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# Myosin V: Chemomechanical-coupling ratchet with load-induced mechanical slip

## Chemomechanical network modelling of molecular motor myosin V

### Mechanical-step transitions

As one of the extraordinary motor properties of myosin V, high-speed processive backward stepping induced by superstall loading has been observed. Although the velocity of the load-assisting forward stepping obviously increases along with ATP concentration, that of the load-induced backward stepping does not greatly depend upon this concentration27. Thus, we equally take into account all possible mechanical transitions between states in which both heads are strongly bound to the AF, i.e. the transitions between states 8 (DD) and 8 (DD), between states 1 (ED) and 4 (DE), and between states 7 (EE) and 7 (EE), in addition to the force-generating mechanical transitions. The 8-to-8 and 7-to-7 transitions do not directly couple with the chemical states and thus would play an important role in the ATP-independent high-speed backward motility under superstall loading.

## Results

### Load dependence of the motor properties

Figure 1c shows the ratio between the numbers of forward and backward steps as a function of external load at the ATP concentration of 100 µM. The nine-state model can quantitatively describe the external-load dependence of the ratio of forward steps to backward steps. To achieve the high ratio around 60 under the load-free condition, the main forward-stepping cycle at 100-µM ATP would have to avoid states 8 (DD) and 7 (EE). It is because, if states 8 (DD) and 7 (EE) frequently appeared in the main forward cycle, the 8-to-8 and 7-to-7 transitions that cause symmetric random walk in the absence of external load would significantly lower the ratio of forward steps to backward steps. The main forward-stepping cycle will be discussed in detail later on.

Figure 2a,b show ATP-concentration dependences of the motor velocity v for forward and backward stepping forced by forward and backward loads of (a) 5 pN and (b) 10 pN. We find that the forward velocity at high ATP concentrations is not affected by the strength of the forward loading in comparison between the black solid and red dotted lines shown in Fig. 2a,b; however, it does strongly depend upon ATP concentration, as shown by the black solid lines in these figures. Concerning the ATP-dependent velocity under the assisting loading, the main transition pathway causing forward motility depends upon ATP concentration, as shown in Fig. 2c,d: the main mechanical transition working at 100-µM ATP is the 2-to-5 transition [Fig. 2c], while the largest and second-largest contributions to the total velocity at 1-µM ATP are the 3-to-6 and 2-to-5 transitions, respectively [Fig. 2d]. These mechanical transitions directly couple with the nucleotide state of the heads, because these nucleotide states are exchanged by the mechanical step [see Fig. 1a]. Thus, the rate-limiting process of the load-assisting forward cycle is basically attributable to ATP-hydrolysis processes. This is why the forward velocity strongly depends upon the ATP concentration but is hardly affected by the assisting load.

Figure 3a shows ATP-concentration dependences of the motor velocity v at Pi concentrations of 0 and 40 mM. As ATP concentration increase, the motor velocity v rapidly increases until becoming saturated at ATP concentrations above about 100 µM10,35. In both the experiment by Zhang et al.23 and the theoretical results, the motor velocity v is reduced by the addition of 40-mM Pi. Figure 3b shows the ATP-concentration dependences of the mean run length until myosin V detaches from an AF. This mean run length is almost constant at the Pi concentration of 40 µM, but decreases under the Pi-free condition with increasing ATP concentration23, even although the motor velocity v monotonically increases as ATP concentration increases [Fig. 3a]. This is not the case for kinesin36; its mean run length increases according to an increase in the motor velocity with increasing ATP concentration. These results indicate that the detachment rate of myosin V would be increased by the increase in ATP concentration, as shown in Fig. 3c. We examined an eight-state model formed by omitting state 9 (TT) from the nine-state model [see Fig. 1a] to describe the experimental data shown in Figs 1, 2 and 3. The results were almost comparable to those of the nine-state model in that the experimental data except for several crucial motor properties could be described by the eight-state model, whereas it was found that the decrease in the mean run length as shown in Fig. 3b and the increase in the detachment rate shown in Fig. 3c could not be described by the eight-state model. Here, it is expected that states 9 (TT), 2 (TD) and 5 (DT) would frequently appear at high ATP concentrations and that myosin V staying in these states would likely detach from the AF; it is because, it would have at least one ATP-binding head that is weakly bound to the AF. These observations imply that, as the probabilities of states 9 (TT), 5 (DT) and 2 (TD) increase along with ATP concentration, so too does the frequency of detachment of myosin V via those states.

