Zhisong
Wang
^{ab}
^{a}Department of Physics, National University of Singapore, Singapore 117542, Singapore. E-mail: phywangz@nus.edu.sg
^{b}NUS Graduate School for Integrative Sciences and Engineering, National University of Singapore, Singapore 117542, Singapore

Received
17th August 2017
, Accepted 9th December 2017

First published on 14th December 2017

Many motor proteins achieve high efficiency for chemomechanical conversion, and single-molecule force-resisting experiments are a major tool to detect the chemomechanical coupling of efficient motors. Here, we introduce several quantitative relations that involve only parameters extracted from force-resisting experiments and offer new benchmarks beyond mere efficiency to judge the chemomechanical optimality or deficit of evolutionary remote motors on the same footing. The relations are verified by the experimental data from F_{1}-ATPase, kinesin-1, myosin V and cytoplasmic dynein, which are representative members of four motor protein families. A double-fitting procedure yields the chemomechanical parameters that can be cross-checked for consistency. Using the extracted parameters, two generic maps of chemomechanical optimality are constructed on which motors across families can be quantitatively compared. The maps reveal two chemomechanical coupling regimes, one conducive to high efficiency and high directionality, and the other advantageous to force generation. Surprisingly, an F_{1} rotor and a kinesin-1 walker belong to the first regime despite their obvious evolutionary gap, while myosin V and cytoplasmic dynein follow the second regime. This analysis also predicts the symmetries of directional biases and heat productions for the motors, which impose constraints on their chemomechanical coupling and are open to future experimental tests. The verified relations, six in total, present a unified fitting framework to analyze force-resisting experiments. The generic maps of optimality, to which many more motors can be added in future, provide a rigorous method for a systematic cross-family comparison of motors to expose their evolutionary connections and mechanistic similarities.

## Insight, innovation, integrationThis study introduces new quantitative benchmarks to judge the chemomechanical optimality or deficit of motor proteins across evolutionarily remote families. Two generic maps of optimality are constructed, and analysing four motor families on the maps reveals two distinct chemomechanical coupling regimes. Overall, this study presents a unified fitting framework for the systematic analysis of force-resisting experiments on many different motor proteins, with the extracted parameters revealing their revolutionary connection and mechanistic similarities. |

A fully experiment-based, quantitative comparison across motor protein families on the same footing offers a rigorous method to expose evolutionary connections and mechanistic similarities between many motor proteins. Previous studies^{31,32} suggested that motors from kinesin and myosin families, and perhaps also G proteins, share a common ancestor in natural evolution as they possess similar mechanisms for ATP-induced conformational changes and for direction rectification.^{13,18} On the other hand, motor proteins are presumably optimized through natural evolution, but quantitative evidence of chemomechanical optimality is hardly beyond the single quantity of the measured efficiency.

In this study, we introduce several quantitative relations that allow an experiment-based cross-family comparison of the chemomechanical optimality or deficit of motor proteins on the same footing beyond mere efficiency. These relations are fully expressed with chemomechanical parameters extracted from single-molecule force-resisting experiments, which are a major tool to detect the chemomechanical coupling of efficient motors. These relations are applied to analyze an F_{1}-ATPase rotor and the processive walkers kinesin-1, myosin V and cytoplasmic dynein, which are representative members of four motor protein families. A double-fitting procedure yields the chemomechanical parameters that can be cross-checked for consistency. All the four motors pass the consistency check, confirming the wide applicability of these relations. Using the extracted chemomechanical parameters, two generic maps of optimality are constructed on which virtually any motor can be quantitatively compared. The maps indicate the existence of two chemomechanical coupling regimes and expose a surprising mechanistic similarity between an F_{1} rotor and a kinesin-1 walker.

A motor of optimal efficiency satisfies four quantitative relations in terms of directionality (D), maximum work (W_{max}), maximum efficiency (η_{max}), and energy utilization (ΔG_{η}). All the quantities can be obtained from force-resisting experiments. No kinetic rates are involved (hence they are free of complication from load dependences of catalytic rates that are difficult to measure in experiments).

