Ns 1Ps 1 us

We only carry out a linear stability analysis here to determine parameter domains for pattern formation, to highlight the key elements necessary for the system to generate spatial patterns and to get some idea of the patterning potential of the mechanism.

We linearize (8.11) about the uniform steady state (8.12) by setting n = 1 + n, p = 1 + p, u = u and, omitting the bars for convenience, obtain the linearized system nt + V • ut = D0V2 (n + 1V - u) ,

V • [x1€t + X20tI + e + vOI + tm] = sut, (8.13)

P = -vO, where n, p,u now denote the small perturbations from the steady state (1,1, 0) and

1 (1 + a)2 In the usual way, we now look for solutions in the form

where a and k are respectively the growth rate and the wave vector. After some messy algebra we obtain the Jacobian matrix (here 4 x 4 because of the two equations for the components of u) from which we get the characteristic equation for a given by

where x = X1 + X2, k2 = | k |2 and b(k2) = xDk4 + (sD + 1 + v - T1)k2, c(k2) = Dk4(1 + v - \t1). (8.16)

So that we do not lose sight of what the parameters represent, in dimensional terms the parameters in (8.15) and (8.16) are given by s(1 + v)L2 Tn0(1 + v)(1 - an2)

Depending on the sign of

the solutions given by (8.14) can be real or complex. Pattern Formation and Parameter Domains

A fairly comprehensive picture of the pattern formation potential of the mechanism can be obtained from a detailed study of the roots a given by (8.18). To be able to generate coherent spatial patterns we require that a(k2 = 0) < 0 and that there exist wavenumbers k2 > 0 such that Rla(k2) > 0; refer to Chapter 6 for a full description of the justification. Basically they ensure that the homogeneous steady state is stable but it is unstable to spatial perturbations some of which initially grow exponentially. Certainly when k2 = 0 the first condition is clearly satisfied from (8.15). Let us now briefly consider the second condition.

The largest root is ai in (8.18) so we can therefore concentrate on it. From (8.16) it is clear that both b(k2) and c(k2) can be positive or negative depending on the relative size of r1, the other parameters and the wavenumber k. Here we only discuss a few of the several cases, leaving a full analysis and quantification of the parameter spaces for pattern formation as a pedagogical, biologically useful and practical, exercise.

Let us first consider the special case in which the cells do not diffuse, that is, D = 0. Using (8.16) in (8.18) we see that the largest a is then ai = _ b(k2) = (x1 _1 - v)k2 (8.20) yk2 + s yk2 + s and so, if the cell traction parameter r1 > 1 + v, a1 > 0 for all k2 > 0 and hence all spatial modes will grow exponentially (in this linear theory) according to (8.14). Since all modes are unstable the final pattern will depend intimately on the initial conditions. As we show in the numerical simulations presented below the patterns formed are generally fairly random but with a gross coherent structure. The interesting implication here is that patterns are generated solely by the interaction of the various mechanical forces present, namely, the cell traction, the matrix and dish resistance to movement and how they move the cells around via convection. It is an often held belief that diffusion is crucial, if not essential, to create spatial patterns of relevance.

Let us consider a1 from (8.18) together with the expressions in (8.16) with k2 > 0; now, of course, D = 0. If c(k2) < 0 then a1 > 0 and the uniform steady state is linearly unstable. So, if r1 > 2(1 + v) all wavenumbers k2 > 0 are unstable and again the final pattern depends on the initial conditions.

In dimensional terms, using (8.17), we have the result that small perturbations in the cell density from the uniform steady state can initiate matrix instabilities and start to form spatial patterns in the cell and matrix densities if t wq(1 + v)(l - anp) > 2(\ + —V—)

In this parameter domain all wavenumbers are unstable irrespective of the values of D and s. A typical dispersion relation in this parameter range is illustrated in Figure 8.2(a).

i so

Figure 8.2. (a)-(c) Typical dispersion relations a(k2), giving the linear growth rate of spatial instabilities. (a) An infinite range of unstable wavenumbers with o(k2) real. Parameter values: s = 10, a = 0.01, f = l, v = 0.2, t = 5, D = 0.001. (b) The unstable growth is more complicated (and more interesting) with a finite part of the domain exhibiting growing oscillatory solutions. Parameter values: s = l0,a = 0.0l, f = 0.3, v = 0.2, t = l.4, D = 0.001. (c) A finite range of unstable growing oscillations. Parameter values: s = 10, a = 0.0l,f = 0.3, v = 0.2, t = 2.4, D = 0.001. (From Manoussaki 1996) (d) Example of the parameter domains with different solution behaviour for v = l. The regions I, II, III and IV correspond to the solution domain behaviour summarized in Table 8.2.

