Specific Adhesion of Soft Elastic Materials - Structural Interfaces and Attachments in Biology - page 150

substrate with a surface profile h ( x ) representing ECM. The Young's modulus and
Poisson's ratio are E 1 , n 1 for the cell body and E 2 , n 2 for the substrate. Following the
convention in contact mechanics, we define a reduced modulus E * such that 1 =E

¼ð 1 n

2 Þ=E 2 . The distribution of interfacial stress is governed by
classical elasticity equations in contact mechanics while the rupture and rebinding
of molecular bonds obey stochastic equations in molecular mechanics. The effects
of elastic moduli, adhesion size, and bond rebinding rate on the cluster lifetime and
strength can be studied in the presence of strongly non-uniform distribution of
interfacial stress.

1 Þ=E 1 þð 1 n

8.4 Adhesion of Rigid Media

For a limiting case, adhesion of rigid media via ligand-receptor bonds has been well
studied [ 32 ], in which case molecular bonds are uniformly loaded. Note that the
bond breaking/reforming are discrete Markov events; we have the one-step master
equation for the number of closed bonds n as [ 42 , 43 ]

d p n = d t ¼ g n 1 p n 1 þ r nþ 1 p nþ 1 ðr n þ g n Þp n (8.1)

where p n ðtÞ represents the probability that n bonds are closed at time t , and r n and
g n are the dissociation and association rates, respectively. A quantity of large
interest is the average property of closed bonds. Define the k th moment as

> ¼ X

N t

<n k n k p n ðtÞ

(8.2)

n¼ 0

, if we multiply the master equation by
n and sum both sides over n , we can obtain [ 44 , 45 ],

For the average number of the bonds <n>

d
d t <n> ¼ <g n > <r n >

(8.3)

As r n is usually a nonlinear function of n , the deterministic equation of <n> is

usually achieved by approximately expanding <r n > around n ¼ <n> . For example,

a first order approximation can lead to the Bell equation [ 32 ]

d<n>= d t ¼ k 0 gðN t <n>Þk 0 <n> exp ðP=<n>F b Þ (8.4)

where r n ¼ k 0 n exp ðP=nF b Þ , g n ¼ k 0 gðN t nÞ , N t is the total bond number, P=n is
the force applied to each bond, and k 0 and F b are the rate and force constants,
respectively.