Alan Turing was neither a biologist nor a chemist, and yet the paper he published in 1952, ‘The chemical basis of morphogenesis’, on the spontaneous formation of patterns in systems undergoing reaction and diffusion of their ingredients has had a substantial impact on both fields, as well as in other areas as disparate as geomorphology and criminology. Motivated by the question of how a spherical embryo becomes a decidedly non-spherical organism such as a human being, Turing devised a mathematical model that explained how random fluctuations can drive the emergence of pattern and structure from initial uniformity. The spontaneous appearance of pattern and form in a system far away from its equilibrium state occurs in many types of natural process, and in some artificial ones too. It is often driven by very general mechanisms, of which Turing's model supplies one of the most versatile. For that reason, these patterns show striking similarities in systems that seem superficially to share nothing in common, such as the stripes of sand ripples and of pigmentation on a zebra skin. New examples of ‘Turing patterns' in biology and beyond are still being discovered today. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.

Abstract The sudden death of Alan Turing on 7 June 1954 deprived mathematics and science of a great original mind at the height of its power. After some years of scientific indecision, since the end of the war, Turing had found, in his chemical theory of growth and form, a theme that gave the fullest scope for his rare combination of abilities, as a mathematical analyst with a flair for machine computing, and a natural philosopher full of bold original ideas. The preliminary report of 1952, and the account that will appear posthumously, describe only his first rough sketch of this theory, and the unfulfilled design must remain a painful reminder of the loss that his early death has caused to science. Alan Mathison Turing was born in London on 23 June 1912, the son of Julius Mathison Turing, of the Indian Civil Service, and of Ethel Sara Turing (néeStoney). The name ‘Turing’ is of Scottish, perhaps ultimately of Norman origin, the final g being an addition made by Sir William Turing, of Aberdeenshire, in the reign of James VI and I. The Stoneys, an English-Irish family of Yorkshire origin, produced some distinguished physicists and engineers in the nineteenth century, three of whom became Fellows of the Society; and Edith A. Stoney was one of the early women equal-to-wranglers at Cambridge (bracketed with 17th Wrangler, 1893). Alan Turing’s interest in science began early and never wavered. Both at his preparatory schools and later at Sherborne, which he entered in 1926, the contrast between his absorbed interest in science and mathematics, and his indifference to Latin and ‘English subjects’ perplexed and distressed his teachers, bent on giving him a well-balanced education.

When patterns are set during embryogenesis, it is expected that they are straightly established rather than subsequently modified. The patterning of the three mouse molars is, however, far from straight, likely as a result of mouse evolutionary history. The first-formed tooth signaling centers, called MS and R2, disappear before driving tooth formation and are thought to be vestiges of the premolars found in mouse ancestors. Moreover, the mature signaling center of the first molar (M1) is formed from the fusion of two signaling centers (R2 and early M1). Here, we report that broad activation of Edar expression precedes its spatial restriction to tooth signaling centers. This reveals a hidden two-step patterning process for tooth signaling centers, which was modeled with a single activator-inhibitor pair subject to reaction-diffusion (RD). The study of Edar expression also unveiled successive phases of signaling center formation, erasing, recovering, and fusion. Our model, in which R2 signaling center is not intrinsically defective but erased by the broad activation preceding M1 signaling center formation, predicted the surprising rescue of R2 in Edar mutant mice, where activation is reduced. The importance of this R2-M1 interaction was confirmed by ex vivo cultures showing that R2 is capable of forming a tooth. Finally, by introducing chemotaxis as a secondary process to RD, we recapitulated in silico different conditions in which R2 and M1 centers fuse or not. In conclusion, pattern formation in the mouse molar field relies on basic mechanisms whose dynamics produce embryonic patterns that are plastic objects rather than fixed end points.

Turing patterns have morphed from mathematical curiosities into highly desirable targets for synthetic biology. For a long time, their biological significance was sometimes disputed but there is now ample evidence for their involvement in processes ranging from skin pigmentation to digit and limb formation. While their role in developmental biology is now firmly established, their synthetic design has so far proved challenging. Here, we review recent large-scale mathematical analyses that have attempted to narrow down potential design principles. We consider different aspects of robustness of these models and outline why this perspective will be helpful in the search for synthetic Turing-patterning systems. We conclude by considering robustness in the context of developmental modelling more generally. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

Local activation and long-range inhibition are mechanisms conserved in self-organizing systems leading to biological patterns. A number of them involve the production by the developing cell of an inhibitory morphogen, but how this cell becomes immune to self-inhibition is rather unknown. Under combined nitrogen starvation, the multicellular cyanobacterium <i>Nostoc</i> PCC 7120 develops nitrogen-fixing heterocysts with a pattern of one heterocyst every 10-12 vegetative cells. Cell differentiation is regulated by HetR which activates the synthesis of its own inhibitory morphogens, diffusion of which establishes the differentiation pattern. Here, we show that HetR interacts with HetL at the same interface as PatS, and that this interaction is necessary to suppress inhibition and to differentiate heterocysts. <i>hetL</i> expression is induced under nitrogen-starvation and is activated by HetR, suggesting that HetL provides immunity to the heterocyst. This protective mechanism might be conserved in other differentiating cyanobacteria as HetL homologues are spread across the phylum.

