Stripes and Shadow Stripes on the Alligator

As noted above the pigmentation first starts in the head region and then proceeds towards the tail. Thus, the first requirement of the mechanism is to be able to generate a travelling wave of stripe patterning from the head. Numerical simulation of the cell-chemotaxis mathematical model above with appropriate parameter values, can generate such a sequential laying down of a regular stripe pattern; Figure 4.2(a) shows a typical case with x = 0 corresponding to the head region and the pattern proceeding towards the tail, that is, in the positive x-direction.

In Figure 4.2(a), the peaked stripe pattern is in cell density; there is a qualitatively similar one for the chemoattractant concentration. The observed white stripe we associate with the troughs in cell density; that is, there are insufficient melanocytes to produce a pigmented area of any significance. This is in line with the observation (Table 1 in Murray et al. 1990) that when more stripes can be accommodated the tail tip is often white. A corollary of this hypothesis is that the default colour of the embryo is the dark pigmented form in that if the pattern formation mechanism were not activated there would be a uniform cell density over the integument all of which could become competent to produce melanin, probably lighter than that found between the white stripes in the normal situation. This lack of pattern would result in a melanistic form which is a very rare occurrence (Ferguson 1985).

If we hypothesise that the time it takes the mechanism to generate pattern is small compared with the time for significant embryonic growth, then the mechanism would produce a regular evenly spaced pattern on the back of the alligator as illustrated schematically in Figures 4.2(b) and in 4.7(a). On the other hand if the mechanism operates over a period of several days during which competency of the cells to generate pattern slowly decreases the striping can be less regular. We suggest that these patterns are the prepatterns for the observed pigmentation. The actual wavelength, that is, the distance between the stripes, is determined by the parameters. What is striking about these specific wavelike patterns in Figure 4.2(a) is their sharpness, which is in keeping with those found on A. mississippiensis. Other mechanisms mentioned above can also produce similar sequential pattern formation but it is less easy to obtain such sharpness in the peaks.

It is reasonable to assume that the mechanism is activated during a specific stage in development. If the parameter values remain fixed, the number of stripes depends principally on the length of the embryo at that stage in development. A given length of the embryo can accommodate a specific number of wavelengths, so the longer the embryo is when the mechanism is operative the greater the number of stripes. Thus the parameter values and size of the embryo determine the number of stripes.

Let us now relate these mathematical results to the experimental results in the previous sections. The first point to recall is that a higher temperature accelerates the growth of the embryo and the time to reach a given stage. So at the higher temperature the embryo is larger when the mechanism is activated. Although temperature can also affect the parameter values in a pattern generator, we do not expect this to be very significant with only a 3°C difference; we come back to this point below.

Once the cell pattern has been formed, such as in Figure 4.2 and Figure 4.7(a), a certain time must elapse for the pigmentation process and hence for the stripes to become visible. Thus the key time in development for stripe patterning must be before

Figure 4.7. (a) Proposed evenly spaced stripe pattern of cell density on the embryo at the time the mechanism generates the pattern which we associate with the cell density stripes in Figure 4.2(a). (b) The appearance of 'shadow stripes' (cf. Figure 4.1) predicted by the model as a consequence of embryonic lateral growth in the trunk during the pattern formation process. (From Murray et al. 1990)

Figure 4.7. (a) Proposed evenly spaced stripe pattern of cell density on the embryo at the time the mechanism generates the pattern which we associate with the cell density stripes in Figure 4.2(a). (b) The appearance of 'shadow stripes' (cf. Figure 4.1) predicted by the model as a consequence of embryonic lateral growth in the trunk during the pattern formation process. (From Murray et al. 1990)

the stripes are visible. There is approximately an eight-day difference between embryos incubated at the two different temperatures. As also noted above, at a time prior to the pattern becoming discernible, embryos were significantly larger when incubated at 33°C than those at 30°C. Thus it is in keeping with the above patterning process that males (eggs incubated at 33°C) should have more stripes than females (30°C incubation) and that it is simply length and not sex which determines the number of stripes. As already mentioned the number of genes is limited and to use them up, for example, in prescribing the precise number and position of stripes on male and female alligator embryos would be an unbelievably inefficient and unnecessary use.

