A morphogen gradient model for pattern regulation. I. Formation of non-repetitive and repetitive structures

A model for pattern formation is proposed based on two concentration gradients S and Sigma. S is a local morphogen generated by a reaction-diffusion mechanism while Sigma is a by-product of the S-decomposition. Under certain conditions S is well approximated by S(x,L) = alpha(L)f(x L ), where alpha(L) is a scaling function of the length L and f(x L ) is a monotonie function of the relative distance x L from the origin. Sigma degradates and diffuses in the field, reaching a stable L-dependent homogeneous distribution. An allosteric protein P with several active sites reacts with S and Sigma and separates the field into segments. To every segment a corresponding active state of P is dominant. Pattern regulation is automatically achieved since the compartmerttal separation depends explicitly only on x L . For the case of repetitive patterns, a supplementary Gierer-Meinhardt mechanism is introduced for activator X and inhibitor Y. The level of Sigma can affect the decomposition rate of X or Y, e.g. via a second order degradation reaction, hence making the chemical wavelength lambda size-dependent. For a particular decay scheme of Y, a variation of L induces a change of lambda so that finally the number of repetitive structures becomes independent of the field size.