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.

The debate about what causes the generation of form and structure in embryological development goes back to antiquity. Most recently, it has focused on the divergent views as to whether the generation of patterns and form in development is a largely self-organized process or is mainly determined by the genome, in particular, complex developmental gene regulatory processes. This paper presents and analyzes pertinent models of pattern formation and form generation in a developing organism in the past and the present, with a special emphasis on Alan Turing's 1952 reaction-diffusion model. I first draw attention to the fact that Turing's paper remained, at first, without a noticeable impact on the community of biologists because purely physical-chemical models were unable to explain embryological development and often also simple repetitive patterns. I then show that from the year 2000 and onwards, Turing's 1952 paper was increasingly cited also by biologists. The model was updated to include gene products and now seemed able to account for the generation of biological patterns, though discrepancies between models and biological reality remained. I then point out Eric Davidson's successful theory of early embryogenesis based on gene-regulatory network analysis and its mathematical modeling that not only was able to provide a mechanistic and causal explanation for gene regulatory events controlling developmental cell fate specification but, unlike reaction-diffusion models, also addressed the effects of evolution and organisms' longstanding developmental and species stability. The paper concludes with an outlook on further developments of the gene regulatory network model.

In a 1952 paper, Alan Turing showed that spatially distributed chemical reactions evolving in time by local kinetic rate laws and in space by unbiased molecular diffusion can develop a stable, time-independent, spatial pattern from a spatially homogeneous, steady state solution subjected to small, spatially periodic perturbations over a critical range of wavelengths. He proposed this mechanism as a potential chemical basis for biological morphogenesis. Although his proposal was initially ignored and remains controversial, Turing's idea still plays a major role in any discussion of spontaneous pattern formation in biological and chemical systems. Nevertheless, it is safe to say that few people have carefully studied his 1952 paper, which is notoriously difficult to read. For this reason, I am 'revisiting' Turing's paper to help new investigators to understand and appreciate his remarkable contribution to mathematical biology. Along the way, we shall resolve several 'peculiarities' of Turing's reaction mechanisms and numerical simulations, and place his work in context with later textbook examples. The relation of stationary Turing patterns to time-dependent traveling waves of chemical activity is also described.

Despite having advanced a reaction-diffusion model of ODE's in his 1952 paper on morphogenesis, reflecting his interest in mathematical biology, Alan Turing has never been considered to have approached a definition of Cellular Automata. However, his treatment of morphogenesis, and in particular a difficulty he identified relating to the uneven distribution of certain forms as a result of symmetry breaking, are key to connecting his theory of universal computation with his theory of biological pattern formation. Making such a connection would not overcome the particular difficulty that Turing was concerned about, which has in any case been resolved in biology. But instead the approach developed here captures Turing's initial concern and provides a low-level solution to a more general question by way of the concept of algorithmic probability, thus bridging two of his most important contributions to science: Turing pattern formation and universal computation. I will provide experimental results of one-dimensional patterns using this approach, with no loss of generality to a n-dimensional pattern generalisation.

Possible mechanisms of pattern formation in biological systems include the classical reaction–diffusion model proposed by Alan Turing and the more-recently-found The science of pattern formation deals with the visible, (statistically) orderly outcomes of self-organization and the common principles behind similar patterns in nature. In developmental biology, pattern formation refers to the generation of complex organizations of cell fates in space and time. The role of genes in pattern formation is an aspect of morphogenesis, the creation of diverse anatomi Biological patterns such as animal markings, the segmentation of animals, and phyllotaxis are formed in different ways. Phyllotaxis is the regular arrangement of leaves, flowers or seeds around a plant stem often forming spiral patterns found in many plants such as sunflowers, pinecones etc. In developmental biology, pattern formation describes the mechanism by which initially equivalent cells in a developing tissue in an embryo assume complex forms and functions. Embryogenesis, such as of the fruit fly Drosophila, involves coordinated control of cell fates. Pattern formation is genetically controlled, and often involves each cell in a field sensing and responding to its position along a morphogen gradient, followed by short-distance cell-to-cell communication through cell-signaling pathways to refine the initial pattern. In this context, a field of cells is the group of cells whose fates are affected by responding to the same set of positional information cues. This conceptual model was first described as the French flag model in the 1960s. More generally, the morphology of organisms is patterned by the mechanisms of evolutionary developmental biology, such as changing the timing and positioning of specific developmental events in the embryo. Possible mechanisms of pattern formation in biological systems include the classical reaction–diffusion model proposed by Alan Turing and the more-recently-found elastic instability mechanism which is thought to be responsible for the fold patterns on the cerebral cortex of higher animals, among other things. Cellular automata and neural network models have both been proposed to explain the patterns on some marine mollusc shells such as those of the genus Conus.