A theory of biological pattern formation

Abstract

One of the elementary processes in morphogenesis is the formation of a spatial pattern of tissue structures, starting from almost homogeneous tissue. It will be shown that relatively simple molecular mechanisms based on auto- and cross catalysis can account for a primary pattern of morphogens to determine pattern formation of the tissue. The theory is based on short range activation, long range inhibition, and a distinction between activator and inhibitor concentrations on one hand, and the densities of their sources on the other. While source density is expected to change slowly, e.g. as an effect of cell differentiation, the concentration of activators and inhibitors can change rapidly to establish the primary pattern; this results from auto- and cross catalytic effects on the sources, spreading by diffusion or other mechanisms, and degradation.

Employing an approximative equation, a criterium is derived for models, which lead to a striking pattern, starting from an even distribution of morphogens, and assuming a shallow source gradient. The polarity of the pattern depends on the direction of the source gradient, but can be rather independent of other features of source distribution. Models are proposed which explain size regulation (constant proportion of the parts of the pattern irrespective of total size). Depending on the choice of constants, aperiodic patterns, implying a one-to-one correlation between morphogen concentration and position in the tissue, or nearly periodic patterns can be obtained. The theory can be applied not only to multicellular tissues, but also to intracellular differentiation, e.g. of polar cells.

The theory permits various molecular interpretations. One of the simplest models involves bimolecular activation and monomolecular inhibition. Source gradients may be substituted by, or added to, sink gradients, e.g. of degrading enzymes. Inhibitors can be substituted by substances required for, and depleted by activation.

Sources may be either synthesizing systems or particulate structures releasing activators and inhibitors.

Calculations by computer are presented to exemplify the main features of the theory proposed. The theory is applied to quantitative data on hydra — a suitable one-dimensional model for pattern formation — and is shown to account for activation and inhibition of secondary head formation.