  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.