First, the maximum work of such an optimally efficient motor is related to its directionality at zero load (D_{0}),

W_{max} = 2γ_{d}k_{B}T × ln[(1 + D_{0})/(1 − D_{0})]. | (1) |

Second, the motor's maximum efficiency is determined by the step partition γ_{d} and another constant γ_{e} characterizing energy partition,

(2) |

Third, the motor's directionality under a finite opposing load (F) has a distinct load dependence,

D(F) = tanh[(Δμ_{D} − Fd_{D})/4k_{B}T]. | (3) |

ΔG_{η}(F) = Δμ_{D} + Fd_{in}. | (4) |

Fitting the D-versus-F data with eqn (3) results in a curve whose shape yields the direction-pertinent displacement d_{D}, and thereby the step partition γ_{d}. Extrapolating the fitting curve yields D_{0} and Δμ_{D} at F → 0, and yields the stall force F_{s} = Δμ_{D}/d_{D} at D → 0. The fitted Δμ_{D} value leads to a value of γ_{e}via the measured Δμ, and the fitted F_{s} leads to W_{max} and η_{max}via the measured d and Δμ. The resultant D_{0}, γ_{d}, γ_{e} and F_{s} can be used to test the W_{max}–D_{0} relation (eqn (1)) and the η_{max}-partition relation (eqn (2)).

Similarly, fitting the ΔG_{η}-vs.-F data with eqn (3) results in a curve whose slope yields the displacement d_{in}, and thereby the step partition γ_{d} again. Extrapolating the fitting curve at F → 0 yields the Δμ_{D} value again. The second fit differs essentially from the first fit, and offers a consistency check: the two independent fits must yield the same Δμ_{D} value, and must yield either d_{D} or d_{in} displacement that adds to match the total step size (d = d_{D} + d_{in}).

Hence a double-fitting to the stepping ratio-vs.-load data from two different aspects (eqn (3) and (4)) yields self-consistent chemomechanical parameters that fully determine a motor's maximum work and efficiency. With the extracted parameters D_{0}, γ_{d}, and γ_{e}, eqn (1) and (2) define two generic maps of optimality against which different motors, even evolutionarily remote ones across motor protein families, can be quantitatively assessed for their extent of chemomechanical optimization.

In addition to the above four relations with no rates explicitly involved, two more quantitative relations involving catalytic rates exist for a motor that is optimal not only in efficiency but also in power. There is a power–efficiency relation, which is specified by four parameters (i.e., ATP turnover rate, two rate ratios, and heat production, see eqn (5) in the Methods section). The power-vs.-efficiency data for a motor can be obtained from the speed-vs.-load data measured at a controlled Δμ value. The data for a motor with a tight chemomechanical coupling can be fitted by the power–efficiency relation to yield the load dependence of the motor's ATP turnover. There is also a quantitative relation between the maximum efficiency of an optimal motor and its speed at zero load (eqn (7), see the Methods section). This efficiency–speed relation is useful for analyzing the efficiency–speed trade-off of different motors with evolutionarily conservative ATPase activity. In total, this study presents six quantitative relations that form a unified fitting framework for the analysis of force-resisting experiments, yielding multiple parameters characterizing a motor's chemomechanical coupling.

Fig. 1–3 analyze the load-dependent directionality and energy utilization of the four motors. A good fitting to the experimental data is obtained and the condition of consistency is well satisfied by all the four motors. Based on the double-fitting procedure, the two independent fittings yield the same Δμ_{D} value for each motor, and yield either d_{D} or d_{in} displacement that adds to match the total step size (d = d_{D} + d_{in}, with the fitted values for d_{D} and d_{in} given in the captions of Fig. 1–3 for the four motors, together with their d values reported from literature). The consistency between the extracted parameters and the good overall fitting suggest that all the four motors follow the load dependences displayed in eqn (3) and (4).

