30 min period by time-lapseDIC images acquired every 10 s (Supplemental Videos 1-3 found online). Error bars represent thestandard error of the mean (SEM). (e) The average MSD at a time interval of 150 s and corresponding
63
The MSDs for cells migrating on Fgn-, Col IV-, or Fn-coated substrates are plotted
in Fig. 3.2 d. The corresponding MSDs and RMS speed at a time interval of 150 s are
shown in Fig. 3.2 e. This time interval corresponds to constant speed ballistic motion
under all conditions studied and is appropriate for comparing the overall relative motion of
cells migrating toward the chemotactic source. Cells migrating on Fn- and Col IV coated
substrates have mechanosensitive changes in MSD, with cells showing greater displacements
on softer substrates. Cells migrating on Fgn-coated surfaces show the largest displacements,
which are independent of substrate stiffness.
3.3. The TAD mechanosensitivity of neutrophil chemotaxis toward fMLP is
ligand-dependent. Prior studies from this lab demonstrated mechanosensitive differences
in TAD of neutrophils migrating on 10 kPa and 100 kPa Fn-coated substrates [48]. The TAD
during cell migration is determined by measuring the angle of displacement with respect
to the previous step at a given time interval (shown in Fig. 3.2 f for a 10-s time interval).
The TAD for cells migrating as a function of ligand and substrate stiffness at τ = 60 s is
shown in Fig. 3.2 g. When analyzing TAD, a comparison of histogram height at 0◦, which
represents the percentage of turning angles that fall between ±5◦, enables a measure of
migration persistence relative to treatment conditions (Fig. 3.2 g, inset bar graphs; denoted
TAD at 0◦). Consistent with our previous data [48], cells migrating on Fn-coated gels
demonstrate mechanosensitive TAD, with cells migrating on 100 kPa substrates showing
a significantly greater persistence than cells migrating on 10 kPa substrates. In contrast,
cells migrating on Fgn-coated surfaces show strongly directed motion with no significant
root mean squared speed (denoted RMSS) for each condition. Error bars on MSD represent SEM,whereas RMS speed data are plotted in quartiles by Box and Whisker Plot. P<0.55 and P<0.05versus Fgn-coated surfaces of all stiffnesses. (F) Turning angles during cell migration are definedby the angles between subsequent displacement vectors that make up the migration trajectory. TheTAD implicitly depends on the sampling interval used to determine the displacement vectors. (g)TADs at a 60-s sampling interval of cells migrating on Fgn-coated substrates (left), Col IV-coatedsubstrates (middle), and Fn-coated substrates (right) are shown. (Insets) The percentage of turningangles between ±5o for each coating, denoted TAD at 0o. Error bars represent SEM. P<0.05, 10kPa Fn-versus 100 kPa Fn-coated surfaces. nd, No difference.
64
dependence on substrate stiffness by this measure. In contrast, cells migrating on Col IV-
coated substrates demonstrated a TAD independent of substrate stiffness and migration
trajectories significantly less directed than all but cells migrating on the softest Fn-coated
substrate.
3.4. The tortuosity of neutrophil chemotaxis toward fMLP is ligand-dependent
and independent of substrate stiffness. Another measure of migration persistence is
tortuosity, calculated as a ratio of the net distance a cell migrates toward the chemotactic
point source to the total migration path length (Fig. 3.3 a). With the use of a 10 s sampling
interval to calculate the total migration path, tortuosity values for neutrophils migrating
on Fgn-, Col IV-, or Fn-coated gels showed no mechanosensitive variation (Fig. 3.3 b). The
value of tortuosity depended on ligand coating and varied significantly between ligands,
with cells on Fgncoated substrates migrating with the highest tortuosity, followed by cells
migrating on Fn-coated substrates, and with cells on Col IV coated substrates migrating
with the lowest tortuosity. This differs from our TAD results. We attribute these incon-
sistencies in persistence to the sampling interval dependence of TAD and tortuosity and
an incomplete representation of the data based on using a single sampling interval for the
characterization of persistence. TAD and tortuosity depend on the sampling interval, as
the turning angles and total migration path length, respectively, depend on the sampling
interval. The subjectivity of choosing a single sampling interval introduces inconsistency,
making comparisons across current persistence measurements potentially inaccurate. This
variability led us to develop a novel, sampling interval-independent metric, called direction-
ality time to quantify directionality without the subjective user input of a sampling interval.
Directionality time is measured by fitting the MSD in log-log coordinates to the exponential
function given in Eq. 3.1. The fit parameters td (directionality time), β0 and β∞, charac-
terize the migration paths based on data at all possible sampling intervals, instead of just
one. Directionality time, which corresponds to the decay time of the exponential function,
65
10s
20s 30s
40s 50s
60s
70s
0
0.2
0.4
0.6
0.8
Col
10 100 10 100
Fn
10 100
Fgn
a
btortuosity
nd1.0
50 50 50
*
tortuosity = Displacement / Path
*
nd
nd
*
Figure 3.3. Tortuosity of neutrophil chemotaxis toward fMLP isligand-dependent but independent of substrate stiffness. (a) Tortu-osity is calculated as the ratio of total displacement (dotted gray line) to totalpath (solid black line), as shown in the schematic, and represents the overalldirectionality of the entire migration pathway. Tortuosity depends on time in-terval associated with the displacement vectors that make up the path. (b) Theaverage tortuosity for neutrophils migrating on Fgn-, Col IV-, and Fn-coatedgels toward a fMLP point source is shown as a Box and Whisker Plot, show-ing mean and quartile data. Tortuosity did not vary significantly by substratestiffness among gels of a given protein coating. tortuosity did vary significantlyby protein coating with cells on Fgn-coated gels migrating most directedly, fol-lowed by Fn-coated gels, and Col IV-coated gels migrating the least directedly.P<0.05 versus all other protein coatings.
describes the minimum observation time necessary to determine that chemotactic motion
up the fMLP gradient is directional. All things equal, a larger value for td indicates more in-
herent “wiggle” to a cell’s migration path. We further characterize the paths by calculating
parameters β− and β+, which are time averages of βexp at time intervals τ < td and τ > td,
respectively. These parameters give a measure for the amount of randomness (values near
one) of the migration path at short time scales compared with the amount of directedness
(values near two) in the migration path at large time scales (see Materials and Methods).
66
3.5. Neutrophils migrating on Fgn-coated substrates toward fMLP show
mechanosensitive changes in td and β−. Cells migrating on Fgn-coated substrates
toward a fMLP point source show no mechanosensitive difference in MSD (Fig. 3.4 a and
Supplemental Video 1 found online) or similarly, RMS speed (Fig. 3.2 e). Values of td , β−,
and β+ were derived from the β(τ) over time-interval curves (Fig. 3.4 b). For cells migrating
on Fgn-coated gels, we have identified a mechanosensitive shift in td, with cells migrating
on 100 kPa substrates showing significantly more directional behavior than cells migrating
on 10 kPa substrates (Fig. 3.4 c). Values of β−, and β+ are represented in Fig. 3.4 d as
the directionality transition. As we increase the time interval of observation beyond td, the
essential directedness of migration becomes apparent for all conditions that we have studied,
with the common β+ of approximately 1.8. When looking at time intervals smaller than td,
we see a measure of how closely the migrating cell stays fixed on its path. This value shows
significant differences between conditions studied and may represent the interplay between
a “homing” and a “surveying” phenotype. Cells on Fgn-coated substrates have significantly
more directionality on 100 kPa substrates at time intervals shorter than td than do cells
migrating on 10 kPa substrates (Fig. 3.4 d).
3.6. Neutrophils migrating on Col-coated substrates toward fMLP show a
td independent of substrate stiffness. Cells migrating on Col IV-coated substrates
toward a fMLP point source show a mechanosensitive difference in MSD at 150 s, with
cells migrating on softer substrates covering a greater distance per time interval (Fig. 3.5 a
and Supplemental Video 2 found online). Measurements of td, β−, and β+, derived from
the β curves (Fig. 3.5 b), plotted in Fig. 3.5, c and d, show that migration directionality
is independent of substrate stiffness. Cells migrating on Col IV-coated substrates show td
values significantly larger than cells migrating on Fgn-coated substrates, indicating that
cells show more locally diffusive motion.
67
time interval, τ (s)
1000
1000 0
100
10
101
100
MSD
(µm
)2
300
MSD at 150s
**
time interval, τ (s)
time
aver
aged
β(τ
)
log-
log
MSD
slo
pe, β
(τ)
directionality time, td (s)
a b
c d
τ < t d τ > t d
β− β+
directionality transition
Figure 3.4. Cells migrating on Fgn-coated gels toward fMLP have sim-ilar MSD and RMS speed but show a mechanosensitive change in tdand β−. Human primary neutrophils migrating on Fgn-coated gels of 10 kPa,50 kPa, or 100 kPa stiffness toward a fMLP point source were tracked over a 30-min period (Supplemental Video 1 found online). (a) The MSD of neutrophilmigration paths is plotted as a function of time between steps. (Inset) Theaverage MSD at τ = 150 s for each condition. MSD and RMS speed for cellsmigrating on Fgn-coated gels are independent of substrate stiffness. (b) Thelog-log MSD slope of each condition is plotted over time interval, τ . (c) Meanvalues of confidence intervals on measurements of directionality time, td. Cellsmigrating on Fgn-coated gels show a stiffness-dependent change in td, with cellsmigrating on 100 kPa gels transitioning to directed motion at a significantlyshorter time interval than those on 10 kPa gels. (d) Values of β− and β+, whichquantify the degree of randomness in the migration path over short time in-tervals τ < td and the degree of directedness in the migration path over longtime intervals τ > td, respectively. These data are plotted as the directionalitytransition. Cells migrating on 100 kPa gels have significantly more directed β−than cells on 10 kPa. β+ of cells on Fgn-coated gels is independent of substratestiffness. Error bars represent SEM. P<0.05, 10 kPa versus 100 kPa.
68
1000
1000 0
100
10
101
100
MS
D (
µm
)2
300
MSD at 150s
*
directionality time, td (s)
time interval, τ (s) time interval, τ (s)
log
-lo
g M
SD
slo
pe
, β(τ
)
a b
c d
tim
e a
ve
rag
ed
β(τ
)
τ < t d τ > t d
β+
directionality transition
Figure 3.5. Neutrophils migrating on Col IV-coated substrates towardfMLP show td, β−, and β+ independent of substrate stiffness. Humanprimary neutrophils migrating on Col IV-coated gels of 10 kPa, 50 kPa, or 100kPa stiffness toward a fMLP point source were tracked over a 30-min period(Supplemental Video 2 found online). (a) The MSD of neutrophil migrationpaths is plotted as a function of time between steps. (Inset) The average MSDat a τ = 150 s for each condition. Cells migrating on 10 kPa Col IV-coated gelsshow a significant increase in RMS speed over cells on 50 kPa or 100 kPa gels.P<0.05, 10 kPa versus 50 kPa and 100 kPa. (b) The log-log MSD slope of eachcondition is plotted over time interval, τ . (c) Mean values of confidence intervalson measurements of directionality time, td. The td of cells migrating on Col IV-coated gels was independent of substrate stiffness. (d) Values of β− and β+,which quantify the degree of randomness in the migration path over short timeintervals τ < td and the degree of directedness in the migration path over longtime intervals τ > td, respectively. These data are plotted as the directionalitytransition. Cells migrating on Col IV-coated gels have no significant differencesin β− or β+. Error bars represent SEM.
3.7. Neutrophils migrating on Fn-coated substrates toward fMLP show
mechanosensitive differences in td and β−. Cells migrating on Fn-coated substrates
toward a fMLP point source show a mechanosensitive difference in MSD at 150 s, with
69
cells migrating on stiff substrates covering the least distance per time interval (Fig. 3.6 a
and Supplemental Video 3 found online). Again, βexp was fit for each stiffness (β plotted
in Fig. 3.6 b) to calculate td (Fig. 3.6 c), and both β− and β+ (Fig. 3.6 d). Cells mi-
grating on 100 kPa Fn-coated substrates show significantly smaller td compared with those
migrating on 10 kPa substrates (Fig. 3.6 c). Cells migrating on 100 kPa substrates also
showed significantly larger β−. Taken together, cells on stiffer Fn-coated gels migrate more
directionally.
3.8. The mechanosensitive component of directionality for neutrophils mi-
grating on Fn-coated substrates toward fMLP is dependent on β2 integrins. For
cells migrating on Fn-coated surfaces that engage β1 and β2 integrins, we were able to gain
some insight into a β2-integrin-dependent modulation of chemotactic parameters under our
experimental conditions. Neutrophils were pretreated on ice for 30 min with 10 µg/ml
β2-blocking antibody (TS1/18, which binds the I-domain and blocks the function of all
β2-integrins) or isotype control (data not shown), which was maintained for the duration of
the experiment. Cells migrating on Fn-coated substrates toward a fMLP point source after
β2-integrin blocking show significant differences in MSD (Fig. 3.7 a), with cells on both
stiffnesses showing slower relative motion. Directionality measures, td (Fig. 3.7 b and Sup-
plemental Videos 3 and 4 found online) and β− (Fig. 3.7 c), demonstrate significantly less
persistent migration after β2-integrin blocking that is statistically indistinguishable from
that of untreated cells migrating on Col IV-coated substrates. This shift in directedness
also holds for our other measures of persistence, tortuosity (Fig. 3.7 d) and TAD (data
not shown). Cells migrating on Col-coated substrates toward a fMLP point source after
β2-integrin blocking show no difference in relative motion or persistence measures (Sup-
plemental Fig. 2 found online). Additionally, blocking β2 integrins of cells on Fn-coated
gels did not significantly change their spread areas, 120 s after cell adhesion (data not
shown). Cells pretreated with the β1-blocking antibody (P5D2) and allowed to migrate on
70
1000
1000 0
100
10
101
100
MSD
(µm
)2
300
MSD at 150s
*
**
τ < t d τ > t d
directionality time, td (s)
time interval, τ (s) time interval, τ (s)
log-
log
MSD
slo
pe, β
(τ)
a b
c d
time
aver
aged
β(τ
)
β+
directionality transition
Figure 3.6. Neutrophils migrating on Fn-coated substrates towardfMLP show mechanosensitive differences in MSD, td, and β−. Hu-man primary neutrophils migrating on Fn-coated gels of 10 kPa, 50 kPa, or 100kPa stiffness toward a fMLP point source were tracked over a 30-min period(Supplemental Video 3 found online). (a) The MSD of neutrophil migrationpaths is plotted as a function of time between steps. (Inset) The average MSDat τ = 150 s for each condition. Cells migrating on 100 kPa Fn-coated gels showa significant decrease in MSD when compared with cells on 50 kPa or 10 kPagels. P<0.05 100 kPa versus 50 kPa or 10 kPa. (b) The log-log MSD slope ofeach condition is plotted over time interval, τ . (c) Mean values of confidenceintervals on measurements of directionality time, td. The td of cells migrating onFn-coated gels show a stiffness-dependent change in td, with cells migrating on100 kPa gels transitioning to directed motion at significantly shorter time scalescompared to cells on 10 kPa gels. (d) Values of β− and β+, which quantify thedegree of randomness in the migration path over short time intervals τ < td andthe degree of directedness in the migration path over long time intervals τ > td,respectively. These data are plotted as the directionality transition. Cells mi-grating on 100 kPa gels were significantly more directed than cells on 10 kPa.The β+ of cells on Fn-coated gels was independent of substrate stiffness. Errorbars represent SEM. P<0.05, 10 kPa versus 100 kPa.
71
Fgn-coated surfaces showed no significant changes in MSD (Supplemental Fig. 3A found
online), td (Supplemental Fig. 3B), or β (Supplemental Fig. 3C). Cells pretreated with the
β1-blocking antibody (P5D2) neither adhere nor migrate on Col IV- or Fn-coated surfaces
(data not shown). These data suggest that for human neutrophils on compliant surfaces,
β1 integrins are driving adhesion and migration on Fn, whereas β2-integrin engagement is
directing a mechanosensitive enhancement of migration directionality. Neutrophils do not
regularly encounter ECM ligands in isolation. By necessity, cells are incorporating input
of varying ligand ratios and substrate compliance. The interplay of β1-integrin mechano-
tactic shifts in RMS speed and β2-integrin-dependent modulation of directionality suggests
a mechanism by which the cellular mechanosensing apparatus can be tuned to fine, incre-
mental changes in the microenvironment through which the cell is migrating.
These different patterns of substrate-dependent mechanotactic sensitivity are particu-
larly interesting in that they point to the fine interpretation of extracellular signals required
to regulate a neutrophil during chemotaxis: integrating mechanical properties of the sub-
strata, nature, and concentration of chemotactic signals, as well as the composition of the
ECM and ligand density. For example, regulation of the migratory response may be more
subject to mechanical cues under conditions of suboptimal chemokine concentrations rather
than under conditions of saturating chemokine concentrations. Alternatively, an examina-
tion of different chemotactic signals, such as IL-8 or C5a, might yield more pronounced
differences in migration between the stiffnesses of Fgn-coated substrates examined here.
Perhaps Fgn, as a nondiffusible cue of an inflamed matrix, and the chemotactic stimulus of
the fMLP gradient may coordinate to increase efficiency of neutrophil targeting to a degree
that overwhelms mechanotactic input. Mechanotactic shifts in RMS speed may prove to be
dependent on β1-integrin engagement.
Our work reported here, using the predominantly β1-integrin ligands Fn and Col IV
and the β2-restricted ligand Fgn, identifies an extraordinary level of complexity in the
72
1000
1000 0
100
10
10
1
100
MS
D (
µm
)2
300
MSD at 150s
*
*
*
*
0
0. 2
0. 4
0. 6
0. 8
Col
10 kPa
Fn
1. 0*
FnFn-β2
Fn-β2Col
100 kPa
*
directionality time, td (s)
directionality transition
time interval, τ (s)
tort
uo
sity
a
b
c
dti
me
av
era
ge
d β
(τ)
sub
stra
te s
ti!
ne
ss (
kPa
)τ < t d τ > t d
β+
Figure 3.7. The mechanosensitive component of directionality for neu-trophils migrating on Fn-coated substrates toward fMLP is dependenton β2 integrins. Human primary neutrophils migrating on Fn-coated gels of10 kPa or 100 kPa stiffness toward a fMLP point source were treated with β2-blocking antibody and tracked over a 30-min period (Supplemental Videos 3 and4 available online). (a) The MSD of neutrophil migration paths is plotted as afunction of time between steps. (Inset) The average MSD at a 150-s time intervalfor each condition. Blocking β2 integrins significantly decreased MSD and RMSspeed on Fn-coated gels of 10 kPa and 100 kPa. (b) Log-log MSD slope, β(τ)was fit to the βtextupexp-model yielding mean values of td that represent thetime interval at which migration transitions from random to directed. Blockingβ2 integrins increased td on 100 kPa Fn-coated gels. β2-Integrin-blocked cellsmigrating on Fn-coated gels have a td equivalent to untreated cells migratingon Col IV-coated gels. (caption continued on pg. 74)
73
characteristic measured metric Fgn Col Fn Fn+β2-blockmorphology spread area X X X X
relative motion MSD or RMSS • X X X
sampling interval dependent TAD • • X •measures of path “wiggle” tortuosity • • • •
sampling interval independent td X • X •measures of path “wiggle” β− X • X •
β+ • • • •
Table 3.1. Mechanosensitive parameters of human neutrophil chemo-taxis toward fMLP. Chemotactic parameters found to be mechanosensitivefor different coating and treatment conditions. A check mark represents param-eter that varies significantly by stiffness under the condition indicated. A bulletrepresents a parameter that does not vary significantly with stiffness.
mechanical regulation of neutrophil migration dependent on the nature of the matrix ligand
(Table 3.1). The RMS speed and directionality of neutrophil migration are both affected
by substrate stiffness on Fn. Persistence alone is affected by surface stiffness on the β2-
restricted ligand Fgn, whereas only the RMS speed alone is altered by surface stiffness
on the β1-restricted ligand Col IV. Therefore, under the conditions studied, differences
between the parameters of migration dynamics demonstrate a selectivity among integrin
ligands that allows for differential regulation of MSD and directionality in response to
mechanotactic cues. Additionally, the metrics that we derived to examine the time scale of
the transition from diffusive to directed migration reveal a previously unrecognized intricacy
in the directionality of neutrophil migration that can be mechanodirected by β2-integrin
engagement, underscoring the need for sensitive quantitative measures of chemotaxis.
(Fig. 3.7 caption continued) (c) Values of β− and β+, which quantify the degree of randomness in themigration path over short time intervals τ < td and the degree of directedness in the migration pathover long time intervals τ > td, respectively. These data are plotted as the directionality transition.Blocking β2 integrins significantly changes β− and alters the directionality transition profile to oneindistinguishable from untreated cells on Col IV-coated gels. (d) Cells migrating on Fn-coated gelsafter β2-integrin blocking have significantly lower tortuosities that are statistically equivalent tothose of cells migrating on Col-coated gels. The average tortuosity for neutrophils migrating on Fn-and Col IV-coated gels toward a fMLP point source is shown as a Box and Whisker Plot, showingmean and quartile data. Error bars represent SEM. P<0.05 untreated versus β2-integrin block onFn.
74
Other investigations into mechanosensing focus on the lower range of stiffness, 0.3-3
kPa [49], which fits well with reported measures of endothelial cell stiffness. The 10- and
100-kPa substrates used in these studies represent the low and high end of physiological
stiffness reported in diverse tissue types [44, 119, 120]. In particular, we are interested in
discovering how mechanotactic changes in tissue, such as in formation of some tumors, com-
bine with ligand composition and chemotactic influences to progress or support a pathologic
or diseased tissue environment. This is likely a key step in mediating chronic inflammation
and fibrosis, where the inflammatory response itself changes the tissue environment. In
the context of a resolving site of inflammation, where matrix deposition is dense, the local
environment is also stiffened. It is plausible that increased stiffness may affect matrix re-
modeling, such as through increased matrix metalloproteinase release, but this hypothesis
may be confounded by the inability to dissociate the effect of matrix stiffness from ligand
density. Moreover, this may vary among tissues, where the response to injury may result
in deposition of different matrix components. Under conditions where an injury response
includes progressive tissue stiffening, such as Acute Respiratory Distress Syndrome, tumor
growth, or fibrosis, the changes in rigidity would be temporally regulated. The mechanosen-
sitive response of the extravasated neutrophil would be expected to vary according to the
time in the host response that the cell entered the site of injury. The synchronicity of
β1- and β2-integrin engagement may additionally afford a transition between surveying and
homing phenotypes in chemotaxing cells. β1-integrin engagement supports decreased cell
motility on stiffer substrates responding to the tonicity of a tissue as a marker of increased
injury or inflammation, while maintaining the local diffusivity of a surveying phenotype.
With β1- and β2-integrin engagement, we see both modulation of RMS speed and direction-
ality, with β2-integrin engagement supporting a homing phenotype of enhanced cell motility
and increasing directionality. Our data suggest that Fgn may be serving as a biologically
75
significant CR3 ligand in this context by acting as a nondiffusible cue of the inflamed ma-
trix that aids neutrophil targeting by increasing β2-integrin engagement and by extension,
migration directionality up a fMLP gradient. These data combine to show that neutrophil
mechanosensitivity is not only ligand-dependent but also that mechanosensitive shifts in
migration dynamics are not binary. Even at this simple level, the parameters of neutrophil
response are finely tunable with environmental cues. For example, it may be that cells
entering a site of progressive stiffness, resulting from a wound, are mechanosensitive for Col
IV, whereas cells encountering Fn and/or Fgn deposition are likely to be associated with
an acute injury before scarring.
Current techniques to parameterize the trajectory of a chemotaxing cell most com-
monly pair migration speed with some measure of persistence/directionality [48, 49, 90].
The root MSD, divided by time, gives RMS speed over a particular time interval, whereas
persistence/directionality is typically characterized by measuring the TAD and/or tortuos-
ity. TAD and tortuosity implicitly depend on an arbitrarily chosen time interval, causing
such measures to skew potentially from report to report. Furthermore, a particular time
interval used to calculate TAD or tortuosity for one experiment may not be applicable to
other experiments if image-acquisition frequency is constrained. The distinction of true
randomness in the migration path from randomness caused by uncertainty in determining
the centroid positions is challenging. As our data represent here, migration dynamics can
vary significantly by ligand coating, and it follows that the optimal time interval by which to
analyze and compare migration conditions may also vary. To address this concern, we intro-
duce directionality time, a sampling-interval invariant measure of directionality motivated
by MSD fitting that incorporates trajectory characteristics for all time intervals. Direc-
tionality time characterizes the time scale at which migration transitions from random to
directed. Conceptually, it measures the time it takes for a cell that has veered off of its
directed course to reorient itself to its chemotactic path. A larger value for td suggests that
76
there is more inherent wiggle to a cell’s migration path evocative of a surveying phenotype.
Conversely, small td suggests a more directed homing phenotype. All else equal, migration
trajectories are more persistent when td decreases. As a result of its incorporation of data
across all time intervals, td enables a global, less subjective characterization of migration
directionality that can uncover subtle shifts in migration dynamics not reflected accurately
using TAD or tortuosity.
In conclusion, to migrate effectively, neutrophils must integrate many divergent signals,
including chemotactic, mechanotactic, and substrate context, to initiate an appropriate cel-
lular response to injury or inflammation. This migration necessarily incorporates elements
of speed, direction, and persistence of motion. Neutrophils are mechanosensitive but may
acquire mechanosensitivity by different mechanisms while migrating on different matrices.
The understanding of the underlying mechanisms that regulate directed neutrophil migra-
tion dynamics, the teasing apart of the constituent contributions, and the knowledge of
how they interact can provide insight into comprehending how subtle shifts in migration
dynamics and overall phenotype caused by environmental cues lead to significant shifts
in cellular behavior and clinical outcome. Sensitive analytical methods introduced in this
work that are capable of capturing these fine changes in cellular behavior may be key to the
identification of novel nodes for clinical intervention and immune modulation, which can be
targeted, not just to cell type but also to the relevant tissue microenvironment.
77
Part 2
Integrating Cell Kinematics with Cell
Mechanics
CHAPTER 4
Cell Mechanics and Motility: Springs and Stick-Slip
Adhesion
Portions of this chapter have been published, AJ Loosley and JX Tang. Phys Rev E
86, 031908 (2012), Copyright 2012 by the American Physical Society. All adaptations of
text and figures from this publication have been done with permission from the American
Physical Society.
79
Forward. Part 1 covered cell migration and mechanosensing. Cell mechanics were
discussed as possible mechanisms for the migration characteristics that were observed. In
part 2, the scope of part 1 is appended to include a direct investigation of the role of
cell mechanics in cell motility. This investigation begins here with a set of multi-element
toy models that give insight into the mechanical interactions and subsequent cell shape
dynamics and contractile forces that arise as part of the cell motility process.
1. Introduction
Cells are the building blocks of life and their migration is crucial to the biological
functions that sustain life. For example, tissue and nervous system formation depends on
the coordinated migration of pre-differentiated stem cells [121–123], whereas host immune
response depends on leukocyte migration to sites of infection and injury [8, 124]. Thus,
understanding the mechanisms of cell migration is important to the field of biology as well
as to advancing the frontier of medicine.
Many cells migrate by crawling along a particular substratum. The mechanisms that
generate cell crawling dynamics can be generally described in two steps. Step one, actin
polymerization occurs at the leading edge of a cell (lamellipodium) causing the cell to
protrude forward [6, 7]. New adhesion sites form at the leading edge during this process.
Step two, contractile forces generated within the cytoskeleton act to pull the rear of the cell
body forward in concert with graded adhesion between the cell and substratum [6]. Cell
shape may also play a role in cell crawling. The subject of how cell shape is determined
based on intra- and extracellular factors has been studied extensively both experimentally
and mathematically [21, 44, 48, 125–127]. There are also studies of the reverse problem in
the context of how cell shape affects focal adhesion site formation, traction forces, and cell
polarization [54, 128, 129], but the specific effects of cell shape on locomotion are as of yet
poorly understood.
80
A variety of cell shape dynamics can occur depending on the type of crawling cells as well
as intra- and extracellular factors [21, 130, 131]. For example, leukocytes and fibroblasts
exhibit fairly nondeterministic ruffling- and bubbling-like shape dynamics [48, 132, 133].
Other cells, particularly fish epithelial keratocytes, exhibit shape dynamics that appear
periodic and coherent [20, 31, 125, 134]. Such dynamical periodicity and regularity over
many cell lengths of migration make the latter cell type, fish keratocytes, a prototypical
system for studying cell shape dynamics and motility [21, 135, 136].
Periodic shape dynamics observed in crawling fish keratocytes are caused by alternating
stick-slip motions localized at opposite sides of the cell’s broad trailing edge [31, 137]. In
fast moving keratocytes, ones that move roughly 0.1µm/s or faster, these sticking and
slipping cycles are often observed to be coherent but opposite in phase [135]. Hence, one
side of the trailing edge sticks while the other slips in what is known as bipedal locomotion.
Barnhart et al. recently introduced a two dimensional mechanical spring model with stick-
slip adhesion to capture the dynamics of bipedal locomotion in fish keratocytes [135]. This
model consists of four point-like elements located at regions of prevalent shape dynamics of
the cell. One element represents the cell leading edge, a region where forces responsible for
cell migration are generated by complex cytoskelatal processes such as actin polymerization
and retrograde flow [138–141]. Two elements represent opposite sides of the cell trailing
edge, regions that exhibit periodic sticking and slipping motions. These three elements are
connected by a particular spring configuration that incorporates a fourth element in the
central region of the cell. The springs represent either cytoskeletal elasticity or coupling
between the cytoskeleton and the nucleus and act to restore overall cell shape in response
to mechanical perturbation.
We build on the model by Barnhart et al. by analyzing different possible spring con-
figurations that recapitulate the shape dynamics of crawling fish keratocytes and use the
results of this analysis to determine how these dynamics are dependent on cell elasticity,
81
size, and aspect ratio. The central element is now interpreted to be the cell nucleus and
we compare its motion to experimentally observed nucleus lateral displacements. Based on
assumptions such as symmetry about the axis of motion and confinement of the nucleus to
the central region of the cell, we determine that there are only four viable spring config-
urations, including the one studied in the previous work. We analyze the dynamics of all
four configurations and choose one deemed most mechanically representative of the real cell
that also generates realistic dynamics. Using this configuration, we identify three principal
parameters representing lamellipodial elasticity, cell length, and cell width that are signif-
icant determinants of the amplitude and period of cell shape oscillations. Varying these
principal parameters over a realistic range, we show that this simple spring model generates
shape dynamics corresponding to coherent bipedal, coherent non-bipedal, and decoherent
crawling cells.
2. Methods
2.1. Model overview. Similar to the previous work by Barnhart et al. [135], we model
the fish keratocyte in 2-D using four elastically coupled point-like elements representing dif-
ferent dynamic regions of the cell. To introduce the assumptions and physics underlying this
elastic coupling model, we begin with a demonstrative 1-D version illustrated in Fig. 4.1 a.
In this version, the front end, represented by x1, extends forward with velocity vf (dashed
line indicates cell protrusion). It is assumed that this forward propulsion is maintained by
the formation of new adhesions to the substrate. The trailing edge of the cell, at position
x2, is coupled to the front by a spring of equilibrium length L0 and stiffness K that is
representative of the cell length and elasticity of the actin cytoskeleton, respectively. The
assumption of a linearly elastic cytoskeleton is justified under physiologically normal strains
[51, 142].
