A reverse stroke characterizes the force generation of cardiac myofilaments, leading to an understanding of heart function

Significance

Force measurements on cardiac myosin molecules and polymerized myofilaments demonstrated that single molecules transition back and forth between three molecular conformations when bound to an actin filament that is experiencing a resistive load, as would be the case in a contracting heart. These distinct positions correspond to a pre–power stroke state and two post–power stroke states. The experimental data are coupled with modeling to predict ensemble behavior and collectively demonstrate that the reversal stroke is a key feature of cardiac muscle that allows for the maintenance of the high force production necessary for cardiac contraction and rapid cardiac relaxation in the heart. Thus, the study provides evidence that the reverse stroke may be a critical feature for normal cardiac function.

Abstract

Changes in the molecular properties of cardiac myosin strongly affect the interactions of myosin with actin that result in cardiac contraction and relaxation. However, it remains unclear how myosin molecules work together in cardiac myofilaments and which properties of the individual myosin molecules impact force production to drive cardiac contractility. Here, we measured the force production of cardiac myofilaments using optical tweezers. The measurements revealed that stepwise force generation was associated with a higher frequency of backward steps at lower loads and higher stall forces than those of fast skeletal myofilaments. To understand these unique collective behaviors of cardiac myosin, the dynamic responses of single cardiac and fast skeletal myosin molecules, interacting with actin filaments, were evaluated under load. The cardiac myosin molecules switched among three distinct conformational positions, ranging from pre– to post–power stroke positions, in 1 mM ADP and 0 to 10 mM phosphate solution. In contrast to cardiac myosin, fast skeletal myosin stayed primarily in the post–power stroke position, suggesting that cardiac myosin executes the reverse stroke more frequently than fast skeletal myosin. To elucidate how the reverse stroke affects the force production of myofilaments and possibly heart function, a simulation model was developed that combines the results from the single-molecule and myofilament experiments. The results of this model suggest that the reversal of the cardiac myosin power stroke may be key to characterizing the force output of cardiac myosin ensembles and possibly to facilitating heart contractions.

Cardiac β-myosin is a mechanoenzyme that displaces actin filaments to produce contraction forces of the heart, by using the free energy of ATP hydrolysis to undergo a power stroke. Since more than 300 pathogenic hypertrophic cardiomyopathy (HCM) mutations in β-cardiac myosin (1) have been reported, numerous single-molecule studies have been conducted to investigate the biophysical and pathophysiological aspects of cardiac myosin and various HCM mutant myosin molecules (2, 3). In particular, the size of the power stroke (i.e., the displacement caused by a conformational change in the myosin head during force generation), the attachment time to actin filaments and the corresponding load dependencies, and dose-dependent effects of small molecules (e.g., omecamtiv mecarbil) have been rigorously investigated using single-molecule approaches (3–7). In addition, the reverse stroke, which is the power stroke performed in reverse, of force-generating myosin has been identified at the level of a single molecule exclusively from cardiac myosin (6) and myosin V (8). Despite these results obtained from different research groups, it is still unclear how the properties of a single cardiac myosin molecule affect the functions of the cardiac myosin ensemble and heart, where cardiac myosin forms thick filaments and collectively generates force.

A comprehensive understanding of the molecular properties of cardiac myosin underlying its collective behavior is important to gain insight into the molecular mechanisms of heart contraction. Forces generated by ensembles of cardiac myosin molecules have been measured in vitro using motility assay–based approaches (9–11), but these assays lack the native spatial organization of the myosin molecules within myofilaments. The native geometry of myosin molecules within myofilaments is retained in single muscle fiber studies (12), but the ensemble properties obtained from structure complex limits quantification of the essential properties of single-molecule interactions. To overcome this problem, we have examined the dynamics of short synthetic fast skeletal myofilaments, composed of 4 to 20 myosin molecules, interacting with a single actin filament, with high spatiotemporal resolution. This allowed us to measure the contribution of individual myosin molecules to the ensemble force generated by myofilaments, closer to that which would occur in vivo. We reported that force generation among fast skeletal myosin molecules is synchronized, and in combination with simulations, this appears to be an essential property for efficient muscle contractions (13, 14). Despite the similarity of amino acid sequences (15) and mechanochemical pathways (6, 13) (Fig. 1A), several studies have reported that the mechanochemical properties of cardiac myosin differ from those of fast skeletal myosin in single-molecule (5, 6, 16), single muscle fiber (17), and bulk assays (2). These results imply that cardiac myosin has unique molecular and ensemble properties, as compared to fast skeletal myosin.

Fig. 1.

Measurement of force generated by cardiac myosin-rod cofilaments. (A) Kinetic scheme for the myosin and actin interaction cycle. A, M, T, D, and P represent actin, myosin, ATP, ADP, and inorganic phosphate, respectively. The mechanochemical transitions in the shaded region represent a major pathway based on a kinetic scheme designed for fast skeletal myosins (13). In chemical step number three, the AMDP state (Inset) was conceptually included but assumed to be negligible in the transition between the MDP and ADP** states ($k_{+3}$ and $k_{-3}$) since the transition between MDP and AMDP states is known to reach equilibrium rapidly. The scheme for cardiac myosins was modified by adding the MD state to represent the results from our single-molecule measurement experiments. The steps for phosphate release/binding are depicted between the AMDP and AMD** states based on the results from our single-molecule measurements. (B) Schematic diagram of the optical tweezer assay on a cardiac myosin-rod cofilament. (C) Time course of bead displacement/force generated by cardiac myosins at 1 mM ATP. (Inset) Corresponding curve for fast skeletal myosin with similar numbers of interacting molecules from our previous study (13). (D) Stepwise displacements of actin filaments. Arrowheads indicate observed backward steps. The black solid lines represent steps detected by the step-finding algorithm (49). (E and G) Mean forward and backward step sizes of cardiac and fast skeletal (13) myofilaments as a function of the load as obtained from experiments at 1 mM (3,452 steps from 4 trials) and 10 μM ATP (850 steps from 11 trials). The diamonds show the step size corresponding to the smaller peak of Gaussian fits of the step size histogram at the lowest load range, and the corresponding circles show the sum of the two step sizes corresponding to the higher peaks of Gaussian fits (SI Appendix, Fig. S2 A and D). (F and H) Mean dwell times for forward and backward steps as a function of the load at 1 mM and 10 μM ATP. (I) Relationship between the stepping ratio and load for cardiac and fast skeletal (13) myofilaments. The shaded regions indicate the 95% CIs for the regression lines. These regions (green and gray) do not overlap at either the ordinates or abscissas, supporting the ANOVA test results showing that the stepping ratios at no load and the stall forces are significantly different between cardiac and skeletal myosins at 1 mM ATP. Error bars represent the SEM in E through I; most of the error bars are smaller than the symbols. (E, F, and I) Reprinted from ref. 13, which is licensed under CC BY 4.0.

