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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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 · 5.29 Impact Factor
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ABSTRACT: Visual simulation can be efficiently performed using programmable graphics hardware. However, in utilizing hardware to maximize throughput, it is important not to constrain interactivity. We present a method of using the graphics hardware while maintaining full interactivity during simulation exploration. This interactivity involves: temporal exploration, data probing and modification, simulation model modification, and user defined visual metadata. Results are shown using our application for exploring a reaction-diffusion simulation.Proceedings of SPIE - The International Society for Optical Engineering 03/2005; DOI:10.1117/12.588115 · 0.20 Impact Factor
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ABSTRACT: Turing mechanism explains the pattern formation in a uniform field in which two substances (e.g. activator and inhibitor) interact locally and diffuse randomly. Two-dimensional Turing models can generate stationary spatial patterns either with stripes or with spots, and have been adopted to explain the skin pattern formation of animals. We first discuss the effect of the choice of reaction terms on pattern selection, whether spots or stripes are formed. It is shown that the relative distance of the equilibrium level of activator between the upper and lower limitations has a very strong effect on the pattern selection. Secondly, we focus on the direction of the stripes generated by Turing model with anisotropic diffusion in order to explain the directionality of stripes on fish skin in closely related species. Relative magnitude of anisotropy of the two substances is shown to determine whether stripes are vertical or horizontal.01/2003; 18:3-18.