Abstract
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.
- Received 10 October 2017
- Revised 23 March 2018
DOI:https://doi.org/10.1103/PhysRevX.8.021071
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
In 1952, Alan Turing formulated a groundbreaking theory that explained how spatial organization emerged in embryonic tissues. This theory has since been proposed to explain the emergence of spatial organization in systems as diverse as chemical reactions, cardiac arrhythmias, semiconductors, and even urban criminal activity. However, the original theory imposes requirements that are rarely met in reality, casting its relevance into doubt. We present a new theory that explains how the limitations in Turing’s theory arise and how to eliminate them altogether.
In its original form, a Turing system consists of two molecular species that must diffuse at very different rates to produce a pattern. Such a difference is typically not found in real systems. Also, most Turing models require a level of fine-tuning that prevents them from being a robust mechanism for any real patterning process. Researchers have found partial solutions to these limitations but lack a definitive understanding of their source.
Our theory demonstrates how the topology of a Turing system determines these limitations. Further, we prove that an appropriate topological arrangement of the feedback between the system’s components can eliminate these limitations altogether. We also discover the rules that determine the spatial overlap of the different Turing species (e.g., the different molecules) in the pattern.
These findings should dispel objections made against the role of Turing patterns in biological development. Our results should help to design synthetic Turing systems and to engineer tissues with specific genes coexpressed.
