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

A chemomechanical-network model for myosin V is presented on the basis of both the nucleotide-dependent binding affinity of the head to an actin filament (AF) and asymmetries and similarity relations among the chemical transitions due to an intramolecular strain of the leading and trailing heads. The model allows for branched chemomechanical cycles and takes into account not only two different force-generating mechanical transitions between states wherein the leading head is strongly bound and the trailing head is weakly bound to the AF but also load-induced mechanical-slip transitions between states in which both heads are strongly bound. The latter is supported by the fact that ATP-independent high-speed backward stepping has been observed for myosin V, although such motility has never been for kinesin. The network model appears as follows: (1) the high chemomechanical-coupling ratio between forward step and ATP hydrolysis is achieved even at low ATP concentrations by the dual mechanical transitions; (2) the forward stepping at high ATP concentrations is explained by the front head-gating mechanism wherein the power stroke is triggered by the inorganic-phosphate (Pi) release from the leading head; (3) the ATP-binding or hydrolyzed ADP.Pi-binding leading head produces a stable binding to the AF, especially against backward loading.

have to apply the second law of thermodynamics based on both the equality and inequality to characterize irreversible cycles of nanomachine in nonequilibrium steady state 2 . According to the formulation presented by Liepelt and Lipowsky 1 , let's start with the conservation of energy during the completion of individual cycle. A change in the internal energy ΔU C ν + ( ) along any directed cycles C ν + satisfies where E chem C ν + ( ) is an energy input given by chemical reactions, W mech C ν + ( ) ; a mechanical work performed by mechanical transitions against an external load, and Q C ν + ( ) ; heat released from the system, during the completion of the directed cycle C ν + .
In this study, we assume that the motor dynamics is described by a continuous-time Markov process with transition rates from state i to state j. The statistical entropies ΔS C ν + ( ) and ΔS C ν − ( ) , which are produced in the steady state during the completion of directed cycles C ν + and C ν − , are provided by 1,3 where the summation is performed over all edges along the directed cycle C ν + . Now we apply the second law of thermodynamics to relate the statistical entropy ΔS C ν + ( ) to the heat released from the ω ij ij system: where the equality holds if the directed cycle C ν + is reversible. Using Eqs. (S1)-(S3), we obtain the following equation: which can be regarded as an extended steady state balance condition, where only the inequality in Eq. (S4) is different from that originally presented by Liepelt and Lipowsky 1 . The irreversibility of the directed cycles resulting in the inequality of Eq. (S4) should be attributable to the non-zero excess flux ΔJ ν + for the directed cycle C ν + or the non-zero ΔJ ν + for its reversed cycle C ν − . In the case of molecular motor myosin V, the large difference between the main forward-stepping cycle and the main backward-stepping cycle seems to reflect the irreversibility of the transduction process of ATP-hydrolysis free energy into mechanical work.
Here the chemical free-energy input per directed cycle can be expressed as where l is the size of the mechanical step and and are respectively the numbers of the mechanical forward and backward steps that are contained in one directed cycle of . Substituting Eqs.
(S5) and (S7) into Eq. (S4), we obtain If the excess fluxes ΔJ ν + s for all the directed cycles are equal to zero, the system is in thermodynamic equilibrium, thus all the directed cycles are reversible and then the equality in Eq. (S9) should hold. As a result, we obtain which corresponds to the detailed balance conditions in equilibrium presented by Liepelt and Lipowsky 1 . It has been pointed out by them that the detailed balance conditions in equilibrium are fully satisfied, if Eq.
(S10) is applied to all the fundamental cycles.
Next, we substitute Eqs.
where the equality holds in reversible process. In the case that all the directed cycles in the system include the mechanical transitions-a single cycle system is a typical simplest case-at a special stall load condition where ΔJ ν + s for all the directed cycles are equal to zero, the system should be in thermal and mechanical equilibrium. In this case, the directed cycles are reversible and thus the equality in Eq. (S11) should hold. As a result, we obtain which is equivalent to a part of the steady state balance condition presented by Liepelt and Lipowsky 1 , and gives a relation 1 F ij = l k B T by substituting Eqs. (5) and (6) into Eq. (S12).
In contrast, in the case that the system contains directed cycles without mechanical transitions, in other words, possible futile chemical cycles, even at the stall load conditions so that ΔJ ν + s for the directed cycles including the mechanical transitions could be equal to zero, the futile chemical cycles that contain no mechanical transitions [for instance the cycles such as |2392> and |3473> seen in Figs. 5g and 5h] would be driven by the free-energy input. However, this situation seems to be rather special case at which all of ΔJ ν + s for the directed cycles including the mechanical transitions become simultaneously zero under the stall load condition if the system has a lot of mechanical transitions like myosin V. This is because the stall load condition is basically achieved if the sum of all the velocity components arising from the mechanical transitions becomes zero [for instance, in the nine-state model for myosin should become zero under the stall load condition, whereas each velocity component does not have to necessarily become zero], indicating that ΔJ ν + for each directed cycle is not necessary being zero at the stall load condition. In these cases discussed above as possible examples, the system is not in equilibrium and then the directed cycles are irreversible; thus the inequality in Eq. (S11) should hold. As a result, we obtain the following condition: (S13) In our model, Eq. (S13) is satisfied by a condition 1 F ij < l k B T that is provided by substituting Eqs. (5) and (6) into Eq. (S13). To satisfy the extended steady state balance condition provided by Eq. (S4), we impose the detailed balance conditions in equilibrium, i.e. Eq. (S10), to the zero-force transition rates ω ij 0 given by Eq. (4) and also apply the condition 1 F ij < l k B T that is provided by Eq. (S13) to the force-dependent factor given by Eqs. (5) and (6).   transitions, which would arise under high backward loading and at low ATP concentrations, respectively.

