The embryonic dentition development in the alligator offers a model system to study many facets of tooth formation. As we have said we only focus here on the teeth pri-mordia initiation process. We use the known biological facts as a guide in constructing a mathematical model mechanism which describes the initiation and spatial patterning of the teeth primordia. The preliminary goal is to reproduce the observed spatial pattern of the first seven teeth primordia in the jaw of Alligator mississippiensis; seven because after this these teeth begin to be reabsorbed (but in a systematic way as we described above).

We begin by discussing how the biological data let us make certain quantitative assumptions which are part of the model. The seminal experimental work of Westergaard and Ferguson (1986, 1987, 1990) forms the basis for the development of the model mechanism. The recent biological investigations and experimental studies in mice supplement the biological database. Although some experimental results have yet to be shown in alligators, we reasonably assume certain similar characteristics. We attempt to incorporate as much of the known biological data as possible.

Of crucial importance for the model mechanism, is the incorporation of the physical growth of the jaw based on the known biological data. The initiation of the first seven teeth primordia occurs during the first one-third of the incubation period of the alligator. As noted, the number of teeth primordia seems to follow an exponential (Figure 4.12(a)) relationship for the first several primordia and a Gompertz-type growth (Figure 4.12(b)) for the full set of primordia during incubation.

To begin with we have to quantitatively characterise the growth of the jaw and then incorporate this into the model system of equations. We then describe how these chemical based model equations attempt to capture the physical process by which the biological mechanisms initiate a tooth primordium and the subsequent spatial patterning of teeth primordia. Since we have considerable experience with such chemically based systems, namely, reaction diffusion systems from earlier chapters, we shall only describe briefly how the system forms the heterogeneous spatial patterns which we hypothesise give rise to the cell condensations (the placodes) in the epithelium which mark the tooth initiation sites.

Although we make certain biological assumptions, the goal of the model is to capture the essential components of the biological mechanism and as much of the known biological data as possible. From the experimental work of Westergaard and Ferguson (1986), a comparison of tooth initiation sequences and positions between left and right sides of the jaw for both the same and other specimens of A. mississippiensis show no evidence of significant differences. So, we assume a symmetry in the initiation processes between the left and right sides of the jaw. The region in the jaw on which the primordia form is very thin compared with its length as is evident from Figure 4.13(a) so we consider that the teeth primordia form along a one-dimensional row. We construct a line from the posterior (back) of the jaw to the anterior (front) (Figure 4.14). This posterior-anterior one-dimensional axis is further justified by the experimental results (Westergaard and Ferguson 1986, 1987, 1990) which show that tooth initiation sites have very little lateral shift from an imaginary line drawn from posterior to anterior along the jaw epithelium.

Biologically, it is unknown how the signal to start tooth initiation is switched on. It is believed that this signal is controlled by neural crest cells in the mesenchyme which somehow send a message to the epithelium to start condensing.

The cell condensations in the epithelium mark the sites of teeth primordia initiation. Since this initiating source is also unknown, we make the assumption that there is a source of chemical at the posterior end of the jaw which starts the initiation process. How the source is switched on is unspecified, which is in keeping with what we know about the biology. The role of the chemical source diminishes quickly after the first tooth primordium is formed.

The experimental identification of certain components involved in tooth initiation and formation revealed epidermal growth factor, bone morphogenetic protein (BMP-4) and certain homeobox genes (Msx1 and Msx2). This suggests a chemical mechanism for the initiation of a primordium, where certain chemical concentrations stimulate an area of the epithelium to form a placode. So, we consider a reaction diffusion system but with some very different features from those in earlier chapters.

Model Equations

The aim then is to show that the proposed class of mechanisms for the initiation of the teeth primordia, which we now construct, following Kulesa (1995), Kulesa and Murray (1995), Murray and Kulesa (1996), and Kulesa et al. (1996a,b), are sufficient to explain the pattern of tooth sites in A. mississippiensis. To construct a dynamic pattern formation system, we must incorporate the physical growth of the jaw into a system capable of forming pattern. To do this, we combine the aspects of a static pattern formation mechanism, which is mediated by a control chemical, with the physical growth of the jaw domain. The result is a dynamic patterning mechanism.

Experimental evidence requires that the pattern arises dynamically as a consequence of jaw growth and not as the result of a prepattern of tooth initiation sites and so is a crucial element in the mechanism. From the experimental evidence of exponential jaw growth (Figure 4.12(a)), we assume that the jaw length, L = L (t), grows at a constant strain rate, r, according to

dt where, with the nondimensionalisation we use, L0 = 1. The experimental data let us obtain good estimates for the parameters. Basically the growing domain dilutes the chemical concentrations.