82
The trailing edge element experiences two types of drag forces, adhesion (sticking) and
viscous shear (slipping). Adhesion occurs due to stochastic binding and unbinding of ad-
hesion proteins between the cell and its substrate (c.f. Chapter 3, Sec. 1.3: integrin borne
adhesion complexes) [134, 143, 144]. The associated free energy landscape that influences
the adhesion proteins is modelled by quadratic potential wells with minima corresponding
to binding sites on the substrate [143, 144]. Equivalently, transient attachments of adhesion
proteins between the cell and its substrate can be thought of as springs (see Fig. 4.1 a
overlay). If the average spring constant for each adhesive bond is κ, then the force against
the direction of motion due to a particular adhesion bond that forms at time tbindi is ap-
proximately
Fi ≈ κx2(t− tbindi ), (4.1)
where x2 is the trailing edge speed, t is time, and index i refers to the ith adhesion bond.
Equation 4.1 is valid only between the binding time, tbindi , and some particular unbinding
time, tunbindi , when the spring detaches. Times tbindi and tunbindi are stochastic variables
with distributions that depend on the trailing edge velocity [143]. Upon summation over
all binding events, the time averaged adhesive drag force is found to scale linearly with x2
and κ at low trailing edge speed. At sufficiently high trailing edge speed, the adhesive force
vanishes because adhesion proteins do not spend enough time within the capture region of
conjugate binding sites to form bonds. At high trailing edge speed, the drag force is also
thought to scale linearly with x2 due to the hydrodynamics of low Reynolds number viscous
shear. However, the constant of proportionality is much smaller than that associated with
adhesion.
One can define two drag coefficients: α for slipping, and β for sticking, which incorpo-
rates κ. The overall stick-slip drag force as a function of trailing edge velocity is modelled
83
by
Fd[x2] =
−βx2 , x2 < v1
v1−x2v1−v2
(βv1 − αv2)− βv1 , v1 < x2 < v2
−αx2 , v2 < x2
(4.2)
Here, sticking occurs when x2 < v1 (stick domain) due to adhesion bonds, slipping occurs
when x2 > v2 (slip domain), and some combination of sticking and slipping occurs when
v1 < x2 < v2 (transition domain). In the transition domain, the drag force is modelled by
linear interpolation (Fig. 4.1 b) though the shape of the curve in this transition region has
little effect on the resulting dynamics.
An additional consideration taking into account the time it takes the cell to switch
from sticking to slipping, and vice versa, is captured by a small inertia-like parameter, g,
the physical meaning of which is fully discussed in Ref. [135]. Including this g-factor, the
equations of motion for the one dimensional model are
x1 = vf (4.3)
gx2 − Fd [x2]−K(L− L0) = 0 (4.4)
where L = x1 − x2, and Fd[x2] is the stick-slip drag force given in Eq. 4.2.
Solutions to Eqs. 4.3 and 4.4 are limit cycles in the phase space of scaled cell-length,
L−L0K , and trailing edge velocity, x2. Fig. 4.1 c shows two such shape-cycle trajectories
plotted in this phase space. When the inertia-like term is removed (g = 0), spring force
must be balanced by drag force. In the stick domain (x2 < v1 < vf ), the trailing edge
velocity is less than the velocity of the extending leading edge. Consequently, the spring
representing cell length extends, increasing the forward force applied to the trailing edge
element. As this force increases, so too does the trailing edge velocity. When x2 increases to
be infinitesimally greater than v1, the sticking drag force is insufficient to balance against
the forward force of the spring. The trailing edge therefore accelerates instantaneously
84
a
b
c
0 v1 vf v2
0
sca
led
ce
ll le
ng
th
velocity of trailing edge x2
βv1
αv2g = 0 nN s2/µm
g = 0.3 nN s2/µm
L-L0
K
0
0
α, slipping
sticking
transition
α<<β
β
v2v1 vf
Fd(v)
βv1
αv2
x2velocity of trailing edge
dra
g fo
rce
K,L0
κ
vf
L
x2 x1
Figure 4.1. 1-D crawling cell model and stick-slip adhesion definitions.(a) The leading edge, x1, moves forward with constant velocity, vf , representinga region where the lamellipodium extends forward. The trailing edge, x2, iselastically coupled to the leading edge by a spring of elasticity K and rest-lengthL0 representing cytoskeletal elasticity and extension, respectively. A stick-slipdrag force underneath the trailing edge is modelled by many small springs withaverage spring constant κ. (b) Drag force-velocity curve. At low trailing edgevelocity, x2 < v1 (stick domain), drag force is generated by adhesion complexesforming between the cell membrane and substrate. To good approximation, suchadhesion generated drag force scales linearly with velocity characterized by dragcoefficient, β. At high trailing edge velocity, x2 > v2 (slip domain), adhesioncomplexes no longer form. The drag force in this domain is purely viscousin nature and characterized by the relatively small linear drag coefficient, α(α � β). At intermediate velocities, v1 < x2 < v2 (transition domain), dragforce is generated by a mixture of the sticking and slipping mechanisms. Theoverall drag-velocity curve is continuous in all domains. (c) Cell length-velocityphase space trajectories with and without the inertia term, g. Data pointsare separated by a constant time step equal to one fiftieth of the limit cycleperiod (T/50). Therefore, rapid changes in velocity and cell length are notedby relatively large distances between consecutive data points.
until force balance is re-established by the slipping drag force (cyan trajectory). When the
inertia-like term is applied, e.g. g = 0.3 s2 nN/µm, force balance is not instantaneously
85
required and abrupt acceleration does not occur. Hence, the limit cycle trajectory in phase
space appears rounded (green trajectory). In both cases, the shape of the drag force-velocity
curve in the transition domain (Eq. 4.2, v1 < x2 < v2) has negligible effect on the resulting
dynamics because the dynamical variable x2(t) remains within this domain over a duration
that is negligible compared to the limit cycle period.
This model is extended into 2-D as shown in Fig. 4.2. The trailing edge, where bipedal
locomotion occurs, is represented by two elements located at ~xl and ~xr. The drag force
in Eq. 4.2 is vectorially applied to both elements in the opposite direction of motion. The
nucleus is represented by an element located at ~xn. Drag on this element is intermediate
between sticking and slipping drags associated with the trailing edge. The front element
that drives the system forward is now represented by location vector ~xf , instead of ~x1 as
in the 1-D model. Later in this work, we replace the front element with a rod-like element
that better represents the wide extent of the protruding edge of the lamellipodium. Spring
and drag forces are combined into a set of 2-D equations of motion. The 2-D equations of
motion and a discussion about initial conditions is provided in the Supplementary Material
(Sec. refsec:sprMdlEqOfMotion). A reference diagram for this model is shown in Fig. 4.2 b
listing the spring constants, spring lengths, and drag coefficients. Cell lengths ∆yl and
∆yr are two of the dynamical variables used to characterize bipedal locomotion. They
are defined as the distance from the front element to the left and right trailing elements,
respectively, projected onto the axis of forward motion (y-axis). Fig 4.2 b also defines a
cell width, ∆x, as the distance between the trailing edge elements projected on the axis
perpendicular to forward motion (x-axis).
2.2. Simulation methods and criteria for characterizing dynamics. Solutions
to the 2-D equations of motion (Eqs. 1-4, 7, 8 in the Supplementary Material) were found
by numerical integration using the Runge-Kutta algorithm built into MATLAB R2010b
(The Mathworks, Natick, MA). A dynamical solution was considered periodic if the left
86
vf
∆yl(t)∆yr(t)
∆x(t)
∆zlam
KN, N
KL, L
K L, L
K D, D
KD , D
vf
, Lxn
xf
xl
xr
KW, W
z
y
xx
y
z
α, β, g
α, β, g
γ
a b
Figure 4.2. Schematic of four element elastic coupling model in 2-D.(a) Side profile of the cell, ∆zlam is the lamellipodium thickness. (b) Top downreference diagram of the 2-D elastic coupling model. Elements, depicted byovals, are located at ~xf , ~xl, ~xr, and ~xn. The front element moves with constantvelocity ~vf . Spring lengths and elasticity are indicated next to each spring.Element specific drag coefficients are shown in rectangular boxes. The cell ismodelled symmetrically about the axis of forward motion, ~vf . Cell lengths ∆yland ∆yr are defined as the distance between ~xf and either ~xl or ~xr respectively,projected onto the axis of forward motion.
and right side cell length dynamics stabilized into periodic motion within 800 s. For a typi-
cal limit cycle period of 40 , this equates to 20 periods. Fourier transformation was used to
measure frequency. For solutions deemed periodic, phase differences between ∆yl and ∆yr
were calculated. Frequency was determined by locating the first harmonic of the Fourier
transform while phase was determined by the complex argument of the Fourier transform at
this harmonic. Fourier transforms were calculated using the MATLAB fast Fourier trans-
form algorithm. We also measured amplitudes of cell length modulation and nucleus lateral
displacement, which is defined as the distance of the central element, ~xn, from the axis
of forward motion. Simulation dynamics were considered bipedal if cell length oscillations
were periodic and the phase difference between ∆yl and ∆yr was between 0.45 and 0.55
periods. Dynamics were otherwise labelled as either periodic or irregular. Bipedal dynam-
ics are said to be realistic if the following three conditions are satisfied, which are based
on experimental observations of fish keratocyte dynamics discussed in the Supplementary
Material (Figs. 4.6 and 4.7):
87
(1) amplitude of cell length modulation from 1 to 3µm;
(2) amplitude of nucleus lateral displacement from 0.3 to 1.2µm;
(3) period of limit cycle from 30 to 70 s.
Throughout this paper, simulation results are benchmarked against experimental analysis
of keratocytes discussed in the Supplementary Material.
2.3. Choice of model parameters. Parameter values for α, β, v1, and v2 were chosen
based on estimates made from the previous work [135]. A summary of the parameters used
in this model, including numerical ranges based on measurements of cell size, aspect ratio,
and other dynamical quantities, is shown in Table 4.1. The elastic modulus of a keratocyte,
E, has been measured to be between 10 and 150 nN/µm2 [142, 145, 146], and is thought
to increase from anterior to posterior. The model was analyzed over this range of E by
varying the stiffness of springs that correspond to different regions of the cell. These spring
stiffnesses were calculated using the relation,
k =ES
d, (4.5)
where S is the cross section area and d is the spring length. For example, to calculate
KD, we set S ≈ L∆zlam and d = D (see Fig. 4.2). Using ∆zlam ∼ 0.1µm [146, 147]
and L/D ∼ 0.5 − 1 based on keratocyte cell shapes measurements (see Supplementary
Material) [21, 125, 135] yields a spring constant range KD ∼ 0.5− 15nN/µm. The viability
of this model was tested using spring constants varied from 0 to 10 nN/µm. Spring lengths
were chosen in conjunction with spring constants so that simulated cell width and length
corresponded to the shapes of fish keratocyte cells observed in previous publications (see
Supplementary Material), though cell shape range need not have been restricted in this
manner.
88
parameter meaning range units refsα slipping drag coefficient
0.15 - 0.5 nN s/µm(viscous shear)
β sticking drag coefficient20 - 100 nN s/µm
(adhesion under trailing edge)
γ nuclear drag coefficient1 - 20 nN s/µm
(adhesion under cell nucleus)
g inertia term0 - 0.8 nN s2/µm [135](sets switching time-scale
between sticking and slipping)
v1 critical sticking velocity0.08 µm/s
(upper limit of the sticking domain)
v2 critical slipping velocity1 µm/s
(lower limit of the slipping domain)
vf leading edge velocity0.2 µm/s
(stick-slip dynamics require vf > v1)
KN
spring constants 0 - 10 nN/µmKD [142, 145]
KL [135, 146]
KW
N 1 - 20
µmD spring lengths 5 - 35 [21, 125, 135]
L (determines cell shape) 10 - 30 [148, 149]
W 18 - 60
R handle bar rod length 0-30 µm
Table 4.1. List of parameters and physiologically viable values for the2D stick-slip model. Parameter ranges correspond to experimentally observedcell velocity, elasticity, etc. as determined by estimation or measurements re-ported in previous work. Each parameter range is justified by the referencesgiven here, except for rod length R, which we scale with the width of the cell’sperceived leading edge. Some spring constants are not applicable depending onwhich configuration is used. Here, spring lengths were selected to permit propercell shape.
2.4. Simulation benchmarking. Phase contrast movies of eleven motile fish kerato-
cytes, five undergoing bipedal locomotion, were analyzed to measure cell sizes, aspect ratios,
and other dynamical quantities used to benchmark simulation dynamics (Fig. 4.6). These
movies were obtained from the supplementary materials of Refs. [21, 125, 135, 150]. Movies
were converted to image sequences using Virtual Dub (Avery Lee) or MPEG Streamclip
(Squared 5) depending on file format. Custom MATLAB software was used to determine
89
image by image cell symmetry axes and trajectories of the leading edge, trailing edge, and
nucleus centroid. We measured nucleus lateral displacement to be the distance from the
nucleus centroid to a line of best fit (Fig. 4.7). Experimental cell length oscillations were
measured as the distance between the center of the leading edge and either of the trailing
edge elements, projected onto the cell symmetry axis. There are minor discrepancies be-
tween these measurements and simulated cell length oscillations because the experimental
symmetry axis does not always correspond to the axis of forward motion. An example of
cell length oscillation measurements is shown in Movie S1.
3. Results
3.1. Viable spring configurations. There are several ways to elastically couple the
elements that make up the two dimensional model (Fig. 4.2 b), in particular by adding or
removing springs to form different spring configurations. By assuming symmetry about the
axis of forward motion and by requiring the cell to maintain a reasonable shape with width
and length comparable to observations, the number of possible configurations is constrained
to the four illustrated in Fig. 4.3 a. Config. 1 is the simplest possible configuration that can
generate bipedal locomotion whereas Configs. 2 through 4 generate bipedal locomotion with
one added element that represents the cell nucleus. The dynamics of all four configurations
are discussed at length in the Supplementary Material (Fig. 4.8).
Briefly, Config. 1 can generate dynamics that are similar to bipedal locomotion, though
the single direct coupling between trailing edge elements through the KW spring leads to
aberrant motions at the trailing edge. Specifically, slipping of one trailing edge element
extends the KW spring causing momentary aberrant slipping of the opposite trailing edge
element. There is also no possibility for Config. 1 to describe the observed lateral displace-
ment of a keratocyte nucleus. Adding a central element allows for indirect elastic coupling
between the trailing edge elements that supplements the direct KW connection. Config. 2
is like Config. 1 except a central element is added and all four elements are directly coupled
90
to each other. This configuration can generate bipedal locomotion and realistic nucleus
lateral displacement if one interprets the central element to be the nucleus. However, such
dynamics are not robust under parameter variation compared to configurations with fewer
springs. Config. 2 works best near the KD → 0 or KW → 0 limits, i.e. Config. 3 or Config. 4,
respectively.
Config. 3 is the spring arrangement considered by Barnhart et al. They found that
stable bipedal locomotion occurs over a range of KW and g-values. During bipedal locomo-
tion, the central element, ~xn, oscillates in the lateral direction entrained to the bipedal limit
cycle. Although Config. 3 produces realistic bipedal dynamics, we have no physical inter-
pretation of a spring directly coupling the trailing edge elements. In contrast, we introduce
an alternative configuration, Config. 4, and use a spring orientation argument to suggest
that it better captures the mechanical properties of the actin cytoskeleton. Config. 4 is dif-
ferent from Config. 3 by the removal of the KW spring (KW = 0), and the addition of two
springs that couple each trailing edge element to the leading edge element (KD > 0). In this
model, the KD springs tend to orient with angles similar to the known orientation angles
of actin filaments that make up the lamellipodial actin network in keratocytes [151, 152].
Specifically, actin filaments in keratocytes under physiological conditions show long range
orientation order with angles between ±25o and ±45o with respect to the direction of lead-
ing edge protrusion. The KD springs capture the anisotropy of network elasticity [153] in
the direction parallel to filament orientation. Cell width is now maintained by both the KD
and KL springs, instead of spring KW as in Config. 3. Springs, KL, coupling the trailing
edge to the nucleus can be interpreted in the context of the contractile actin-myosin bundle
at the rear of the cell [154], though a KW spring can also be interpreted in the same way.
Configuration 4 generates dynamics similar to those of Config. 3, in some cases with
a slightly larger nucleus lateral displacement closer to experimentally observed values. An
example time lapse showing the dynamics of Config. 4 is shown in Fig. 4.3 b. Cell length
91
po
siti
on
(μ
m)
position (μm) position (μm) position (μm)
po
siti
on
(μ
m)
0
5
10
15
20
25 26 s 33 s 40 s
0
5
10
15
20
25
-10 -5 0 5 10
47 s
-10 -5 0 5 10
54 s
-10 -5 0 5 10
61 s
dc
b
a
-1
0
1
time (seconds)100806040200n
ucl
eu
s la
tera
l
dis
pla
cem
en
t (μ
m)
time (seconds)0 20 40 60 80 100
1012141618
cell
len
gth
, ∆y
(μ
m)
xn
xxl r
xf
xf
xl xr l r
xn
x xl r
xf
∆y
l r
xn
xrlx
xf
Con!guration 1 Con!guration 2
Con!guration 3 Con!guration 4
Figure 4.3. Viable 2-D spring configurations and the dynamics of Con-fig. 4. (a) Diagrams of the four viable 2-D spring configurations. Each viableconfiguration maintains reasonable cell shape and is symmetric about the axisof forward motion. Config. 4 is the preferred configuration proposed in thiswork. (b) Time lapse of the simulated dynamics of Config. 4. showing bipedallocomotion. The time lapse is shown in 7 s increments and corresponds to timeplots of cell length and nucleus lateral displacement shown in (c) and (d). Thissimulation corresponds to a cell with a time averaged length and width of 13and 19µm, respectively. The amplitude of length oscillations is 3.0µm with aperiod of 40.5 s. The amplitude of nucleus lateral displacement is 0.6µm. A con-tinuous motion time lapse of this simulation is found in Movie S2. Parameterscorresponding to this simulation are listed in Fig. 4.8.
92
and nucleus lateral displacement time plots corresponding to the time lapse are shown in
Fig. 4.3 c and d, respectively. In this example, the nucleus undulates laterally in a series
of exponential decays that have a period of 40.5 s, and an amplitude of 0.6µm, consistent
with observations. Nucleus lateral displacement can be made more sinusoidal if the drag
force (parameter γ) is reduced compared to the spring forces acting on the nucleus, though
it is unclear if this would be more realistic.
To assess the viability of Config. 4, we investigated how its bipedal dynamics changed
in response to varying mechanical parameters, KD, KL, and KN , drag parameters, α, β,
γ, and g, and cell shape parameters, D, L, and N . Fig. 4.4, a and b, are phase diagrams
of the dynamical responses plotted in the g-KD and g-KL spaces, respectively. Hatched
areas indicate regions of realistic bipedal dynamics for two choices of drag coefficient γ.
The dynamical response is characterized by amplitudes of cell length and nucleus lateral
displacement oscillations (blue curves), and the overall limit cycle period (green curve)
shown in Fig. 4.4, c and d. Bipedal locomotion occurred for spring stiffnesses, KD &
0.4 nN/µm and KL & 0.5 nN/µm. The model fails at lower spring stiffnesses because there
is not enough rigidity between elements to maintain normal cell shape. In the case of lower
elasticity, KL < 0.5 nN/µm, element ~xl can unrealistically swing over from the left side to
the right side of the cell (and vice versa for element ~xr). In the normal cell shape regime,
response characteristics changed very little under variation of KL, in contrast to variation of
KD. Therefore, we identify lamellipodial spring stiffness, KD, as a principal parameter that
tunes the length- and time-scales of limit cycle behaviour, more so than other mechanical
parameters in this model. Fig. 4.9 in the Supplementary Material shows that response
characteristics are also sensitive to variation of sticking coefficient, β, though less sensitive
to variation of drag coefficient, γ, and inertia-like parameter, g.
Not shown in Fig. 4.4 or in the Supplementary Material is the effect of spring elasticity
KN on the dynamics of Config. 4. Spring KN in conjunction with drag on the nucleus
93
can be used to fix the average displacement between the front element and the nucleus
element. Spring elasticity KN is required to be greater than 1 nN/µm in order to maintain
cell shape. As KN increases, the nucleus element is drawn toward the front of the cell,
thus also drawing the trailing edge elements inwards. Bipedal locomotion still occurs with
slightly altered period and amplitudes.
0 1 2 3 40
1
2
3
4
0
15
30
45
60
75
KL (nN/µm)
pe
riod
(s)
NLD
cell length
period
0 1 2 3 40
1
2
3
4
0
15
30
45
60
75
am
plit
ud
e (µ
m)
KD (nN/µm)
cell lengthperiodNLD
g (
nN
s2/µ
m)
KD (nN/µm)0 1 2 3 4
0
0.1
0.2
0.3
0.4
KL (nN/µm)0
0
0.1
0.2
0.3
0.4
1 2 3 4
amplitude of celllength oscillations, both ∆yl(t) and ∆yr(t)
amplitude of nucleus lateral displacement(NLD), ∆xn(t)
period of oscilations
realistic bipedal, γ = 5 nN s/µmbipedal
non bipedal realistic bipedal, γ = 2 nN s/µm
a
c
b
d
vf
∆xn(t)
∆yl(t) ∆yr(t)KD KDKN
KLKL
Figure 4.4. Characterized dynamical responses of Config. 4 with re-spect to parameters KD, KL and g. The green shaded regions of the g−KD
(a) and g−KL (b) parameter spaces indicate bipedal dynamics and hatch pat-terns indicate realistic bipedal dynamics for two choices of γ. (c, d) Amplitudesof cell length and nucleus lateral displacement (NLD) oscillations (blue curves),as well as the limit cycle period (green curve), plotted against mechanical pa-rameters KD and KL. Lamellipodium spring elasticity, KD, significantly alterslimit cycle amplitudes and periods, whereas trailing edge spring elasticity, KL,does not. The model fails when either spring constant is too low (. 0.5 nN/µm)where then point-like elements delocalize leading to a loss of normal shape. Ineach chart, parameters that are not varied are listed in Fig. 4.8 under Config. 4.
94
3.2. Crawling dynamics depend on cell aspect ratio. Using Config. 4, we system-
atically varied lamellipodial elasticity, cell size, and cell aspect ratio to analyze their effects
on crawling dynamics. Cell width was varied by changing spring length L (width ≈ 2L), and
lamellipodial elasticity was varied via the KD parameter, which is the principle mechanical
parameter that tunes the dynamics of this model. Two different cell lengths were studied
based on the experimental cell length distribution in Fig. 4.6 a: short, 〈∆y(t)〉 = 11µm
(N = 8µm), and long, 〈∆y(t)〉 = 16µm (N = 12µm). Mathematically, diagonal spring
length D was made functionally dependent on L, N , and KD in order to hold the cell length
constant under variation of the dependent parameters (Fig. 4.2).
The results of this analysis are shown as cases 1 and 2 in Fig. 4.5. Case 3 is a modification
of Config. 4 where a rod-like element is used at the leading edge instead of point-like element,
~xf . In the sense that the KD springs represent a center of mean elasticity on the two sides of
the lamellipodial actin network, it is likely more realistic that these springs should couple to
two points at the leading edge that are symmetrically displaced from the axis of symmetry,
instead of to a point at ~xf . A rod-like element allows us to modify the endpoints of springs
KD in just this way. The rod also better aligns the KD springs with the known long range
angular orientation of the cytoskeleton discussed above. The modified equations of motion
for this configuration, which we call the handlebar model, are found in the Supplementary
Material.
Diagrams of the three cases discussed above are shown in Fig. 4.5 a. For each case, we
investigated the effects of varying cell width and lamellipodium elasticity. Dynamical output
is characterized by amplitude of cell length oscillations and dynamical behaviour as shown
in Fig. 4.5 b and c, respectively. The amplitude maps can be broken down into three key
regions. Regions of red indicate relatively large amplitudes (3-10µm). These amplitudes are
greater than those of most coherent keratocytes and occur when lamellipodium elasticity
is low, causing the trailing edge to stick longer. Regions of dark blue indicate relatively
95
small amplitudes (0-1µm). These amplitude describe most smooth gliding keratocytes
that have small stick-slip events observed at the trailing edge (for example, in Movie S1).
Regions of both light blue and yellow indicate amplitudes of realistic bipedal locomotion (1-
3µm). In cases 1 and 2, there appear to be “anomalous” amplitude variations for wide cells
when KD & 6 nN/µm. The phase diagrams indicate that these anomalous regions of the
amplitude maps correspond to irregular behaviour (Fig. 4.5 c). Such regions could represent
the phase space for fast moving decoherent cells. Smooth gliding cells are described by all
dynamical behaviours with small amplitudes. Exceptions shown here are cells that fail
to maintain proper cell shape, which are indicated in white at the top-right corner of the
phase diagram for case 1, although there are other examples beyond these parameter ranges.
Realistic bipedal locomotion occurs in regions of overlap between those labelled bipedal on
the phase diagrams and those where amplitudes of cell length oscillations fall between 1
and 3µm. Overall, one can use the phase diagrams in conjunction with the amplitude maps
to characterize the dynamical responses of the model. These diagrams describe how cell
crawling dynamics are dependent on cytoskeletal elasticity, extension, and cell aspect ratio.
4. Discussion
Cell motility models typically consist of a set of dynamical equations that describe the
biochemistry (i.e., diffusion and flow of biomolecules that regulate myosin motors, actin
polymerization, etc.) and/or the biomechanics (i.e., adhesion between cell and substrate,
cortical tension, etc.) of a system to varying degrees of complexity [126, 127, 135, 140, 155–
158]. That a simple mechanical model involving only four elements coupled by passive
springs is able to significantly recapitulate the motion of these highly complicated systems
is surprising. There are alternative models describing the shape dynamics of keratocytes,
such as one proposed by Ziebert et al. [158], in which cell length oscillations result from
filament orientation and overall cell shape. In another model proposed by Barnhart et al.
[127], the implications of substrate adhesion strength on keratocyte motility were studied
96
Amplitude of Cell Length Oscillations
L (µm)10 15 20 25 30
10
8
6
4
2
periodic
bipedal
10 15 20 25 30L (µm)
10
8
6
4
2
irregular
bipedal
periodic
10 15 20 25 30L (µm)
2
4
6
8
10
KD (µ
m)
irregular
bipedal
periodic
2
4
6
8
10
3025201510L (µm)L (µm)
2
4
6
8
10
3025201510L (µm)
2
4
6
8
10
3025201510
KD (µ
m)
0
2
4
6
8
10
(µm)
Phase Diagrams
<∆y>=16µm
Case 3
<∆y>=16µm
Case 2
<∆y>=11µm
Case 1
∆yr(t)∆yl(t)
b
a
c
Figure 4.5. Dynamical responses of Config. 4 and the handlebar modelwith respect to lamellipodial elasticity, cell size, and cell aspect ratio.(a) Diagrams indicating average cell length, 〈∆y〉, and mechanical model: eitherConfig. 4 (first and second columns) or the handlebar model (third column).Cell width is varied by changing spring length L. Cell length is varied bychanging middle spring length, N , and then choosing lamellipodial spring lengthD = D(L,KD, N) such that 〈∆y(t)〉 remains constant. Parameter KD sets thelamellipodial elasticity. In case 3, the length of the handle bar is set equal toL. (b) Color maps indicating the amplitude of cell length oscillations undervariation of KD and L. Saturated red (larger amplitudes occurring when KD
is relatively small) indicates amplitudes greater than those typically observed.Dark blue (smaller amplitudes occurring when KD is relatively large) indicatesamplitudes that are small and difficult to measure experimentally. Light blueand yellow (light gray shades) indicate amplitudes corresponding to realisticbipedal dynamics. (caption continued on pg. 98)
by considering the interplay between actin polymerization, myosin II transport, myosin II
generated actin retrograde flow, and linear adhesion forces between the cell and its substrate.
Their model generates the characteristic cell shapes and migration speeds recorded in fish
97
keratocytes, but it does not account for bipedal locomotion. Our model describes general
cell crawling dynamics and its dependency on mechanical properties but does not account
for the specific effects of adhesion strength because the physics of adhesion at the leading
edge and other more complicated factors such as actin polymerization and actin retrograde
motion are all contained in the self-propulsion parameter, ~vf , which prescribes the cell’s
locomotion speed. We ignore the finer details of these factors in exchange for a simple
model to study cell crawling dynamics with a focus distinct from that of Ref. [127].
In our analysis of cell crawling movies from previous publications, it is apparent that
the leading edge velocity of bipedally crawling cells fluctuates, although these fluctuations
are not as large as stick-slip induced velocity fluctuations at the trailing edge. As such,
assuming a constant velocity, ~vf , at the leading edge is an oversimplification. Instead, one
could model the forward propulsion by introducing a propulsion force in place of velocity
~vf . Such a force would have to be anchored by adhesion formation under the ventral surface
of the cell that would lead to a net forward spring force large enough to propel the trailing
edge elements into the slipping domain without allowing the cell to stall. The force at
the leading edge could be calculated from empirical force-velocity relations such as the one
recently measured by Heinemann et al. [147] using slow crawling keratocytes. In this case,
the load force would be assumed to scale with the total spring force acting against leading
edge and the force-velocity relation would then be used to calculate the leading edge velocity.
At this time, however, it is unknown how the force-velocity curves measured by Heinemann
et al. are different from those of fast crawling keratocytes, which have distinctly different
leading edge characteristics. Measurements and models of force-velocity curves for other
systems such as listeria [140, 159] are strikingly different from those in Ref. [147]. In listeria,
(Fig. 4.5 caption continued) Amplitude maps can be interpreted in conjunction with the corre-sponding phase diagrams beneath in (c). The dynamical response of the system is categorized intothree behaviours: bipedal, periodic, and irregular. Bipedal regions includes both realistic and otherbipedal locomotion. Both periodic and bipedal regions correspond to coherent gliding-like kerato-cytes, whereas irregular dynamics with anomalously large amplitudes may correspond to decoherentcells. The white region (case 1) indicates solutions where the cell fails to maintain a reasonableshape.
98
the protrusion velocity is nearly independent of load under high loading conditions, whereas
the opposite is true for slow crawling keratocytes. Given the variability of force-velocity
relationships in the literature, we stay with the constant velocity approximation as a first
step to mechanically modelling cell dynamics, instead of invoking a protrusion velocity that
requires knowledge of a force velocity curve.
Beyond investigating the stick-slip dynamics of the keratocyte trailing edge, our analy-
sis also probes the role of the cell nucleus. It is well known that epithelial keratocytes are
complex systems with mechanical properties that depend on the actin cytoskeleton [160–
162], but here we find that coupling to a central element is required to generate realistic
nucleus lateral displacement. This implies a mechanical landscape where the trailing edge
and lamellipodium are both elastically coupled to the cell nucleus. The coupling scheme
made up of all possible spring connections among the four elements (Config. 2) can generate
bipedal locomotion; however, the system is not robust under parameter variation. Because
this configuration is made up of more than the minimal number of springs required to main-
tain proper cell shape, the trailing edge elements are more sensitive to sudden motions that
propagate through multiple spring pathways during stick-slip transitions. Hence, adding
additional springs over-constrains the system.
