Did Turing really "explain" pattern formation?

 Exactly seventy years ago, Alan Turing published a seminal article, in which he proposed a simple reaction-diffusion model for pattern formation in biological systems. The basic idea is that there are two molecules (morphogens) that react with one another chemically and also diffuse through the system.

The potential relevance of the model can be seen by comparing the lower panels below. The left panel is a real fish and the right panel shows the results of a simulation. The figure above is taken from a beautiful review article published a decade ago.

Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation  Shigeru Kondo and Takashi Miura

The authors state that the model is not accepted by many experimental biologists and hope their review will lead to a greater engagement with it. Some of the reasons are related to issues in the philosophy of science and how to model complex systems. What is an explanation? What is the role of simple models for complex systems that ignore so many details?

Kondo and Miura point out the universality of the reaction-diffusion model in the sense that a similar model can be derived where the "molecules" are instead a circuit of cellular signals. Diffusion can be replaced by a relay of signals between cells. Alfred Gierer and  Hans Meinhardt in 1972 showed that all that is required is a network with "a short-range positive feedback [competing] with a long-range negative feedback."

A short video from the Sante Fe Institute also provides a helpful introduction including some simulations.

 

There is another problem with Turing's model that is succinctly described in the opening paragraph a recent PRL. In a system with two molecular species, patterns only form when there is a large difference between the diffusivity of the two molecules. However, this seems unrealistic because one expects the molecules to have comparable diffusivities.

Turing’s Diffusive Threshold in Random Reaction-Diffusion Systems 

Pierre A. Haas and Raymond E. Goldstein 

 In 1952, Turing described the pattern-forming instability that now bears his name [1]: diffusion can destabilize a fixed point of a system of reactions that is stable in well-mixed conditions. Nigh on threescore and ten years on, the contribution of Turing’s mechanism to chemical and biological morphogenesis remains debated, not least because of the diffusive threshold inherent in the mechanism: chemical species in reaction systems are expected to have roughly equal diffusivities, yet Turing instabilities cannot arise at equal diffusivities [2,3]. It remains an open problem to determine the diffusivity difference required for generic systems to undergo this instability, yet this diffusive threshold has been recognized at least since reduced models of the Belousov–Zhabotinsky reaction [4,5] only produced Turing patterns at unphysically large diffusivity differences.

I first became aware of this paper through a commentary by Changbong Hyeon, at the Journal Club for Condensed Matter. It is also helpful because it explains the simple mathematics behind the threshold value of the model parameters for pattern formation.