# Lifetime and strength of periodic bond clusters between elastic media under inclined loading.

**ABSTRACT** Focal adhesions are clusters of specific receptor-ligand bonds that link an animal cell to an extracellular matrix. To understand the mechanical responses of focal adhesions, here we develop a stochastic-elasticity model of a periodic array of adhesion clusters between two dissimilar elastic media subjected to an inclined tensile stress, in which stochastic descriptions of molecular bonds and elastic descriptions of interfacial traction are unified in a single modeling framework. We first establish a fundamental scaling law of interfacial traction distribution and derive a stress concentration index that governs the transition between uniform and cracklike singular distributions of the interfacial traction within molecular bonds. Guided by this scaling law, we then perform Monte Carlo simulations to investigate the effects of cluster size, cell/extracellular matrix modulus, and loading direction on lifetime and strength of the adhesion clusters. The results show that intermediate adhesion size, stiff substrate, cytoskeleton stiffening, and low-angle pulling are factors that contribute to the stability of focal adhesions. The predictions of our model provide feasible explanations for a wide range of experimental observations and suggest possible mechanisms by which cells can modulate adhesion and deadhesion via cytoskeletal contractile machinery and sense mechanical properties of their surroundings.

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**ABSTRACT:**Cells sense and respond to the elasticity of extracellular matrix (ECM) via integrin-mediated adhesion. As a class of well-documented mechanosenors in cells, integrins switch among inactive, bound, and dissociated states, depending upon the variation of forces acting on them. However, it remains unclear how the ECM elasticity directs and affects the states of integrins and, in turn, their cellular functions. On the basis of our recent experiments, a biomechanical model is proposed to reveal the role of ECM elasticity in the state-switching of integrins. It is demonstrated that a soft ECM can increase the activation level of integrins while a stiff ECM has a tendency to prevent the dissociation and internalization of bound integrins. In addition, it is found that more stable focal adhesions can form on stiffer and thinner ECMs. The theoretical results agree well with relevant experiments and shed light on the ECM elasticity-sensing mechanisms of cells.Journal of biomechanics 01/2014; · 2.66 Impact Factor -
##### Article: A stochastic description on the traction-separation law of an interface with non-covalent bonding

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**ABSTRACT:**We formulate a stochastic description about the mechanical response of an interface composed of non-covalent bonds. In such interfaces, the evolution of bonding probability in response to deformation plays the central role in determining their traction-separation behavior. The model connects atomistic and molecular level bonding properties to meso-scale traction-separation relationship in an interface. In response to quasi-static loading, the traction-separation of a stochastic interface is the resultant of varying bonding probability as a function of separation, and the bonding probability follows the Boltzmann distribution. The quasi-static stochastic interface model is applied to understand the critical force while detaching a sphere from an infinite half space. We further show the kinetics of interfacial debonding in the context of the Bell model (1978) and two of its derivatives – the Evans-Richie model (1997) and the Freund model (2009). While subjected to constant force, an interface creeps and its separation–time curve shows typical characteristics seen during the creep of crystalline materials at high temperature. When we exert constant separation rate to an interface, interfacial traction shows strong rate-sensitivity with higher traction at faster separation rate. The model presented here may supply a guidance to bring the stochastic nature of interfacial debonding into theories on cracking initiation and growth during fatigue fracture.Journal of the Mechanics and Physics of Solids 10/2014; 70. · 4.29 Impact Factor - SourceAvailable from: Roger D KammGang Bao, Yuri Bazilevs, Jae-Hyun Chung, Paolo Decuzzi, Horacio D Espinosa, Mauro Ferrari, Huajian Gao, Shaolie S Hossain, Thomas J R Hughes, Roger D Kamm, Wing Kam Liu, Alison Marsden, Bernhard Schrefler[Show abstract] [Hide abstract]

