Complexity, Chaos, and Biological Evolution pp 323-331 | Cite as

# Reaction-Diffusion Prepatterns (Turing Structures): Supercomputer Simulation of Cytokinesis, Mitosis and Early *Drosophila* Morphogenesis

Chapter

## Abstract

The spontaneous formation of complex patterns and form in biological systems is largely unexplained. Turing(1952) demonstrated however that autocatalytic biochemical reactions coupled to internal diffusion, but without external control, could break up from the original homogeneous state and form stable well defined inhomogeneous concentration gradients and patterns. General reaction-diffusion systems may be described by where c is the concentration vector, spontaneous pattern formation may occur if the rates and diffusion constants satisfy certain inequalities. In the case (a) one speaks of an activation-inhibition system, as

$$\partial c = \partial t = F(c) + D\Delta c$$

(1)

**F**is the chemical kinetics rate vector and the last term describes Fickian diffusion. If the Jacobian**J**= ∂**F**/∂**c**, evaluated at the homogeneous steady state, is of one of the forms (a) or (b):$$
J = \left( {\begin{array}{*{20}c}
+ & - \\
+ & - \\
\end{array} } \right) (a) J = \left( {\begin{array}{*{20}c}
+ & - \\
+ & - \\
\end{array} } \right) (b)
$$

*c*_{1}activates both its own formation and that of*c*_{2}, and*c*_{2}inhibits both rates. The second class (b) was introduced by Sel’kov (1968) and studied by the socalled Brussels group, the leader of which, I. Prigogine, got the Nobel price in 1977. Their work demonstrates that Turing structures are fully compatible with the second law of thermodynamics since living systems are open systems and they showed (1974) with bifurcation theory that the patterns found in computer simulations are genuine solutions to the nonlinear partial differential equations above. References to early work on such*spontaneous symmetry breaking*in biochemical systems are found in (Nicolis and Prigogine, 1977).## Preview

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© Plenum Press, New York 1991