1386 Integr. Biol., 2012, 4, 1386–1397 This journal is c The Royal Society of Chemistry 2012

Cite this: Integr. Biol., 2012, 4, 1386–1397

Integrating focal adhesion dynamics, cytoskeleton remodeling, and actin

motor activity for predicting cell migration on 3D curved surfaces of the

extracellular matrixw

Min-Cheol Kim,*ab Choong Kim,a Levi Wood,b Devin Neal,b Roger D. Kammabc

and H. Harry Asadaab

Received 26th June 2012, Accepted 21st August 2012

DOI: 10.1039/c2ib20159c

An integrative cell migration model incorporating focal adhesion (FA) dynamics, cytoskeleton and

nucleus remodeling and actin motor activity is developed for predicting cell migration behaviors on

3-dimensional curved surfaces, such as cylindrical lumens in the 3-D extracellular matrix (ECM).

The work is motivated by 3-D microfluidic migration experiments suggesting that the migration speed

and direction may vary depending on the cross sectional shape of the lumen along which the cell

migrates. In this paper, the mechanical structure of the cell is modeled as double elastic membranes of

cell and nucleus. The two elastic membranes are connected by stress fibers, which are extended from

focal adhesions on the cell surface to the nuclear membrane. The cell deforms and gains traction as

transmembrane integrins distributed over the outer cell membrane bind to ligands on the ECM, form

focal adhesions, and activate stress fibers. Probabilities at which integrin ligand–receptor bonds are

formed as well as ruptures are affected by the surface geometry, resulting in diverse migration behaviors

that depend on the curvature of the surface. Monte Carlo simulations of the integrative model reveal

that (a) the cell migration speed is dependent on the cross sectional area of the lumen with a maximum

speed at a particular diameter or width, (b) as the lumen diameter increases, the cell tends to spread

and migrate around the circumference of the lumen, while it moves in the longitudinal direction as the

lumen diameter narrows, (c) once the cell moves in one direction, it tends to stay migrating in the same

direction despite the stochastic nature of migration. The relationship between the cell migration speed

and the lumen width agrees with microfluidic experimental data for cancer cell migration.

Introduction

Cells generate traction forces at focal adhesion (FA) sites,

triggered by chemotaxis and haptotactic responses from the

extracellular environment,1 and contractile forces through

myosin II motor activity in actin stress fibers by an intracel-

lular signalling cascade involving the RhoA small GTPase.2

a BioSystem & Micromechanics IRG, Singapore MIT AllianceResearch Technology, Singapore, 117543, Singapore.E-mail: mincheol@mit.edu; Tel: +656516-8603

bDepartment of Mechanical Engineering, Massachusetts Institute ofTechnology, Cambridge, MA 02139, USA

cDepartment of Biological Engineering, Massachusetts Institute ofTechnology, Cambridge, MA 02139, USAw Electronic supplementary information (ESI) available. See DOI:10.1039/c2ib20159c

Insight, innovation, integration

To understand three-dimensional crawling behaviour of

a single stalk cell on the surface of the angiogenic lumen,

we developed an integrated cell migration model incorporat-

ing three key mechanisms of cell biology, consisting of

remodelling of the cell and nuclear membranes, focal adhe-

sion dynamics, and actin motor activity. After we success-

fully compared our model with an existing experimental

work of spontaneous cancer cell migration in a confined

microfluidic device, we predicted stalk cell migration in

various circular tubes. When the cell migrates on the wall

of a wide lumen, the cell tends to stretch out along the

circumference, where the radius of curvature is smaller

than that of the longitudinal direction, resulting in a high

probability for transverse migration. The new cell migration

model can be further developed toward a more complex

model with the inclusion of cell–cell interactions to predict

emergent behaviours of collective cell migrations in various

geometries.

Integrative Biology Dynamic Article Links

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This journal is c The Royal Society of Chemistry 2012 Integr. Biol., 2012, 4, 1386–1397 1387

Studies indicate that actomyosin activity in cell migration is

critical for many biophysical phenomena, including angiogen-

esis, tumor growth, metastasis, and wound healing in multi-

cellular phenomena.3–5 Additionally, myosin II motor activity

is essential for cell adhesion to the substratum and for changes

in cell morphology through mechanical force balances

between the generation of traction force at the leading edge

and the release of FAs in the rear of the cell.6

A study reports that cells contain at least three different

types of contractile actin-based stress fibers: transverse arcs,

dorsal stress fibers, and ventral stress fibers;7 transverse arcs

are curved actomyosin bundles that are not directly associated

with focal adhesion on either end, dorsal stress fibers consist of

one end bound to a focal adhesion site and the other end

attached to the nucleus or transverse arcs, whereas ventral

stress fibers bridge two focal adhesions on both ends. Among

the different types of actin based stress fibers, actin stress fibers

connected to the nuclear membrane, in particular, appear to

play an important role in cell migration and contraction. For

example, the recent experiments by Chancellor et al.8 demon-

strate that the actomyosin tension is balanced in part by

the nucleus through mechanical links mediated by nesprin-1

(nuclear membrane proteins that bind to F-actin). Interestingly,

when its connection to the nucleus is inhibited, the number of

ventral SFs is rapidly increased, which results in the reduction

of cell migration speed. In the recent literature,9 the reduction of

cell migration speed due to the disconnection of actin stress

fibers to the nuclear membrane was also verified using a

computational cell migration modeling approach.

Previous experimental works mostly involve 2-dimensional

cell migration on planar surfaces. However, 2-D cell migration

on the 3-D extracellular matrix (ECM) surface, e.g. stalk cell

migration in angiogenesis, remains poorly understood. When

migrating on a 3-D curved surface, e.g. the inside surface of a

tunnel produced by matrix degradation of a lead or tip cell,

interactions between the transmembrane integrins and the

surrounding ECM create complex spatiotemporal dynamics

in forming focal adhesions and stress fibers, leading to

complex migratory behaviors strikingly different from the

2-D migration we observe in traditional gel surface experi-

ments.10 The objective of the present work is to build a

computational model to predict cell migration on 3-D curved

surfaces of ECM by integrating multifaceted mechanisms.

