A mathematical model for embryonic cell division based on a surface “cleavage field”

Abstract

The process whereby a fertilized egg divides to give rise to an embryo, i.e. the process of cleavage—which can be considered, in some sense, as the early phase of embryonic differentiation—exhibit in many species a precise geometry. Such a geometry may be altered within certain limits, as was done in various classical experiments, and yet normal differentiation may occur. However, since the pattern of cleavage is clearly under genetic control, any model of cleavage should incorporate some device apt to produce a specific geometry. In this paper, a model for embryonic cell division based on a surface “cleavage field” is described. This surface field may be interpreted, for instance, as the surface density of sources of active transport of ions which diffuse into the cell, although other interpretations —such as the surface density of specific binding sites or functional membrane receptors, etc.—are possible. Assumptions relating the geometry of cleavage to the geometry of the level surfaces of the ionic concentration are given together with a discussion of the change in the surface field due to cleavage. Finally, a simplified two-dimensional version of the model is presented which develops interesting patterns of “cleavagerd”, calculated by computer, similar in many ways to those of real threedimensional embryos.