Sea-shells and Tigers

‘Each week I plot your equations dot for dot, xs against ys in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God’s truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose?’ So says Lady Thomasina Coverly, the heroine of Tom Stoppard’s Arcadia, to her tutor Septimus Hodge. Her question was echoed a century after her (fictitious) life by the unorthodox biologist D’Arcy Wentworth Thompson, whose mathematical investigations of the living world were collected in On Growth and Form, published in 1917. Although they have excited the admiration of some important thinkers, including Alan Turing and the biologists John Tyler Bonner and Stephen Jay Gould, Thompson’s ideas do not figure prominently in the biological curriculum or the mainstream of research.

By contrast, that mainstream takes very seriously an unguarded remark of the youthful Francis Crick, who once announced to the lunchtime crowd at the Eagle in Cambridge: ‘Jim and I have discovered the secret of life.’ Watson and Crick’s identification of the molecular structure of DNA was followed by further insights, as they and their successors explained the replication of genetic material, the nature of the genetic code, the mechanisms of gene regulation and of protein synthesis, as well as devising the techniques of manipulating DNA, of splicing, cloning and sequencing that are currently transforming biomedicine. Contemporary molecular biologists foresee the possibility of laying bare the fine structure of the major processes that occur in living things, and it is easy to understand their high rhetoric of ‘quests’, ‘the grail’, and ‘the secret of life’. Easy also to see why they might think that Thompson’s tentative answers are the quaint curiosities of a bygone age.

Ian Stewart thinks that such dismissals are wrong, the product of an obsession with biochemical detail that can become myopic. Life’s Other Secret is intended to display the many possibilities for the mathematical analysis of nature, and pays homage to D’Arcy Thompson in the quotations from On Growth and Form that serve as epigraphs to the 12 chapters. In part, Stewart, a creative mathematician who is also an exceptionally lucid expositor, wants to publicise lines of research that he considers exciting. But his particular examples are there as evidence for the thesis that we can’t appreciate all the important features of life by focusing simply on the details of what happens in a particular embodiment of ‘life’, that based on DNA: ‘DNA is just the trick that earthly life uses to exploit the deep rules – it’s not the rules themselves.’ The ‘deep rules’ are what biologists ought to be trying to fathom.

Stewart’s account is wide-ranging, entertaining and informative. In early chapters, he considers how general studies of symmetry-breaking (as when you pile weights on a vertical spring, which eventually buckles on one side or the other) can illuminate our understanding of biological processes, and explores the idea that shapes – of cells, tissues or even whole organisms – can sometimes be understood as consequences of minimisation principles (in the simplest case, already studied by Thompson, the emergent structures are those that require minimum energy). He goes on to explain recent ventures in artificial life, particularly attempts to simulate organic evolution on a computer: much of the excitement here comes from discovering that features of living things that we might have seen as improbable accidents – social behaviour or mass extinctions, for example – turn out to be almost inevitable. Later chapters explore the mathematics behind sexual selection (focusing on the peahen’s preferences for gaudy, but clumsy tails), the patterns of mammalian gaits, the generation of spiders’ webs from simple rules, and the possibility of modelling ecological competition.

At the centre of Life’s Other Secret are two chapters that connect most directly with the Thompson tradition and are most persistent in Stewart’s quarrel with the hegemony of molecular biology. Both are concerned with development and the patterns that emerge: ‘Flowers for Fibonacci’ focuses on pattern formation in plants, ‘Morphogens and Mona Lisas’ on the generation of spots, stripes and other markings, particularly in sea-shells and mammals.

One of the major achievements of the study of plant growth is the use of abstract formal systems that show the relationship between large-scale structure and the form of smaller parts: the ‘uncanny resemblance’ between the overall shape of a tree and the pattern of one of its twigs. The mathematical biologist Astrid Lindenmayer (whose work has been continued by the computer scientist Przemyslaw Prusinkiewicz) introduced an approach that seems initially more suited to the study of grammar than to explaining the living world. Lindenmayer systems (L-systems) are defined by an alphabet of symbols, an initial string of symbols from that alphabet and a collection of production rules, which tell you how to replace individual symbols with other symbols or groups of symbols. You obtain a developmental sequence within an L-system by starting with its initial string and applying all the production rules you can, then you apply all the production rules you can to the result, and keep going in this way for as many steps as you choose. So, if the initial string is a and the rules tell you to replace a with ab and to replace b with ba, then a developmental sequence would lead you in one step to ab, in two steps to abba, in three steps to abbabaab and so on. L-systems can be much more complicated than this, involving more refined instructions, rules that depend on the size of the string, and rules that are probabilistic in character (so that what you get at each stage may depend on the outcome of a random process). As Stewart points out, the more complicated L-systems have been very successful in generating computer images of mature plants and of the stages through which they grow.