Recently, the development of a three-dimensional cell

culturing technique has provided a method to investigate a

novel mechanism of cell migrations in the ECM which has

been rarely observed using two-dimensional substrates.11–14 In

the 3D ECM environment, cells can interact with ligands of

ECM proteins with many microscopic properties including

fiber density, fiber strength, degree of cross-linking, filament

length and constitutive deformability of the scaffold through

integrin receptors, which lead to activation of signalling net-

works and cytoskeleton remodeling.15 Several studies have

emphasized the differences between 3-D cell migration and

2-D cell migration.16–18 Interestingly, it has been recently

reported that the higher activation level of Rac GTPase is

observed in 2D than 3D, which promotes cell spreading

and inhibits uniaxial migration phenotype observed in 3D.16

Another recent study addresses, unlike in 2D, the number of

focal adhesions in cells fully embedded in collagen 3D matrices

is smaller than that in cells on curved matrix surfaces (2.5D)

because cells tend to migrate toward stiffer 3D gel regions.18

On the other hand, current studies on angiogenic sprouting

using 3-D microfluidic assays present clear observations of

focal adhesions along the surface of the lumen, which differ

from such embedded 3-D cell experiments since the cells

migrate on curved surfaces.

This study is motivated by two experimental works; one on

cancer cell migration through conduits with diverse cross-

sections,19 and the other on angiogenic sprouting using 3-D

microfluidic assays.14 Both experiments have indicated that

3-D interactions between a cell’s cytoskeleton and a curved

surface directly affect the migration speed and direction.

Furthermore, when proceeding through a narrow conduit,

the cell exhibits a unique deformation pattern; not only does

its cytoskeleton conform to the geometric constraint, but its

nucleus deforms elastically and changes its location relative to

the rest of the cell.20,21 The specific goal of the present work is

to build a computational model to elucidate and predict these

experimentally observed migratory behaviors in relation to the

geometry of migration surfaces. This entails (a) deformation

mechanics of both cytoskeleton and nucleus, (b) 3-D inter-

actions between transmembrane integrins and ECM ligands,

leading to focal adhesion formation, and (c) stress fiber

formation and traction generation. Integration of these key

mechanisms is pivotal for elucidating the aforementioned

migration behaviors.

Here we describe 3-dimensional spatiotemporal dynamics of

cell migration by incorporating focal adhesion dynamics,

cytoskeleton and nuclear remodeling and actin motor activity,

all interfaced with a curved ECM surface. In the following,

experimental observations of cell migration on cylindrical

surfaces of the angiogenic lumen are first discussed, an integrated

migration model on such geometries is then presented, and

numerical simulation experiments demonstrate the diverse migra-

tion behaviors in relation to the geometry of migrating surfaces.

Experimental observations

To gain insights into cell migration on a curved surface,

experiments using 3-D microfluidic assays were conducted

(Fig. 1D(1)). The results showed cell behaviors significantly

different from traditional on-the-gel experiments.10 Fig. 1A

shows the schematic of a 3-D microfluidic assay for angiogenic

sprouting experiments of human Micro Vascular Endothelial

Cells (hMVECs). The cells seeded on one side of the collagen

gel are exposed to vascular endothelial growth factor (VEGF)

while a concentration gradient is created across the gel matrix.

Sprouts are formed from the monolayer of the seeded cells that

extend towards the higher concentration of VEGF. Holes are

created in the gel matrix by tip cells that cleave the gel with

matrix-degrading enzymes (matrix metalloproteinases, or

MMPs)22–24 (Fig. 1D(1)). Stalk cells14,25 migrate along the

hole created by the tip cell, crawling on the curved surface of

the conduit. Fig. 1B shows 3-D images of a stalk cell migrating

along a narrow conduit (15 mm). The cell is stretched out

parallel to the axis of the conduit, while the nucleus (shown

in blue) deforms to fit the narrow conduit. Focal adhesions

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(red spots) are distributed 3-dimensionally across the curved

conduit surface. Typically, focal adhesions are more highly

concentrated on the front of the migrating cell, as shown in

Fig. 1B(2), the top slice of the 3-D image. The middle slice

Fig. 1 Experimental observations of focal adhesion sites and actin stress fibers on the lumen in a 3-D collagen matrix: (A) 3-D microfluidic assay

with hMVECs seeded on one side of the collagen gel.14 Higher concentration of VEGF is supplied to channel A and lower concentration of VEGF

to channel B so that a gradient of VGEF is created across the gel. (B) Stalk cells migrating into the gel are observed; (1) collapsed confocal 3-D

image (120�) showing a stalk cell migration along a narrow lumen, and slices at selected heights of (2) z= 6.3 mm (top), (3) 10.08 mm (middle), and

(4) 16.38 mm (bottom); nucleus and focal adhesion sites are stained with Hoechst (blue) and vinculin (red), respectively. (C) Actin stress fibers in a

larger lumen with a magnification of 120�; sectional slices showing stress fibers at selected heights; (1) z = 0 mm, (B) 0.76 mm, (3) 5.32 mm and (4)

6.84 mm; nucleus and actin stress fibers are stained with Hoechst (blue) and Rhodamine phalloidin (red), respectively. (D(1)) A reflectance

microscopy image showing the creation of a hole by a migrating tip cell in the 3-D collagen matrix; nucleus and vascular endothelial (VE)-

cadherins are stained with Hoechst (blue) and anti-VE cadherin (green), (D(2)) collapsed confocal 3D image (120�); nucleus and actin stress fibers

are stained with Hoechst (blue) and Rhodamine phalloidin (red), respectively, (D(3)) the longitudinal cross sectional view across line aa0 shown in

D(2); short actin stress fibers are seen beneath the nucleus, (D(4)) the longitudinal cross sectional view across line bb0 shown in D(2); long stress

fibers (yellow arrows) connected to the nucleus extend towards the leading edge of the migrating cell. (E) Quantification of the aspect ratio of the

nucleus under two conditions when the cell migrates into the lumen with a diameter less than 25 mm and larger than 25 mm. Data are means � SD.

