Energetics of the Spontaneous Beating of Motor-Driven Filaments

Abstract

Cilia and flagella beat through the collective action of molecular motors, driving fluid transport and cell motility, and are an attractive template for engineered actuators. The follower force model captures spontaneous beating in minimal reconstituted systems, but treats motor activity as a uniform, constant force density and so cannot resolve how individual motors supply and dissipate the energy that sustains oscillation. We simulated a microtubule (MT, 5–30 μm long) clamped at its minus end and driven by randomly distributed surface-immobilized kinesins (5–50 motors/μm²). Each motor was modelled as a linear spring (k = 100 pN/ μm) with load-dependent velocity (unloaded 0.8 μm/s, stall force 5 pN) and force-dependent detachment (detachment force 7 pN). Dynamics were obtained by solving overdamped equations, with hydrodynamics included only as local isotropic drag. Beating emerged from a localized buckling instability once the motor force exceeded a critical threshold set by MT length and motor density, with frequencies of 30–50 mHz. Bending waves travelled from the clamped end to the free end at ~0.8 μm/s, the unloaded kinesin velocity. Frequency scaled with motor density with an exponent of 0.27 ± 0.01, much weaker than the 4/3 predicted by the follower force model; we attribute this discrepancy to the mechanical interaction between the filament and kinesin. Amplitude scaled with an exponent of −0.27 ± 0.02, consistent with the scaling of the buckling wavelength. These results show that motor protein mechanics govern the energetics of wave-like beating in actively driven filaments and can guide the design of micro- and nanoscale devices for fluid pumping and mixing.