In Fig. 3d, we find a decrease in the motor velocity v as ADP concentration increases. Figure 3f shows that the theoretical result for the detachment rate decreases with increasing ADP concentration and that the experimental results either slightly decrease or stay almost constant. In contrast to the ATP-concentration dependence of the mean run length, the decrease in the mean run length with the increase in ADP concentration seen in Fig. 3e is attributable to the decrease in the motor velocity v shown in Fig. 3d.

### Efficiency of the transduction of ATP-hydrolysis free energy into mechanical work

Figure 4a shows external-load dependences of the ratio of the number of forward steps to the amount of ATP hydrolysis, ΔJ(Step)/ΔJ(hydrolysis), i.e. chemomechanical-coupling efficiency, at several ATP concentrations in the absence of ADP and Pi, where ΔJ(Step) and ΔJ(hydrolysis) are defined as the sums of ΔJ ij [shown by Eq. (2)] that are related to the mechanical-step transitions and ATP-hydrolysis transitions, respectively. Except for the lowest ATP concentration of 1 µM, the ratios under zero loading are larger than 0.8, indicating that a high ratio of transduction from the hydrolysis of one ATP molecule to one forward step is achieved, even at the low ATP concentration of 10 µM. In addition, these load-dependent curves at ATP concentrations between 10 µM and 1 mM almost overlap with each other. These results show that myosin V has robustness against changes in ATP concentration. As later be discussed based on the main working cycles, the high chemomechanical-coupling efficiency under the load-free condition would be attributable to the dual force-generating mechanical transitions, i.e. the 2-to-5 and 3-to-6 transitions.

Figure 4b shows load dependences of chemomechanical-transduction efficiencies, η, at ATP concentrations of 1 mM, 10 µM and 1 µM under the constant ADP and Pi concentration of 10 µM. The chemomechanical-transduction efficiency η is defined by $$\eta =Fv/[\Delta \mu \Delta J({\rm{hydrolysis}})]$$, where Δμ is the difference between the chemical potentials for one ATP and for one ADP plus one Pi, i.e. $$\Delta \mu ={k}_{{\rm{B}}}T\,\mathrm{ln}\{{K}_{eq}[{\rm{ATP}}]/[{\rm{ADP}}][{\rm{Pi}}]\}$$ 37, with the equilibrium constant K eq  = 4.9 × 1011 µM38. Under the case that the ADP and Pi concentrations are 10 µM, the maximum efficiency of myosin V at 1-mM ATP is obtained at a backward load around 0.8 pN, which is lower than the value for kinesin about 3.5 pN28, while the maximum η value for myosin V, about 0.13, is almost identical to that for kinesin. The loss in the transduction of the hydrolysis free-energy Δμ into the mechanical work should be attributable to both heat that irreversibly dissipates through the viscous friction of the probe39 and that irreversibly dissipates from the molecular motor due to chemical and mechanical slip cycles. A larger maximum value of η at 100-µM ATP than that at 1-mM ATP is due to the smaller value of Δμ at the ATP concentration of 100 µM than that at 1 mM, because Δμ, which appears in the denominator of $$\eta =Fv/[\Delta \mu \Delta J({\rm{hydrolysis}})]$$, is decreased by the decrease in ATP concentration. In the same manner, the maximum value of η can be slightly increased by appropriately increasing ADP and/or Pi concentrations, because Δμ is decreased by an increase in the concentrations of ADP and/or Pi if the motor velocity v is not mostly decreased by the increase in the ADP and/or Pi concentrations.