The detailed balance conditions for the nine-state model
In the graph theory, a fundamental cycle basis of an undirected graph is given by a set of simple cycles that forms a basis of the cycle space of the graph. The number of fundamental cycles N fc in a given connected graph is provided as N e -N v +1 where N e is the number of edges and N v is the number of vertices 8 . In the case of nine-state chemomechanical network model [ Fig. 1 (a)], the number of the fundamental cycles is obtained as N fc = 13 because of N e =21 and N v =9. Therefore, thirteen independent equations among the transition rates are provided by the detailed balance condition 1 . Thirteen fundamental cycles are displayed in Fig. S1. Each fundamental cycle provides the following equations: (1) Cycle <234716592> k 23 k 34 k 47 k 47k71k71k16k65k59 k 92k32k29 k 95 k 56 k 61 k 17k74k43 = 1. (S14) We can use these thirteen equations to reduce the number of unknown transition rates. (1) (3) (4)

Asymmetries and similarity relations between transition rates
There are 42 (= 2 N e ) transitions among nine states in the nine-state network model except for the 7-to-7 and 8-to-8 transitions. Based on the effect on the chemical-transition rates caused by the intramolecular strain of the leading and trailing heads, in other words, the difference between the post-recovery-stroke and pre-recovery-stroke conformations [see Fig. S4] for the leading and trailing heads, respectively, we introduce asymmetries and similarity relations between transition rates to reduce the number of unknown transition rates systematically. For example, we here consider the chemical transitions from state 9 (TT) to state 2 (TD) and from state 9 (TT) to state 5 (DT). Although these transitions are ATP-hydrolysis reactions, the 9-to-2 transition is ATP hydrolysis on the leading head that is pulled backward by the partner trailing head, while the 9-to-5 transition is ATP hydrolysis on the trailing head that is pulled forward by the partner leading head.
This observation indicates that the rates for these transitions would be affected by the intramolecular strain of the leading and trailing heads and thus would be different each other. This asymmetry between these chemical-transition rates would be attributable to the different conformations of the catalytic domains in the leading and trailing heads with post-recovery and pre-recovery conformations, respectively. [see Fig. S4].
On the other hand, for instance, we can find a similarity relation between the chemical transitions from state If we substitute these 12 relations to Eqs. S14-S26 that are provided by the detailed balance condition, we finally obtain the following ten independent detailed balance conditions: (2) Cycle <2952> respectively. As a result, the number of unknown transition rates is reduced to 20 from 30.
Here, the transition from state 3 (TE) to state 9 (TT) is the ATP binding to the leading head and this leading head has a pre-power-stroke or post-recovery-stroke conformation [see Fig. S4], whereas the partner trailing head with ATP binding is weakly bound to the AF. During the transition from state 4 to state 5 and from state 7 to state 6, the leading head to which ATP is going to bind also has a pre-power-stroke or post-recovery-stroke conformation, although the partner trailing head is strongly bound to the AF. Based on the pre-power-stroke or post-recovery-stroke conformation of the leading head to which