In each case 35 cells were analysed.

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(Fig. 1B(3)) indicates that focal adhesions are also created

along the side wall of the conduit, while Fig. 1B(4) shows the

bottom slice where the nucleus is located (Movie S1, ESIw).zSignificant deformations of both the cytoskeleton and the

nucleus in the longitudinal direction are observed along with

the 3-D distribution of FAs.

In a larger diameter conduit (B30 mm) the migrating cell

stretches more along the circumferential direction (Fig. 1C).

The migration speed in the axial direction was substantially

lower than that for the narrower conduit (15 mm). Actin stress

fibers stained in red are distributed across the wall of the

conduit. The nucleus appears to be pulled down by many short

stress fibers (approximately 2 mm) (Movie S2, ESIw). On the

other hand, in a narrow conduit (B15 mm) the migrating cell

elongates more along the longitudinal direction (Fig. 1D).

Actin stress fibers are distributed parallel to the longitudinal

direction. Longitudinal cross-sectional views of lines aa0 and

bb0 in Fig. 1D(2) are shown in Fig. 1D(3) and D(4), respec-

tively. The white arrows in Fig. 1D(3) indicate short stress

fibers beneath the nucleus, while yellow arrows in Fig. 1D(3)

and D(4) indicate long stress fibers connected to the nucleus.

These long stress fibers extend towards the front side of the

migrating cell. The nucleus also appears to be elongated along

the longitudinal direction; the aspect ratio of semi-major and

semi-minor is B2. The aspect ratio of nuclear size was

quantified and contrasted between cells in lumen o25 mm(1.91 � 0.26), and in lumen >25 mm (1.37 � 0.19) (Fig. 1E).

The aspect ratio of nuclear size is progressively increased as

the diameter of the lumen becomes narrower.

In summary, the experimental observations suggest that:

� The nucleus of a migrating cell deforms depending on the

geometry of a contacting surface;

� The cell cytoskeleton stretches out when passing a narrow

conduit;

� The cell stretches out in the circumferential direction when

the conduit diameter is large; and

� Short stress fibers are formed beneath the nucleus, pulling

it down towards the cytoskeleton, and long stress fibers can be

observed between the nucleus and focal adhesions at the front

side of the cytoskeleton.

These experiments support the hypothesis that migrating

cell behaviors are different depending on the surface geometry.

The results are aligned with the in vitro cancer cell migration

experiments by Irimia and Toner.19 They showed that the

speed of migrating cancer cells differs significantly depending

on the cross-sectional area of the microfluidic channel through

which the cell spontaneously migrates.yThe direction of

migration and its persistency, too, differ depending on the

width of the channel.19

Finally, the recent experiments by Chancellor et al.8 demon-

strate that the actomyosin tension is balanced in part by the

nucleus through mechanical links mediated by nesprin-1

(nuclear membrane proteins that bind to F-actin). We have

also observed SFs connected to the nucleus, as shown

in Fig. 1C and D. At the zero optical height, short SFs

(2–3 mm) are formed and connected to the nuclear membrane

on the bottom lumen that appear to tether the nucleus to the

cell membrane.26 Long SFs are also formed that point towards

the nucleus. These SFs connected to the nucleus play an

important role in cell migration. In ref. 8 the authors also

demonstrated that nesprin-1 depleted endothelial cells showed

decreased migration speed with no SFs connected to the

nuclear membrane. Furthermore, Khatau et al.26 highlighted

the interplay between cell shape, nuclear shape, and cell

adhesion mediated by the perinuclear actin cap.

Based on these experimental observations, we constructed a

3-dimensional cell migration model incorporating mechanical

links between FAs and the nucleus through contractile acto-

myosin motor activity, and formation of FAs through inter-

actions between transmembrane integrins and ligands on 3D

curved ECM surfaces. Our motivation is to investigate how

these mechanisms are coordinated to create migratory motion

on a curved surface, how a migrating cell deforms and spreads

out over a curved substrate, and how the migration speed and

direction are affected by the geometry of the curved surface.

Simulated migration behaviors will be compared to the experi-

mental results in terms of migration speed.

Methods

We aim to build a computational model to investigate cell

migration behaviors on a curved surface, as described in the

Experimental section. Specifically, we simulate haptotactic

migration due to a gradient in ligand density on a curved

surface. We simulate binding kinetics between integrin recep-

tors and extracellular matrix protein ligands (e.g. collagen,

fibronectin and laminin), model the formation of stress fibers,

and predict how the forces acting on the cell deform the

nucleus and the cytoskeleton, resulting in diverse patterns of

the cell profile and migratory motion.

Integrated cell migration model in the curved surface

We have built an integrated cell migration model incorporat-

ing focal adhesion (FA) dynamics, cytoskeleton and nucleus

remodeling and actin motor activity, and detailed modeling of

cell migration in the 2-D planar surface and fibronectin coated

patterns.9 The integrated cell migration model was further

extended to simulate cell migration on 3-D curved surfaces.

The simulation utilizes a Lagrangian approach with the time

dependent motions of nodes in double membranes of cell and

nucleus. The geometry of this double mesh structure is shown

in Fig. 2A; the outer mesh representing the cell membrane and

the inner mesh representing the nuclear membrane. Each mesh

consists of N nodes connected elastically to adjacent nodes,

forming a double elastic membrane. The inner and outer mesh

nodes may be connected when Stress Fibers (SFs) that are

formed between membranes of the nucleus and the cell.8,27

Multiple transmembrane integrins are bundled together and

placed at each node on the outer mesh. They can bind to

ligands on the matrix substrate, forming focal adhesions, to

which stress fibers are connected (Fig. 3A).

z In the recent literature18 the formation of focal adhesions in cellsembedded in collagen 3-D matrices has been addressed. The presentexperiments, however, differ from such embedded 3-D cell experimentssince the cells migrate on curved surfaces. Focal adhesions were clearlyobserved along the surfaces.y This will be discussed in detail later in the results and discussionsection. See Fig. 4A.

