Chemical principles of instability and self-organization in reacting and diffusive systems

TL;DR

This paper analyzes the instability sources and self-organization conditions in reaction-diffusion systems, revealing that simple activator-inhibitor mechanisms can produce patterns without complex interactions, and higher dimensions influence patterning principles.

Contribution

It derives new conditions for biological pattern formation, showing that single AI mechanisms suffice for self-organization and that higher dimensions alter patterning dynamics.

Findings

Single AI mechanisms can self-organize without II involvement.

Cross diffusion reduces the need for self-enhancement and diffusion differences.

Higher dimensions change patterning principles.

Abstract

How patterns and structures undergo symmetry breaking and self-organize within biological systems from initially homogeneous states is a key issue for biological development. The activator-inhibitor (AI) mechanism, derived from reaction-diffusion (RD) models, has been widely believed to be the elementary mechanism for biological pattern formation. This mechanism generally requires activators to be self-enhanced and diffuse more slowly than inhibitors. Here, we identify the instability sources of biological systems and derive the self-organization conditions through solving eigenvalues (dispersion relation) of the generalized RD model for two chemicals. We show that both the single AI mechanisms with long-range inhibition and activation are enough to self-organize into fully-expressed domains without the involvement of the inhibitor-inhibitor (II) mechanism, through singly enhancing the difference in self-proliferation rates of activators and inhibitors or weakening the coupling degree between them. When cross diffusion involves, both the self-enhancement and the difference in diffusion coefficients of chemicals are no longer necessary for self-organization, and the patterning mechanism can be extended to semi-inhibitor and II mechanisms. However, we show that the single activator-activator (AA) mechanism is generally unable to self-organize, even if biological domain growth is additionally involved. Moreover, adding an II system after an AI one can produce discrete and bi-stable patterns. We also observe that a higher dimensional space can solely alter the patterning principles derived from a lower dimensional space, which may be due to the instability driven by the higher degree of spatial freedom. Such results provide new insights into biological pattern formation.