Enhanced Diffusion of Molecular Motors in the Presence of Adenosine Triphosphate and External Force

Enhanced Diffusion of Molecular Motors in the Presence of Adenosine Triphosphate and External Force

Ryota Shinagawa and Kazuo Sasaki E-mail: sasaki@camp.apph.tohoku.ac.jp
\abst

The diffusion of a molecular motor in the presence of a constant external force is considered on the basis of a simple theoretical model. The motor is represented by a Brownian particle moving in a series of parabolic potentials placed periodically on a line, and the potential is switched stochastically from one parabola to another by a chemical reaction, which corresponds to the hydrolysis or synthesis of adenosine triphosphate (ATP) in motor proteins. It is found that the diffusion coefficient as a function of the force exhibits peaks. The mechanism of this diffusion enhancement and the possibility of observing it in -ATPase, a biological rotary motor, are discussed.

1 Introduction

Diffusion is a fundamental phenomenon in nonequilibrium physics and has been of research interest since Einstein [1, 2] figured out that the Brownian motion of a mesoscopic particle is caused by the fluctuation in the bombardment of fluid molecules on it. One area of recent interest is the diffusion enhancement of a Brownian particle moving in a periodic potential. The diffusion of such a particle should be reduced compared with free diffusion (diffusion without a potential) because the potential barrier suppresses the meandering of the particle. The enhancement is possible when the system is driven away from the thermal equilibrium state, for example, by applying an external force [3, 4, 5, 6] or by switching the potential (an on-off ratchet mechanism) [7, 8]. The diffusion coefficient can be increased considerably by tuning the strength of a constant external force [3, 4, 5], the frequency of an unbiased ac force [6], or the rate of potential switching [7, 8].

In the case of a particle in a one-dimensional periodic potential in the presence of a constant external force (a particle in a tilted periodic potential), the diffusion coefficient as a function of exhibits a peak at a value close to the maximum slope of the potential [3, 4, 5]; note that in the absence of thermal noise, the particle remains stationary at a force-balanced location for while it continues to run in one direction (“running state”) for . The reason for this diffusion enhancement is that the behavior of the particle, whether to remain in a potential well or to move to the adjacent well, is quite sensitive to thermal noise for close to , which results in a large dispersion of the particle displacement and hence a large diffusion coefficient. This diffusion enhancement was experimentally confirmed for a colloidal particle in a potential created by periodically arranged optical traps [9, 10, 11]. More recently, enhanced diffusion of this type has been observed for a rotary motor protein, -ATPase, in the absence of ATP (a fuel molecule required for the motor to work) and in the presence of a constant external torque [12]; from the dependence of the rotational diffusion coefficient on the torque, the potential barrier for the rotor of this motor was estimated for the first time.

In this work, we investigate the diffusion of molecular motors in the presence of ATP and a constant external force on the basis of a simple theoretical model schematically shown in Fig. 1 (see Sect. 2 for a detailed description). It is found that the diffusion coefficient shows peaks in its dependence on the force; the mechanism of the diffusion enhancement at a high ATP concentration is similar to that for a particle in a tilted periodic potential, while an alternative mechanism applies when the ATP concentration is low. We also analyze the data on the torque dependence of the rotation rate for -ATPase [13] using our model, and predict that the diffusion enhancement of both types can be observed experimentally.

Figure 1: (Color online) Model for motor protein that undergoes stepwise motion. A rotary motor (a) or a “two-legged” linear motor (b) is modeled as a Brownian particle moving in a series of potential valleys (c) arranged periodically (period ) on the -axis. Here, represents the rotation angle of the rotor (thick arrow) in (a) or the displacement of the joint (big dot) connecting the legs in (b), and corresponds to the step size. See the text for a detailed explanation.

The paper is organized as follows. In Sect. 2, our model for the molecular motor is defined, and the methods of calculating the average velocity and the diffusion coefficient of the motor are explained in Sect. 3. The result for the dependence of the diffusion coefficient on the external force is presented and the mechanism of the diffusion enhancement is discussed in Sect. 4. Section 5 is devoted to the analysis of -ATPase based on our model, and Sect. 6 provides concluding remarks.