**ABSTRACT:**Over decades, the theoretical and applied mechanics community has developed sophisticated approaches for analysing the behaviour of complex engineering systems. Most of these approaches have targeted systems in the transportation, materials, defence and energy industries. Applying and further developing engineering approaches for understanding, predicting and modulating the response of complicated biomedical processes not only holds great promise in meeting societal needs, but also poses serious challenges. This report, prepared for the US National Committee on Theoretical and Applied Mechanics, aims to identify the most pressing challenges in biological sciences and medicine that can be tackled within the broad field of mechanics. This echoes and complements a number of national and international initiatives aiming at fostering interdisciplinary biomedical research. This report also comments on cultural/educational challenges. Specifically, this report focuses on three major thrusts in which we believe mechanics has and will continue to have a substantial impact. (i) Rationally engineering injectable nano/microdevices for imaging and therapy of disease. Within this context, we discuss nanoparticle carrier design, vascular transport and adhesion, endocytosis and tumour growth in response to therapy, as well as uncertainty quantification techniques to better connect models and experiments. (ii) Design of biomedical devices, including point-of-care diagnostic systems, model organ and multi-organ microdevices, and pulsatile ventricular assistant devices. (iii) Mechanics of cellular processes, including mechanosensing and mechanotransduction, improved characterization of cellular constitutive behaviour, and microfluidic systems for single-cell studies.Journal of the Royal Society, Interface / the Royal Society. 08/2014; 11(97).

Page 1

Lifetime and Strength of Periodic Bond Clusters between Elastic Media

under Inclined Loading

Jin Qian,†Jizeng Wang,†Yuan Lin,‡and Huajian Gao†*

†Division of Engineering, Brown University, Providence, Rhode Island; and‡Department of Mechanical Engineering, The University

of Hong Kong, Hong Kong, China

ABSTRACT

understand the mechanical responses of focal adhesions, here we develop a stochastic-elasticity model of a periodic array of

adhesion clusters between two dissimilar elastic media subjected to an inclined tensile stress, in which stochastic descriptions

of molecular bonds and elastic descriptions of interfacial traction are unified in a single modeling framework. We first establish

a fundamental scaling law of interfacial traction distribution and derive a stress concentration index that governs the transition

between uniform and cracklike singular distributions of the interfacial traction within molecular bonds. Guided by this scaling

law, we then perform Monte Carlo simulations to investigate the effects of cluster size, cell/extracellular matrix modulus, and

loading direction on lifetime and strength of the adhesion clusters. The results show that intermediate adhesion size, stiff sub-

strate, cytoskeleton stiffening, and low-angle pulling are factors that contribute to the stability of focal adhesions. The predictions

of our model provide feasible explanations for a wide range of experimental observations and suggest possible mechanisms by

which cells can modulate adhesion and deadhesion via cytoskeletal contractile machinery and sense mechanical properties of

their surroundings.

Focal adhesions are clusters of specific receptor-ligand bonds that link an animal cell to an extracellular matrix. To

INTRODUCTION

Most animal cells cannot survive in isolation and must

adhere to adjacent cells or extracellular matrix (ECM)

through the formation of focal complexes (FXs) or focal

adhesions (FAs) (1). FXs are small primordial adhesion clus-

ters of specific membrane-bound receptors and their comple-

mentary ligands on ECM formed close to the edge of

advancing membrane protrusions of a cell, whereas FAs

are more mature, stable, micron-sized bond clusters linking

cells to ECM (2). FAs are usually exposed to forces induced

by external physical interactions such as blood flow, as well

as those generated by cell’s own contractile machinery as

stress fibers made of bundles of actin filaments and myosin

II motors actively pull FAs at an inclined angle with respect

to the cell-ECM interface (3). These forces are known to

exert significant influences on cell shape, cytoskeleton orga-

nization, and intracellular processes such as cell growth,

differentiation, motility, and apoptosis (4). Cells react to

mechanical signals by producing a series of biochemical

processes within the FAs. The feedback loop between

mechanical stimuli and biochemical responses is critical to

the regulation of cell adhesion.

