Inside cells of living organisms, actin filaments and microtubules
self-assemble and dissemble dynamically by incorporating actin or tubulin from
the cell plasma or releasing it into their tips' surroundings. Such
reaction-diffusion systems can show diffusion- or reaction-limited behaviour.
However, neither limit explains the experimental data: while the offset of the
linear relation between growth speed and bulk tubulin density contradicts the
diffusion limit, the surprisingly large variance of the growth speed rejects a
pure reaction limit. In this Letter, we accommodate both limits and use a
Doi-Peliti field-theory model to estimate how diffusive transport is perturbing
the chemical reactions at the filament tip. Furthermore, a crossover bulk
density is predicted at which the limiting process changes from chemical
reactions to diffusive transport. In addition, we explain and estimate larger
variances of the growth speed