Both the third and fourth configurations generate realistic bipedal dynamics including
nucleus lateral displacement oscillations. In the fourth configuration, the KD springs cap-
ture the elasticity of the lamellipodial actin cytoskeleton in the direction parallel to overall
filament orientation. There is a general robustness in terms of parameter ranges over which
realistic bipedal locomotion occurs, though some parameters (KD and β) significantly mod-
ify the corresponding time-scales and amplitudes. Cell shape dynamics are highly dependent
on spring constant KD, whereas they are far less dependent on spring constant KL. This
implies that locomotion is sensitive to elasticity of the actin cytoskeleton, but not elasticity
of couplings between the trailing edge and the nucleus. Blebbistatin, a myosin II inhibitor,
99
is known to inhibit actin network flow at the rear of keratocytes [154] but does not greatly
change keratocyte stick-slip dynamics [135]. This result is consistent with our model, which
predicts that cell crawling dynamics are relatively insensitive to variation of elasticity at
the rear of the cell. By the same logic, the model does not contradict findings suggesting
that the rear of a keratocyte is stiffer than its lamellipodium [146].
When interpreting Fig. 4.5, one should consider how KD scales with cell size. Holding
the elastic modulus, E, constant, one can estimate KD as a function of cell width parameter,
L, using Eq. 4.5. This function is obtained by solving KD ≈ EL∆zlamD(L,KD) , where D(L,KD) is
the spring length necessary to maintain constant cell length when L and KD are varied.
Solutions to this equation show that many limit cycles with abnormally large amplitudes in
Fig. 4.5, cases 1 and 2, are not possible given realistic values of E and ∆zlam. This mechanics
argument gives insight into why keratocytes are not observed with these excessively large
stick-slip cycles.
In addition to recapitulating the dynamics of fast moving keratocytes, the model is also
applicable to slow moving ones. The transition between slow and fast is set by parame-
ter v1, where in this work we have analyzed cells modelled by vf > v1. Figure 4.6 shows
distributions representing the wide variety of cell speeds, sizes, and stick slip amplitudes
recorded in fast moving keratocyte. Observed crawling dynamics can are categorized into
three groups: Coherent bipedal, coherent non-bipedal, and decoherent locomotion. The
model presented in this work reproduces these observations. We have shown how lamel-
lipodial elasticity, cell size, and cell aspect ratio can determine crawling behaviour even
before consideration for more complicated biological mechanisms. These findings suggest
the existence of mechanically preferred cell shapes for cells that need to move quickly and
efficiently. The mechanical model presented in this work should be applicable to other
fan shaped cells such as gliding human fibrosarcoma cells [132] and the ameboid sperm of
ascaris [163]. More complicated cell shapes and shape dynamics are possible by adding
100
more stick-slip elements to the model. Therefore, this model may also be applicable to cells
such as leukocytes and fibroblasts that undergo more complicated, highly variable, shape
dynamics.
5. Supplementary Material
5.1. Analysis of fish keratocyte movies. Movies of crawling fish keratocytes from
previous publications [21, 125, 135] were analyzed as discussed in the methods section to
determine realistic ranges of dynamical variables that could be used to test and benchmark
our model. Cell aspect ratio, speed, and oscillation amplitude distributions derived from
the analysis are plotted in Fig. 4.6. Additionally, each keratocyte was categorized into
one of two main groups: decoherent and coherent. Decoherent cells typically crawl slowly
(vf < 0.1µm/s) with a blebbing or irregular leading edge. Coherent cells often crawl
faster than decoherent cells and maintain relatively smooth leading edge. Coherent cells
were further categorized as either bipedal or non bipedal. Bipedal cells exhibited stick-
slip events that alternated on both sides of the trailing edge, whereas these events do
not alternate in nonbipedal cells. The majority of cells were 20 to 50µm wide and 10
to 20µm long, crawled with cell length oscillation amplitudes from 0 to 4µm, and with
speeds from 0.1 to 0.4µm/s. Movie S1 shows an example of the data analysis we did on
a coherent non-bipedal keratocyte movie published in Ref. [21]. The cell in this movie
changes its direction of motion to the right (clockwise), causing the cell length to be greater
on the left than on the right (Movie S1 B). This cell corresponds to the data point at
[width,∆yamp] = [40.4± 2.6µm, 0.8± 0.8 µm] in Fig. 4.6 b. We used the length-width
distribution to determine a realistic range of cell aspect ratios that were tested with our
spring model to answer the question: How is cell crawling dependent on cell aspect ratio?
5.2. Shape dynamics of fast crawling keratocytes. A time lapse of a bipedally
crawling fish keratocyte migrating with a leading edge velocity of 0.17µm/s is shown in
101
1 2 3 40.1
0.2
0.3
0.4
20 40 600.1
0.2
0.3
0.4
5030
1 2 3 40
1
2
3
4
20 40 600
1
2
3
4
30 50
10
15
20
25
30
20 40 605030
coherent non-bipedaldecoherent
values used for model
coherent bipedal
(a)
(b)
(d) (e)
(c)
aver
age
cell
leng
th,
<∆y>
(µm
)le
adin
g ed
ge
spee
d, v
f (µm
/s)
ampl
itude
of c
ell l
engt
h os
cilla
tions
, ∆y
amp (
µm)
cell width (µm) [cell width] / <∆y>
vf
width
∆yl(t) ∆yr(t)
Figure 4.6. Cell aspect ratio, speed, and amplitude of cell length oscil-lation distributions measured from previously published movies of fishkeratocytes. Cells are categorized into two groups, coherent and decoherent(dark green squares) as described in this work. Coherent cells are subcatego-rized as either bipedal (light green circles) or non-bipedal (blue triangles). (a)Cell length versus width distribution. Data points and error bars represent av-erages and standard deviations of measurements made at five randomly chosenframes. Most cells were 20 to 50 µm wide and 10 to 20µm long. Our modellingwork covers two cell length, 11 and 16µm, in addition to this entire range of cellwidths. (b and c) Cell length oscillation amplitude is plotted against cell widthand the width / 〈∆y〉 ratio. Horizontal error bars represent standard deviationssimilar to those in (a). (caption continued on pg. 103)
Fig. 4.7 a. Periodic retractions on opposite sides of the trailing edge are manifestations of
cycling between sticking and slipping. Here, these periodic retractions are opposite in phase
102
indicative of a cell undergoing bipedal locomotion. Bipedal sticking and slipping on both
sides of the trailing edge occur every 50 to 65 s. Cell outlines drawn in 25 s increments depict
this behaviour, with black arrows indicating regions of distinguishable sticking occurring
over successive time frames.
The alternating stick and slip events at the trailing edge lead to nucleus undulation
perpendicular to the direction of migration and phase-locked to stick-slip cycle. Red dots
in Fig. 4.7 b indicate locations of the nucleus centroid corresponding to the time points
shown. The trajectory of the nucleus is fit to a third order polynomial (Fig. 4.7 c). Here,
the amplitude of nucleus lateral displacement ranges from 0.3 to 1.2µm with an average
of 0.7µm (Fig. 4.7 d). The period of nucleus lateral displacement ranges from 50 to 65 s
(Fig. 4.7 e), same as the stick-slip period. Of the bipedally crawling keratocytes we analyzed,
nucleus lateral displacement amplitudes varied greatly and it was often unclear whether
lateral undulations were closer to a periodic exponential relaxation, or a sinusoidal-like
curve as plotted in Fig.4.7 d. The average amplitude of nucleus lateral displacement was
0.98µm with a standard deviation of 0.60µm. This data is compared against the results of
our modelling work.
5.3. The two dimensional model. In two dimensions, we analyze the shape dynam-
ics for a variety of spring configurations that describe the general model depicted in Fig. 4.2.
In the notation of this paper, ~xi is the position of the ith element where index i denotes
one of the following abbreviations: f for the leading edge element, n for the nuclear region
element, l for the left trailing edge element, and r for the right.
(Fig. 4.6 caption continued) Vertical error bars are estimated based on the standard deviation in celllength change caused by individual stick-stip events, and based on the average cell length uncertainty,measured frame by frame. The amplitudes of coherent bipedal cells are largest, whereas the othercells have small amplitudes with large error bars. (d and e) Leading edge speed versus cell widthand width / 〈∆y〉 ratio.
103
10μm
1007550250
1007550250
0 10 20 30 4005
10152025
x (µm)
y (µ
m)
(c) (d) (e)
(b)
(a)
0 100 200-3
0
3
time (seconds)nucl
ear l
ater
aldi
spla
cem
ent (µm
)
0 0.02 0.0400.20.40.60.8
frequency (Hz)
Four
ier a
mpl
itude
Figure 4.7. Bipedal locomotion and lateral oscillations of the nucleusobserved in fish keratocytes. (a) Time-lapse of a fish keratocyte moving fromtop-left to bottom-right, shown in 25 s increments. Cell outlines in fading greyrepresent the same cell 25 and 50 s prior to current position. Periodic sticking(arrows) and slipping alternate across opposite sides of the trailing edge. (b)Time lapse of the same cell in A, now showing the trajectory of the nucleus aswell as cell outlines for every time increment. (c) Nucleus position of the cellanalyzed in A (red) fit to a third order polynomial (grey). (d) Lateral motionof the nucleus calculated as the distance between nucleus position and fit in thedirection normal to the fit. Lateral motion is plotted with respect to time. (e)Fourier transform of (d) shows a frequency corresponding to a period between50 to 65 s.
5.3.1. Equations of Motion. The general equations of motion for the two dimensional
model are
~xf = vf y (4.6)
γ~xn −∑i=f,l,r
Kin(|~xn − ~xi| − Lin,0)~xn − ~xi|~xn − ~xi|
= 0 (4.7)
g~xl − Fd[|~xl|]~xl
|~xl|−∑
i=f,n,r
Kil(|~xl − ~xi| − Lil,0)~xl − ~xi|~xl − ~xi|
= 0 (4.8)
g~xr − Fd[|~xr|]~xr
|~xr|−∑i=f,n,l
Kir(|~xr − ~xi| − Lir,0)~xr − ~xi|~xr − ~xi|
= 0 (4.9)
104
where γ is the drag coefficient of the central element, and matrix elements Kij and Lij,0
are the spring elasticity and equilibrium length, respectively, for the spring connecting the
ith and jth elements. Assuming symmetry about the axis of forward motion, the spring
elasticity and equilibrium length matrix elements are given by
Kij =
f n l r
0 KN KD KD
KN 0 KL KL
KD KL 0 KW
KD KL KW 0
ij
(4.10)
and
Lij,0 =
f n l r
0 N D D
N 0 L L
D L 0 W
D L W 0
ij
(4.11)
Different spring configurations are analyzed by varying elements of the spring matrix and
a range of cell shapes can be analyzed by varying both Kij and Lij . A spring is removed
by setting its elasticity coefficient to zero. A movie of one spring configuration in which
KW = 0 is shown in Movie S2. This particular configuration is discussed in detail in the
main article and is referred to as Config. 4.
5.3.2. Handlebar model. In the standard 2-D model above, the diagonal springs (KD)
couple both trailing edge elements to a point-like leading edge element along the symmetry
axis of the cell. In the sense that the KD springs represent a center of mean elasticity of
the actin cytoskeleton in the lamellipodium, it is likely more realistic that the KD springs
105
should couple to two points at the leading edge symmetrically displaced from the axis of
symmetry. We therefore introduce a handlebar model in which a rod perpendicular to the
axis of symmetry represents the cell leading edge instead of point-like element ~xf . The KD
springs couple to the ends of the rod and the KN spring couples to center, (see Fig. 4.5 a,
right diagram). Equations of motion 4.6 and 4.7 remain unchanged while Eqs. 4.8 and 4.9
become
g~xl − Fd[|~xl|]~xl
|~xl|−
∑i=f+,n,r
Kil(|~xl − ~xi| − Lil,0)~xl − ~xi|~xl − ~xi|
= 0 (4.12)
g~xr − Fd[|~xr|]~xr
|~xr|−
∑i=f−,n,l
Kir(|~xr − ~xi| − Lir,0)~xr − ~xi|~xr − ~xi|
= 0, (4.13)
where ~xf± = ~xf ± R2 x. This model is discussed in association with Fig. 4.5 of Sec. 3.2.
5.3.3. Initial Conditions. In general, the solutions of a dynamical system depend on
initial conditions. In the context of this work, choosing initial conditions is akin to deter-
mining the initial cell shape and regional velocities. Depending on the choice of parameters,
many solutions converge to bipedal limit cycles. To speed up this convergence, we chose
initial conditions that were asymmetric about the axis of symmetry. For instance, when
solving the equations of motion for Config. 4, the following initial conditions were applied:
[xf , yf ] |t=0 = [0, 0]
[xn, yn] |t=0 = [0,−N ]
[xl, yl] |t=0 = [−L,−(N + 1)]
[xr, yr] |t=0 = [+L,−(N − 1)]
[xl, yl] |t=0 = [xr, yr] |t=0
(4.14)
In the special case where fully symmetric initial conditions are applied (i.e., xl|t=0 =
−xr|t=0 and xl|t=0 = −xr|t=0), cell length modulation that would otherwise be bipedal is
in-phase across the trailing edge elements. Such an in-phase limit cycle is unstable in that
any small driving force applied to either trailing edge element causes the system to shift
106
from the in-phase limit cycle to the bipedal limit cycle. These unstable states are ruled out
and not discussed further.
5.4. Comparing 2-D Elastic Configurations. There are several ways to elastically
couple the elements that make up the two dimensional model. Four viable configurations
that capture the prevalent shape dynamics of fish keratocytes are illustrated in the top
row of Fig. 4.8. In the previous work [135], it was shown that elastic coupling Config. 3
generates realistic bipedal dynamics over a range of spring elasticities, KW , and inertia-like
parameter, g. We investigate the dynamics of three alternative elastic coupling configura-
tions to determine if they reproduce realistic bipedal locomotion of crawling cells and lateral
displacement of the cell nucleus (Fig. 4.7 d).
5.4.1. Configuration 1. The simplest possible configuration that can produce bipedal
locomotion consists of two elastically coupled trailing edge elements both coupled to the
leading edge (Config. 1). The trailing edge elements, ~xl and ~xr, undergo bipedal dynamics,
while the front element, ~xf , pulls the cell forward at a constant velocity. A representative
dynamical response of this configuration is shown in the first column of Fig. 4.8. The
dynamics of cell length, ∆y, follow a bipedal limit cycle with a modulation amplitude of
1.9µm and a period of 35.3 s.
There are, however, three notable deviations between the simulation dynamics of Con-
fig. 1 and the actual bipedal dynamics of keratocytes. First, an aberrant secondary slip
occurs during the sticking phase on one side of the trailing edge coincident with slipping
of the opposite side. As the slipping side element, say ~xl, is propelled forward, increased
spring force through the KW spring causes momentary slipping of ~xr, which would other-
wise remain in the sticking phase. This secondary slip is a result of direct coupling between
the two trailing edge elements. More precise measurements of the dynamics of keratocyte
shape are necessary to determine whether these aberrations occur in real cell crawling.
107
Second, stable bipedal limit cycles occur over a range of physiologically relevant param-
eters only when the ratio of spring equilibrium lengths, W/D, is near 2 or greater. For
W/D < 1.6, the range of parameters corresponding to stable bipedal limit cycles is signif-
icantly reduced if existing at all. In Fig. 4.8, the corresponding ratio is W/D = 1.6. The
average cell length in this simulation is approximately 10µm, less than that of a typical
keratocyte, and cannot be increased without widening the model cell as required to main-
tain the ratio of W/D ≥ 1.6. Alternatively, to generate bipedal locomotion for a cell of this
length without increasing cell width, one can increase the drag force coefficient of sticking,
β, or decrease spring constant, KD. Although these variations allow for a more realistic
overall cell aspect ratio, the amplitude of cell length modulation becomes unrealistically
large. After translating W and D into cell width and length, this configuration generally
does not reproduce the dynamics of a bipedally crawling cell unless the cell width to length
ratio is at least 1.5. We measure width to length ratios of bipedally crawling cells to range
from 1 to 3 (Fig. 4.6). Thus, while it is possible to simulate bipedal locomotion of wider fish
keratocytes, there are many aspect ratios that are not appropriately modelled by Config.
1.
Finally, this spring configuration does not recapitulate the smooth periodic nucleus
lateral displacement observed. Unlike the other configurations, Config. 1 has no centrally
coupled element representative of the cell nucleus. At best, one could interpret the nucleus
centroid to be the location of a fictitious spot marked at the middle of the posterior spring.
Mathematically, this is ~xn = (~xl+~xr)/2. The lateral motion of this spot is plotted as lateral
displacement for Config. 1 in Fig. 4.8. In this interpretation, the nucleus jumps laterally
from one side to the other every time one of the trailing edge elements slips. Without
unrealistically overdamping the system, smooth sinusoidal-like or exponential relaxation-
like nucleus lateral displacement cannot be generated.
108
xn
xxl r
xfxf
xl xr l r
xn
x xl r
xf
la
tera
l d
isp
lace
me
nt
(μm
) ce
ll le
ng
th, ∆
y (
μm
)
∆y
Con!guration 1 Con!guration 2 Con!guration 3 Con!guration 4
l r
xn
xrlx
xf
0.01 0.02 0.03 0.040
0.25
0.5
0.75
1
frequency (Hz)
0.01 0.02 0.03 0.040
0.25
0.5
0.75
1
frequency (Hz)
0.01 0.02 0.03 0.040
0.25
0.5
0.75
1
frequency (Hz)
0 20 40 60 80 1008
10
12
14
16
18
time (seconds)
0 20 40 60 80 1008
10
12
14
16
18
time (seconds)
Fo
uri
er
am
plit
ud
e
0 20 40 60 80 1008
10
12
14
16
18
time (seconds)
0.01 0.02 0.03 0.040
0.25
0.5
0.75
1
frequency (Hz)
0 20 40 60 80 1008
10
12
14
16
18
time (seconds)
-1.5-1
-0.50
0.51
1.5
time (seconds)
100806040200-1.5
-1-0.5
00.5
11.5
time (seconds)
100806040200-1.5
-1-0.5
00.5
11.5
time (seconds)
100806040200-1.5
-1-0.5
00.5
11.5
time (seconds)
100806040200
left ∆yl
∆yrright
g α β γ KN KD KL KW N D L W
Config. (nN s2
µm) (nN s
µm) (nN s
µm) (nN s
µm) ( nN
µm) ( nN
µm) ( nN
µm) ( nN
µm) (µm) (µm) (µm) (µm)
1 0.1 0.15 12 n/a 0 1 0 1 n/a 15 n/a 242 0.1 0.5 50 5 1 0.9 1 1 8 13 12 203 0.1 0.5 25 5 1 0 1 10 3 n/a 10 204 0.3 0.5 50 5 1 0.7 1 0 8 13 10 n/a
Figure 4.8. Representative dynamical responses of four select config-urations of the 2-D cell crawling model. Config. 1 is the simplest arrange-ment of springs necessary for bipedal locomotion. This configuration lacks thefourth element necessary to represent smooth oscillations of the nucleus in thelateral direction. Configuration 2 is the most general spring arrangement whena central element is added to the model. Configurations 3 and 4 are specialcases of Config. 2, where particular springs are removed by setting their corre-sponding spring constants to zero. Oscillations in cell length, ∆y (see secondrow), are shown for both the left side (green) and right side (blue) trailing edgeelements. Fourier transforms (third row) correspond to cell length oscillations.Lateral displacement (fourth row) corresponds to the average position of ~xl and~xr in the case of Config. 1, and the central element in the cases of Configs. 2-4.
5.4.2. Configuration 2. To address the discrepancies between actual keratocyte dynam-
ics and the dynamics of Config. 1, a central element (location ~xn) is introduced to the 2-D
109
crawling cell model. The central element allows for alternative coupling pathways between
the two trailing edge elements, rather than the direct coupling through spring KD that
leads to aberrant secondary slips in the cell length dynamics. Config. 2 is the case where all
elements are elastically interconnected. Configurations 3 and 4, discussed below, are special
cases of Config. 2 each corresponding to the removal of particular springs. We assume sym-
metry about the axis of forward motion and require the cell to maintain a reasonable shape
with width and length comparable to our benchmarks. This shape requirement is achieved
by the appropriate assignment of equilibrium spring lengths and elasticities. Configura-
tions 2-4, illustrated in Fig. 4.8, are the only three spring configurations that satisfy these
constraints. Unsuitable Configs. include those that do not have the necessary springs to
maintain cell width {KD = KW = 0, KL = KW = 0, or KN = KW = 0}. For each Config.
in this set, the nuclear and trailing edge elements move from their initial locations to the
axis of forward motion indicating an unrealistic collapse of cell shape into one dimension.
Config. 2 generates bipedal locomotion over many regions of parameter space. For ex-
ample, choosing parameters similar to those of Configs. 1, 3 or 4 results in a comparable
dynamical response despite the additional elastic coupling. In Fig. 4.8, Config. 2 parameters
are chosen similar to those listed for Config. 4 (discussed below) and the result is realistic
bipedal dynamics with cell length modulation and nucleus lateral displacement amplitudes
of 2.9 and 0.53µm, respectively, and a period of 40.0 s. Unlike cell length limit cycle shown
for Config. 1, here there is no secondary slip. Due to the maximal number of springs in
this configuration, however, there are many cases of aberrated bipedal limit cycles associ-
ated with Config. 2 despite this representative result in Fig. 4.8 not being one such case.
Compared to configurations with fewer springs, the dynamical behaviours produced with
Config. 2 are more sensitive to parameter variation, especially spring length variations. In
110
the KD → 0 and KW → 0 limits, we found that Config. 2 robustly generates bipedal lo-
comotion without secondary aberrations. With this insight, we analyze the KD = 0 and
KW = 0 cases in more detail, which correspond to Configs. 3 and 4, respectively.
5.4.3. Configuration 3. Config. 3 is the spring arrangement considered by Barnhart et
al. [135]. They found that stable bipedal locomotion occurs for choices of spring elasticity
KW > 3 nN/µm (0.01 < g < 10 nN s2/µm), consistent with benchmark values. Not dis-
cussed in the previous work, one finds that the central element, ~xn, smoothly oscillates in
the lateral direction entrained to bipedal limit cycle. The amplitude of these oscillations
is 0.55µm, comparable to experimental observation (Fig. 4.7). This amplitude can be ad-
justed by varying the nuclear drag coefficient, γ, but due to the interplay of spring and
nuclear drag forces, cannot be increased much greater than 0.6µm.
An interesting transition in dynamics occurs when the nuclear drag coefficient, γ, is
increased to make the nucleus insensitive to the spring forces through the KL springs. One
gets a crude estimate of this critical value, γc, by using the dynamics shown in Config. 3 to
determine the drag force necessary to counterbalance these spring forces. Here, two springs
of elasticity KL ∼ 1 nN/µm apply forces to the central element, which in turn laterally
traverses ∼ 1µm every half period ∼ 20 s, of the bipedal limit cycle. Combining these
numbers gives γc ≈ 10 nN s/µm. When the nucleus is immobilized with respect to forces
from the KL springs, the system should behave like Config. 1 in which case element ~xn
assumes the role of the leading edge element, ~xf . As predicted, we find that the dynamical
responses of Config. 3 approach those of Config. 1 when γ > 7 nN s/µm. In essence, Config. 3
can be considered an extension of Config. 1 in which an extra spring at the front is added
to extend the cell length. Whereas cell shape dynamics produced by Config. 1 were not
robustly bipedal when W/D < 1.6, the front spring length of Config. 3 can be specifically
tuned to account for proper cell length.
111
5.4.4. Configuration 4. Although Config. 3 produces realistic bipedal dynamics, we pro-
pose Config. 4 because the springs are easier to interpret with respect to the framework of
the actin cytoskeleton. As discussed in the main article, we determined that this spring con-
figuration generates realistic crawling dynamics that are sensitive to changes in KD, but not
KL (Fig. 4.4 b). Sensitivity with respect to drag force parameters β, γ, and g is investigated
here. Drag force on either trailing edge element is pointed in the direction opposing motion,
which at any given time, primarily opposes spring forces from the KD springs. Therefore,
tuning the sticking coefficient β is similar to tuning the spring elasticity KD, in terms of
changes to the dynamical behaviour of Config. 4. These dynamical behaviours are plotted
in Fig. 4.9 a. As stick drag force is scaled up by increasing β, the trailing edge elements
stick longer giving rise to monotonically increasing amplitudes of cell length modulation,
nucleus lateral displacement (NLD), and period. Fig. 4.9 b shows how increasing nucleus
drag coefficient, γ, decreases nucleus lateral displacement. Finally, Fig. 4.9 c shows that
the dynamics are relatively insensitive to variation of parameter g, which sets the timescale
between sticking and slipping.
Movie Legends (Movies Online)
Movie S1. Live tracking of the shape dynamics of a coherent non-bipedalkeratocyte. This movie is representative of our analysis of previously published moviesof fish keratocytes. (a) Overlaid on raw footage of the crawling cell are the perceivedcoordinates of the left and right side trailing edge elements, the nucleus element, and theleading edge element. The line approximates the axis of symmetry. The scale bar is 20µm.Cell aspect ratio, width, cell length oscillations, and cell speed are measured based on thepositions and movements of these elements. (b) Plot of the left and right side cell lengthdynamics. Left cell length is greater than right cell length because the cell is turning to theright with respect to its direction of motion. Despite noisy data, small stick slip events aredetected in this plot including one caused by collision with a fragment at 285 s.
Movie S2. Two dimensional simulation movie representative of a fish kerato-cyte undergoing bipedal locomotion. (a) Animation showing the four element model(Config. 4). Dashed lines indicate springs. (b) Cell length limit cycles for the trailing left(green) and right (blue) elements corresponding to (a). Cell length, ∆y, is defined as thevertical distance between the front element and either one of the trailing edge elements.This simulation corresponds to the timelapse and dynamics shown in Fig. 4.8, Config. 4,and Fig. 4.3. Parameter values are listed in Fig. 4.8.
112
0 0.2 0.4 0.6 0.80
1
2
3
4
0
15
30
45
60
75a
mp
litu
de
(µ
m)
g (nN s2/µm)
NLD
cell length
period
20 40 60 80 1000123456
0
15
30
45
60
75
am
plit
ud
e (µ
m)
β (nN s/µm)
NLD
cell length
period
0 4 8 12 16 200
1
2
3
4
0
15
30
45
60
75
γ (nN s/µm)
pe
riod
(s)
NLD
cell length
period
(a)
(c)
(b)
realistic bipedal,
γ = 5 nN s/µm
amplitude of celllength oscillations, both ∆yl(t) and ∆yr(t)
amplitude of nucleus lateral displacement(NLD), ∆xn(t)
period of oscilations
vf
∆xn(t)
∆yl(t) ∆yr(t)KD
KLKL
KD
α,β,gα,β,g
γ
Figure 4.9. Drag force sensitivity analysis of Config. 4. Amplitudes ofcell length and nucleus lateral displacement (NLD) oscillations (blue curves),as well as the limit cycle period (green curve), are plotted with respect to dragparameters β (a), γ (b), and g (c). Cell length oscillation amplitude and periodmonotonically increase with the increase of all three drag parameters, but aremost sensitive to the stick drag coefficient, β. Dynamics are insensitive to freeparameter g, which sets the time of switching between sticking and slipping. Pa-rameters not varied in each chart are kept fixed at the values listed for Config. 4in Fig. 4.8.
113
CHAPTER 5
Immune System Modifier β-Glucan Regulates Motility
Through the Actin Cytoskeleton
The contents of this chapter are intended to be part of a future manuscript.
114
Abstract and Forward. Neutrophil motility is an integral component of the innate
immune system. As they migrate, neutrophils exert cytoskeletal driven contractile forces
on their environment through integrin borne adhesions, but the relationship between these
mechanical outputs and immune system potentiating biological response modifiers has yet
to be determined. We use live cell imaging and traction force microscopy to study the
relationship between neutrophil motility, force exertion, and the biological response modifier
β-glucan, a substance that has shown promise as a clinic grade therapeutic. Subsequently,
we find that β-glucan modifies the mechanical output of neutrophils on soft, 5 and 25 kPa
2D substrates, decreasing both the contractile force and energy applied as the concentration
of β-glucan is increased over a clinically relevant range. Despite these changes in mechanical
output, contractile energy is conserved across substrate stiffness for all concentrations of
β-glucan tested. We also report an overall inverse correlation between migration speed
and mechanical output. While there has been some speculation that a specific force range
exists for facilitating optimal cell migration, we conclude that optimal cell migration is not
associated with a specific magnitude of contractile force in the 1-100 nN range. Instead,
increased motility is associated with diminished contraction.
1. Introduction
1.1. Neutrophils. Neutrophils are the first responders to injury and infection and
are integral components of the innate immune system [8]. To reach sites of injury and
infection, neutrophils must transmigrate through the endothelial barrier of the blood vessel
and migrate within the extracellular matrix (ECM). During migration, neutrophils interact
mechanically with their surrounding environment through integrin borne adhesions [164].
Forces exerted by neutrophils on their substrate are a fundamental component of the sub-
strate stiffness dependent regulation of migration [48, 50, 165], which ultimately affects
wound healing [84] and other host defense functions [166–168].
115
1.2. Traction Forces. The net force required to propel a neutrophil in typical cell
media one cell length per minute without friction is on the order of 10−6 to 10−5 nN.1 Yet
the forces that many motile neutrophils exert on their substrate in vitro in 2D and 3D
range from 1 to 100 nN in magnitude [48, 50, 165], up to 8 orders of magnitude larger than
the frictionless propulsion force. There are two main reasons for such large forces on the
substrate. First, cells apply force to their environment as a means of probing its rigidity
[29], which in turn regulates many cell characteristics from morphology to migration speed
[48, 49]. This detection of the extracellular rigidity is termed mechanosensing. Second,
as adhesion complexes form, a significant amount of friction must be overcome to sustain
motility (c.f. Fig. 4.1). Several models of cell motility would indicate that an intermediate
range of contractile force exertion is optimal for facilitating cell migration. Relatively large
forces are thought to indicate motility-impairing levels of adhesion, whereas relatively small
forces are thought to indicate an inability to form mature adhesion complexes [41, 42, 169].
1.3. Force Motility Relationships. Bangasser et al. have recently compiled the
migration speeds and contractile forces of several motile cell types crawling on soft substrates
[42]. Their analysis suggests that many cell types exhibit a biphasic speed-stiffness and
force-stiffness dependence. They adapt a motor-clutch model [170] to fit the latter relation.
Missing is a detailed analysis for the force/energy-speed dependence. With neutrophils
migrating on fibronectin coated polyacrylamide gels with stiffnesses between 3 and 13 kPa,
Stroka et al. have shown that migration speed peaks on stiffnesses near 7 kPa, depending
on fibronectin density [41]. The peak root mean squared (RMS) speed asymptote was
roughly vrms(∞) = 0.05-0.1µm/s (see Materials and Methods for definition). In similar
experiments, Oakes et al. and O’Brien et al. have shown that the neutrophil RMS speed
asymptote is vrms(∞) = 0.15-0.2µm/s on gels of 10 kPa stiffness, and drops monotonically
1The net force required to propel a neutrophil without friction is estimated here using Stokes’ law,F = 6πµRv. Plugging in order of magnitude estimates corresponding to neutrophil motility in atypical cell media, µ 10−3 kg/m/s, R 10−5m, v 3 ·10−7 m/s, gives F ≈ 10−6 nN, more than six ordersof magnitude smaller than the forces reliably measured with traction force microscopy.
