Critical motor number for fractional steps of cytoskeletal filaments in gliding assays

In gliding assays, filaments are pulled by molecular motors that are immobilized on a solid surface. By varying the motor density on the surface, one can control the number Formula: see text] of motors that pull simultaneously on a single filament. Here, such gliding assays are studied theoretically using Brownian (or Langevin) dynamics simulations and taking the local force balance between motors and filaments as well as the force-dependent velocity of the motors into account. We focus on the filament stepping dynamics and investigate how single motor properties such as stalk elasticity and step size determine the presence or absence of fractional steps of the filaments. We show that each gliding assay can be characterized by a critical motor number, Formula: see text]. Because of thermal fluctuations, fractional filament steps are only detectable as long as Formula: see text]. The corresponding fractional filament step size is Formula: see text] where Formula: see text] is the step size of a single motor. We first apply our computational approach to microtubules pulled by kinesin-1 motors. For elastic motor stalks that behave as linear springs with a zero rest length, the critical motor number is found to be Formula: see text], and the corresponding distributions of the filament step sizes are in good agreement with the available experimental data. In general, the critical motor number Formula: see text] depends on the elastic stalk properties and is reduced to Formula: see text] for linear springs with a nonzero rest length. Furthermore, Formula: see text] is shown to depend quadratically on the motor step size Formula: see text]. Therefore, gliding assays consisting of actin filaments and myosin-V are predicted to exhibit fractional filament steps up to motor number Formula: see text]. Finally, we show that fractional filament steps are also detectable for a fixed average motor number Formula: see text] as determined by the surface density (or coverage) of the motors on the substrate surface.