With the above heuristic scenario we are now ready to simulate the complete model mechanism and solve for the spatial and temporal sequence of the first seven teeth pri-mordia; full details of the numerical analysis and discussion of the results are given by Kulesa (1995) and these and others in Kulesa and Murray (1995), Murray and Kulesa (1996) and Kulesa et al. (1995, 1996a,b). We first have to estimate a set of model parameters. Some we can obtain from experiment but others must be determined to ensure the appropriate spatial pattern in u. We can estimate the crucial growth rate parameter r from Westergaard and Ferguson (1986, 1990). We can get real biological estimates for the diffusion coefficient p since it is related to the epidermal growth factor. Since the degradation constant 8 is also related to epidermal growth factor we also have estimates for it. For the others, we have to use the equations and the ultimate pattern we require. This was done using a straightforward linear analysis of a simplified form (4.6) and (4.7), namely the system (4.14). We do not repeat the analysis since it is exactly the same as that carried out in detail in Chapter 2, Sections 2.4 and 2.5 where we derived the patterning space for this simplified system. We simply chose the set for the appropriate mode 2-like pattern in u to be initated when the system was diffusion-driven unstable. There is a systematic, and simple to use, logical numerical procedure for determining the parameter set for specific patterns in pattern generator mechanisms in general (Ben-til and Murray 1991) when it is not possible to do it analytically which is more the norm. The boundary conditions are given by (4.9) and (4.10) and initial conditions:

b u(x, 0) = u0 = h + b, v(x, 0) = v0 =--, c(x, 0) = a exp(-kx), (4.16)

where a and k are positive parameters. A representative continuous function of time for the source of inhibitor at the anterior end of the jaw, (4.16), was taken as c(0, t) = co(t) = —m tanh(t — f )/g + j, (4.17)

where m, f, g and j are constant parameters: this gives a smooth step function form for the initial switching on of the patterning process. A small random spatial perturbation was then introduced to u0 and v0 and the full nonlinear model system (4.6)-(4.8) solved numerically for the spatial and temporal sequence of the first seven teeth primordia of the lower jaw. A finite difference scheme based on the Crank-Nicholson method was used with Ax = 0.01, At = (Ax)2. All parameter values are given in the legend to Figure 4.19 where the results are compared with experiment.

Figure 4.17 shows the numerical simulation for u as far as the time of formation for the first five teeth together with the solution for the inhibitory chemical c.

Figure 4.17. (a-c) The computed concentration profiles for the substrate u and the inhibitor c for times up to tooth formation time tj where i denotes the i th tooth. Tooth 1 is the dental determinant. The spatial positioning of teeth primordiais given by X{, the positions where u reaches the threshold u^. Solutions are given on [0,1] but the actual domain size at each time is given by [0, exp rt]. The parameter values for all of the simulations are given in the legend of Figure 4.19.

Figure 4.17. (a-c) The computed concentration profiles for the substrate u and the inhibitor c for times up to tooth formation time tj where i denotes the i th tooth. Tooth 1 is the dental determinant. The spatial positioning of teeth primordiais given by X{, the positions where u reaches the threshold u^. Solutions are given on [0,1] but the actual domain size at each time is given by [0, exp rt]. The parameter values for all of the simulations are given in the legend of Figure 4.19.

In Figure 4.17(a) the time ti is the time u reaches the threshold uth which initiates a tooth primordium at x1 and this switches on a source of the inhibitor c at this position; at x1 there is then a zero flux barrier. At this stage a new simulation, with the u,v and c distributions from the first calculation, was started in the two regions 0 < x < x1 and x1 < x < 1 including the new source of c at x1. This simulation was carried through until c again decreased sufficiently over a large enough region and the spatial pattern in u again reached uth at time t2 at some place in one of the domains at a position denoted x2 thereby fixing the position of the next tooth, tooth 2. The next simulation was then carried out in a similar way but now in the three regions 0 < x < x2 and x2 < x < x1 and x1 < x < 1 and the position was determined where u next crossed the threshold uth at time t3 at position x3. The sequence of simulations was then carried out to determine the position and time for each tooth primordium position. All of the results shown in Figure 4.17 are plotted on the domain [0,1] but the actual domain size is [0, exp rt1] where r is the growth rate parameter of the jaw.

We can represent the results in Figure 4.17 in a different way in Figure 4.18 to make it clearer where and when the teeth primordia are determined to form as the jaw grows. In Figure 4.18(a) we show the three-dimensional evolution of the substrate concentration u as a function of distance along the jaw (normalised to 1) and of time in days. The dimensionless time in the simulations is related to the actual time in days via the estimation of the expontial growth rate parameter r from the experimental data reproduced graphically in Figure 4.12: values are given in Figure 4.19. Figure 4.18(b) plots the time of incubation in days against the length of the simulated jaw with the order of teeth primordia superimposed.