Fig. 1 Directionality and utilized energy of kinesin-1 and F_{1}-ATPase. The symbols are the results extracted from the forward-to-backward stepping ratio data reported in ref. 62 and 63 (for kinesin) and in ref. 34 (for F_{1}, various colors for data collected from different individual F_{1} motors). The species producing the motor samples and the temperature (T) at which the data were collected are indicated. The lines are fits with eqn (3) (panels A and C) and with eqn (4) (panels B and D). The fits yield d_{in} = 0, d_{D} = d, and γ_{d} = 1.0 for both motors, with the resultant d_{D} values matching the measured step size of d = 120° for F_{1}^{33,34} and d = 8.2 nm for kinesin.^{60,62} The two fits also yield the same Δμ_{D} = 15.75k_{B}T, D_{0} = 99.924%, and F_{s} = 31.2 pN nm rad^{−1} for F_{1} and the same Δμ_{D} = 14.0k_{B}T, D_{0} = 99.818%, and F_{s} = 7 pN for kinesin. |

Fig. 2 Directionality and utilized energy of myosin V. The symbols are extracted from the forward-to-backward stepping ratio data obtained from ref. 64 (collected at T = 20 °C). The fit with eqn (3) (the line in panel A) yields d_{D} = 19 nm, Δμ_{D} = 9.4k_{B}T; the fit with eqn (4) (the line in panel B) yields d_{in} = 17 nm and the same Δμ_{D} value. The d_{D} and d_{L} values add to match the measured step size^{42} of d = 36 nm and yield γ_{d} = 1.9. The fit in panel A also yields D_{0} = 98.197% and F_{s} = 2 pN. |

Fig. 3 Directionality (panel A) and utilized energy (B) of cytoplasmic dynein. The symbols are the results extracted from the experimental data reported in ref. 35 (collected at T = 25° for dynein from a species of yeast). The ΔG_{η}(F) data in panel B are obtained using the load-dependent step size (d = 9.3 nm for 0 and 1 pN, 8.5 nm for 3 pN, 7.3 nm for 6 pN, and 6.0 nm for 7 pN as counted from the data of step-size distribution reported in ref. 35). The fit with eqn (3) (the line in panel A) yields d_{D} = 1.19 nm and Δμ_{D} = 2.0k_{B}T, F_{s} = 7 pN; the fit with eqn (4) (the line in panel B) yields d_{in} = 5.1 nm and the same Δμ_{D}. The sum of d_{D} and d_{in} values is 6.3 nm, which is near to the measured step of 6.0 nm under the stall load. Hence, W_{max} = 7 pN × 6.3 nm = 44.1 pN nm. The fits yield D_{0} = 46.212%, γ_{d} = 7.8 at the zero load, and yield γ_{d} = 5.0 at the stall load. |

Fig. 4 shows the comparison of the extracted step partition, zero-load directionality and maximum work for the four motors. The step partition is a load-independent constant for F_{1}, kinesin, and myosin V in line with their load-independent step size. The step partition for cytoplasmic dynein is load-dependent as its step size^{35} is reduced by increasing load. The extracted γ_{d} value is 1.0 for F_{1} and kinesin, 1.9 for myosin V, and between 5.0 and 7.8 for cytoplasmic dynein (Fig. 4A). While the γ_{d} value rises from F_{1} to kinesin, myosin V and then dynein, their zero-load directionality and the corresponding energy consumption (Δμ_{D}) both drop successively (Fig. 4B and C). However, the work done over the direction-decoupled displacement d_{in} increases from the virtually zero value for F_{1} and kinesin to the finite values for myosin V and dynein (Fig. 4C). In the order of increasing γ_{d}, the four motors appear to shift their balance of energy consumption from sustaining a robust direction to force generation. This is in line with the fact that myosin V and cytoplasmic dynein are evolutionarily related to force-generating motors (muscle myosin and axonemal dynein), but kinesin 1 and F_{1} are optimal motors for directional transport and energy conversion.