The theoretical prediction and resulting experimental verification described here clearly show that the number of stripes is just a question of embryonic length and size when the mechanism is activated. It is a simply a direct application of the results derived in Chapter 2 where we showed that the number of stripes in a one-dimensional domain was directly related to the length of the domain. In other words if, for example, two stripes can be formed on a domain by a typical pattern formation mechanism, whether reaction-diffusion, reaction-chemotaxis or mechanical, then four stripes will be formed if the domain is twice as long.

At the time of pattern formation a stripe (defined as from the front of a white stripe to the front of the next white stripe) must occur, on the trunk and tail, on average every L mm where tail length + nape to rump length (at pattern initiation) number of stripes at hatching

Although the size of the stripes at hatching is not uniform, being wider towards the tail tip, this can be accounted for simply by the very different growth rates of the trunk and tail during embryogenesis. On the basis that the pattern generation by the mechanism is relatively fast compared to embryonic growth, and that the mechanism is activated just prior to stripe visibility, the data in Table 2 of Murray et al. (1990), from which Figures 4.3 to 4.6 were obtained, show that for embryos incubated at

33°C: Day 32 L = (56.25 + 39.50)/20.35 = 4.71 mm, 30°C: Day 40 L = (53.60 + 37.20)/18.55 = 4.89 mm.

The extra 7.75 mm in length, 33°C versus 30°C, at this stage would allow for a mean of 1.64 (7.75/4.71) more stripes on male embryos. This is qualitatively in line with the observations on alligator hatchlings reported in Murray et al. (1990).

Note also that the size of the stripe, at formation, on the male is slightly smaller than on the female. Although this small size variation may not be significant it could be due to the temperature effect on the model parameters.

Some effects of temperature on pigment pattern, albeit in a very different situation, were reported by Nijhout (1980a) for the spread of an eyespot on the wing of the butterfly Precis coenia: the temperature difference in his experiments was 10°C. In Chapter 3 we investigated this temperature effect using a reaction diffusion model and saw that the size variation in the wing spot could be accounted for by parameter variations which were not inconsistent with those to be expected in diffusion coefficients and reaction rate constants of possible biochemicals with such a temperature change. The results of Harrison et al. (1981) on the effect of temperature on hair spacing in regenerating Acetabularia, which we also discussed in Chapter 3 tend to lend substance to this hypothesis: they also used a reaction diffusion model.

Shadow Stripes

We assumed that the time to generate the stripe pattern is fast compared to the time for significant embryonic growth. If we relax this assumption it implies that the patterning is taking place on a growing domain. The implication is that the cells remain competent to secrete chemoattractant, albeit with a decreasing efficiency, but nevertheless with the ability to create patterns for longer. In this case it means that the size of the integument wherein the mechanism acts is both longer and, on the trunk particularly, wider. The mathematical model implies that the type of pattern which will appear when a long thin domain becomes slightly wider, after the main stripes have been initiated, consists of less distinct 'shadow stripes' positioned between the principal stripes: see Figures 4.1 and 4.7(b). This is as we would expect from our knowledge of the effect of scale and geometry on pattern (Chapter 2). A further consequence of increasing the time of pattern formation is that since the pattern is initated from the head the density of the stripes towards the tail is less sharp in colour definition, a feature which is also often observed. Thus the mechanism, or even just knowledge of how pattern is formed by these pattern generators, can offer explanations for the more complex patterning typical of A. missis-sippiensis.

As we commented before, similar patterns can be generated by any of the models mentioned above, so the conclusions are independent of the actual mechanism involved, as long as it can form the pattern relatively quickly. So, the reaction diffusion and mechanochemical models are also candidate mechanisms. As also mentioned what distinguishes one mechanism from another are the experiments each suggests. With mechanisms which directly involve real biological quantities, such as those used here and the Murray-Oster mechanochemical models, actual cells are directly involved. It is considerably easier to manipulate embryonic cell density (a key parameter here) than it is to manipulate unknown chemicals in a reaction diffusion theory. In general it is easier to disprove theories which involve cells and tissue directly but in the process of doing so we generally increase our understanding of the patterning processes. One example of this comes from the development of cartilage patterns in the vertebrate limb discussed in detail later in the book.

