Markov modelling of run length and velocity for molecular motors

Molecular motors are nanometer scale proteins involved in various in-tracellular processes such as cargo transport, muscle contraction and cell division. The steps in the chemo-mechanical cycle that use ATP hydrolysis to generate motion along the cytoskeletal tracks can be characterized by stepping rates depending only on the current state, permitting modeling of the cycle as a Markov process. To learn more about the nature of motor mo-tion, cell researchers have conducted in vitro experiments in which molecular motors pull beads along filaments attached to glass slides and their veloc-ity and run length until detachment from the microtubule are recorded. In this article a formula is derived for distance traveled until detachment. Combining this with a standard result for time to absorption for Markov processes gives a formula for velocity. Two kinesins, and two variants of myosin VI, for which there are run length and velocity measurements in the literature are considered. In each case the derived formulas for run length and velocity have a generalized Michaelis-Menten form as functions of ATP concentration. The degrees of the numerator and denominator polynomials increase with the complexity of chemo-mechanical cycle. The degrees of the Michaelis-Menten form determine the maximum number of of reaction rates that can be determined from experimental data on run length and velocity, however consistency conditions may reduce this number and restrict which rates can be determined.