Length Distributions of a Model of Filament Growth with Side-Branching

Branching structures are ubiquitous in biological systems at a variety of scales, from cytoskeletal networks to plant roots. Here, we present an analytical study of deterministically elongating filaments that undergo stochastic branching in the absence of interactions. We calculate the length distributions of subpopulations of fixed and actively growing segments for tip bifurcations and side-branching processes. In the absence of side-branching, these subpopulations exhibit identical distributions. However, the inclusion of side-branching breaks this symmetry, producing distinct universality classes (functional forms) of length distributions depending on whether branching occurs on fixed and/or growing filaments. Furthermore, we find that the ratio of typical branch lengths of fixed and active filaments is confined to a finite interval when side branching acts on growing filaments. Thus, experimental measurements deviating from the predicted finite interval would suggest the involvement of branching mechanisms beyond those included in this model. All analytical results are verified by numerical simulations.