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Gilbert SF. Developmental Biology. 6th edition. Sunderland (MA): Sinauer Associates; 2000.

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# Mathematical Modeling of Development

Developmental biology has been described as the last refuge of the mathematically incompetent scientist. This phenomenon, however, is not going to last. While most embryologists have been content trying to analyze specific instances of development or even formulating some general principles of embryology, some researchers are now seeking the laws of development. The goal of these investigators is to base embryology on formal mathematical or physical principles (see Held 1992; Webster and Goodwin 1996). Pattern formation and growth are two areas in which such mathematical modeling has given biologists interesting insights into some underlying laws of animal development.

## The mathematics of organismal growth

Most animals grow by increasing their volume while retaining their proportions. Theoretically, an animal that increases its weight (volume) twofold will increase its length only 1.26 times (as 1.263 = 2). W. K. Brooks (1886) observed that this ratio was frequently seen in nature, and he noted that the deep-sea arthropods collected by the Challenger expedition increased about 1.25 times between molts. In 1904, Przibram and his colleagues performed a detailed study of mantises and found that the increase of size between molts was almost exactly 1.26 (see Przibram 1931). Even the hexagonal facets of the arthropod eye (which grow by cell expansion, not by cell division) increased by that ratio.

D’Arcy Thompson (1942) similarly showed that the spiral growth of shells (and fingernails) can be expressed mathematically (r = aθ), and that the ratio of the widths between two whorls of a shell can be calculated by the formula r = e2πcotθ (Figure 1.17; Table 1.1). Thus, if a whorl were 1 inch in breadth at one point on a radius and the angle of the spiral were 80°, the next whorl would have a width of 3 inches on the same radius. Most gastropod (snail) and nautiloid molluscs have an angle of curvature between 80° and 85°.* Lower-angle curvatures are seen in some shells (mostly bivalves) and are common in teeth and claws.

#### Figure 1.17

Equiangular spiral growth patterns. (A) A ram's horn and the shell of a chambered nautilus both show equiangular spiral growth. The nautilus shell (below) is cut in cross section. (B) René Descartes’ analysis of an equiangular spiral, (more...)

#### Table 1.1

Constant angle of an equiangular spiral and the ratio of widths between whorls.

Such growth, in which the shape is preserved because all components grow at the same rate, is called isometric growth. In many organisms, growth is not a uniform phenomenon. It is obvious that there are some periods in an organism's life during which growth is more rapid than in others. Physical growth during the first 10 years of person's existence is much more dramatic than in the 10 years following one's graduation from college. Moreover, not all parts of the body grow at the same rate. This phenomenon of the different growth rates of parts within the same organism is called allometric growth (or allometry). Human allometry is depicted in Figure 1.18. Our arms and legs grow at a faster rate than our torso and head, such that adult proportions differ markedly from those of infants. Julian Huxley (1932) likened allometry to putting money in the bank at two different continuous interest rates.

#### Figure 1.18

Allometry in humans. The embryo's head is exceedingly large in proportion to the rest of the body. After the embryonic period, the head grows more slowly than the torso, hands, and legs. Human allometry has been represented in Western art only since the (more...)

The formula for allometric growth (or for comparing moneys invested at two different interest rates) is y = bx a/c, where a and c are the growth rates of two body parts, and b is the value of y when x = 1. If a/c > 1, then that part of the body represented by a is growing faster than that part of the body represented by c. In logarithmic terms (which are much easier to graph), log y = log b + (a/c)log x.

One of the most vivid examples of allometric growth is seen in the male fiddler crab, Uca pugnax. In small males, the two claws are of equal weight, each constituting about 8% of the crab's total weight. As the crab grows larger, its chela (the large crushing claw) grows even more rapidly, eventually constituting about 38% of the crab's weight (Figure 1.19). When these data are plotted on double logarithmic plots (the body mass on the x axis, the chela mass on the y axis), one obtains a straight line whose slope is the a/c ratio. In the male Uca pugnax (whose name is derived from the huge claw), the a/c ratio is 6:1. This means that the mass of the chela increases six times faster than the mass of the rest of the body. In females of the species, the claw remains about 8% of the body weight throughout growth. It is only in the males (who use the claw for defense and display) that this allometry occurs.

#### Figure 1.19

Male specimens of the fiddler crab, Uca pugnax. Allometric growth occurs only in one of the male's claws. In females (not shown), both claws retain isometric growth. (Photograph courtesy of Swarthmore College Marine Biology laboratory.)