Here, we applied a comprehensive approach to reveal unique molecular properties of cardiac myosin that underlie the collective behaviors of the cardiac myosin ensemble and potentially contractile functions in the heart. We used an optical-tweezer–based assay to measure the forces generated by both single cardiac myosin molecules and myofilaments with ∼15 molecules, with microsecond temporal resolution, and combined these data with simulations. We demonstrated that cardiac myosin molecules interacting with a single actin filament have a greater propensity for backward steps at low loads and exhibit higher stall forces than those of fast skeletal myofilaments. Single-molecule experiments further demonstrated that cardiac myosin undergo intermittent back-and-forth displacements under load. These displacements, where characterized by three distinct distances, occur between the pre– and post–power stroke positions, consistent with a two-step power stroke mechanism, while fast skeletal myosin stayed primarily in one position. A Monte Carlo–based mechanochemical simulation model was developed based on these results obtained from single-molecule experiments and computed force generation of actomyosin interactions in single cardiac myofilaments and sarcomeres. Combining these experimental results with results from simulations showed that the reverse stroke is key to enhancing the force output of cardiac myosin ensembles and possibly facilitating efficient heart contractions, contributing to stable systolic pressure output, rapid relaxation of end-systolic pressure, and low ATP consumption rate.

Results

Frequent Backward Steps and High Stall Forces of Cardiac Myosin Ensembles.

A mean myosin-rod cofilament length of 960 ± 120 nm (mean ± SD, n = 4 and 11 for 1 mM and 10 µM ATP, respectively) was selected for force measurements (Materials and Methods) such that 15 ± 2 myosin molecules were estimated to interact with a single actin filament (Fig. 1B). In this experiment, experimental data from cardiac myofilaments were compared with those for fast skeletal myofilaments observed in our previous study (13). Cardiac myofilaments continuously displaced actin filaments and generated forces beyond 50 pN at 1 mM (Fig. 1C) and 10 µM ATP (SI Appendix, Fig. S1), while the forces of fast skeletal myofilaments were ∼35 pN (Fig. 1C) obtained from our previous work (13), even at a similar number (∼17) of interacting molecules. Bead displacements of cardiac myofilament showed stepwise behavior (Fig. 1D), as observed from fast skeletal myosin filaments (13, 14). Histograms of the observed forward step size at low loads were fit well with two Gaussian functions with peaks at 5.0 and 9.0 nm at 1 mM (Fig. 1E and SI Appendix, Fig. S2A) and 5.5 and 8.0 nm at 10 µM ATP (Fig. 1G and SI Appendix, Fig. S2D), suggesting a first power stroke of 5 to 6 nm followed by a second power stroke of 3 to 4 nm (13). In contrast, histograms of the forward step size at moderate-to-high loads were fit well with a single Gaussian function, and their peak positions gradually decreased with increasing load at both 1 mM and 10 µM ATP (Fig. 1 E and G and SI Appendix, Fig. S2 B and E). The stepping ratio, which is the ratio between the number of forward and backward steps at no load, was lower for 1 mM ATP (7:1) than for 10 µM ATP (37:1) (Fig. 1I), indicating that backward steps were more frequent at 1 mM ATP. The corresponding stepping ratio was 20:1 for fast skeletal myofilaments at 1 mM ATP (13) (Fig. 1I), suggesting that backward steps occurred more frequently for cardiac myofilaments, while the stall force estimated from the abscissa of the fitting curves was higher for cardiac myofilaments (55 pN) than for fast skeletal myofilaments (36 pN) (13) (Fig. 1I). Therefore, frequent backward steps at low loads and high stall forces are unique properties for ensembles of cardiac myosin.

Direct Observation of the Reverse Stroke in Single Cardiac Myosin Molecules.

The observed backward steps could be potentially caused by detachment and/or reverse strokes of force-generating myosin. Therefore, we performed single-molecule experiments (Materials and Methods) to elucidate the dynamic behaviors of the myosin head under various loads. Cardiac or fast skeletal myosin and rods were mixed at a molar ratio of 1:1,500, guaranteeing that either zero or one myosin molecule could interact with actin filaments. To further ensure interaction with a single myosin molecule after measurements, beads were manipulated upwards to check if single unbinding events were observed. Gelsolin-conjugated beads were attached to the barbed ends of actin filaments so that opposing loads could be applied to myosin by displacing the position of the optical tweezers (Fig. 2A). To observe the dynamic responses of strongly bound myosin heads over a certain period of time, bead positions were detected during loading (7 to 18 pN) in the absence of ATP but with 1 mM ADP and 0, 1, 5, or 10 mM phosphate. The bead positions appeared to switch occasionally between three discrete positions for cardiac myosin (Fig. 2 B and C) but showed no such displacements in the absence of nucleotides (Fig. 2C). The histograms of bead positions obtained from experiments under all conditions clearly showed three distinctive peaks for cardiac myosin, and the populations at these peaks differed at different loading ranges (Fig. 2D), while fast skeletal myosin stayed predominantly in a single conformational position (Fig. 2D). These conformational positions potentially indicated three force-generating states, that is, the post–second power stroke (AMD), the post–first power stroke (AMD*), and the pre–power stroke (AMD**) states (Fig. 2D). Myosin initially resides in the AMD position (Fig. 2C) before switching to other conformational positions, indicating that myosin preferentially positioned in the AMD state in the presence of ADP at no load. When actomyosin interactions were disrupted, cardiac myosins detached mainly in the AMD** state, while fast skeletal myosins detached mainly in the AMD state (Fig. 2C, white arrowheads), implying that different main detachment pathways exist in cardiac and fast skeletal myosin.

Fig. 2.