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Figure 8.2. (a)-(c) Typical dispersion relations a(k2), giving the linear growth rate of spatial instabilities. (a) An infinite range of unstable wavenumbers with o(k2) real. Parameter values: s = 10, a = 0.01, f = l, v = 0.2, t = 5, D = 0.001. (b) The unstable growth is more complicated (and more interesting) with a finite part of the domain exhibiting growing oscillatory solutions. Parameter values: s = l0,a = 0.0l, f = 0.3, v = 0.2, t = l.4, D = 0.001. (c) A finite range of unstable growing oscillations. Parameter values: s = 10, a = 0.0l,f = 0.3, v = 0.2, t = 2.4, D = 0.001. (From Manoussaki 1996) (d) Example of the parameter domains with different solution behaviour for v = l. The regions I, II, III and IV correspond to the solution domain behaviour summarized in Table 8.2.

Since the ratio of the cell density used in experiments to the confluent density (where crowding effects come into play) is of the order of 0.1-0.4, (1 + a^)2/(1 — an2) ~ 1 and the last condition becomes simply

We immediately get some qualitative results on the role of key parameters in the experiments, such as the larger the seeding density (n0) the greater the likelihood of pattern formation. Other qualitative results on how the patterns form are obtained from the further detailed analysis below.

Parameter Domain t1 < 2(1 + v), sD > 1 + v: Region II

In this range, c(k2) > 0 for all k2 > 0 so instability, that is, Rla1 > 0, can only obtain if b(k2) < 0. If sD > 1 + v the coefficient of the O (k2) term is always positive and so is b(k2) and hence the uniform steady state is always stable and no pattern formation is possible.

Parameter Domain t1 < 2(1 + v), sD < 1 + v: Region III If sD < 1 + v then, from (8.19) with (8.16) the roots of A are k2 = 0, k2, kf = + v — sD ±yT1 [2(1 + v) — tx]) . (8.22)

With these we can then deduce (after a little algebra) that if the parameters satisfy tx < 2(1 + v), sD < 1 + v,

the dispersion relation, a1, from (8.18) is complex for wavenumbers k\ < k2 < k|. In detail we have

Rla1(k2) > 0, 0 < k2 < k2, Rlax(k2) < 0, k2 < k2 < to,

Im ax (k2) > 0, kf < k2 < k2, Im ax (k2) = 0,0 < k2 < kf, k2 > k^, where k2 is the value of k2 where the curve for Rla (k2) = 0. A typical dispersion form in this parameter range is shown in Figure 8.2(b). Note that in this case there is a finite range of unstable modes with a fastest growing mode which can be derived from the expression for a(k2). There is also the possiblity of oscillatory growing perturbations in this case since for k| < k2 < k2, Rlax > 0, Im ax = 0.

Parameter Domain t1 < 2(1 + v), sD < 2(1 + v): Region IV

By a similar analysis we get the quantitative behaviour of the dispersion relation for the parameters in the ranges r1 < 2(1 + v), sD < 2(1 + v),

1 + v + sD < t1 < 1 + v + VsD[2(1 + v) — sD], and in this case we get the dispersion relation behaviour

Rla1(k2) > 0, 0 < k2 < k2, Rla1(k2) < 0, k2 < k2 < to,

Im a1(k2) > 0, 0 < k2 < k2, Im a1(k2) = 0, k2 < k2, (8.26)

k2 = (t1 — 1 — v — sD)/ixD, with k2 given by (8.22).

In this case there is a finite range of unstable modes and all of them grow in an oscillatory manner. An example of this dispersion relation is given in Figure 8.2(c).

We now have a fairly complete picture of how the parameters determine whether or not the uniform steady state is unstable and whether or not the mechanism will produce a spatial pattern. Figure 8.2(d) shows the r1 — sD parameter space divided into the various regions covered by the parameter domains I, II, III and IV given above. The parameter domain is in fact 6-dimensional in the dimensionless parameters D,/j>1, fi2, v, t1 and s.

Effect ofParameter Variation on the Pattern Formation

The linear analysis provides a means of classifying the type of instabilities we can expect—exponential growth, growing exponential oscillations or a mixture of both— when the parameters are in the various parameter domains and the relative growth of the unstable wavenumbers.

Table 8.2 gives the type of growing instability for various parameter regions. The effect in terms of the dimensional parameters is then given by using the definitions

Table 8.2. Conditions on the parameters for the formation of spatial patterns with regions I, II, III and IV denoted in Figure 8.2. The parameter T\ = t(1 — an0)/(1 + an0)2, Ta = 2(1 + v), Tb = 1 + v + V^-D[2(1 + v) - jD], Tc = 1 + v + sD.

Region

Parameter Range

Linear Solution Behaviour

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