Consider the scalar reaction diffusion equation ct = Dct$ + Yf (c) (4.2)

on a growing domain where D, the diffusion coefficient, and y , the scale factor, are constants and f (c) is a reaction term, a function of the concentration c. Let s = quantity of reactant in length l ^ c = s/1.

Then, in the time interval s s + As

l l + Al which implies that change in the concentration s + As s s + lf (c)At s

Ac = —---= ^ Jy '---= y f (c)At — rcAt, l + Al l l + rlAt l J

which implies that

Ac lim — = y f (c) — rc At^0 At and so (4.2) becomes

The domain grows at an model it is easiest to change Here we set ct = Dcçç + y f (c) — rc

exponential rate. For ease of numerical simulation of the the variable so that the equations are on a fixed domain.

x = f e-rt ^ ct = De-2rtCxx + Yf (c) - rc, xe[x, L], L fixed. (4.4)

The reaction diffusion equation (4.3) on a growing domain becomes a non-autonomous reaction diffusion equation in a fixed domain with a diffusion coefficient decreasing exponentially with time.

For the chemical patterning mechanism, we take, by way of example, a basic di-mensionless reaction diffusion system, namely, (2.32) from Chapter 2, which we studied in some depth in Sections 2.4 and 2.5. We modify it, guided by the above discussion of the biology, by combining it with an equation for a chemical c, which controls the substrate u, in a simple inhibitory way to get d U

d u dx2

dv dt

Here u(x, t) and v(x, t) represent the respective concentrations of a substrate and an activator with y the usual scale factor (see Chapter 3), b and h constants, 8 the assumed first-order kinetics decay of c, d the usual ratio of the activator's diffusion coefficient to that of the substrate and p the ratio of diffusion coefficient of c to that of the substrate u.

We assume the source of u is controlled by c, an inhibitor related to the epidermal growth factor, EGF. That is, we assume the existence of an inhibitory substance whose concentration decreases as the concentration of EGF increases and vice-versa. It comes from the source at the posterior end of the jaw.

If we now carry out the scale transformation in (4.3) and a nondimensionalisation to make the domain fixed to be 1 the resulting nondimensionalised equations for the substrate, u, the activator, v, and the inhibitor, c, are, on a scaled domain 0 < x < 1:

dv dt

Each equation (4.6)-(4.8) has a dilution term, due to jaw growth, and a time-dependent diffusion coefficient which arises from the coordinate transformation to the scaled domain in x . These equations govern the variables on the growing domain of the jaw and were studied in depth by Kulesa (1995). Relevant boundary conditions are where c0(t) is a decreasing function corresponding to a source term at the posterior end as discussed above. The condition (4.9) implies zero flux of u and v at either end of the domain, while for c there is zero flux only at the anterior end (4.10). Recall that we have taken only half of the jaw to scale to 1 which means that at the anterior end it is the symmetry condition which gives the condition at x = 1 in (4.10).

From the analyses in Chapter 2, specifically Sections 2.4 and 2.5, for a range of parameter values and a domain size larger than some minimum, the reaction diffusion system given by the first two equations of (4.5) with c a constant, is capable of producing steady state spatial patterns in u and v. By varying one or more of the parameters in these equations the system can select a stable heterogeneous state or a specific regular spatial pattern. When the inhibitor, c, in the first of (4.6) is above a threshold value, pattern formation in u and v is inhibited. For c below this threshold, the pattern formation mechanism is switched on, via diffusion-driven instability and a spatial pattern forms in u and v when the subthreshold portion of the domain is large enough. The interplay between the parameters and the domain scale is important in determining the specific pattern formed with certain spaces giving rise to specific patterns; refer specifically to Figure 4.21 below where the parameter spaces are given specifically for the hump-like patterns we require here. The analysis there is for hc a constant parameter (equivalent to the 'a' there) and for a fixed domain size. The situation we consider here has a variable domain and c varies in space and time. It is not easy to carry out an equivalent analysis in such a case. We can however get an intuitive feel for what is going to happen as we show below.

The representation of the tooth primordium becoming a source of inhibitor c(x, t) is strongly suggested by the biology. The experimental studies (Westergaard and Ferguson 1986,1987,1990, Osborn 1971) led us to postulate a zone around a newly formed tooth which inhibits subsequent teeth primordia from forming in the local region. Westergaard and Ferguson (1986) noted that if a new primordium was forming in between older primordia, the new primordium would form closer to the older neighbour. Mathematically, we characterise this zone of inhibition by allowing each new tooth primordium to become a source of chemical growth factor, whose role is to inhibit tooth primordium formation in the local region of a newly formed primordium. We now define a new tooth site where the substrate, u(x, t), crosses a threshold on a subdomain of 0 < x < 1. This turns on a new tooth source, c, of c at this site, which simulates a zone of inhibition. The concentration of this tooth source we model according to logistic growth,

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