A breakthrough in natural pattern formation

[Marcelo Malheiros]

Hello!

My name is Marcelo and I recently finished my PhD thesis. I'm eager to share my findings widely.

First, I would like to thank the Ready team for such a wonderful tool. It helped to ignite my initial interest in Reaction-Diffusion systems, to learn the algorithmic detail of the simulations and to make many experimentations. Thank you very much for all your effort!!

In summary, I have proposed and implemented a generic model that combines continuous mechanisms (like diffusion) with discrete events (like reaction or cell division). My aim was to explore possible pattern formation mechanisms using the smallest set of features, at the same time trying to maintain biological plausibility.

In this search I eventually discovered that just a minor modification to standard reaction-diffusion equations makes possible to reproduce many complex patterns, like the ones found on the skin of several species. Even more interestingly, those patterns are amenable to growth: that is, if the domains grows like living tissue, the overall pattern is maintained while being enlarged.

Of course the word "breakthrough" is quite strong, but the results are striking and many generated patterns have never been achieved before. Moreover, most realistic patterns are very easy to reproduce. I have attached a mosaic with a few comparison images from my work.

I have made my complete implementation available as open source, called Pattern Explorer. The source code is hosted at Github, and I also have built precompiled versions for Windows:

https://github.com/mgmalheiros/pattern-explorer

The best results are shown in my published paper and in the supplementary material, along with the exact experiment files that created them. The supplementary material also describes the simple syntax of the experiments (which are simple ASCII text files):

https://github.com/mgmalheiros/pattern-explorer/tree/master/paper

If anyone is interested, my PhD thesis "The mechanochemical basis of pattern formation" is also online:

http://hdl.handle.net/10183/169104

I will soon post some Ready files ready to play (pun intended :-). I'd love to have feedback and know about interesting patterns found by the community.

My best regards,

  Marcelo Malheiros

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TECHNICAL DETAILS

I use a slight variation of Alan Turing’s classic reaction-diffusion (RD) non-linear equations [1], as proposed by Turk [2]. Using the variable names from Ready we have:

da/dt = s (16 − a b) + D_a ∇²a

db/dt = s (a b − b − 12) + D_b ∇²b

Analysis of the problem of maintaining the pattern on a growing tissue signaled a missing behavior, which I extensively researched. As I have implemented a very general simulation model, I was able to evaluate many different hypotheses that combined cell division, reaction, diffusion, conditional chemical production and consumption.

I have found that chemical saturation plays a significant role when coupled with the non-linear Turing RD equations, being able to provide enough heterogeneity to match existing pigmentation patterns of several animal species.

Moreover, chemical saturation adds much more variety to the outcome, which helps explain (at least on a theoretical level) the emergence of many complex visual results.

It is particularly striking that the extension to the non-linear Turing RD equations is trivial: we just enforce upper concentration limits, here named L_a and L_b, to reagent concentrations at the end of each simulation iteration.

next_a = min(a, L_a)

next_b = min(b, L_b)

Moreover, we should keep them non-negative too. When saturation is coupled with tissue growth, more complex behavior emerges. In particular, the spontaneous appearance of the Leopard Rosettes, which have long been pursued by academics.

My simulation model also makes possible the definition of distinct diffusion rates for cells and the use of cell polarity to induce anisotropic diffusion. Most examples run over a square lattice, but I am also able to run the simulation on arbitrary arrangement of cells.

A technical conference paper was presented at Graphics Interface 2017, focusing on the conceptual and implementation details of our simulation model. This paper and related materials are on the Github project.

Most of my research focused on the non-linear Turing model. However, I have also briefly evaluated the effects of saturation on other RD models, and a similar behavior appears. Particularly interesting is that the specific Gray-Scott parameters f=0.042 k=0.059 gives almost identical results as the saturated non-linear Turing model.

My research is ongoing on several fronts: further exploration of growth patterns, simulation over a 3D mesh, accelerated GPU implementation, interactive web version and generation of contrasting fur coatings using a single differential equation.

[Torolf Sauermann]

Thank you for sharing but ready remains my favorite :)

sincerely
Torolf

PS: https://goo.gl/photos/u9hffPWbe2vTy5ti8