2 Model

We consider the following model of a molecular motor, which can be a rotary motor consisting of a rotor and a stator [Fig. 1(a)] or a ‘two-legged’ motor walking along a linear track [Fig. 1(b)]. Our model is analogous to the ones introduced in Refs. \citenzimmermann12, watanabe13, kawaguchi14. Let be the rotation angle of the rotor for a rotary motor or the displacement of the joint of the two legs for a linear motor; hereafter, will simply be referred to as the displacement of the motor. We assume that the stator has a structure of -fold symmetry for the rotary motor ( for F-ATPase) or the track has a periodic structure for the linear motor. The motor changes its state, denoted by an integer index , upon a chemical reaction catalyzed by the motor; is changed to by a forward reaction and to by a backward (reverse of the forward) reaction. Let be the mechanical energy of the motor in state , which represents the interaction between the rotor and the stator or the elastic energy of the legs together with the interaction between the motor and the track. The potentials are assumed to satisfy the “periodicity condition”

(1)

where is for the rotary motor or the period of the track for the linear motor [Fig. 1(c)]. It is assumed that as .

The rates of transitions, , from state to are assumed to depend on and possess the same periodicity as Eq. (1), i.e., . Let be the free energy released by the forward reaction (the free energy of the environment is decreased by upon the reaction). Then, and should be related to each other through the detailed balance condition

(2)

where is the Boltzmann constant and is the temperature of the environment.

We assume that a constant external force is applied to the motor in addition to the force due to the interaction potential . The fluid surrounding the motor also exerts forces on the motor: the drag force , with being the frictional coefficient, and the random force, which is modeled as the Gaussian white noise. Then, the Fokker–Planck equation for the probability distribution of in state at time , , reads

(3)

In this work, we restrict ourselves to a particular case of

(4)

for simplicity, where , , and are positive constants. One of the reasons why the exponential dependence of on in Eq. (4) has been chosen is that a similar dependence was observed for F-ATPase [17, 18]. From Eqs. (2) and (4), we find that also depends exponentially on . It should be remarked that a model equivalent to our model with and given by Eq. (4) was studied in Ref. \citenkawaguchi14 for a different purpose.

A comment concerning the modeling of the linear motor may be in order. In motor proteins, such as kinesin and myosin V, which walk on linear tracks, the forward step is associated with ATP hydrolysis, but the backward step is usually not caused by its reverse reaction (ATP synthesis) [19, 20, 21]. Hence, our model as it is may not be applicable to these motor proteins. However, since the diffusion enhancement to be discussed in this work can occur in the situation where the motor takes only forward steps, we consider that the model can be appropriate for linear motors under restricted conditions.

3 Velocity and Diffusion Coefficient

The average velocity and the diffusion coefficient of the motor are defined by

(5)

where is the displacement of the motor at time and the angular brackets indicate the statistical average.

The average velocity can be obtained from the solution to the Fokker–Planck equation (3) for the steady state () as follows. Note that the steady-state distribution has the same periodicity, , as and . Let be the rescaled such that . Then we have

(6)

To calculate the diffusion coefficient, we need to solve a differential equation, which is related to the Fokker–Planck equation for the steady state [23, 22]:

(7)

Note that if the right-hand side of this equation were set to zero, the resulting equation with replaced by would be identical to the equation for the steady-state distribution obtained from Eq. (3). From a solution of Eq. (3), the diffusion coefficient is obtained as

(8)

where

(9)

is the diffusion coefficient due to the Einstein relation. It is remarked that if is a solution to Eq. (3), then , with an arbitrary constant, is also a solution. However, this ambiguity in does not affect the result for .

4 Diffusion Enhancement

It is convenient to work with dimensionless variables and parameters to present the results succinctly. By introducing the dimensionless time and displacement defined by

(10)

and the dimensionless external force and potential defined by

(11)

the Fokker–Planck equation (3) with Eq. (4) is rewritten as

(12)

where the dimensionless transition rates are given by

(13)

with

(14)

and

(15)

It is not difficult to see from Eqs. (6)–(15) that the dimensionless velocity and diffusion coefficient defined by

(16)

depend on only four parameters, , , , and .

It is remarked that and do not depend explicitly on the free energy change associated with the chemical reaction. This is because a particular set of the potential and transition rate given in Eq. (4) has been chosen. In this situation, can be absorbed into the dimensionless displacement , the effective external force , and the effective rate constant as given in Eqs. (10), (11), and (14), respectively. This helps us to reduce the number of parameters to work with.

