Phenomenological Analysis of ATP Dependence of Motor Proteins

In this study, through phenomenological comparison of the velocity-force data of processive motor proteins, including conventional kinesin, cytoplasmic dynein and myosin V, I found that, the ratio between motor velocities of two different ATP concentrations is almost invariant for any substall, superstall or negative external loads. Therefore, the velocity of motors can be well approximated by a Michaelis-Menten like formula , with the step size, and the external load dependent rate of one mechanochemical cycle of motor motion in saturated ATP solution. The difference of Michaelis-Menten constant for substall, superstall and negative external load indicates, the configurations at which ATP molecule can bind to motor heads for these three cases might be different, though the expression of as a function of might be unchanged for any external load . Verifications of this Michaelis-Menten like formula has also been done by fitting to the recent experimental data.

In this study, by phenomenological comparison of the velocityforce data of different ATP concentrations, I found that the velocity of processive motor proteins can be described by a Michaelis-Menten like formula V~½ATPk(F )L=(½ATPzK M ), but might with different constant K M for substall, superstall and negative external loads. The motor velocity in saturated ATP solution is V Ã~k (F )L, and generally, the velocity of motor can be obtained by multiplying V Ã by a constant [ATP]/([ATP]+K M ).

Results
For the sake of comparison, the velocity-force data of kinesin, dynein and myosin are plotted in Figs. 1, 2 and 3(a). In Fig. 1(a), the thick dashed line V 1 is the velocity-force data of kinesin for [ATP] = 1 mM obtained by Nishiyama et al [6], and the solid line V 2 is for [ATP] = 10 mM. One can easily see that there is only little difference between the lines V 2 and V 1 =3:3. Similar phenomena can also be found for the velocity-force data of dynein and myosin obtained in [8,10,32], see Figs. 1(b,c,d). Meanwhile, for negative and superstall force cases, one can find the similar results, but the ratio constants might be different from the positive substall force case, see Figs. 2 and 3(a) for data of kinesin obtained in Refs. [4,7,9]. For the kinesin data in [9], the ratio constant is about 2.6 for Fv0, about 7.1 for 0ƒF ƒ7 pN, and about 2.3 for F w7 pN [see Fig. 2(a)]. For the data in [7], the ratio constant is about 16 for F v0, and about 29 for 0ƒFƒ5 pN [see Fig. 2(b)]. But for the kinesin data measured in [4], the constant 3.6 works well for both substall and negative external load [see Fig. 3(a)].
From the above observations about the experimental data plotted in Figs. 1 and 2, one can see that the velocity-force relation of motor proteins satisfies V (F ,½ATP)~f (½ATP)V Ã (F ). Where V Ã~VÃ (F) is the velocity-force relation at saturated ATP concentration, and obviously V Ã can be written as V Ã (F )~k(F )L with L the step size of motor proteins, and k(F ) the force dependent rate to complete one ATP hydrolysis cycle (coupled with one mechanical cycle). The function f (½ATP) increases with [7,[33][34][35]. Finally, the velocity formula can be written as V (F,½ATP)1 ATPk(F)L=(½ATPzK M ).
To verify the above velocity-force formula, the force dependent expression of rate k(F ) should be given firstly. Usually, the mechanical coupled cycle of ATP hydrolysis includes several internal states, here, as demonstrated in the previous mechanochemical model [27], I assume that, in each cycle, there are two internal states, denoted by state 1 and state 2 respectively.
Let u i ,w i be the forward and backward transition rates at state i, then the steady state rate k(F ) can be obtained as follows [27,36] The force dependence of rates u i ,w i are assumed to be [27] Where k B is the Boltzmann constant, T is the absolute temperature, and h + i are load distribution factors which satisfy For this two-state model, one can easily get the following formula of motor velocity  The fitting results of the above velocity-force formula to kinesin data measured in [9] are plotted in Fig. 3(b). In which, the Michaelis-Menten constant K M~1 5:8 mM for F v0, K M~3 9:2 mM for 0ƒF ƒ7 pN, and K M~1 1:9 mM for F w7 pN, other parameter values are listed in Tab. 1. Meanwhile, the fitting results to the dynein data measured in [10] and myosin data measured in [5] are plotted in Fig. 4(a) and Fig. 4    . Experimental data for cytoplasmic dynein obtained in [10] and myosin obtained in [5] (see also [32] for the method to get the present values), and the theoretical prediction using the Michaelis-Menten like formula V~½ATPk(F)L=(½ATPzK M ). constant between V 1 and V 2 is 6.6, but 6.5 is used in Fig. 1(b). The different values of K M (or R M in Fig. 2) for F v0, 0vF vF s and F wF s means the possible motor configurations, at which ATP can bind to motor head, might be different for these three force regimes. But in each configuration, the ATP binding rate to motor heads might be the same, i.e. it is independent of the ways used (or time spent) by the motor to get to this configuration. But the time spent by motor proteins to get to such configurations depends on external force F . Note, the step size used in the calculations is L~8:2 nm for motor proteins kinesin and dynein, but L~36 nm for myosin V. Certainly, the same fitting process can also be done to other experimental data. The plots in Figs. 3(b) and 4 indicate that, the experimental data of motor proteins can be well reproduced by the Michaelis-Menten like formula (4), so the phenomenological analysis about the ATP dependence of motor motion is reasonable.

Discussion
In summary, in this study, the ATP dependence of motor proteins is phenomenologically discussed. Based on the recent experimental data and numerical calculations, I found the motor velocity can be well described by a Michaelis-Menten like formula V~½ATPk(F)L=(½ATPzK M ) with force dependent rate k(F ) at saturated ATP. The different values of K M for substall, superstall and negative external load imply, the ATP binding rate to motor heads might be different for these three cases, though the basic mechanism in each mechanochemical cycle (either forward or backward) might be the same. An obvious conclusion from the Michaelis-Menten like formula is that the stall force, under which the mean motor velocity is vanished, is independent of ATP concentration [9,10,12]. Finally, to describe the ADP concentration dependence of motor velocity, the formula V~½ATPk(F)L=(½ATPzK M ) should be changed correspondingly, such as V~½ATPk(F )L=(½ATPzK M (1z½ADP=K 1 )) with K 1 a new parameter [37].

Author Contributions
Conceived and designed the experiments: YZ. Performed the experiments: YZ. Analyzed the data: YZ. Contributed reagents/materials/analysis tools: YZ. Wrote the paper: YZ.