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Fig. 2B shows the free body diagram of the i-th node of the

cytoskeleton, called the i-th integrin node, where a bundle of

integrins is formed. Acting on this node are force vectors due

to frictional dissipative FD,ic(= �Ccvi

c) where Cc is a friction

coefficient (0.001 N s m�1)28–30 associated with the energy

dissipation at the integrin node and vic is the velocity of the i-th

integrin node, focal adhesion force FFA,ic, elastic energy force

FE,ic, and stress fiber force FSF,i

c. Similarly, acting on the i-th

node of the nucleus are force vectors due to frictional dis-

sipative force FD,in(= �Cnvi

n) where Cn is a friction coefficient

(0.001 N s m�1)28–30 associated with the energy dissipation at

the nuclear node and vin is the velocity of the i-th nuclear node,

elastic energy force FE,in and stress fiber force FSF,i

n. The

equation of motion for each integrin node is given by

mic dvi

c

dt¼ FD;i

c þ FFA;ic þ FE;i

c þ FSF;ic; i ¼ 1; . . . ;N:

ð1Þ

where vic is the velocity vector of the i-th integrin node.

Similarly, the equation of motion for each node of the nucleus

is given by

min dvi

n

dt¼ FD;i

n þ FE;in þ FSF;i

n; i ¼ 1; . . . ;N ð2Þ

where FD,in, FE,i

n and FSF,in are frictional dissipative force,

elastic energy force and SF force at the i-th nuclear node,

respectively, and vin is the velocity of the i-th nuclear node. It

should be noted that sum of forces at any node is zero because

their motions are very slow and in quasi-equilibrium in time,

thus motion equations in terms of the velocities vic and vi

n

dxic

dt¼ vi

c; dxin

dt¼ vi

n� �

can be simplified to the following six

ordinary differential equations:

Ccdxi

c

dt¼ FFA;i þ FE;i

c þ FSF;ic; i ¼ 1; . . . ;N ð3Þ

Cndxi

n

dt¼ FE;i

n þ FSF;in; i ¼ 1; . . . ;N ð4Þ

where xic and xi

n represent coordinates of the i-th integrin node and

the i-th nuclear node (Fig. 3A), respectively. Most of the frictional

dissipative term FD,ic arises from the rupture of stretched ligand–

receptor bonds (Fig. 3B) in the focal adhesion dynamics; when they

rupture, the stored strain energy is released and dissipated. Simi-

larly, FD,in also arises from the energy stored in SFs that, when

F-actins are depolymerized, the stored strain energy is released and

dissipated. The focal adhesion force FFA,i acts between the i-th

integrin node and the point on the curved ECM surface where the

extension of the unit normal vector, n̂R,i, intersects with the curved

ECM surface.9 From Fig. 3B this intersection position, that is, the

root location of receptor and ligand bonds (xL,i), is given by

xL;i ¼ xic þ Lbn̂R;i ¼ xi

c � hpn̂R;i

n̂w � n̂R;ið5Þ

where Lb is the bond length, n̂w is the unit normal vector of the

ECM surface, and hp is the gap between the i-th integrin node and

the curved ECM surface, as shown in Fig. 3B. These expressions

are valid only when n̂w�n̂R,io 0 and the gap hp is less than a critical

height (hc) of 300 nm: hpo hc. The latter condition is to restrict the

formation of receptor–ligand bonds within the upper limit hc. The

focal adhesion force of the i-th integrin node FFA,i is computed as

FFA,i = nb,i kLR (Lb � l)n̂R,i (6)

where nb,i is the number of ligand–receptor bonds at the i-th

integrin node, kLR is an effective spring constant for a single

ligand–receptor bond (B1 pN nm�1),31 and (Lb � l) repre-sents averaged stretched distance from the equilibrium length

(l; 30 nm32) (see Fig. 3C). The detailed explanations of focal

adhesion dynamics using Monte Carlo simulation methods

can be found in the literature.9

Forces due to actin SFs’ motor activity at the i-th integrin

and j-th nuclear nodes are given by

FSF;ic ¼ � kSF

NSF;iðdSF;i �NSF;iLSF;i

1Þ @dSF;i@xi c

ð7aÞ

FSF,jn = �FSF,i

c (7b)

where kSF ¼ ESFASF

LSF;i1

� �is stiffness of a SF which is variable

depending on Young’s modulus of SFs (ESF; 230 kPa33), average

cross-sectional area of SFs (ASF; 250 nm in radius34) and a length

of a single compartment of the i-th SF (LSF,i1),NSF is the number

of compartments of the i-th SF, and dSF,i is full length of SFs

under tension (see Fig. 3D). In particular, considering myosin II’s

slide on F-actin filaments with a sliding rate of vm at both

ends,35–37 LSF,i1 can be calculated with following discretized form

of equation at every time step (Dt):

dLSF;i1

dt¼ �2vm ð8aÞ

LSF,i1 = LSF,i

0 �2vmDt (8b)

where LSF,i0 indicates the length of a single unit of the i-th SF

at the previous time (t � Dt).These forces are generated when focal adhesions have been

formed and F-actin filaments are fully polymerized. It has

Fig. 2 (A) Integrated cell migration model consisting of the cytoskeleton,

the nucleus, N integrin nodes on the surface of cytoskeleton, N nucleus

nodes on the surface of nucleus, and actin stress fibers which connect the

integrin node to the nucleus node; (A) a top view of the model showing

triangular mesh network of double membranes of cytoskeleton and

nucleus. (B) The free body diagram of the i-th integrin node in the circle

marked in (A) where five external forces are acting. Note that, while

showing in 2-D, the force balance exists in 3-D.

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been known that SF assembly occurs over several min-

utes,38–40 but SFs disassemble rapidly within seconds.41–44 In

addition, it takes several minutes to form FAs from focal

complexes (FCs). These observations suggest that myosin

motor activities in SFs are switched off during the remodelling

of the actin cytoskeleton (polymerization) and SFs turnover.

In our simulations, time for full formation of F-actin is set

to be Tp = 180 s, and time for the complete disassembly

of F-actin is set to 1 s, based on the above reference

information.