Fig. 4 Step partition (panel A), zero-load directionality (panel B), maximum work done over the two portions of the step (panel C), and directionality–work relation (panel D) for the four motors as extracted from Fig. 1–3. In panel B, the R_{0} data are the zero-load forward-to-backward stepping ratio obtained from the D_{0} data (inset) following the relation^{7,26}R_{0} = (1 + D_{0})/(1 − D_{0}). The lines in panel D extracted using eqn (1) for the γ_{d} values of the motors (for dynein, the γ_{d} value extracted near the stall load is used for relevance to the maximum work). The filled square, sphere, star and diamond are the D_{0} and W_{max} obtained from Fig. 1–3 for kinesin, F_{1}-ATPase, myosin V, and dynein, respectively. The empty triangles are for kinesin too but obtained from stepping data reported in ref. 65 by the same method as for Fig. 1A (downward, upward, leftward and rightward triangles for bovine brain kinesin 1 at temperature 7, 14, 25 and 35 °C, respectively). |

Despite their very different γ_{d} and D_{0} values, the four motors all satisfy the D_{0}–W_{max} relation of eqn (1) (Fig. 4D). This is also true for cytoplasmic dynein considering its step partition extracted near the stall load (γ_{d} = 5 value), which is relevant to the discussion of the maximum work.

The four motors also follow the efficiency relation of eqn (2) but to a different extent (Fig. 5A). For F_{1} and kinesin, their actual efficiency and energy partition extracted from experiments lie on the η_{max}–γ_{e} curve predicted from eqn (2) for their corresponding step partition. Myosin V deviates slightly; dynein deviates the most. As for the optimal γ_{d}–γ_{e} relation necessary for 100% efficiency, F_{1} follows it exactly, kinesin and myosin V differ from it slightly, and dynein even more (Fig. 5B).

Fig. 5 Efficiency–energy partition relation (panel A) and energy partition versus step partition (panel B) of the four motors. The lines in panel A are extracted using eqn (2) for the γ_{d} values of the motors, the line in panel B is the γ_{e}–γ_{d} relation necessary for 100% efficiency. The solid symbols are the motors’ actual efficiency (η = W_{max}/Δμ) and energy partition (γ_{e} = Δμ/Δμ_{D} − 1) estimated with the W_{max} and Δμ_{D} extracted from Fig. 1–3, and with the Δμ values at which the data were obtained. The Δμ value for the F_{1} data is 65.2 pN nm reported in ref. 34 (obtained at 0.4 μM ATP, 4 μM ADP and 1 mM phosphate). No measured Δμ is available for kinesin-1, myosin V and dynein data shown in Fig. 1–3. A kinesin-1 study^{62} suggests Δμ ∼ 20k_{B}T (equivalent of 82 pN nm for T = 25 °C), which is used to estimate the η and γ_{e} values for the three translational motors (solid symbols). |

