The best-known theory used to study self-organized pattern
formation in biology is the Alan Turing’s reaction-diffusion equa-
tions [2]. It consists of two coupled reacting species:∂X 1
∂t¼r 1 ðÞþX 1 ;X 2 D 1 ∇^2 X 1 ð 35 Þ∂X 2
∂t¼r 2 ðÞþX 1 ;X 2 D 2 ∇^2 X 2 ð 36 Þwherer 1 andr 2 are reaction terms, andD 1 andD 2 are diffusion
coefficients of activator-repressor speciesX 1 andX 2 , respectively.
The diffusion terms are key for a biochemical system, in far
from equilibrium conditions, to undergo symmetry breaking and
form macroscopic stationary patterns. Such pattern formations are
often referred to as the Turing patterns, named after the scientist
who was first known to have developed it in 1952 [2]. Subse-
quently, there have been several types of activator-repressor reac-
tion terms that have been developed to model various spatial
patterns, according to the type of patterns (Fig.13).Fig. 13Spatio-temporal patterns using reaction-diffusion equations that mimic life patterns. (a) Simulations
from Turing [2], Gierer-Meinhardt [35], and Gray-Scott models, (b) actual experimental patterns found on fish
skin and vascular mesenchymal cells. Figures adapted from [36, 37]
Complex Biological Responses Using Simple Models 195