Stripes, spots, or reversed spots in two-dimensional Turing systems
ABSTRACT Two-dimensional Turing models can generate stationary striped patterns or spotted patterns, and are used to explain the body pattern formation of animals. We studied the effects of the choice of reaction terms on pattern selection, i.e., which pattern is likely to be formed. We examined in detail a model with linear reaction terms and additional constraint terms that confine two variables within a finite range. In the one-dimensional model, a periodic stationary pattern can be formed only when the activator level is constrained both from below and from above. In the two-dimensional model, the relative distance of the equilibrium level of the activator between the upper and lower limitations determines the pattern selection. Striped patterns are produced when the equilibrium is equally distant from the upper and the lower limitations, but spotted patterns are produced when the equilibrium is clearly closer to one than to the other of two limitations. We then examined models with nonlinear reaction terms, including both activator-inhibitor and activator-depletion substrate type models; we attempted to explain the pattern selection of these nonlinear models based on the results of linear models with constraints. The distribution of the activator level is skewed positively and negatively for spotted patterns and reversed spotted patterns, respectively. In contrast, the skew of the distribution of the activator level was close to zero in the case of striped patterns. This observation provides a heuristic argument of how the location of the equilibrium between the constraints leads to pattern selection.
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- "Up till now, heuristic criteria had been solely proposed, with restrictions on reactive terms (e.g. [14,15]). So, the main purpose of our paper is to propose an analytic selection criterion aimed at predicting patterns for general reaction–diffusion systems, depending on the nonlinearities involved in the reaction terms. "
ABSTRACT: Alan Turing's work in Morphogenesis has received wide attention during the past 60 years. The central idea behind his theory is that two chemically interacting diffusible substances are able to generate stable spatial patterns, provided certain conditions are met. Ever since, extensive work on several kinds of pattern-generating reaction diffusion systems has been done. Nevertheless, prediction of specific patterns is far from being straightforward, and a great deal of interest in deciphering how to generate specific patterns under controlled conditions prevails. Techniques allowing one to predict what kind of spatial structure will emerge from reaction-diffusion systems remain unknown. In response to this need, we consider a generalized reaction diffusion system on a planar domain and provide an analytic criterion to determine whether spots or stripes will be formed. Our criterion is motivated by the existence of an associated energy function that allows bringing in the intuition provided by phase transitions phenomena. Our criterion is proved rigorously in some situations, generalizing well-known results for the scalar equation where the pattern selection process can be understood in terms of a potential. In more complex settings it is investigated numerically. Our work constitutes a first step towards rigorous pattern prediction in arbitrary geometries/conditions. Advances in this direction are highly applicable to the efficient design of Biotechnology and Developmental Biology experiments, as well as in simplifying the analysis of morphogenetic models.Theoretical Biology and Medical Modelling 01/2014; 11(1):7. DOI:10.1186/1742-4682-11-7 · 0.95 Impact Factor
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- "The formation or direction of stripes does not critically depend on the choice of reaction terms (Shoji et al. 2002). However, spots, reverse spots, or irregular stripes are produced depending on the shape of nullclines of reaction terms (Shoji et al. 2003a). The shape of the nullcline depends on the upper and lower limits of activation level constraints, which can be manipulated by the β parameter in Equation 1b. "
ABSTRACT: Species in the suborder Serpentes present a powerful model for understanding processes involved in visual signal design. Although vision is generally poor in snakes, they are often both predators and prey of visually oriented species. We examined how ecological and behavioral factors have driven the evolution of snake patterning using a phylogenetic comparative approach. The appearances of 171 species of Australian and North American snakes were classified using a reaction-diffusion model of pattern development, the parameters of which allow parametric quantification of various aspects of coloration. The main findings include associations between plain color and an active hunting strategy, longitudinal stripes and rapid escape speed, blotched patterns with ambush hunting, slow movement and pungent cloacal defense, and spotted patterns with close proximity to cover. Expected associations between bright colors, aggressive behavior, and venom potency were not observed. The mechanisms through which plain and longitudinally striped patterns might support camouflage during movement are discussed. The flicker-fusion hypothesis for transverse striped patterns being perceived as uniform color during movement is evaluated as theoretically possible but unlikely. Snake pattern evolution is generally phylogenetically conservative, but by sampling densely in a wide variety of snake lineages, we have demonstrated that similar pattern phenotypes have evolved repeatedly in response to similar ecological demands.Behavioral Ecology 08/2013; 24(5-5):1237-1250. DOI:10.1093/beheco/art058 · 3.18 Impact Factor
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- "Species-specific carpal and tarsal structures of the wrist and ankle need to be investigated further. These may form because of transitions from stripe-like to spot-stripe patterns which can occur in this class of equations (Shoji et al. 2003). We suggest that integration of knowledge of cellular and biochemical processes of development with dynamical modelling, geometry and computational strategies will prove useful in understanding limb skeletal development at a more detailed level, as well as other aspects of organogenesis. "
ABSTRACT: We describe a 'reactor-diffusion' mechanism for precartilage condensation based on recent experiments on chondrogenesis in the early vertebrate limb and additional hypotheses. Cellular differentiation of mesenchymal cells into subtypes with different fibroblast growth factor (FGF) receptors occurs in the presence of spatio-temporal variations of FGFs and transforming growth factor-betas (TGF-betas). One class of differentiated cells produces elevated quantities of the extracellular matrix protein fibronectin, which initiates adhesion-mediated preskeletal mesenchymal condensation. The same class of cells also produces an FGF-dependent laterally acting inhibitor that keeps condensations from expanding beyond a critical size. We show that this 'reactor-diffusion' mechanism leads naturally to patterning consistent with skeletal form, and describe simulations of spatio-temporal distribution of these differentiated cell types and the TGF-beta and inhibitor concentrations in the developing limb bud.Proceedings of the Royal Society B: Biological Sciences 09/2004; 271(1549):1713-22. DOI:10.1098/rspb.2004.2772 · 5.05 Impact Factor