“Eppur (non) si muove”: why cellular movements may not be essential to the formation of Turing patterns in biology.

D. Bullara* and Y. De Decker
*domenico.bullara@mail.com

When Catarina Vicente (Community Manager of “The Node”) proposed us to write a post about our recent paper on pattern formation in zebrafish [Bullara2015] we were very glad for the opportunity she was giving us to tell the background story about our work in this blog. We are not biologists (we are two theoretical chemists working in the field of nonlinear chemistry and self-organization) and we took Dr. Vicente’s invitation as an opportunity to present our outsiders’ point of view on a quite debated question related to morphogenesis. We therefore very much hope to gain inspiring feedbacks from your comments.

Following this spirit, we initially wrote our post including a number of theoretical details and comments, in the hope to bridge the gap between the typical jargon and assumed basic knowledge of theoretical nonlinear chemistry and experimental developmental biology. We however realized that the final manuscript was too long to fit the scopes of this blog. So – following Dr. Vicente’s advice – we decided to leave the full version in a separate file (which can be downloaded here) for the interested reader, and summarize what we think may be the more interesting paragraphs in the following post.

— Alan Turing and the reaction-diffusion mechanism

Morphogenesis and nonlinear chemistry share a special bond since the British mathematician Alan Turing published his seminal paper “On the chemical basis of morphogenesis” [Turing1952], which set the basis for a theoretical development of both disciplines. The basic question that Turing wanted to answer was: How can a system with such a high degree of symmetry as an egg cell (essentially a sphere) develop organisms with a much lower degree of symmetry (i. e. living beings)?

The pivotal idea of Turing is that “a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium [NOTE1], which is triggered off by random disturbances” [Turing1952]. This mechanism has since then being referred to as the “reaction-diffusion (RD) mechanism”, and the corresponding stationary patterns as “Turing patterns”.

From a molecular point of view, a chemical reaction is essentially an exchange of atoms between molecules, or between molecular segments of a single molecule. But theoretical approaches to reactive systems are often based on a much coarser level of description: one usually divides the whole space into a collection of infinitesimal volumes (or points), within which chemical reactions are considered as local processes. In this framework, diffusion is a physical mechanism which allows molecules to migrate from one point in space to another in a Brownian motion. Both concepts can straightforwardly be extended to non-chemical systems, as long as one can define local events taking place between the units composing a system whose outcome is to change the number of units (reactions) and Brownian motions of these units (diffusion). From this point of view a wolf killing a rabbit or a cell undergoing mitosis may be both considered as “reactions”, although not chemical ones.

Precise mathematical requirements involving the parameters of the system must be fulfilled in order for a RD system to undergo the kind of dynamical instability described by Turing. When this happens, one says that a “Turing instability” or “Turing bifurcation” occurs, and stationary patterns with an intrinsic wavelength can be generated. The original model proposed by Alan Turing as well as several other pattern-generating models undergo precisely this type of instability, which led in practice to an identification of the terms “Turing instability”, “Turing patterns” and “reaction-diffusion mechanism”. It is important however to understand that these terms express separate concepts, and therefore a Turing instability (as well as a Turing pattern) is not limited to reaction-diffusion mechanisms.

— Turing patterns without diffusion? The riddle of the zebrafish stripes

Our interest in zebrafish patterning began in 2012, when we discovered the experimental work of Shigeru Kondo and coworkers [Yamaguchi2007, Nakamasu2009, Inaba2012, Hamada2014]. In their experiments the zebrafish skin patterns exhibit a dynamics which closely resemble what can be observed in typical RD equations schemes forming Turing patterns. Moreover, the experiments clearly shows that the stripes of the zebrafish possess an intrinsic wavelength, which is recovered even after total ablation of the pattern. Both these results would strongly suggest that a RD mechanism could be behind the observed pattern formation. We believe however that this idea should be ruled out for several reasons, among which the following two stand as the most important.

