The Effect of Growth and Curvature on Pattern Formation

Abstract

Based on first principles, we derive a general model to describe the spatio-temporal dynamics of two morphogens. The diffusive part of the model incorporates the dynamics, growth and curvature of one-and two-dimensional domains embedded in R 3 . Our generalized diffusion process includes spatio-temporal varying diffusion coefficients, advection, and dilution terms. We present specific examples by analyzing a third order activator-inhibitor mechanism for the kinetic part. We carry out illustrative numerical simulations on two-dimensional growing domains having different geometries. Comparisons with former results on fixed domains show the crucial role of growth and curvature of pattern selection. Evidence is given that both effects might be biologically relevant in explaining the selection of some observed patterns and in changing or enhancing their stability.

Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains

“…Firstly, we have shown that pattern initiation is in general sensitive to, and changes with, even slow domain growth. This is quite distinct from the commonly reported result that domain growth enhances the robustness of pattern selection (Plaza et al 2004). Nonetheless, despite its effects, slow domain growth will not prohibit the Turing mechanism as a means of driving a symmetry breaking bifurcation nor will slow domain growth induce oscillations at the bifurcation, as required for biological pattern.…”

“…In most cases, as the first step in considering the Turing diffusively-driven instability analysis on growing domains, the reaction-diffusion equations (RDEs) are transformed into RDEs on fixed domains, but with time-dependence in the diffusion and dilution terms (Crampin et al 1999;Plaza et al 2004). These non-autonomous terms however typically invalidate standard linear stability analysis via plane wave decompositions, even with the common simplification that the domain growth is assumed to be isotropic.…”

“…(8) have been extensively studied in papers by Crampin et al ( , 1999, Plaza et al (2004), Madzvamuse (2005), and Madzvamuse and Maini (2007).…”

“…In contrast to the case of a fixed domain, numerous different stripe and spot patterns occur depending on perturbations of the initial conditions for the model. Further examples of studies of RDEs illustrating the role of domain growth can be found in papers by Varea et al (1999), Chaplain et al (2001), Liaw et al (2001), Painter et al (1999), Crampin et al ( , 1999, Oster and Bressloff (2006), Madzvamuse et al (2003Madzvamuse et al ( , 2005, Madzvamuse (2005) and for a review see Plaza et al (2004).…”

“…Despite the above-mentioned studies and the simplifying assumption of isotropy, there is only limited analytical work investigating the impact domain growth has on pattern formation (see, for example Plaza et al (2004)). Hence, this paper aims to derive and present the Turing diffusively-driven instability conditions for RDEs on continuously deforming growing domains restricted to the case of slow growth only.…”