On the force-velocity relationship of a bundle of rigid bio-filaments

In various cellular processes, bio-filaments like F-actin and F-tubulin are able to exploit chemical energy associated with polymerization to perform mechanical work against an obstacle loaded with an external force. The force-velocity relationship quantitatively summarizes the nature of this process. By a stochastic dynamical model, we give, together with the evolution of a staggered bundle of N_f rigid living filaments facing a loaded wall, the corresponding force-velocity relationship. We compute the evolution of the model in the infinite wall diffusion limit and in supercritical conditions (monomer density reduced by critical density ρ^1>1), and we show that this solution remains valid for moderate non-zero values of the ratio between the wall diffusion and the chemical time scales. We consider two classical protocols: the bundle is opposed either to a constant load or to an optical trap setup, characterized by a harmonic restoring force. The constant load case leads, for each F value, to a stationary velocity Vstat(F;Nf,ρ^1) after a relaxation with characteristic time τ_micro(F). When the bundle (initially taken as an assembly of filament seeds) is subjected to a harmonic restoring force (optical trap load), the bundle elongates and the load increases up to stalling over a characteristic time τ^OT. Extracted from this single experiment, the force-velocity VOT(F;Nf,ρ^1) curve is found to coincide with Vstat(F;Nf,ρ^1), except at low loads. We show that this result follows from the adiabatic separation between τ_micro and τ^OT, i.e., τ_micro ≈ τ^OT.