The dynamics of angular homeostasis: I. General principles.

A general model is proposed casting aspects of ontogeny in quantitative terms amenable to genetic analysis. Its primordial construct is a chain of cells (termed a "pursuer") growing under the influence of a signal towards a fixed structure termed a "target." There is provision for graduated correction of the direction of growth of the pursuer. The determinants of scale include the size of the cells and the distance from the target. The minimum number of parameters is two: the initial angle of growth; and the force of the correction of errors of direction. Both are potentially of genetic interest. The impact of variation in these factors on the path of growth is studied. These findings are readily translated into biological terms, notably in congenital defects of the heart. Besides the primordial purposes, there are other objectives to the process. Some membranes require free edges, or large curvatures, or circular arrays. These secondary qualities require that the cells never reach the target. The target then becomes simply a construction point: that is, while remaining a center of attraction, it is no longer a true goal. If, because of undercorrection, the cell line misses the target at the first pass, it assumes a permanent orbit about it. The orbit rapidly comes to lie on a circle, with a radius independent of the initial angle of growth but related to the cell size and the restoration constant. From this property, several kinds of structures other than a simple bridge may result, especially when a series of lines of growth together form a tissue: a cusp, a free-floating membrane, or a circular membrane to fill a gap.