Partial Differential Equations - Modelling and ... - ResearchGate

Cell Adhesion and Detachment in Shear Flow 211
receptor
ligan
Fig. 1. Model geometry of cell and surface. The surface is covered by ligans and
the cell is rigid and covered by receptors distributed randomly.
(FBs) between the members of the integrin family and corresponding ECM
proteins, such as collagen type-II and fibronectin [Loe93, GHR04].
To model cell adhesion, Hammer et al. in [KH01, CH96] have developed
an adhesive dynamics algorithm, in which adhesion molecules are modeled
as linear, Hookean springs, distributed randomly over the particle surface as
shown in Figure 1. For chondrocytes, which have microvilli on the cell surface
[CKGA03], the randomly distributed receptors as shown in Figure 1 still can
be used. The adhesive dynamics algorithm is as follows:
1. All free adhesion molecule receptors in the contact area are tested for
formation of binding with the substrate ligand against the probability
P f =1− exp(−k f n l τ),
where k f is the forward reaction rate, n l is the density of ligans, and the
time step is τ. If the generated random number is less than P f ,abondis
established at this time step.
2. All of the currently bound receptors are tested for breakage against the
probability
P r =1− exp(−k r τ),
where k r is the reverse reaction rate. If the generated random number is
less than P r the bond breaks at this time step.
3. Each existing bond is characterized by the vector x b and the force imparted
by the spring on the cell is F b = σ(|x b |−λ)u b with the Hookean
spring constant σ, equilibrium length λ and unit directional vector u b =
x b /|x b |.
4. A summation of the forces from each spring and associated torques is
the information that needs to be included in the Newton–Euler equations
to study cell interaction with the Navier–Stokes flow discussed in the
following section.