Abstract:
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.
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 noncommutativity
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. |