The plant hormone auxin is transported with a well defined velocity through many tissues. To explain this, one type of theory proposes that a polar mechanism operates at the interface between two cells. I show that, if auxin diffuses freely through the interior of cells, then there is an upper limit to the velocity that can be achieved by such a mechanism. This is compatible with the observed velocities provided that the diffusion constant for auxin within a cell is not much less than that measured for auxin in aqueous media. Cytoplasmic streaming, unless specially organized, would not assist the movement of auxin. This is because rapid diffusion between streams will cancel out any directed motion. I also show that the permeability that characterizes the forward movement between cells must exceed a certain limit. If auxin moves mainly through the cytoplasm, which occupies only a small part of the volume of a cell, then the permeability per unit area of membrane needed to achieve a given velocity is much reduced. Transport would be channelled through the cytoplasm if the membrane bounding the vacuole were relatively impermeable to auxin. The theory that I develop leads to predictions about, for example, the route of auxin and its concentration gradients within cells, and the dependence of velocity on cell length.
Grioli’s Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations