Introduction to Cell Mechanics and Mechanobiology

Christopher R. Jacobs

Hayden Huang

Ronald Y. Kwon

GS Garland Science Taylor & Francis Group

NEW YORK AND LONDON

Preface v CHAPTER 2: Fundamentals in Cell Biology 19

PART 1: PRINCIPLES 1 2.1 Fundamentals in cell and molecular

CHAPTER 1: Cell Mechanics as a Framework 3 biology 19

1.1 Cell mechanics and human disease 4 Proteins are polymers of amino acids 20

Specialized cells in the ear allow you to hear 5 DNA and RNA are polymers of

nucleic acids 22 Hemodynamic forces regulate endothelial Polysaccharides are polymers of sugars 24

cells 6

To keep bone healthy, bone cells need Fatty acids store energy but also form

structures 24 mechanical stimulation 6

The cells that line your lungs sense stretch 7 Correspondence between DNA-to-RNA-

to-protein is the central dogma of Pathogens can alter cell mechanical properties 7 modern cell biology 25 Other pathogens can use cell mechanical Phenotype is the manifestation

structures to their advantage 7 of genotype 27 Cancer cells need to crawl to be metastatic 8 Transcriptional regulation is one way Viruses transfer their cargo into cells they that phenotype differs from genotype 28

infect 8 Cell organelles perform a variety 1.2 The cell is an applied mechanics grand of functions 29

challenge 8 2.2 Receptors are cells' primary chemical Computer simulation of cell mechanics sensors 30

requires state-of-the-art approaches 9 Cells communicate by biochemical signals 30 1.3 Model problem: micropipette aspiration 9 Signaling between cells can occur

What is a typical experimental setup for through many different mechanisms 31 micropipette aspiration? 9 Intracellular signaling occurs via small

The liquid-drop model is a simple model molecules known as second messengers 32 that can explain some aspiration results 11 Large molecule signaling cascades

The Law of Laplace can be applied to a have the potential for more specificity 34 spherical cell 12 Receptors use several mechanisms to

Micropipette aspiration experiments can initiate signaling 35 be analyzed with the Law of Laplace 12 2.3 Experimental biology 36

How do we measure surface tension Optical techniques can display cells clearly 37 and areal expansion modulus? 13 Fluorescence visualizes cells with lower

Why do cells "rush in"? 15 background 38 Cells can behave as elastic solids Fluorophores can highlight structures 39

or liquid drops 16 Fluorophores can probe function 40 Key Concepts 16 Atomic force microscopy can elucidate Problems 17 the mechanical behavior of cells 40 Annotated References 17 Gel electrophoresis can separate molecules 41

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Visualizing gel-separated products employs Torsion of a solid cylinder can be a variety of methods 42 modeled as a torsion of a series of

PCR amplifies specific DNA regions shells of increasing radius 61

exponentially 43 Kinematics, equilibrium, and constitutive

2.4 Experimental design in biology 46 equations are the foundation of solid

Reductionist experiments are powerful mechanics 62

but limited 46 Kinematics in a beam are the strain-

Modern genetics has advanced our displacement relationship 62

ability to study in situ 48 Equilibrium in a beam is the stress-

Bioinformatics allows us to use vast moment relationship 64

amounts of genomic data 49 The constitutive equation is the

Systems biology is integration rather stress-strain relationship 65

than reduction 49 The second moment of inertia is a

Biomechanics and mechanobiology are measure of bending resistance 65

integrative 49 The cantilevered beam can be solved

Key Concepts 50 from the general beam equations 66

Problems 50 Buckling loads can be determined from

Annotated References 52 the beam equations 67

Transverse strains occur with axial loading 68

CHAPTER 3: Solid Mechanics Primer 53 The general continuum equations can be

3.1 Rigid-body mechanics and free-body developed from our simple examples 68

diagrams 53 Equilibrium implies conditions on stress 69

What is a "rigid" body? 53 Kinematics relate strain to displacement 71

One of the most powerful, but underused, The constitutive equation or stress-strain

tools is a free-body diagram 53 relation characterizes the material behavior 73

Identifying the forces is the first step in Vector notation is a compact way to express

drawing a free-body diagram 54 equations in continuum mechanics 74

Influences are identified by applying the Stress and strain can be expressed as matrices 76

equations of motion 54 In the principal directions shear stress is zero 76

Free-body diagrams can be drawn for parts 3.3 Large deformation mechanics 78

of objects 55 The deformation gradient tensor

3.2 Mechanics of deformable bodies 55 describes large deformations 78

Rigid-body mechanics is not very Stretch is another geometrical measure

useful for analyzing deformable bodies 55 of deformation 79

Mechanical stress is analogous to pressure 56 Large deformation strain can be defined

Normal stress is perpendicular to the in terms of the deformation gradient 80

area of interest 56 The deformation gradient can be

Strain represents the normalized decomposed into rotation and

change in length of an object to load 57 stretch components 82

The stress-strain plot for a material 3.4 Structural elements are defined by

reveals information about its stiffness 57 their shape and loading mode 83

Stress and pressure are not the same Key Concepts 84

thing, because stress has directionality 58 Problems 84

Shear stress describes stress when forces Annotated References 87

and areas are perpendicular to each other 59

Shear strain measures deformation CHAPTER4: Fluid Mechanics Primer 89 resulting from shear stress 59 4.1 Fluid statics 89

Torsion in the thin-walled cylinder can Hydrostati.c pressure results from be modeled with shear stress relations 60 gravitational forces 89

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Hydrostatic pressure is isotropic 91 5.1 Internal energy 120

Resultant forces arising from hydrostatic Potential energy can be used to make pressure can be calculated through predictions of mechanical behavior 120 integration 92 Strain energy is potential energy stored

4.2 Newtonian fluids 92 in elastic deformations 122

Fluids obey mass conservation 93 Equilibrium in continuum mechanics is

Fluid flows can be laminar or turbulent 94 a problem of strain energy minimization 123

Many laminar flows can be solved analytically 95 Changes in mechanical state alter

Many biological fluids can exhibit internal energy 123

non-Newtonian behavior 97 5.2 Entropy 124

4.3 The Navier-Stokes equations 98 Entropy is directly defined within

Derivation of the Navier-Stokes equations statistical mechanics 124

begins with Newton's second law 99 Microstates, macrostates, and density

Constitutive relations and the continuity of states can be exemplified in a

equation are necessary to make Navier's three-coin system 124

equations solvable 102 Microstates, macrostates, and density

Navier-Stokes equations: putting it all of states provide insight to macroscopic

together 103 system behavior 127

4.4 Rheological analysis 103 Ensembles are collections of microstates

The mechanical behavior of viscoelastic sharing a common property 127

materials can be decomposed into Entropy is related to the number of microstates

elastic and viscous components 104 associated with a given macrostate 127

Complex moduli can be defined for 5.3 Free energy 128

viscoelastic materials 106 Equilibrium behavior for thermodynamic

Power laws can be used to model frequency- systems can be obtained via free energy

dependent changes in storage and minimization 128

loss moduli 108 Temperature-dependence of end-to-end

4.5 Dimensional analysis no length in polymers arises out of

Dimensional analysis requires the competition between energy and entropy 129

determination of base parameters llO 5.4 Microcanonical ensemble 131

The Buckingham Pi Theorem gives the The hairpinned polymer as a non-

number of dimensionless parameters interacting two-level system 132

that can be formed from base parameters ll1 A microcanonical ensemble can be

Dimensionless parameters can be found used to determine

through solving a system of equations 1ll constant energy microstates 132

Similitude is a practical use of Entropy can be calculated via

dimensional analysis 113 combinatorial enumeration of the

Dimensional parameters can be density of states 133

used to check analytical expressions ll4 Entropy is maximal when half the

Key Concepts 115 sites contain hairpins 133

Problems ll6 S(W) can be used to predict equilibrium

behavior 133 Annotated References ll7 The number of hairpins at equilibrium is

dependent on temperature 134 CHAPTER 5: Statistical Mechanics Primer 119 Equilibrium obtained via the

Statistical mechanics relies on the use microcanonical ensemble is identical to

of probabilistic distributions ll9 that obtained via free energy minimization 135

Statistical mechanics can be used to 5.5 Canonical ensemble 136 investigate the influence of random Canonical ensemble starting from the molecular forces on mechanical behavior 119 microcanonical ensemble 136

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Probability distribution from the canonical 6.2 Measurement offorces produced by cells 160 ensemble gives Boltzmann's law 138 Traction force microscopy measures the forces

The free energy at equilibrium can be exerted by a cell on its underlying surface 160 found using the partition function 139 Cross-correlation can be used for

The internal energy at equilibrium can be particle tracking 160 determined using the partition function 141 Determining the forces that produced a

