Extending the absorbing boundary method to fit dwell-time distributions of molecular motors with complex kinetic pathways

Abstract

Dwell-time distributions, waiting-time distributions, and distributions of pause durations are widely reported for molecular motors based on single-molecule biophysical experiments. These distributions provide important information concerning the functional mechanisms of enzymes and their underlying kinetic and mechanical processes. We have extended the absorbing boundary method to simulate dwell-time distributions of complex kinetic schemes, which include cyclic, branching, and reverse transitions typically observed in molecular motors. This extended absorbing boundary method allows global fitting of dwell-time distributions for enzymes subject to different experimental conditions. We applied the extended absorbing boundary method to experimental dwell-time distributions of single-headed myosin V, and were able to use a single kinetic scheme to fit dwell-time distributions observed under different ligand concentrations and different directions of optical trap forces. The ability to use a single kinetic scheme to fit dwell-time distributions arising from a variety of experimental conditions is important for identifying a mechanochemical model of a molecular motor. This efficient method can be used to study dwell-time distributions for a broad class of molecular motors, including kinesin, RNA polymerase, helicase, F(1) ATPase, and to examine conformational dynamics of other enzymes such as ion channels.

Fig. 1.

Dwell events obtained from single-molecule experiments with myosin V and myosin VI. (a) Dwells in mechanical stepping of a double-headed myosin VI. (b) Dwells of actin binding and unbinding for single-headed myosin V. The portions of large oscillations correspond to the actin unbound states, whereas the portions of small fluctuations correspond to the actin bound states. (c) The movement of an actin filament (top arrow) driven by the power stroke of a single-headed myosin V attached to a bead. The arrows on the left of myosin indicate the cyclic behavior of binding, power stroke, unbinding, and recovery.

Fig. 2.

Statistics of dwell events. A time trajectory (top trace) is split into individual dwell events. The ensemble average of all these events is proportional to the population of states remaining in the dwell. The population of absorbing boundary states, shown in y, is obtained by subtracting the population remaining in dwells from the total population. Because dwell-time distributions reported in experiments are histograms of events binned within certain time intervals, it is important to also consider calculated results in time intervals. The number of events that exit a dwell during the time interval (t, t + dt) is equal to the number of events that exited dwells before time t + dt excluding those events that exited dwells before time t, or y(t + dt) − y(t). These are the number of dwell events with their dwell time between the time interval t and t + dt. In the limit, the dwell-time distributions at time t are proportional to dy/dt.

Fig. 3.

Agreement of dwell-time distributions results from the EAB method and other methods. (a) The results from the EAB method (red line), the analytical solution (blue dashed line) and the histogram obtained from the Monte Carlo simulation for Scheme 1. The curves from the EAB method and the analytical solution overlap almost perfectly. (b) The results from the EAB method and the histogram from the Monte Carlo simulation for Scheme 10.

Fig. 4.

Fitting of dwell-time distributions under different conditions using only one kinetic scheme with Boltzmann force factors and the predicted effects of applied forces on dwell-time distributions of single-headed myosin V. Experimental data (dots) are from Purcell et al. (2). (a) Fitting of dwell-time distributions in [ATP] = 10 μM and a forward force 2 pN. (b) Fitting of dwell-time distributions in [ATP] = 10 μM and a reverse force 2 pN. (c) Fitting of dwell-time distributions in [ATP] = 1 mM and a forward forces 2 pN. (d) Fitting of dwell-time distributions in [ATP] = 1 mM and a reverse forces 2 pN. (e) Fitting of dwell-time distributions in [ATP] = 1 mM, [ADP] = 1 mM, and a forward force 2 pN. (f) Fitting of dwell-time distributions in [ATP] = 1 mM, [ADP] = 1 mM, and a reverse force 2 pN. (g) Predictions of dwell-time distributions under different reverse forces when [ATP] is 10 μM.