Size adaptation of turing prepatterns
Spontaneous pattern formation may arise in biological systems as primary and secondary bifurcations to nonlinear parabolic partial differential equations describing chemical reaction-diffusion systems. Such Turing prepatterns have a specified geometry as long as D/R2 (the diffusion coefficient of the morphogen D divided by the square of a characteristic length) is confined to a (usually) limited interval. As real biochemical systems like cleaving eggs or early embryos vary considerably in size, Turing prepatterns are unable to maintain a specified prepattern-geometry, unless D/R2 is varied as well. We show, that actual biochemical control systems may vary Dapp/R2, where Dapp(k) is an apparent diffusion constant, dependent on enzyme regulated rate constants, and that such simple control systems allow Turing structures to adapt to size variations of at least a factor 103 (linearly), not only in large connected cell systems, but in single cells as well.