PHYSICAL REVIEW X 8, 021071 (2018) Key Features of Turing Systems are Determined Purely by Network Topology Xavier Diego,1,2,3,* Luciano Marcon,4,5 Patrick Müller,4 and James Sharpe1,2,6,3 1Center for Genomic Regulation, Barcelona Institute for Science and Technology, 08003 Barcelona, Spain 2Universitat Pompeu Fabra, 08003 Barcelona, Spain 3European Molecular Biology Laboratory, Barcelona Outstation, 08003 Barcelona, Spain 4Friedrich Miescher Laboratory of the Max Planck Society, 72076 Tübingen, Germany 5Centro Andaluz de Biología del Desarrollo, Consejo Superior de Investigaciones Científicas, Universidad Pablo de Olavide, 41013 Seville, Spain 6Institucio Catalana de Recerca i Estudis Avancats, 08010 Barcelona, Spain (Received 10 October 2017; revised manuscript received 23 March 2018; published 20 June 2018) Turing’s theory of pattern formation is a universal model for self-organization, applicable to many systems in physics, chemistry, and biology. Essential properties of a Turing system, such as the conditions for the existence of patterns and the mechanisms of pattern selection, are well understood in small networks. However, a general set of rules explaining how network topology determines fundamental system properties and constraints has not been found. Here we provide a first general theory of Turing network topology, which proves why three key features of a Turing system are directly determined by the topology: the type of restrictions that apply to the diffusion rates, the robustness of the system, and the phase relations of the molecular species. DOI: 10.1103/PhysRevX.8.021071 Subject Areas: Biological Physics Complex Systems, Nonlinear Dynamics I. INTRODUCTION smoothing spatial heterogeneities and tends to generate uniform distributions. In 1952, Alan Turing proposed a theory of bio- Doubts about the relevance of Turing’stheorywere logical morphogenesis to explain the emergence of spatial compounded by the lack of conclusive evidence of their organization in the embryo [1]. The influence of this ground-breaking theory has been phenomenal, as it has existence in real-world examples for almost 40 years after he subsequently been invoked to explain patterning processes proposed it. Analysis of simple Turing systems suggested in chemical reactions [2], nonlinear optical systems [3], two main reasons why they may be hard to find in nature or to semiconductor nanostructures [4], galaxies [5], predator- create artificially. Firstly, in its original form, a Turing system prey models in ecology [6], vegetation patterns [7], cardiac requires two species that diffuse at significantly different arrhythmias [8], and even crime spots in cities [9]. Yet, rates [10], whereas, in reality, many species that might especially in the field for which Turing developed his ideas conceivably form Turing patterns have very similar diffusion —developmental biology—it still suffers from concerns rates. Secondly, there appeared to be only extremely about its validity, which arose early on in its history. narrow ranges of the reaction parameters that would be Turing’s original model consisted of two molecular compatible with a Turing pattern. Achieving a pattern would, species that would diffuse through the embryonic tissue therefore, require setting these parameters with unrealistic and chemically react with each other. He proved that, under precision [11]. This property seemed to make Turing patterns certain conditions, such a simple reaction-diffusion system an unlikely phenomenon and intrinsically unrobust. should be able to spontaneously create periodic patterns in A partial solution to these severe limitations has been space. The idea that a self-organized symmetry breaking of found experimentally. The first experimental confirmation molecular concentrations would be driven by diffusion was of the existence of Turing patterns, found in the chlorite– counterintuitive, as diffusion normally has the effect of iodide–malonic acid (CIMA) chemical reaction [12,13], suggested a method to circumvent the diffusion constraints. *[email protected] In the CIMA reaction, a color indicator was included to act as a visual read-out of the chemical patterns produced. This Published by the American Physical Society under the terms of indicator bound reversibly to one of the reactants and the Creative Commons Attribution 4.0 International license. effectively slowed down its diffusion [2,14]. A refinement Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, of this method [15,16] has been followed in the design of and DOI. almost all new chemical systems producing Turing patterns 2160-3308=18=8(2)=021071(23) 021071-1 Published by the American Physical Society DIEGO, MARCON, MÜLLER, and SHARPE PHYS. REV. X 