## Physical Models of Ligand Mediated Cell Adhesion

Cell capture by the hollow fibre substrate is related to the number of receptor-ligand bonds that are formed. This process is referred to as ligand-mediated cell adhesion. One of the first attempts to model this process was by Bell . The reverse rate constant was related to bond stress, the range of interaction and temperature using the following equation:

where k0 is the unstressed reverse rate constant, X the range of interaction of the bond, fB the force per bond, kB the Boltzmann constant and T the temperature. If a bond is stressed by application of a detachment force, then the expression XfB is the reversible work done on the bond and the increase in free energy of the receptor-ligand complex. The numbers of bonds that have enough thermal energy for dissociation follow the Maxwell-Boltzmann distribution.

Hammer and Lauffenburger  developed a dynamic model of this process. It was assumed that for attachment the bonds are unstressed:

dh a where C is the number of receptor-ligand complexes, ta unstressed surface contact time (attachment), k0 forward rate constant (unstressed), k0 reverse rate constant (unstressed), RT total number of receptor molecules and NL number of ligand molecules. It was assumed that the number of receptor molecules was small in comparison to the number of ligand molecules (Nl » Rt).

The process of detachment by stressing bonds incorporates Bell's expression for the reverse rate constant:

At steady state = oj the rate of bond formation is balanced by the dissociation rate:

The numbers of bonds that form (C) depend on the applied bond force (Ft). If Ft is greater than a critical detachment force, then Eq. 4 which is a convex function (Fig. 1), will have no roots, ^ is negative, and all bonds will dissociate. Below the critical detachment force, there will be two roots. A stable equilibrium will form with the bond number equal to the upper root if the initial bond number is greater than the lower root. If the initial bond number is less than the lower root, ^ is negative, and all bonds will dissociate. This model predicts that cell adhesion should be an all-or-none phenomenon, depending on the detachment force.

Referring to Fig. 1, the critical force and bond number for cell detachment is found at the stationary point. Transforming Eq. 4 using non-dimensional quantities:

C k° àFt e = — , î = k\$NLtd, K = 7ôf- and « = —i-. (5)

Rt 1 k0 Nl kBTRT Fig. 1 Influence of detachment force on bond number (Hammer and Lauffenburger ) at steady state. Definition of critical detachment force

The stationary point is found by solving || = 0 and -j^ = 0 with respect to the bond number (non-dimensional):

BT de

Thus the critical detachment forces is related to the critical bond number by the following relation:

1 + ac where the subscript c denotes critical values for the detachment force and bond number.

The cell detachment is an "all-or-none" model and does not account for the stochastic nature of fracture kinetics. Cells detach over a range of applied shear stresses . Heterogeneity in cell detachment kinetics is likely related to variation in the applied forces, which depend on the shape of the cell and the method of applying a detachment force. The drag and torque applied to a sphere in contact with a plane wall in a slow uniform shear field (Re < 1) are directly related to the cube and square of the sphere radius, respectively . So biological variation in cell shape and radius will result in even larger vari ations in the detachment force resulting from application of fluid shear stress. Receptor number will also vary from cell to cell, and this will be reflected by a random distribution of critical detachment forces.

Cozen-Roberts proposed a probabilistic model of cell adhesion, where stochasticity was predicted purely on the basis of small numbers of receptor-ligand bonds (< 10 000). McQuarrie has calculated the statistical fluctuations that result from reaction between small numbers of molecules using a Markov chain model . Cozen-Roberts and co-workers have adopted this kinetic model to describe the stochastic nature of cell detachment phenomena [12, 13].