Fig. 6 Power output versus efficiency of F_{1}-ATPase. (A) Maximal power predicted using eqn (5) for the various values of fuel energy (Δμ). The following parameters are used: the energy dissipation ΔQ_{in} is obtained from ref. 7 as 0.75k_{B}T; the rate ratios (a_{1} and a_{2}) are assumed to be load-independent and estimated from zero-load rates^{66} (the ATP hydrolysis rate k_{1,2} = 140 s^{−1}, the phosphate release rate k_{2,3} = 210 s^{−1}, the ADP dissociation rate k_{4,1} = 380 s^{−1}, and the ATP binding rate k_{3,4} = 4 × 10^{7} s^{−1} M^{−1} × [ATP] with ATP concentration [ATP] = 10 μM used to predict the curves shown in panel A). (B and C) Absolute power for various Δμ values but a fixed ATP concentration (10 μM) (B), and for a constant Δμ (75 pN nm) but different ATP concentrations (C). The power-vs.-efficiency data (symbols) are extracted from the speed-vs.-load data reported in ref. 34, where the Δμ values were controlled using various ATP, ADP and phosphate concentrations as Δμ = Δμ_{0} + k_{B}T × ln([ATP]/[ADP][Pi]). For data in panel B, [ATP] = 10 μM, [ADP] and [Pi] are 2 μM and 100 μM (for Δμ = 91 pN nm), 10 μM and 100 μM (for Δμ = 84 pN nm), 10 μM and 1 mM (for Δμ = 75 pN nm), 50 μM and 1 mM (for Δμ = 68 pN nm), and 100 μM and 2 mM (for Δμ = 62 pN nm). For data in panel C, [Pi] = 1 mM, [ATP] = [ADP] were changed simultaneously as indicated in the figure. The lines are eqn (5) (for the same values of ΔQ_{in} and the rates as for panel A) times the ATP turnover rate (k_{tot}), which is assumed to follow a load-dependent function^{36}k_{tot}(F) = k_{tot}(F = 0)/(p + qe^{Fδ/kBT}) with p + q = 1. The parameters k_{tot}(F = 0), δ and p are used as fitting parameters. The fitting for all the data yields δ = 36° (near a substep of ∼40° reported in ref. 67). The k_{tot}(0) and p values from the fitting are 63 s^{−1} and 0.965 (for triangle data), 63 s^{−1} and 0.94 (diamond), 63 s^{−1} and 0.94 (sphere), 55 s^{−1} and 0.90 (star), 44 s^{−1} and 0.89 (square) for panel B. The k_{tot}(0) and p values are 62 s^{−1} and 0.94 (triangle), 65 s^{−1} and 0.92 (sphere), 68 s^{−1} and 0.89 (diamond), 43 s^{−1} and 0.94 (star), 12 s^{−1} and 0.94 (square) for panel C. |

A study by Purcell et al.^{40} suggested that compared to myosin V, a group of other myosin motors partition more energy to promote the rate-limiting^{41} ADP dissociation for higher speed, resulting in less energy for work. According to the consensus chemomechanical coupling pattern^{42} for myosin V and other motors from the myosin family (see Fig. 8D), the ADP dissociation is not involved in the kinetic pathways for the energy consumption Δμ_{D} (this energy produces direction at zero load but does work at finite loads). Hence, the other myosin motors likely differ from myosin V by higher γ_{e} values and higher speeds. For myosin V, which is a processive dimeric member of the myosin family, the double-fitting procedure yields γ_{e} ∼ 1 and γ_{d} ∼ 2 (Fig. 5), which imply a potential capacity for ∼100% efficiency according to eqn (2). Any higher γ_{e} results in a lower efficiency for the value of step partition (γ_{d} ∼ 2) that is likely conserved within the myosin family. This rationalizes the negative correlation found by Purcell et al.^{40} between the measured efficiency and the zero-load speed for nine myosin motors, including processive myosin V, myosin XI and seven non-processive muscle myosins. Fig. 7 shows that the efficiency–speed data of the nine processive and non-processive myosins can be quantitatively reproduced with the η_{max}–v_{max}(F = 0) relation predicted using catalytic rates slightly adjusted from the reported values for myosin V (see the Methods section for details).

Fig. 7 Efficiency versus zero-load speed for multiple motors from the myosin family. The data cover processive myosins (circles, myosin V from ref. 10 and 64, myosin XI from ref. 68) and non-processive muscle myosins (squares, from tortoise,^{69} chicken (anterior latissimus dorsi),^{70} frog,^{69} chicken (posterior latissimus dorsi),^{70} mouse (soleus),^{71} drosophila (adult indirect flight muscle),^{72} and mouse (extensor digitorum muscle)^{71}). The lines are prediction from eqn (7) and (2) (see details in the Methods section). The dashed line is obtained using Δμ = 20k_{B}T and the rates of myosin V as reported in ref. 41: the ADP dissociation rate k_{1,2} = 12 s^{−1}, the ATP binding rate k_{2,3} = 2000 s^{−1} for saturating ATP concentration [ATP] = 2 M, the hydrolysis rate, k_{3,4} = 250 s^{−1}, and the phosphate release rate k_{4,1} = 750 s^{−1}. The solid line is obtained with two rates adjusted: k_{1,2} = 6 s^{−1} and k_{4,1} = 2000 s^{−1}. |