Using the canonical approach may be displacement is an inverse problem 163 preferable for analyzing thermodynamic Microfabricated micropillar arrays can systems 142 be used to measure traction forces directly 165

5.6 Random walks 143 Surface modification can help determine A simple random walk can be how a cell interacts with its surroundings 166

demonstrated using soccer 143 6.3 Applying forces to cells 167

The diffusion equation can be derived Flow chambers are used for studying from the random walk 145 cellular responses to fluid shear stress 167

Key Concepts 147 The transition between laminar and

Problems 148 turbulent flow is governed by the

Annotated References 149 Reynolds number 168

Parallel plate flow devices can be designed for low Reynolds number shear flow 168

CHAPTER 6: Cell Mechanics in the Fully developed flow occurs past the

Laboratory 151 entrance length 169

6.1 Probing the mechanical behavior of cells Cone-and-plate flow can be used to

through cellular micromanipulation 151 study responses to shear 170

Known forces can be applied to cells through Diverse device designs can be used to

the use of cell-bound beads and an study responses to fluid flow 171

electromagnet 152 Flexible substrates are used for

The dependence of force on distance subjecting cells to strain 172

from the magnet tip can be calibrated Confined uniaxial stretching can lead through Stokes' law 152 to multiaxial cellular deformations 172

Magnetic twisting and multiple-pole Cylindrically symmetric deformations micromanipulators can apply stresses generate uniform biaxial stretch 172 to many cells simultaneously 153 6.4 Analysis of deformation 173

Optical traps generate forces on particles Viscoelastic behavior in micromanipulation through transfer of light momentum 153 experiments can be parameterized

Ray tracing elucidates the origin of through spring-dashpot models 173

restoring forces in optical tweezers 154 Combinations of springs and dash pots can

What are the magnitudes of forces in be used to model viscoelastic behavior 174

an optical trap? 155 Microscopy techniques can be adapted

How does optical trapping compare to visualize cells subject to mechanical

with magnetic micromanipulation? 156 loading 177

Atomic force microscopy involves the Cellular deformations can be inferred

direct probing of objects with a small from image sequences through

cantilever 157 image correlation-based approaches 178

Cantilever deflection is detected using Intracellular strains can be computed

a reflected laser beam 157 from displacement fields 179

Scanning and tapping modes can be 6.5 Blinding and controls 181

used to obtain cellular topography 158 Key Concepts 182

A Hertz model can be used to estimate Problems 183

mechanical properties 158 Annotated References 184

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PART II: Practices 187 Force is the gradient of free energy in thermodynamic systems 208

CHAPTER 7: Mechanics of Cellular The behavior of polymers tends Polymers 189 toward that of an ideal chain in

7.1 Biopolymer structure 189 the limit oflong contour length 209 Microfilaments are polymers composed 7.5 Freely jointed chain (FJC) 210

of actin monomers 189 The FJC model places a limit on F-actin polymerization is influenced by polymer extension 210

the molecular characteristics of G-actin 189 The force-displacement relation for the Microtubules are polymers composed FJC can be found by the canonical

of tubulin dimers 191 ensemble 211 MT polymerization is affected by Differences between the ideal chain

polarity and GTP /GDP binding 191 and the FJC emerge at large forces 213 Intermediate filaments are polymers 7.6 Worm-like chain (WLC) 214

with a diverse range in composition 192 The WLC incorporates energetic Intermediate filaments possess a coiled-coil effects of bending 214

structure 192 The force-displacement relation for Intermediate filaments have diverse the WLC can be found by the

functions in cells 192 canonical ensemble 216 7.2 Polymerization kinetics 194 Differences in the WLC and FJC emerge

Actin and MT polymerization can when they are fitted to experimental data be modeled as a bimolecular reaction 195 for DNA 217

The critical concentration is the only Persistence length is related to Kuhn concentration at which the polymer length 218 does not change length 195 Key Concepts 219

Polarity leads to different kinetics on Problems 220 each end 196 Annotated References 221

Polymerization kinetics are affected by ATP/ADP in actin and GTP/GDP binding in tubulin 197 CHAPTER 8: Polymer Networks and

Subunit polarity and ATP hydrolysis lead to the Cytoskeleton 223 polymer treadmilling 197 8.1 Polymer networks 223

7.3 Persistence length 198 Polymer networks have many degrees Persistence length gives a measure of freedom 223

of flexibility in a thermally fluctuating Effective continuums can be used to polymer 198 model polymer networks 223