116
as stiffness increases from 5 to 100 kPa [1, 48]. Furthermore, the average traction stress
of neutrophils was shown to increase with substrate stiffness over the 5 to 50 kPa range.
Taken together with other neutrophil speed and traction measurements from [49] and [50],
it is possible to reconstruct a rudimentary force-speed distribution. But, since only the
ensemble averages are available, much of the force-speed distribution is missing, and there
is no way to determine the energy-speed distribution. These distributions would be highly
useful for modeling cell motility. In this manuscript, we measure the force-speed and energy-
speed distribution explicitly for neutrophils migrating on compliant substrates with Young’s
moduli of 5 and 25 kPa.
1.4. β-Glucan. To further examine the connection between cell mechanics and speed
in a context relevant to the immune system, we investigate the effect of the pathogen
associated molecular pattern (PAMP), β-glucan (BG) [76]. BG is a complex polysaccharide
ubiquitous in the cell walls of bacteria and fungi such as Saccharomyces cerevisiae [171, 172].
Several forms of BG are also the main element of fiber in grains such as oats and barley [173].
BG from yeast can be purified into a clinical grade, water soluble form called PGG-glucan
[69], which has been shown to be a biological response modifier in vivo [174], improving
wound healing in mice [70], showing promise as an anti-cancer agent [71–73], and enhancing
chemotactic honing of neutrophils on glass [75, 76]. Several clinical trials have shown that
0.5-2.25 mg/kg intravenous dosages of PGG-glucan to humans with colorectal cancer or post
surgical infections significantly improved recovery, highlighting its potential as a therapeutic
drug for humans [74].
While BG has potential to be a positive pharmacological intervention that improves the
function of the innate immune system, the role of cell mechanics as part of this positive
intervention and immunological response has yet to be studied. BG is known to bind and
allosterically regulate the affinity of Complement Receptor 3 (CR3, also known as αMβ2,
Mac-1, and CD11b/CD18) [77, 79], an integrin expressed on the surface of neutrophils that
117
plays a significant role in mechanosensitive neutrophil behavior [1, 48]. Little is known about
the link between biological response modifiers and the cell mechanics of immune cells, but it
is rational to propose that BG (and possibly other biological response modifiers) regulates
their mechanical output. This hypothesis is tested below using traction force measurements
of motile neutrophils treated with soluble BG.
2. Materials and Methods
2.1. Reagents. 40% acrylamide solution, 2% bis-acrylamide solution, N,N,N’,N’ -
tetramethylethylenediamine (TEMED) and ammonium persulfate (APS) were purchased
from Bio-Rad Laboratories (Mississauga, ONT, USA). Dulbecco’s PBS, Lebovitz’s L-15
medium, Hanks Balance Salt Solution without Ca2+/Mg2+ (HBSS−/−), and 0.5µm Fluo-
spheres carboxylate-modified microspheres were purchased from Invitrogen (Carlsbad, CA,
USA). Polymyxin B and sulfo-SANPAH were purchased from Pierce Biotechnology. Human
fibronectin isolated from plasma (>95% purity), histopaque 1077, dextran for neutrophil
prep (average molecular weight, 400-500 kDa), dextran for osmolarity control experiments
(Mw 5000), formyl-methionyl-leucyl-phenylalanine (fMLP), and sterile dimethyl sulfoxide
(DMSO) were purchased from Sigma-Aldrich. Pharmaceutical-grade purified, endotoxin
free, soluble yeast β-glucan (Imprime PGG) in citrate buffer was generously provided by
Biothera (Eagan, MN, USA). β-glucan is referred to as BG throughout this work.
2.2. Substrate Preparation. Polyacrylamide gel (PAG) substrates with Young’s
moduli of 5 and 25 kPa were prepared as described previously [48]. Acrylamide and bis-
acrylamide solutions (Bio-Rad) were mixed with 0.5µm fluorescent spheres (Invitrogen)
and polymerized using TEMED and APS (Bio-Rad). Gels were cast on silanized Bioptechs
Delta-T glass dishes inside air-tight molds made from gene frames and plastic coverslips.
Immediately after catalyzing PAG polymerization and sealing the air-tight mold (before
polymerization), gel mixtures were centrifuged at 1000 g at 4◦C for 25 minutes to sediment
118
fluorescent spheres along a subapical layer at the top surface of the PAG (Fig. 5.1 a). After
centrifugation, the gels were allowed to polymerize at room temperature. The final size
of the gels was ∼ 1 cm× 1 cm× 300µm. At ∼ 300µm thick, the underlying glass has no
bearing on the stiffness detected by cells plated on top [175], nor do applied traction forces
in the cellular force range of 1 to 100 nN penetrate through the gel down to the glass [37].
Gel stiffness measurements and substrate functionalization with fibronectin (Fig. 5.1 a) was
done exactly as described in Ref. [1] .
2.3. Neutrophil Preparation. Under the approval and guidelines of the Rhode Is-
land Hospital Institutional Review Board, neutrophils were isolated from healthy human
volunteers by collection into EDTA-containing Vacutainer tubes (BD Biosciences, San Jose,
CA, USA). Histopaque 1077 (Ficoll Histopaque) was used for the initial cell separation,
followed by gravity sedimentation through 3% dextran (average molecular weight, 400-
500 kDa). Contaminating erythrocytes were removed by a single hypotonic lysis, yielding
a neutrophil purity of >95%. Neutrophils were suspended in HBSS−/− on ice until use in
the experiments. All reagents contained <0.1 pg/ml endotoxin, as determined by Limulus
amoebocyte lysate screening (BioWhittaker, Walkersville, MD, USA). When indicated, neu-
trophils were pretreated on ice for 30 min with 7µL (∼ 1 mg/mL) of monoclonal anti-CR3
antibody, clone 44abc, isolated from hybridoma cells (American Type Culture Collection,
Manassas, VA, USA) per 106 cells (total volume = 500µL).
For traction-migration assays, approximately 106 neutrophils were resuspended at 37◦C
in 2 mL L-15 with 2 mg/mL glucose added. When indicated, this media was modified
to contain 10µg/mL β-glucan (BG10), 50µg/mL β-glucan (BG50), or 50µg/mL dextran
(Dex50) in citrate buffer. These doses roughly correspond to the BG weight to blood volume
ratios of humans intravenously dosed with BG during recent phase III clinical trials [74].
Neutrophils were allowed to settle for at least 5 min before introducing fMLP (10 nM final
concentration) to activate and induce migration.
119
2.4. Microscopy. A Nikon TE-2000U inverted microscope (Nikon, Melville, NY,
USA) coupled to an iXonEM + 897E back-illuminated Electron Multiplying CCD cam-
era (Andor, Belfast, UK), outfitted with a Bioptechs (Butler, PA, USA) stage heater, a
Nikon 40× CFI S Plan Fluor ELWD objective, and a FITC dichroic optical cube (Nikon)
was used for all live cell imaging experiments. DIC and fluorescent bead field images were
captured over 80-100 min in 10 s intervals using the Elements program (Nikon). To minimize
photo damage to the cell, fluorescent images were captured with long exposures (> 1 s) at
low illumination intensity rather than short exposures at high illumination intensity. All
data were analyzed using ImageJ (U.S. National Institutes of Health, Bethesda, MD, USA),
and MATLAB (MathWorks, Natick, MA, USA) computational software.
2.5. Cell Tracking and Migration Analysis. Perceived cell boundaries were traced
from DIC images using custom MATLAB software and a stylus and touch screen (Fig. 5.1 b,
white boundaries). Sampling intervals after manual tracing were either 20, 40, 80, or 160 s
(usually the latter). We found this manual tracing method more accurate compared to
using automatic tracking software. From these cell boundary measurements, the regionprops
algorithm in MATLAB was used to calculate cell centroid ~r(t), spread area, and aspect ratio.
The turning angles between successive displacements (e.g. the angle between ~r(t+∆t)−~r(t)
and ~r(t)−~r(t−∆t), where ∆t is the sampling time interval) were also tracked. Persistence
fractions were calculated as the fraction of all turning angles between ±30◦ calculated with
a sampling interval of 160 s.
Mean squared displacement was defined in the usual way [1, 48, 95],⟨r2c (τ)
⟩=⟨
|~rc(t+ τ)− ~rc(t)|2⟩c, where the overline indicates a time average over t, τ is the time
interval over which displacements are measured, c = 1, 2, 3, ... is the index of the migration
trajectory of a specific cell, and 〈...〉c indicates an ensemble average of all cell trajectories,
c. To minimize nonergodic contributions to MSD and increase the comparability of this
statistic, all trajectories were truncated at t = 1200 s (20 min) before calculation.
120
2.6. Traction Force Microscopy. Images of bead fields were compared to a reference
image before cells adhered to the substrate. Corresponding displacement field ~u(~x, t), and
traction stresses ~T (~x, t), were calculated in 2D space (~x) and time (t) using digital area
correlation and Fourier-transform traction cytometry [33] as described previously [48].
0.5µmpolyacrylamide gel
10−25µmintegrin�bronectinactin cytoskeleton�uorescent sphere
a
b
c
5kPa
BG0 BG50
25kPa
BG0 BG50
10µm
100nN
10µm
stress islands
20nN
traction stress (10-3nN/µm2) traction stress (10-3nN/µm2)0 375 750 1125 15000 125 250 375 500
Figure 5.1. Traction force microscopy diagram and representative DICand traction field frames of cells on soft and stiff (5 kPa and 25 kPa)substrates with β-glucan concentrations of 0 (BG0) and 50µg/mL(BG50). (a) A neutrophil on the surface of a 2D polyacrylamide gel substratecoated with fibronectin (triangles). Contractile forces on the substrate are facil-itated by integrins (rectangles) that couple the substrate to the retrograde flowof the actin cytoskeleton (inward cyan arrows). (b) Representative images ofneutrophils and (c) corresponding traction stress maps show examples of stressislands (outlined in black) and their corresponding forces (black arrows). Colorindicates the absolute value of traction stress, whereas small white arrows showtraction stress direction. Traction stresses were larger on the 25 kPa substrates(note the change in color bar scale).
2.7. Force and Energy Measurements. To minimize the measurement of noise,
traction stresses below 0.1 and 0.3 nN/µm2 noise thresholds were ignored on the soft and
121
stiff gels (5 and 25 kPa), respectively. Stress islands were defined either as distinct closed
regions of traction stress above the noise threshold, or partitioned subregions from one closed
region above the noise threshold. Examples of both cases can be seen in Fig. 5.1 c. The
5 kPa BG50 shows three distinct stress islands whereas the other panes show one traction
region partitioned into either two or three stress islands. Where appropriate, partitions
were made along lines of local traction stress minima.
At times, stress islands encompassed areas outside the perceived cell boundary. Nonva-
nishing stress measurements outside the perceived boundary occur for three reasons: First,
the perceived cell boundary differs from the actual cell boundary as visualized in Fig. 5.7
with a plasma membrane stain. Second, the displacements are Gaussian filtered, which
slightly blurs the stress islands. While blurring slightly extends the area of each stress
island, the integrated force is not affected. Third, we determine the traction stress field by
solving the inverse Bousinesq equation, which assumes point-like traction sources [33]. It
has been shown that the assumption of finite sized traction sources on the order of the area
of a small integrin cluster leads to stress regions that are more contained within the per-
ceived cell boundary [35], but this analytical method was not applied here. Taken together,
we take a justified as-is approach to analyze the stress field.
Once stress islands were identified, the force from the nth island was calculated as
~Fn(t) =∫n~T (~x, t)d~x, where ~T is the 2D traction stress field, ~x is the position in 2D, and
t is time. The average stress island magnitude, denoted as contractile force throughout
this work, was calculated as⟨|~Fn(t)|
⟩n, where 〈...〉n is an average over each stress island.
Contractility time was defined as the time that contractile forces were above thresholds
of 8 nN on a 5 kPa substrate and 25 nN on a 25 kPa substrate. The energy required to
stress the substrate (denoted contractile energy throughout this work) was calculated as
E =∑n
1
2
∫n
~T (~x) · ~u(~x)d~x ≈∑n
∆z
2Y
∫n
∣∣∣~T (~x)∣∣∣ d~x, where ~u is the 2D displacement field.
122
2.8. Dipole Moment Analysis. Similar to the procedure outlined in [176], dipole
moments were calculated from the stress dipole tensor, Mij =∫S xiTjdS, where S is a
80µm× 80µm bounding box centered at the cell centroid that captures the relevant stresses,
and xi is the ith element of the position vector (i = 1, 2), and Tj is the jth element of the
traction stress vector (j = 1, 2). The diagonal terms M11 and M22 correspond to linear
extensile or contractile stress in the x1 and x2 directions, respectively. The off-diagonal
terms correspond to off-axis stress. In the absence of net torque within the bounding
box, the dipole tensor is symmetric and diagonalizable with orthogonal eiganvectors. The
principle dipole moments, denoted ~µM and ~µm for the major and minor dipole moments,
respectively, were calculated by diagonalizing the dipole tensor, Mij , and multiplying each
unit eigenvector by their corresponding eigenvalue. The major dipole moment was the
vector with the larger magnitude.
2.9. Statistics. Significance has been determined by assessing the overlap of standard
error measurements. All shaded regions on time plots represent ± one standard error of
the mean drawn about the mean. All error bars represent standard error of the mean.
Pearson product-moment correlation coefficients (PCCs) between two paired variables, for
example, A1 and A2, were calculated as ρ = 〈A1A2〉−〈A1〉〈A2〉σA1
σA2, where σ2
Ai= 〈A2
i 〉−〈Ai〉2 is the
variance (i = 1, 2). PCCs characterize the linear correlation between two paired variables,
but not the nonlinear correlation. To measure the nonlinear correlation, alternating condi-
tional expectation (ACE) [177] calculations were made using an ACE algorithm described
in Ref. [178].
3. Results
3.1. Spread Area and Motility Biphasically Dependent on BG Concentra-
tion. To tease out the force-speed and energy-speed relationships, we tracked the spread ar-
eas, migration trajectories, and traction stresses of fMLP induced chemokinetic neutrophils
123
(example shown in Movie S1). Ensemble averages of the MSD, spread area, contractile
energy, and contractile force are plotted in Fig. 5.2. Zero time, t = 0, refers to the moment
fMLP was added, approximately five minutes after neutrophils were plated. Notationally,
BG0, BG10, and BG50 refer to soluble BG concentrations of 0, 10, and 50µg/mL, respec-
tively. BG is shown to alter spread area and MSD in a biphasic manner. Specifically, MSD
and spread area were smallest with BG10 cells compared to the other BG concentrations,
irrespective of substrate stiffness (Fig. 5.2, a and b). Interestingly, while BG10 cells showed
decreased cell speed with increasing substrate stiffness, this trend reversed with the BG50
cells. Also interesting, substrate stiffness did not affect cell migration speed under the BG0
condition. In Ref. [48], speed was affected by substrate stiffness but the difference they
found may be attributed to not accounting for nonergodicity in their MSD calculations.
Finally, there was a small relative increase in spread area as substrate stiffness increased,
irrespective of BG concentration.
3.2. Mechanotactic Conservation of Energy and Contractility Time. Sum-
ming up the total energy required to strain the substrate at a given point in time, the strain
energy (also denoted contractile energy) applied by neutrophils significantly decreased with
increasing concentrations of BG (Fig. 5.2 c). Contractile force, measured by integrating
traction stress over discrete stress islands and averaging the magnitudes of those forces (see
Materials and Methods), also decreased with increasing concentrations of BG. Osmolar-
ity control experiments using 50µg/mL dextran showed that decreases in the mechanical
output of neutrophils was not attributable to the added osmolarity of adding BG (Suppla-
mentary Material, Fig. 5.8). Futhermore, measurements of adhesion rates, percent active,
and percent of active cells contractile, showed no change between the BG0 and BG50 con-
centrations (Supplamentary Material, Fig. 5.9). In context, the mean contractile force of
BG0 neutrophils on 5 kPa substrates varied from 20 to 30 nN, values remarkably similar
to the reported mean RMS force of 28 ± 10 nN associated with neutrophils migrating on
124
25 kPa5 kPa
a
c
b
d
e f
50
100
150
200
250
300
350
spre
ad
are
a (µ
m2)
time, t (min) time, t (min) frequency
time interval, τ (s) time interval, τ (s) frequency
0 10 20 300
20
40
60
80
100
120
con
tra
ctile
fo
rce
(n
N)
0 10 20 30
MS
D (µ
m2)
102 103
102
103
102 103
0
2
4
6
8
10
12
con
tra
ctile
en
erg
y (
nNµ
m)
0 5 10
0
50
100
150
0 5 10 150
10
20
30
0 10 20
200
400
600
500
300
100
0 20 40
0
2000
4000
6000BG0
BG10
BG50
5 kPa 25 kPa0
10
20
30
40
50
con
tra
ctili
ty t
ime
(m
in)
0
10
20
30
40
5 kPa 25 kPa
pe
rsis
ten
ce f
ract
ion
(%
)
Figure5.2. Neu-trophil motil-ity and me-chanical out-put depend onBG concen-tration andsubstrate stiff-ness. (captionon pg. 126)
125
9 kPa PAG substrates coted with a mixture of E-selectin and ICAM-1 [165]. Contractile
forces on the 25 kPa substrates were less than five times larger than their 5 kPa counter-
parts (Fig. 5.2 d), indicating that neither the magnitudes of the substrate displacements,
nor the corresponding magnitudes of the contractile forces, were conserved across substrate
stiffness. This result is consistent with the force-stiffness observations of several other cell
types [42, 51] including Madin-Darby canine kidney cells [47], mouse embryonic fibroblasts,
and NIH3T3 fibroblasts [109]. Other than decreasing the absolute magnitudes of contrac-
tile forces, BG did nothing to alter the relative force-stiffness dependence as there was a
three-fold relative increase in contractile force magnitude going from 5 to 25 kPa substrates,
irrespective of BG concentration.
Most interestingly, the contractile energy was conserved across substrate stiffness, and
this conservation held for all concentrations of BG. The conservation of contractile energy
across substrate stiffness was first demonstrated with NIH 3T3 fibroblasts very recently by
Oakes et al. [179]. This conservation has never been demonstrated with neutrophils or with
a biological response modifier.
(Fig. 5.2 caption continued) For all time series plots, the shaded regions correspond to the meanplus and minus the standard error of the mean. The notation BG0, BG10, and BG50 refers to 0,10 and 50µg/mL concentrations of soluble BG. (a) Mean Squared Displacement (MSD) plots inlog-log coordinates. To minimize nonergodic contributions to MSD and ensure the contributions ofeach time average were unbiased, all migration trajectories were truncated at 20 min for the purposeof calculating MSD. All log-log MSD slopes range from 1 to values less than 1.5 indicating slightlysuper diffusive chemokinetic motion at asymptotic time intervals. Cell speed changes biphasicallyas a function of BG concentration on both soft and stiff substrates, with stiffness slightly tuningthe biphasic response. (b) Spread area plotted against time. As with MSD, the spread area alsobiphasically responds to BG concentration. (c) Strain energy, the potential energy loaded into thesubstrate by the cell at a particular point in time. Strain energy is conserved across substratestiffnesses, regardless of BG concentration. Increasing BG leads to decreased strain energy and(d) contractile force. In each of (a-d), histograms to the right correspond to each distribution ofdata at 560 s on the 25 kPa substrate. (e) Contractility time is the time contractile force remainsabove 8 nN on the 5 kPa substrates, and above 42 nN on the 25 kPa substrates. No significantchange in contractility time was observed between the BG0 and BG10 cells, but contractility timesignificantly decreased with 50µg/mL BG concentrations compared to either BG0 or BG10 cells,on both substrate stiffnesses. (f) Persistence fraction is the fraction of turning angles between±30◦. Like MSD and spread area, persistence time trends biphasically with BG concentration.n = 19, 21, 20, 18, 29, 23 for the 5 kPa BG0, BG10 and BG50, and the 25 kPa BG0, BG10, andBG50 conditions, respectively.
126
Contractility time is defined to be the amount of time a cell contracts its substrate (see
Materials and Methods). No significant change in contractility time was observed between
the BG0 and BG10 cells, but contractility time significantly decreased going from either
BG0 or BG10, to BG50 cells (Fig. 5.2 e). Contractility time, like contractile energy, was
conserved across substrate stiffness, irrespective of BG concentration.
The ideal measure of the persistence of chemokinetic trajectories in 2D is persistence
time [56, 60, 62], the average time a cell crawls before changing direction. However, the
persistence times of our trajectories were not resolvable with confidence because they were
much smaller than the sampling time. To get a measure of the relative persistence, we
instead calculate turning angle distributions. Turning angles in our context were defined
as the angles between successive trajectory tangent vectors measured in 160 s increments
(see Materials and Methods), and the persistence fraction was defined as the fraction of all
turning angles within ±30◦ (Fig. 5.2 f). These persistence fractions are not a measure of how
long a cell is likely to travel before turning, but rather the appearance of directionality at
160 s time scales. The relative values of persistence fraction showed a biphasic relationship to
BG concentration similar to that of the MSD and spread area. As with contractile energies
and contractility times, persistence fractions were conserved across substrate stiffnesses,
irrespective of BG concentration.
The correlations between spread area, motility characteristics, and the mechanical out-
put are graphed in Fig. 5.3. The six nodes correspond to variables that were simultaneously
measured for each cell (ensemble averages plotted in Fig. 5.2). Weighted lines connecting
each node indicate Pearson product-moment correlation coefficients (PCCs), which measure
the linear correlation between the pairs of variables. Negative PCCs (red lines) indicate
negative correlations and positive PCCs (black lines) indicate positive correlations. MSD,
spread area, and persistence fraction were all positively correlated, irrespective of stiffness.
Essentially, persistent cells traveled further than non persistent cells in a given amount of
127
time, and the greater the spread area, the more persistent the cell was. PCCs indicate
correlation, not causality. However, we suspect that cells with greater spread areas are
more persistent because of increased integrin engagement and the accompanying requisite
to break more integrin bonds to change direction. Negative correlations exist between BG
concentration and energy, energy and MSD, MSD and contractility time, and contractility
time and BG. In the following section, we study these correlations in more detail to deter-
mine the links between energy and motility. One final note about PCCs, PCCs reflect only
the linear portion of correlation between pairs of variables. To complement the PCCs, alter-
nating conditional expectation (ACE) values are tabulated in the Supplementary Material
(Table 5.1). ACE values reflect the probability that any function, linear or nonlinear, can
be used to describe the relationship between pairs of variables. For example, whereas the
PCC between spread area and contractile energy on the 25 kPa substrate may indicate no
linear relationship between the two variables, the corresponding ACE value indicates that
a nonlinear relationship may exist.
3.3. Optimal Motility Correlated Inversely With Mechanical Output. The
PCC values graphed above indicate the possibility of a correlation between motility and
energy/force exertion. At present, little is known about the energy/force-speed dependency
and its elucidation would be useful for cell motility modeling. We find that the contractile
energy of faster moving neutrophils (MSD at 560 s> 500µm2) is significantly lower than
the contractile energy of slower moving cells (MSD at 560 s≤ 500µm2, Fig. 5.4, a and b).
The exception to this finding is with BG50 cells on 5 kPa substrates, where there were very
few significantly motile cells in the ensemble. Insets (panel b) show the entire distribution of
time average contractile energy against MSD at 560 s. The inverse speed energy correlation
is also borne out by the PCC values shown as ρ in the inset. As with the conservation of
energy across substrate stiffnesses, the inverse correlation between energy and speed was
also conserved in the absence of BG. The corresponding force-speed distributions are plotted
128
beta glucan
contractile energy
MSD at 560 s
contractility time
persistence fraction
spread area
0.200.20
0.05
0.05
-0.2
7-0
.27
0.0
8
0.3
8
-0.08-0.08
-0.290.06
0.090.09
-0.07-0.07
-0.26
0.16
0.290.29
0.030.03
beta glucan
contractile energy
contractility time
persistence fraction
spread area
-0.07-0.07
0.12
0.12
-0.1
9-0
.19 -0
.03
-0.15-0.15
-0.22-0.0
5
0.610.61
-0.06-0.060.070.07
-0.12
0.490.6
2-0.18-0.18
0.39
MSD at 560 s
25 kPa5 kPa ba
Figure 5.3. Two graphs showing the Pearson product-moment corre-lation coefficients (PCCs) between variables describing the motilityand mechanical output of neutrophils. Each graph corresponds to onestiffness, (a) 5 kPa on the left, and (b) 25 kPa on the right. Each node cor-responds to a variable measured in Fig. 5.2. Values of MSD, spread area, andcontractile energy correspond to measurements at 560 s. PCCs have values from-1 to 1 estimating the linear portion of the correlation between the two variables.Line thickness is proportional to the magnitude of the PCC, with red denot-ing a negative correlation (PCC< 0), and black denoting a positive correlation(PCC> 0).
in Fig. 5.4 c. Contractile forces were determined by taking the average magnitude of vector
stress integrated over each stress island, whereas contractile energies were calculated by
integrating scalar traction stresses. Insomuch that one quantity is a vector sum and the
other is a scale sum, there was no reason to expect the force-speed distributions to mirror
the energy-speed distributions. However, the force-speed distributions were, overall, very
similar to the energy-speed distributions, showing the same inverse correlations with similar
PCC values. Overall, optimal motility correlated inversely with mechanical output, even
when the mechanical output was diminished by the biological response modifier, soluble
BG.
3.4. Dipole Case Study: Forces Perpendicular to the Direction of Motion. To
examine the spatiotemporal distribution of contractile forces that occured during migration,
129
0
2
4
6
8
10
12
cont
ract
ile e
nerg
y (n
Nµm
)
25 kPa5 kPa
fastBG0
slowBG0
0 10 20 300 10 20 300
2
4
6
8
10
12
cont
ract
ile e
nerg
y (n
Nµm
)
slowBG50fastBG50
time (min) time (min)
a
b
c
cont
ract
ile e
nerg
y (n
Nµm
)MSD at 560s
(103µm2)
cont
ract
ile e
nerg
y (n
Nµm
)
cont
ract
ile fo
rce
( nN
)
MSD at 560s (103µm2)
0 1 2 3 4 50
4
8
12 ρ = -0.24ρ = -0.24ρ = -0.52ρ = -0.52
0 1 2 3 4 50
4
8
12 ρ = -0.24ρ = -0.24ρ = -0.08ρ = -0.08
1 2 3 50 4MSD at 560s (103µm2)
ρ = -0.20ρ = -0.20ρ = -0.49ρ = -0.49
0
20
40
60
80
100
120
140
1 2 3 50 4MSD at 560s (103µm2)
ρ = -0.26ρ = -0.26ρ = -0.10ρ = -0.10
Figure 5.4. Slower moving cells exert more energy contracting theirsubstrate than faster moving cells. Slower cells were gated from faster cellsusing a threshold MSD of 500µm2 (measured at 560 s). The ensemble averagedcontractile energies are plotted (a) in the absence of BG (BG0), and (b) inthe presence of 50µg/mL BG (BG50). Shaded (fast moving cells) and striped(slow moving cells) regions indicate the mean plus and minus one standard errorof the mean. Except with BG50 cells on 5 kPa substrates, slower moving cellsexerted significantly more energy than faster moving cells. With the BG50 cellson 5 kPa substrates, no significant difference in contractile energy was observed.(caption continued on pg. 132)
130
dipole moment analysis was applied to the traction stress fields of two motile neutrophils
on 5 kPa subtrates, one with the BG0 concentration (Movie S1) and one with the BG50
concentration (Movie S2). Major and minor dipole moments (µM and µm, respectively)
were determined from principal moment analysis of the dipole tensor (see Materials and
Methods). Two variables were used to characterize the spatial distribution of forces: First
the relative angle between the major dipole moment and the direction of motion (angle θ,
Fig. 5.5 a). Second, the difference in the absolute magnitudes of the major and minor dipole
moments, |~µM | − |~µm|. Small differences indicate a relatively isotropic traction stress field
(such as an annulus, see Chapter 6, Fig. 6.7 e for an example), whereas large differences
indicate an anisotropic stress field that is principally oriented along the axis of the major
dipole moment vector, ~µM (as in Fig. 5.5 a). Forces on the BG0 cell were situated toward
the uropod, just as reported in [48, 50, 165], with the principal stress orientation often
nearly perpendicular to the direction of motion (Fig. 5.5, b and c, see Movie S1). As
reported previously, we observed reorganization of the stress island centers prior to changes
in direction [165]. However, once the force centers had stabilized into a perpendicular
mode, changes to the dipole axis trailed changes to the direction of motion. Therefore,
contractile force reorganization appears to be the driving force behind sharp changes in
direction, whereas smooth, shallow changes in direction are not the result of contractile
force reorientation. This observation is characterized in Fig. 5.5 b. Here the cell crawls with
an overall counterclockwise migration trajectory (Movie S1). The major dipole orientation
was most often between -50 and -70◦ with respect to the direction of motion, indicating that
changes to the dipole moment orientation lagged behind changes to the direction of motion.
With the exception of anomalous dipole calculations between 0 and 10 min (Fig. 5.5 d),
and during the time of significant traction stress reorientation close to the 15 min mark,
the traction stress distribution was primarily anisotropic in nature, as seen by the nearly
131
three-fold difference between the major and minor dipole magnitudes over the 19 to 35 min
range.
We also analyzed a BG50 cell on a 5 kPa substrate. As reflected by the ensemble av-
eraged plots (Fig. 5.2), the BG50 cell migrated less persistently, had smaller spread area,
contractile forces, and contractile energies, compared to the BG0 cell (BG50 cell in Movie S2
compared with BG0 cell in Movie S1). With these differences though, the major dipole mo-
ment was still primarily either perpendicular or parallel to the direction of motion (Fig. 5.5, e
and f). Dipole moments became less pronounced ahead of turns made. As the persistence
of this BG50 neutrophil was much smaller, it was difficult to determine whether stabilized
dipole moments preceded or proceeded changes in direction. Overall, the spatial distri-
bution of the stresses was less organized, and the three-fold difference between major and
minor dipole moment magnitude was not observed (Fig. 5.5 g).
3.5. Estimating Active Contractile Force and Cytoskeletal Stiffness. Marcq
et al. recently introduced a simple 1D spring model to interpret the contractile forces ex-
erted by cells on soft substrates [51]. In their model, the cell is represented by a spring
with stiffness kC , and the substrate is represented by two surrounding springs of stiffness
ksub (Fig. 5.6 a). Inward “active” forces, ~FA, represent the total myosin mediated con-
tractile force transduced from the actin cytoskeleton to the substrate. Note, the force
measured by detecting substrate displacements is only a fraction of the total active force
because of the cytoskeletal spring. Using this model and converting the spring stiffnesses to
Young’s moduli, the magnitude of the measured contractile force follows the Hill function
F = FA
(Ysub
YC+Ysub
), where YC is the estimated Young’s modulus of the cytoskeleton, and
Ysub is the Young’s modulus of the substrate. Fitting the model using contractile force
(Fig. 5.4 caption continued) Insets show scatter plots of time averaged contractile energy versusMSD at 560 s for BG0 (yellow triangles) and BG50 (blue circles) cells on 5 kPa (left) and 25 kPa(right) substrates. The striped region in the scatter plots corresponds to slower moving cells belowthe MSD threshold. (c) Scatter plots of time averaged contractile force versus MSD at 560 s. Colorsand shapes correspond to the insets in (b).
