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

Abstract

In various cellular processes, biofilaments like F-actin and F-tubulin are able to exploit chemical energy associated to polymerization to perform mechanical work against an external load. 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 systematically the simplified evolution of the model in supercritical conditions ${\rho }_1=U_0/W_0>1$ at $\epsilon =d^2{W}_0/D=0$, where $d$ is the monomer size, $D$ is the obstacle diffusion coefficient, $U_0$ and $W_0$ are the polymerization and depolymerization rates. Moreover, we see that the solution at $\epsilon =0$ is valid for a good range of small non-zero $\epsilon$ values. We consider two classical protocols: the bundle is opposed either to a constant load or to an optical trap set-up, characterized by a harmonic restoring force. The constant force case leads, for each $F$ value, to a stationary velocity $V^{stat}(F;N_f,{\rho }_1)$ after a relaxation with characteristic time ${\tau }_{micro}(F)$. 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 (equilibrium) over a characteristic time ${\tau }^{OT}$. Extracted from this single experiment, the force-velocity $V^{OT}(F;N_f,{\rho }_1)$ curve is found to coincide with $V^{stat}(F;N_f,{\rho }_1)$, except at low loads. We show that this result follows from the adiabatic separation between ${\tau }_{micro}$ and ${\tau }^{OT}$, i.e. ${\tau }_{micro}\ll {\tau }^{OT}$.

19 pages, 5 figures