Direct observation of the power and reverse strokes of single cardiac myosins. (A) Schematic diagram for detecting the dynamics of single cardiac myosins by using optical tweezers. (B) Time course of bead displacement at a load of ∼8 pN. (Inset) Expanded plot of displacement curves with steps (red) detected by using the vbFRET algorithm (18). (C) Typical time course and histogram of bead displacement at a load of ∼11 pN. (D) Cumulative histogram of bead displacement obtained at 1 mM ADP for cardiac myosin at loads of 6 to 8.5 pN (9 trials) in (a), 8.5 to 12 pN (8 trials) in (b), and 12 to 17 pN (9 trials) in (c) and for fast skeletal myosins at loads of ∼8.5 to 12 pN (15 trials) in (d). The black solid lines are triple Gaussian functions. The pink, green, and blue arrowheads indicate the peak positions of these functions, which correspond to three force-generating states, that is, the AMD, AMD*, and AMD** states (Fig. 1A). A, M, and D represent actin, myosin, and ADP, respectively. Skeletal myosins occasionally stayed in the AMD* state, as shown in C (gray); however, no Gaussian function can fit this position in D, d. In each histogram, the peak positions for the AMD state were defined as 0 nm.

The distances between two positions represent the size of the power/reverse stroke. Here, we applied the vbFRET algorithm (18) to detect the three stable conformational positions of the cardiac myosin head (Fig. 2B, red line). However, for fast skeletal myosin, it was difficult to identify individual steps due to the rapid reactions buried in experimental noise caused by the compliance of the protein–bead complex (SI Appendix, Fig. S3). The mean sizes of the first power stroke and reverse stroke were 6 and −6 nm, respectively, and those of the second power stroke and reverse stroke were 3 and −3 nm, respectively (Fig. 3A). The stroke sizes were nearly independent of the load (Figs. 2D and 3A) and [Pi] (Fig. 4G). The power or reverse stroke rates calculated from the dwell times (Materials and Methods and Fig. 3 B and C) decreased or increased with increasing load (Fig. 3 D and E), consistent with results from previous studies (12, 13, 19, 20). Intriguingly, the rates of the second reverse stroke were much more sensitive to loading than those of the first reverse stroke (Fig. 3E). Our single-molecule experiments demonstrated that cardiac myosin switched between three conformational positions by power strokes/reverse strokes, while fast skeletal myosin stayed primarily in the post–power stroke position, suggesting that the reverse stroke is a unique feature of cardiac myosin.

Fig. 3.

Step size, dwell time, and power/reverse stroke rate of single cardiac myosins. (A) Mean power/reverse stroke sizes as a function of the load obtained from single-molecule experiments (3,156 steps from 81 trials). (B and C) Mean dwell times as a function of the load for the first/second power stroke in B and the first/second reverse stroke in C. (D and E) Mean transition rates calculated for the first/second power stroke in D and the first/second reverse stroke in E. All the rates except for the first power stroke rate were independent of Pi concentration (Fig. 4) and thus calculated from all the data from the experiments at 1 mM ADP and 0, 1, 5, and 10 mM Pi, while the first power stroke rate was calculated from the data at 1 mM ADP only. Error bars represent the SEM.

Fig. 4.

Pi dependency of the step size, dwell time, and transition rate obtained from single-molecule experiments. (A–C) Mean dwell times as a function of Pi concentration for the first power/reverse stroke in A, the second power/reverse stroke in B, and detachment from AMD** in C (0 mM Pi: 867 steps from 15 trials; 1 mM Pi: 771 steps from 15 trials; 5 mM Pi: 680 steps from 15 trials; and 10 mM Pi: 538 steps from 16 trials). (D–F) Mean transition rates as a function of Pi concentration for the first power/recovery stroke in D, the second power/recovery stroke in E, and detachment from AMD** in F. The orange line in F is a linear fit $k_{detach}=k_{+7}+k_{-3}^0\times \left[Pi\right]$, where $k_{-3}^0$ is the second-order rate constant for Pi binding to transition from AMD** to the MDP state via the AMDP state, and $k_{+7}$ is the load-dependent detachment rate from AMD** to the MD state (Fig. 1A). (G) Mean step sizes as a function of Pi concentration. All data at loads of 8 to 14 pN were combined for these analyses. Error bars represent the SEM.

Effect of the Reverse Stroke on Force Generation of Cardiac Myosin Ensembles.

To understand how the reverse stroke affects the force generation of cardiac myosin ensembles, we developed a Monte Carlo–based simulation model based on a fast skeletal myosin model (13). A cycle of actomyosin interaction was divided into seven mechanochemical states: four force-generating states (AMD**, AMD*, AMD, and AM [actin–myosin]) and three detachment states (MT [myosin–ATP], MDP [myosin–ADPPi], and MD [Myosin–ADP]) (Materials and Methods, Figs. 1A and 5). Unlike our previous model for fast skeletal myosin (13), the detachment pathway from the AMD** state to the MD state was incorporated (SI Appendix, Implementation of the MD state in the cardiac myosin model) for two reasons: first, cardiac myosins primarily disrupted actin filaments in the AMD** state in our single-molecule experiments (Fig. 2C, arrowhead); and second, sudden drops in tension observed during the rising phase of the force curve in cardiac myofilaments (Fig. 1C) cannot be reproduced without the detachment pathway to the MD state (SI Appendix, Implementation of the MD state in the cardiac myosin model and Fig. S4). The power/reverse stroke rate functions were determined empirically by fitting model outputs to the stepping ratio–load relationships obtained from myofilament force measurements (Fig. 1I). Then, these rate functions were found to be reasonably consistent with the hypothetical transition rates calculated from single-molecule experiments (Fig. 5A), for which the load is assumed to be half of the measured load. Other rates were taken from previous studies (2, 5, 13) or estimated/determined empirically by fitting to experimental data (Materials and Methods and Table 1). The sizes of the first and second power/reverse strokes were 6 and 3 nm, respectively, based on the results from both single-molecule experiments and myofilament force measurements (Figs. 1 E and G and 3A). The high reverse stroke (HRS) rate model, based on these parameter adjustments, outstandingly reproduced the force generation and stepping behaviors observed in cardiac myofilaments (Fig. 6). To accentuate the effect of the second reverse stroke, which was found to be more sensitive to loading than the first reverse stroke (Fig. 3E), on force generation, only the second reverse stroke rate was modified to be less sensitive to loading in the low reverse stroke (LRS) rate model by setting a higher energy gap between the AMD* and AMD states (Materials and Methods and Fig. 5 A and B). Intriguingly, in contrast to the HRS rate model associated with frequent backward steps and high stall forces as observed from cardiac myofilaments, the LRS rate model generated skeletal myosin-like forces and stepping behaviors despite much slower reaction rates than those of fast skeletal myosin (Fig. 6 A and G).