The first reason is that the cell-to-cell interactions, which are at the core of the pattern formation mechanism, cannot be considered as local events like reactions in RD systems. They involve instead two specific distances. When two skin pigment cells of different colors (the yellow xantophore and the black melanophore) are in close contact, they mutually inhibit each other’s growth. However, xantophores can also increase the rate at which melanophores appear (and their survivability) at long distance. The nonlocal character of the cellular interactions makes it impossible to cast them into chemical-like reactive terms, which (as said before) are supposed to act locally at each point in space.

The second – and perhaps even more important – evidence is that the pigment cells do not diffuse across the skin of zebrafish. They do exhibit some degree of mobility, but their movement – which has been characterized in vitro as a “run-and-catch” motion [Yamanaka2014] – cannot be represented as a Brownian motion. Even more importantly, this motion is very limited and is not enough to induce by itself a migration of pigment cells into separate domains [Mahalwar2014] [NOTE2]. In other words, cells are in a first approximation almost immobile.

The patterns on the skin of zebrafish thus look like RD patterns, but cannot be explained by reaction and diffusion. In order to solve the riddle posed by these patterns, we took inspiration from nonlinear nanochemistry. When chemical reactions are described at the nanoscale they cannot be interpreted as local processes, but as “propagating” in space. In mathematical terms, this effect translates into virtual diffusion terms [DeDecker2004] even if the molecules are immobile, because the reaction itself can induce a redistribution of the molecular populations in space. We thus thought that a similar effect could also exist for pigment cells on the skin of zebrafish.

— A new mechanism: differential growth

The question we wanted to answer was essentially the following: Is the the nonlocal character of the short-range and long-range interactions able to create a “virtual movement” of cells across the zebrafish skin, and to generate in such a way a pattern with an intrinsic wavelength?

To test our hypothesis, we needed a simple mathematical model which is also biologically relevant. In order to test whether the observed pattern formation can be explained only in terms of non-local interactions between xantophores and melanophores, we decided to completely remove any form of cellular motion from our model. For the same reason, we did not explicitly include a third type of pigment cell (iridophores), which was shown to be of some importance in the pattern formation on the body of the fish [Singh2014], but not in the fins [Patterson2013]. We then introduced the short-range and long-range interactions as stochastic processes occurring with different probabilities, opting again for the simplest possible implementation: pairwise cell-to-cell interactions. For the sake of completeness we also included the spontaneous differentiation and death of both pigment cells on the skin of the fish [NOTE3]. Finally we further simplified our model by finding a mathematically simple yet biologically representative set of parameters which would trigger pattern formation.

Numerical simulations showed that patterns with an intrinsic wavelength could be formed with our model. We moreover observed that the morphology and the periodicity of the patterns resemble those of the experiments. An analytical study of the evolution equations also showed that the patterns emerge from a Turing bifurcation, despite the absence of cellular motion, thanks to the non-local cellular interactions. This mechanism is intrinsically different from the reaction-diffusion mechanism proposed by Turing although, in our opinion, the patterns thus generated may still be called Turing patterns, because they result from a Turing bifurcation generated by nonequilibrium processes. The key ingredient to form the patterns is that cells can “be born” and die with different rates – or in more mathematical words can have different growth rates – depending on their surrounding. In order to give a unambiguous connotation to this mechanism and distinguish it from others, we proposed to call it “differential growth”. Differential growth promotes a non-trivial redistribution of cells across space by combining short-range and long-range cellular interactions in an appropriate way. In such situations cellular migration becomes accessory to pattern formation, so one cannot rule out the possibility of having Turing patterns solely because of lack of extensive cellular movement.

As a final note, we would like to mention a related, very interesting article which has recently been published in Development [Hiscock2015]. The authors propose a way to rationalize the different patterns-generating mechanism under a common mathematical framework, and try to derive simple rules for the control parameters which can be used as a guide to design experiments. It is interesting to note that the only mechanism for which the authors could not calculate a simple parametric constraint is precisely the type of mechanism we consider here. For reaction-diffusion systems, classical toy models can be used to derive the general rule that “the inhibitor must diffuse faster than the activator”. For the class of systems which fall under the differential growth mechanism, our model suggests that “the inhibitor must grow faster than the activator”, provided that the growth of the former is controlled by a long-range positioning of the latter.

