Discrete step sizes of molecular motors lead to bimodal non-Gaussian velocity distributions under force

Fluctuations in the physical properties of biological machines are inextricably linked to their functions. Distributions of run-lengths and velocities of processive molecular motors, like kinesin-1, are accessible through single molecule techniques, yet there is lack a rigorous theoretical model for these probabilities up to now. We derive exact analytic results for a kinetic model to predict the resistive force ($F$) dependent velocity ($P(v)$) and run-length ($P(n)$) distribution functions of generic finitely processive molecular motors that take forward and backward steps on a track. Our theory quantitatively explains the zero force kinesin-1 data for both $P(n)$ and $P(v)$ using the detachment rate as the only parameter, thus allowing us to obtain the variations of these quantities under load. At non-zero $F$, $P(v)$ is non-Gaussian, and is bimodal with peaks at positive and negative values of $v$. The prediction that $P(v)$ is bimodal is a consequence of the discrete step-size of kinesin-1, and remains even when the step-size distribution is taken into account. Although the predictions are based on analyses of kinesin-1 data, our results are general and should hold for any processive motor, which walks on a track by taking discrete steps.