Biomedical Engineering Reference
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the adhesion domain. In particular, we note that the stiffness of both cell and
substrate needs to be sufficiently large in order to keep a small.
For any instantaneous bond configuration during the stochastic process of bond
breaking/reforming, one can determine the interfacial traction using the appropriate
elastic Green's functions [ 59 ]. The force acting on each closed bond and surface
separation at each open bond are critical factors to determine the bond reaction rates
at any instant. We consider a cluster with all bonds closed initially. The bonds
undergo stochastic breaking or rebinding described by the master equation ( 8.1 ). If
all of the n closed bonds equally share the total applied load P , then r n ¼ n e P=n and
g n ¼ gðN t . However, elastic deformation of the system makes the load P to be
shared non-uniformly among closed bonds, the total dissociation rate becomes
r n ¼ X
n
X
n
e P i
;
P i ¼ P
(8.11)
1
1
Thus, the behavior of stochastic-elasticity coupling of the adhesion system can
be governed by ( 8.1 ), ( 8.8 ), and ( 8.11 ). Monte Carlo simulations have been
conducted by Qian et al. [ 36 , 40 ] to solve these equations, where each bond location
x i is considered as an independent reaction site where the next event will be bond
rupture at rate k off ðx i Þ¼r n =k 0 if the bond is currently closed, and bond rebinding at
rate k on ðx i Þ¼g n =k 0 if the bond is currently open. The reaction rates, k on ðx i Þ and
k off ðx i Þ , are computed from the elastic solution of forces at closed bonds and surface
separations at open bonds. The so-called first reaction method of Gillespie's
algorithm is used in the simulations [ 36 , 40 ]. When the binding state of any bond
(open vs. closed) has undergone a change, an update of the force and surface
separation at all bonds is re-calculated using the associated elastic Green's function,
and the results are used to determine the next reaction events. This coupling
between elasticity and stochastic events starts at an initial state when all bonds
are assumed to be closed and the process proceeds until all bonds within the
adhesion domain become open. The total elapsed time t is recorded as the lifetime
of the cluster.
Figure 8.4 shows the normalized lifetime of the cluster as a function of the
cluster size for different values of the reduced elastic modulus E * and rebinding
rate g . The simulation results indicate that a size-window exists for stable adhesion.
In all cases, the traction distribution along the adhesion interface is non-uniform
and the failure becomes increasingly crack-like 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 * decrease the
SCI toward the regime of uniform interfacial traction. The concept of a size-
window for stable adhesion should be a general feature of molecular adhesion
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