Metric dynamics for membrane transformation through regulated cell proliferation

This study develops an equation for describing three-dimensional membrane transformation through proliferation of its component cells regulated by morphogen density distributions on the membrane. The equation is developed in a two-dimensional coordinate system mapped on the membrane, referred to as the membrane coordinates. When the membrane expands, the membrane coordinates expand in the same manner so that the membrane is invariant in the coordinates. In the membrane coordinate system, the transformation of membrane is described with a time-derivative equation for metric tensors. By defining relationships between morphogen density distributions and the direction and rate of cell division, trajectories of membrane transformation are obtained in terms of the morphogen distributions. An example of the membrane transformation is shown numerically.