So what? How are these formal games relevant to serious biology? The point is not simply to produce pretty pictures as pedagogical devices. Lindenmayer and Prusinkiewicz have tried to give substance to the common metaphor that organisms undergo a ‘developmental programme’. We should think of the initial string as standing for some particular type of cell or cluster of cells. The production rules tell us that there are biological processes that will replace particular kinds of cells, or clusters of cells, with other cells and clusters – thus, one of the symbols might stand for a node and a rule might tell us that, whenever there’s a node, it’s replaced, at the next developmental stage, by a branch. L-systems abstract completely from the details of how these processes work, revealing how the recursive sequence of processes generates the patterns found in algae, ferns, leaves, flowers and trees.

In turning from plants to the stripes of sea-shells and tigers, we descend a bit from the heights of abstraction. Just before his death, Alan Turing proposed that patterns in groups of cells could be understood as the effects of ‘diffusion-driven instability’. His idea was that local production of an ‘activator’ molecule might foster local increases in the concentration of that molecule, even though an ‘inhibitor’ molecule would also be produced and would prevent the broad diffusion of the ‘activator’. As Turing showed, the dynamics of the system are governed by (partial differential) equations, and, by looking at the solutions as the parameters in these equations take on different values, it’s possible to see quite different patterns. Following his lead, Hans Meinhardt has shown how ‘reaction diffusion’ equations can generate a wide variety of stripes, bands, whorls and ridges in seashells. Meinhardt’s analyses, displayed in some beautiful simulations, do not identify the particular molecules that play a role in forming the stripes. Instead, they tell us that if a growing shell has cells containing molecules which conform to a particular system of equations, then it will show the pigmentation pattern distinctive of a particular species.

Another mathematical biologist, James Murray, has also drawn inspiration from Turing. Taking a set of equations whose solutions depend on the size of the cluster of cells in which the diffusion reaction occurs, he has shown how to generate the kinds of pattern found in mammals – in tigers and leopards, giraffes and cheetahs. In particular, Murray has demonstrated a remarkable theorem. The same equations that yield stripes when the domain of the diffusion is smaller give rise to spots when the size of that domain increases sufficiently. So if the diffusion process goes on in different areas of a mammal’s coat, some larger and some smaller, it can generate spots in the bigger areas and stripes in the smaller ones. In this way, Murray can account for a curious biological regularity: the fact that there are spotted mammals with stripes on their tails but no striped mammals with spots on their tails.

Stewart reviews these examples, while acknowledging that there are currently limits to our ability to apply Turing’s idea. The successes, however, show clearly what is wrong with insisting that all biological analysis should begin with molecules and work up. What we sometimes want to understand about the organic world is the software of life. Instead of grubbing in the molecular details, we hope for an account that reveals the structure of processes common to different mammals, or even to mammals and sea-shells. Exhaustive biochemical analyses that show one combination of molecules acting here, another there, don’t bring out the overarching ‘program’ that the organism is running. By contrast, when we see how quite different molecular processes are governed by the same equations, and how a whole family of molecular processes conform to related equations, we recognise important regularities in the organic world.

In fact, we need analyses at various levels. Lindenmayer’s highly abstract approach enables us to give different interpretations to the same production rules working in different plants. As we probe the mechanical details, we might discover another type of unity in the fact that some of the growth processes are governed by similar sets of equations. Finally, we might focus on a case of particular interest and identify the molecular bases of those processes. Each of the three levels would offer different insights into the plants studied; none could be discarded without loss.

Molecular biologists can (and do) complain that biologists who play with mathematical models easily lose touch with biological reality. Given sufficient ingenuity, the computer programmer can produce a picture that looks remarkably like an actual shell or an actual sunflower, but this is only clever make-believe until someone has shown that the growth of the shell or the flower in question involves chemical reactions that conform to the hypothetical equations or rules.

The molecular biologists have a point, but one that is easily overstated. The fruitfulness of the relationship between modellers and experimentalists has been demonstrated again and again in the history of science. Take the discovery of Neptune. Recognising anomalies in the predicted orbit of Uranus, astronomers committed to Newtonian gravitational theory speculated that the perturbations might be produced by a hitherto unknown planet, and their theory-based calculations told them where to point their telescopes. In much the same way, mathematical models of developmental processes can guide molecular biologists as they deploy powerful tools in the search for particular kinds of molecules in particular cells. By the same token, the enterprise of model-building ought to be informed by the details that molecular studies disclose. The right response is a division of labour between mathematical and molecular studies and a pooling of insights that transforms both.

In his choice of title, as well as in some of his formulations, Stewart misleadingly suggests either that the mathematical biology that fascinates him is the real key to biological understanding (the source of the deep rules) or that its insights are independent of the fashionable focus on molecules. His central chapters, however, reveal the need for studies at various stages of abstraction. Although he never quite makes the point, he comes close to it, as when he concludes his discussion of development: ‘DNA alone does not control development – nor do dynamics alone. Development requires both, interacting with each other, like a landscape that changes shape according to the traffic that passes through it,’ a metaphor which doesn’t altogether succeed in identifying the ideal structure of a future developmental biology. Despite its (minor) failings this is an admirable book and my only real complaint is that, fearful perhaps of overtaxing its readers, it forfeits the opportunity to show the real power of some of its mathematical approaches.