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where a, P and 8 are positive constants. The form of R(c) is typically ^-shaped as shown in Figure 6.25(a): if 4p82 < a2 there are two linearly stable steady states at c = 0 and c = c3 and an unstable steady state at c = c2.

The release of calcium can also be triggered by straining the cytogel, a phenomenon known as 'stretch activation.' We can model this by including in the kinetics R(c) a term yd, where y is the release per unit strain and d is the dilation. Figure 6.25(b) shows the effect of such a term and how it can trigger calcium release if it exceeds a certain threshold strain. (Certain insect flight muscles exhibit this phenomenon in that stretching induces a contraction by triggering a local calcium release.)

Calcium, of course, also diffuses so we arrive at a model conservation equation for calcium given by d C 2

= dv2c + ^ - Sc + yQ, where D is the diffusion coefficient of the calcium. We have already discussed this equation in detail in Chapter 3, Section 3.3 (cf. also Chapter 6, Exercise 3) and have shown it gives rise to excitable kinetics. We should emphasise here that the kinetics in (6.72) is simply a model which captures the qualitative features of the calcium kinetics. The biochemical details of the process are not yet completely understood.

The mechanochemical model for the cytogel consists of the mechanical equilibrium equation (6.70), and the calcium conservation equation (6.72). They are coupled through the calcium-induced traction term t(c) in (6.70) and the strain-activation term yQ in (6.72). In the subsequent analysis we shall take E(Q), the viscosities , i = 1, 2 and the density p to be constants.

We nondimensionalise the equations by setting r c u r * = —, t * = St, c* = —, u* = —, L C3 L

E SC3

where L is some appropriate characteristic length scale and c3 is the largest zero of R(c) as in Figure 6.25(b). Substituting these into (6.70) and (6.72) and omitting the asterisks for notational simplicity, we have the dimensionless equations for the cytogel continuum as

V ■ {*i£t + *2®tI + £ + V'QI + t(c)I} = su, dc 2 ac2 2 (6.74)

— = DV2c + 2 - c + yQ = DV c + R(c) + yQ. d t 1 + j3 c2

The boundary conditions depend on the biological problem we are considering. These are typically zero flux conditions for the calcium and periodic or stress-free conditions for the mechanical equation.

The linear stability of the homogeneous steady state solutions of (6.74), namely, u = Q = 0, c = Ci, i = 1,2, 3, (6.75)

0 0

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