Confined Turing patterns in growing systems

Abstract

In this paper we address the problem of pattern formation in confined Turing systems in two dimensions, when one assumes the enhancement of the concentration of one of the chemicals at some of the confining surfaces. This model is suitable to study biological systems, such as the skin patterns shown by some marine fish. We also study numerically the dynamical growth of the system by changing the size of the confined region while dynamical diffusion and reaction phenomena take place. This idea is tested in two different models. This allows one to estimate the robustness of stripe formation. ͓S1063-651X͑97͒06807-4͔ PACS number͑s͒: 47.54.ϩr, 82.40.Bj, 82.20.Mj In 1952, Turing established the basis to explain biological patterns using two interacting chemicals ͓1͔. The experimental observation of a ''Turing pattern'' occurred in a chemical system nearly 40 years after their prediction by Turing ͓2,3͔, but it was not until very recently that the example of a Turing pattern in a biological system was confirmed in skin patterns of the angelfish ͑Pomacanthus͒ by Kondo and Asai ͓4͔. In this work the authors propose and solve a system of two reaction-diffusion equations in a growing onedimensional domain to explain the insertion of new stripes between the older ones during the growth of Pomacanthus semicirculatus and the rearrangement of the stripe pattern of Pomacanthus imperator.Kondo and Asai's interpretation was subject to criticism by Höfer and Maini ͓5͔, who did not find enough evidence to say that reaction-diffusion systems could provide a mechanistic basis for the strip-doubling phenomenon. In particular, they claim that a two-dimensional simulation would be a more realistic representation of the fish skin than a onedimensional domain. Höfer and Maini argue that a mechanism that sets the distance between adjacent stripes and some kind of ''memory'' that conserves the location of old stripes is needed in order to explain the patterning dynamics of the Pomacanthus. Accordingly, in this work we shall show that a reaction-diffusion system is capable of describing the main features of the phenomena observed in the Pomacanthus skin. For that goal, we consider two sets of Turing equations known to form different kinds of patterns; these are solved in a two-dimensional spatial domain that simulates the fish shape, with zero flux boundary conditions. The key feature of our simulation is the enforcement of an enhanced source of the activator along some of the boundaries of the domain. This idea has close parallels with the mechanism of stripe formation in the Drosophila embryo where the pattern of the anteroposterior ͑head-tail͒ segmentation is caused by a high concentration of the Bicoid protein along the anterior ͑head͒ side ͓6͔.To see how the boundary conditions and domain shape affect the stationary patterns from an initially homogeneous state, we study a simplified version of a model for glycolisis as the specific reaction mechanism, which has been extensively studied numerically by Dillon et al. ͓7͔ in one dimension...

A Two-dimensional Numerical Study of Spatial Pattern Formation in Interacting Turing Systems

“…Observe that the stripes align along the source edges. These patterns are very similar to the ones published in Varea et al (1997), but the model is different. Hence, one can conclude that the important factor here is the presence of the source, and not the peculiarities of the model used.…”

“…We find that there is no detectable difference with the patterns already shown. Next, we consider the rôle of boundary conditions, as it is known that these can have a pronounced effect on the pattern exhibited by simple Turing systems (Dillon et al, 1994;Varea et al, 1997). In this case, we find that the existence of a source of chemical at the borders dramatically changes the patterns.…”

Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

“…In the model domain growth plays a central role in establishing the order in which tooth precursors appear. Recently, stemming from work by Kondo and Asai (1995), various authors have considered the effect of growth in numerical simulations of reaction-diffusion models for the skin patterns on species of fish during their development from juvenile to adult forms (Varea et al, 1997;Painter et al, 1999). Here the pattern change is commensurate with growth and shape change, and many different pattern behaviours are observed in response to the growing domain before the final domain geometry is achieved.…”

Mathematical Modelling of Juxtacrine Patterning

“…For example, Kauffman et al (1978) presented one of the first practical applications to early segmentation of the embryo of the fruit fly Drosophila, while Murray (1993) applied reaction-diffusion systems to animal coat patterns at the beginning of the 1980s. Recent work by Varea et al (1997) has looked at the applications to skin patterns of some marine fish, considering a confined Turing system on a growing domain.…”