Fig. 5.

Simulation model for cardiac myosin–actin interaction. (A) Load-dependent rate functions for transitions between two states, such as the AMD** and AMD* states in (a), AMD* and AMD states in (b) and AMD and AM states $\left(k_{+6}\right)$ and AMD** and MD states $\left(k_{+7}\right)$ in (c). These rate functions, except for $k_{-5}$, are common for both HRS and LRS rate models. Open symbols indicate the hypothetical transition rates calculated from single-molecule experiments (Fig. 3 D and E), for which the loads are assumed to be half of the measured loads. Error bars represent the SEM. (B) Total energy profiles of AMD**, AMD*, and AMD states calculated from the nonlinear elasticity of the myosin head as a function of the strain in the myosin head. The total energy curve of the AMD state for the LRS rate model was shifted down by 4.8 k_BT. (C) Schematic diagram of simulation for 15 myosin molecules interacting with a single actin filament connected to the spring, with the spring constant of 0.28 pN ⋅ nm⁻¹ used in force measurements.

Table 1.

Parameter values in the simulation model

	Parameter values	Reference values and sources
k+1	1.5 × 106 M−1· s−1	1.6 × 106 M−1· s−1 (2)
k+2	12.8 s−1	k+2 + k-2 = 14 s−1 (2) and
k-2	1.28 s−1	K2 = k+2/k-2 = 10
k+3	18 s−1	60 s−1 (13)
$k_{-3}^0$	8.0 × 106 M−1· s−1	Calculated by fitting to SM data (Fig. 4F)
$l_4$ (AMD**→AMD*)	6 nm	Measured from SM (Fig. 3A)
$k_4^0$	3.0 × 106 s−1	Determined by fitting to MF data
$l_5$ (AMD*→AMD)	3 nm	Measured from SM (Fig. 3A)
$k_5^0$	2.1 × 104 s−1	The value of $k_5$ is calculated to be 900 s−1 at 1.5 pN in HRS model, while it was ∼900 s−1 at 1.5 pN from SM (6)
$\mathrm{\Delta }{U}_{AMD\ast \ast }$	28.5 kBT	28.5 kBT (13)
$\mathrm{\Delta }{U}_{AMD\ast }$	18.5 kBT	18.2 kBT (13)
$\mathrm{\Delta }{U}_{AMD}$ (HRS)	15.0 kBT	Determined by fitting to MF data†
$\mathrm{\Delta }{U}_{AMD}$ (LRS)	10.0 kBT	9.8 kBT (13)
$k_{+6}^0$	80 s−1	71 s−1 (5)
d+6	0.8 nm	1.0 nm (5)
$k_{+7}^0$	0.1 s−1	Determined by fitting to MF data
d+7	2.0 nm	1.8 nm (42)
k+8	1,000 s−1	Determined by fitting to MF data

SM: single-molecule experiment; MF: myofilament force measurement.

^†$\mathrm{\Delta }{U}_{AMD\text{*}}-\mathrm{\Delta }{U}_{AMD}=3.5k_{B}T$ for the HRS rate model and 1.3$k_{B}T$ for human cardiac myosin obtained from the equilibrium constant (2).

Fig. 6.

Simulation results for cardiac myofilaments. (A) Time course of actin displacement for the HRS and LRS rate models. (B) Stepwise displacements computed by these models. The red solid lines represent steps detected by the step-finding algorithm (49). Arrowheads indicate observed backward steps. (C and E) Mean forward and backward step sizes of the HRS and LRS rate models as a function of the load at 1 mM ATP in C (HRS rate model: 2,902 steps from seven trials; LRS rate model: 2,856 steps from seven trials) and at 10 μM ATP in E (HRS rate model: 1,364 steps from 10 trials). (D and F) Mean dwell times of the HRS rate model and LRS rate model for forward and backward steps as a function of the load at 1 mM ATP in D and at 10 μM ATP in F. (G) Relationship between stepping ratio and load for the HRS rate model and the LRS rate model at 1 mM ATP and the HRS rate model at 10 μM ATP. (H and I) The duty ratio was obtained as a ratio of total binding time to ATP hydrolysis cycle time calculated at each cycle of ATP hydrolysis for each motor and accounts for the load-dependent kinetics and the effect of other force-generating motors on its kinetics in an ensemble of 15 myosin motors (51, 52) in H and load per myosin in I as a function of the load for the HRS rate model and the LRS rate model at 1 mM ATP. Error bars represent the SEM in C through I; most of the error bars are smaller than the symbols.

Discussion

Measurements of single myosin dynamics at 1 mM ADP and 0 to 10 mM Pi showed that cardiac myosin heads preferentially position among three discrete conformational positions (Fig. 2). These states potentially correspond to the AMD, AMD* and AMD** states (Figs. 1A and 2D), being similar to the multiple ADP states suggested by previous studies (2, 5, 21–25). The bead movement distance associated with the transition between the AMD and AMD* states represents the size of the second power stroke or reverse stroke (i.e., 3 nm) and that associated with the transition between the AMD* and AMD** states is the size of the first power stroke or reverse stroke (i.e., 6 nm) (Fig. 2D). One might argue that back-and-forth displacements of a single myosin molecule observed in response to a load are potentially caused by detachment/reattachment of one of the myosin heads. In that case, backward displacements would be caused by stretching of one myosin head that remains bound to actin after detachment of another head. Accordingly, the sizes of the backward displacements from the AMD state would be proportional to the additional load exerted on a bound myosin head. However, our data showed nearly constant step sizes of 3 nm (Fig. 3A), eliminating this possibility.

The step size distribution obtained from force measurements of myofilaments showed two peaks at 5 to 6 and 8 to 9 nm for low loads (Fig. 1 E and G), suggesting that the first power stroke size of 5 to 6 nm is followed by a second power stroke size of 3 nm (13). The sizes of the power/reverse strokes were independent of the load in the single-molecule experiments (Fig. 3A), while the step sizes obtained from myofilaments decreased with increasing load (Fig. 1 E and G) (6, 13, 14). These discrepancies can be explained as follows. In our single-molecule experiments, the elastic portion of the myosin head is prestretched and kept stretched throughout measurements and thus does not limit the size of the power stroke, while the full size of the power stroke is taken up progressively by stretching of the elastic portion as the load increases in myofilament experiments (6, 13, 14) (SI Appendix, Fig. S5). Therefore, the sizes and load dependency of the first and second power strokes obtained from the two independent experiments are similar.