Several key experiments have suggested that mechanical

forces and cell/ECM elasticity play an essential role in FA

growth and maintenance. Mature FAs cannot grow unbound-

edly and are usually subjected to a size limit up to a few

microns (5). Stable FAs only form on sufficiently rigid

substrates, and cells tend to migrate toward stiffer region

when cultured on an elastically nonhomogeneous substrate

(6,7). The elastic modulus of cytoskeleton can change over

several orders of magnitude in response to different levels

of myosin-II-driven contractility (8,9), whereas inhibition

of the contractile stress leads to dissolution of cytoskeleton

and disappearance of FAs (10). When myosin II activity is

suppressed, application of an external force, irrespective of

its physical origin, is found to stimulate growth of FAs in

the direction of the force (11). Experiments have also shown

that the size of mature FAs can reversibly increase or

decrease in response to the applied force, with force per unit

area (stress) maintained near a constant value at ~5.5 kPa

irrespective of the cell type (12,13).

A number of theoretical studies have been performed to

investigate how mechanical stimuli and cell/ECM properties

affect the behaviors of cell adhesion. Deshpande et al. (14)

have proposed a model of cellular contractility that accounts

for dynamic reorganization of cytoskeleton, with prediction

that stress fibers are more effectively developed by multiple

activation signals than a single prolonged signal. Bruinsma

(15) has described the regulation of cytoskeletal force which

is generated along actin filaments during the growth stage

from initial contacts to FXs. Nicolas et al. (16) have devel-

oped a model to investigate the distribution of shear stress

along cell-substrate interface, with the postulate that stress

gradient in FAs may control their growth or shrinkage. Smith

et al. (17) considered force-induced adhesion strengthening

aided by the lateral mobility of molecular receptors.

Erdmann and Schwarz (18,19) studied the stochastic effects

of a cluster of uniformly stressed molecular bonds transiting

between open and closed states under the influence of

thermal fluctuation. Based on the solutions to a one-step

master equation, Erdmann and Schwarz demonstrated that

Submitted July 7, 2009, and accepted for publication August 24, 2009.

*Correspondence: huajian_gao@brown.edu

Editor: Alexander Mogilner.

? 2009 by the Biophysical Society

0006-3495/09/11/2438/8$2.00

doi: 10.1016/j.bpj.2009.08.027

2438 Biophysical JournalVolume 97November 20092438–2445

Page 2

clusters below a critical size behave like a single molecular

bond with a finite lifetime whereas those above the critical

size survive over a much-prolonged time due to the collec-

tive effect of clustering. Therefore, adhesion size can play

a very important role in the stability of a bond cluster: small

clusters can easily switch between adhesion and de-adhe-

sion, similar to FXs, which are subjected to frequent turn-

over, whereas large clusters tend to have a much longer

lifetime similar to stable FAs. Qian et al. (20) have extended

the work of Erdmann and Schwarz to include the effect of

elasticity and nonuniform stress distribution on the stability

of a single adhesion cluster under a tensile load applied

perpendicular to the cell-ECM interface. The results predict

a size-dependent transition between uniform and cracklike

distributions of interfacial traction as well as a window of

cluster size for relatively stable adhesion and an optimal

size for maximum strength. Recent analysis by Lin and

Freund (21), based on a direct analogy between focal adhe-

sions and periodic cracks, also predicts an optimum cluster

size for maximum strength.

Despite these fascinating studies, the physical mecha-

nisms of focal adhesions and mechanosensitivity in general

are still a subject of intense speculation and debate. There

exist few theoretical studies to synergize different experi-

mental observations of FAs into a coherent understanding.

Motivated by the existing experiments and seemingly

complex interplay among cluster size, cell/ECM elasticity,

receptor-ligand binding/unbinding, and cytoskeletal con-

tractile forces, here we develop a stochastic-elasticity model

of a periodic array of adhesion clusters between two

dissimilar elastic media subjected to an inclined tensile

stress, in which stochastic descriptions of molecular bonds

and continuum elastic descriptions of interfacial traction

distribution are unified in a single modeling framework.

This model can be subjected to rigorous mathematical and

numerical analysis to address several important questions

including:

Why is there a micron-scale size limit on FAs?

Why do cells prefer stiffer substrates?

Why are cytoskeletal contractile forces necessary to stabi-

lize FAs?

How do lifetime and strength of FAs depend on the stress

fiber orientation?