### Chemical-transition pathways at different ATP concentrations and external loads

One of the most crucial results obtained in this study are local fluxes on the network at various ATP concentrations and external loads. The nine-state model based on the full-network representation can provide a unified description on the chemomechanical-transition pathways of the working cycle under all chemical and mechanical conditions. Figure 5 shows diagrams of the main local fluxes for high (1 mM) and low (1 µM) ATP concentrations under zero loading [Fig. 5a,b], a forward-assisting load of 5 pN [Fig. 5c,d], and backward loads of 1 pN [Fig. 5e,f] and 5 pN [Fig. 5g,h], respectively. The red arrows indicate normalized local fluxes larger than 0.10, while the blue arrows indicate other normalized fluxes larger than 0.07 and the green arrow in Fig. 5g indicates a normalized flux with 0.03, where these normalized local fluxes are defined in the steady state (st) as the value of the local flux $${J}_{ij}^{{\rm{st}}}={P}_{i}^{{\rm{st}}}{\omega }_{ij}$$ [see Method section] divided by the sum of all the local fluxes. The perpendicular axis for each diagram indicates the ATP-concentration level; thus states located at higher places on this axis would be more probable at high ATP concentrations. The red and blue letters F and B indicate the forward- and backward-stepping transitions, respectively. Although the direction of the mechanical-step transitions from state 2 to state 5, from state 3 to state 6, from state 4 to state 1 is shown to be pointing towards the opposite direction in comparison with Fig. 1a, it is noted that these transitions indicate the forward-stepping transitions.

In contrast, the main forward cycle |17361> that is obtained by the nine-state model at the low ATP concentration of 1 µM [Fig. 5b] goes through state 7 (EE), wherein both heads are the empty (E) state; it is because ATP binding is the rate-limiting process and the probability of state 7 (EE) becomes highest as the ATP level decreases. It is remarkable that this cycle |17361> includes the 3-to-6 mechanical transition, since this transition has been newly introduced into the network model as a force-generating mechanical-step transition on the basis of the experimental observation by Kodera et al. using the AFM technique30. The 3-to-6 mechanical transition enables the nine-state model to reproduce the high forward velocity [Figs 1b, 2a,b] and the high chemomechanical-coupling efficiency on converting the hydrolysis of one ATP molecule into one forward step [Fig. 4a] even at low ATP concentrations.

In Fig. 5c, we find that the main forward cycle at the high ATP concentration of 1 mM is the same as that without the assisting load shown in Fig. 5a. The comparison between Fig. 5d,b shows that the load-assisted main-forward cycle at the low ATP concentration of 1 µM is almost the same as that without the assisting load, except that the 7-to-7 forward-stepping transition indicated by the blue arrow in Fig. 5d, which does not directly couple with ATP hydrolysis, is caused by the assisting load. The slight increase in forward velocity at the low ATP concentration of 1 µM with increasing assisting load (as seen in Fig. 1b) and the large effect of assisting load upon the forward velocities at low ATP concentrations (as shown in Fig. 2a,b) are attributable to the load-induced 7-to-7 forward-stepping transition. In any case, the ATP-concentration-dependent forward velocities under the assisting load observed in Figs 1b, 2a,b are basically governed by the main forward cycles driven by ATP hydrolysis indicated as the red arrows in Fig. 5c,d.

In Fig. 5g,h, we find that the backward-stepping cycle is completely different from the reversal of the forward-stepping cycle seen in Fig. 5a,b. This observation indicates that the transduction processes of the free energy by ATP hydrolysis to the mechanical work are irreversible nonequilibrium processes, even if each chemical and mechanical transition on the network is reversible. The irreversibility of the chemomechanical cycles should be attributed to the branched pathways given by the network representation. At the high ATP concentration of 1 mM [Fig. 5g], the main local flux is a futile hydrolysis cycle, |2392>, located on the upper-right-hand side of the network, while the main backward-stepping cycle is due to the 8-to-8 mechanical-slip transition. At the low ATP concentration of 1 µM [Fig. 5h], we find both the 7-to-7 backward mechanical-slip transition and a futile hydrolysis cycle |3473> on the bottom-right-hand side as the main local fluxes. These futile hydrolysis cycles observed at both the ATP concentrations basically go through states at which at least the leading head is strongly bound to the AF, except for state 9 (TT). The results imply that the strong binding to the AF of the leading head inhibits the detachment of myosin V from the AF during load-induced backward stepping. It is noted that these futile hydrolysis cycles would be observed even without accounting for the 8-to-8 and 7-to-7 transitions, since these mechanical-slip transitions have no effect upon the steady-state local fluxes and probabilities. Furthermore, it is hard to describe the sufficiently high velocity of backward stepping under superstall loading using only the reversed 2-to-5 and 3-to-6 transitions, i.e. without accounting for the mechanical-slip transitions. These arguments and the results shown in Figs 5g and 5h indicate that the 8-to-8 and 7-to-7 mechanical-slip transitions and 1-to-4 load-induced mechanical transitions play a crucial role in the high-speed backward-stepping motility of myosin V.

