Why Are There No 3-Headed Monsters? Mathematical Modeling in Biology

Abstract

A lan Turing's crucial intelligence work in the Second World War is well known. His contribution to the interdisci-plinary field of mathematics and the biological sciences is less so. Turing published only one paper related to biology, " The chemical basis of morphogenesis " , in 1952 [64], which has been seminal in several areas of spatial patterning modeling in development, ecology, and other biological areas since its rediscovery in the 1960s. He did not apply his model to any specific biological situation. Basically Turing showed how, in a system of reacting chemicals where the chemicals can also diffuse, the system can generate a steady-state heterogeneous spatial pattern of chemical concen-trations. He called these chemicals morphogens. He hypothesized that these morphogenetic prepat-terns could cue cell differentiation and result in observed spatial patterns. His model is encapsu-lated in the coupled system of reaction diffusion equations, of the general form (1) ∂u ∂t = γf (u, v) + D u ∇ 2 u, ∂v ∂t = γg(u, v) + D v ∇ 2 v, where the functions f (u, v) and g(u, v) denote the reaction kinetics associated with the chemicals u, called the activator, and v, called the inhibitor,