The experimental data (Westergaard and Ferguson 1990) give the sequence of teeth appearance for both the upper and lower jaw of A. mississippiensis and it was from these data that Figure 4.12 was obtained. From Figure 4.12(a) we see that the upper jaw varies slightly for the first several primordia, namely primordia 6 and 7 are in different spatial locations. When compared to the lower jaw data, it can also be seen (Figure 4.12(a)) that the upper jaw grows at a slightly increased rate. For the simulations described we used exactly the same parameter set for both the upper and lower jaw except for the slightly higher growth rate for the upper jaw, specifically r = 0.34/day as compared with r = 0.31/day. The experimental data also let us directly relate simulation time T to real time t as described in the figure legend.

The numerical simulations, with the parameter sets, for the sequence of teeth in both the upper and lower jaw are presented in Figure 4.19 together with the experimental data for comparison. The model mechanism reproduces the correct spatial and temporal sequence for the first 8 teeth in the lower jaw and the correct sequence for the first 6 teeth in the upper jaw. The results are encouraging.

Kulesa et al. (1996a) also relate the numerical results with the teeth primordia appearance on the actual jaw; again the agreement is very good. The spatial positions from the experimental data were obtained from a Cartesian xy-coordinate system attached to Figure 4.13(a). A line in this figure lying along the teeth primordia, represented by a parabola f (x), was fit to a curve of the experimental data ((f (x) = b1 x2 + b2x + b3; b1 = -0.256, b2 = 0, b3 = 7.28). The numerical spatial positions were then related by using the length l, of f (x) from x = 0 to x = x* (where f (x*) = 0), as the nondimensional length of the anterior-posterior axis in their simulations.

Tooth 2 Toolh 3

Length of Simulated Jaw

Figure 4.18. (a) The simulated concentration profile for the substrate u as a function of time and distance along the jaw. The positioning of the teeth is determined by the positions in the jaw where u reaches a threshold value. (b) The jaw length and appearance of the tooth primordia as they relate to the incubation time in days. All the parameter values are given in the legend of Figure 4.19. (From Kulesa 1995)

Figure 4.19. Comparison of the numerical versus experimental data for the teeth primordia initiation sequence in both the upper and lower half-jaws of A. mississippiensis. Upper jaw: '*' denotes the numerical data with a solid line: N(t) = N3 exp(r3t); N3 = 0.0042, r3 = 0.35/day. The dash-dot line (-----) and '+'

denote the experimental data: N (t) = N4exp(r4t); N4 = 0.0047, r4 = 0.34/day. Lower jaw: 'x' denotes the numerical data with a dashed line (---): N(t) = N\ exp(rjt); N\ = 0.012, r\ = 0.28/day. The dotted line (• • •) and 'o' denote the experimental data: N(t) = N2exp(r21); N2 = 0.0066, r2 = 0.31/day. Time t (days) was scaled to time T (simulation) using T = ait + a2; ai = 27.06 and a2 = -286.6. The model parameters are (refer to (4.6)-(4.11), (4.16) and (4.17)): h = 1, b = 1, d = 150, r = 0.01, p = 0.5, y = 40, a = 0.2, k1 = 0.3, k2 = 1.0, a = 2.21, k = 0.9 and c0(t) = c(0, t) = -m tanh[(t - f )/g] + j; with m = 0.65, f = 200, g = 34, j = 1.5. (From Kulesa et al. 1996a)

Figure 4.19. Comparison of the numerical versus experimental data for the teeth primordia initiation sequence in both the upper and lower half-jaws of A. mississippiensis. Upper jaw: '*' denotes the numerical data with a solid line: N(t) = N3 exp(r3t); N3 = 0.0042, r3 = 0.35/day. The dash-dot line (-----) and '+'

denote the experimental data: N (t) = N4exp(r4t); N4 = 0.0047, r4 = 0.34/day. Lower jaw: 'x' denotes the numerical data with a dashed line (---): N(t) = N\ exp(rjt); N\ = 0.012, r\ = 0.28/day. The dotted line (• • •) and 'o' denote the experimental data: N(t) = N2exp(r21); N2 = 0.0066, r2 = 0.31/day. Time t (days) was scaled to time T (simulation) using T = ait + a2; ai = 27.06 and a2 = -286.6. The model parameters are (refer to (4.6)-(4.11), (4.16) and (4.17)): h = 1, b = 1, d = 150, r = 0.01, p = 0.5, y = 40, a = 0.2, k1 = 0.3, k2 = 1.0, a = 2.21, k = 0.9 and c0(t) = c(0, t) = -m tanh[(t - f )/g] + j; with m = 0.65, f = 200, g = 34, j = 1.5. (From Kulesa et al. 1996a)

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