132
BG0
BG50
g
f
d
c
b
e
a
5% 11% 17% 23% 29%0% 90−90
0relative dipoleangle, θ(deg)
-375 > µM-250 > µM > -175-125 > µM > -250
µM > -125
major dipole moment (nN µm)
0 10 20 30−500
−250
0
250
time (min)
dipo
le m
omen
t (n
N µ
m)
µm
µM
−90−60−30
0306090
rela
tive
dipo
le
angl
e, θ
(deg
)
θ
minor dipole, µM
major dipole,µ
m
−90−60−30
0306090
rela
tive
dipo
le
angl
e, θ
(deg
)
90−90
0relative dipoleangle, θ(deg)
4% 8% 13% 17% 21%0%
dipo
le m
omen
t (n
N µ
m)
0 10 20 30−500
−250
0
250
time (min)
µM
µm
Figure 5.5. Dipole moment analysis of a motile BG0 cell migratingon a 5 kPa substrate. (a) Schematic of the major (red) and minor(blue)dipole moment axes. Stress island force vectors are shown for comparison. Theangle between the major dipole moment and the direction of motion, known asthe relative dipole angle, is defined to be θ. This PMN corresponds to movieS1 at 21 min. The scale bar illustrates 5µm. (b) Histogram of the relativedipole angles. On average, the dipole moment aligns nearly perpendicular tothe direction of motion. (c) Relative dipole angle plotted over time. (d) Majorand minor dipole moments plotted over time. Negative moments correspond tocontractile motion. Large differences between the major and minor momentsindicate a definitive axis of contractility, whereas small differences indicate anannular like traction field.
measurements on both 5 and 25 kPa substrates, the estimated cytoskeletal Young’s modu-
lus was determined to be within a range from 10 to 20 kPa, without significant dependence
on BG concentration (Fig. 5.6 b). Comparatively, the cytoskeletal moduli was one order
133
0
10
20
30
40
50
0
50
100
150
BG10 BG50BG0
kC
FA FA
ksubksub
cyto
ske
leta
l mo
du
lus,
EC (
kPa
)
act
ive
fo
rce
, FA (
nN
)
BG10 BG50BG0
a
b c
Figure 5.6. Estimating total myosin mediated contractile force andcell stiffness. (a) A schematic of a 1D spring model proposed by Marcq et al.to estimate the intracellular contractile force, FA, and stiffness, kC , given theobserved contractile forces on a range of substrate stiffnesses, ksub. Solving thespring model at varying concentrations of BG show that, (b) the cytoskeletalstiffness does not change significantly with BG, and (c) the active contractileforce decreases with increasing BG. Error bars represent the standard errors ofthe mean.
of magnitude smaller than that predicted from Madin-Darby Canine Kidney fibroblasts
(∼ 100 kPa) [51], comparable to the cytoskeletal stiffness of fish keratocytes (10-150 kPa)
[2, 135, 180] but intermediate of Young’s moduli estimates for cortical actin (0.01-0.1 kPa)
[145] and stress fibres (∼ 1000 kPa) [142]. Increasing BG concentration decreased the ac-
tive contractile force (Fig. 5.6 c), suggesting either a loss of coupling strength between the
cytoskeleton and the substrate, or a decrease in myosin generated contraction. The mea-
surement of contractile forces on a wider range of substrate stiffnesses would improve the
confidence placed on these numbers, YC and FA.
4. Discussion
We have shown that contractile energy is conserved across substrate stiffness, and this
conservation remains with the addition of the biological response modifier, BG. The energy
and force per integrin can be estimated by assuming that roughly 300 integrins per µm2
134
populate a stress island [29], and that each stress island ranges in area from 30-50µm2
(∼ 100µm2 total average stress island area). The result is an energy per integrin on the
order of 10 pNnm, and a forces per integrin of approximately 2.5 and 5 pN for neutrophils on
5 and 25 kPa substrates, respectively. These force per integrin estimates are slightly smaller
than values of 13-50 pN obtained by measuring the rupture force of VLA5 to fibronectin
[181, 182]. Assuming a similar rupture force for CR3 (both VLA5 and CR3 bind fibronectin),
then there is a significantly higher capacity for integrin borne force generation than what
we have observed. Indeed, a very small fraction of the neutrophil ensemble was able to
maintain forces as much as 4 to 5 times the mean. Another explanation for the small force
per integrin estimate is that only a sub-population of integrins bind stably and transduce
force [79]. Thus, the force per stably bound integrin would be greater, as would the energy
per integrin.
For neutrophils in suspension with human serum, the amount of active CR3 and total
CR3 expression of BG50 treated neutrophils was approximately 1.26±0.18 times greater
than BG0 neutrophils [79]. If this result can also be shown for neutrophils treated with
both fMLP and BG, then one can rule out a reduction in active integrin as a mechanism
for BG induced decreased mechanical output. Instead, reduction of mechanical output
with increasing BG concentration may be the effect of intracellular factors leading to a
decrease in total myosin contraction, or a disruption of the proteins that bind integrins to
the cytoskeleton (e.g. talin, vinculin, etc.). Regardless of the mechanism, such data would
shed light on the amount of force and energy transduction per integrin, and may set a new
lower bound on the force per integrin required for mechanosensors within the neutrophil to
detect substrate stiffness.2
In addition to experiments probing for differential integrin expression as BG concentra-
tion is ramped up, a causal link between BG and changes in the mechanical output would
2These experiments can be done by flow cytometry, and should be completed shortly.
135
be better established by imaging the actin cytoskeleton. The organization of the actin cy-
toskeleton may be suggestive of a mechanism to explain the differences in BG mediated
mechanical output (Fig. 5.2 c), or differences in the total active myosin generated contrac-
tile force determined from the Marcq model (Fig. 5.6 c). It has recently become possible to
introduce primary human neutrophils with a small fluorophore fused peptide called LifeAct
that binds F-actin without inhibiting actin dynamics [183], thus allowing for real time imag-
ing of the actin cytoskeleton. The most informative experiment would be to visualize the
actin cytoskeleton in real time using LifeAct, while doing TFM, at varying concentrations
of BG. However, there is yet to be compelling data showing that LifeAct does not alter
neutrophil motility and function. Therefore, TFM followed by cell fixation and staining
of the actin cytoskeleton with fluorophore fused phalloidin (a compound that binds at the
interface between F-actin subunits and prevents depolymerization [184]) is likely the most
feasible course of action at this time.3
We have also shown numerous correlations between variables characterizing morphology,
motility, and mechanical output. Correlation does not imply causality, it merely implies
the possibility of causality. Some of the correlations and possible causalities are discussed
here. As discussed above, spread area was positively correlated to persistence. We suspect
that cells with greater spread areas were more persistent because of increased integrin
engagement. Cells with greater spread likely require more energetically costly bond rupture
events in order to change direction, as speculated by Oakes et al. who also observed a
positive correlation between spread area and persistence. O’Brien et al., have shown that
persistence of neutrophils crawling on 2D substrates decreases, irrespective of substrate
stiffness and cell spread area, when substrate engagement is restricted to β1 integrins [1].
Conversely, fibronectin used in this study engages both the β1 integrin VLA5 and the β2
integrin CR3. Thus, we can hypothesize that persistence in 2D motility is likely causally
correlated to CR3 ligation that is increased by greater spread area. This hypothesis would
3These simple experiments will be completed shortly.
136
be best supported using integrin labeling and adhesion detection from a TIRF microscope.
However, integrin labeling with primary cells such as neutrophils is extremely difficult and
a PAG substrate does not have an optical index of refraction suitable for TIRF microscopy.
This experiment could instead be done with neutrophil-like DMSO differentiated HL-60
transfected with GFP-CR3, plated onto a silicon gel optically corrected for TIRF microscopy
[40].
Other than its dependence on spread area, persistence fraction appears to be indepen-
dent of all other variables, indicating that neither mechanical output nor the addition of
BG affects persistence on soft substrates. However, BG has been shown to improve 2D
neutrophil honing towards weak chemoattractant sources on glass [75]. TFM experiments
have shown evidence of a principal force orientation from perpendicular the direction of
motion to parallel the direction of motion that occurs in chemotactic cells (force orientation
data shown in Chapter 6). This, in contrast to the dipole moment analysis only showing
perpendicular principal force orientation, suggests that chemotactic honing is facilitated by
a mode of mechanical output that does not occur during chemokinesis. Taken together with
the significant effect of BG on mechanical output, it is reasonable to suggest that honing
and mechanical output are linked, and that changes in mechanical output caused by BG are
either the cause of, or the result of, differential honing, thus also explaining why BG appears
to modify neutrophil honing, but not persistence. These observations can be confirmed us-
ing dipole moment analysis to determine the principal orientation of chemotactic forces and
should be combined with directionality time analysis, which gives an unbiased measure of
the extent of honing during directed cell migration [185] (c.f. Chapter 2). Higher order
multipole moments such as quadrupoles have also been shown to be indicative of direction-
ality in dictyostelium discoideum motility [176], and may be informative in the analysis of
neutrophil motility.
137
In summary, the chemokinetic contractile forces of neutrophils were localized towards
the posterior of the cell, as reported previously [48, 165]. Dipole moment analysis rigorously
indicated that the principal orientation of these contractile forces were perpendicular to the
direction of motion. This contractile force orientation is similar to that of keratocytes [5, 31],
but differs from that of other eukaryotic cells such as fibroblasts which generate forces at
the leading edge that typically align parallel the direction of motion [32]. We did not find
that motility was optimally dependent on an intermediate range of contractile force, as
suggested by Bangasser et al. [42]. Instead, we found the cell speed scaled inversely with
both contractile force and energy, and that the strength of this correlation was the same on
both 5 and 25 kPa substrates. As also shown previously, neither contractile displacements
nor contractile forces were conserved against substrate stiffness [48]. We show, for the
first time, evidence that contractile energy is a substrate stiffness conserved quantity for
neutrophils. The addition of the biological response modifier BG, to model the role of cell
mechanics in pharmacological intervention, decreased contractile energy, yet energy was
still conserved across substrate stiffness. In cases where BG increased cell motility (25 kPa
substrate), there was also a significant inverse correlation between contractile energy and
cell speed. Taken together with the conserved inverse correlation to cell speed, it appears
that consideration for total contractile energy should be an important component of a cell
motility model.
5. Supplementary Material
5.1. Supplementary Movie Legends.
Movie S1. BG0 neutrophil undergoing chemokinesis on a 5 kPa substrate. Time-
0 corresponds to the addition of fMLP (10 nM). (top-left) Cell migration trajectory and
spread area were tracked from differential contrast images shown here. The white outline is
the perceived cell boundary, and scatter points are centroids calculated from these outlines.
138
Scatter point color indicates time. The white arrow originating from the cell centroid shows
the instantaneous velocity of the cell time averaged over an interval enclosing 1 min before
and after the time shown. The instantaneous velocity legend (white arrow, top-left) cor-
responds to 0.125µm/s. (middle-left) Spread area calculated based on the perceived cell
boundaries. (bottom-left) The average of stress island force magnitudes are plotted here
above the total contractile energy. (right) Stress maps of the neutrophil. Color describes
traction stress (color bar shown in Fig. 5.1 c). The white outline is again the perceived cell
boundary. Black arrows originating from the center of stress islands correspond to the stress
island force. The stress island force legend (black arrow, top-left) corresponds to 10 nN.
139
Movie S2. BG50 neutrophil undergoing chemokinesis on a 5 kPa substrate.
Time-0 corresponds to the addition of fMLP (10 nM). (top-left) Cell migration trajectory
and spread area were tracked from differential contrast images shown here. The white
outline is the perceived cell boundary, and scatter points are centroids calculated from
these outlines. Scatter point color indicates time. The white arrow originating from the
cell centroid shows the instantaneous velocity of the cell time averaged over an interval
enclosing 1 min on before and after the time shown. The instantaneous velocity legend
(white arrow, top-left) corresponds to 0.125µm/s. (middle-left) Spread area calculated
based on the perceived cell boundaries. (bottom-left) The average of stress island force
magnitudes are plotted here above the total contractile energy. (right) Stress maps of
the neutrophil. Color describes traction stress (color bar shown in Fig. 5.1 c). The white
outline is again the perceived cell boundary. Black arrows originating from the center of
stress islands correspond to the stress island force. The stress island force legend (black
arrow, top-left) corresponds to 10 nN.
5.2. Supplementary Tables and Figures.
5 kPa r 25 kPa A CT E PF MSDA × 0.71 0.87 0.80 0.81
CT 0.75 × 0.68 0.73 0.69E 0.81 0.72 × 0.72 0.76
PF 0.93 0.67 0.78 × 0.83MSD 0.94 0.71 0.78 0.99 ×
Table 5.1. Table of Alternating Conditional Expectations (ACE) val-ues. ACE values range from 0 to 1. The larger the value, the greater theprobability there exists a function (either linear or nonlinear) that describes thedistribution between the two variables. ACE values are tabulated in matrixform to correlate spread area (A), contractility time (CT), contractile energy(E), persistence fraction (PF), and mean squared displacement (MSD). Thebottom-left off-diagonal terms correspond to ACE values on 5 kPa substrates.The top-right off-diagonal terms correspond to ACE values on 25 kPa substrates.This table is meant to be used in relation to the Pearson product-moment cor-relation coefficients graphed in Fig. 5.3.
140
DIC DiI (PM)DIC (trace)
a b c
Figure 5.7. The perceived cell boundary may differ from the true cellboundary. The perceived cell boundary of a neutrophil captured by (a) dif-ferential interference contrast microscopy and (b) traced by eye. The trace isbased not only on the current, but also on prior and subsequent images to makea best guess of the perceived cell boundary. (c) The same cell, showing DiIplasma membrane (PM) staining shows that the perceived cell boundary candiffer from the true cell boundary.
a
b
time (min) time (min)
0 10 20 30
BG0
Dex50
BG50 + CR3 block
BG0
Dex50
BG50 + CR3 block
0 10 20 300
20
40
60
80
100
120
con
tra
ctile
fo
rce
(n
N)
0
2
4
6
8
10
12
con
tra
ctile
en
erg
y (
nNµ
m)
25 kPa5 kPa
Figure 5.8. The mechanical output during control experiment for BGassociated change in osmolarity. (a) Contractile energy and (b) contrac-tile force measurements of neutrophils migrating in purified 50µg/mL dextran(Dex50) in citrate buffer show that the osmolarity and sugar concentration ofthe BG50 condition does not account for the changes in mechanical output.The mechanical output was nearly completely blocked when BG50 cells werepretreated with CR3 blocking antibody. n = 20, 15 for Dex50 5 and 25 kPasubstrates, respectively.
141
0
50
100
5kPa 25kPa
BG0
5kPa 25kPa
BG50
5kPa 25kPa
BG50+CR3block
percent
P(adhere) P(active|adhere) P(contractile|active)
Figure 5.9. Neutrophil adhesion rate, activity rate, and contractilityrate. The percentage P (adhere) is the percent of neutrophils that contact andadhere to the substrate divided by all cells the make contact with the substrate.The conditional percentage P (active|adhere) is the percent of adhered cells thatappear to be active, meaning they are extending pseudopodia and/or motile(determined by eye). The conditional percentage P (contractile|active) is thepercent of all active cells that apply visible displacements to the substrate. BG0and BG50 refer to 0 and 50µg/mL β-glucan, respectively, and BG50+CR3blockrefers to BG50 cells pretreated with CR3 integrin blocking antibody. These ratesact as internal controls indicating similar neutrophil preparation and proteincoating conditions across experiments. CR3 block treated cells show a significantdecrease in adhesion-, activity-, and contractility rate. Error bars representstandard error of the mean.
142
CHAPTER 6
Future Directions and Closing Remarks
143
Forward. The contents of this chapter discuss the preliminary data of two ongoing
projects that show strong potential as avenues of future study. The first is an extension
of the findings in Chapter 5 regarding the orientation of traction forces during chemotaxis.
Dipole moment analysis revealed that traction forces during chemokinesis were principally
oriented perpendicular to the direction of motion. In another data set described in this
chapter, we measured the traction forces of directionally migrating neutrophils crawling up
a chemoattractant gradient toward a point source. In agreement with previous findings, the
contractile forces during chemotaxis were localized toward the cell posterior [48, 50, 165].
What has not been well characterized before is the spatiotemporal orientation of these con-
tractile forces. We found that forces during chemotaxis begin small and perpendicular to the
direction of motion. Over time, they build in magnitude and transition (at approximately a
magnitude of 20 nN) from perpendicular to parallel. A model that describes these contrac-
tile force observations is proposed in Sec. 1, and we find that this same model may also be
able to mechanistically account for measurements of directionality time. A computational
algorithm with sample solutions that can be replicated by a beginning to intermediate level
scientific programmer is also contained below.
The second ongoing project address the basis of a reported chemotactic defect of neu-
trophils in patients with sepsis that may be associated with pathological sequela and mortal-
ity. Again, the perpendicular to parallel force transition is observed, but cells isolated from
patients with sepsis also show large isotropic stress distributions. These findings, discussed
briefly in Sec. 2, go hand in hand with our BG finding indicating that biological response
modifiers may act through the modulation of cell mechanics and motility.
This chapter finishes with a short closing remark section that ties together findings
presented in this dissertation and places them in the context of the current state of the
field.
144
Receptor
Chemoattractant
Filaments
NucleusPseudopod
Figure 6.1. Schematic showing the chemical signaling of chemotaxisand the downstream cellular components that drive cell migration.This figure has been borrowed from Ref. [186].
1. Future Project 1: Signaling During Early Stage Chemotaxis
1.1. Introduction. Chemotaxis is the process in which a cell aligns its migration path
to the concentration gradient of a specific external chemoattractant. Unlike most prokary-
otic cells which detect concentration gradients over time, eukaryotic cells including neu-
trophils detect chemoattractant gradients spatially over the extent of the cell body. This
detection occurs when chemoattractant molecules (and/or other external chemical cues)
bind to specific receptors on the surface of the cell membrane. Figure 6.1 shows an exam-
ple of chemoattractant (green circles) binding transmembrane receptors (blue horns). In
Chapter 5, the chemoattractant was fMLP and the receptors were formyl peptide receptor 1
(FPR1). Chemical signaling mechanisms downstream of bound receptors lead to processes
such as actin filament polymerization and myosin contraction that drive pseudopod forma-
tion, ultimately leading to cell migration. In the context of a chemoattractant gradient, the
distribution of bound receptors about the cell membrane is asymmetric and this asymmetry
determines the direction of chemotactic transmigration.
The chemotaxis process can be divided into three steps. The first step is quantifying
signal detection. What is the spatial distribution of bound receptors given a distribution of
chemoattractant molecules? Statistical mechanics provides the tools to analytically model
receptor occupation [186]. The second and third steps are signal processing and the imple-
mentation of biomechanical processes that drive cell migration. A good signal processing
145
mechanism must be able to separate the average chemoattractant concentration, S, from
the concentration gradient, ∇S, measured over the length of the cell, l. Robust directed
cell migration like that of neutrophils requires that the cell respond over a broad range of
average chemoattractant concentrations, S. Essentially, the cell needs to adapt to different
average chemoattractant concentrations. Consider a test model of a cell with motility ma-
chinery coupled to the extracellular chemoattractant concentration by a factor of ξ ∝ l∇SS
.
In a linear gradient, the motility machinery would change its state as the cell migrates and
S increases (ξ decreases). Using this test model, the conditions under which chemotaxis
occurs would be severely restricted. Therefore, this test model is not suitable for describing
chemotaxis under physiological conditions. A better model of chemotaxis signal processing
leaves a large range of S decoupled from the motility machinery. If ξ is completely decoupled
from S, it is called perfect adaptation.
Several mathematical models exist for predicting the distribution of signaling proteins
associated with cell migration. Most of these models achieve something close to perfect
adaptation using a design principle called local excitation and global inhibitor (LEGI).
A LEGI model can be described in terms of four chemical components, the signal, two
factors turned on by the signal that activate and inhibit a downstream response, and a
downstream response that activates the machinery of cell motility. The activator is locally
confined and fast acting whereas the inhibitor is diffusible and slow acting. Each component
effective represents signaling proteins and other compounds associated with chemotaxis in
cells. Because the response is what drives the motility machinery, it is the response that
must perfectly adapt to the chemoattractant signal. To achieve perfect adaptation, the
signal up-regulates the fast activator, which causes the response to increase. The signal
also up-regulates the slow inhibitor, which eventually causes the response to relax back to
its original equilibrium value. The magnitude and timescale of this transitory response spike
depends on the reaction rates between the catalytic reactions that regulate the response.
146
However, at steady state, the response does not depend on average signal concentration
across the cell, S. Thus perfect adaptation is achieved. We hypothesize that the duration
of the transitory response spike in reaction to a change in chemotactic cue can be used to
mechanistically model results found using our directional motility metric.
As amoeboid cells migrate, they apply contractile forces to their substrate. In our recent
experiments using traction force microscopy to measure the forces applied by neutrophils
during chemotaxis, we observed an initial period of small contractile forces that were prin-
cipally oriented perpendicular to the direction of migration. After a period of one to two
minutes during which time the magnitude of contractile forces increased, the principal force
orientation transitioned from a perpendicular mode to a parallel mode (Fig. 6.2). Although
the average scalar traction and RMS contractile force of chemotactic neutrophils have been
characterized in the past [48, 50, 165], the orientational dynamics of these force modes has
yet to be studied. Due to the timing and transient nature of the perpendicular force mode
as we observe it, and the premise that cells are designed to be adaptive to chemotactic sig-
nals [187], it is plausible that our observations can also be explained by a process featuring
perfect adaptation. With this in mind, we solve two LEGI models below in order to study
the nature of the transient response factor. In neutrophils, the elements of several LEGI
models have been suggested to correspond to particular signaling proteins such as phospho-
inositide 3-kinase (PI3K), and small G-protein [188]. A solution to this model provides both
an experimentally testable mechanism for the perpendicular to parallel force transition, and
should give mechanistic insights into why contractile force exertions are significantly altered
from early to later stages of cell adhesion [189].
1.2. Computational Model.
1.2.1. Reaction Diffusion Equations. There are several LEGI models in existence but
here we use one chosen one by Levchenko and Iglesias [190]. The model is described as
follows: A local activator of concentration A found along the cell membrane catalyzes the
147
1 min 2 min 3 min 4 min
a
b
c
d
0 100 200 300 400 500traction (Pa)
0 1 2 3 402460
20
40
60
0
1
time (min)
en
erg
y
(pJ)
forc
e
(nN
)
dire
ctio
na
lity
ind
ex [γ]
Figure 6.2. Classification of neutrophil traction stress fields during 2Dchemotaxis. (a) Traction field time-lapse of a chemotactic neutrophil in oneminute increments. White lines indicate the cell determined from correspond-ing DIC images (not shown here). Black dashed lines outline traction islands.Black vectors represent the forces calculated by integrating over the local trac-tion island region. Two modes are used to describe the orientation of islandforces: perpendicular (⊥) and parallel (‖). Perpendicular and parallel modessignify traction forces oriented nearly perpendicular or parallel to direction ofmotion, respectively. (b) Average traction island force and directionality indexplotted against time. After a period of low force output from 0 to 1 minute,island forces increase in magnitude and the traction force mode transitions fromperpendicular to parallel by 2 minutes. (c) Work done on the substrate plot-ted against time (neglecting energy exerted to stress the substrate in the 3rddimension). (d) Schematic showing the general force mode progression (newunpublished result). Scale bars show 10µm.
conversion of an inactive response protein, R, to an active response protein, R∗ (Fig. 6.3).
The presence of active response protein leads to cell locomotion in a manner not described
here. The active response protein can also return to its inactive state as catalyzed by a
globally diffusing inhibitor, I. Below is a brief derivation of reaction kinematics similar to
the derivation in Ref. [190]. First consider the enzymatic reactions,
R+Akc1ku1
U1ka1⇀ R∗ (6.1)
148
R∗ + Ikc2ku2
U2ka2⇀ R. (6.2)
Corresponding reaction rates are derived using the law of mass action as follows,
dR
dt= −kc1RA+ ku1U1 + ka2U2 (6.3)
dR∗
dt= −kc2R∗I + ku2U2 + ka1U1 (6.4)
dU1
dt= −kc2R∗I + ku2U2 + ka1U1 (6.5)
dU2
dt= −kc2R∗I + ku2U2 + ka1U1 (6.6)
where U1 and U2 are substrate-enzyme reaction intermediates and the k’s are rate constants
[191]. A more general treatment for the reaction rate equations 6.3-6.6 imposes Hill functions
to account for the total free enzyme concentration, A and I. Here, we have simplified the
reaction equations by assuming that the fraction of bound enzyme is very small compared
to the amount of total enzyme. Assuming a quasi-steady state solution for the reaction
intermediates, Eqn. 6.4 simplifies to,
dR∗
dt= −k−RIR∗ + kRAR. (6.7)
Applying similar logic (see Ref. [190]), the reaction rates for activator A and inhibitor I
are,
dA
dt= −k−AA+ kAS (6.8)
dI
dt= −k−II + kIS. (6.9)
Finally, Eqns. 6.7-6.9 are simplified by non-dimensionalization and scaling using the fol-
lowing parameters: τ ≡ k−At, a ≡ (kR/k−A)A, i ≡ (kRkAk−I/kIk2A)I, r ≡ R∗/Rtot,
α ≡ k−I/k−A, and β ≡ (kIk−Rk−A/k−IkRkA). The resulting equations define the reaction
rates of the response protein fraction r and scaled enzyme concentration a and i localized
149
to the cell membrane at location m:
da(m)
dτ= −(a− s) (6.10)
di(m)
dτ= −α(i− s) (6.11)
dr(m)
dτ= −βir + a(1− r). (6.12)
Allowing for global diffusion of inhibitor i, Eqn. 6.11 is a boundary condition for the global
diffusion equation,
∂i(~x, t)
∂τ=
D
k−A∇i(~x, τ) (6.13)
where D is the diffusion coefficient of the diffusible inhibitor in cytosol, and ~x is the 2-D
spatial coordinate within the cell.
I
S S(t)
R*/Rtot(t)
t
t
R
R*
A
a b
Figure 6.3. The LEGI model. (a) Chemical reaction circuit diagram. Thesignal, S, activates a response activator, A, and a response inhibitor, I, whichcatalyze the forward and reverse reactions between an inactive and active re-sponse protein, R and R∗, respectively. In this model, each element is confinedto the cell membrane except the inhibitor, I, which can diffuse through thecellular cytosol. Active response factor, R∗, up-regulates cell motility. (b) Per-fect Adaptation. In this LEGI model, when the reaction rates for the reactionsS ⇀ A and S ⇀ I have similar form in their dependence on signal concen-tration S, the equilibrium fraction of active response does not change when aspatially homogeneous signal is altered. In response to the signal change, thereis a transient increases in active response protein. The magnitude and durationof this response factor spike depend on the ratio between inhibitor productionand activator production. Colors: cyan indicates low signal concentration, redindicates high signal concentration.
150
1.2.2. Finite Difference Equations - Cartesian Coordinates. The model can be solved
on a disk of radius P in Cartesian coordinates (x, y) using the forward time centered space
(FTCS) method. The coordinate system is discretized using xp = p∆x and yq = q∆x, both
with dimensions of µm and p, q = −P∆x ,
−P∆x + 1, ..., 0, ..., P∆x − 1, P∆x such that the domain is
a square with side length 2P . Time is discretized as τn = n∆τ , with dimensions of k−1−A.
In this coordinate system, the diffusion equation is approximated by the finite difference
equation,
i(xp, yq, τn+1) = i(xp, yq, τn) (1− 4ζ) +
+ζ [i(xp+1, yq, τn) + i(xp−1, yq, τn) + i(xp, yq+1, τn) + i(xp, yq−1, τn)](6.14)
where ζ ≡ D∆τk−A(∆x)2 .1 The initial condition for the inhibitor is zero everywhere on the
domain. The disk boundary defined by the circle p2 + q2 = P 2/(∆x)2. In discretized space,
a list of boundary coordinates is generated by populating the set
{p, q} ∈ round(√
p2 + q2)
=P
∆x(6.15)
This is the boundary where differential Eqns. 6.10-6.12 are applied (using 2nd order Runge
Kutta algorithm). Equation. 6.14 is solved everywhere on the domain before i is updated
at the boundary using a Runge Kutta step based on Eqn. 6.11.
1.2.3. Finite Difference Equations - Polar Coordinates. The model can be solved on a
disk of radius P using the FTCS method. The disk is discretized in polar coordinates (ρ, θ)
such that ρx = x∆ρ and θy = y∆θ, with dimensions of µm and degrees ( ◦) respectively.
Time is discretized again as τn = n∆τ , with dimensions of k−1−A. In these polar coordinates,
1As a side note, using ζ = 14 gives the Jacobi iterative method for solving the Poisson equation (i.e.
D = 1, ∆τ = 0.01, and ∆x = 0.2).
151
the diffusion equation for inhibitor i is
∂i(ρ, θ, τ)
∂τ=
D
k−A
[∂2i
∂ρ2+
1
ρ
∂i
∂ρ+
1
ρ2
∂2i
∂θ2
]. (6.16)
Using the definition ζ ≡ D∆τk−A(∆ρ)2 , Eqn. 6.16 corresponds to the following FTCS finite
difference equation,
i(ρx, θy, τn+1) = i(ρx, θy, τn)
[1− 2ζ − 2ζ
(x∆θ)2
]+
+ [i(ρx−1, θy, τn) + i(ρx+1, θy, τn)] (ζ +ζ
2x) +
+[i(ρx, θ(y−1), τn) + i(ρx, θ(y+1), τn)
] ζ
(x∆θ)2. (6.17)
valid for x = 1, 2, 3, ..., P−∆ρ∆ρ , y = 0, 1, 2, ..., 360◦−∆θ
∆θ , and n = 1, 2, 3, ...,τf∆τ . Notice that the
center of the disk must be treated carefully due to the 1ρ∂i∂ρ and 1
ρ2∂2i∂θ2 terms in Eqn. 6.16.
While ∂2i∂θ2 is identically zero, 1
ρ∂i∂ρ must be bound in the limit that ρ → 0. So long as any
mathematical source of i is bound throughout the disk of radius P , the following boundary
condition can be derived at the pole ρ = 0 using arguments found in Ref. [192],
∂i
∂ρ(0+, θ, τ) = − ∂i
∂ρ(0−, θ, τ) (6.18)
which corresponds to the following finite difference equation,
I(0, θ, τn+1) =1
2
[i(∆ρ, 0, τn+1) + i(∆ρ,
180◦
∆θ, τn+1)
](6.19)
In [186], a Neumann boundary condition for inhibitor i is cited in lieu of the boundary
condition in Eqn. 6.11. That condition is
D
P
∂i(m)
∂n= kis(m)− k−ii(m) (6.20)
152
where n represents the direction normal to the outer boundary (cell membrane). The
corresponding finite difference equation is
i(P, θy, τn+1) = i(P −∆ρ, θy, τn+1) +τ1ki
1 + τ1k−is(P, θy, τn+1) (6.21)
using the definition τ1 ≡ P∆ρ/D. The diffusion equation is solved above using polar
coordinates instead of Cartesian coordinates in order to simplify the Neumann boundary
condition. Assuming that diffusion of the inhibitor occurs along the membrane, one can
plug the Neumann boundary condition, Eqn. 6.21, into the diffusion reaction equation to
get an initial value problem with tangential diffusion at the boundary. Using this equation,
a source term for i can be defined, in addition to a continuity equation that accounts for
inhibitor flux at the boundary. This avenue of mathematical analysis of the model is not
pursued in this work since my derivation of the model could not recapitulate the Neumann
boundary condition in Eqn. 6.20.