The Fokker–Planck equation (12) for the steady state and the dimensionless version of Eq. (3) were solved numerically by the finite-difference method with a grid spacing of or smaller.

Our primary interest is in the dependence of on and how this dependence changes with , , and . Figure 2 shows examples of such dependence for and with several choices of . In each example, we observe two peaks, one for positive and one for negative . For the external force around these peaks, the diffusion is enhanced () compared with the diffusion expected from the Einstein relation (). It is also noted that is almost symmetric about for and is asymmetric for other cases; the asymmetry is more prominent for smaller . This asymmetry arises from the asymmetry between the forward and backward transition rates ; if , then we have the symmetry relation and it is not difficult to see that holds in this particular case. We expect that unless . The reason why is symmetric for large even if will be discussed in Sect. 4.1.

Figure 2: (Color online) Dependence of the diffusion coefficient on the external force for and for different choices of rate constant as indicated in the figure. The dimensionless quantities , , , , and are defined in Eqs. (11), (14), and (16).

Figure 3 shows how the position and the height of the peak observed in Fig. 2 vary with : the value of at which reaches a maximum is denoted by , and the corresponding maximum value is denoted by . The peak for in Fig. 2 is referred to as branch 1, and the one for is referred to as branch 2. In the limit of large , and tend to certain limiting values for both branches 1 and 2. The limiting values of for branches 1 and 2 are the same in their magnitudes and opposite in their signs, whereas those of are identical. These properties are consistent with the symmetry of for large mentioned above. As we will show in Sect. 4.3, in the limit of small , the asymptotic behavior of can be expressed as

(17)

where the upper and lower signs are for branches 1 and 2, respectively, and is a function of and ; the dashed lines in Fig. 3(a) represent this expression with for branch 1 and for branch 2. On the other hand, converges to definite values in this limit. These properties will be discussed in Sects. 4.2 and 4.3. In the intermediate region of , we notice peculiar behaviors of and for branch 1: the graph of has inflection points and that of has a dip. We do not have intuitive explanations for these behaviors.

  

Figure 3: (Color online) (a) Position and (b) height of the peaks in the graphs shown in Fig. 2 plotted against . The dashed lines indicate the asymptotic behaviors of expressed as Eq. (17).

4.1 Limit of large

Let us discuss the case of large . If the transition rates are large enough, the motor settles in a “chemical equilibrium” before the displacement changes appreciably. In this limiting case, the motor at displacement is in state with probability

(18)

where is the dimensionless potential defined in Eq. (11). Hence, the motor can be described as a Brownian particle moving in an effective potential[16]

(19)

which is a periodic function of period 1 and is symmetric, . An example of is depicted in the inset of Fig. 4. Note that is almost identical to one of the , if is large (), except for small intervals of width on the order of around () where the graphs of and intersect. This observation indicates that the transitions occur mainly when the motor is in these small intervals, where we have . Therefore, the condition for the description in terms of the effective potential to be appropriate should be as follows [16]: these rates are large compared with the inverse of the time () for the particle to traverse an interval of length by diffusion or by drift with velocity (due to or at ), i.e.,

(20)

Figure 4: (Color online) Diffusion coefficient (solid line) as a function of external force for a particle moving in the effective potential , defined in Eq. (19), for shown in the inset (solid line). The data from Fig. 2 for and 10 are also shown (open circles and dashed line, respectively). The vertical arrows indicate the locations of , where is the maximum slope of the effective potential. The dashed lines in the inset are the potentials () given in Eq. (11).

The diffusion of a Brownian particle moving in a one-dimensional periodic potential under a constant external force was studied thoroughly by Reimann et al. [4, 5], and a closed-form expression for the diffusion coefficient was obtained [4, 5, 24, 25]. In the present context, it is given by

(21)

where

(22)

The dependence of on given by Eq. (21) for is plotted in Fig. 4 (solid line). Note that is symmetric about , which is due to the symmetry of mentioned above. As explained in the introduction [4, 5], has peaks near (indicated by vertical arrows in Fig. 4), where is the maximum slope of the effective potential . In Fig. 4, the dependence of on shown in Fig. 2 for and 10 is also plotted (open circles and dashed line, respectively). We see that the data for is very close to that predicted by Eq. (21), and that the result for (data not shown) is indistinguishable from the latter. Let us check whether these numerical results are consistent with the condition (20) for the description in terms of the effective potential to be valid. The right-hand side of this inequality is about 130 for and . Considering the fact that the result for is very close to the result of the effective-potential approximation, we suggest that the condition (20) can practically be replaced by

(23)

From all these observations, we conclude that the mechanism of the diffusion enhancement observed in Fig. 2 for large is essentially the same as that for the diffusion enhancement of the Brownian particle in a tilted periodic potential.

