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This study develops an equation for describing three-dimensional membranetransformation through proliferation of its component cells regulated bymorphogen density distributions on the membrane. The equation is developed in atwo-dimensional coordinate system mapped on the membrane, referred to as themembrane coordinates. When the membrane expands, the membrane coordinatesexpand in the same manner so that the membrane is invariant in the coordinates.In the membrane coordinate system, the transformation of membrane is describedwith a time-derivative equation for metric tensors. By defining relationshipsbetween morphogen density distributions and the direction and rate of celldivision, trajectories of membrane transformation are obtained in terms of themorphogen distributions. An example of the membrane transformation is shownnumerically.

Metric dynamics for membrane transformation through regulated cell proliferation