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-to-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 ${\tau }_{+}$ and ${\tau }_{-}$ prior to forward and back steps, respectively, are normally unequal although the dwell times ${\tau }_{++}$ and ${\tau }_{--}$ between 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.

Figure 1

(Color online) Schematic graphs illustrating why the full-cycle interpretation is not adequate for describing stepping data: Plot (a) depicts a hypothetical time series of forward steps for an $(N=4)$ cycle with $M=2$. The (0,1)’s indicate that the states ${\left[0\right]}_l$ and ${\left(1\right)}_l$ are both certain to occur at least once between the pairs of states ${\left(3\right)}_{l-1}$ and ${\left(2\right)}_l$. (See the text.) Then (b) represents a similar time series but with one major back step. The question marks between the forward-back and back-forward steps indicate that one does not know if the motor has ever passed through the states ${\left[0\right]}_l$ and/or ${\left(1\right)}_l$ in these intervals. Finally, (c) and (d) depict two equivalent time series for a motor with $N=3$ and $M=1$ but with different noise levels. The first plot allows one to identify only major forward and backward transitions at times $t_k$ (marked on the axis) while the states ${\left[0\right]}_l$ and corresponding substeps are hidden in the noise. However, the second plot reveals all the transitions and substeps, so that the $t_k^0$, marking the beginning (or end) of each full cycle, can be determined. These schematic examples demonstrate that detectable transitions do not necessarily correspond to a full biochemical cycle so that a proper statistical analysis must take account of substeps hidden in the noise. Notice, indeed, that the three major transitions identified as forward and back steps at times $t_3$, $t_4$, and $t_5$ (on the left) are associated with only a single complete cycle from $t_2^0$ to $t_3^0$.

Figure 2

(Color online) Schematic plots illustrating substeps and their counting rules (see text) in the $N=4$ case with $M=3$ for (++), (+−), (−+), and (−−) stepping intervals. Thus, for example, from the first plot one confirms $n_{0,+}^{++}=n_{1,-}^{++}+1$ and, correspondingly, that the motor visited the state [0] three times in this realization; and, likewise, $n_{1,+}^{++}+n_{1,-}^{++}=n_{0,+}^{++}+n_{2,-}^{++}$ is, here, equal to 4 so that, correspondingly, the motor spent time in the substate (1) on four occasions before undergoing the forward major transition $\left(3\right)\to (4\equiv 0)$. Similarly, the other plots serve to establish the frequency relations (76, 77, 78, 79, 80, 81, 82, 83, 84, 85).

Figure 3

(Color online) Plots illustrating different dependencies on the normalized force, $F∕\mid F_S\mid$, of (a) the normalized velocity $V\left(F\right)∕V\left(0\right)$; (b) the forward splitting fraction ${\pi }_{+}$; (c) the logarithm of the splitting fraction ratio $({\pi }_{+}∕{\pi }_{-})$; and (d) the ratio of mean prior backward to mean total dwell time, ${\tau }_{-}\left(F\right)∕\tau \left(F\right)$. The individual plots, labeled (1, 2, 3, 4, 5), correspond to ($N=2$, $M=1$) models under scalar loading, $\mathbf{F}=(F,0,0)$, with the selected parameter values presented in Table .

Figure 4

Schematic depiction of individual biochemical states, $\left(i\right)$, organized into mechanical levels, $A$, $B$,$\dots$, $J-1$, $J$,$\dots$ when there are $K>1$ detectable substeps or major transitions with substep $J$ going forward from state $\left(M_J\right)$ to state $(M_J+1)$.