A further study of the holoblastic cleavage field

The division of holoblastically cleaving eggs was modelled by Goodwin & Trainor in terms of a minimization principle applied to the generalized surface free energy of a spherical surface, expressed in terms of an order parameter. Their calculations led to a cumulative description of the sequence of horizontal and vertical cleavage lines or planes in terms of the nodal lines of spherical harmonics. In this paper we give a complete treatment, first of the numerical regularities exhibited by holoblastic cleavage resulting in a group-theoretic formulation, and secondly of the calculations which arise from Goodwin & Trainor's model. It is shown mathematically that there is no other cumulative description than the one they produced but that it suffers from certain inaccuracies. We prove that there is an essentially unique non-cumulative description which is more accurate and which predicts the loss of global order after a particular stage which corresponds to that beyond which global biological order itself fails to persist. In the discussion, this is interpreted to mean that the initial global field is superseded by a more local one, in accordance with the biological evidence as observed in the midblastula transition.