On the two maps, motor proteins across families can be systematically compared for evolutionary optimality and mechanistic connections. Motor proteins are presumably optimized by natural evolution. This biological optimality is based on the chemomechanical optimality permitted by physical laws but can deviate from the latter due to extra requirements from diverse cellular functions of the motors. Indeed, most motor proteins do not necessarily optimize towards ∼100% efficiency; they instead sacrifice their efficiency for speed, force generation or other functional preferences. The two optimality maps (and also the relations of eqn (3) and (4)) provide new quantitative standards beyond mere efficiency to judge a motor's actual extent of chemomechanical optimality/deficit and functional compromise. As can be seen in Fig. 4D and 5A, F_{1} and kinesin-1 fall on the W_{max}–D_{0}–γ_{d} and the η_{max}–γ_{e}–γ_{d} curves, but cytoplasmic dynein shows an apparent deviation from the η_{max}–γ_{e}–γ_{d} curves (also from the γ_{e}–γ_{d} relation necessary for ∼100% efficiency, Fig. 5B). These results indicate, in a rigorous quantitative way, that the biological optimization and the chemomechanical optimality converge for F_{1} and kinesin-1 but not for cytoplasmic dynein.

The two regimes differ substantially in energy utilization over the partitioned displacements d_{D} and d_{in}. The double-fitting procedure for myosin V finds d_{in} ∼ 17 nm that matches an experimentally detected ∼16.5 nm on-track sliding motion.^{43} The value of d_{in} ∼ 5 nm found for cytoplasmic dynein also matches a detected on-track sliding.^{44} These results suggest that the d_{in} portion of a motor's step is likely an on-track downhill displacement that is decoupled from a motor's direction but still does work against a load. This amounts to an advantage in force generation, which is consistent with the fact that the two processive motors are evolutionarily linked to muscle myosins and axonemal dyneins.

The nature of energy use over the d_{D} portion is better exposed by the γ = 1 motors F_{1} and kinesin-1: both motors have a flat ΔG_{η}versus F, i.e., ΔG_{min}(D(F)) + Fd = constant from near-zero load up to stall load despite typical fluctuations of single-motor measurements (Fig. 1B and D). Thus, the total work rises and ΔG_{min}(D) drops by an equal amount when D is reduced by an increasing F. This implies a peculiar channel that uses the same energy to produce either direction or work.

A motor from regime I consumes the entire ATP energy (Δμ) by this channel of dual energy utilization. As a consequence, such a motor may use all the energy supply to attain near-unity D_{0} at zero load and then achieve ∼100% efficiency near the stall load. This is proven by the F_{1} rotor that has a D_{0} of ∼99.92% and ∼100% efficiency. Kinesin-1 is another motor from regime I with a high D_{0} (∼99.82%) and a reasonably high efficiency (∼80%). This finding that F_{1} and kinesin belong to the same chemomechanical regime conducive to high efficiency plus high directionality implies a profound mechanistic similarity in energy conversion and direction rectification between the two seemingly unrelated motors in evolution. Motors from regime II unlikely access high directionality and high efficiency together, as suggested by the lower D_{0} of myosin V and cytoplasmic dynein (Fig. 4B). Besides, the two regimes require different energy partitions for a motor to approach ∼100 efficiency: γ_{e} = 0 for regime I; γ_{e} = γ_{d} − 1 > 0 for regime II (Fig. 5B).

