Can a reptile compute? In one species of lizard, Timon lepidus, the colour and pattern of its scales evolve in a manner akin to a discrete rule-based computation called a cellular automaton. See Letter p.173
In 1902, Rudyard Kipling wrote the Just So Stories, which provided fanciful accounts of how, for example, the leopard got its spots. More than 80 years later, the mathematician James D. Murray suggested a mathematical mechanism that could explain this spotted pattern formation1. On page 173, Manukyan et al.2 tell an even more intriguing tale. The authors describe a strikingly beautiful biological pattern-forming system that spans the development of the ocellated lizard (Timon lepidus), which changes from a drab brown youngster with white polka dots (ocelli) to an adult whose skin is a rich black and green tapestry (Fig. 1). The authors call this patterning system a living cellular automaton.
Originally conceived in the 1940s, a cellular automaton3,4 is a system of spatially discrete but interconnected units that switch between different states depending on their own state and the states of their neighbours. Cellular automata have been used to probe theoretical concepts in computer science (such as a universal Turing machine5), study complex patterns in nature6, produce startling moving patterns based on simple rules (the 'Game of Life'7) and model biological systems8 and a panoply of discrete systems that are too numerous to list.
The quest to understand pattern-forming biological systems has spanned more than half a century. It originated with the seminal work of Alan Turing, who published a much-cited paper9 on the chemical basis of morphogenesis — the developmental emergence of shape and form in living organisms. Similar ideas were proposed two decades later10, and have since been popularized in many papers and books. These theories established that interacting and diffusing chemicals can create spontaneous patterns of concentration: mountains and valleys in a chemical landscape. Such an uneven chemical distribution can direct morphogenesis. This concept has led to new scientific fields in mathematics, biology, chemistry and physics.
How patterns form in living systems remains hotly debated and much discussed. The chemical signalling that underlies morphogenesis is fully understood in just a few systems. Moreover, tracking the fates of individual microscopic cells in a living tissue is a difficult problem that has been addressed only in the past two decades, albeit in various systems, including the transparent zebrafish embryo11,12.
Enter the team of Manukyan and colleagues. The authors present a case study in which a living pattern can be observed in detail as it unfolds. The skin pattern of T. lepidus is mesoscopic (a length scale between microscopic and macroscopic) and involves microscopic interactions of single pigment cells. These cells collectively give rise to the colour of the skin scales, and therefore the speckled pattern of the adult lizard's skin.
The authors devised a remarkable way to track the skin pattern of an individual lizard over 3–4 years of its growth, matching skin-scale for skin-scale as the lizard's length increased fourfold. They used 3D scans of the lizard's skin and corrected for the curvature of its body and irregular surface texture to identify the centres of its scales. They then used a slight but sophisticated adjustment to map these points to the centres of hexagons; the resulting hexagonal array becomes a flattened 'tiling pattern', in which each tile represents one scale. The authors were able to track the skin scales because their number and relative position are maintained throughout the reptile's growth. The level of detail achieved by Manukyan et al. makes this study innovative in terms of providing empirical data with which to drive a theory of pattern formation.
At the macroscopic scale, pattern formation is usually described as a smooth process that is continuous in space and time, and it is modelled by a set of reaction–diffusion equations — mathematical equations that describe how chemicals redistribute over space and time. The authors follow this convention, but with a key difference: they show that the boundaries between T. lepidus scales constrict during morphogenesis, and argue that this creates partial barriers to the diffusion of cells and chemicals between adjoining scales. As a result, the scales form discrete spatial units that each take on a uniform colour (black or green on the lizard's back) in a way that depends on the states of their neighbours. Formally, then, the biological pattern-forming system resembles the output of a cellular automaton. The scale pattern evolves by obeying a set of rules that transform one configuration of scale colours into another, with certain probabilities. It is in this sense that the authors describe T. lepidus as a 'living' cellular automaton.
The authors determined empirically the probabilities of scales changing colour for distinct colour configurations of scales and their nearest neighbours. They then linked the reaction–diffusion and cellular-automaton approaches in a theoretical model. They found good agreement between the patterns that evolve over years on their reptilian subjects and the solutions to their model based on neighbourhood-dependent rules obtained from empirical data. This agreement is surprising and adds to the novelty of the authors' approach.
What underlying mechanism drives this pattern formation? According to the authors, a system consisting of pigment cells (melanophores and xanthophores) interacting with a long-range, rapidly diffusing chemical suffices to explain the pattern formation — taking into account the partial diffusion barriers between adjoining scales that form during morphogenesis. The authors' model, which is modified from a pre-existing zebrafish pigmentation model13, reproduces the black and green labyrinthine pattern of the adult lizard's skin. Future work in which the chemicals and cellular interactions are identified in more detail (possibly in related but simpler in vitro experimental systems, such as cells and chemicals interacting in a tissue culture) would provide an opportunity for manipulating the patterns experimentally, and therefore allow us to learn more about the underlying cellular and molecular pattern-forming mechanisms.
As the authors conclude, a cellular automaton is not just an abstract concept, but corresponds to a process generated by biological evolution. Nearly 80 years after its conception, the cellular automaton has come of age — it has matured from an abstract concept in the 1940s to in silico realizations since the 1960s and, finally, to a pattern-forming mechanism that has biological relevance and can be observed in reptilio.