The first and second power stroke sizes have been measured by optical tweezers (4, 5, 13, 21), the quantum dot assay (26), and atomic force microscopy (27). The first power stroke size was reported as 4.7 to 5.2 nm, which is similar to our estimated first step size of 6 nm. For the second power stroke size, several studies have reported sizes of 2.5 to 4.0 nm (4, 13, 26, 27), consistent with our estimated value of 3 nm, while other studies reported smaller sizes of 1.3 to 1.9 nm (5, 21). The difference in the reports of the second power stroke size is likely due to differences in experimental setup and sample preparation. In our experiments, myosins embedded in myofilaments were suitably oriented along the longitudinal direction of myofilament and actin filaments, while myosins were oriented randomly in the optical tweezer studies (5, 21), which might result in the power stroke size being underestimated (28, 29). Moreover, Capitanio’s group measured the size of power stroke for myosin subfragment 1 (21), potentially associated with a smaller power stroke than that of two-headed myosins (30). Even with consideration for discrepancies in stroke size, the previous findings from other groups support our assumption of three conformational positions accompanied by two steps of power stroke observed in single-molecule measurements.

The rates of the first and second power/reverse strokes calculated from our single-molecule measurements were found to be load dependent (Fig. 3 D and E), consistent with previous models (13, 19, 20, 22, 31, 32), as the power or reverse stroke rates decreased or increased with increasing load. In addition, the relative amplitudes of the population at three myosin conformational positions changed at different loading ranges (Fig. 2D), suggesting changes in the preferential conformations of the myosin head at different loads. These results further suggest that the energy gap between the AMD and AMD* states is high enough to prevent the execution of the reverse stroke at low loads; however, this gap is progressively reduced as the load increases. Therefore, our single-molecule experiments successfully obtained direct observations of two steps of both power and reverse strokes at the single-molecule level and enabled evaluation of the corresponding load-dependent kinetics, although these experiments were conducted in the absence of ATP (1 mM ADP + 0 to 10 mM Pi).

It is important to investigate the temporal relationship between Pi release and the force generation of myosin to understand the mechanochemical cycle of actomyosin interaction. It has been reported that the first power/reverse stroke may be executed upon (32–35) or prior to Pi release/binding (6, 36). Moreover, it has been suggested that Pi release is not tightly coupled with the working stroke (37, 38) and may be slow in the pre–power stroke state and rapid in the postrigor state (39). Thus, the appropriate method to determine the timing of Pi release relative to the power stroke event remains controversial. A striking result of our single-molecule measurements is that the first power/reverse stroke events were observed in the presence of ADP but the absence of ATP and Pi (Fig. 2), implying that Pi release may occur prior to power stroke execution. Although the power/reverse stroke rates were almost independent of [Pi] (Fig. 4 D and E), the detachment events were strongly influenced by [Pi] (Fig. 4F) and were mainly detected from the pre–power stroke AMD** position (Fig. 2C) in the single-molecule experiments. These results also support our idea that Pi release may occur prior to the first power stroke execution. In addition, the stepping ratio at no load obtained from cardiac myofilament force measurements was 7.6 at 1 mM ATP and 10 mM Pi (SI Appendix, Fig. S6) and, similarly, 7.7 at 1 mM ATP and 0 Pi (Fig. 1I), suggesting that Pi does not influence the kinetics of the first power/reverse stroke in the presence of ATP at low loads. Taken together, our single-molecule and myofilament force measurements suggest that Pi release/binding may occur prior to the first power/reverse stroke event (Fig. 1A).

In contrast to the cardiac myosin histograms, which showed peaks at three conformational positions, a histogram of fast skeletal myosin showed one prominent peak at the post–second power stroke AMD position with one minor peak at the pre–first power stroke AMD** position (Fig. 2D, gray). One might argue that the dynamics of single fast skeletal myosin were not fully detected because of the limited temporal resolution of our measurements (∼80 µs) and the vbFRET algorithm (SI Appendix, Fig. S3). However, according to single muscle fiber measurements, the power stroke rate is 3 to 4 times faster for fast skeletal myosin than for cardiac myosin (12). Thus, the dwell times for the AMD state or AMD* state of fast skeletal myosin might be ∼300 µs under similar loading conditions. Hence, the temporal resolution of our system is sufficient to collect distributions of fast skeletal myosin dynamics, although individual displacements of fast skeletal myosin are not detectable due to the limited step detection ability of our measurements (SI Appendix, Fig. S3). Bulk kinetics measurements (2, 23) indicated a lower energy gap between the AMD and AMD* states for cardiac myosin than fast skeletal myosin, implying that cardiac myosin has a higher chance of executing the second reverse stroke. An ultrafast force clamp system with tens of microseconds resolution showed no clear sign of a reverse stroke in single skeletal myosin (16), while reverse stroke events were observed in single cardiac myosin by using the same system (6). A single muscle fiber mechanical measurement also showed a minimal change in isometric tension with the same skeletal muscle fiber stiffness with the addition of ADP to rigor solution (40). Recently, Swank’s group demonstrated an enhancement of stretch activation in response to a rapid active stretch at higher [Pi] in slow skeletal muscle fibers but not in fast skeletal muscle fibers (41). They proposed that the reverse stroke of slow skeletal myosin enables quick rebinding to actin and enhances force production, while fast skeletal myosin detaches from actin. The myosin heavy chain gene (MYH7) is the same for the slow skeletal and β-cardiac myosin heavy chains, and thus, the reverse stroke may be essential for slow skeletal and cardiac myosin but not for fast skeletal myosin. Combination of our data with previous findings shows that the reverse stroke is likely an essential feature for cardiac myosin but not for fast skeletal myosin.