4.2 Limit of small

Now we consider the case of small . If the transition rates are small enough, the displacement of the motor in state will acquire the equilibrium distribution

(24)

before a transition to state or takes place. In Eq. (24), defined by

(25)

is the displacement of the motor at which the external force is balanced with the force due to the potential defined in Eq. (11). In Fig. 5(a), the probability density functions (pdfs) in the steady state, with , are plotted together with given by Eq. (24) for in the case of , , and . In this example, is almost identical to , and therefore the “quasi-equilibrium” approximation discussed above is likely to be appropriate.

Figure 5: (Color online) Probability density functions (bold solid lines) in state for , , and with the dimensionless force (a) , (b) , and (c) . The open circles in (a) represent the equilibrium distribution function in state in the absence of transitions to other states; the dashed lines in (b) and (c) are multiplied by and , respectively. The thin solid lines in (b) and (c) are the total distribution functions defined by Eq. (28).

In the quasi-equilibrium situation, the effective rates, , of transitions from state are given by

(26)

and the dynamics of the motor is equivalent to a random walk on a lattice of lattice spacing 1 with forward and backward hopping rates and , respectively. Then, the average velocity and the diffusion coefficient can be expressed as and , respectively. Therefore, increases exponentially with increasing ; the rate of increase is determined by the factor , and is larger for than for in the case of and shown in Fig. 2, because is much larger than in this case. This prediction is consistent with the results for small ( and ) presented in Fig. 2. Note that this exponential increase in arises from the fact that the transition rates increase exponentially with ; see Eq. (13).

The condition for the quasi-equilibrium approximation for to be valid may be obtained as follows. The relaxation time for the equilibrium in to be reached in the parabolic potential is on the order of  [26]. If the rate of this relaxation is much larger than the rate of potential switching in the quasi-equilibrium approximation, then this approximation should be appropriate. Since we have for from Eq. (26), this condition may be expressed as

(27)

For example, the right-hand side of this inequality is about 14.6 for and , and therefore and satisfy this condition.

Since increases exponentially with , the quasi-equilibrium approximation will no longer be valid for a large enough : the forward (or backward) transition will occur before the equilibrium in is reached, and therefore the motor will move continuously in one direction without pauses around the force-equilibrium locations, given in Eq. (25); the diffusion coefficient will be small in such a running state. The pdfs shown in Fig. 5(c) indicate that the running state seems to be realized for , for example, in the case of , , and : the function is well separated from to the left, which suggests that the switching of potential from to occurs before the motor reaches the force-equilibrium position , and hence the driving force continues to push the motor in one direction. As a result, the total pdf

(28)

shown by the thin solid line in Fig. 5(c), does not vary much with , which implies a more or less smooth flow of the motor and a small diffusion coefficient.

From the above arguments, it is expected that in an intermediate range of , the probability of the motor remaining around a force-equilibrium position and that of switching the potential before the motor arrives at a force-equilibrium position are comparable. The pdfs shown in Fig. 5(b) seem to support this idea. In such a situation, the dispersion of the motor displacement will increase with time more rapidly than in the quasi-equilibrium state and in the running state. Therefore, the diffusion enhancement is anticipated for intermediate values of .

According to this scenario, the values of around which the diffusion enhancement occurs may roughly be estimated as follows. The rate of relaxation for the equilibrium in to be reached in the parabolic potential is on the order of  [26] as mentioned earlier. If the rate of potential switching is comparable to this rate of relaxation, the diffusion will be enhanced. Taking either of the effective transition rates in Eq. (26) in the quasi-equilibrium approximation as the switching rate, we would have the condition for the diffusion enhancement. This means that the diffusion coefficient as a function of would have peaks around at , where

(29)

The dependence of on is similar to Eq. (17) for the peak positions for small . This qualitative agreement of Eq. (29) with Eq. (17) seems to support the above scenario of the diffusion enhancement for small : the crossover from the quasi-equilibrium state to the running state by the force-assisted increase in the transition (switching) rate causes the diffusion enhancement.

