  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.
