Neurofilament Diffusive Search for Randomly Distributed Microtubules Exhibits Distinct Power-Law Behavior

Neurofilaments (NFs) are space-filling protein polymers and one of the most abundant organelles in the neuronal axon. They are synthesized in the cell body and transported into the axon along microtubule (MT) tracks via slow axonal transport, characterized by a stop-and-go movement, i.e., rapid movement interrupted by long pauses. We assume that during these pauses, NFs detach from their MT tracks and radially diffuse in search for another MT to bind to and resume their transport. In this study, we model the two-dimensional diffusive search of NFs for randomly distributed MT targets within the axon. Since the effective size of the NFs can vary due to factors such as phosphorylation, we investigate the dependence of the average binding rate more broadly on the density and size of MTs and NFs. We derive analytical expressions for the binding rates and test them through direct numerical simulations over a wide range of MT densities. Our results suggest that, governed by target and searcher size, the density dependence of the binding rates of NFs transitions between two distinct power-laws: from an exponent of ~1.1 at low densities to an exponent of 3/2 at higher densities, a range relevant to axonal transport. This nonlinearity suggests that the axon caliber is highly sensitive to changes in NF kinetics, directly impacting neuronal function. This sensitivity may also contribute to potential accumulations of NFs in certain neurodegenerative diseases.