4.3 Effects of and

So far, we have discussed how the diffusion enhancement is affected by . Now we turn our attention to the effects of and . In the case of large , the diffusion enhancement is essentially the same as in the case of a Brownian particle in a tilted periodic potential, as explained in Sect. 4.1: the result does not depend on , and the peak position in is close to the value of where the effective potential has the maximum slope. Since it is easy to see that the maximum slope in approaches as , we expect that for large . In the case of small , however, it is not clear how the diffusion enhancement is affected by and . We investigate this problem in this subsection.

As discussed in the preceding subsection, under the condition that the diffusion enhancement occurs in the case of small , the backward transition rate is negligibly small compared with the forward transition rate for branch 1 (positive ) or vice versa for branch 2 (negative ). Therefore, we employ an approximation in which either or is neglected. Then, the Fokker–Planck equation (12) is simplified as

(30)

where and with the plus and minus signs being for branch 1 and branch 2, respectively. The transition rate in Eq. (30) depends on two parameters,

(31)

as

(32)

Note that in this approximation, the model is characterized by three parameters, , , and , whereas the original model is characterized by four parameters, , , , and , as remarked in the first paragraph of Sect. 4. From this observation, we conclude that the dimensionless velocity and diffusion coefficient depend on only these three parameters: the external force is now absorbed into .

Figure 6 shows the dependence of the diffusion coefficient on the scaled force for with several choices of . It appears that the peak height of diffusion enhancement increases with . Let be the value of at which reaches the maximum for given and , and be the corresponding maximum value of . Then, from the definition of in Eq. (31), it is deduced that can be expressed as Eq. (17) if either or can be neglected. For example, we have for and from the result shown in Fig. 6, which provides the dashed line for branch 1 in Fig. 3(a). Note that depends on and but not on in the present approximation; this is the reason why in Fig. 3(b) tends to a constant value in the limit of small . The peak height of for in Fig. 6, for example, is identical to the limiting value of for branch 1 in Fig. 3(b).

Figure 6: (Color online) Dependence of the diffusion coefficient on the scaled external force defined by Eq. (31) for with different choices of as indicated in the figure.

Now we analyze how and depend on and . In Fig. 7(a), is plotted against and in Fig. 7(b), is plotted against for , 5, and 10. It is seen that both and increase with for any value of , and decrease with increasing for any . The asymptotic behaviors of and for large turn out to be expressed in simple mathematical forms as

(33)

where coefficients and depend on . The lines in Fig. 7 indicate these asymptotic expressions with correction terms: we have used the expression

(34)

for , where and are determined so that this expression fits the data of in Fig. 7(a), and the expression

(35)

for , where all the data in Fig. 7(b) are used to determine , , and . It may be worth pointing out that the result (29) obtained from the qualitative argument correctly gives the logarithmic dependence of on obtained numerically for large .

Figure 7: (Color online) Dependences of (a) and (b) on for , 5, and 10. Here, denotes the value of at which reaches the maximum, , for given values of and . The lines indicate the asymptotic expressions with correction terms, Eqs. (34) and (35) for and , respectively.

We were not able to find simple mathematical expressions for the dependences of and in Eq. (33) on . The numerically obtained values of and are shown as a function of in Figs. 8(a) and 8(b), respectively. These data are fitted to polynomials of degree three in , and the results are indicated by the solid lines in these figures. Both and are monotonically decreasing functions of and appear to tend to nonzero values as . Extrapolation using the data for large yields and for the limiting values of and , respectively.

Figure 8: (Color online) Dependences of coefficients and in the asymptotic expressions in Eq. (33) on . The lines represent the fitting by polynomials of degree three in .

5 Application to -ATPase

Our model may be too simple for describing real motor proteins. Nevertheless, it would be tempting to speculate whether the enhancement of diffusion predicted for our model can be observed in -ATPase. The rotational diffusion coefficient has not been measured for this molecular motor in the presence of both ATP and external torque, although the dependence of the rotation rate on the external torque was observed in the presence of ATP [27, 13]. Here, we determine the values of the parameters in our model so that the theoretical result for agrees with the experimental data, and then calculate the diffusion coefficient using these parameters.

