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Back-stepping, hidden substeps, and conditional dwell times in molecular motors

Processive molecular motors take more-or-less uniformly sized steps, along spatially periodic tracks, mostly forwards but increasingly backwards under loads. Experimentally, the major steps can be resolved clearly within the noise but one knows biochemically that one or more mechanochemical substeps remain hidden in each enzymatic cycle. In order to properly interpret experimental data for back/forward step ratios, mean conditional step-to-step dwell times, etc., a first-passage analysis has been developed that takes account of hidden substeps in $N$-state sequential models. The explicit, general results differ significantly from previous treatments that identify the observed steps with complete mechanochemical cycles; e.g., the mean dwell times $τ_+$ and $τ_-$ prior to forward and back steps, respectively, are normally {\it unequal} although the dwell times $τ_{++}$ and $τ_{--}$ between {\it successive} forward and back steps are equal. Illustrative (N=2)-state examples display a wide range of behavior. The formulation extends to the case of two or more detectable transitions in a multistate cycle with hidden substeps.