Periodic sequential kinetic models with jumping, branching and deaths

Motivated by recent applications to experiments on processive molecular motors (i.e., protein molecules that drag loads along linear, periodic molecular filaments), the analysis of Derrida (J. Stat. Phys. 31 (1983) 433–450) is extended to obtain exact, closed form expressions for the velocity, V, and dispersion (or diffusion constant), D, of discrete one-dimensional nearest-neighbor kinetic hopping models with arbitrary forwards and backwards periodic rate constants of general period N, which include, in addition: (i) direct jumps between sites, l=kN (k=0,±1,…) and sites (k±1)N; (ii) the possibility of finite, periodically arranged, but otherwise arbitrary, side branches at each site l; and (iii) an arbitrary (but periodic) probability rate of death at each site (which describes the motor protein detaching from the track). General expressions, following from previously developed theoretical principles, are given for the forces predicted by the corresponding extended motor models. The results are illustrated by plots of “randomness” (∝D/V) for more-or-less realistic N=2 molecular-motor models as a function of the load force, F.