The data we use to determine the model parameters are those shown in Fig. 9, which were obtained by Toyabe et al[13]. Parameters other than , , and are determined as follows. We have from the threefold symmetry in the structure of -ATPase. The experimental conditions were , , and , where [ATP], for example, stands for the ATP concentration. From these concentrations of substances may be estimated. However, the estimations of are somewhat different among those published in the literature. We used the largest values listed in Table S1 of Ref. \citentoyabe11: and for and , respectively. This is because, with these choices of , good agreement is obtained between the stall torque (the value of torque at which the rotation rate vanishes) predicted by our model, , and the stall torque obtained from the data in Fig. 9 for . The value of depends on the size of the probe (bead of diameter in Ref. \citentoyabe11) attached to the rotor ( subunit) and the geometry of how it is attached, which is difficult to control, and it was estimated for every measurement as described in Refs. \citentoyabe10 and \citentoyabe12 to obtain and for and , respectively [30].

Figure 9: (Color online) Rotation rate of -ATPase as a function of external torque. The lines are theoretical results fitted to the experimental data (symbols) obtained by Toyabe et al. [13] (the error bars indicate the standard deviation). Note that the scales of the vertical axes for and are different.

In determining , , and , note that the forward transition rate in the present model represents the following series of events in -ATPase [31]: (i) the binding of ATP to the motor accompanied by the release of ADP from the motor, (ii) the hydrolysis of ATP catalyzed by the motor, and (iii) the release of Pi from the motor. Therefore, the rate constant in Eq. (4) is expected to increase as the ATP concentration increases. Here, we assume that for is large enough for the effective-potential description discussed in Sect. 4.1 to be appropriate; the validity of this assumption will be discussed at the end of this section. Since the rotation rate and diffusion coefficient are independent of and in this approximation, we can determine by comparing the theoretical result with the data for in Fig. 9. By trial and error, it was found that the data fits well to the theory with as shown in Fig. 9. This is the same strategy as the one used by Kawaguchi et al. [16] for their harmonic-potential model. However, the value that they obtained is different from ours. The cause of this discrepancy is likely to be the different values of used in the analysis: they used , which was reported in Ref. \citentoyabe12; this value is different from that for the probe used to obtain the data [13] in Fig. 9.

We tried to determine the values of the remaining parameters and from the data for in Fig. 9, but were unsuccessful in finding a unique set of these values. For any value of smaller than about , it is possible to fit the data to the theory by adjusting the value of . Four examples of such fittings are presented in Fig. 9 for , 1, 2, and with the adjusted values of listed in Table 1. Although these lines lie close to each other for , they become separated as increases for (assisting torque): the rotation rate becomes saturated for , while it keeps increasing for the other cases (the larger the value of , the larger the rate of increase). In an earlier experiment [27], the rotation rate was observed to increase with assisting torque under various conditions, which indicates that the value of should not be too small.

() () Eq. (27) Eq. (20)
0 10 3.94 19.7
1 8.8 1.26 18.6
2 7.2 0.37 15.8
3 6.4 0.12 12.0
Table 1: The value of in the second column is determined so that the experimental data on the rotation rate for shown in Fig. 9 fits the theoretical result for each choice of listed in the first column. Then, and the right-hand side (RHS) of inequality (27) are estimated for . Furthermore, and RHS of inequality (20) are estimated for , by assuming that .

Other data on the values of are available in the literature. First, the rate of ATP binding was measured [17, 18] as a function of the rotation angle of the rotor. Under low ATP concentrations, the ATP-binding event is rate-limiting [32] and therefore the rate of this event may be identified, in the first approximation, with the rate of forward transition in the present model in analyzing the data for . With this identification, we obtain and from the result in Ref. \citenwatanabe12; these values are not too far from those given in the last two lines in Table 1. Second, the theoretical investigation of Ref. \citenkawaguchi14 revealed that the asymmetry parameter , which corresponds to in the present model, should be close to zero so that the theory is consistent with the experimental finding [28] that the heat dissipation through the internal degrees of freedom in -ATPase is negligibly small. Note that corresponds to , and therefore our choice of listed in Table 1 seems consistent with the result of Ref. \citenkawaguchi14 that should be small.

From the consideration given above, we conclude that and should be appropriate for the present model to describe the rotational dynamics of -ATPase. By using these parameters, the values of defined in Eq. (14) and the right-hand side of inequality (27) are estimated and listed in Table 1. Note that the condition for the quasi-equilibrium approximation to be valid [Eq. (27)] is satisfied for for any choice of in the range given above.