— Notes

[NOTE1] Intended as the mathematical equilibrium of the set of equations describing the dynamics of the system, or in other words any reference homogeneous steady state solution of the latter.

[NOTE2] One of our initial guesses was that the short-range movement shown by the pigment cells could have been important in shaping the fine details of the stripes, more particular the small gap observed between two adjacent stripes. Because of the nature of our model, we could not test this hypothesis ourselves, but we recently discovered a preprint paper by A. Volkening and B. Sandstede titled “Modeling stripe formation in zebrafish: an agent-based approach” which independently proves this hypothesis true with a different modelling approach.

[NOTE3] To this regard, we feel like we should somehow apologize to the biology community for the choice of jargon we made in our paper: we there call “birth” what should more correctly be called “differentiation”. The reason of this choice is that the name commonly used in the stochastic mechanics literature for the class of processes we used is “birth/death” processes, so we felt that the model could be more easily understood by a broader audience of also non-biological scientists if we stuck to these names.

— References

[Bullara2015] Bullara, D., & De Decker, Y. (2015). Pigment cell movement is not required for generation of Turing patterns in zebrafish skin Nature Communications, 6 DOI: 10.1038/ncomms7971

[DeDecker2004] De Decker Y, Tsekouras GA, Provata A, Erneux T, & Nicolis G (2004). Propagating waves in one-dimensional discrete networks of coupled units. Physical review. E, Statistical, nonlinear, and soft matter physics, 69 (3 Pt 2) PMID: 15089388

[Hamada2014] Hamada, H., Watanabe, M., Lau, H., Nishida, T., Hasegawa, T., Parichy, D., & Kondo, S. (2013). Involvement of Delta/Notch signaling in zebrafish adult pigment stripe patterning Development, 141 (2), 318-324 DOI: 10.1242/dev.099804

[Hiscock2015] Hiscock, T., & Megason, S. (2015). Mathematically guided approaches to distinguish models of periodic patterning Development, 142 (3), 409-419 DOI: 10.1242/dev.107441

[Inaba2012] Inaba M, Yamanaka H, & Kondo S (2012). Pigment pattern formation by contact-dependent depolarization. Science, 335 (6069) PMID: 22323812
[Mahalwar2014] P. Mahalwar, B. Walderich, A.P. Singh and C. Nüsslein-Volhard, Local reorganization of xantophores fine-tunes and colors the striped pattern of zebrafish, Science 345:1362-1364 (2014).

[Nakamasu2009] Nakamasu, A., Takahashi, G., Kanbe, A., & Kondo, S. (2009). Interactions between zebrafish pigment cells responsible for the generation of Turing patterns Proceedings of the National Academy of Sciences, 106 (21), 8429-8434 DOI: 10.1073/pnas.0808622106

[Patterson2013] Patterson, L., & Parichy, D. (2013). Interactions with Iridophores and the Tissue Environment Required for Patterning Melanophores and Xanthophores during Zebrafish Adult Pigment Stripe Formation PLoS Genetics, 9 (5) DOI: 10.1371/journal.pgen.1003561

[Singh2014] Singh, A., Schach, U., & Nüsslein-Volhard, C. (2014). Proliferation, dispersal and patterned aggregation of iridophores in the skin prefigure striped colouration of zebrafish Nature Cell Biology, 16 (6), 604-611 DOI: 10.1038/ncb2955

[Turing1952] Turing, A. (1952). The Chemical Basis of Morphogenesis Philosophical Transactions of the Royal Society B: Biological Sciences, 237 (641), 37-72 DOI: 10.1098/rstb.1952.0012

[Yamaguchi2007] Yamaguchi, M., Yoshimoto, E., & Kondo, S. (2007). Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism Proceedings of the National Academy of Sciences, 104 (12), 4790-4793 DOI: 10.1073/pnas.0607790104

[Yamanaka2014] Yamanaka, H., & Kondo, S. (2014). In vitro analysis suggests that difference in cell movement during direct interaction can generate various pigment patterns in vivo Proceedings of the National Academy of Sciences, 111 (5), 1867-1872 DOI: 10.1073/pnas.1315416111

(6 votes)

Loading...