Modelling Concepts Determining the Time of Stripe Formation

Although we do not know the actual mechanism the basic concept of essentially all pattern generation models can be couched intuitively in terms of short range activation and long range inhibition as we point out in Chapter 2. There we discussed in detail reaction-diffusion-chemotaxis mechanisms and later in the book we introduce other pattern generator systems, such as neural network models and the Murray-Oster mechanochemical theory of biological pattern formation. Any of them at this stage of our knowledge of the detailed biological processes involved could be a candidate mechanism for generating the stripes on the alligator. However, from the experimental evidence from skin histological sections given in Murray et al. (1990) and briefly described above there is some justification in taking (although really by way of example) a cell-chemotaxis-diffusion sytem in which the cells create their own chemoattractant. How such a mechanism generates spatial heterogeneity was discussed in detail in Chapter 2 and is dealt with in much more detail in Chapter 5. When the aggregative effects (chemotaxis) are greater than the dispersal effects (diffusion), pattern evolves. Of particular relevance here, however, is the sequential laying down of a simple stripe pattern by a travelling wave as given in Chapter 2, Section 2.6. The reason for this is that from observation the pattern of stripes on the alligator embryo appear first at the nape of the neck and progress down the body in a wavelike manner.

The ability of a model mechanism to generate a specific pattern is no indication as to its relevance to the biological problem under study. However, different models usually suggest different experiments. A major drawback in checking or trying out any theory, is that experimentalists generally do not know when the actual patterning takes place. In the case of the alligator the actual stripe pattern, as in Figure 4.1, becomes visually evident around 40 days through the gestation period of approximately 70 days. It is almost certain that the pattern is laid down much earlier. Thus, before it is possible to determine what mechanism is actually producing the pattern it is clearly essential to know when in gestation the mechanism is active.

Pigment deposition may depend on the number of cells present in a region. Three possible ideas can explain white stripes: either (i) melanocytes are absent, or (ii) all melanocytes produce melanin to the same extent but the concentration of cells in an area is too low for the region to look dark, or (iii) the formation of melanin by cells is dependent on cell number; for example, a threshold in cell number within a certain area has to be reached before melanin is produced. In alligator embryos, white stripes appear to be due to a low number of melanocytes in white stripes which do not produce melanin in large quantities (Murray et al. 1990).

Development generally begins at the anterior end of the embryo; the extent of differentiation at the head is always much greater than at the tail. Thus in the process of melanin deposition it is the head which shows the first significant signs of pigmentation. The pattern of white stripes is first seen on the body and gradually moves down the tail.

Whether or not the skin develops a pigmented patch depends on whether pigment cells produce melanin in sufficient quantity. The specimen cell-chemotaxis-diffusion model can certainly produce a series of stripes, or spots, depending on the geometry and scale, in which the cell concentration is high. The idea then is that at the time the mechanism is activated to produce pattern the embryo is long and thin, essentially a one-dimensional domain, and so, if the mechanism starts at one end, the nape of the neck, stripes are laid down in a sequential manner as illustrated in Figure 4.2(a) via a wavelike pattern generator which then appears as shown in Figure 4.2(b). This waveform is a one-dimensional solution (Myerscough amd Murray 1992) of a basic cell-chemotaxis-diffusion system (in dimensionless form) given in the legend of Figure 4.2(a) with the parameter values given in the figure. We discuss the model in more detail later in the chapter. Here n denotes the cells, which diffuse with coefficient D (relative to the diffusion coefficient of the chemoattractant) and which produce their own chemoattractant,

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