MODEL

The system under investigation involves a periodic array of

adhesion clusters of molecular bonds between two dissimilar

elastic media subjected to a tensile stress sNapplied at an

inclined angle q with respect to the cell-ECM interface, as

shown in Fig. 1. Both cell and substrate are modeled as

semi-infinite elastic media with Young’s modulus and Pois-

son’s ratio EC, nC, and ES, nS, respectively. It will be conve-

nient to define a reduced elastic modulus E* according to the

convention of contact mechanics (22)

1

E?¼1 ? n2

C

EC

þ1 ? n2

S

ES

:

(1)

We consider the situation that interfacial adhesion arises

solely from the receptor-ligand bonds modeled as Gaussian

chains having a finite stiffness kLRand zero rest length.

All bonds are assumed to be closed at the initial state

and subsequently can statistically transit between open

(broken) and closed (linked) states as described by Bell

(23). The bonds are grouped in adhesion clusters of size

2a, which are periodically distributed at a period of 2c

along the interface. Within each cluster, the bonds are dis-

tributed uniformly at spacing b, corresponding to a bond

density of rLR¼ 1=b2. The average bond density along

the interface is rLR¼ arLR=c. For simplicity, we consider

a slice of the system with out-of-plane thickness b, corre-

sponding to the so-called plane strain problem in the theory

of elasticity.

We note that the present problem can be considered

a combination of the bond dynamics obeying one-step

master equation (24) and the periodic crack model in interfa-

cial fracture mechanics (25). In the absence of molecular

bonds, our model is reduced to a periodic array of cracks

between two elastic media and in the limit of rigid elastic

media—i.e., the type of cluster model discussed by Erdmann

and Schwarz (18,19).

Due to the periodic nature of the problem, we focus our

attention on one cluster and adopt a set of coordinates

(x, z) with directions shown in Fig. 1 and origin located at

the center of the cluster. In our plane strain model, the total

number of bonds within the cluster is Nt¼ 2a/b. To under-

stand how interfacial traction is distributed within the

adhesion domain, let us first consider the initial state when

all bonds are closed. In this case, the tangential and normal

FIGURE 1

between two dissimilar elastic media under an inclined tensile stress.

Schematic illustration of a periodic array of adhesion clusters

Biophysical Journal 97(9) 2438–2445

Adhesion Model of Periodic Bond Clusters2439

Page 3

components of interfacial traction, t(x) and s(x), are related

to displacement discontinuities across the interface as

?uC

sðxÞ ¼ rLRkLR

where uxand uzare displacements in x and z directions,

respectively. Superscripts C and S here denote cell and

substrate. Using the elastic Green’s function for semi-infinite

media (22), it can be shown that t(x) and s(x) obey the inte-

gral equations

?Za

vsðxÞ

vxa2

?a

tðxÞ ¼ rLRkLR

xðxÞ ? uS

?uC

xðxÞ?

zðxÞ ? uS

zðxÞ?;

(2)

vtðxÞ

vx

¼

a

a2

a

?a

tðsÞcot

?pðx ? sÞ

?pðx ? sÞ

2c

?

?

ds þ 2cbsðxÞ

?

? ;

¼

?Za

sðsÞcot

2c

ds ? 2cbtðxÞ

(3)

where

a ¼arLRkLR

E?

;

(4)

b ¼1

2

?ð1 ? 2nCÞð1 þ nCÞ

The global force balance along the interface requires that

Ra

EC=E?

?ð1 ? 2nSÞð1 þ nSÞ

ES=E?

?

:

(5)

?atðxÞdx ¼ 2csNsinqcosq

Ra

?asðxÞdx ¼ 2csNsin2q

:

(6)

Equation 3 shows that the interfacial traction is governed by

twodimensionlessparametersaandb,wherebcanberecog-

nized as one of Dundurs’ constants (26) for elasticity prob-

lems in a bi-material system (The present problem is in

some sense a tri-material system). It has been pointed out

beforethatbplaysaratherminorroleininterfacialcrackprob-

lems (27). Biological materials are often modeled as incom-

pressible and the Poisson’s ratio would be near one-half,

in which case b z 0. Therefore, taking b z 0, we identify

a as the unique controlling parameter to determine how the

interfacial traction is distributed within the adhesion clusters.

