Specific Adhesion of Soft Elastic Materials - Structural Interfaces and Attachments in Biology - page 151

In the stochastic description of cluster adhesion, cluster lifetime can be identified
by the time at which the last bond ruptures. This is exactly the concept of mean first
passage time from an initial state with number of closed bonds n ¼ N t to the state
without closed bonds, n ¼ 0. Analytical solutions of mean first passage time for
different boundaries are available in Honerkamp's book [ 42 ]. On the other hand,
numerical results can be obtained by Monte Carlo stochastic simulations as
demonstrated by Erdmann and Schwarz [ 32 ] for the bond clusters with equally
shared loading. In detail, for each set of parameter values of N t , r n , and g n , they
compute many different trajectories. Thus, averaging for given time t over the
different simulation trajectories yields the desired results. Following this analysis,
Erdmann and Schwarz [ 32 ] have demonstrated that, under a given value of the
applied force, there exists a critical cluster size beyond which the system behaves
like a macroscopic adhesion patch with a much-prolonged lifetime; below this
critical size, the cluster behaves as a single molecular bond with a finite lifetime for
a given force.

8.5 Adhesion of Elastic Media

8.5.1 Interfacial Bond Distribution

The uniform distribution of adhesion molecules along the adhesion interface is
intrinsically unstable. In order to answer why molecular bonds need to be clustered
to stabilize the cellular adhesion, we consider an elasticity-diffusion description of
the adhesion system, which consists of two elastic half-spaces, each covered with a
lipid membrane, joined together by mobile molecular bonds that diffuse, along with
unbonded binder molecules, in the interface under the combined action of a layer of
glycocalyx repellers and an applied stress. We assume that r t is the total number of
available molecular sites per unit area, which is distributed among r b of closed
bonds, r L of free ligands, r R of free receptors, and r t r b r L r R of empty sites.
The glycocalyx repellers are assumed to be immobile with a constant area density
equal to r g .
As the characteristic time scale for certain ligand-receptor binding/dissociation
can be on the order of 100 s [ 31 ], which is two orders of magnitude larger than that
of protein molecules in a lipid membrane to diffuse a distance on the order of 1 m

by Brownian motion [ 44 - 46 ], we assume that the total number of ligand-receptor
bonds remains constant in the analysis of bond clustering due to diffusion.
We induce small perturbations to the densities of molecular bonds and free
binders [ 41 ], which cause non-uniform elastic deformation in the two elastic solids.
The perturbation-induced energy change of the system per unit area of interface
consists of three terms: the strain energy in the two solids and two membranes,
the binding energy in the stressed molecular bonds and repeller molecules, and the

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