Specific Adhesion of Soft Elastic Materials - Structural Interfaces and Attachments in Biology - page 153

Fig. 8.2 The critical
perturbation wavelength, l c ,
as a function of r 0 =r t . Other
parameter choices: E 1 ¼ E 2 ,
x g ¼ 0 : 01 pN/nm, x b ¼ 0 : 25
pN/nm, n 1 ¼ n 2 ¼ 0

E*=17.4 KPa, κ=0

E*=17.4KPa, κ=20k B T

E*=57.9 kPa, κ=0
E*=57.9 kPa, κ=20 k B T

10 1

37,
r t ¼ 10000/ m m 2 , l b ¼ 11 nm,
l g ¼ 30 nm, T ¼ 300 K

:

m 2 ρ

g =10000/

μ

10 0

10 -2

10 -2 10 -1

ρ 0 /ρ t

length about 11 nm [ 51 ]. The Young's modulus of cells has been reported between
4 and 230 kPa [ 52 ] and Poisson's ratio around 0.36-0.38 [ 53 ].
We have demonstrated in [ 41 ] that the condition 0

1 is satisfied if the
number of receptors in cell membrane as observed in experiments is 50-500/

<b<

m 2
[ 54 - 56 ], corresponding to the range of relative bond density of 0.01-0.1 in which
clustering instability occurs. For this range of r 0 =r t values, Fig. 8.2 plots the critical
wavelength l c as a function of the relative bond density r 0 =r t for different values of
Young's modulus with or without including the membrane deformation, i.e., k ¼ 0
or k ¼ 20 k B T . Figure 8.2 shows that, for bond density values in the instability
range, the critical wavelength is generally on the order of 1

m

m for realistic values
of cell modulus. This length scale is in broad agreement with experimentally
observed adhesion plaques in cell adhesion [ 57 ].

m

8.5.2 Stochastic-Elasticity Coupling

Consider the adhesion of molecular bonds grouped in an adhesion cluster of size 2 a
between two dissimilar elastic media subjected to a tensile force P , as shown in
Fig. 8.1 . We focus on the situation that interfacial adhesion arises solely from the
ligand-receptor bonds which are assumed to have a finite stiffness x and can
statistically transit between open and closed states, as described by Bell [ 58 ].
For simplicity, we consider a slice of the system with out-of-plane thickness b ,
corresponding to the so-called plane-strain problem in elasticity. Within each
cluster, individual molecular bonds are periodically distributed at a spacing of b ,
corresponding to a bond density of r 0 ¼ 1 =b 2 and a total number of N t ¼ 2 a=

. The
normal interfacial traction, sðxÞ , is related to the surface separation u as sðxÞ¼r b
xu . Using the elastic Green's functions for semi-infinite media [ 59 ], it can be shown
that u obeys the following integral equations [ 35 ]