The effects of a can be immediately understood from the

solutions to Eqs. 3 and 6 in extreme cases. In the limit when

a / 0, the solution is

tðxÞ ¼sNsinqcosq

a=c

; sðxÞ ¼sNsin2q

a=c

(7)

within the adhesion domain jxj % a, indicating a uniform

distribution of interfacial traction independent of the bond

location x. In this limit, the interfacial traction is equally

shared among all bonds in the adhesion domain. In the oppo-

site limit when a / N, the solution becomes

tðxÞ ¼

sNsinqcosq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1? cos2pa

2c=cos2px

2c

r

; sðxÞ¼

sNsin2q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1? cos2pa

2c=cos2px

2c

(8)

r

for jxj % a, which is the classical singular solution for a peri-

odic array of interfacial cracks (28). For the intermediate

range 0 < a < N, the maximum traction generally occurs

at the edge of adhesion and the minimum traction occurs at

the center. Fig. 2 shows that the interfacial traction is nearly

uniform for a-values <0.1, while a cracklike stress concen-

tration emerges near the adhesion edge for a-values >1.

Therefore, we shall refer to a as the stress concentration

index. Equation 4 shows that a is linearly proportional to

the adhesion size, the bond stiffness, and the density, and

inversely proportional to the reduced elastic modulus of

cell and substrate. These factors all play an essential role

in controlling the distribution of interfacial traction within

the adhesion domain. In particular, we note that the elastic

modulus of both cell and substrate needs to be sufficiently

large to keep a small.

A similar concept of the stress concentration index has

been developed by Qian et al. (20) for a single adhesion

cluster between two elastic media. This analysis generalizes

this concept to a periodic array of clusters subject to inclined

stretching.

From a microscopicpoint ofview, themolecular bondsare

subjectedtostochasticeventsofdissociation/association.The

bond reaction (dissociation or association) rates are governed

bytheforcesactingonclosedbondsandsurfaceseparationsat

open bonds, which can be determined for any instantaneous

bondconfigurationusingtheappropriateelasticGreen’sfunc-

tion given in the Appendix. In particular, the dissociation rate

koffofaclosedbondisassumedtoincreaseexponentiallywith

force F acting on the bond as (23,29–32)

koffðxiÞ ¼ k0exp

?FðxiÞ

Fb

?

:

(9)

FIGURE 2

different values of the stress concentration index a ¼ arLRkLRðð1 ? n2

ECþ ð1 ? n2

a transition between uniform and cracklike singular distributions of interfa-

cial traction.

The distributions of the normalized interfacial traction at

CÞ=

SÞ=ESÞ while taking b ¼ 0 and c/a ¼ 2. The results indicate

Biophysical Journal 97(9) 2438–2445

2440Qian et al.

Page 4

Here k0is the spontaneous dissociation rate in the absence

of an applied force and Fbis a force scale typically in the pN

range. For receptor-ligand bonds in focal adhesions, k0?1

falls in the range from a fraction of a second to ~100 s

(33,34). In our model, F depends on the bond location xi,

which is generally larger near the adhesion edge than at

the center due to stress concentration.

The association rate konof an open bond is assumed to

decrease with the surface separation d as (20,35,36)

konðxiÞ ¼ k0

on

lbind

Z

exp

?kLRdðxiÞ2

2kBT

!

;

(10)

where kBis Boltzmann’s constant, T is the absolute tempera-

ture(kBTz4.1pN$nmatphysiologicaltemperature),kon0is

a reference association rate when the receptor-ligand pair are

withinabindingradiuslbind,andZisthepartitionfunctionfor

a receptor confined in a harmonic potential between zero and

d. The surface separation d is generally larger near the edge

than at the center. Consequently, rupture is more likely while

rebindingislesslikelyneartheadhesionedge.Understrongly

nonuniform distribution of interfacial traction, the failure

process is expected to be similar to crack propagation.

