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.
- SourceAvailable from: Hans Meinhardthttp://www.eb.tuebingen.mpg.de/departments/former-departments/h-meinhardt/82-book/Bur82.htm. 01/1982;
- [show abstract] [hide abstract]
ABSTRACT: The formation of stripe patterns in animal skin has been explained by the reaction-diffusion (RD) system, a hypothetical chemical reaction proposed by A. Turing. Although animal stripes usually have directionality, the RD model alone cannot explain how the direction is specified. To investigate the mechanism regulating the direction of stripes, we studied stripe pattern formation in two species of Genicanthus during sexual conversion. These species share almost identical morphologic properties, except for their stripe direction. In both species, spots transiently arise at random positions and then combine and rearrange to form directional stripes. Computational analysis has shown that diffusion anisotropy is very effective at specifying the direction of stripes formed by the RD system. Model simulations reproduce the transient dynamics of directional pattern formation observed in fish as well as the resulting stripes. In cases where the magnitude and direction of diffusion anisotropy of the substances are identical, the resulting stripes are not directional. However, if they differ, stripes become directional. As only a small difference in anisotropy is required for this effect, any kind of structure with directional conformation might cause a marked change in stripe direction. Scales are the most likely candidate structure for generating anisotropic interactions in skin.Developmental Dynamics 05/2003; 226(4):627-33. · 2.59 Impact Factor
Article: Mathematical Sociology?01/1952;