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

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.