Bifurcations and nonlinear dynamics of the follower force model for active filaments

Biofilament-motor protein complexes are ubiquitous in biology and drive the transport of cargo vital for many fundamental life processes at the cellular level. As they move, motor proteins exert compressive forces on the filaments to which they are attached. If the filament is clamped or tethered in some way, this force leads to buckling and a subsequent range of dynamics. The follower force model, in which a single compressive force is imposed at the filament tip, is a simple filament model that is becoming widely used to describe an elastic filament, such as a microtubule, compressed by a motor protein. Depending on the force value, one can observe different states including whirling, beating, and writhing, though the bifurcations giving rise to these states are not completely understood. In this paper, we utilize techniques from computational dynamical systems to determine and characterize these bifurcations. We track emerging time-periodic branches and identify quasiperiodic states. We investigate the effect of filament slenderness on the bifurcations and, in doing so, present a comprehensive overview of the dynamics which emerge in the follower force model.