Introduction to Turing Systems

A question presented to Turing: "Do you really think that Turing systems can explain the stripes of the zebra?"
Answer: "Stripes are easy, it is the horse part that worries me."
- Alan Turing


Natural systems exhibit an amazing diversity of structures in both living and nonliving systems. Trees and plants growing from a single seed can show extremely complex organization not to mention mammals, whose development begins from a single fertilized egg cell. Mathematical biologists devise and study models that at least qualitatively capture some essential characteristics of the natural mechanisms of growth. In these models the primary interest is not in genes, but in the processes that follow the activation of a gene. The information stored in DNA corresponds to a blueprint, which is put into action by spontaneous physico-chemical processes. DNA is known to have three billion (3,000,000,000) base pairs, whereas an adult human being consists of tens of trillions of cells (~10,000,000,000,000). Thus it cannot contain enough information for determining everything in detail, but it is responsible for the major guidelines for the development of an organism. What could the spontaneous processes generating the structure by following these guidelines be?

More than half a century ago a British mathematician Alan M. Turing addressed the problem. He assumed that genes (or proteins and enzymes) act only as catalysts for spontaneous chemical reactions, which regulate the production of other catalysts or morphogens. Finally, cells differentiate according to the morphogen concentration in their surroundings. There was not any new physics involved in Turing’s idea, but he merely suggested that the fundamental physical laws can account for complex physico-chemical behavior. In 1952 Turing published his seminal article titled The Chemical Basis of Morphogenesis, where he showed that a simple mathematical model describing spontaneously spreading and reacting chemicals could give rise to stationary spatial concentration patterns of fixed characteristic length from a random initial configuration and proposed that reaction-diffusion models might have relevance in describing morphogenesis, the growth of biological form.

Figure: Complex network structures that evolved from a random state in numerical simulations of a Turing system (Gray-Scott model) in two (left) and three (right) dimensions.

From mathematical point of view Turing systems are coupled partial differential equations, which describe the reaction and diffusion behavior of chemicals. Under particular conditions, Turing systems are capable of generating stationary chemical patterns of finite characteristic wave lengths even if the system starts from an arbitrary initial configuration. The characteristics of the resulting dissipative patterns are determined intrinsically by the reaction and diffusion rates of the chemicals, not by external constraints. Turing patterns grow due to diffusion-driven instability as a result of infinitesimal perturbations around the stationary state of the model and exist only under non-equilibrium conditions. Turing systems have been studied using chemical experiments, mathematical tools and numerical simulations.

My work on Turing models has focused on a model called the Barrio-Varea-Aragon-Maini (BVAM), which I have studied by employing both analytical and numerical methods. In addition to the pattern formation in two-dimensional domains, I have also extensively studied the formation of three-dimensional structures. The model can be studied using the standard linear stability analysis, which reveals the parameter sets corresponding to a Turing instability and the resulting unstable wave modes. Then nonlinear bifurcation analysis, on the other hand, reveals the stability of morphologies induced by two-dimensional hexagonal symmetry and various three-dimensional symmetries (SC, BCC, FCC). My main results presented included the study of the Turing pattern selection in the presence of bistability, and the study of the structure selection in three-dimensional Turing systems depending on the initial configuration. Also, my work on the effect of numerous constraints, such as random noise, changes in the system parameters, thickening domain and multistability on Turing pattern formation brings new insight concerning the state selection problem of non-equilibrium physics.