To understand how the reverse stroke affects the force generation of cardiac myosin ensembles, we developed a simulation model (Materials and Methods and Figs. 1A and 5). The power and reverse stroke rate functions in the HRS rate model, with some parameters ($\mathrm{\Delta }{U}_{AMD}$, $k_4^0$) determined by fitting to data from myofilament force measurements, were found to be consistent with the hypothetical transition rates calculated from single-molecule experiments (Fig. 5A), for which the load is assumed to be half of the measured load. This adjustment may be reasonable if both myosin heads are assumed to bind to actin filaments and bear nearly equal loads in solution with 1 mM ADP and 0 to 10 mM Pi, as observed in skeletal heavy meromyosin and subfragment-1 (42), while one of the two heads primarily interacts with an actin filament in the presence of ATP (43). We also performed a sensitivity analysis of critical model parameters, such as the power stroke size, stiffness of the myosin head, and energy gap between force-generating states on force generation in the HRS model (SI Appendix, Effect of the model parameters on the model outputs and Figs. S7–S9), concluding that the sets of parameters used in the HRS model are appropriate to reproduce our experimental data obtained from our single-molecule and myofilament experiments. The mechanical work and efficiency per myosin molecule calculated in the HRS rate model were also found to be within a range of those estimated from single muscle fiber studies (44) (SI Appendix, Mechanical work and efficiency of single cardiac myosin and Fig. S10). To investigate how the load-dependent second reverse stroke rate, which is more prominent for cardiac myosin than skeletal myosin (Fig. 5A), affects the force generation, we ran the LRS rate model, in which the second reverse stroke rate is less sensitive to loads. The model generated fast skeletal myosin-like forces and stepping ratios (Fig. 6 A and G), suggesting that the highly load-dependent second reverse stroke rate is a key property to achieve the frequent backward steps at low loads and high stall forces observed in cardiac myofilaments.

The molecular mechanism responsible for the increased stall force in the HRS rate model is that the high rate of the second reverse stroke shifts the myosin population from the AMD state to the AMD* state (SI Appendix, Fig. S11), limiting the transition to the release of ADP and subsequent detachment. Consequently, a higher number of myosin molecules can remain attached to actin filaments and enhance the duty ratio (Fig. 6H) and thus the force outputs. In addition, the increase in attachment time in response to a load (5) was found to linearly increase the duty ratio up to 50% in the HRS rate model, while that of the LRS rate model increased up to 30% (Fig. 6H), as consistently observed in skeletal myosin (45, 46). Such a difference in the duty ratio between the two models has a significant effect on the force per myosin head, which is less than 6 pN (13, 45) at a load of 30 pN in the HRS rate model but exceeds 7 pN in the LRS rate model (Fig. 6I). Therefore, the results of the HRS rate model may suggest that the higher stall force observed for cardiac myofilaments (Fig. 1I) can be attributed to the presence of myosin molecules in the AMD* state reinforced by frequent reverse stroke events.

We observed a load-independent dwell time for cardiac myosin but not for fast skeletal myosin (Fig. 1F), consistent with the HRS and LRS models (Fig. 6D). Thus, the difference in the load dependency of the dwell time between cardiac and fast skeletal myosins can be attributed to their characteristics. Under the HRS rate model, both the first and the second load-dependent reverse stroke rates led to frequent reverse stroke events at high loads (Fig. 5B), frequently associated with detachments from the AMD** state. Thus, this reaction counteracts the slowing effect of the forward transition caused by a load-dependent power stroke and ADP release rates (5, 8), resulting in a nearly constant dwell time with increasing load as observed in cardiac myofilaments at 1 mM ATP (Figs. 1F and 6D), while the dwell times increased with increasing load in the LRS rate model (Fig. 6D), in which the second reverse stroke rate ($k_{-5}$, Fig. 1A) is set to be less sensitive to a given load than in the HRS rate model (Fig. 5A). Therefore, the load-independent dwell time observed in cardiac myofilaments may be due to the highly load-dependent second reverse stroke.

Finally, to understand the physiological meaning of the reverse stroke for heart contraction, HRS and LRS rate models were used to simulate single-actin dynamics in sarcomeres during isometric contraction by increasing the number of interacting molecules from 15 to 75, corresponding to the approximate number of myosin molecules interacting with a single actin filament in sarcomeres (13) (SI Appendix, Simulation of single-actin dynamics in sarcomeres). The stiffness of a spring connected to the actin filament was set to be 50 pN/nm to represent the stiffness of the actin–Z-disk connection (13). One cycle of isometric contraction was regulated by a cyclic change in the myosin attachment rate, representing Ca²⁺ oscillation in sarcomeres. The tension curve developed by the HRS rate model showed stable tension for ∼500 ms, followed by a rapid reduction in tension to 0 in 20 to 30 ms in every 1-s cycle of changes in Ca²⁺ concentration (SI Appendix, Fig. S12), while the LRS rate model could not readily maintain the tension curve. The difference in the force output between the two models implies that the second reverse stroke is the key to producing stable tension even at low Ca²⁺ concentrations followed by a rapid reduction in tension. The characteristics of the tension curve developed by the HRS model appear to be similar to the left ventricular pressure curve in the heart. Despite the simulation model being oversimplified to test sarcomere dynamics during the systole phase (SI Appendix, Simulation of single-actin dynamics in sarcomeres), the analyses of these simulation results imply a potential role of the reverse strokes in heart contractions, such as the maintenance of stable systolic pressure and the rapid relaxation of pressure in the latter systole phase. Moreover, these catastrophic events possibly allow force-generating myosin to rebind and regenerate tension without consuming ATP (17) and thus slow the cycle of ATP hydrolysis (SI Appendix, Fig. S12D). One of the main causes of HCM has been suggested to be hypercontraction associated with impaired relaxation and excessive ATP consumption (1). Furthermore, one of the major HCM point mutants, R453C, shows a higher energy gap between the AMD and AMD* states than the wild-type (47), implying that some HCM point mutants may limit the reverse stroke performance, which may merit future investigation.

Materials and Methods

Protein Preparation.

Cardiac β-myosin was purified from the left ventricle of pig heart, and G-actin was purified from rabbit psoas muscle (48). Myosin and tetramethylrhodamine isothiocyanate (TRITC)-labeled cardiac rods were mixed at a molar ratio of 4:1 for force measurements of myofilaments, 1:1,500 for single-molecule experiments, and 1:7 for unbinding measurements to count the number of molecules interacting with actin. To make synthetic myosin filaments, the mixing solution was rapidly diluted two times to decrease the concentration of KCl from 600 to 50 mM by adding standard assay buffer (20 mM Pipes pH 7.2, 5 mM MgSO₄, and 1 mM EGTA).

Force Measurements of Cardiac Myofilaments.