Figure 5,f show that the parts of the local fluxes observed under both zero load [Fig. 5,b] and 5-pN-backward load [Fig. 5e,f] at each ATP concentration appear as the main local fluxes at the backward load of 1 pN.

## Summary

In this study, we presented a systematic modelling of molecular-motor myosin V on the basis of a chemomechanical network theory22,24. A nine-state-network model introduced in this study is based on a full-network representation for two-headed molecular motors with one catalytic domain per head, where one of three different chemical states, i.e. ATP-binding, ADP-binding and empty states, are assigned to each head. In this nine-state model [Fig. 1a], we took into account five different mechanical transitions: between 2 (TD) and 5 (DT), between 3 (TE) and 6 (ET), between 1 (ED) and 4 (DE), between 8 (DD) and 8 (DD) and between 7 (EE) and 7 (EE). Here, the first two transitions produce a forward-biased mechanical step, since the leading head is strongly bound to an actin filament (AF) and the trailing head is easily unbound from the AF, while the last three transitions mainly cause passive mechanical steps when external loading is applied, since both heads are strongly bound to the AF.

The probabilities of the state at which one head or two heads are occupied by ATP generally increase along with ATP concentration, while the probabilities of the state at which one head or two heads are occupied by no ATPs increases with a decrease in ATP concentration [Fig. 5a,b]. These ATP-dependent state probabilities are not only related to the ATP-concentration dependence of the main working cycle mentioned above in (i), but also lead to qualitatively different ATP-concentration dependences of the mean run lengths for myosin V and kinesin: the mean run length of myosin V is somewhat decreased by an increase in ATP concentration, although the motor velocity is increased [Fig. 3a]; on the other hand, the mean run length of kinesin is increased according to an increase in the motor velocity with increasing ATP concentration. This abnormal decrease in the mean run length of myosin V is attributable to an increase in the detachment rate of myosin V from the AF [Fig. 3c], because the probability of having at least one ATP-binding head (whose binding affinity to the AF is weaker than the other nucleotide states) is increased by the increase in ATP concentration.

## Methods

### Motor dynamics

The dynamics of a molecular motor can be described by a continuous-time Markov process. Thus, the probability $${P}_{i}(t)$$ of finding the molecular motor in state i at time t is governed by the following continuous-time master equations:

$$\frac{d}{dt}{P}_{i}(t)=-\sum _{j}\Delta {J}_{ij}(t),$$
(1)
$$\Delta {J}_{ij}(t)={J}_{ij}(t)-{J}_{ji}(t)={P}_{i}(t){\omega }_{ij}-{P}_{j}(t){\omega }_{ji},$$
(2)

where ω ij is a transition rate from state i to state j, i.e. the number of transitions from i to j per unit time, J ij (t) and ΔJ ij (t); the local flux and local excess flux due to the transition from state i to state j. In general, the transition rate, ω ij , depends on both the external load parallel to the actin filament, F, and the molar concentrations [X], where X denotes the molecular species ATP, ADP, or Pi (inorganic phosphate). Thus the transition rates can be given by

$${\omega }_{ij}={\omega }_{ij}^{0}{\Phi }_{ij}(F),$$
(3)

where $${\omega }_{ij}^{0}$$ is a zero-force transition rate and $${\Phi }_{ij}(F)$$ is a force-dependent factor with $${\Phi }_{ij}(F=0)=1$$. Moreover, $${\omega }_{ij}^{0}$$ for the binding of an X-molecule depends on the molar concentration [X] as

$${\omega }_{ij}^{0}=\{\begin{array}{c}{\hat{k}}_{ij}[X]\,\\ {k}_{ij}\end{array}\begin{array}{c}{\rm{for}}\,X-\mathrm{binding}\\ {\rm{for}}\,X-\mathrm{release}\end{array},$$
(4)