ATP is going to bind, the transition rate from state 3 (TE) to state 9 (TT) would be much more similar to the transition rate from state 4 to state 5 and from state 7 to state 6 than that from state 6 (ET) to 9 (TT); it is because, the trailing head to which ATP binds during the transition from state 6 (ET) to 9 (TT) has a pre-recovery-stroke or post-power-stroke conformation. In the same manner, the transition rate from state 6 (ET) to state 9 (TT) would be much more similar to the transition rate from state 1 to state 2 and from state 7 to state 3 than that from state 3 (TE) to state 9 (TT). Therefore, we can suppose the following approximate relations: (1) ATP binding to/release from the leading head with a pre-power-stroke or post-recovery-stroke conformation: ω 93 ≈ ω 54 = ω 67 . (S44b) (2) ATP binding to/release from the trailing head with a pre-recovery-stroke or post-power-stroke conformation: ω 96 ≈ ω 21 = ω 37 . (S45b) In addition, in the same manner, we can suppose the following approximate relations: (3) ATP hydrolysis/synthesis on the leading head with a pre-power-stroke or post-recovery-stroke conformation: ω 29 ≈ ω 85 = ω 16 .

Detachment of myosin V from an actin filament
It has been experimentally demonstrated that the head in state D or E is strongly bound to an AF while the head in state T is weakly bound to an AF 10 . Therefore, we expect that the detachment of myosin V from an AF would more frequently take place via states at which either one head or both heads are occupied by ATP.
This observation implies that the detachment of myosin V from the AF is increased with increasing ATP concentration. In addition, even if both heads are strongly bound to the AF, the detachment via the states strongly binding to the AF would occasionally be observed if the residence time in these states is longer than that in the other states. Based on these considerations, we took into account the detachment transitions from where k i 0 is a zero-force detachment rate and Ω i 0 F ( ) is a force-dependent factor for the detachment transition under load, Here, F i 0 is a force scale with respect to the force dependence of the detachment transition via state i under load.
According to the Hill's method for the mean time to absorption 11,12 , the mean time of binding or unbinding rate can be calculated using a modified diagram in which each absorption state is replaced by a one-way cycle back to a starting state. In this study, we employed states 8 (DD) as the starting state to determine the mean time of binding.
The transition rates obtained using the global fitting by the nine-state model to available

experimental data on myosin V's motor properties
The transition rates obtained using the nine-state model should be regarded as optimal values for describing the motor dynamics in nonequilibrium steady states, and those values could be different from the rate constants that have been determined by biochemical experiments under thermodynamic equilibrium.
Nevertheless, in the point of view of the asymmetries and similarity relations between the transition rates on the leading and trailing heads as discussed above, the obtained values would be qualitatively consistent with that determined by biochemical experiments under thermodynamic equilibrium. Table S1. The chemical transition rates in the nine-state network model. Load distribution on the 7-to-7 transition 0.0001 Force dependence on the 7-to-0 unbinding 6.90 pN