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We know from Section 2.4 that with y, b, d and ha in the appropriate parameter domain there is a minimum length below which no pattern can form and above which it can with the mode depending on the domain length (and the other parameters, of course). We also saw that there is a parameter space (with the dimension equal to the number of parameters) in which patterns can form. It is exactly the same with this model system except that we cannot determine it analytically as we did in Chapter 2. Examples of two-dimensional cross-sections of the real parameter space are reproduced in Figure 4.21 below.

Consider now Figure 4.15(a) which schematically shows a hypothetical parameter space for some group of parameters plotted against another group of parameters and that there is a subspace in which mode 1-like and another in which mode 2-like patterns will start to grow in a large enough domain. Now consider Figure 4.15(b) which illustrates a typical solution of the third equation of (4.5) with a source of c at x = 0 and zero flux at x = 1. Now as c decreases (because of the —8c term in the equation) there will be

parameter grouping

parameter grouping

Figure 4.15. (a) Schematic parameter space for which patterns will grow if the parameters lie in the appropriate space and specific dominant patterns if they lie in their respective subspaces. (b) Typical qualitative solution c of the third equation of the sytem (4.5) with a source at x = 0. (c) Schematic mode 2 solution of the full system (4.5) for some critical Lc.

Figure 4.15. (a) Schematic parameter space for which patterns will grow if the parameters lie in the appropriate space and specific dominant patterns if they lie in their respective subspaces. (b) Typical qualitative solution c of the third equation of the sytem (4.5) with a source at x = 0. (c) Schematic mode 2 solution of the full system (4.5) for some critical Lc.

a value of Lc such that the average c (or rather, approximately the average) for Lc < x < 1 is such that the system (4.14) can generate a spatial pattern, specifically a mode 2-like pattern as in Figure 4.15(b) if the parameters are in the appropriate subspace. With c = a, a constant, we can certainly calculate what this critical Lc will be for the mode 2 pattern to be formed with the methods of Chapter 2. With c(x, t) a solution of its own reaction diffusion equation (the third of (4.5)) it is considerably less easy since the full system must be solved. Nevertheless it is intuitively clear for the full system (4.5) that for some length Lc a mode 2-like solution as in Figure 4.15(b) will start to form and that u will increase. It is again intuitively clear that a somewhat similar patterning scenario will occur for the model system (4.6)-(4.8) in a growing domain and that a critical Lc exists for a mode 2-like pattern to start to form in Lc < x < 1. All of these behaviours are confirmed by the numerical simulation of the full system as we see below.

Let us now return to the full model set of equations (4.6)-(4.8) with (4.9) and (4.10) on the fixed domain 0 < x < 1; recall that it is a fixed domain because of the scale transformation. We assume there is an initial source of epidermal growth factor, c, at the posterior end of the jaw (x = 0). This chemical diffuses through the jaw epithelium, is degraded and diluted by growth according to (4.8). As the jaw grows, c decreases further towards the anterior end x = 1 until it crosses below the critical threshold on a sufficiently large subdomain Lc < x < 1 to drive the substrate-activator system unstable (through diffusion-driven instability more or less in the usual way). The specific mode that starts to grow depends on the parameters. We choose parameter values such that the single hump (mode 2) spatial pattern is the first unstable mode. So, when the subdomain, on which c is below the threshold, has grown large enough, a single mode spatial pattern in u and v will start to grow like the mode 2 pattern in Figure 4.15(b). Eventually, the substrate concentration, u, crosses an upper threshold which triggers initiation of a placode (tooth primordium) fixing the spatial position of tooth 1: this is the dental determinant; see Figure 4.16(a).

As mentioned, the experimental evidence (Westergaard and Ferguson 1986) suggests that the dental determinant and each subsequent tooth primordium become a source of inhibitor thus simulating an inhibition zone. So, in our model, when u grows above a certain threshold, we make the location of the peak in u a source of c, the inhibitory substance. Mathematically, this is equivalent to an internal boundary condition at each tooth c(xi, t) = c (t) (4.15)

for x = xi and t > ti, where c (t) is the solution of (4.11). So, with the appearance of the dental determinant there are now two sources of inhibitor, one at the posterior end of the jaw and the other at the dental determinant position xi. Now, as the jaw grows, c eventually drops below the critical threshold in the region between the two sources and another hump-like pattern in u starts to appear in the posterior end of the jaw. The second primordium forms in the region where u again crosses the patterning threshold, and the tooth that is initiated becomes another source of c as illustrated in Figure 4.16(b). In this way, tooth development proceeds: c(x, t) dips below a threshold, causing a local pattern to form when the domain size is large enough. In forming the pattern, u crosses a threshold, and creates a source of c, hence another tooth primordium is created. Subsequent primordia appear in an analogous manner. Based on the tooth formation scenario in Figure 4.10 the placode, which we assume forms where u crosses the threshold, induces cell aggregation, the papilla, in the mesenchyme. The exact order of which comes first, the placode or the papilla is still not generally agreed. At this stage in our modelling we do not address this question; we consider it in Chapter 6 when discussing the mechanical theory of pattern formation.

Figure 4.16. (a) Representation of the formation of the first tooth primordium: t = t\ is the time u = uth-The jaw length at this stage is approximately 0.6 mm. (b) Representation of the formation of the second tooth primordium: t = tx is the time when u = uth again. According to the developmental scenario described above in Figure 4.10 the placode, initiated by the substrate u, and the cell aggregation in the mesenchyme, the papilla, appear at the same place.

Figure 4.16. (a) Representation of the formation of the first tooth primordium: t = t\ is the time u = uth-The jaw length at this stage is approximately 0.6 mm. (b) Representation of the formation of the second tooth primordium: t = tx is the time when u = uth again. According to the developmental scenario described above in Figure 4.10 the placode, initiated by the substrate u, and the cell aggregation in the mesenchyme, the papilla, appear at the same place.

It is always helpful, however complicated the equation system, to try and get an intuitive feel of what the solutions will or could look like.

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