Persistence length is related to 8.2 Scaling approaches 225 flexural rigidity for an elastic beam 200

Cellular solids theory implies Polymers can be classified as stiff, flexible, scaling relationships between

or semi-flexible by the persistence length 202 effective mechanical properties and 7.4 Ideal chain 203 network volume fraction 225

The ideal chain is a polymer model Bending-dominated deformation results for flexible polymers 203 in a nonlinear scaling of the elastic

The probability for the chain to have different modulus with volume fraction 225 end-to-end lengths can be determined Deformation dominated by axial strain from the random walk 204 results in a linear scaling of the

The free energy of the ideal chain can be elastic modulus with volume fraction 227 computed from its probability The stiffness oftensegrity structures distribution function 207 scales linearly with member prestress 228

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8.3 Affine networks 229 The fluid mosaic model of the cell membrane

Affine deformations assume the describes its physical properties 252

filaments deform as if they are 9.2 Phospholipid self-assembly 252 embedded in a continuum 229 Critical micelle concentration depends

Flexible polymer networks can on amphiphile molecular structure 253 be modeled using rubber elasticity 230 Aggregate shape can be understood

Anisotropic affine networks can be from packing constraints 254 modeled using strain energy approaches 233 9.3 Membrane barrier function 255

Elastic moduli can be computed The diffusion equations relate from strain energy density 233 concentration to flux per unit area 256

Elastic moduli of affine anisotropic networks Pick's second law shows how spatial can be calculated from appropriate concentration changes as a function of time 257 strain energy density and angular

9.4 Membrane mechanics 1: In -plane shear distribution functions 235

and tension 259 8.4 Biomechanical function and

Thin structures such as membranes cytoskeletal structure 236

can be treated as plates or shells 260 Filopodia are cross-linked bundles

Kinematic assumptions help describe of actin filaments involved in cell motility 236 deformations 260

Actin filaments within filopodia can be A constitutive model describes

modeled as elastic beams undergoing material behavior 262

buckling 236

The membrane imparts force on the The equilibrium condition simplifies

ends of filopodia 238 for in-plane tension and shear 262

The maximum filopodium length before Equilibrium simplifies in the case of

buckling in the absence of cross-linking is shear alone 265

shorter than what is observed in vivo 238 Equilibrium simplifies in the case of

Cross-linking extends the maximum equibiaxial tension 266

length before buckling 238 Areal strain can be a measure of

Is the structure of the red blood cell's biaxial deformation 267

cytoskeleton functionally advantageous? 239 9.5 Membrane mechanics II: Bending 267

Thin structures can be analyzed using the In bending the kinematics are

two-dimensional shear modulus and governed by membrane rotation 268

the areal strain energy density 240 Linear elastic behavior is assumed

Sixfold connectivity facilitates resistance for the constitutive model 269

to shear 242 Equilibrium places conditions on

Fourfold connectivity does not resultant forces and moments 269

sustain shear as well as sixfold 245 Which dominates, tension or bending? 272

Key Concepts 246 9.6 Measurement of bending rigidity 272

Problems 246 Membranes undergo thermal

Annotated References 247 undulations similar to polymers 272

Membranes straighten out with tension 273

CHAPTER 9: Mechanics of the Cell Key Concepts 275

Membrane 249 Problems 275

9.1 Membrane biology 249 Annotated References 277

Water is a polar molecule 249

Cellular membranes form by interacting CHAPTER 10: Adhesion, Migration, and with water 250 Contraction 279

The saturation of the lipid tails determines 10.1 Adhesion 279

some properties of the membrane 251 Cells can form adhesions with the substrate 279

The cell membrane distinguishes inside Fluid shear can be used to measure and outside 251 adhesion strength indirectly 281

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ll.4 Alteration of cellular function 330 Mechanical stimulation can induce

Intracellular calcium increases in extracellular matrix remodeling 333

response to mechanical stress 330 Cell viability and apoptosis are altered

Nitric oxide, inositol triphosphate, and by different processes 334

cyclic AMP, like Ca2+, are second Key Concepts 334

messenger molecules implicated in Problems 334 mechanosensation 331 Annotated References 335

Mitogen-activated protein kinase activity is altered after exposure to mechanical stimulation 332 Abbreviations 337

Mechanically stimulated cells exhibit List of variables and units 338 prostanglandin release 332 Index 343

Mechanical forces can induce morphological changes in cells 332