1.3. Results.
1.3.1. Cartesian Coordinate Model. Solutions to linear algebraic equation 6.14 were it-
erated and 2nd order Runge Kutta was used to solve Eqns. 6.10-6.12 along the circular
boundary. The solutions for a, i, and r, solved using Runge-Kutta at FTCS are plotted
against time in Fig. 6.4. Here, a periodic signal, s(θ) = cos2( 2πθ720◦ ), is initially turned on at
τ = 0.5. Later, the signal is made homogeneous and increased in two steps at τ = 25 and
τ = 50. During the period signal, the response reaches a positive steady state along the
entire membrane except where these is no signal. As the signal increases further, perfect
adaptation is observed. The duration of the transient response before steady state varies
inversely with k−I/k−A (not shown here). Parameters used to generate this solution are
found in the figure caption.
1.3.2. Polar Coordinate Model. Solutions to linear algebraic equation 6.17 were iter-
ated. Along the cell membrane, ρx = P , the initial value problems for activator a(θ, τ)
153
time (1/k-A) time (1/k-A)
con
cen
tra
tio
n (
AU
)
inhibitor
an
gle
(θ
) /
de
ga
ng
le (θ
) /
de
ga
ng
le (θ
) /
de
g
activator
response
c
d
e
b
a
P
θ
90270
180
0
Pyq
xp
-P-P
-P
Figure 6.4. LEGI model results using Cartesian coordinate system.Modeling Dynamics for inputs of P = 10µm, kA = 10.0 s−1, k−A = 10.0 s−1,kI = 1.0 s−1, k−I = 1.0 s−1, kR = 1.0 s−1, k−R = 5.0 s−1, D = 10.0µm2 s−1,s(θ, τ < 0.5) = 0, s(θ, 0.5 ≤ τ < 25.) = cos2
(2πθ720◦
), s(θ, 25. ≤ τ < 50.) =
2, s(θ, 50. ≤ τ) = 2.5, and all initial conditions are zeroed. The integrationparameters are ∆x = 0.5µm and ∆τ = 0.01. (a) Schematic of the 3-D (2-space,1-time) model cell in discretized Cartesian coordinate. The coordinate systemshown here is left handed to be consistent with Python matrices (this should bechanged in any future work). (b) Time course of the concentrations of signal,activator, inhibitor, and response protein at ρ = P (membrane), and θ = 0◦.After an initial steady state response is established after the onset of signal(τ = 20−25), perfect adaptation is maintained. (c), (d), and (e) The activator,inhibitor, and response along the membrane are plotted, a(θ, τ), i(ρ = P, θ, τ)and r(θ, τ) respectively. Color represents concentration in arbitrary units.
and response r(θ, τ) (Eqns. 6.10 and 6.11) were solved using 2nd order Runge-Kutta and
Euler methods, respectively. Figure 6.5 shows a solution to the model in which a signal,
s(θ) = cos2( 2πθ720◦ ), is turned on at τ = 0.5 (dashed blue line). Following the onset of the
signal, the activator exponentially relaxes toward its steady state. The response variable
transiently increases before relaxing back to zero. While this is indicative of perfect adap-
tation, this adaptation is achieved by the means described in section 1.2.1. Instead, the
154
inhibitor dynamically blows up to infinity after the diffusive inhibitor reaches the origin,
and adaptation is achieved by inhibitor saturation (Fig. 6.5, b and d, and Movie 2). The
inhibitor saturation is the result of computationally solving the diffusion equation in polar
coordinates and not realistic. Interestingly though, inhibitor saturation is a perfectly fair
means of decoupling the response from the average signal concentration. Unfortunately, an
inhibitor saturated cell also loses the ability to detect the signal gradient.
In an attempt to “fix” the inhibitor instability at the origin (see Movie 2), the diffusion
operator was redefined at the origin two different ways. First, a no flux condition was
applied, ∂i∂ρ |ρ=0 = 0. Second, the treatment in Eqn. 6.19 was applied. Both treatments lead
to the same instability.2
1.4. Discussion. In this work, the LEGI model showed that adaptations to changes in
chemoattractant signal could be achieved by two mechanisms. First (Section 1.3.1), a finite
valued concentration of inhibitor competing against an activator led to perfect adaptation.
Second (Section 1.3.2), inhibitor saturation gave perfect adaptation because the response
was completely cut off from the signal. The latter mechanism was generated by accident
because of a computational pole instability that should be fixed by anyone continuing this
project.
Hidden in the way the Cartesian coordinate model was presented above, the diffusion
equation was not coupled to the boundary. While the inhibitor from the membrane was
allowed to diffuse into the bulk, bulk inhibitor did not interact with membrane inhibitor.
A more realistic model would couple the globally diffusing inhibitor to the membrane using
a no flux boundary condition,
∂i
∂n= −α(i− s). (6.22)
2An applied mathematician suggested the boundary condition ∂i∂t = 2D
[∂2i∂ρ2 + 1
ρ2∂2i∂θ2
]at the origin.
This may fix the r = 0 singularity.
155
b
con
cen
tra
tio
n (
AU
)
time (1/k-A)
inhibitor
time (1/k-A)
an
gle
, θ (
de
g)
an
gle
, θ (
de
g)
an
gle
, θ (
de
g)
activator
response
c
d
e
a
ρx
P
θy
90270
180
0
Figure 6.5. LEGI model results using polar coordinate system. Mod-eling Dynamics for inputs of P = 10µm, kA = 1.0 s−1, k−A = 10.0 s−1,kI = 1.0 s−1, k−I = 1.0 s−1, kR = 1.0 s−1, k−R = 1.0 s−1, l = 1.0µm,D = 1.0µm2 s−1, s(θ, τ < 0.5) = 0, s(θ, τ ≥ 0.5) = cos2
(2πθ720
), and all initial
conditions to zero. The integration parameters are ∆ρ = 0.5µm, ∆θ = 1.0 ◦,and ∆τ = 0.01. (a) Schematic of the 3-D (2-space, 1-time) model cell in dis-cretized polar coordinates. (b) Time course of the concentrations of signal,activator, inhibitor, and response protein at ρ = P (membrane), and θ = 25 ◦.Though the system of partial differential equations is well-posed, the system isunstable in i the way it is currently coded, after the signal is turned on, i blowsup to infinity. Nonetheless, the response has the expected form after the signalis turned on, initially increasing before relaxing back to zero. This is adaptationby inhibitor saturation. (c), (d), and (e) The activator, inhibitor, and responsealong the membrane are plotted, a(θ, τ), i(ρ = P, θ, τ) and r(θ, τ) respectively.Color represents concentration in arbitrary units. All color bars saturate at aconcentration of 1 AU. The dashed white lines correspond to the plots in (a).
Here the right hand side is the source, and n represents the direction normal to the boundary.
When the source at the boundary is zero, the inhibitor flux, u = −Dn · ∇i = 0, as desired.
Therefore, the Neumann boundary condition presented previously [186] is correct. Taken
together, the Cartesian coordinates model solved above (Fig. 6.4) is valid but the diffusion of
156
the inhibitor did not play a role in the response. In the polar coordinates model (Fig. 6.5),
inhibitor diffusion was properly coupled to the membrane but this model had a pesky
computational instability at the pole that needs to be fixed.
In the future, one should also consider the mechanics of the cell membrane. It has been
proposed [186] that membrane speed, vn, couple to the LEGI model as follows:
dvndt
= f(R)− γκ− λvn. (6.23)
Here, f(R) is some function of the response protein, R, described by Eqn. 6.11, γ is the
membrane bending rigidity, κ is the membrane curvature, and λ is a frictional coefficient
[193]. More generally, any number of terms can be added to Eqn. 6.23 to capture the physics
of the system (See Ref. [126] for examples).
Implementation of the LEGI model will be highly useful because it can lead to an
experimental design for testing the signaling mechanism underlying the recently observed
orientation transitions in chemotactic neutrophils. Furthermore, the adaptation time of the
response in this model may make it possible to mechanistically explain measurements of
directionality time [1, 185] (c.f. Chapter 2). One of the data sets we have collected involves
neutrophil chemotaxis on soft substrates with the inhibition of motility activator PI3K. The
LEGI model is ideally suited for modeling the findings made from this data set.
1.5. Pseudocode. The following pseudocode is intended to help a beginning to inter-
mediate computational physicist solve the LEGI model. Sample code written in Python is
also available online (see Chapter 7, Sec. 1 for a link to my github account).
• preamble, import
– math, numpy, scipy, matplotlib
– progressbar (so I can see how efficiently the program is running)
• initialize BVP object
– declare variables
157
– set default parameters
– set default initial conditions
– set boundary location equation (I set the boundary to the circumference of a
disk in Cartesian coordinates)
• Euler forward step and initial value problem iterations (i.e. Eqns. 6.14, 6.17,etc)
until time t reaches end-time tf
– diffusion equation solved using C++ code embedded into Python via the
scipy.weave package (originally coded in Python without C++ but the pro-
gram took way too long to complete)
– update order for each iteration
(1) update a by 2nd order Runge Kutta (RK)
(2) update i by FTCS everywhere
(3) update i on the boundary (membrane) by 2nd order RK (Linear update
in polar coordinate code)
(4) update r by 2nd order RK (Euler method update used in polar coor-
dinate mode)
(5) update τ = τ + ∆τ
• output a progress bar during iteration algorithms (C++ about 45-50 times faster
than straight Python)
• if necessary, convert i from polar coordinates to Cartesian coordinates and make
a colorplot movie of i(2-D space) vs. time
• save files
• other plots
158
2. Future Project 2: Neutrophils From Patients With Sepsis Show Novel
Stress Distributions During Chemotaxis
2.1. Introduction. Gram negative bacterial sepsis is caused in part by an overabun-
dance of lipopolysaccharide (LPS), an endotoxin found in the outer cell membrane of Gram
negative bacteria [194]. LPS activates several immune system pathways including the pro-
duction of proinflammatory cytokines that elicits neutrophil diapedesis (Fig. 6.6, lower left).
In this manner, high levels of LPS from a Gram negative bacterial infection can lead to a
systemic inflammatory response, known as sepsis. In this case, tissue damage and systemic
organ failure resulting from excessive neutrophil activity is often more harmful than the
initial infectious insult [195].
Several treatments focusing on the blockage of LPS induced extracellular cues that
mediate neutrophil activity have been clinically unsuccessful [195, 196], and the mortality
rate of patients with sepsis remains an issue [197, 198]. New treatments for sepsis are
necessary and the effect of sepsis on the neutrophil itself may elucidate new treatment
possibilities. It has been reported that neutrophil chemotaxis is impaired in patients with
sepsis, and that this impairment correlates to expected rate of survival. In the following
section, we analyze the migration paths of chemotactic human neutrophils from both healthy
donors and patients with sepsis. To better understand the differences in migration and
intracellular signaling between the two populations, the cellular traction forces were also
measured using traction force microscopy (All relevant Materials and Methods can be found
in Chapter 5, all substrates are 5 kPa in stiffness).
2.2. Results. In the previous chapter, we showed that the traction forces of chemoki-
netic neutrophils were localized at the uropod and oriented primarily perpendicular the
direction of motion during motility. This was no surprise because traction forces at the uro-
pod of the neutrophil perpendicular to the direction of motion had been reported previously
[50, 165]. With chemotactic neutrophils, we found that overtime from the initial onset of
159
lipopolysaccharide (LPS)neutrophil chemotaxis
to sources of infection
Figure 6.6. Neutrophil activation, endothelium transmigration, andchemotaxis through the extracellular matrix. Lipopolysaccharide in-duces proinflammatory cytokine production that leads to neutrophil activationand transmigration through the endothelium. Once in the extracellular matrix,neutrophils migrate by chemotaxis to sites of infection and injury. This figurewas adapted from Ref. [199].
a chemotactic signal, the magnitude of contractile force increased to a point where a force
orientation transitioned from a perpendicular mode to a parallel mode. An example of this
phenomenon is shown in the TFM time lapse depicted in Fig. 6.7 a (perpendicular force
mode denoted by ⊥, parallel force mode denoted by ‖). Here, the contractile force magni-
tude increased until it peaks near 25 nN, at which time the force orientation transitioned
from perpendicular to parallel (Fig. 6.7 b, blue data points). Directionality index, defined
as the average cosine between stress island forces and direction of motion, is also plotted
to quantify the force directionality.3 Directionality index transitioned from an intermediate
value between 0 and 1, to 1 at approximately 2 min after the initial onset of chemoattractant
signal (Fig. 6.7 b, green data points). The contractile energy is shown in Fig. 6.7 c, and a
schematic of the force orientation transition is shown in Fig. 6.7 d.
3In the future, this should be reanalyzed using dipole moment analysis instead of directionality indexto obtain an unbiased measure of the principal force orientation.
160
1 min 4 min 7 min 10 min
1 min 2 min 3 min 4 min
Healthy
Septic
{ {
a
b
c
e
f
g
i
h
d
0 100 200 300 400 500
traction (Pa)
cell outline
stress island and
force vectorforce
directionalityindex
en
erg
y
(nN
. µm
)fo
rce
(n
N)
dire
ctio
na
lity
ind
ex [γ]
time (min)0 2 4 6 8 10 12
0
5
0
20
40
60
0
1
en
erg
y
(nN
. µm
)fo
rce
(n
N)
dire
ctio
na
lity
ind
ex [γ]
time (min)0 2 4 6 8 10 12
0
5
0
20
40
60
0
1
isotropic
Figure 6.7. Differential Traction Force Mapping of representative neu-trophils from healthy (H) and septic (S) donors. Healthy donor: (a)Traction field time-lapse of a migrating healthy cell in 1 min increments. Whitelines indicate the cell outline determined from corresponding DIC images (notshown). Black dashed lines outline stress islands. Black vectors represent theforces corresponding to local traction islands. Orientation of island forces areperpendicular (⊥) or parallel (‖) relative to direction of motion. (caption con-tinued on pg. 162)
161
In chemotactic neutrophils from patients with sepsis, contractile forces again begin ori-
ented perpendicular to the direction of motion and slowly increased in magnitude. After an
initial period in the perpendicular mode, the force orientation transitioned to a mode de-
scribed as a cycling between the parallel mode and an additional isotropic force mode (6.7 e,
isotropic mode denoted �). Because neutrophils effectively produce zero net force (and
torque) on their substrate during migration [176] (c.f. Chapter 5, Sec.refsec:mechTFM),
measurements of the stress island force magnitude during periods of the isotropic mode are
not applicable (Fig. 6.7 f, grey shaded periods). Beyond the differential force mode, the
contractile energies of neutrophils from septic donors were significantly greater than the
contractile energies of neutrophils from healthy donors (Fig. 6.7, g compared to c). Neu-
trophils from septic donors also spread more, migrated slower, and honed less effectively
than neutrophils from healthy donors (Fig. 6.8).
2.3. Discussion. Combining the chemotaxis results shown here with the chemokinesis
results from Chapter 5, one can describe the cell mechanics of neutrophil motility as follows
(Fig. 6.9): First, cells apply contractile forces even when they are not motile. When motile
under chemokinetic conditions, the contractile forces are remarkably smaller compared to
cells that contract but are not motile. Motile neutrophils show forces localized at the uro-
pod as reported previously [48, 50, 165], and these forces are perpendicular to the direction
(Fig. 6.7 caption continued) (b) Average traction island force and directionality index plotted againsttime. After a period of low force output from 0 to 1 min, island forces increase in magnitude whilethe directionality of the traction force mode transitions from perpendicular to parallel after 2 min.(c) Energy exerted by a migrating neutrophil from a healthy donor. (d) Schematic showing theforce mode transitions from perpendicular to parallel direction. Septic Donor: (e) Traction fieldtime-lapse of a migrating neutrophil from a septic donor shown in 4 min increments. Markings areas in (a), with the addition of another force mode, isotropic (�). The isotropic mode correspondsto a single donut shaped traction island with negligible net force. (f) Forces oscillate and tend tozero during the isotropic mode. (g) The energy exerted by a migrating neutrophil from a septicdonor is greater than healthy cell. (h) Schematic showing the general force mode progression ofa neutrophil from a septic donor. Perpendicular force mode abates and the cell cycles betweenparallel and isotropic force exertion modes. (i) Legend including traction field color bar. Scale barsare 10µm. Similar findings were found from 11 neutrophils collected from 3 patients with sepsis,and 9 cells from 3 healthy donors.
162
are
a [
A]
(µm
2)
Spread Area
Migration Speed
Mean Square Displacement
rati
o [
x/y
]Aspect Ratio
Directionality Time
spe
ed
[Ι∆dΙ/∆
t]
(µm
/min
)ti
me
[t p
] (m
in)
xy
A
∆d
t
t+∆t
time (min)
MS
D
exp
on
en
t
0 5 100123
td
0
2
4
6
1
1.5
2
100
200
300
5
10
H S
H S
a
d
e
b
c
20 40 80 160
101
102
103
SH
squ
are
dis
pla
cem
en
t
[<d(t,∆t)2>
t] (µ
m2)
time increment [∆t] (s)
t
t+∆t
t+2∆t
Figure 6.8. Neutrophils fromhealthy (H) and septic (S)donors differ in morphol-ogy and migration speed,and honing. (a) S-neutrophilsspread more than H-neutrophils.(b) H-neutrophils trend towardsa greater range of aspect ra-tios than S-neutrophils. (c) and(d) S-neutrophils move slowerthan H-neutrophils. (e) Thedirectionality time was greaterfor S-neutrophils compared to H-neutrophils, indicating less hon-ing efficiency by S-neutrophils.n=11 neutrophils from 3 donorswith sepsis; 9 cells from 3 healthydonors.
of motion under chemokinetic conditions. Also reported previously, the spatial force profile
undergoes significant reorientation prior to sharp changes in direction [165]. Not reported
previously, dipole moment analysis quantitatively demonstrates that the principal force ori-
entation is perpendicular to the direction of motion during chemokinesis. Further, changes
163
in force orientation lag behind changes in motility direction during smooth turns, in con-
trast to changes in force that occur prior to sharp turns. We present arguments suggesting
that persistence is likely dependent on spread area and not mechanical output.
With neutrophils undergoing chemotaxis, the early spatio-orientational distribution of
the contractile forces resembles that of chemokinetic cells. Initially, forces are small and
perpendicular to the direction of motion. However, the similarities between chemokinetic
mechanical output and chemotactic mechanical output only last on the order of minutes.
Over time from the onset of the chemotactic gradient, contractile forces are observed to
increase and a transition from perpendicular to parallel force orientation occurs. Perhaps
the mechanical output of a chemokinetic cell would resemble the mechanical output of a
chemotactic cell if the chemokinetic cell was highly persistent and moved in one direction
long enough for the adhesion centers to mature. With neutrophils from septic donors, a
parallel force mode was also observed following an initial perpendicular force mode. But
neutrophils from septic donors also displayed an isotropic force mode characterized by rel-
atively large contractile energies. A significant difference in honing efficiency was observed
between neutrophils from septic and healthy donors. Based on our observation that parallel
force modes are associated with chemotaxis, but not chemokinesis, and we hypothesize that
honing, not persistence, is highly dependent on the strength, distribution, and orientation
of contractile forces.
3. Closing Remarks
Cell motility is required for many important processes in nature including inflammation,
wound healing, sexual reproduction, embryonic development, cancer metastasis, and more.
Determining the mechanisms that underlie cell motility is important for advancing the
frontier of medicine, but this is not a simple task. The cell is an extremely complex system
with cell motility being a spontaneous emergent behavior that is the culmination of many
stochastic molecular processes occurring within this system. Further complicating matters,
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force modes
changing direction
smooth turns sharp turnsnon motile
chemokinetic
(with and without BG)
chemotactic
(healthy)
chemotactic
(septic) { {cell outline
migration trajectory
stress island and force vector
force modes
Figure 6.9. Schematics summarizing all the force modes of motile andnon motile neutrophils. Symbols ∴, ⊥, ‖, and � indicate anomalous forceorientations (anomalous mode), orientations perpendicular to the direction ofmotion (perpendicular mode), orientations parallel to the direction of motion(parallel mode), and isotropic orientation (isotropic mode), respectively. Inchemokinesis, when a neutrophil makes a smooth turn, force reorientation lagsbehind changes in direction. In a sharp turn, force orientation is lost prior tothe sharp turn before being re-established.
motility also couples to extracellular cues. To gain a full understanding of the processes
underlying cell motility would require the tracking of millions of molecular reactions between
an incredibly large number of signaling proteins. The experimental apparatus to fully
elucidate a system as complex as the cell simply does not exist at this time.
On a whole, this dissertation is a combination of experimental analysis of cell motility
and the implementation of tools from classical physics and random walk theory to create
models that predict and/or characterize the motile behaviors of cells. The models we
165
presented were simple in nature, reducing the complex cell to a small set of elements that
capture the essential biomechanical and biochemical factors that affect cell motility.
Cellular motion has been routinely characterized by parameters such as speed, tor-
tuosity, and persistence time. However, most of these parameters are not reproducible
because their numerical values depend on technical parameters like sampling interval and
measurement error. In Part 1, Chapter 2 of this dissertation, we addressed the need for a
reproducible metric that did not depend on technical parameters by analytically deriving a
novel metric called directionality time. Directionality time was interpreted as the minimum
observation time required to identify motion as directionally biased, or similarly, the time
required for a cell to adapt to and hone towards an external cue. Measured based on fit-
ting mean squared displacement in log-log coordinates, we showed that the corresponding
fit function was approximately model invariant and applicable to a variety of directionally
biased motions, including processes that were nonergodic. Simulations were used to further
show the robustness of directionality time measurements and their decoupling from mea-
surement errors. Finally, we demonstrated, step-by-step, how to measure the directionality
time of noisy, nonergodic experimental data. Directionality time should have broad appli-
cability across many fields of inquiry as a robust metric for characterizing the motions of
single particles, cells, animals, and other motile systems.
In Chapter 3 of this dissertation, the focus then switched to using the directionality
time metric to characterize human neutrophil chemotaxis. Neutrophil chemotaxis is well
understood to occur in response to biochemical stimuli such as chemoattractants binding
cognate receptors and leading to directional motility. We showed the physical properties of
the underlying substrate also contribute significantly to the regulation of human neutrophil
chemotaxis by characterizing directed migration toward fMLP on fibrinogen (Fgn), type IV
collagen (Col-IV), and fibronectin (Fn)-coated gels of varying stiffness. Using directionality
time together with MSD to quantify migration speed, we found that mechanoregulation
166
of migration varied significantly by ligand-coating. We showed that neutrophils on the β1
integrin ligand Col-IV demonstrated decreasing migration speed but no change in direc-
tionality time as substrate stiffness increased. In contrast, neutrophils on the β2 integrin
ligand Fgn demonstrated no change in migration speed but decreased directionality time as
substrate stiffness increased. For neutrophils on Fn, recognized by both β1 and β2 integrins,
both speed and directionality time decreased with increasing substrate stiffness leading us
to hypothesize that β1 and β2 integrin engagement were necessary for the mechanosensitive
regulation of migration speed and honing efficiency, respectively. Further substantiating
our hypothesis, blocking β2 integrins of cells migrating on Fn altered migration dynamics,
with directionality time becoming statistically indistinguishable from those of untreated
cells migrating on the β1 integrin ligand Col-IV. These data demonstrate that individ-
ual components of the neutrophil chemotactic response are tunable with different ligand,
mechanotactic, and chemotactic cues and that this tuning is integrin dependent.
In Part 2 of this dissertation, we explored the role of cell mechanics in determining
the dynamics of motile cells. Crawling motile cells in particular exhibit a variety of cell
shape dynamics ranging from complex ruffling and bubbling to oscillatory protrusion and
retraction. For example, periodic shape changes during cell migration have been recorded
in fast moving fish epithelial keratocytes where sticking and slipping at opposite sides of the
cell’s broad trailing edge generate bipedal locomotion. In Chapter 4, we modeled crawling
cells using a 2D construction of elements representing linkages between the cytoskeleton and
the underlying substrate. The mechanical properties of the cell were modeled as follows:
First, the elements were connected with springs representing the actin cytoskeleton. Second,
a front element in the lamellipodial region was propelled in the direction of motion. Third,
stick slip adhesion, generating frictional forces opposing the direction of motion, was applied
to elements at the posterior of the cell. By benchmarking the cell shape dynamics of
four spring configurations, each representing a possible elastic configuration of the actin
167
cytoskeleton, against the cell shape dynamics of crawling keratocytes, our analysis showed
that elastic coupling to the cell nucleus was necessary to generate the observed motion.
Based on this finding, we select a configuration to study the effects of cell elasticity, size, and
aspect ratio on crawling dynamics. This configuration predicted that shape dynamics are
highly dependent on the lamellipodial elasticity, but less sensitive to elasticity at the trailing
edge. The model could be used to predicted a wide range of dynamics, from coherent bipedal
shape dynamics as seen in crawling keratocytes, to decoherent chaotic shape dynamics as
observed in amoeboid cells. In general, this model could also predict the traction force
applied by cells on their substrate. This work highlights how the dynamical behavior of
crawling cells can be derived from a mechanical properties through which biochemical factors
may operate to regulate cell motility.
Finally, in Chapter 5, we combined our tools for characterizing cell motility from Part 1,
to study the traction forces of both chemokinetic and chemotactic neutrophils. In addition
to seeking the effect of neutrophil traction forces as a function a substrate stiffness, we
sought to measure traction forces in the context of a biological response modifier to assess
the role of cell mechanics in pharmacological intervention, something that had not been
studied previously. β-glucan was used as a model biological response modifier, a substance
that has shown promise as a clinic grade therapeutic. We found that β-glucan modified the
mechanical output of human neutrophils on soft, 5 and 25 kPa 2D substrates, decreasing
both the contractile force and energy applied as the concentration of β-glucan was increased
over a clinically relevant range. Despite these changes in mechanical output, contractile
energy was conserved across substrate stiffness for all concentrations of β-glucan tested. We
also reported an overall inverse correlation between migration speed and mechanical output.
While there has been some speculation that a specific force range existed for facilitating
optimal cell migration, we concluded that optimal cell migration was not associated with
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a specific magnitude of contractile force in the 1-100 nN range. Instead, increased motility
was associated with diminished contraction.
The work presented in this dissertation underscores the synergistic connection between
physics and biology. Here, data from an otherwise biology oriented experimental project
studying cell motility was analyzed using a novel metric developed from random walk theory.
Meanwhile, the principals of fluid and continuum mechanics were merged to measure cellular
traction forces and determine the interdependencies between contractile forces and motility.
As we discuss in the Future Directions section, the link between physics and biology is
further strengthened in that the cellular traction forces of neutrophils from sepsis patients
may indicate new avenues of treatment for sepsis. Indeed, there are many interesting and
clinically relevant opportunities for physicists to apply their skill set to research projects in
the field of biology.
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CHAPTER 7
Extras
1. Cell Trajectory Simulator Class (MATLAB)
Part of Ph.D. training for a modeler and data analyst is learning code, but one quickly
discovers that being a good coder does not necessarily translate into being a good researcher.
Graduate students are not often motivated to write humanly understandable code, often
instead opting to use a “slap it together until it works” approach. As such, reading through
just about any graduate student’s computer code is a surefire way to induce excessive
cringing. To demonstrate what I hope is one exception to this rule, this appendix contains
publishable quality computer code of a MATLAB object class that when instantiated, gen-
erates simulated cell trajectories based on a 2-D persistent biased random walk. Sampling
error and sampling rate are free parameters that make the simulated data output mimic
experimental data. This code was instrumental for work on the directionality time model
and is generically useful for studying cell motility from the perspective of biased random
walk theory.
This modelling tool along with the LEGI model (c.f. Chapter 6), cell tracking, traction
force microscopy, and other modeling tools (all written in either MATLAB or Python) can
be found at, http://www.github.com/aloosley.
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classdef cellTrajSim < handle
% cellTrajSim: Simulates a biased random walk representative of something
% that moves randomly by nature and is biased by an external cue to move in
% a particular direction. An example is a cell undergoing chemotaxis.
%
% The biased random walker choose to change direction based on a Poissonian
% process with magnitude lambda [1/s]. When a reorientation decision is made,
% the orientation angle is drawn from a von Mises distribution with
% directionality factor, kappa. The random walk is sampled every tSample
% [s] with an error factor representing centroid measurement error [um].
% The direction of external bias is set by angMean, and can change using
% the angMeanChgRate [1/s] variable. The simulation time interval is tInt [s],
% and simulations continue until tEnd [s].
%
% Usage
% -----
% **default constructor**
% cellTrajSim(lambda, kappa, speedInit, speedChgRate, centMeasErr,...
% angMean, angMeanChgRate, tInt, tSample, tEnd)
%
% Arguments (input)
% ----------------
% lambda - the average frequency of deciding to change direction [1/s]
% kappa - directional factor (inverse width of von Mises Distribution)
% speed - instanteous speed during random walk [um/s]
% centMeasErr - centroid measurement error [um]
% angMean - mean bias angle
% angMeanChgRate - mean bias angle change rate [1/s]
% tInt - simulation time interval [s]
% tSample - sampling time interval [s]
% tEnd - end simulation time [s]
%
% Public Methods List
% -------------------
% simulateTraj(); % Determines simulation trajectory,
% % coordinates saved to xyArray, sampled
% % coordinates with centroid measurement error
% % are saved to xySampArray
% calcSqDisp(); % Calculates squared displacement from xySampArray
% fitTASDexp(fitTime); % fits TASD to exponential model, stopping fit
% % a fitTime [s]
% fitTASDbeta(fitTime); % fits TASD to beta model, stopping fit a fitTime [s]
% fitTASDbetaCME(fitTime); % fits TASD modified to acount for centroid
% % measurement error (CME),to beta model,
% % stopping fit a fitTime [s]
%
%
% Examples
% --------
% sim = cellTrajSim % Instantiates random walk simulation object with
% % default parameters
%
% sim = cellTrajSim(lambda, kappa, speedInit, speedChgRate, centMeasErr,...