A motor's directionality is determined by two complementary directional biases. Take a bipedal motor for example: one bias is preferential dissociation of the motor's rear leg over the front leg from the track, which is quantified by the back–front dissociation probability ratio (α). The other bias is preferential binding of the dissociated leg to the front site over the back site, and is quantified by the forward–backward binding probability ratio (β). The two biases produce a motor's directionality as D = (αβ − 1)/[(α + 1)(β + 1)] (see the Methods section for derivation). A motor attains the optimal directionality when both biases are equal, i.e., α = β. This bias symmetry is ensured by equal heat productions along the motor's kinetic pathway for the rear leg dissociation and reverse binding and along the pathway for the forward binding and reverse dissociation. The bias symmetry and the heat symmetry are necessary conditions for the optimality relations of eqn (1) and (2), which are matched by the four motors analyzed in this study (Fig. 1–3).

Considering the equal biases for F_{1}, kinesin-1 and myosin V (a reasonable assumption for their high D_{0} values, see Fig. 4D), their zero-load biases are α = β ∼ 2500:1, 1000:1, and 100:1, respectively. These predictions might be verified by future experiments in which the steps are resolved^{45} to measure the biases. For F_{1} and kinesin-1, the heat production from internal friction has been successfully measured.^{29,30} New experiments detecting heat–bias correlation might help identify the molecular origin of chemomechanical deficiency, and probably also test the predicted heat symmetry. Besides, it is interesting to note that the level of biases for myosin V has been achieved by artificial bipedal nanomotors.^{46,47}

A subtlety exists for cytoplasmic dynein, whose D_{0} (∼50%) coincides with a general limit^{23} of D ≤ 50% for motors missing either bias. This motor's low directionality might be explained if it possesses a single high bias, or if it possesses two weak biases (∼3:1 for equal biases). This uncertainty may be resolved by future experiments. F_{1}, kinesin and myosin V are apparently above the directionality limit (Fig. 4B), which is consistent with their energy consumption for direction (Δμ_{D}) being apparently beyond that for the limit, ΔG_{min}(D = 50%) = 2.2k_{B}T (dynein's Δμ_{D} is near the value, see Fig. 4C). These results suggest the presence of two biases in the three motors, which is in line with previous studies on F_{1},^{3} kinesin^{13,48,49} and myosin V.^{18} Two complementary biases are also achieved in artificial nanomotors,^{47,50–55} with an autonomous bipedal motor^{47} reaching a D value (above ∼59%) already better than that of dynein.

A recent study by Belyy et al.^{56} reports that the binding of a dynactin complex to mammalian dynein activates its processive motility and raises its force output. According to eqn (1), a motor's maximum work (equivalent of force generation for a fixed step size) depends on both directionality and step partition. Data analysis in Fig. 3 indicates a low directionality (D_{0} < 0.5) but a high step partition (γ_{d} ∼ 5 near stall load) for dynein from yeast. For mammalian dynein, the activation of processive motility by dynactin probably suggests an increase in directionality, which likely results in a higher force. This possibility can be examined by measuring the stepping ratio for dynactin-bound mammalian dynein, which might be considered for future experiments. An analysis of the stepping ratio-vs.-load data with the fitting equations given by this study will elucidate whether the dynactin complex can modify dynein's directionality or/and step partition, and thereby switch the chemomechanical coupling regime of this motor.

This study provides quantitative evidence (extracted from experimental data) that the directionality-based chemomechanical optimality is a real and important element of the evolutionary optimization of motor proteins across families. This does not mean that all motor proteins follow the directionality-based chemomechanical optimality, as biological optimization often is not single-dimensional but involves a balance of multiple aspects. Understanding of the complex optimization benefits from the understanding of individual dimensions. Depending on their cellular functions, many motor proteins can deviate from the chemomechanical optimality quantified in this study. The deviation carries important information about the motor, and can be quantified using the fitting equations and optimality maps, which are presented in a format easy for experimenters to use. The extent to which a motor matches or deviates from the directionality-based chemomechanical optimality can be systematically extracted from load-dependent stepping experiments, which can be done on many more motors in future (desirably with controlled chemical potentials).

