Pattern formation in reaction-diffusion models with nonuniform domain growth

Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction-diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain. In this paper we generalize our recent results for reaction-diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary (apical) growth are unified in a consistent manner. The results have implications for all reaction-diffusion type models subject to underlying domain growth.     

Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model

 Reaction-diffusion equations are ubiquitous as models of biological pattern formation. In a recent paper [4] we have shown that incorporation of domain growth in a reaction-diffusion model generates a sequence of quasi-steady patterns and can provide a mechanism for increased reliability of pattern selection. In this paper we analyse the model to examine the transitions between patterns in the sequence. Introducing a piecewise linear approximation we find closed form approximate solutions for steady-state patterns by exploiting a small parameter, the ratio of diffusivities, in a singular perturbation expansion. We consider the existence of these steady-state solutions as a parameter related to the domain length is varied and predict the point at which the solution ceases to exist, which we identify with the onset of transition between patterns for the sequence generated on the growing domain. Applying these results to the model in one spatial dimension we are able to predict the mechanism and timing of transitions between quasi-steady patterns in the sequence. We also highlight a novel sequence behaviour, mode-tripling, which is a consequence of a symmetry in the reaction term of the reaction-diffusion system. Received: 19 December 2000 / Revised version: 24 May 2001 / Published online: 7 December 2001     

Reaction-diffusion pattern in shoot apical meristem of plants

A fundamental question in developmental biology is how spatial patterns are self-organized from homogeneous structures. In 1952, Turing proposed the reaction-diffusion model in order to explain this issue. Experimental evidence of reaction-diffusion patterns in living organisms was first provided by the pigmentation pattern on the skin of fishes in 1995. However, whether or not this mechanism plays an essential role in developmental events of living organisms remains elusive. Here we show that a reaction-diffusion model can successfully explain the shoot apical meristem (SAM) development of plants. SAM of plants resides in the top of each shoot and consists of a central zone (CZ) and a surrounding peripheral zone (PZ). SAM contains stem cells and continuously produces new organs throughout the lifespan. Molecular genetic studies using Arabidopsis thaliana revealed that the formation and maintenance of the SAM are essentially regulated by the feedback interaction between WUSHCEL (WUS) and CLAVATA (CLV). We developed a mathematical model of the SAM based on a reaction-diffusion dynamics of the WUS-CLV interaction, incorporating cell division and the spatial restriction of the dynamics. Our model explains the various SAM patterns observed in plants, for example, homeostatic control of SAM size in the wild type, enlarged or fasciated SAM in clv mutants, and initiation of ectopic secondary meristems from an initial flattened SAM in wus mutant. In addition, the model is supported by comparing its prediction with the expression pattern of WUS in the wus mutant. Furthermore, the model can account for many experimental results including reorganization processes caused by the CZ ablation and by incision through the meristem center. We thus conclude that the reaction-diffusion dynamics is probably indispensable for the SAM development of plants.     

Changing clothes easily: connexin41.8 regulates skin pattern variation

The skin patterns of animals are very important for their survival, yet the mechanisms involved in skin pattern formation remain unresolved. Turing's reaction-diffusion model presents a well-known mathematical explanation of how animal skin patterns are formed, and this model can predict various animal patterns that are observed in nature. In this study, we used transgenic zebrafish to generate various artificial skin patterns including a narrow stripe with a wide interstripe, a narrow stripe with a narrow interstripe, a labyrinth, and a 'leopard' pattern (or donut-like ring pattern). In this process, connexin41.8 (or its mutant form) was ectopically expressed using the mitfa promoter. Specifically, the leopard pattern was generated as predicted by Turing's model. Our results demonstrate that the pigment cells in animal skin have the potential and plasticity to establish various patterns and that the reaction-diffusion principle can predict skin patterns of animals.     

Reaction-diffusion microtubule concentration patterns occur during biological morphogenesis.

Reaction-diffusion processes can lead to a macroscopic concentration pattern from an initially homogeneous solution, and thus provide a physical-chemical mechanism for biological pattern formation and morphogenesis. The central prediction of reaction-diffusion theory is that the patterns contain periodic concentration variations in some of the reactives. Microtubules assembled in vitro spontaneously self-organise and form stationary striped macroscopic structures. In agreement with reaction-diffusion theory. Here we show, in agreement with reaction-diffusion theory, that these preparations contain substantial microtubule concentration variations. Similar striped microtubule patterns arise during Drosophila embryogenesis. A characteristic of these patterns is their dependence on sample dimensions. In Drosophila eggs shortened by ligation, we found that the microtubule pattern varied with egg fragment length in the same way as the in vitro microtubule pattern varied with sample length, and as expected from theory. This is evidence that reaction-diffusion structures occur during Drosophila morphogenesis.     

Zebrafish leopard gene as a component of the putative reaction-diffusion system

It has been suggested, on a theoretical basis, that a reaction-diffusion (RD) mechanism underlies pigment pattern formation in animals, but as yet, there is no molecular evidence for the putative mechanism. Mutations in the zebrafish gene, leopard, change the pattern from stripes to spots. Interestingly each allele gives a characteristic pattern, which varies in spot size, density and connectivity. That mutations in a single gene can generate such a variety of patterns can be understood using a RD model. All the pattern variations of leopard mutants can be generated in a simulation by changing a parameter value that corresponds to the reaction kinetics in a putative RD system. Substituting an intermediate value of the parameter makes the patterns similar to the heterozygous fish. These results suggest that the leopard gene product is a component of the putative RD mechanism.     

Mode Transitions in a Model Reaction–Diffusion System Driven by Domain Growth and Noise

Pattern formation in many biological systems takes place during growth of the underlying domain. We study a specific example of a reaction-diffusion (Turing) model in which peak splitting, driven by domain growth, generates a sequence of patterns. We have previously shown that the pattern sequences which are presented when the domain growth rate is sufficiently rapid exhibit a mode-doubling phenomenon. Such pattern sequences afford reliable selection of certain final patterns, thus addressing the robustness problem inherent of the Turing mechanism. At slower domain growth rates this regular mode doubling breaks down in the presence of small perturbations to the dynamics. In this paper we examine the breaking down of the mode doubling sequence and consider the implications of this behaviour in increasing the range of reliably selectable final patterns.     

Taxis-driven pattern formation in a predator-prey model with group defense

We consider a reaction-diffusion(-taxis) predator-prey system with group defense in the prey. Taxis-driven instability can occur if the group defense influences the taxis rate (Wang et al., 2017). We elaborate that this mechanism is indeed possible but biologically unlikely to be responsible for pattern formation in such a system. Conversely, we show that patterns in excitable media such as spatiotemporal Sierpinski gasket patterns occur in the reaction-diffusion model as well as in the reaction-diffusion-taxis model. If group defense leads to a dome-shaped functional response, these patterns can have a rescue effect on the predator population in an invasion scenario. Preytaxis with prey repulsion at high prey densities can intensify this mechanism leading to taxis-induced persistence. In particular, taxis can increase parameter regimes of successful invasions and decrease minimum introduction areas necessary for a successful invasion. Last, we consider the mean period of the irregular oscillations. As a result of the underlying mechanism of the patterns, this period is two orders of magnitude smaller than the period in the nonspatial system. Counter-intuitively, faster-moving predators lead to lower oscillation periods and eventually to extinction of the predator population. The study does not only provide valuable insights on theoretical spatially explicit predator-prey models with group defense but also comparisons of ecological data with model simulations.     

A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves

In this paper, we employ the novel application of a reaction-diffusion model on a growing domain to examine growth patterns of the ligaments of arcoid bivalves (marine molluscs) using realistic growth functions. Solving the equations via a novel use of the finite element method on a moving mesh, we show how a reaction-diffusion model can mimic a number of different ligament growth patterns with modest changes in the parameters. Our results imply the existence of a common mode of ligament pattern formation throughout the Arcoida. Consequently, arcoids that share a particular pattern cannot be assumed, on this basis alone, to share an immediate common ancestry. Strikingly different patterns within the set can easily be generated by the same developmental program. We further show how the model can be used to make quantitatively testable predictions with biological implications.     

Reaction-Diffusion Patterns in Plant Tip Morphogenesis: Bifurcations on Spherical Caps

We study a chemical reaction-diffusion model (the Brusselator) for pattern formation on developing plant tips. A family of spherical cap domains is used to represent tip flattening during development. Applied to conifer embryos, we model the chemical prepatterning underlying cotyledon (“seed leaf”) formation, and demonstrate the dependence of patterns on tip flatness, radius, and precursor concentrations. Parameters for the Brusselator in spherical cap domains can be chosen to give supercritical pitchfork bifurcations of patterned solutions of the nonlinear reaction-diffusion system that correspond to the cotyledon patterns that appear on the flattening tips of conifer embryos.