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% angMean, angMeanChgRate, tInt, tSample, tEnd)
% sim.simulateTraj()
% % Instantiates random walk simulation object and simulates
% % trajectory with argument parameters
%
% Variables Returned
% ------------------
% All output variables are publicly accessable. For example, to access
% the raw simulated trajectory (2D x-y coordinates) use,
% simTraj sim.xyArray
%
% List of public object variables:
% tArray - N by 1 array of time coordinates, where N is the simulation
% length, given by tEnd/tInt
% xyArray - N by 2 array of x-y coordinates
% speedArray - N by 1 array of speed over time
% tSampArray - M by 1 array of sampled time coordinates, where M is
% tEnd/tSample
% xySampArray - M by 2 array of sampled coordinates. Noise has been
% added to each sampled coordinate using a radial Gaussian
% with a width given by argument centMeasErr, which
% corresponds to real world centroid measurement error
% angleArray - N by 1 array of the orientation of the cell at each timestep
% turnsMAde - N by 1 array tracking whether or not a decision to turn has
% been made at each timestep
%
%
% Author: Alex J Loosley
% e-mail address: [email protected]
% Release: 2
% Release date: 6/12/14
% Please acknowledge the usage of this code
properties(GetAccess = ’public’, SetAccess = ’private’)
% **Default Simulation Parameters**
lambda = 0.5; % [1/s] Average number of pseudopod events per second
speedInit = 0.2033; % [um/s] Speed at time = 0
speedChgRate = 0.00001 % [um/s] Linear acceleration over time
angMean = 0*(pi/180); % [rad] Overall direction [rad] (model invariant to choice)
angMeanChgRate = 0*(pi/180)/300; % [rad/s] Rate
kappa = 2; % Unitless factor, kappa, in Von Mises distribution
centMeasErr = 2; %[um]
% Simulation Discretization and Timing
tInt = 1; % time interval [s]
tSample = 10; % experimental sampling interval [s]
tEnd = 12*60; % [s]
% Tracking
turnsMade
tArray
tSampArray
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xyArray
xySampArray % trajectories sampled from simulation (i.e. every tSample = 10 seconds)
angleArray
speedArray % new as of 02/09/2014
% Tracking Event timing (Poisson Dist double checked)
evDur % Event Duration Timer
evDurOut % Formatted for output (zeros truncated off end)
% Dynamic Variables (time, spatial coordinates, walker orientation
% angle, speed)
t
x
y
angle
speed
% Indices
tIdx
tSampIdx
evIdx
% SD, TASD, MSD measures
% (SD = squared displacement, TA = time averaged)
sqDispTime
sqDisp
TAsqDisp
TAsqDisp_StE
sqDispExp
TAsqDispExp
% SD and TASD fitting and GOF
expFit
expFitGOF
expFitOutput
expFitRChi2
betaFit
betaFitGOF
betaFitOutput
betaFitRChi2
betaFit_dirTime
expFit_dirTime
expFit_asymp
expFit_asympOff
TASDexpFit
TASDexpFitGOF
TASDexpFitOutput
TASDexpFitRChi2
TASDbetaFit
TASDbetaFitGOF
TASDbetaFitOutput
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TASDbetaFitRChi2
TASDbetaFit_dirTime
TASDexpFit_dirTime
TASDexpFit_asymp
TASDexpFit_asympOff
TASDbetaFitCME
TASDbetaFitCMEGOF
TASDbetaFitCMEOutput
TASDbetaFitCMERChi2
TASDbetaFitCME_dirTime
% Other
tempAngles % see comment below - work around for Von Mises distribution error
end
methods
% Class Constructor
function self = cellTrajSim(lambda, kappa, speedInit, speedChgRate,...
centMeasErr, angMean, angMeanChgRate, tInt, tSample, tEnd)
if nargin > 0
self.lambda = lambda;
self.kappa = kappa;
self.speedInit = speedInit;
self.speedChgRate = speedChgRate;
self.centMeasErr = centMeasErr;
self.angMean = angMean;
self.angMeanChgRate = angMeanChgRate;
self.tInt = tInt;
self.tSample = tSample;
self.tEnd = tEnd;
end
self.turnsMade = zeros(round(self.tEnd/ self.tInt)+1,1);
self.tArray = zeros(round(self.tEnd/ self.tInt)+1,1);
self.xyArray = zeros(round(self.tEnd/ self.tInt)+1,2);
self.angleArray = zeros(round(self.tEnd/ self.tInt)+1,1);
self.xySampArray = zeros(round(self.tEnd/ self.tSample),2);
self.speedArray = zeros(round(self.tEnd/ self.tSample),1);
end
end
methods(Access = private)
% Sets ICs to the origin of space and time (initial angle)
function initialize(self)
self.t = 0;
self.x = 0;
self.y = 0;
self.angle = self.angMean;
self.speed = self.speedInit;
self.tArray(1) = self.t;
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self.xyArray(1,1) = self.x;
self.xyArray(1,2) = self.y;
self.angleArray(1) = self.angle;
self.speedArray(1) = self.speed;
self.tIdx = 1;
self.tSampIdx = 1;
self.evIdx = 1;
end
function addGaussNoiseXY(self)
self.xySampArray(self.tSampIdx,1) = self.x + normrnd(0,self.centMeasErr);
self.xySampArray(self.tSampIdx,2) = self.y + normrnd(0,self.centMeasErr);
self.tSampIdx = self.tSampIdx + 1;
end
% trajectory Step based on a Poissonian process
function trajStep(self)
r = rand(1,1);
if r <= self.lambda*self.tInt % CHANGE DIRECTION
self.turnsMade(self.tIdx) = 1; % Notes that a turn was made
self.evDur(self.evIdx) = self.evDur(self.evIdx) + self.tInt;
% The above line: Minimum event time is dt
self.evIdx = self.evIdx + 1; % Initializes next event
self.angMean = atan2(-self.y, 200-self.x);
self.tempAngles = vmrand(self.angMean + self.angMeanChgRate*self.t,...
self.kappa,2,1);
self.angle = self.tempAngles(1);
% Above line: I requested 2 random numbers at a time because
% vmrand errors from time to time when only one is requested
% (cause of error currently unknown)
self.speed = random(’exp’, self.speedChgRate*self.t + self.speedInit);
% check angle limit - not necessary using Von Mises distribution,
% "vmrand", uncomment for a pseudowrapped normal distribution.
%{
chk = 1;
while chk == 1
if abs(self.angle)> pi
tempAngles = normrnd(self.angMean + self.angMeanChgRate*self.t, self.kappa);
self.angle = tempAngles(1); % I requested 2 random numbers at a time because vmrand errors from time to time when only one is requested (cause of error unknown)
else
chk = 0;
end
end
%}
% self.angle = mod(self.angle+pi,2*pi)-pi; % sets angle between -pi and pi
self.x = self.x + self.speed*cos(self.angle)*self.tInt;
self.y = self.y + self.speed*sin(self.angle)*self.tInt;
self.xyArray(self.tIdx+1,1) = self.x;
self.xyArray(self.tIdx+1,2) = self.y;
self.angleArray(self.tIdx+1) = self.angle;
self.speedArray(self.tIdx+1) = self.speed;
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else % Otherwise DON’T CHANGE DIRECTION (or speed)
self.evDur(self.evIdx) = self.evDur(self.evIdx) + self.tInt;
% The above line updates event interval duration timer
self.x = self.x + self.speed*cos(self.angle)*self.tInt;
self.y = self.y + self.speed*sin(self.angle)*self.tInt;
self.xyArray(self.tIdx+1,1) = self.x;
self.xyArray(self.tIdx+1,2) = self.y;
self.angleArray(self.tIdx+1) = self.angle;
self.speedArray(self.tIdx+1) = self.speed;
end
% Update time counters
self.t = self.t + self.tInt;
self.tArray(self.tIdx + 1) = self.t;
self.tIdx = self.tIdx + 1;
end
end
methods
function simulateTraj(self)
self.initialize()
self.evDur = zeros(round(self.tEnd/self.tInt),1);
while self.t < self.tEnd
if mod(round(self.t/self.tInt),round(self.tSample/self.tInt)) == 0
self.addGaussNoiseXY() % If the above is true, this samples
% the trajectory point and adds Gaussian noise
end
self.trajStep()
end
self.evDurOut = self.evDur(1:find(self.evDur==0,1)-1);
self.tSampArray = 0:self.tSample:self.tEnd;
end
% Square displacement (SD, TASD) and square displacement
% exponent calculations
function calcSqDisp(self)
nSamp = size(self.xySampArray,1);
self.sqDispTime = (self.tSample : self.tSample : (nSamp-1)*self.tSample)’;
% Square Displacement, reiterate simulation to get
% ensemble average
self.sqDisp = NaN(nSamp-1,1);
for k_int = 1:nSamp - 1
xd = self.xySampArray(1+k_int,1)-self.xySampArray(1,1);
yd = self.xySampArray(1+k_int,2)-self.xySampArray(1,2);
self.sqDisp(k_int) = xd^2 + yd^2;
end
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% Time Averaged Square Displacement
self.TAsqDisp = NaN(nSamp-1,1);
self.TAsqDisp_StE = NaN(nSamp-1,1);
for k_int = 1:nSamp-1
rSq = NaN(nSamp-k_int,1);
for k =1:nSamp-k_int
xd = self.xySampArray(k+k_int,1)-self.xySampArray(k,1);
yd = self.xySampArray(k+k_int,2)-self.xySampArray(k,2);
rSq(k)=xd^2+yd^2;
end
self.TAsqDisp(k_int)=nanmean(rSq);
self.TAsqDisp_StE(k_int)=nanste(rSq);
end
% meanSqDisp Exponent calculated by taking the forward
% difference derivative of MSD vs time in loglog space
self.sqDispExp = NaN(size(self.sqDisp,1)-1,1);
self.TAsqDispExp = NaN(size(self.TAsqDisp,1)-1,1);
for kk = 1:size(self.sqDisp,1)-1
self.sqDispExp(kk) = log(self.sqDisp(kk+1)/self.sqDisp(kk)) / log((kk+1)/kk);
self.TAsqDispExp(kk) = log(self.TAsqDisp(kk+1)/self.TAsqDisp(kk)) / log((kk+1)/kk);
end
end
% Fits the simulated MSD using data points up to time tEndFit (seconds)
function fitTASDexp(self,tEndFit)
model = fittype(’a-b*exp(-time/dirTime)’,’independent’,’time’,...
’coefficients’,{’a’,’b’,’dirTime’});
Value = ’NonlinearLeastSquares’;
opts = fitoptions(’method’, Value);
opts.StartPoint=[2,1,20];
opts.Upper = [2.5,2.5,tEndFit];
opts.Lower = [1.2,0.1,1 ];
[self.TASDexpFit, self.TASDexpFitGOF, self.TASDexpFitOutput] =...
fit(self.sqDispTime(1:round(tEndFit/self.tSample)), ....
self.TAsqDispExp(1:round(tEndFit/self.tSample)), model, opts);
self.TASDexpFit_dirTime = self.TASDexpFit.dirTime;
self.TASDexpFit_asymp = self.TASDexpFit.a;
self.TASDexpFit_asympOff = self.TASDexpFit.b;
self.TASDexpFitRChi2 = self.TASDexpFitGOF.rmse^2;
end
function fitTASDbeta(self,tEndFit)
model = fittype(’(1+2*(time/dirTime))/(1+(time/dirTime))’,...
’independent’,’time’,’coefficients’,{’dirTime’});
Value = ’NonlinearLeastSquares’;
opts = fitoptions(’method’, Value);
opts.StartPoint= [20];
opts.Upper = [tEndFit];
opts.Lower = [1 ];
[self.TASDbetaFit, self.TASDbetaFitGOF, self.TASDbetaFitOutput] =...
fit(self.sqDispTime(1:round(tEndFit/self.tSample)), ...
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self.TAsqDispExp(1:round(tEndFit/self.tSample)), model, opts);
self.TASDbetaFit_dirTime = self.TASDbetaFit.dirTime;
self.TASDbetaFitRChi2 = self.TASDbetaFitGOF.rmse^2;
end
function fitTASDbetaCME(self,tEndFit)
model = fittype(’(1+2*(time/dirTime))/(1+(time/dirTime))’,...
’independent’,’time’,’coefficients’,{’dirTime’});
Value = ’NonlinearLeastSquares’;
opts = fitoptions(’method’, Value);
opts.StartPoint= [20];
opts.Upper = [tEndFit];
opts.Lower = [1 ];
[self.TASDbetaFitCME, self.TASDbetaFitCMEGOF, self.TASDbetaFitCMEOutput] =...
fit(self.sqDispTime(1:round(tEndFit/self.tSample)),...
self.TAsqDispExp(1:round(tEndFit/self.tSample)).*...
(self.TAsqDisp(1:round(tEndFit/self.tSample))./...
(self.TAsqDisp(1:round(tEndFit/self.tSample))-self.centMeasErr^2)),...
model, opts);
self.TASDbetaFitCME_dirTime = self.TASDbetaFitCME.dirTime;
self.TASDbetaFitCMERChi2 = self.TASDbetaFitCMEGOF.rmse^2;
end
function fitExpIn(self,tIn,yIn,tEndFit)
model = fittype(’a-b*exp(-time/dirTime)’,’independent’,’time’,...
’coefficients’,{’a’,’b’,’dirTime’});
Value = ’NonlinearLeastSquares’;
opts = fitoptions(’method’, Value);
opts.StartPoint=[2,1,20];
opts.Upper = [2.5,2.5,tEndFit];
opts.Lower = [1.2,0.1,1 ];
[self.expFit, self.expFitGOF, self.expFitOutput] =...
fit(tIn(1:round(tEndFit/self.tSample)), yIn(1:round(tEndFit/self.tSample)),...
model, opts);
self.expFit_dirTime = self.expFit.dirTime;
self.expFit_asymp = self.expFit.a;
self.expFit_asympOff = self.expFit.b;
self.expFitRChi2 = self.expFitGOF.rmse^2;
end
function fitBetaIn(self,tIn,yIn,tEndFit)
model = fittype(’(1+2*(time/dirTime))/(1+(time/dirTime))’,...
’independent’,’time’,’coefficients’,{’dirTime’});
Value = ’NonlinearLeastSquares’;
opts = fitoptions(’method’, Value);
opts.StartPoint=[20];
opts.Upper = [tEndFit];
opts.Lower = [1 ];
[self.betaFit, self.betaFitGOF, self.betaFitOutput] =...
fit(tIn(1:round(tEndFit/self.tSample)), yIn(1:round(tEndFit/self.tSample)),...
model, opts);
178
self.betaFit_dirTime = self.betaFit.dirTime;
self.betaFitRChi2 = self.betaFitGOF.rmse^2;
end
end
end
2. How to Save $150 On A New Mercury Lamp
Earlier this year, our fluorescent mercury arc lamp began flickering at the far too
young age of 1144 hours. After a long back and forth with the manufacturer about
honoring the 1200 hour warranty, a stalemate ensued because the warranty was only
valid if the bulb failed to ignite completely. Regardless of the warranty conditions, a
flickering arc lamp greatly impaired our ability to do experiments so it was time to
bring out the big guns (Fig. 7.1), an intensity kymograph that led to a well deserved
$165 discount on a new bulb. Small victories.
time length along yellow line in
KymographBright
frames 58 & 59
frames 37 & 38
Dark
NOTESa) There �uctuations a persistentb) These �uctuations began happening like this sometime around Dec 10th, there was no smooth transition from proper lamp output to this behaviourc) A di�erent lamp (same model) with 800hrs instead of 1144 hrs works perfectly in the same unit, everything else equald) This is one movie, of manye) In all cases, lamp has been left on >15 min after ignitef ) Fluctuations are not at the frequency of AC �uctuations, they last 10-30sg) Intensity drops are 10-20% (see arrows above right of Kymograph)h) Fluctuations clearly visible through multiple cameras and by eye through microscope eyepiece
Intensity Fluctuation
10µm
2min
Figure 7.1. The $165 kymograph. Still frames (left) and corresponding pixelintensity kymograph (right) showing flickering of our too young to die fluorescentbulb. Arrows along the y-axis of the kymograph correspond to decreased bulboutput.
179
Bibliography
[1] X. M. O’Brien, A. J. Loosley, K. E. Oakley, J. X. Tang, and J. S. Reichner,
“Technical advance: Introducing a novel metric, directionality time, to quantify
human neutrophil chemotaxis as a function of matrix composition and stiffness,”
J Leukocyte Biol, vol. 95, pp. 993–1004, 2014.
[2] A. J. Loosley and J. X. Tang, “Stick-slip motion and elastic coupling in crawling
cells,” Phys Rev E, vol. 86, p. 031908, 2012.
[3] M. Amercrombie, “The croonian lecture, 1978: The crawling movement of meta-
zoan cells,” Proc Roy Soc London Ser B, vol. 207, pp. 129–147, 1980.
[4] G. Li and J. X. Tang, “Accumulation of microswimmwers near a surface me-
diated by collision and rotational brownian motion,” Phys Rev Lett, vol. 103,
p. 078101, 2009.
[5] T. Oliver, M. Dembo, and K. Jacobson, “Traction forces in locomoting cells,”
Cell Motil Cytoskel, vol. 31, pp. 225–240, 1995.
[6] T. J. Mitchison and L. P. Cramer, “Actin-based cell motility and cell locomo-
tion,” Cell, vol. 84, pp. 371–379, 1996.
[7] T. D. Pollard and G. G. Borisy, “Cellular motility driven by assembly and
disassembly of actin filaments,” Cell, vol. 112, pp. 453–465, 2003.
[8] C. Nathan, “Neutrophils and immunity: challenges and opportunities,” Nat Rev
Immunol, vol. 6, pp. 173–182, 2006.
[9] P. Martin, “Wound healing – aiming for perfect skin regeneration,” Science,
vol. 276, pp. 75–81, 1997.
[10] B. Eckes, R. Nischt, and T. Krieg, “Cell-matrix interactions in dermal repair
and scarring,” Fibro Tissue Repair, vol. 3, p. 4, 2010.
180
[11] P. Martin and J. Lewis, “Actin cables and epidermal movement in embryonic
wound healing,” Nature, vol. 360, pp. 179–183, 1992.
[12] X. Trepat, M. R. Wasserman, T. E. Angelini, E. Millet, D. A. Weitz, J. P.
Butler, and J. J. Fredberg, “Physical forces during collective cell migration,”
Nat Phys, vol. 5, pp. 426–430, 2009.
[13] X. Serra-Picamal, V. Conte, R. Vincent, E. Anon, D. T. Tambe, E. Bazellieres,
J. P. Butler, J. J. Fredberg, and X. Trepat, “Mechanical waves during tissue
expansion,” Nat Phys, vol. 8, pp. 628–634, 2012.
[14] G. A. Dunn and G. E. Jones, “Cell motility under the microscope: Vorsprung
durch technik,” Nat Rev Mol Cell Bio, vol. 5, pp. 667–672, 2004.
[15] O. Shimomura, F. H. Johnson, and Y. Saiga, “Extraction, purification and prop-
erties of aequorin, a bioluminescent protein from the luminous hydromedusan,
aequorea,” J Cell Compar Physl, vol. 59, pp. 223–239, 1962.
[16] V. Lecaudey and D. Gilmour, “Organizing moving groups during morphogene-
sis,” Curr Opin Cell Biol, vol. 18, pp. 102–107, 2006.
[17] B. Akiyoshi, K. K. Sarangapani, A. F. Powers, C. R. Nelson, S. L. Reichow,
H. Arellano-Santoyo, T. Gonen, J. A. Ranish, C. L. Asbury, and S. Big-
gins, “Tension directly stabilizes reconstituted kinetochore-microtubule attach-
ments,” Nature, vol. 468, pp. 576–579, 2010.
[18] F. Julicher, K. Kruse, J. Prost, and J.-F. Joanny, “Active behavior of the cy-
toskeleton,” Phys Rep, vol. 449, pp. 3–28, 2007.
[19] F. Gittes, B. Mickey, J. Nettleton, and J. Howard, “Flexural rigidity of micro-
tubules and actin filaments measured from thermal fluctuations in shape,” J
Cell Biol, vol. 120, pp. 923–934, 1993.
[20] J. Lee, A. Ishihara, J. A. Theriot, and K. Jacobson, “Principles of locomotion
for simple-shaped cells,” Nature, vol. 362, pp. 167–171, 1993.
[21] K. Keren, Z. Pincus, G. M. Allen, E. L. Barnhart, G. Marriott, A. Mogilner,
and J. A. Theriot, “Mechanism of shape determination in motile cell,” Nature,
vol. 453, pp. 475–480, 2008.
181
[22] J. L. Ross, M. Y. Ali, and D. M. Warshaw, “Cargo transport: Molecular motors
navigate a complex cytoskeleton,” Curr Opin Cell Biol, vol. 20, pp. 41–47, 2008.
[23] J. R. Bamburg, H. E. Harris, and A. G. Weeds, “Partial purification and char-
acterization of an actin depolymerizing factor from brain,” FEBS Lett, vol. 121,
pp. 178–182, 1980.
[24] M. V. Troys, L. Huyck, S. Leyman, S. Dhaese, J. Vandekerkhove, and C. Ampe,
“Ins and outs of adf/cofilin activity and regulation,” Eur J Cell Biol, vol. 87,
pp. 649–667, 2008.
[25] A. Huttenlocher and A. R. Horwitz, “Integrins in cell migration,” Cold Spr
Harbor Perspec Biol, vol. 3, p. a005074, 2011.
[26] E. Puklin-Faucher and M. P. Sheetz, “The mechanical integrin cycle,” J Cell
Sci, vol. 122, pp. 179–186, 2009.
[27] M. L. Gardel, B. Sabass, L. Ji, G. Danuser, U. S. Schwarz, and C. M. Waterman,
“Traction stress in focal adhesions correlates biphasically with actin retrograde
flow speed,” J Cell Biol, vol. 183, pp. 999–1005, 2008.
[28] Y. Li, P. Bhimalapuram, and A. R. Dinner, “Model for how retrograde actin flow
regulates adhesion traction stresses,” J Phys-Condens Mat, vol. 22, p. 194113,
2010.
[29] S. W. Moore, P. Roca-Cusachs, and M. P. Sheetz, “Stretchy proteins on stretchy
substrates: The important elements of integrin-mediated rigidity sensing,” Dev
Cell, vol. 19, pp. 194–206, 2010.
[30] A. K. Harris, P. Wild, and D. Stopak, “Silicone rubber substrata: a new wrinkle
in the study of cell locomotion,” Science, vol. 208, pp. 177–179, 1980.
[31] J. Lee, M. Leonard, T. Oliver, A. Ishihara, and K. Jacobson, “Traction forces
generated by locomoting keratocytes,” J Cell Biol, vol. 127, pp. 1957–1963,
1994.
[32] M. Dembo and Y. L. Wang, “Stresses at the cell-to-substrate interface during
locomotion of fibroblasts,” Biophys J, vol. 76, pp. 2307–2316, 1999.
182
[33] J. P. Butler, I. M. Tolic-Nørrelykke, B. Fabry, and J. J. Fredberg, “Traction
fields, moments, and strain energy that cells exert on their surroundings,” Am
J Physiol Cell Physiol, vol. 282, pp. C595–C605, 2002.
[34] B. Sabass, M. L. Gardel, C. M. Waterman, and U. S. Schwarz, “High resolution
traction force microscopy based on experimental and computational advances,”
Biophys J, vol. 94, pp. 207–, 2008.
[35] J. Huang, X. Peng, L. Qin, T. Zhu, C. Xiong, Y. Zhang, and J. Fang, “Determi-
nation of cellular tractions on elastic substrate based on an integral boussinesq
solution,” J Biomech Eng, vol. 131, p. 061009, 2009.
[36] J. Toyjanova, E. Bar-Kochba, C. Lopez-Fagundo, J. Reichner, D. Hoffman-
Kim, and C. Franck, “High resolution, large deformation 3d traction force mi-
croscopy,” PLoS One, vol. 9, p. e90976, 2014.
[37] S. A. Maskarineca, C. Franck, D. A. Tirrell, and G. Ravichandran, “Quantifying
cellular traction forces in three dimensions,” Proc Natl Acad Sci USA, vol. 106,
pp. 22108–22113, 2009.
[38] J. C. del Alamo, R. Meili, B. Alvarez-Gonzalez, B. Alonso-Latorre, E. Bastou-
nis, R. Firtel, and J. C. Lasheras, “Three-dimensional quantification of cellular
traction forces and mechanosensing of thin substrata by fourier traction force
microscopy,” PLoS One, vol. 8, p. e69850, 2013.
[39] C. Lopez-Fagundo, E. Bar-Kochba, L. Livi, D. Hoffman-Kim, and C. Franck,
“Traction forces of schwann cells on compliant substrates,” J R Soc Interface,
vol. accepted, 2014.
[40] E. Gutierrez, E. Tkachenko, A. Besser, P. Sundd, K. Ley, G. Danuser, M. H.
Ginsberg, and A. Groisman, “High refractive index silicone gels for simultaneous
total internal reflection fluorescence and traction force microscopy of adherent
cells,” PLoS One, vol. 6, p. e23807, 2011.
[41] K. M. Stroka and H. Aranda-Espinoza, “Neutrophils display biphasic relation-
ship between migration and substrate stiffness,” Cell Motil Cytoskeleton, vol. 66,
pp. 328–341, 2009.
183
[42] B. L. Bangasser, S. S. Rosenfeld, and D. J. Odde, “Determinants of maximal
force transmission in a motor-clutch model of cell traction in a compliant mi-
croenvironment,” Biophys J, vol. 105, pp. 581–592, 2013.
[43] C.-M. Lo, H. Wang, M. Dembo, and Y. li Wang, “Cell movement is guided by
the rigidity of the substrate,” Biophys J, vol. 79, pp. 144–152, 2000.
[44] T. Yeung, P. C. Georges, L. A. Flanagan, B. Marg, M. Ortiz, M. Funaki, N. Za-
hir, W. Ming, V. Weaver, and P. A. Janmey, “Effects of substrate stiffness
on cell morphology, cytoskeletal structure, and adhesion,” Cell Motil Cytoskel,
vol. 60, pp. 24–34, 2005.
[45] A. J. Engler, S. Sen, H. L. Sweeney, and D. E. Discher, “Matrix elasticity directs
stem cell lineage specification,” Cell, vol. 126, pp. 677–689, 2006.
[46] V. Vogel and M. P. Sheetz, “Local force and geometry sensing regulate cell
functions,” Nat Rev Mol Cell Bio, vol. 7, pp. 265–275, 2006.
[47] M. Ghibaudo, A. Saez, L. Trichet, A. Xayaphoummine, J. Browaeys, P. Sil-
berzan, A. Buguin, and B. Ladoux, “Traction forces and rigidity sensing regu-
late cell functions,” Soft Matter, vol. 4, pp. 1836–1843, 2008.
[48] P. W. Oakes, D. C. Patel, N. A. Morin, D. P. Zitterbart, B. Fabry, J. S. Reichner,
and J. X. Tang, “Neutrophil morphology and migration are affected by substrate
elasticity,” Blood, vol. 114, pp. 1387–1395, 2009.
[49] R. A. Jannat, G. P. Robbins, B. G. Ricart, M. Dembo, and D. A. Hammer,
“Neutrophil adhesion and chemotaxis depend on substrate mechanics,” J Phys-
Condens Mat, vol. 22, p. 194117, 2010.
[50] R. A. Jannat, M. Dembo, and D. A. Hammer, “Traction forces of neutrophils
migrating on compliant substrates,” Biophys J, vol. 101, pp. 575–584, 2011.
[51] P. Marcq, N. Yoshinaga, and J. Prost, “Rigidity sensing explained by active
matter theory,” Biophys J, vol. 101, pp. L33–L35, 2011.
[52] S.-Y. Tee, J. Fu, C. S. Chen, and P. A. Janmey, “Cell shape and substrate
rigidity both regulate cell stiffness,” Biophys J, vol. 100, no. 5, pp. L25–L27,
2011.
184
[53] S. Ghassemi, G. Meacci, S. Liu, A. A. Gondarenko, A. Mathur, P. Roca-
Cusachs, M. P. Sheetz, and J. Hone, “Cells test substrate rigidity by local
contractions on submicrometer pillars,” Proc Natl Acad Sci USA, vol. doi,
p. 1119886109, 2012.
[54] A. R. Houk, A. Jilkine, C. O. Mejean, R. Boltyanskiy, E. R. Dufresne, S. B.
Angenent, S. J. Altschuler, L. F. Wu, and O. D. Weiner, “Membrane tension
maintains cell polarity by confining signals to the leading edge during neutrophil
migration,” Cell, vol. 148, no. 1, pp. 175–188, 2012.
[55] P. W. Oakes, Y. Backham, J. Stricker, and M. L. Gardel, “Tension is required
but not sufficient for focal adhesion maturation without a stress fiber template,”
J Cell Biol, vol. doi, p. 10.1083/jcb.201107042, 2012.
[56] G. A. Dunn, “Characterising a kinesis response: time averaged measures of cell
speed and directional persistence,” Agents Actions Suppl, vol. 12, pp. 14–33,
1983.
[57] R. T. Tranquillo, D. A. Lauffenburger, and S. H. Zigmond, “A stochastic model
for leukocyte random motility and chemotaxis based on receptor binding fluc-
tuations,” J Cell Biol, vol. 106, pp. 303–309, 1988.
[58] C. L. Stokes, D. A. Lauffenburger, and S. K. Williams, “Migration of individual
microvessel endothelial cells: Stochastic model and parameter measurement,” J
Cell Sci, vol. 99, pp. 419–430, 1991.
[59] M. R. Parkhurst and W. M. Saltzman, “Quantification of human neutrophil
motility in three-dimensional collagen gels. effect of collagen concentration,”
Biophys J, vol. 61, pp. 306–315, 1992.
[60] E. A. Codling, M. J. Plank, and S. Benhamou, “Random walk models in biol-
ogy,” J R Soc Interface, vol. 5, pp. 813–834, 2008.
[61] C. A. Reinhart-King, M. Dembo, and D. A. Hammer, “Cell-cell mechanical
communication through compliant substrates,” Biophys J, vol. 95, pp. 6044–
6051, 2008.
185
[62] P.-H. Wu, A. Giri, S. X. Sun, and D. Wirtz, “Three-dimensional cell migration
does not follow a random walk,” Proc Natl Acad Sci USA, vol. 111, pp. 3949–
3954, 2014.
[63] T. R. Mempel, S. E. Henrickson, and U. H. von Andrian, “T-cell priming by
dendriticcells in lymph nodes occurs in three distinct phases,” Nature, vol. 427,
pp. 154–159, 2004.
[64] G. Shakhar, R. L. Lindquist, D. Skokos, D. Dudziak, J. H. Huang, M. C. Nussen-
zweig, and M. L. Dustin, “Stable t cell-dendritic cell interactions precede the
development of both tolerance and immunity in vivo,” Nat Immunol, vol. 6,
pp. 707–714, 2005.
[65] R. G. A. Gautreau, “Quantitative and unbiased analysis of directional persis-
tence in cell migration,” Nat Protoc, vol. 9, p. 19311943, 2014.
[66] E. Batschelet, Circular statistics in biology. London, UK: Academic Press Inc,
1981.
[67] T. Nattermann and L.-H. Tang, “Kinetic surface roughening. i. the kardar-
parisi-zhang equation in the weak-coupling regime,” Phys Rev A, vol. 45,
pp. 7156–7161, 1992.
[68] N. Gal, D. Lechtman-goldstein, and D. Weihs, “Particle tracking in living cells:
a review of the mean square displacement method and beyond,” Rheol Acta,
pp. 1–19, 2013.
[69] M. Gawronski, J. T. Park, A. S. Magee, and H. Conrad, “Microfibrillar structure
of pgg-glucan in aqueous solution as triple-helix aggregates by small angle x-ray
scattering.,” Biopolymers, vol. 50, pp. 569–578, 1999.
