Computational biology: Turing’s lessons in simplicity

Abstract

Biophysical modeling of development started with Alan Turing. His two-morphogen reaction-diffusion model was a radical but powerful simplification. Despite its apparent limitations, the model captured real developmental processes that only recently have been validated at the molecular level in many systems. The precision and robustness of reaction-diffusion patterning, despite boundary condition-dependence, remain active areas of investigation in developmental biology.

Significance

Computer models of development have played a big part in the progress of developmental biology. They owe much to Alan Turing and his 1952 work, which directly and explicitly addressed the self-organization of embryogenesis. His simple and ingenious model remains a powerful tool in understanding the molecular networks involved.

Main text

2021 saw the introduction of a new design of the UK’s £50 banknote; the famous person depicted was Alan Turing. Many people know that Alan Turing was a mathematician who helped crack the Nazi Enigma code in World War Two, saving lives and perhaps shortening the war itself. Fewer know that before the war, he provided the mathematical theory behind machine computation—modern digital computers—and after the war pioneered the practical design and programming of computers and founded the field of artificial intelligence. If anyone can be called a genius, then surely Alan Turing must be one. 2022, however, sees the 70th anniversary of what Turing himself considered to be one of his greatest achievements since his mathematical breakthroughs of the 1930s. In 1952, a work entitled “The Chemical Basis of Embryogenesis” was published in the Philosophical Transactions of the Royal Society (1). It is a remarkable work by any standards. It opens with a paragraph that displays Turing’s wonderful combination of bravado and wit (yes, he did have a strong sense of humor, unlike the caricature of him in the movie “The Imitation Game”); he begins by announcing that what follows is “a simplification and an idealisation and therefore a falsification.” It goes on to describe an idea so original that the work itself has only seven references. It was so ahead of its time that it was virtually forgotten for 20 years. For every biologist interested in spatial organization in biology at any scale and with even the slightest interest in computational biology—which should be all biologists—this work is essential reading, even if one skips the more mathematical parts.

So, what does “The Chemical Basis of Embryogenesis” actually describe? Firstly, it sets up the problem of how embryonic anatomy arises from seemingly uniform or nearly uniform eggs. In short, how does the “miracle” of developmental biology happen? It then answers this question by describing a mathematical model system of two diffusible chemicals, for which Turing coined the term “morphogens” (“shape makers”), that, through purely physicochemical interactions, can self-organize in space to patterns of stripes or spots of high concentrations of one or other of the morphogens. (For more details, see (2).) This reversal of the normal mixing and homogenization of chemicals through diffusion was achieved in the model by having one morphogen trigger both more production of itself and production of the second morphogen, which inhibits the first. By having the morphogens diffuse at different rates, the counterintuitive outcome is a spatial pattern of stripes or spots, with a spacing defined by activation, inhibition, and diffusion parameters. This provided a proof of principle that pure physics and chemistry could lead to self-organization independent of all but the most minimal of initiating cues. Implicit was the notion that this process could be used to divide an initially homogeneous embryo into two different cell types and that the process could occur iteratively to subdivide the embryo into finer and finer territories where different tissues could arise and make the diversity of structures and forms we see in adult animals. Turing had been recently thinking not only about computers and the mind but also the “hardware” of brains. His experience was that making real computers (as he had done—he was not just a theoretician) had informed the theory of computation. One might speculate that just as he had reduced computation to its simplest possible physical form (the so-called Turing Machine), his radical simplification of the brain problem led to an interest in embryology.

So far, so remarkable. Yet, there were problems. Although Turing patterns robustly generate patterns whose spacing (or wavelength) is tightly defined by the parameters, the orientation and even straightness of the stripes or the precise locations of the spots depend completely on the initiating minimal inhomogeneity and boundaries within which the pattern is formed. This led the leading embryologist of the era, C. H. Waddington, to be rather dismissive of Turing’s work as describing a mechanism good for “mere stipplings and dapplings.” In other words, it could apply to spots on a leopard or stripes on a tiger, which vary from individual to individual despite a consistent spacing, but was not going to explain normal development, which robustly puts a head at one end of an animal and a tail at the other (among many other highly precise localizations). As it happens, the slightly imprecise type of patterning is more common and important than perhaps Waddington appreciated at the time. Yes, dapplings and stipplings are present in skin from zebras to zebrafish, but actually there are many other instances in which the spacing—the periodicity or texture of a pattern—but not the precise localization of structures is important. These include the distribution of hairs and feathers, the spacing of branches in branching glands, and quite likely the distribution of nerves and blood vessels. Still, Waddington had a point; sometimes precise location does matter in development.