Previously, a number of numerical algorithms have been

developed for studying bond kinetics in cell adhesion

(37,38). In our Monte Carlo simulations, each bond location

xiis considered an independent reaction site where the next

event will be bond rupture at rate koff(xi) if the bond is

currently closed, and bond rebinding at rate kon(xi) if the

bond is currently open (20). The reaction rates, kon(xi) and

koff(xi), are determined from the computed forces on closed

bonds and surface separations at open bonds. The first-reac-

tion method of Gillespie’s algorithm (39,40) is used to deter-

mine when and where the next reaction will occur through

random number generation (18–20). When the binding state

of any bond (open versus closed) has undergone a change, an

update of the force and surface separation at all bonds is per-

formed using the associated elastic Green’s function, and the

results are used to determine the subsequent events. This

coupling between elastic analysis of interfacial traction/sepa-

ration and stochastic events starts at the initial state when all

bonds are closed and the process proceeds until all bonds

within the adhesion domain become open. The total elapsed

time T (real time normalized by k0?1) is recorded as the life-

time of the adhesion. The statistical lifetime is obtained from

an average of 200 independent simulation trajectories. For

relevant physical/biological parameters used in the simula-

tion, we adopt the following typical values: b ¼ 32 nm,

c/a ¼ 2, kLR¼ 0.25 pN/nm, Fb¼ 4 pN, kon

and lbind¼ 1 nm, unless stated otherwise.

0/k0¼ 3200,

RESULTS

The lifetime T of the periodic clusters is shown in Fig. 3 as

a function of the cluster size Ntfor different values of the

reduced elastic modulus E* between 1 kPa and 300 kPa.

The loading angle q is fixed at 45?. The simulation results

indicate that a size-window exists for stable adhesion. In

all cases, the traction distribution along the cell-ECM inter-

face is nonuniform and the failure becomes increasingly

‘‘cracklike’’ at increasing cluster size. Very small clusters

resemble single molecule behavior with limited lifetime

and large clusters fail by severe stress concentration near

the adhesion edge. Increasing the reduced elastic modulus

tends to stabilize and strengthen the adhesion by alleviating

stress concentration within the FAs domain. We observe that

the size-window of stable adhesion shifts and broadens as

the cell and substrate stiffen, which can be understood

from the point of view that large values of E* decreases

the stress concentration index a toward the regime of

uniform interfacial traction. The concept of a size-window

forstableadhesionissimilartoourpreviousstudy onasingle

cluster under normal tensile load and should be a general

feature of molecular adhesion clusters between elastic

media because stochastic effects are expected to dominate at

small scales and cracklike failure dominates at large scales.

FIGURE 3

the cluster size Ntfor different values of the reduced elastic modulus E*.

The pulling angle q is fixed at 45?. The selected values of E* are (a) 1 kPa;

(b) 10 kPa; (c) 100 kPa; and (d) 300 kPa.

The lifetime T of periodic adhesion clusters as a function of

Biophysical Journal 97(9) 2438–2445

Adhesion Model of Periodic Bond Clusters2441

Page 5

Increasing adhesion size or decreasing cell/ECM modulus

tends to increase a toward the regime of cracklike stress

concentration, hence reducing the lifetime and stability of

adhesion.

Increasing the applied load eventually destabilizes the

adhesion. For an example system with Nt¼ 40, E* ¼ 10 kPa,

and q ¼ 45?, Fig. 4 a plots some representative simulation

trajectories at three different stress levels. The average life-

time T of 200 simulation trajectories is plotted as a function

of the applied stress sNin Fig. 4 b. It is seen that the lifetime

asymptotically approaches infinity as the applied stress is

reduced to below a critical value. Stresses larger than this

critical value dramatically reduce the lifetime and destabilize

the adhesion. This suggests that we can define the critical

stress at which the cluster lifetime asymptotically approaches

infinity as the adhesion strength. To investigate the depen-

dence of the adhesion strength on the pulling angle q, we

have further performed a series of Monte Carlo simulations

by varying the value of q. For a given magnitude of the

applied stress, smaller pulling angles with respect to the

cell-ECM interface always lead to more stable adhesion.

Fig. 4 c shows the adhesion strength as a function of q for

different reduced moduli E* ranging from 1 kPa to 300 kPa.

Here the cluster size Ntis fixed at 40 bonds. Stiffening cell

and ECM leads to dramatic increases in adhesion strength.

