The polar co-ordinate model for pattern regulation in epimorphic fields: a critical appraisal.

Abstract

Abstract The polar co-ordinate model for pattern regulation in epimorphic fields, as first proposed by French, Bryant & Bryant (1976), belongs to a general class of models employing two co-ordinates for describing the field value distribution over a monolayer of cells. I first analyse the constraints imposed by the use of polar co-ordinates for biological modelling. These constraints entail: distinction of the actual and derived spaces for the field with their specific and characteristic mapping modes; representation of the field value or a measure of the morphogenetic activity at various positions in the actual space by two polar co-ordinates—the phase or angle and the radius—in the derived space; and attribution of a singular property to the origin in the derived space, which ought to be mapped onto a putative field centre in the actual space. Upon reformulating the polar co-ordinate model while adhering to these constraints, one realizes that the choice of the shortest route between two points in the derived space applies only to the angular, and not to the radial, co-ordinate. Hence the shortest route between two points cannot be conceived as a straight line connecting them. If we assume that cells have certain limits in resolving differences in the field value, the field will virtually fall into territories of distinctive field values, each of which might encompass more than one cell; cells that lie within a single territory should be mutually interchangeable for prospective developmental pathways which may themselves diversify. The central postulate of the model is that disparate field values brought in apposition will induce a field value intercalation through cell proliferation to restore an unbroken sequence of field values for the newly emerging array of territories. This is assumed universally for the angular component of the field value but it need not apply to the radial component. It follows that angular territories meeting at the field centre will produce a central discontinuity of the field value, leading to a perpetual angular intercalation unless some theoretical provision is made. This may be resolved by any ad hoc hypotheses but also by introducing the concept of degeneracy towards the field centre of the angular co-ordinate down to three distinctive values at the innermost level of radial territories. One can then deduce and generalize various empirically recognized behaviours of epimorphic fields. These include the shortest intercalation rule (which may be modified into the more generally applicable, minor-in-major rule), and the distal transformation and so-called complete circle rules (which may now be unified into the more feasible distalization rule proposed by Bryant, French & Bryant, 1981). The models are assessed against various experimental observations mainly for theoretical consistency. A series of experimental tests for various theoretical alternatives are suggested. This work has simply explored the internal logic of the polar co-ordinate model and no suggestion is made, at this stage, to distinguish different families of alternative models for epimorphic systems on experimental grounds.