The model we consider here involves actual cell movement. Pattern formation models which directly involve cells are potentially more amenable to related experimental investigation. There is also some experimental justification from the evidence on pige-ment cell density variation observed in histological sections which we described above. Also, Le Douarin (1982) speculated that chemotaxis may be a factor in the migration of pigment cells into the skin. Heuristically we can see how chemotaxis could well be responsible for the rounding up and sharpening of spots and stripes. In the model, we propose that chromatoblasts both respond to and produce their own chemoattractant. Such a mechanism can promote localisation of differentiated cells in certain regions of the skin which we associate with the observed patterns on the snake integument. The cells, as well as responding chemotactically, are assumed to diffuse. It is the interaction of the cell mitosis, diffusion and chemotaxis which can result in spatial heterogeneity. The relatively simple mechanism we propose is d n 0
— = Dn V2n - aV • (nVc)v - rn(N - n), (4.19) d t d c 0 Sn
— = DcV c + —— - Yc, (4.20) d t p + n where n and c denote the cell and chemoattractant densities respectively; Dn and Dc are their diffusion coefficients. We have taken a simple logistic growth form for the cell mitotic rate with constant linear mitotic rate r and initial uniform cell density N. The chemotaxis parameter a is a measure of the strength of the chemotaxis effect. The parameters S and Y are measures, respectively, of the maximum secretion rate of the chemicals by the cells and how quickly the chemoattractant is naturally degraded; p is the equivalent Michaelis constant associated with the chemoattractant production. This is the specific model discussed by Oster and Murray (1989) in relation to developmental constraints. In spite of its relative simplicity it can display remarkably complex spatial pattern evolution, particularly when varying chemotactic response and growth are allowed to take place during the pattern formation process. We first nondimensionalise the system by setting x* = [Y/(Dcs )]1/2x, t * = y t/s, n* = n/p, c* = y c/S,
N * = N/p, D* = Dn/Dc, a *= a S/(y Dc), r *= rp/Y, where s is a scale factor. We can think of s = 1 as the basic integument size, carry out the simulations on a fixed domain size and then increase s to simulate larger integuments. We used this procedure in the last chapter. With (4.21) the nondimensional equations become, on omitting the asterisks for notational simplicity, d n 2
The numerical simulations of these equations (including growth) were carried out on a simple rectangular domain in which length is considerably longer than the width, with zero flux of cells and chemoattractant on the boundaries. The detailed numerical simulations and complex bifurcating pattern sequences which can occur as the parameters vary are given in Winters et al. (1990), Myerscough et al. (1990) and Maini et al. (1991).
The reason we consider a long rectangular domain is that the skin patterns are probably laid down at a stage when the embryo is already distinctly snake-like; that is, it is already long and more or less cylindrical, even if it is in a coiled state. Details of the embryo of the asp viper (Vipera aspis), for example, are given by Hubert and Defaure (1968) and Hubert (1985). Although it would be more realistic to study the model mechanism on the surface of a coiled cylindrical domain the numerical simulation difficulties were already considerable even on a plane domain. Here we are mainly concerned with the variety of patterns that the mechanism can generate so we consider the cylindrical snake integument laid out on a plane. The main features of the patterns on an equivalent cylindrical surface will be similar. We could, of course, equally well have taken periodic boundary conditions. Equations (4.22) have one positive homogeneous steady state no = N, co =
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