[70] B. W. LeBlanc, J. E. Albina, and J. S. Reichner, “The effect of pgg-β-glucan on
neutrophil chemotaxis in vivo.,” J Leukocyte Biol, vol. 79, pp. 667–675, 2006.
[71] B. Li, D. J. Allendorf, R. Hansen, J. Marroquin, C. Ding, D. E. Cramer, and
J. Yan, “Yeast β-glucan amplifies phagocyte killing of ic3b-opsonized tumor
cells via complement receptor 3-syk-phosphatidylinositol 3-kinase pathway,” J
Immunol, vol. 177, pp. 1661–1669, 2006.
186
[72] F. Hong, J. Yan, J. T. Baran, D. J. Allendorf, R. D. Hansen, G. R. Ostroff,
P. X. Xing, N. K. Cheung, and G. D. Ross, “Mechanism by which orally admin-
istered β-1,3-glucans enhance the tumoricidal activity of antitumor monoclonal
antibodies in murine tumor models,” J Immunol, vol. 173, pp. 797–806, 2004.
[73] C. Qi, Y. Cai, L. Gunn, C. Ding, B. Li, G. Kloecker, K. Qian, J. Vasilakos,
S. Saijo, Y. Iwakura, J. R. Yannelli, and J. Yan, “Differential pathways regulat-
ing innate and adaptive antitumor immune responses by particulate and soluble
yeast-derived β-glucans,” Blood, vol. 117, pp. 6825–6836, 2011.
[74] Biothera; Biothera. Study of Imprime PGGr in Combination With Cetuximab
in Subjects With Recurrent or Progressive KRAS Wild Type Colorectal Cancer
(PRIMUS). In: ClinicalTrials.gov [Internet]. Bethesda (MD): National Library
of Medicine (US). 2000-2014. Available from: http://clinicaltrials.gov/
show/NCT01309126 NLM Identifier: NCT01309126. Several of these clinical trial
records can be found by searching the ClinicalTrials.gov database for “imprime
PGG”.
[75] M. B. Harler, E. Wakshull, E. J. Filardo, J. E. Albina, and J. S. Reichner,
“Promotion of neutrophil chemotaxis through differential regulation of β1 and
β2 integrins,” J Immunol, vol. 162, pp. 6792–6799, 1999.
[76] L. M. Lavigne, J. E. Albina, and J. S. Reichner, “Beta-glucan is a fungal de-
terminant for adhesion-dependent human neutrophil functions,” J Immunol,
vol. 177, pp. 8667–8675, 2006.
[77] Y. Xia and G. D. Ross, “Generation of recombinant fragments of cd11b ex-
pressing the functional β-glucan-binding lectin site of cr3,” J Immunol, vol. 162,
pp. 7285–7295, 1999.
[78] H. S. Goodridge, A. J. Wolf, and D. M. Underhill, “Beta-glucan recognition by
the innate immune system,” Immunol Rev, vol. 230, pp. 38–50, 2009.
[79] X. M. OBrien, K. E. Heflin, L. M. Lavigne, K. Yu, M. Kim, A. R. Salomon,
and J. S. Reichner, “Lectin site ligation of cr3 induces conformational changes
and signaling,” J Biol Chem, vol. 287, pp. 3337–3348, 2012.
187
[80] K. Guevorkian and J. James M. Valles, “Swimming paramecium in magnetically
simulated enhanced, reduced, and inverted gravity environments,” Proc Natl
Acad Sci USA, vol. 103, pp. 13051–13056, 2006.
[81] B. L. Taylor, I. B. Zhulin, and M. S. Johnson, “Aerotaxis and other energy-
sensing behavior in bacteria,” Ann Rev Microbiol, vol. 53, pp. 103–128, 1999.
[82] C.-M. Lo, H.-B. Wang, M. Dembo, and Y. li Wang, “Cell movement is guided
by the rigidity of the substrate,” Biophys J, vol. 79, pp. 144–152, 2000.
[83] J. B. McCarthy, S. L. Palm, and L. T. Furcht, “Migration by haptotaxis of a
schwann cell tumor line to the basement membrane glycoprotein laminin,” J
Cell Biol, vol. 97, no. 3, pp. 772–777, 1983.
[84] R. F. Diegelmann and M. C. Evans, “Wound healing: An overview of acute,
fibrotic and delayed healing,” Front Biosci, vol. 9, pp. 283–289, 2004.
[85] C.-L. Lin, R. M. Suri, R. A. Rahdon, J. M. Austyn, and J. A. Roake, “Den-
dritic cell chemotaxis and transendothelial migration are induced by distinct
chemokines and are regulated on maturation,” European Journal of Immunol-
ogy, vol. 28, no. 12, pp. 4114–4122, 1998.
[86] D. R. Levitan and C. Petersen, “Sperm limitation in the sea,” Trends Ecol Evol,
vol. 10, pp. 228–231, 1995.
[87] G. Valentin, P. Haas, and D. Gilmour, “The chemokine sdf1a coordinates tissue
migration through the spatially restricted activation of cxcr7 and cxcr4b,” Curr
Biol, vol. 17, no. 12, pp. 1026–1031, 2007.
[88] E. C. Woodhouse, R. F. Chuaqui, and L. A. Liotta, “General mechanisms of
metastasis,” Cancer, vol. 80, pp. 1529–1537, 1997.
[89] C. R. Cooper and K. J. Pienta, “Cell adhesion and chemotaxis in prostate cancer
metastasis to bone: a minireview,” Prostate Cancer P D, vol. 3, pp. 6–12, 2000.
[90] F. Lin, C. M.-C. Nguyen, S.-J. Wang, W. Saadi, S. P. Gross, and N. L. Jeon,
“Effective neutrophil chemotaxis is strongly influenced by mean il-8 concentra-
tion,” Biochem Bioph Res Co, vol. 319, pp. 576–581, 2004.
188
[91] L. F. Richardson, “The problem of contiguity: An appendix to statistic of
deadly quarrels,” Gen Syst, vol. 61, pp. 139–187, 1961.
[92] B. Mandelbrot, “How long is the coast of britain? statistical self-similarity and
fractional dimension,” Science, vol. 156, no. 3775, pp. 636–638, 1967.
[93] H. G. Othmer, S. R. Dunbar, and W. Alt, “Models of dispersal in biological
systems,” J Math Biol, vol. 26, pp. 263–298, 1988.
[94] L. M. Marsh and R. E. Jones, “The form and consequences of random walk
movement models,” J Theor Biol, vol. 133, pp. 113–131, 1988.
[95] W. Luo, C. han Yu, Z. Z. Lieu, J. Allard, A. Mogilner, M. P. Sheetz, and A. D.
Bershadsky, “Analysis of the local organization and dynamics of cellular actin
networks,” J Cell Biol, vol. 202, pp. 1057–1073, 2013.
[96] S. Goldstein, “On diffusion by discontinuous movements, and on the telegraph
equation.,” J Mech Appl Math, vol. 6, pp. 129–156, 1951.
[97] M. Evans, N. Hastings, and B. Peacock, Statistical Distributions. Wiley, 3 ed.,
2000.
[98] D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear pa-
rameters,” J Soc Ind Appl Math, vol. 11, pp. 431–441, 1963.
[99] S. M. Sze and M.-K. Lee, Semiconductor Devices: Physics and Technology.
Wiley, 3 ed., 2012.
[100] T. N. Mayadas and X. Cullere, “Neutrophil β2 integrins: moderators of life or
death decisions,” Trends Immunol, vol. 26, pp. 388–395, 2005.
[101] R. J. Pelham and Y.-L. Wang, “Cell locomotion and focal adhesions are reg-
ulated by substrate flexibility,” Proc Natl Acad Sci USA, vol. 94, pp. 13661–
13665, 1997.
[102] L. A. Flanagan, Y.-E. Ju, B. Marg, M. Osterfield, and P. A. Janmey, “Neurite
branching on deformable substrates,” Neuroreport, vol. 13, pp. 2411–2415, 2002.
[103] A. J. Engler, M. A. Griffin, S. Sen, C. G. Bonnemann, H. L. Sweeney, and
D. E. Discher, “Myotubes differentiate optimally on substrates with tissue-like
stiffness : pathological implications for soft or stiff microenvironments,” J Cell
189
Biol, vol. 166, pp. 877–887, 2004.
[104] A. J. Engler, M.Sheehan, H. L. Sweeney, and Dennis.E.Discher, “Substrate
compliance versus ligand density in cell on gel responses.,” Biophys J, vol. 86,
pp. 617–628, 2004.
[105] B. Qi, N. Tessier-Doyen, and J. Absi, “Youngs modulus evolution with tem-
perature of glass/andalusite model materials: Experimental and numerical ap-
proach,” Comp Mater Sci, vol. 55, pp. 44–53, 2012.
[106] F. Coussen, D. Choquet, M. P. Sheetz, and H. P. Erickson, “Trimers of the
fibronectin cell adhesion domain localize to actin filament bundles and undergo
rearward translocation,” J Cell Sci, vol. 115, pp. 2581–2590, 2002.
[107] A. Saez, A. Buguin, P. Silberzan, and B. Ladoux, “Is the mechanical activity
of epithelial cells controlled by deformations or forces?,” Biophys J, vol. 89,
pp. L52–L54, 2005.
[108] F. M. Hameed, A. J. Loosley, D. van Noort, and M. P. Sheetz, “Cells test rigidity
of nanopillars by local contractions to uniform displacements.” submitted to
Biophys J 2013, 2013.
[109] A. K. Yip, K. Iwasaki, C. Ursekar, H. Machiyama, M. Saxena, H. Chen,
I. Harada, K.-H. Chiam, and Y. Sawada, “Cellular response to substrate rigidity
is governed by either stress or strain,” Biophys J, vol. 104, pp. 19–29, 2013.
[110] Y. Sawada, M. Tamada, B. J. Dubin-Thaler, O. Cherniavskaya, R. Sakai,
S. Tanaka, and M. P. Sheetz, “Force sensing by mechanical extension of the
src family kinase substrate p130cas,” Cell, vol. 127, pp. 1015–1026, 2006.
[111] A. del Rio, R. Perez-Jimenez, R. Liu, P. Roca-Cusachs, J. M. Fernandez, and
M. P. Sheetz, “Stretching single talin rod molecules activates vinculin binding,”
Science, vol. 323, pp. 638–641, 2009.
[112] M. A. Williams and J. S. Solomkin, “Integrin-mediated signaling in human
neutrophil functioning,” J Leukocyte Biol, vol. 65, pp. 725–736, 1999.
[113] R. O. Hynes, “Integrins: Bidirectional, allosteric signaling machines,” Cell,
vol. 110, pp. 673–687, 2002.
190
[114] S. Gehler, M. Baldassarre, Y. Lad, J. L. Leight, M. A. Wozniak, K. M. Riching,
K. W. Eliceiri, V. M. W. abd David A. Calderwood, and P. J. Keely, “Filamin
aβ1 integrin complex tunes epithelial cell response to matrix tension,” Mol Biol
Cell, vol. 20, pp. 3224–3238, 2009.
[115] J. B. Litzenberger, J.-B. Kim, P. Tummala, and C. R. Jacobs, “β1 integrins
mediate mechanosensitive signaling pathways in osteocytes,” Calcif Tissue Int,
vol. 86, pp. 325–332, 2010.
[116] D. C. Altieri, R. Bader, P. M. Mannucci, and T. S. Edgington, “Oligospecificity
of the cellular adhesion receptor mac-1 encompasses an inducible recognition
specificity for fibrinogen,” J Cell Biol, vol. 107, pp. 1893–1900, 1988.
[117] O. Mueller, H. E. Gaub, M. Baermann, and E. Sackmann, “Viscoelastic moduli
of sterically and chemically cross-linked actin networks in the dilute to semidi-
lute regime: measurements by oscillating disk rheometer,” Macromolecules,
vol. 24, pp. 3111–3120, 1991.
[118] D. E. Discher, P. A. Janmey, and Y. li Wang, “Tissue cells feel and respond to
the stiffness of their substrate,” Science, vol. 310, pp. 1139–1143, 2005.
[119] T. Matsumoto, H. Abe, T. Ohashi, Y. Kato, and M. Sato, “Local elastic mod-
ulus of atherosclerotic lesions of rabbit thoracic aortas measured by pipette
aspiration method,” Physiol Meas, vol. 23, pp. 635–648, 2002.
[120] A. L. McKnight, J. L. Kugel, P. J. Rossman, A. Manduca, L. C. Hartmann,
and R. L. Ehman, “Mr elastography of breast cancer: preliminary results,” Am
J Roentgenol, vol. 178, pp. 1411–1417, 2002.
[121] E. M. Leise, “Modular construction of nervous systems: A basic principle of
design for invertebrates and vertebrates,” Brain Res Rev, vol. 15, pp. 1–23,
1990.
[122] N. M. L. Douarin and M.-A. M. Teillet, “Experimental analysis of the migra-
tion and differentiation of neuroblasts of the autonomic nervous system and of
neurectodermal mesenchymal derivatives, using a biological cell marking tech-
nique,” Dev Biol, vol. 41, no. 1, pp. 162 – 184, 1974.
191
[123] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular
biology of the cell, pp. 969–982. New York: Garland Science, fourth ed., 2002.
[124] V. Witko-Sarsat, P. Rieu, B. Descamps-Latscha, P. Lesavre, and L. Halbwachs-
Mecarelli, “Neutrophils: Molecules, functions and pathophysiological aspects,”
Lab Invest, vol. 80, pp. 617–653, 2000.
[125] C. I. Lacayo, Z. Pincus, M. M. VanDuijn, C. A. Wilson, D. A. Fletcher, F. B.
Gertler, A. Mogilner, and J. A. Theriot, “Emergence of large-scale cell mor-
phology and movement from local actin filament growth dynamics,” PLoS Biol,
vol. 5, no. 9, p. e233, 2007.
[126] M. Herant and M. Dembo, “Form and function in cell motility: from fibroblasts
to keratocytes,” Biophys J, vol. 98, no. 8, pp. 1408–1417, 2010.
[127] E. L. Barnhart, K.-C. Lee, K. Keren, A. Mogilner, and J. A. Theriot, “An
adhesion-dependent switch between mechanisms that determine motile cell
shape,” PLoS Biol, vol. 9, no. 5, p. e1001059, 2011.
[128] C. A. Lemmon and L. H. Romer, “A predictive model of cell traction forces
based on cell geometry,” Biophys J, vol. 99, no. 9, pp. L78–L80, 2010.
[129] A. D. Rape, W. Guo, and Y. Wang, “The regulation of traction force in relation
to cell shape and focal adhesions,” Biomaterials, vol. 32, no. 8, pp. 2043–2051,
2011.
[130] R. W. Carthew, “Adhesion proteins and the control of cell shape,” Curr Opin
Genet Dev, vol. 15, no. 4, pp. 358 – 363, 2005.
[131] M. T. Cabeen and C. Jacobs-Wagner, “Bacterial cell shape,” Nat Rev Microbiol,
vol. 3, pp. 601–610, 2005.
[132] S. Paku, J. Tovari, Z. Lorincz, F. Timar, B. Dome, L. Kopper, A. Raz, and
J. Timar, “Adhesion dynamics and cytoskeletal structure of gliding human fi-
brosarcoma cells: a hypothetical model of cell migration,” Exp Cell Res, vol. 290,
no. 2, pp. 246–253, 2003.
[133] C. Rotsch, K. Jacobson, and M. Radmacher, “Dimensional and mechanical
dynamics of active and stable edges in motile fibroblasts investigated by using
192
atomic force microscopy,” P Natl Acad Sci USA, vol. 96, no. 3, pp. 921–926,
1999.
[134] C. W. Wolgemuth, “Lamellipodial contractions during crawling and spreading,”
Biophys J, vol. 89, no. 3, pp. 1643–1649, 2005.
[135] E. L. Barnhart, G. M. Allen, F. Julicher, and J. A. Theriot, “Bipedal locomotion
in crawling cells,” Biophys J, vol. 98, pp. 933–942, 2010.
[136] T. M. Svitkina, A. B. Verkhovsky, K. M. McQuade, and G. G. Borisy, “Analysis
of the actin-myosin ii system in fish epidermal keratocytes: mechanism of cell
body translocation,” J Cell Biol, vol. 139, no. 2, pp. 397–415, 1997.
[137] M. Dembo, T. Oliver, A. Ishihara, and K. Jacobson, “Imaging the traction
stresses exerted by locomoting cells with the elastic substratum method,” Bio-
phys J, vol. 70, pp. 2008–2022, 1996.
[138] M. Welch, A. Mallavarapu, J. Rosenblatt, and T. Mitchison, “Actin dynamics
in vivo,” Curr. Opin. Cell. Bio., vol. 9, pp. 54–61, 1997.
[139] J. A. Theriot and T. J. Mitchison, “Actin microfilament dynamics in locomoting
cells,” Nature, vol. 352, pp. 126–131, 1991.
[140] A. Mogilner and G. Oster, “Force generation by actin polymerization ii: the
elastic ratchet and tethered filaments,” Biophys J, vol. 84, pp. 1591–1605, 2003.
[141] C. Jurado, J. R. Haserick, and J. Lee, “Slipping or gripping? fluorescent speckle
microscopy in fish keratocytes reveals two different mechanisms for generating
a retrograde flow of actin,” Mol Biol Cell, vol. 16, pp. 507–518, 2005.
[142] S. Deguchi, T. Ohashi, and M. Sato, “Tensile properties of single stress fibers iso-
lated from cultured vascular smooth muscle cells,” J Biomech, vol. 39, pp. 2603–
2610, 2006.
[143] A. E. Filippov, J. Klafter, and M. Urbakh, “Friction through dynamical forma-
tion and rupture of molecular bonds,” Phys. Rev. Lett., vol. 92, p. 135503, Mar
2004.
[144] S. Walcott and S. X. Sun, “A mechanical model of actin stress fiber formation
and substrate elasticity sensing in adherent cells,” Proc Natl Acad Sci USA,
193
vol. 107, no. 17, pp. 7757–7762, 2010.
[145] F. Wottawah, S. Schinkinger, B. Lincoln, R. Ananthakrishnan, M. Romeyke,
J. Guck, and J. Kas, “Optical rheology of biological cells,” Phys Rev Lett,
vol. 94, p. 098103, 2005.
[146] V. M. Laurent, S. Kasas, A. Yersin, T. E. Schaffer, S. Catsicas, G. Dietler,
A. B. Verkhovsky, and J.-J. Meister, “Gradient of rigidity in the lamellipodia of
migrating cells revealed by atomic force microscopy,” Biophys J, vol. 89, no. 1,
pp. 667–675, 2005.
[147] F. Heinemann, H. Doschke, and M. Radmacher, “Keratocyte lamellipodial pro-
trusion is characterized by a concave force-velocity relation,” Biophys J, vol. 100,
pp. 1420–1427, 2011.
[148] Z. Pincus and J. A. Theriot, “Comparison of quantitative methods for cell-shape
analysis,” J Microsc, vol. 227, pp. 140–156, 2007.
[149] P. Lenz, K. Keren, and J. A. Theriot, “Biophysical aspects of actin-based cell
motility in fish epithelial keratocytes,” in Cell Motility, Biological and Medical
Physics, Biomedical Engineering, pp. 31–58, Springer New York, 2008.
[150] J. A. Theriot, “Theriot lab movies.” http://cmgm.stanford.edu/theriot/
movies.htm, May 2011.
[151] A. B. Verkhovsky, O. Y. Chaga, S. Schaub, T. M. Svitkina, J.-J. Meister, and
G. G. Borisy, “Orientational order of the lamellipodial actin network as demon-
strated in living motile cells,” Mol Biol Cell, vol. 14, pp. 4667–4675, 2003.
[152] T. E. Schaus, E. W. Taylor, and G. G. Borisy, “Self-organization of actin fil-
ament orientation in the dendritic-nucleation/array-treadmilling model,” Proc
Natl Acad Sci USA, vol. 104, pp. 7086–7091, 2007.
[153] F. Fleischer, R. Ananthakrishan, S. Eckel, H. Schmidt, J. Kas, T. M. Svitk-
ina, V. Schmidt, and M. Beil, “Actin network architecture and elasticity in
lamellipodia of melanoma cells,” New J Phys, vol. 9, p. 420, 2007.
[154] S. Schaub, S. Bohnet, V. M. Laurent, J.-J. Meister, and A. B. Verkhovsky,
“Comparative maps of motion and assembly of filamentous actin and myosin ii
194
in migrating cells,” Mol Biol Cell, vol. 18, no. 10, pp. 3723–3732, 2007.
[155] Y. Lin, “Mechanics model for actin-based motility,” Phys Rev E, vol. 79,
p. 021916, 2009.
[156] K. Larripa and A. Mogilner, “Transport of a 1d viscoelastic actinmyosin strip
of gel as a model of a crawling cell,” Physica A, vol. 372, pp. 113–123, 2006.
[157] A. Mogilner and D. W. Verzi, “A simple 1-d physical model for the crawling
nematode sperm cell,” J Stat Phys, vol. 110, pp. 1169–1189, 2003.
[158] F. Ziebert, S. Swaminathan, and I. S. Aranson, “Model for self-polarization and
motility of keratocyte fragments,” J Roy Soc Interface, vol. (online in adv of
print), pp. 1–9, 2011.
[159] J. L. McGrath, N. J. Eungdamrong, C. I. Fisher, F. Peng, L. Mahadevan, T. J.
Mitchison, and S. C. Kuo, “The force-velocity relationship for the actin-based
motility of listeria monocytogenes,” Curr Biol, vol. 13, pp. 329–332, 2003.
[160] A. F. Straight, A. Cheung, J. Limouze, I. Chen, N. J. Westwood, J. R. Sellers,
and T. J. Mitchison, “Dissecting temporal and spatial control of cytokinesis
with a myosin ii inhibitor,” Science, vol. 299, pp. 1743–1747, 2003.
[161] P. T. Yam, C. A. Wilson, L. Ji, B. Hebert, E. L. Barnhart, N. A. Dye, P. W.
Wiseman, G. Danuser, and J. A. Theriot, “Actin-myosin network reorganiza-
tion breaks symmetry at the cell rear to spontaneously initiate polarized cell
motility,” J Cell Biol, vol. 178, no. 7, pp. 1207–1221, 2007.
[162] M. L. Gardel, J. H. Shin, F. C. MacKintosh, L. Mahadevan, P. Matsudaira,
and D. A. Weitz, “Elastic behavior of cross-linked and bunded actin networks,”
Science, vol. 304, pp. 1301–1305, 2004.
[163] J. E. Italiano, T. M. Roberts, M. Stewart, and C. A. Fontana, “Reconstitution
in vitro of the motile apparatus from the amoeboid sperm of ascaris shows
that filament assembly and bundling move membranes,” Cell, vol. 84, no. 1,
pp. 105–114, 1996.
[164] A. Arnaout, “Structure and function of leukocyte adhesion molecules,” Blood,
vol. 75, pp. 1037–1050, 1990.
195
[165] L. A. Smith, H. Aranda-Espinoza, J. B. Haun, M. Dembo, and D. A. Hammer,
“Neutrophil traction stresses are concentrated in the uropod during migration,”
Biophys J, vol. 92, pp. L58–L60, 2007.
[166] J. A. Swanson, M. T. Johnson, K. Beningo, P. Post, M. Mooseker, and N. Araki,
“A contractile activity that closes phagosomes in macrophages,” J Cell Sci,
vol. 112, pp. 307–316, 1999.
[167] M. Herant, V. Heinrich, and M. Dembo, “Mechanics of neutrophil phagocytosis:
behavior of the cortical tension,” J Cell Sci, vol. 118, pp. 1789–1797, 2005.
[168] M. Herant, V. Heinrich, and M. Dembo, “Mechanics of neutrophil phagocytosis:
experiments and quantitative models,” J Cell Sci, vol. 119, pp. 1903–1913, 2006.
[169] R. Meili, B. Alonso-Latorre, J. C. del Alamo, R. A. Firtel, and J. C. Lasheras,
“Myosin ii is essential for the spatiotemporal organization of traction forces
during cell motility,” Mol Biol Cell, vol. 21, pp. 405–417, 2010.
[170] C. E. Chan and D. J. Odde, “Traction dynamics of filopodia on compliant
substrates,” Science, vol. 322, pp. 1687–1691, 2008.
[171] A. Bacic, G. Fincher, and B. Stone, eds., Chemistry, Biochemistry, and Biology
of 1-3 Beta Glucans and Related Polysaccharides. Academic Press, 2009.
[172] G. C. Chan, W. K. Chan, and D. M. Sze, “The effects of beta-glucan on human
immune and cancer cells,” J Hematol Oncol, vol. 2, p. 25, 2009.
[173] S. Y. Kim, H. J. Song, Y. Y. Lee, K.-H. Cho, and Y. K. Roh, “Biomedical issues
of dietary fiber beta-glucan,” J Korean Med Sci, vol. 21, pp. 781–789, 2006.
[174] G. D. Brown and S. Gordon, “Fungal β-glucans and mammalian immunity,”
Immunity, vol. 19, pp. 311–315, 2003.
[175] A. Buxboim, K. Rajagopal, A. E. X. Brown, and D. E. Discher, “How deeply
cells feel: methods for thin gels,” J Phys-Condens Mat, vol. 22, p. 194116, 2011.
[176] H. Tanimoto and M. Sano, “A simple force-motion relation for migrating cells
revealed by multipole analysis of traction stress,” Biophys J, vol. 106, pp. 16–25,
2014.
196
[177] L. Breiman and J. H. Friedma, “Estimating optimal transformations for multiple
regression and correlation,” J Am Stat Assoc, vol. 80, pp. 580–598, 1985.
[178] H. Voss and J. Kurths, “Reconstruction of nonlinear time delay models from
data by the use of optimal transformations,” Phys Rev A, vol. 234, pp. 336–344,
1997.
[179] P. W. Oakes, S. Banerjee, C. M. Marchetti, and M. L. Gardel, “Geometry
regulates traction stresses in adherent cells,” Biophys J, vol. 107, pp. 825–833,
2014.
[180] V. M. Laurent, S. Kasas, A. Yersin, T. E. Schaffer, S. Catsicas, G. Dietler,
A. B. Verkhovsky, and J.-J. Meister, “Gradient of rigidity in the lamellipodia
of migrating cells revealed by atomic force microscopy,” Biophys J, vol. 89,
pp. 667–675, 2005.
[181] O. Thoumine, P. Kocian, A. Kottelat, and J.-J. Meister, “Short-term bind-
ing of fibroblasts to fibronectin: optical tweezers experiments and probabilistic
analysis,” Eur Biophys J, vol. 29, pp. 398–408, 2000.
[182] F. Li, S. D. Redick, H. P. Erickson, and V. T. Moy, “Force measurements of the
α5β1 integrin-fibronectin interaction,” Biophys J, vol. 84, pp. 1252–1262, 2003.
[183] J. Riedl, A. H. Crevenna, K. Kessenbrock, J. H. Yu, D. Neukirchen, M. Bista,
F. Bradke, D. Jenne, T. A. Holak, Z. Werb, M. Sixt, and R. Wedlich-Soldner,
“Lifeact: a versatile marker to visualize f-actin,” Nat Methods, vol. 5, p. 605,
2008.
[184] J. A. Cooper, “Effects of cytochalasin and phalloidin on actin,” J Cell Biol,
vol. 105, pp. 1473–1478, 1987.
[185] A. J. Loosley, X. M. O’Brien, J. S. Reichner, and J. X. Tang, “Describing
directionally biased paths by a characteristic directionality time,” submitted,
vol. xx, pp. xxx–xxx, 2014.
[186] H. Levine and W.-J. Rappel, “The physics of eukaryotic chemotaxis,” Phys
Today, vol. 66, pp. 24–30, 2013.
197
[187] K. Takeda, D. Shao, M. Adler, P. G. Charest, W. F. Loomis, H. Levine, A. Gro-
isman, W.-J. Rappel, and R. A. Firtel, “Incoherent feedforward control governs
adaptation of activated ras in a eukaryotic chemotaxis pathway,” Sci Signal,
vol. 5, p. ra2, 2012.
[188] D. M. Lehmann, A. M. P. B. Seneviratne, and A. V. Smrcka, “Small mole-
cule disruption of g protein beta gamma subunit signaling inhibits neutrophil
chemotaxis and inflammation,” Mol Pharmacol, vol. 73, pp. 410–418, 2008.
[189] M. A. Fardin, O. M. Rossier, P. Rangamani, P. D. Avigan, N. C. Gauthier,
W. Vonnegut, A. Mathur, J. Hone, R. Iyengar, and M. P. Sheetz, “Cell spreading
as a hydrodynamic process,” Soft Matter, vol. 6, pp. 4788–4799, 2010.
[190] A. Levchenko and P. Iglesias, “Models of eukaryotic gradient sensing: applica-
tion to chemotaxis of amoebae and neutrophils,” Biophys J, vol. 82, pp. 50–63,
2002.
[191] L. Michaelis and M. L. Menten, “Die kinetik der invertinwirkung,” Biochem Z,
vol. 49, pp. 333–369, 1913.
[192] A. Bruno-Alfonso, L. Cabezas-Gomez, and H. A. Navarro, “Alternate treat-
ments of jacobian singularities in polar coordinates within finite-difference
schemes,” World J Model Sim, vol. 8, pp. 163–171, 2012.
[193] I. Hecht, M. L. Skoge, P. G. Charest, E. Ben-Jacob, R. A. Firtel, W. F. Loomis,
H. Levine, and W.-J. Rappel, “Activated membrane patches guide chemotactic
cell motility,” PLoS Comp Biol, vol. 7, p. e1002044, 2011.
[194] E. Lien, T. K. Means, H. Heine, A. Yoshimura, S. Kusumoto, K. Fukase, M. J.
Fenton, M. Oikawa, N. Qureshi, B. Monks, R. W. Finberg, R. R. Ingalls, and
D. T. Golenbock, “Toll-like receptor 4 imparts ligand-specific recognition of
bacterial lipopolysaccharide,” J Clin Invest, vol. 105, pp. 497–504, 2000.
[195] R. S. Hotchkiss and I. E. Karl, “The pathophysiology and treatment of sepsis,”
N Engl J Med, vol. 348, pp. 138–150, 2003.
[196] C. G. Leon, R. Tory, J. Jia, O. Sivak, and K. M. Wasan, “Discovery and devel-
opment of toll-like receptor 4 (tlr4) antagonists: A new paradigm for treating
198
sepsis and other diseases,” Pharm Res, vol. 25, pp. 1751–1761, 2008.
[197] H. E. Wang, N. I. Shapiro, D. C. Angus, and D. M. Yealy, “National estimates of
severe sepsis in united states emergency departments,” Crit Care Med, vol. 35,
pp. 1928–1936, 2007.
[198] E. K. Stevenson, A. R. Rubenstein, G. T. Radin, R. S. Wiener, and A. J.
Walkey, “Two decades of mortality trends among patients with severe sepsis:
A comparative meta-analysis,” Crit Care Med, vol. 42, pp. 625–631, 2014.
[199] V. Kumar, A. K. Abbas, N. Fausto, and R. Mitchell, Robbins Basic Pathology.
Elsevier/Saunders, 2007.
⊙199