Filament-motor protein system under loading: instability and limit cycle oscillations

We consider the dynamics of a rigid filament in a motor protein assay under external loading. The motor proteins are modeled as active harmonic linkers with tail ends immobilized on a substrate. Their heads attach to the filament stochastically to extend along with it, resulting in a force on the filament, before detaching. The rate of extension and detachment are load-dependent. Here we formulate and characterize the governing dynamics in the mean field approximation using linear stability analysis, and direct numerical simulations of the motor proteins and filament. Under constant loading, the system shows transition from a stable configuration to instability toward detachment of the filament from motor proteins. Under elastic loading, we find emergence of stable limit cycle oscillations via a supercritical Hopf bifurcation with change in activity and the number of motor proteins. The number of motor proteins required at the onset of limit cycle oscillations increases with the increasing stiffness of the elastic loading. Numerical simulations of the system for large number of motor proteins show good agreement with the mean field predictions.