This can be interpreted again from the point of view that

stiffening cell/ECM tends to decrease the stress concentra-

tion index a toward the regime of uniform interfacial

traction. Fig. 4 d shows the q-dependence of the adhesion

strength for different cluster sizes. Here E* is fixed at

100 kPa. As the cluster size is increased from 6 to 20 bonds,

the adhesion strength increases due to the collective effect of

bond clustering. However, as the cluster size is further

increased to 100 bonds, stress concentration effects dominate

and cause the adhesion strength to decrease as a consequence

of cracklike failure.

The lifetime T of the periodic cluster array is plotted as

a function of the pulling angle q at various stress levels in

Fig. 5. Here we fix Nt¼ 40 and E* ¼ 10 kPa in the calcula-

tion. We see that, for a given magnitude of the applied stress

sN, decreasing q tends to stabilize adhesion. In fact, the

adhesion lifetime asymptotically approaches infinity as the

pulling angle is reduced to below a critical threshold. This

is especially interesting in view of the fact that cells generally

flatten when successfully adhering to a substrate, and this

immediately suggests a regulation mechanism by which cells

can switch between long- and short-lived adhesions by

adjusting pulling direction around the critical angle.

DISCUSSION

From the basic scaling parameter a, which governs the inter-

facial traction distribution and detailed Monte Carlo simula-

tions, we observe that the adhesion size, substrate rigidity,

cytoskeleton stiffening, and the direction of pulling forces

all play important or critical roles in the stability of FAs.

In particular, we have shown an elasticity-controlled transi-

tion between uniform and cracklike tractions along the

cell-ECM interface. Although we do not expect that the

present model can fully capture the complexity of real focal

adhesions, it seems that the predictions from this model can

provide feasible explanations for a wide range of experi-

mental observations and also suggest possible cytoskeletal

mechanisms by which cells can control and regulate the

growth and stability of FAs via contractility.

OurmodelsuggeststhatthereasonforFAstolieinanarrow

size range from a few hundred nanometers to a few microns

might be that the growth of FAs eventually leads to cracklike

FIGURE 4

(a) Representative simulation trajectories of the number

of closed bonds k versus time t at three selected stress

levels (Nt¼ 40, E* ¼ 10 kPa, and q ¼ 45?). (b) The adhe-

sion lifetime T as a function of the applied stress sN. The

lifetime asymptotically approaches infinity as the stress is

reduced below a critical value which is defined as the adhe-

sion strength (Nt¼ 40, E* ¼ 10 kPa, and q ¼ 45?). (c) The

adhesion strength as a function of the pulling angle q for

different values of the reduced elastic modulus E* from

1 to 300 kPa (Nt¼ 40). (d) The adhesion strength as a func-

tion of the pulling angle q for different values of the cluster

size Ntfrom 6 to 100 bonds (E* ¼ 100 kPa).

The strength of periodic adhesion clusters.

Biophysical Journal 97(9) 2438–2445

2442Qian et al.