The elastic forces, FE,ic and FE,i

n, are obtained by using the

virtual work theory in structural mechanics.9 It should be noted

that the elastic forces FE,ic and FE,i

n at the i-th node represent

the resultant force acting on the i-th node that is calculated by

vectorial addition of elastic forces from neighbouring nodes. To

compute this, first the coordinates of each node are updated in

each time cycle, and distances from each node to neighbouring

nodes are computed along with the areas of the surrounding

rectangles. The elastic forces are derived from these distances

and areas for individual nodes.

Fig. 3 3-D integrated cell migration model: (A) 2-D schematic representation of a cell migration model on the curved substrate, showing

deformable cell and nuclear membranes, and actin stress fibers, (B) a magnified view in A showing the structure of focal adhesion including the

attachment of the end of stress fibers through an integrin node to the underlying extracellular matrix, (C) a magnified view in B illustrating a

stochastic ligand–receptor bonding process at the focal adhesion site, and (D) a magnified view in A showing the structure of actin stress fibers

connected to a nucleus node.

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Computation of an ‘‘integrated cell migration model’’

Cell migration simulations were carried out using a fourth

order Rosenbrock method45 based on an adaptive time-

stepping technique for integrating ordinary differential equa-

tions with the convergence criterion o10�4. The ordinary

differential equations were solved for the 6 � N (N = 98)

unknown variables associated with the mesh node position

vectors for both cell membrane and nuclear membrane: xic,

xin, i = 1 – N. For cell migration simulation the Rosenbrock

method outperforms the standard Runge–Kutta method

which requires a relatively large number of iterations.45

Furthermore, the Rosenbrock method consumes less comput-

ing time by using adaptive time-step control that ranges from

10�3 s to 10�2 s in the present work. Thus, it is suitable for

simulating transient cell migratory behaviors over 10 hours.

The methods for building geometrical models for the simu-

lation of cell migration have been well documented in the

literature.46,47 See also Fig. S1 (ESIw). One practical issue in

computing finite mesh geometric models is to check geome-

trical compatibility. As the coordinates of cell membrane and

nuclear nodes are updated based on the equations of motion,

geometrically incompatible situations occur occasionally in

the configurations of the cell membrane mesh and that of

the nucleus in relation to the curved ECM surface. For

example, some cell membrane nodes intersect with the conduit,

and the nucleus intersects with the cell membrane. These

incompatible situations must be checked in every computa-

tional cycle, and necessary corrections must be made.

Cell migration in the collagen gel matrix

hMVECs, cell culture media and supplements were purchased

from Lonza. All cell culture work was carried out in sterile

tissue culture hoods and cell culture was carried out in a 5%

CO2 humidified incubator at 37 1C. hMVECs were cultured in

EGM-2MV (Lonza) culture medium, grown in a T75 tissue

culture flask (Corning), and trypsinized when the culture flask

becomes confluent (0.25% with EDTA, Gibco). Afterwards,

75 mL of cell suspension with a cell density of 2� 106 cells mL�1

were loaded into the Micro-fluidic channel B (Fig. 1A) of the

in vitromicrofluidic device for observing the angiogenesis.14 The

pressure difference between channels A and B (Fig. 1A) enables

interstitial flow and seeded cells to adhere on the side surface of

the collagen gel matrix, followed by putting the device into the

5% CO2 humidified incubator for several hours to form a

monolayer on the side of the collagen gel matrix. Once the

confluent monolayer was formed on the surface of channel B,

fresh media with 20 and 40 ng mL�1 of vascular endothelial

growth factor (VEGF; R&D systems) were, respectively,

replenished into channels A and B to generate a gradient.11,14

Stalk cell migration was monitored by confocal microscopy

with an incubator keeping 5% CO2 at 37 1C with a time-

interval of 30 minutes for 12 hours. Immediately after the live

cell imaging, immunofluorescence staining was performed

to visualize the intracellular structure of the stalk cell at the

end-point of the experiment. Cells were fixed with 4%

paraformaldehyde (Sigma-Aldrich) for 15 minutes at room

temperature, and permealized with 0.1% Triton X-100

(Sigma-Aldrich) for 5 minutes. Actin stress fibers, vinculin

and nuclei were stained with Rhodamine-Phalloidin (Sigma-

Aldrich), monoclonal anti-vinculin antibody produced in mouse

(Sigma-Aldrich) and Hoechst (Sigma-Aldrich), respectively.

Results and discussion

Simulations of cell migration were performed for various

conduit geometries and ligand density gradients. Type I

collagen was considered for the ligands on the conduit surface.

Two cases were examined: (a) uniform ligand concentration,

and (b) graded ligand concentration. For case (a), the ligand

concentration was uniformly set to be 0.8 mg mL�1 over a

longitudinal conduit length of 100 mm. Since the molecular

mass of Type I collagen is 350 kDa, the corresponding ligand

surface density was 750 molecules mm�2 using the relationship

between plating concentration and ligand surface density of

type I collagen48 (Fig. 4). For case (b), the high ligand

concentration was varied continuously from 2.60 mg mL�1

to 3.35 mg mL�1 over a longitudinal conduit length of 100 mm.

This created a density gradient of 1.2 ng mm�3, whose ligand

surface density was curve-fitted as 1.25 � 103 molecules mm�2

at the lowest end and 1.55 � 103 molecules mm�2 at the highestend using the relationship between plating concentration

and ligand surface density of type I collagen48 (Fig. S2, ESIwand Fig. 5).

At the initial state of each simulation, both cell and nuclear

membranes were assumed to be round. Since the migration

model is stochastic, simulations were repeated multiple

times under the same initial conditions. Table 1 lists all the

parameters used for the simulations with numerical values and

their sources.

Comparison to experimental data

The first set of cell migration simulations was aimed to

compare the integrated model against the experimental data

published previously. Irimia and Toner19 performed cancer

cell migration experiments in confined microfluidic channels

having a uniform ligand concentration. They have reported

that the observed spontaneous cell migration along conduits

of uniform ligand concentration differs significantly depending

on the conduit cross-section. The simulation results, too,

showed similar behaviours.

