active transport

Auxin is a major plant hormone, and its spatial distribution in plant tissues is a key driver of plant structure and geometry. Auxin transport is a complex process, combining cell-to-cell diffusion and active transport. The latter is mediated by membrane-bound transporters whose inhomogeneous distribution is controlled by auxin itself. The details of this process are still largely unknown, despite numerous recent advances. In this work, the focus is on a mathematical model implementing one of the current biological assumptions, which is that auxin flux is the variable controlling transporters' distribution. We show that identical auxin patterns can be achieved by distinct transporters distributions, and characterize these in graph theoretical terms. Under a condition of regularity of the dependence of transporters on the flux, we can prove that one of these steady states, with zero flux everywhere, is asymptotically stable for any choice of parameters. When the condition of regularity is not satisfied the same steady state may undergo bifurcations and become unstable. In particular, we can observe stable oscillations taking the form of a travelling wave of auxin, on a row of cells.