Page 6

delamination failure near the adhesion edge. From this point

of view, the growth of FAs is self-limiting. Our analyses

show that the optimal size-window for stable adhesion is in

the submicron-to-micron range, depending on the rigidity of

cell and substrate. This is also in qualitative agreement with

the experimental observations that stable and large FAs are

only formed on sufficiently rigid substrates. The fact that

FAsonstiffsubstratesaremorestableprovidesadrivingforce

forcellstomigratetowardthestifferpartofthesubstrate.The

effectsofthereducedelasticmodulusE*onadhesionlifetime

and strength imply that very soft substrates tend to diminish

the adaptive capability of cells, in that cracklike interfacial

traction would persist irrespective of the cytoskeleton stiff-

ness, which may prevent short-lived FXs from maturing

into stable FAs. On hard substrates, the reduced elastic

modulus E* tends to be dominated by the stiffness of cyto-

skeleton. The cytoskeletal contractile forces can stiffen cyto-

skeletonbydecreasingentropicelasticityoftheactinnetwork

and therefore alleviate stress concentration at FAs to achieve

long-term stability. This is consistent with the experimental

observationsthatcytoskeletalcontractile forcesarenecessary

to stabilize cell adhesion. We also demonstrate the depen-

dence of adhesion lifetime and strength on the loading

angle q. Low-angle pulling dramatically increases the adhe-

sion lifetime and strength. Therefore, cell spreading and flat-

teningovera substrate result inlow-anglepullingonFAs and

benefit stable adhesion. All these results are consistent with

experimental observations and suggest multiple mechanisms

by which cells can actively control adhesion and deadhesion

by modulating the cytoskeleton or adjusting the angle of

stress fibers. Generally, rigid substrates, cytoskeleton stiff-

ening, intermediate adhesion size, and low-angle pulling

are factors that contribute to stable focal adhesions whereas

soft substrates, cytoskeleton softening by dissolution of actin

network, extreme adhesion size, and high-angle pulling are

factors that tend to destabilize FAs. Our model provides

auniqueguidingparameteratounderstandthesephenomena.

CONCLUSION

This model aims to bridge elastic descriptions of adhesive

contactatlargescale,andstochasticdescriptionsofmolecular

bonds at small scale. A dimensionless stress concentration

index a has been identified as a controlling parameter to

help us understand the transition between uniform and crack-

liketractiondistributionsalongthecell-ECMinterface.Inthe

Monte Carlo simulations, the choice of parameters is specific

but the essential features of the model, e.g., the observed size

window for stable adhesion and the angle-dependence of

adhesion lifetime and strength, appear to be generic. Finally,

wepointoutsomecriticalassumptionsmadeinourmodelthat

can be improved in future work. We have assumed immobile

receptor-ligand bonds at the cell-ECM interface, whereas

bonddiffusionislikelytobeimportantinrealbiologicalcells.

Cell and ECM are idealized to be purely elastic but in reality

they show nonlinear and viscoelastic behaviors. The loading

conditions at FAs can also be much more complex than

assumed. Despite these limitations, it is encouraging that

the predictions of such an idealized model are essentially

consistent with relevant experimental observations.

APPENDIX

Consideraperiodicarrayofconcentratedforceswithperiod2calongthecell-

substrate interface. We focus on an arbitrary period and adopt (x,z) coordi-

nates with origin located at the adhesion center and directions shown in

Fig. 1. The discontinuity of the displacement u on cell and substrate surfaces

atabondlocationxicausedbyadifferentbondatlocationxj(isj)isgivenby

(41,42)

Du?xi;xj

¼

?

¼ uC?xi;xj

2Fj

pE?bln

?? uS?xi;xj

?

????sinp?xi? xj

?

2c

????;

(11)

where Fjis the bond force at xj. To avoid singularity, the self-displacement

discontinuity at xiinduced by the force array Fi, which is modeled as an

equivalent uniform pressure with half-width a0(a0¼ 5 nm in simulations

(43)), is given as (41,42)

Duðxi;xiÞ ¼ uCðxi;xiÞ ? uSðxi;xiÞ

2Fi

pE?bln

¼

????sinp b

2c

Z0

????

þ

Fi

pE?a0b

?b

ln

????

sinðpðx þ a0Þ=2cÞ

sinðpðx ? a0Þ=2cÞ

????dx:

(12)

The compatibility condition at the interface is

X

k

j¼1

GijFj?Fi

kLR

þ h ¼ 0;

(13)

FIGURE 5

the pulling angle q at various levels of the applied stress (Nt¼ 40 and

E* ¼ 10 kPa).

The lifetime T of periodic adhesion clusters as a function of

Biophysical Journal 97(9) 2438–2445

Adhesion Model of Periodic Bond Clusters 2443

Page 7

where k is the current number of closed bonds within the adhesion domain, h

is the cell-substrate surface separation in the absence of elastic deformation

and

is the elastic Green’s function for a periodic array of forces.

The global force balance requires that

X

k

i¼1

Fi ¼ 2bcsNsinq:

(15)

Once the k þ1 unknowns (F1, F2, $$$, Fk, h) are solved, the surface separa-

tion dibetween the two elastic media at an open bond location xican be

calculated through

di ¼

X

k

j¼1

GijFjþ h:

(16)

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