Fig. 4A and Fig. S2A (ESIw) show sample trajectories of

simulated cell migrations along rectangular conduits of

various widths. The conduit sizes used for the simulations

matched those of the available experimental data; the depth of

the conduit was fixed to 3 mm, while the width was varied to 7

different values: 6 mm, 10 mm, 15 mm, 20 mm, 30 mm, 50 mm and

70 mm. First the total path length of each trajectory was

obtained and was divided by the travelling time, 3 hours, to

obtain the average migrating speed. In the experiments,

the speed of cancer cell migration was monitored in every

6 minutes, and was time averaged over the entire migration

period for each of the channel geometries. Fig. 4B compares

the average migration speed between the experiment and

simulations.

The experimental data show that the cell velocity is the

lowest when migrating in the narrowest conduit, increases with

the increasing cross-sectional width, reaches a maximum value

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at the width of 20 mm, and then decreases as the conduit

becomes wider (Fig. 4B and Fig. S2B, ESIw). The simulated

cell migration speed, too, shows a trend similar to the experi-

ments: slow for a very small cross section, the fastest in the

midrange, then slower again. Both experiments and simula-

tions attain the fastest speed around 20–30 mm of width, or

60–90 � 10�12 m2 of cross-sectional area. As the width became

even wider, the simulated speed decreased modestly while the

experiment speed decreased more rapidly. Overall both show a

good agreement over the range of 6–30 mm of conduit width.

Statistical analysis of linear regression was performed by

comparing the experiment and the simulation in terms of the

‘‘mean values’’ of time-averaged cell migration speed for the

same conduit cross section. As shown in Fig. 4C and Fig. S2C

(ESIw), significant correlations were found between the two

with R2 = 0.771 and R2 = 0.719, respectively. Thereby, cell

migration speeds are strongly dependent on channel’s widths.

Interestingly, as the width becomes narrower than 15 mm, cell

trajectories are almost straight lines since the cell contacts all

four walls of the channel. On the other hand, as the width

becomes wider, the cell tends to wander, following a curved

trajectory until it contacts a side wall of the conduit.

Since the cell’s motility mechanism is stochastic due to the

binding and rupture kinetics of integrin-receptors and ligands,

the cell does not necessarily move in the direction of higher

ligand density. To further examine migration direction, the

same simulations were repeated for conduits having the ligand

concentration gradient, 1.2 ng mm�3, as described previously.

Despite the gradient the cells migrated laterally or backward

as shown in Fig. S2A and Movie S3a (ESIw). Since the liganddensity has a gradient, on average the cell moves towards the

higher ligand density (Fig. S2A, ESIw). But, it may move in

the opposite direction with a certain probability. Occasional

backward migration was observed in our experiments as well

(Movie S3b, ESIw).It is interesting to note that the cell, once committed to

moving in one direction, tends to keep moving in the same

direction. It persists to move in the same direction until

it stochastically switches direction. The duration of the

persistent migration varies depending on the ligand gradient

as well as on the geometry of the conduit. The conduit width

affects this persistency of migration direction. As shown in

Fig. S2A (ESIw), the trajectories for the conduits of widths 6,

10 and 15 mm are almost straight, while the ones for 50 and

60 mm of conduit width are winding. It is noticeable that the

cell membrane profile for the 6 mm conduit is so narrow that

the cell membrane is in contact with the four walls surrounding

the cell, creating a smaller difference in the FA numbers

Fig. 4 (A) Simulated trajectories of cell migrations along seven rectangular conduits with the identical height of 3 mm, and different widths of 6 mm,

10 mm, 15 mm, 20 mm, 30 mm, 50 mm and 70 mm. Cells are initially spherical. The ligand surface density is 750 molecules mm�2 and constant over a

longitudinal conduit length of 100 mm. The black lines indicate trajectories of nuclei for the first three hours, (B) comparison of average cell migration

speeds: the simulation model vs. experimental data by Irimia and Toner.19 Average speed and standard error of mean (N=5) are shown for the seven

different channels, and (C). Linear regression (R2 = 0.771) of simulated migration speed vs. experimental migration speed.

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between the leading and tailing edges. This results in a smaller

traction force and slower migration speed than others. As the

width gets wider, the cell has more room for spreading the

shell-shaped cell membrane on the leading edge, resulting in

higher speeds, as shown in 20 and 30 mm conduits (Fig. S3 and

Movie S3c, ESIw).

Cell migration along cylindrical lumens

Migration simulations under the same ligand gradient along

cylindrical conduits with diverse diameters: 6 mm, 8.8 mm,

12 mm, 15 mm, 20 mm and 30 mm, produced results similar to

the rectangular conduit cross sections. The results show that

migration speed is maximized at a specific diameter of conduit

and that the migration direction was influenced by the conduit

diameter (Fig. 5A–C). The maximum speed was found at a

diameter around 10 mm, which is interestingly equivalent to

the capillary diameter (Fig. S4B, ESIw). As the diameter was

reduced to 6 mm, the migratory speed became very slow

because most of integrin nodes (97%) contacted the cylindrical

wall. This caused the density of FAs to be almost the same

between the leading and tailing edges. Furthermore, the

magnitude of actin SF forces at the tailing edge is relatively

strong because the actin SFs are aligned with the longitudinal

direction of the lumen (Fig. S4C, ESIw). This makes the

backward force larger, resulting in a small net traction (Movie

S4a, ESIw). As the conduit diameter became larger, 8.8 and

12 mm, the density of tailing edge FAs reduced, and SFs

became short and not aligned with the longitudinal direction,

while the leading edge had many long SFs well aligned with the

longitudinal direction, resulting in a large traction force and

fast migration (Movie S4b, ESIw). Fig. 5D shows longitudinal

and transverse cross-sectional views of the cell. In this range of

conduit diameter the cell membrane contacted the entire

circumference of the conduit, thereby plugging the conduit.

