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
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
Focal adhesions are clusters of specific receptor-ligand bonds that link an animal cell to an extracellular matrix. To
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.
Editor: Alexander Mogilner.
? 2009 by the Biophysical Society
2438Biophysical JournalVolume 97November 20092438–2445
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
Why is there a micron-scale size limit on FAs?
Why do cells prefer stiffer substrates?
Why are cytoskeletal contractile forces necessary to stabi-
How do lifetime and strength of FAs depend on the stress
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)
E?¼1 ? n2
þ1 ? n2
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
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
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
components of interfacial traction, t(x) and s(x), are related
to displacement discontinuities across the interface as
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-
tðxÞ ¼ rLRkLR
xðxÞ ? uS
zðxÞ ? uS
?pðx ? sÞ
?pðx ? sÞ
ds þ 2cbsðxÞ
ds ? 2cbtðxÞ
?ð1 ? 2nCÞð1 þ nCÞ
The global force balance along the interface requires that
?ð1 ? 2nSÞð1 þ nSÞ
?atðxÞdx ¼ 2csNsinqcosq
?asðxÞdx ¼ 2csNsin2q
Equation 3 shows that the interfacial traction is governed by
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
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
; sðxÞ ¼sNsin2q
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
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
From a microscopicpoint ofview, themolecular bondsare
bond reaction (dissociation or association) rates are governed
open bonds, which can be determined for any instantaneous
tion given in the Appendix. In particular, the dissociation rate
force F acting on the bond as (23,29–32)
koffðxiÞ ¼ k0exp
different values of the stress concentration index a ¼ arLRkLRðð1 ? n2
ECþ ð1 ? n2
a transition between uniform and cracklike singular distributions of interfa-
The distributions of the normalized interfacial traction at
SÞ=ESÞ while taking b ¼ 0 and c/a ¼ 2. The results indicate
Biophysical Journal 97(9) 2438–2445
2440 Qian et al.
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
where kBis Boltzmann’s constant, T is the absolute tempera-
a reference association rate when the receptor-ligand pair are
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
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.
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
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.
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
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
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.
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.
size range from a few hundred nanometers to a few microns
might be that the growth of FAs eventually leads to cracklike
(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.
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
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-
and therefore alleviate stress concentration at FAs to achieve
long-term stability. This is consistent with the experimental
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
This model aims to bridge elastic descriptions of adhesive
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-
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,
can be improved in future work. We have assumed immobile
receptor-ligand bonds at the cell-ECM interface, whereas
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.
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
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
sinðpðx þ a0Þ=2cÞ
sinðpðx ? a0Þ=2cÞ
The compatibility condition at the interface is
þ h ¼ 0;
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 Clusters2443
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
is the elastic Green’s function for a periodic array of forces.
The global force balance requires that
Fi ¼ 2bcsNsinq:
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
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