8, 021071 (2018) [17] and inspired most of the theoretical efforts to relax the role of the immobile color indicator [20–23]. This design, diffusion constraints [18,19]. shown schematically in Fig. 1, is only a partial solution Recent years have seen a shift in interest from chemical because the relaxation of the diffusion constraints is not systems back to the biological problem for which Turing complete and because it requires a very specific topological conceived the idea—embryo development. In this context, arrangement. Regarding the allegedly intrinsic lack of models of biological pattern formation that can bypass the robustness of Turing systems, early studies observed that diffusion constraints have also been proposed. Again, these the volume of the parameter space varies greatly between models follow the design of the CIMA reaction, including different models [24,25], but little is known about what immobile cell receptors as part of the network to play the determines this property. However, a recent computational study showed that allowing more freedom to the interactions of the immobile species with the rest of the network could result in greater relaxation of the diffusion constraints [26]. This study showed that, in networks of three and four species, with only two diffusible species, it is possible to remove any constraints to their diffusion rates, a novel class of Turing networks that had not been found before. Further, the computational exploration showed that the percentage of networks with relaxed constraints was unexpectedly high. This finding requires a mechanistic explanation and sug- gests that central aspects of Turing networks have to be clarified. Hence, we decided to investigate the relationship between the topology of a Turing network and its behavior using tools from graph theory. This proved to be fruitful, because we find that topology determines three fundamen- tal properties of a Turing system illustrated in Fig. 1. Firstly, we demonstrate how the topology determines the diffusion constraints and the general method to relax or even remove them completely. Secondly, we are able show that Turing systems are not intrinsically unrobust because they do not require unphysical adjustment of parameters. FIG. 1. Each panel illustrates a question about Turing systems Thirdly, our analysis allows us to resolve a question that, that is resolved through analyzing the topology of the underlying surprisingly, has not been addressed before: What deter- network of interacting species. Namely, in a Turing system, how mines the spatial overlap of the species in a Turing pattern? does the topology determine (a) the constraints on diffusion rates, Again, we show that topology determines the spatial phases (b) the size of the patterning parameter region, and (c) the phase of the species and, further, that it is possible to construct a of each species? More specifically, each panel contains three network with any desired pair of species overlapping in explicit networks with large differences in these properties. In all space. panels, edges ended with an arrow and a bar represent activation In addition, our theory can be extended to oscillatory and inhibition, respectively. A wriggled arrow indicates that the patterns and explains a new class of pseudopatterning node corresponds to a diffusible species. The two-species system networks that we call “Turing filters.” The patterns gen- illustrated in the left of (a) requires highly different diffusion rates of u and v for a pattern to exist, whereas the CIMA reaction erated by Turing filters do not have a characteristic wave- schematic in the center can produce patterns with iodine (I−) and length; instead, these networks amplify preexisting spatial − chlorite (ClO2 ) diffusing at the same rate. Finally, the network on heterogeneities if their characteristic wavelength is smaller the right produces patterns with no restriction on the diffusion than a critical threshold, thus acting like a low-pass filter. rates. In (b), we see three three-species models whose Turing Finally, we apply our theory to analyze experimental patterning parameter space are vastly different. The robustness of Turing systems, relevant models of biological pattern a patterning system can be assessed by the volume of the formation, and network designs proposed to genetically parameter sets compatible with reaction stability (shown in engineer Turing systems in cells. yellow) and diffusion-driven instabilities (shown in orange). Overall, these results show that our graph-based theory For the three examples, the ratios of diffusion rates are the same, and the axes show the strength of the feedbacks between species. of Turing networks provides a