This prevented the cell from moving sideways, and directed it

only towards the longitudinal direction; the trajectories of the

nucleus are almost straight lines, as seen in Fig. 5C for

diameters of 6, 8.8, and 12 mm. The cell body stretched out

in the longitudinal direction, and its length is approximately

inversely proportional to the conduit cross-sectional area:

longitudinal body lengths of 23 mm, 15 mm, and 10 mm were

observed for conduit diameters of 6 mm, 8.8 mm, and 12 mm,

respectively (Fig. S4, ESIw). Furthermore, the nucleus is most

elongated when the lumen is narrowest. Similar experimental

observations were reported that cells were restricted and nuclei

were stretched at the widths 10–50 mm of the underlying

FN patterns.26

As the conduit diameter further increases, the cell no longer

plugged the conduit, but tended to move in the transverse

directions. See the conduit diameter of 15, 20 and 30 mm, in

Fig. 5C. The conduit curvature affects cell movement signifi-

cantly; the probability of integrin–ligand bond formation for a

curved surface (transverse direction) is significantly higher

than that of a planar surface (longitudinal direction), since

the distance between an integrin and a ligand becomes shorter

(see Fig. 3B). As a result, the cell stretched out in the

circumferential direction, and tended to move transversely.

See the cross-sectional views in Fig. 5E (Movies S4d and S4e,

ESIw). Spiral movements were also observed for the conduit

diameters of 15 mm and 20 mm. When the diameter of

the lumen was 30 mm, the cell wandered in the transverse

direction, and stretched out along the conduit circumference

(Movie S4f, ESIw). However, the probability of the lateral

cell migration will increase with a shorter distance between

integrin–ligand formation as the diameter increases. However,

note that there is an exceptional case when the radius of

curvature increases largely, the lumen becomes flatter locally.

In this case, the distance between integrin–ligand formation is

uniform such that cell migration becomes increasingly similar

to randomised 2-D cell migration on a planar surface. It is of

interest to identify the critical radius of curvature for the

transition from directed, lateral cell migration to randomised

cell migration. However, this critical radius of curvature for

the transition may be different for different cell types due, for

Fig. 5 (A) Average cell migration speeds and standard error of mean

(N = 5) are shown for cylindrical luminal diameters of 6, 8.8, 12, 15,

20 and 30 mm, 5 hour-trajectories of cell migrations in corresponding

lumens in (B) cross-sectional views, (C) top views, and (D) and (E)

longitudinal and transverse cross sectional views of migrating cells

along a narrow (8.8 mm) and wider (20 mm) cylindrical conduits, and

(F) an example of backward cell migration in the lumen despite a

positive ligand gradient on its surface (diameter of 12 mm).

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example, to the size of the cell relative to the diameter of the

lumen; as the size of the stretched cell increases, the critical

radius of curvature for the transition may be relatively

increased more. We presume the critical radius of curvature

for the transition in current simulations is at the value larger

than our maximum radius of 30 mm.

The stochastic persistence phenomenon was also observed.

Once the cell began to move in one direction, it continued to

move in that direction for some time, but it occasionally

switched the direction of migration (Movie S4c, ESIw).

Computation of dissipative friction coefficients

The frictional dissipation term, FD,ic = �Cc[|vi

c|]vic, in the

equation of motion is a state-dependent, nonlinear term. In the

simulations, a fixed value was used for the coefficient Cc based

on the literature information: Cc = 0.001 N s m�1.28–30 This

coefficient value is now examined below.

The dominant effect of this frictional dissipation comes

from the energy release due to the rupture of integrin

receptor–ligand bonds. The dissipated energy arises from the

tension built up at each focal adhesion, which is released when

the bond ruptures. The probability of bond rupture as well as

bond formation depends on the local geometry of each

integrin and surrounding ligands, the relationship of which

varies depending on the velocity of the integrin node relative

to the substrate, vic. By simulating this process we can obtain

the relationship between the frictional dissipation force and

the velocity.

As shown in the inset of Fig. 6, the cell membrane is moved

at a constant speed |vic| along the bed of ligands, and the

formation and rupture of ligand–receptor bonds are simulated

based on the stochastic computation algorithm described

previously. Since the transmembrane integrins are at nodes

of the elastic mesh structure, the integrins are suspended

elastically, as illustrated in the inset of Fig. 6. Once a bond

is formed, the integrin pulls the ligand as the cell membrane

moves, the tension increases as it elongates, and finally the

bond ruptures. Fig. 6 shows stochastic simulations delineating

the relationship between the cell membrane speed and the

rupture-induced dissipative energy release per unit time, i.e.

power loss. (See ESIw for mathematical derivation.) The figure

also shows how long individual bonds last, i.e. bonding time,

indicated with color-coded dots. There are two strikingly

different groups of data points; one in a lower speed range

and the other at higher speeds. At low speeds, the bonds tend

to last for a longer time, generating a larger force and a larger

amount of energy release, while at higher speeds the bonds

rupture immediately. Each data group can be approximated

by a curved solid line, the slope of which gives the coefficient

Cc (see ESIw for details). These two different values of Cc are

resulted from different speeds of |vic|. For example, as the

Table 1 List of simulation parameters

Parameter Definition Value, (equation) Sources

ASF Averaged SFs’ sectional area/mm2 0.196, (7a) 34Cc Friction coefficients associated with the energy dissipation at the integrin node/N s m�1 0.001, (3) 28–30Cn Friction coefficients associated with the energy dissipation at the nuclear node/N s m�1 0.001, (4) 28–30F Force/N (1)–(4) and (6)–(7)L Length (5), (6) and (7a)Lb Stretched length of bonds between receptors and ligands Variable, (5)LSF,i

