  Among the main actors of organism development there are morphogens, which are
signaling molecules diffusing in the developing organism and acting on cells to
produce local responses. Growth is thus determined by the distribution of such
signal. Meanwhile, the diffusion of the signal is itself affected by the
changes in shape and size of the organism. In other words, there is a complete
coupling between the diffusion of the signal and the change of the shapes. In
this paper, we introduce a mathematical model to investigate such coupling. The
shape is given by a manifold, that varies in time as the result of a
deformation given by a transport equation. The signal is represented by a
density, diffusing on the manifold via a diffusion equation. We show the
non-commutativity of the transport and diffusion evolution by introducing a new
concept of Lie bracket between the diffusion and the transport operator. We
also provide numerical simulations showing this phenomenon.
