  We introduce a general theoretical framework to study the shape dynamics of
actively growing and remodeling surfaces. Using this framework we develop a
physical model for growing bacterial cell walls and study the interplay of cell
shape with the dynamics of growth and constriction. The model allows us to
derive constraints on cell wall mechanical energy based on the observed
dynamics of cell shape. We predict that exponential growth in cell size
requires a constant amount of cell wall energy to be dissipated per unit
volume. We use the model to understand and contrast growth in bacteria with
different shapes such as spherical, ellipsoidal, cylindrical and toroidal
morphologies. Coupling growth to cell wall constriction, we predict a
discontinuous shape transformation, from partial constriction to cell division,
as a function of the chemical potential driving cell-wall synthesis. Our model
for cell wall energy and shape dynamics relates growth kinetics with cell
geometry, and provides a unified framework to describe the interplay between
shape, growth and division in bacterial cells.