1 Length of the i-th single unit of SFs at the present time/nm Variable, (7a) and (8) Current workLSF,i

0 Length of the i-th single unit of SFs at the previous time/nm Variable, (8b) Current workN Number of nodes at each membrane 98 Current workNSF,i Number of contractile compartments in the i-th SFs Variable, (7a) Current workdSF,i Distance between i-th integrin and j-th nuclear nodes/m Variable, (7a) Current workhc Critical height/nm 300, (5) Current workhp Height from the surface to the i-th integrin node/nm Variable, (5) Current workkLR Effective spring constant of ligand-receptor bond/pN nm�1 1.0, (6) 31kSF Effective stiffness of the i-th single unit of SFs/N m�1 Variable, (7a) Current worknb,I Number of bonds between receptors and ligands Variable, (6) Current workn̂R,i Unit normal vector at the i-th integrin node (5) and (6) Current workn̂w Unit normal vector at the local surface of the lumen (5) Current workt Time/s (1)–(4), and (8)v Velocity vector/nm s�1 (3) and (4)vm Sliding rate of non-muscle myosin II on the actin filaments/nm s�1 10, (8) 35–37x Location vector/mm (3)–(5), and (7a)xL,i Location vector of the root of ligand–receptor bonds on the local surface of the lumen/nm (5)l Equilibrium distance of an integrin/nm 30, (6) 32SuperscriptsD Drag or friction (1) and (2)E Elastic (1)–(4)FA Focal adhesion (1), (3) and (6)SF Stress fiber (1)–(4) and (7)c Cytoskeleton (1), (3), (5) and (7)n Nucleus (2), (4) and (7b)i i-th node (1)–(8)0 Previous time or initial state (8b)1 Present time (7a) and (8a)Subscriptsb Bond (5) and (6)

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power loss is lower and |vic| is higher, the value of Cc becomes

lower (equation (A10), ESIw). In the migration simulation, the

fixed coefficient value Cc = 0.001 N s m�1 was used, which

gives the linear relationship shown by the red straight line in

Fig. 6. Note that this value taken from the literature can

approximate the data very accurately. The caveat is that at

higher speeds the coefficient Cc reduces significantly, as shown

by the blue line in Fig. 6. The power loss due to the quick

ruptures is negligibly small compared to that of the red line,

and the occurrence probability of quick ruptures is much

lower than that of the red line. Therefore, it is justifiable to

use the fixed value Cc = 0.001 N s m�1 found in the literature

for migration simulations.

Integrated cell migration model

The current integrated cell migration model has been developed

for the general cell migration on the surface of ECMmolecule or

intercellular adhesion molecule (i.e. VE-cadherin) coated sub-

strates independent of cell type. In order to mimic a specific cell

type simply requires changing the size of the cell, or the numbers

of adhesion molecules per node, or per sectional area of actin

stress fibers etc. For example, in the case of fibroblasts which

have a higher number of focal adhesions, the density of integrin

nodes on the cell membrane (Fig. 2) can be increased. Addi-

tionally, although our model as currently constructed is limited

to migration along a surface, it has the potential to incorporate

effects that would permit the simulation of 3-D migration. This

would require the addition of MMP activity which is ongoing

work. Finally, the model can be further extended to simulate

homogeneous cell–cell interactions as well as heterogeneous

cell–cell interactions to simulate paracellular or transcellular

migration of immune cells across endothelial monolayers.49

Cell migration is a complex multifaceted process, triggered

by chemotaxis and haptotactic responses from the extra-

cellular environment.1 The motion of cell migration model is

initially triggered by strong traction force from the ECM

molecules like a thin lamellipodium protrusion at the leading

edge, followed by the assembly of a number of focal adhesions

between the lamellipodium base and the ECM. Afterwards,

actin stress fibers extend from nascent focal adhesions and

some of which connect to the nucleus. The corresponding

motor activity exerts force on the FAs fore and aft, enabling

the generation of a traction force and the release of FAs in the

rear of the cell, creating the cell body’s forward movement.

The current integrative cell migration can provide new

biological insight into designing a better experiment. For

example, as an extension of cell migration on the curved

surface, it is of interest to predict cell migration on a wavy

surface. The model can predict whether cells migrate perpendi-

cular to the grooves or how the migration direction differs if

the cells are on a concave or convex surface. The model we

proposed will not only provide new insight into better building

experiments, but also such an experiment will allow us to

better validate the model. Thus, as a selected application, we

confirm how this cell migration model may be applied to

the designs of efficient experiment for cell migration and

further experiment for spontaneous cancer cell migration for

a diagnostic assay.

Conclusion

An integrated computational model for predicting cell migra-

tion on 3-D curved surfaces has been presented. The equations

of motion based on the elastic double mesh structure allowed

both cell membrane and nucleus to deform flexibly in accor-

dance with forces acting on each mesh node, including focal

adhesion (FA) force, elastic membrane forces, and frictional

dissipative force. Integrins distributed over the cell membrane

interact stochastically with ligands on the ECM lumen surface,

form FAs, anchor them to the surface, create stress fibers

(SFs), and collectively generate a traction force for migration.

Probabilities for formation and rupture of ligand–receptor

bonds are affected by the surface curvature and lumen

Fig. 6 Dissipative friction due to bond ruptures; a graph of power released by ruptures of integrin receptor–ligand bonds versus jvi cj2 a color map

indicates receptor–ligand bonding time, and a schematic in the inset of the i-th integrin node representing frictional dissipations due to ruptures of

integrin receptor–ligand bonds.

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geometry, resulting in unique migration behaviors, which

cannot be explained by migration models for 2-D flat surfaces.

Specifically, we have found.

(1) The migration speed takes a maximum at a particular

diameter or width of the conduit along which the cell migrates.

The computational model successfully produced the speed vs.

conduit width relationship consistent with the existing micro-

fluidic experimental data of cancer cell migration.

(2) For a narrow lumen, the cell is confined and stretched

along the longitudinal direction, contacting all the circumfer-

ence of the lumen and resulting in a straight movement

directed towards the longitudinal direction. For a wide lumen,

the cell tends to stretch out along the circumference, where

the radius of curvature is smaller than that of the longi-

tudinal direction, resulting in a high probability for transverse

migration.

The model can be extended to a more complex model including

more details about mechanisms of cell–cell interactions.

Acknowledgements

The authors thank the Singapore-MIT Alliance of Research

and Technology for financial support of this work. This

material is based upon work supported by the National

Science Foundation under Grant No. EFRI-0735997 and

Grant No. STC-0902396.

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