## The mathematics of patterning

One of the most important mathematical models in developmental biology has been that formulated by Alan Turing (1952), one of the founders of computer science (and the mathematician who cracked the German “Enigma” code during World War II). He proposed a model wherein two homogeneously distributed solutions would interact to produce stable patterns during morphogenesis. These patterns would represent regional differences in the concentrations of the two substances. Their interactions would produce an ordered structure out of random chaos.

Turing's reaction-diffusion model involves two substances. One of them, substance S, inhibits the production of the other, substance P. Substance P promotes the production of more substance P as well as more substance S. Turing's mathematics show that if S diffuses more readily than P, sharp waves of concentration differences will be generated for substance P (Figure 1.20). These waves have been observed in certain chemical reactions (Prigogine and Nicolis 1967; Winfree 1974).

#### Figure 1.20

Reaction-diffusion (Turing model) system of pattern generation. Generation of periodic spatial heterogeneity can come about spontaneously when two reactants, S and P, are mixed together under the conditions that S inhibits P, P catalyzes production of (more...)

The reaction-diffusion model predicts alternating areas of high and low concentrations of some substance. When the concentration of such a substance is above a certain threshold level, a cell (or group of cells) may be instructed to differentiate in a certain way. An important feature of Turing's model is that particular chemical wavelengths will be amplified while all others will be suppressed. As local concentrations of P increase, the values of S form a peak centering on the P peak, but becoming broader and shallower because of S's more rapid diffusion. These S peaks inhibit other P peaks from forming. But which of the many P peaks will survive? That depends on the size and shape of the tissues in which the oscillating reaction is occurring. (This pattern is analogous to the harmonics of vibrating strings, as in a guitar. Only certain resonance vibrations are permitted, based on the boundaries of the string.)

The mathematics describing which particular wavelengths are selected consist of complex polynomial equations. Such functions have been used to model the spiral patterning of slime molds, the polar organization of the limb, and the pigment patterns of mammals, fish, and snails (Figures 1.21 and 1.22; Kondo and Asai 1995; Meinhardt 1998). A computer simulation based on a Turing reaction-diffusion system can successfully predict such patterns, given the starting shapes and sizes of the elements involved.

#### Figure 1.21

A photograph of the snail Oliva porphyria (left), and a computer model of the same snail (right) in which the growth parameters of the shell and its pigmentation pattern were both mathematically generated. (From Meinhardt 1998; computer image courtesy (more...)

#### Figure 1.22

Pigment patterns of zebrafish homozygous for the wild-type allele (A) and for three different mutant alleles (B–D) of the leopard gene. Computer simulations of the pigment patterns are shown in the bottom row. Changing a single parameter of the (more...)

One way to search for the chemicals predicted by Turing's model is to find genetic mutations in which the ordered structure of a pattern has been altered. The wild-type alleles of these genes may be responsible for generating the normal pattern. Such a candidate is the leopard gene of zebrafish (Asai et al. 1999). Zebrafish usually have five parallel stripes along their flanks. However, in the different mutations, the stripes are broken into spots of different sizes and densities. Figure 1.22 shows fish homozygous for four different alleles of the leopard gene. If the leopard gene encodes an enzyme that catalyzes one of the reactions of the reaction-diffusion system, the different mutations of this gene may change the kinetics of synthesis or degradation. Indeed, all the mutant patterns (and those of their heterozygotes) can be computer-generated by changing a single parameter in the reaction-diffusion equation. The cloning of this gene should enable further cooperation between theoretical biology and developmental anatomy.

### WEBSITE

1.3 The mathematical background of pattern formation. The equations modeling pattern formation are a series of partial derivatives depicting rates of synthesis, degradation, and diffusion of the activator and inhibitor molecules. http://www.devbio.com/chap01/link0103.shtml

### WEBSITE

1.4 How does the zebra get its stripes? No one knows for sure, but adding the Turing equations to what's known about equine embryology allows one to model how each of the three known zebra species acquired its unique striping pattern. http://www.devbio.com/chap01/link0104.shtml

## Footnotes

*

If the angle were 90°, the shell would form a circle rather than a spiral, and growth would cease. If the angle were 60°, however, the next whorl would be 4 feet on that radius, and if the angle were 17°, the next whorl would occupy a distance of some 15,000 miles!

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