Turing’s work and his idea remained a neglected curiosity until in 1972, Gierer and Meinhardt rediscovered it (3) and were able to apply the (by now) much better computers to test more parameters and other variations on the Turing theme. During the 1970s and early 1980s, Meinhardt also made the mathematics much more elegant and provided a much better terminology and explanation for the workings of Turing patterning, for example, naming the morphogens “activator” and “inhibitor.” Meinhardt was able to provide highly plausible models of many patterning events, most of the dappling and stippling class but also some for seemingly more precision-sensitive processes such as the obviously periodic (regularly spaced) segmentation of the insect body. By the mid-1980s, molecular biology and genetics began to be applied to developmental biology, and so the models could begin to be tested. Strikingly, the experimental test of Meinhardt’s model for Drosophila (fruit fly) showed that rather than the segments being formed by a Turing-style mechanism, each segment was formed by a unique combination of overlapping gradients of molecules organized in the first place by the structure of the egg—the opposite of the self-organization process that Turing had described. A landmark commentary in Nature was memorably entitled “Making Stripes Inelegantly” (4). A beautiful theory had been destroyed by ugly facts.

It is important to recognize that models are made to be tested and that most good theoreticians are surprisingly comfortable with falsification of their models. In fact, in the mathematical and physical modeling world, there is a much-recited quotation that “all models are wrong, but some models are useful.” This maxim should not be misunderstood; the “wrongness” of models is that they simplify—the wrongness then, is a feature, not a bug, as it would be futile to have a model that was as complex as the thing it is trying to describe. (There’s another quotation in the modeling world that captures this: “The best model of a cat is a cat, preferably the same cat.”) In the case of Turing’s model, it had at least raised the possibility of modeling embryonic development and was certainly influential for Lewis Wolpert, the leading developmental theoretician after Turing, in formulating his ideas of positional information, the “French Flag Problem,” and positional value (see (2)).

Turing’s model was largely set aside for a second time during the 1990s and early 2000s, but in the mid-2000s, a number of studies were published that reintroduced it as the framework for understanding the actions of signaling molecules such as FGF, Wnt, and Hedgehog in regulating the patterns of hair follicles and feathers in mouse and chicken skin established experimentally. This provided convincing validation of the Turing mechanism, albeit for the stippling and dappling spotted class of patterns. However, in 2012, my group showed that the striped pattern of Shh gene expression that patterns the ridges known as rugae in the roof of the mammalian (mouse) mouth are controlled by a Turing-type mechanism (5). The pattern of rugae is an interesting halfway house between a stippling and dappling pattern and one that is fixed and highly reliable; rugae are largely straight, and there is a characteristic number of them (eight in the mouse, four in humans), but slight variations in both number and straightness do occur sometimes (without significant consequence—these ridges merely help feel and manipulate food). The really striking breakthroughs in appreciating the role of Turing patterning in situations in which precision and reproducibility are more crucial were studies that showed that it establishes left-right pattern in vertebrates (6) and—perhaps most remarkably—the number and spacing of digits of the hands and feet (7). Here is an instance in which the robustness and precision of a pattern is familiar to everyone, contradicting Waddington’s notion that Turing patterning is only good for textures. One could argue that having an extra or missing digit is not life-threatening, and indeed, polydactyly (having extra digit(s)) is not all that rare among birth defects (occurring at ∼1 in 1000 births). This is, perhaps, a case in which the pattern matters a bit less than, say, primary axis patterning, in which failure would be catastrophic for an embryo. But conversely, the very fact that digit patterning is as reliable as it is shows that somehow evolution has found ways of controlling the initiating and boundary conditions of this Turing system to minimize variability. (This type of precise control, which includes scaling the pattern to fit variable tissue size and correcting local ripples, remains an area of active research and of many unknowns.)

All of the above experimental validations of Turing were greatly helped by the existence of the model and the implementation of versions of it to extend the validation by challenging (perturbing) the biological system and seeing whether the model predicted the outcome accurately (e.g., (8); see also (9)). Often, the most interesting use of a model is when it predicts an outcome that is nearly, but not exactly, right; this shows that there is some “missing mass”—a novel factor or parameter that was not included in silico, but one that must therefore be sought and characterized experimentally. Having a model that predicts all experimental results completely is not as useful as a tool for discovery as one that requires adjustment and innovation.

Although simplicity is a great strength of Turing modeling, it is also a weakness. We now know that in real patterning in vivo, there are more (sometimes many more) than two morphogens and certainly more molecules that could serve some morphogen-like function. It is unclear, and in some ways arbitrary, which molecules we should consider as morphogens. For example, it is possible to treat a cell surface receptor as a Turing “morphogen” with zero diffusibility (7). Modeling the regulation of the vast complexity of omics data while keeping the models simple enough to provide explanatory and operational power is a major frontier in computational biology.

Reflecting on the 70th anniversary of “The Chemical Basis of Morphogenesis” highlights the exciting and at times delightful rubbing together of theory and experiment, the back and forth between computer model and laboratory measurement. As an experimental biologist, I have found that working with computational biologists forces me to think more clearly about what my assumptions, prejudices, and even basic ideas about the biology are and to realize that we are all making models in our heads whether we realize them explicitly or not. Computational modeling forces us to decide what we think.

Although it is perhaps ironic that Turing is appearing on banknotes at a time when cashless payments and even cryptocurrencies are becoming the norm, I, for one, am inspired to see him so publicly celebrated. Hopefully, knowing that he was a computational biologist will inspire a few to follow in his footsteps.

Editor: Stanislav Shvartsman.