Using the model parameters thus determined, we calculated the dependence of the diffusion coefficient on the external torque , and the result is shown in Fig. 10. We see diffusion enhancements similar to the ones observed in Fig. 2; the case of for is exceptional, where increases monotonically with and saturates to a limiting value. Since the torque of is in an experimentally accessible range [12], it is suggested that the diffusion enhancement can be observed experimentally for -ATPase: the diffusion enhancement of the type discussed in Sect. 4.1 will be observed under high ATP concentrations (e.g., ), and that of the type discussed in Sect. 4.2 will be observed under low ATP concentrations (e.g., ). However, the diffusion enhancement of the latter type under assisting torque () may not be observed if is too small (smaller than about for ).

Figure 10: (Color online) Dependence of the diffusion coefficient on the external torque predicted for -ATPase.

Unfortunately, we were not able to determine the model parameters and , and we could only provide ranges of these parameters, as mentioned above. The theoretical results shown in Fig. 9 indicate that the measurement of the rotation rate under assisting torque will help determine these parameters. In addition, the measurement of will provide complementary information on , since the position and height of the peak in for at low ATP concentrations are quite sensitive to , as seen in Fig. 10, even though the precision of the diffusion coefficient determined experimentally may not be as good as that of the rotation rate.

Now we discuss the validity of the effective-potential approximation used to analyze the data for . The condition for this approximation to be valid is given by inequality (20). However, we do not know the value of for and hence cannot check whether this condition is satisfied. If the ATP-binding event is rate-limiting, the rate constant is proportional to (since the forward transition rate can be identified with the ATP binding rate, which is proportional to the ATP concentration), and for arbitrary [ATP] can be estimated from the value of for . Here, we tentatively assume that the relation holds even up to , and the values of and the right-hand side of inequality (20) for estimated based on this assumption are listed in Table 1. Although these values do not satisfy the condition (20), they satisfy the weaker condition (23) suggested from the numerical results shown in Fig. 4. Therefore, the effective-potential approximation is likely to work for . For confirmation, we have calculated the rotation rate and the diffusion coefficient without the effective-potential approximation for by using the parameters , , and , and estimated in this way for each choice of . The results (data not shown) are almost identical to those given in Figs. 9 and 10 obtained on the basis of the effective-potential approximation. Therefore, it is plausible that this approximation is appropriate.

6 Concluding Remarks

We have introduced a simple model for molecular motors to investigate the dependence of the diffusion coefficient on the constant external force in the presence of ATP, inspired by a similar work [12] on -ATPase in the absence of ATP. It turns out that the diffusion enhancement occurs, if the force is in certain ranges, in the presence as well as in the absence of ATP. The mechanism of enhancement for high ATP concentrations is essentially the same as that in the case without ATP (i.e., the mechanism of diffusion enhancement in a tilted periodic potential [4, 5]). An alternative mechanism applies for low ATP concentrations, and the diffusion enhancement is sensitive to how the transition rate depends on the displacement (rotation angle) of the motor. It is suggested that both types of diffusion enhancement can be observed for -ATPase in the experimentally accessible range of external torque and that such observations will provide useful insights into the angular dependence of the reaction rate for this motor protein.

Although the present model seems to convincingly explain the dependence of the rotation rate on external torque for -ATPase, the value of we have determined is somewhat smaller than obtained experimentally in Ref. \citentoyabe11 from the angular distribution of the probe attached to the rotor, or estimated in Ref. \citenkawaguchi14 by analyzing the potential profile reconstructed in Ref. \citentoyabe12. One of the reasons for this discrepancy in , we suppose, is that the present model is too simple. For example, there is evidence that the interaction potential is not a simple parabola [29, 16, 33]. Furthermore, it may be necessary to take into account additional (intermediate) chemical states because the ATP-hydrolysis cycle catalyzed by -ATPase proceeds in several steps [31]; accordingly, the rotation of the rotor per hydrolysis cycle breaks up into and substeps [32]. We will investigate a refined model, in the future, to resolve the inconsistency between the present theory and the experiment on -ATPase.

Acknowledgements.
We thank Kumiko Hayashi and Takashi Yoshidome for useful discussions and comments. We also thank Shoichi Toyabe for providing us with the experimental data reported in Ref. \citentoyabe11 and unpublished data on for -ATPase.

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