In summary we have shown that the incubation temperature of A. mississippien-sis significantly affects the number of stripes on the dorsal side of the hatchling with incubation at 30°C resulting in fewer stripes than at 33°C. Although we used a pattern formation mechanism based on the idea that cells secrete a chemoattractant to which the cells react the evidence for it is certainly not conclusive. All we require of a mechanism is that it can generate patterns in a similar way and relate similarly to the constraints of geometry and scale. The time at which the mechanism, which establishes the pattern of stripes, acts at 33°C is advanced compared to 30°C. Although the stage of development at which the mechanism is invoked is also earlier at the higher temperature, the time and developmental stage are combined in such a way that when the mechanism is activated the embryo at 33°C is longer than at 30°C and hence more stripes are laid down. The pattern of stripes is only related to the sex of the animal through temperature; the pattern is not specifically sex linked. All of the results we have found here are independent of the detailed pattern formation mechanism. This case study is another example where the modelling suggested the explanation as to why there is a larger number of stripes on male alligators as compared with female ones. The experiments described above were developed to investigate the model's hypothesis, namely, that it is simply length and not sex as such, which accounts for the difference. The theoretical predictions were confirmed.

Stripe Formation on the Juvenile Angelfish (Pomacanthus)

Another example of what might appear to start out as shadow stripes arises with growing juvenile angelfish (Pomacanthus), an example of which is shown in Figure 4.8. The small fish initially has three stripes but as it grows it adds to this number by inserting new stripes in between those already there: it is a gradual process which repeats itself until the fish is fully grown. However, unlike the shadow stripes on the alligator those on this fish are different in that the interdigitating stripes first appear as faint very narrow stripes unlike the same size stripes that would appear if it were simply an increase in domain space as the fish grew; this is the explanation put forward by Kondo and Asai (1995). There have been various attempts at tinkering with reaction diffusion models but none were able to explain the distinct character of the observed sequential patterning. Painter et al. (1999) obtained growth rates from extant experiments and concluded that all subepidermal tissue cell types undergo mitosis, including pigment cells. They proposed a model which included an equation for diffusional and chemotactic cell movement in which the chemotaxis is modulated by morphogens from an independnt reaction diffusion system. They show that it is the chemotaxis which produces the slow growth of the new stripes. The sequential striping they obtained, in two dimensions,

Figure 4.8. Typical stripe pattern on the angelfish, Pomacanthus circulatis at age 12 months. The number of stripes go from 3 at 2 months to 6 at 6 months to 12 at 12 months. (Photograph from the National Aquarium, Washington D.C. and reproduced with permission)

quantitatively mimics that observed on the angelfish. The patterning challenges on fish pose different problems to those on animal coats and snakes (discussed below). Aragon et al. (1998) investigated the role of boundary conditions, domain growth and the coupling of reaction diffusion models on the patterns that can be formed. They compared their results with the pigmentation patterns on several marine fish.

An interesting article by Denton and Rowe (1998) suggests a very practical use for the stripes on the backs of mackerel (Scomber scombrus L.) other than just for camouflage (or being lost in the crowd) as usually thought. They argue that the stripes are used for precise signalling information about the fish's movement to neighbours in the school. There is a thin layer of reflecting platelets which overlies the central parts of the light and dark stripes on each side of the dorsal surfaces of the fish. Denton and Rowe (1998) show how these reflecting platelets and the distribution of the body stripes can greatly facilitate information communication. When the fish changes its orientation or its velocity with respect to its neighbours there is a change in the patterns of brightness on the dorsal surfaces. They show convincingly that very slight changes in the roll, yaw and pitch immediately produce marked shifts in the observed patterns and they suggest that this is a major role for the stripe arrangement. It is well known that schools of fish coordinate their movement extremely quickly and this could be the means of doing it.

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