Here, $${\hat{k}}_{ij}$$ has dimensions of $$1/(\mu {\rm{Ms}})$$, while $${k}_{ij}$$ has dimensions of 1/s. In the present study, we assume that only the mechanical-transition rates depend upon the external force and that $${\Phi }_{ij}(F)$$ for the forward-step transitions under a backward force has the following form (note that the backward force F is taken to be positive):

$${\Phi }_{ij}(F)=\exp [-F{\theta }_{ij}/{F}_{ij}],$$
(5)
$${\Phi }_{ji}(F)=\exp [F(1-{\theta }_{ij})/{F}_{ij}],$$
(6)

where F ij is a force scale with respect to the force dependence of the mechanical transition and $${\theta }_{ij}$$ is a load-distribution factor with a value of $$0 < {\theta }_{ij} < 1$$.

### Calculation details

In the nine-state model, the motor velocity v is given by $$v={v}_{25}+{v}_{36}+{v}_{41}+{v}_{77}+{v}_{88}$$, where $${v}_{25}=l\Delta {J}_{25}^{{\rm{st}}}$$, $${v}_{36}=l\Delta {J}_{36}^{{\rm{st}}}$$, $${v}_{41}=l\Delta {J}_{41}^{{\rm{st}}}$$, $${v}_{77}=l{P}_{7}^{{\rm{st}}}({\omega }_{77}^{f}-{\omega }_{77}^{b})$$ and $${v}_{88}=l{P}_{8}^{{\rm{st}}}({\omega }_{88}^{f}-{\omega }_{88}^{b})$$, with $$\Delta {J}_{ij}^{{\rm{st}}}$$ and $${P}_{i}^{{\rm{st}}}$$ being the local excess flux and the state probability at the steady state, respectively, and $${\omega }_{ii}^{f}$$ and $${\omega }_{ii}^{b}$$ being the transition rates towards the forward and backward directions, respectively. Here, $${\omega }_{ii}^{f}$$ and $${\omega }_{ii}^{b}$$ are given by $${\omega }_{ii}^{f}={\omega }_{ii}^{0}\exp [-F{\theta }_{ii}/{F}_{ii}]$$ and $${\omega }_{ii}^{b}={\omega }_{ii}^{0}\exp [F(1-{\theta }_{ii})/{F}_{ii}]$$, respectively. There are 42 transitions among the nine states other than the 7-to-7 and 8-to-8 mechanical transitions. Using an extension of the steady state balance condition (provided in the Supplementary Information of this work and of ref.28) and the similarity relations among the chemical transitions, the number of unknown transition rates can be reduced from 42 to 20 (see the Supplementary Information). The determination of unknown parameters for the transition rates was manually performed as follows. First, we applied the nine-state model [Fig. 1a] to the experimental data on the external load dependence of the motor velocity [Fig. 1b] at several ATP concentrations and the step ratio [Fig. 1c]. Then, we examined the validity of the manually determined parameters by applying to both ATP-concentration dependences of the motor velocity under constant loads [Fig. 2a,b] and ATP- and ADP-concentration dependences of motor velocity without external loading [Fig. 3a,d]. If the theoretical results did not agree well enough with all of the experimental data considered in the present study, this procedure was repeated to refine the parameters until sufficient agreement was achieved. After we determined the parameters for the chemical and mechanical transitions, we applied an extended nine-state model, wherein unbinding transitions of myosin V from the AF via states 9 (TT), 5 (DT), 2 (TD), 6 (ET), 8 (DD), 1 (ED) and 7 (EE) were taken into account [see Fig. 1a and the Supplementary Information], for fitting to the experimental data on the mean run length [Fig. 3b,e] and the detachment rate [Fig. 3c,f] as well. As a final procedure, we recalculated all of the motor properties and refined all the parameters again. It was found that the detachment transitions had only a small effect upon the motor velocity.

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## Acknowledgements

This work was supported in part by the Grants-in-Aid for Scientific Research (KAKENHI) from the Ministry of Education, Culture, Sports, Science and Technology of Japan. I would like to thank Professor Stefan Klumpp in Georg-August-Universität Göttingen for the discussion about an extension of the steady state balance condition shown in the Supplementary Information.

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Correspondence to Tomonari Sumi.

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Sumi, T. Myosin V: Chemomechanical-coupling ratchet with load-induced mechanical slip. Sci Rep 7, 13489 (2017). https://doi.org/10.1038/s41598-017-13661-0

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