Flow chambers were prepared using 24 × 32 mm and 18 × 18 mm coverslips with 10-μm thick double-sided tape. The chambers were incubated with 0.2 mg ⋅ ml⁻¹ casein followed by a 3 μg ⋅ ml⁻¹ myosin-rod cofilament solution. The solution was then replaced with a mixture of 400-nm diameter streptavidin-coated polystyrene beads (Polysciences, Inc.), 2 nM biotinylated phalloidin-TRITC–labeled actin, an oxygen scavenger system, 0.2 mg ⋅ ml⁻¹ casein, 1 μM phalloidin, and 10 μM or 1 mM MgATP in the standard assay buffer. Once the bead was trapped and then attached to an actin filament by manipulating optical tweezers with a trap stiffness of 0.28 pN ⋅ nm⁻¹ (1,064 nm, 800 mW, Spectra Physics, Inc.), the actin filament was placed near a myofilament immobilized on a glass surface. All experiments were conducted at 25 °C.

Dark-field images of the beads were projected onto quadrant detector photodiodes and recorded at a sampling rate of 20 kHz. The displacements of beads were corrected by setting the compliances of the bead–protein complex to ∼10% (13) and low-pass filtered at a bandwidth of 5 kHz to detect steps using the step-finding algorithm (49).

Measurement of Single Myosin Dynamics.

A glutaraldehyde-coated coverslip was used to immobilize myosin-rod cofilaments tightly onto a glass surface to reduce noise. The chambers were incubated with 0.05 mg ⋅ ml⁻¹ casein, followed by 3 mg ⋅ ml⁻¹ myosin-rod cofilament solution. The solution was then replaced with an actin-conjugated gelsolin bead, 0.2 mg ⋅ ml⁻¹ casein, 5 mM MgSO₄, 1 mM EGTA, an oxygen scavenger system, and 10 U ⋅ ml⁻¹ hexokinase, which was added to eliminate ATP contamination in ADP solution. A total of 20 mM glucose was sufficient as a substrate for an oxygen scavenger system and hexokinase. We also added 1 mM MgADP x (= 0, 1, 5, 10) mM Pi and 20-x mM PIPES (pH 7.2) to maintain a constant ionic strength at different Pi concentrations. Once an actin bead was maintained in optical tweezers with a trap stiffness of 0.05 to ∼0.1 pN ⋅ nm⁻¹, it was positioned in the vicinity of a myosin-rod cofilament immobilized on the glass surface. When a single myosin attached to an actin filament, loading was applied instantaneously by displacing a trapped bead using a custom-made piezo-assisted mirror (Fig. 2). To confirm a single myosin interaction, the bead was manipulated upward to rupture the interaction and then the number of unbinding events was counted after the measurements.

Calculation of Reaction Rates from Single-Molecule Experiments.

Since a reaction of myosin in each force state is a branching reaction (Fig. 1A), we calculated the associated transition rates as follows:

$$k_i=\frac{1}{{\tau }_i}\frac{N_i}{N_{total}},$$ [1]

where ${\tau }_i$ is the mean dwell time for the ith pathway (i = ± 4, +5 and detach), and $N_i$ and $N_{total}$ are the numbers of ith events detected in the experiments and the total number of events involved in the branching reaction, respectively. Note that $k_{detach}$ is the rate of detachment leading to the MDP and/or MD states.

Simulation Model.

To elucidate the molecular mechanism underlying collective behaviors observed in cardiac myofilaments, a simulation model (HRS rate model and LRS rate model) was developed based on a model for fast skeletal myosin (13) (Fig. 1A). Compared with that previous model (13), the following was primarily changed: 1) the energy gap between the AMD* and AMD states ($\mathrm{\Delta }{U}_{AMD}$ in Table 1); 2) implementation of the MD state; and 3) the characteristics of strain-dependent revere stroke rate functions. The parameter values for $k_4^0$, $\mathrm{\Delta }{U}_{AMD}$, $k_7^0$, and $k_8$ were determined by fitting to data obtained from myofilament force measurements, and other values were slightly modified from previous studies or calculated from our experimental data (Table 1). The rate constants $k_{+1}$, $k_{+2}$, and $k_{-2}$, corresponding to transitions from the AM state to the MDP state via the MT state, were taken as 1.5 × 10⁶ M⁻¹s⁻¹ and 1.28 and 12.8 s⁻¹ (2), respectively. Based on our results from single-molecule experiments (Figs. 2 and 4), we assume that Pi release occurs prior to power stroke execution. The transition between the MDP and AMDP states is known to reach equilibrium rapidly and thus is omitted in the kinetic pathway between the MDP and AMD** states (Fig. 1A). In addition, $k_{+3}$, the rate constant for myosin attachment and Pi release, is assumed to be 18 s⁻¹ (13, 14), while $k_{-3}$, the rate function for Pi binding and myosin detachment, is approximately as follows:

$$k_{-3}=k_{-3}^0\left[\text{Pi}\right],$$ [2]

where $k_{-3}^0$ is the second-order rate constant for Pi binding, estimated to be $8\times {10}^6{\text{\hspace{1em}M}}^{-1}{\text{s}}^{-1}$ by a linear fit of our experimental results (Fig. 4F). The equilibrium constants $K_4$ and $K_5$ are given so that the Boltzmann distribution law is fulfilled at the statistical equilibrium (13):

$$K_i\left(x\right)=\frac{k_i\left(x\right)}{k_{-i}\left(x\right)}=\exp \left(-\frac{\left(W\left(x+l_i\right)+\mathrm{\Delta }{U}_{post}\right)-\left(W\left(x\right)+\mathrm{\Delta }{U}_{pre}\right)}{k_{B}T}\right),$$ [3]

where pre and post refer to AMD** and AMD* for $i=4\hspace{1em}$or AMD* and AMD for $i=5$, respectively, and $l_i$ is the power stroke size of 6 nm for $i=4$ or 3 nm for $i=5$. $\mathrm{\Delta }{U}_{AMD\ast \ast }$, $\mathrm{\Delta }{U}_{AMD\ast }$, and $\mathrm{\Delta }{U}_{AMD}$ are the energies of 28.5 k_BT from AM to AMD**, 18.5 k_BT from AM to AMD*, and 15.0 k_BT from AM to AMD, respectively, in the HRS rate model. $W\left(x\right)$ and $W\left(x+l_i\right)$ are the elastic potential energy of the pre– and post–power stroke myosin head as a function of myosin strain x and are calculated from the nonlinear elasticity of the myosin head (13, 14) (Fig. 5B). The stiffness of myosin head depends on the myosin strain, $x$, and the motor force, $F$, based on the nonlinear elasticity of the myosin, which is expressed as the following:

$$\begin{array}{l}F\left(x\right)=k_{m}x\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(0<x\right)\\ F\left(x\right)=\frac{k_m}{\beta }\left[\exp \left(-\beta x_c\right)-1\right]\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(-x_c<x<0\right)\\ F\left(x\right)=\alpha k_m\left(x+x_c\right)-\frac{k_m}{\beta }\left[1-\exp \left(-\beta x_c\right)\right]\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(x<-x_c\right),\end{array}$$ [4]

where $k_m$, $x_c$, α, and β are the constants of 2.8 pN/nm, 4.35 nm, 0.05, and 0.69, respectively. The load-dependent rate functions of the first power/reverse stroke, $k_{+4}$ and $k_{-4}$, or the second power/reverse stroke, $k_{+5}$ and $k_{-5}$ (Fig. 5A), are expressed by using the concept of energy barrier as follows:

$$k_i\left(x\right)=\text{min}\left[k_i^0\exp \left(\frac{W\left(x\right)+\mathrm{\Delta }{U}_{pre}-\left(W_{barrier}\left(x\right)+\mathrm{\Delta }{U}_{barrier}\right)}{k_{B}T}\right),\hspace{1em}10000\right],$$ [5]

$$\begin{array}{l}{k}_{-i}\left(x\right)=k_i^0\exp \left(\frac{W\left(x+l_i\right)+\mathrm{\Delta }{U}_{post}-\left(W_{barrier}\left(x\right)+\mathrm{\Delta }{U}_{barrier}\right)}{k_{B}T}\right)\hspace{1em}\\ \left(if\hspace{1em}{k}_i\left(x\right)<10000\right),\end{array}$$ [6]

$$k_{-i}\left(x\right)=\frac{k_i\left(x\right)}{K_i}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(if\hspace{1em}{k}_i\left(x\right)=10000\right),$$ [7]

where $k_i^0$ is a constant 3.0 × 10⁶ or 2.1 × 10⁴ s⁻¹ for i = 4 or 5, respectively. $W_{barrier}\left(x\right)+\mathrm{\Delta }{U}_{barrier}$ is the energy barrier between the pre– and post–power stroke states (SI Appendix, Derivation of the power/reverse stroke rate function based on energy barrier and Fig. S13), defined based on the assumption that the energy barrier is positioned midway between the pre– and post–power stroke states (50) as follows:

$$W_{barrier}\left(x\right)=W\left(x+l/2\right),$$ [8]

$$\mathrm{\Delta }{U}_{barrier}=1.1\mathrm{\Delta }{U}_{pre}.$$ [9]

The parameters in Eq. 5, that is, $k_4^0$ and $\mathrm{\Delta }{U}_{AMD}$, were empirically determined by fitting data analyzed from the model outputs to the stepping ratio–load relationships (Fig. 1I). In single-molecule experiments, we found that the power/reverse stroke rates monotonically decreased/increased with increasing loads for cardiac myosins (Fig. 5A). Moreover, the rate functions of the reverse stroke were associated with a single peak in the model for fast skeletal myosin because of the nature of the Boltzmann distribution used in the model (13). Thus, Eqs. 5–9 were modified from a previous model (13) to characterize the monotonous decrease/increase in power/reverse stroke rates by increasing the load as assigned in a whole-heart simulator (50).

The ADP release rate, $k_{+6}$, is expressed as follows:

$$k_{+6}\left(x\right)=k_{+6}^0\exp \left(-\frac{d_{+6}F\left(x\right)}{k_{B}T}\right),$$ [10]

where $k_{+6}^0$ is 80 s⁻¹, and $d_{+6}$ is the characteristic distance of 0.8 nm, as reported in previous single-molecule experiments (5). $F$ is the motor force, which depends on the nonlinear elastic property of the myosin head with the myosin strain, x (Eq. 4) (13, 14).

The load-dependent rupture rate function of the myosin head, $k_{+7}$, is written as follows:

$$k_{+7}\left(x\right)=k_{+7}^0\text{exp}\left(\frac{d_{+7}F\left(x\right)}{k_{B}T}\right),$$ [11]

where $k_{+7}^0$ is 0.1 s⁻¹, and $d_{+7}$ is the characteristic distance of 2.0 nm (42). The value of $k_{+7}$ estimated by the linear fit (Fig. 4F) is 9 s⁻¹, which is reasonably similar to the value of 3 s⁻¹ calculated by Eq. 11 in the HRS rate model at the corresponding mean load of 6 pN, as shown in Fig. 4F. The reattachment rate constant, $k_{+8}$, was set at a constant value of 1,000 s⁻¹.

The LRS rate model was generated by changing the value of $\mathrm{\Delta }{U}_{AMD}$ from 15.0 to 10.0 k_BT, which means that the energy gap between AMD* and AMD was 5.0 k_BT higher for the LRS rate model than the HRS rate model (Fig. 5B), resulting in less load dependence of the second reverse stroke rate (Fig. 5A).

The values for all the parameters are summarized and compared with those of other studies in Table 1.

The state transition was updated by a simple Monte Carlo simulation method (50). In detail, for each myosin molecule, a random number (0 ≤ r ≤ 1) is generated, and the following procedure is applied. If the molecule is in the state $S_i$ and is able to transit to the state $S_{i-1}$ with the rate constant $k_{-\left(i-1\right)}$ or to the state $S_{i+1}$ with the rate constant $k_{+i}$, the state transition is determined as follows:

• transit to $S_{i-1}$ if $r\le \mathrm{\Delta }t{k}_{-\left(i-1\right)}$,

• transit to $S_{i+1}$ if $\mathrm{\Delta }t{k}_{-\left(i-1\right)}<r\le \mathrm{\Delta }t\left(k_{-\left(i-1\right)}+k_{+i}\right)$,

• and stay in $S_i$ if $r>\mathrm{\Delta }t\left(k_{-\left(i-1\right)}+k_{+i}\right)$.

In our numerical simulations, $\mathrm{\Delta }t=1\text{\hspace{1em}μs}$ was applied so that $\mathrm{\Delta }t\left(k_{-\left(i-1\right)}+k_{+i}\right)$ is sufficiently smaller than 1.

Data Availability

All study data are included in the article and/or SI Appendix. Previously published data were used for this work (13).