At last, this study presents a new conceptual framework to understand motor proteins. The new framework does not contradict traditional kinetic modelling; instead, it provides extra restrictions to reduce the number of independent rates for a kinetic model if a motor is experimentally verified to follow the directionality-based optimality. Hence a fruitful area for future study is the integration of this framework with other approaches including kinetic, mechanical and structural models to cover more aspects of the evolutionary optimization of motor proteins.

Fig. 8 A generic kinetic diagram suitable for analyzing efficient motors (panel A), and the mapping of the kinetic diagram to translational motors kinesin-1 (B), myosin V (D), and the rotational motor F_{1}-ATPase (C) according to their consensus chemomechanical coupling patterns from the literature (ref. 67 for F_{1}, ref. 48 for kinesin, and ref. 42 for myosin V). A major feature of the kinetic diagram in panel A is the differentiation of two types of states (station and bridge, see explanation in the main text), which are assigned to the three motors in panels B–D. The forward step of the motors is represented by the transition cycle indicated by an arrowed circle at the center of panel A. In panel B, the motor domains of a kinesin dimer are shown in red, the soft necklinkers and the coiled-coil dimerization domain in blue, α and β tubulins of the microtubule track in grey and dark grey. In panel C, the rotary γ-subunit of F_{1} is shown by arrow, and the stator (αβ) subunits by spheres. In panel D, the motor domains of myosin V are shown by green spheres, the IQ motifs of the rigid necks by small green bars in tandem, and the helical actin track by grey spheres. The labels are T for ATP (adenosine triphosphate), D for ADP (adenosine diphosphate) and Pi for phosphate. The step size of the three motors is shown. For myosin V, an on-track sliding is indicated (a feature shared by cytoplasmic dynein^{44} that is the fourth motor discussed in this study). |

In the kinetic diagram, two states are ‘doorway’ states (marked as states 1 and 2) by which a motor enters or exits a bridge state formed between two adjacent stations. The other two states of the kinetic diagram (states 3 and 4) are doorway states by which the motor enters or exits a station state. For bipedal translational motors, which cover all the three translational motors in this study, their two-leg bound states are bridge states and the one-leg states are station states. For F_{1}, the station and bridge states are assigned to the two long-paused angles of the γ subunit in its full 120° rotational step. As shown in Fig. 8B–D, the four states of the kinetic diagram can be assigned to the experimentally detected mechanical and nucleotide states for kinesin 1, F_{1}, and myosin V according to their consensus chemomechanical cycles. A forward inter-site step is represented by the transition cycle 1 → 2 → 3 → 4 → 1 (arrowed circle in Fig. 8A).

A motor's directionality (D) is defined^{23,24} as the probability for its forward inter-site step minus that for the backward inter-site divided by the total probability for the forward step, backward step, and futile steps that are attempted but failed forward or backward step returning the motor to its previous location. A kinetic–thermodynamic study^{24} found that a motor's D is entirely decided by the entropy productions along the transitions between a station and a bridge. For the kinetic diagram in Fig. 8A, the steady-state solution, which quantifies a motor's sustained operation, readily reproduces the D expression of ref. 24, i.e., D = [exp(ΔS_{2,3}/k_{B}) − 1] × [exp(ΔS_{4,1}/k_{B}) − 1]/[exp(ΔS_{2,3}/k_{B} + ΔS_{4,1}/k_{B}) − 1]. Here k_{B} is the Boltzmann constant, and ΔS_{2,3} and ΔS_{4,1} are entropy productions along the transition pathways 2 ↔ 3 and 4 ↔ 1, which are henceforth called D-pertinent pathways. The intra-station or intra-bridge pathways 1 ↔ 2 and 3 ↔ 4 are called D-uncoupled pathways. A motor may undergo a center-of-mass displacement by the D-uncoupled transition pathways, and this is likely true for myosin V (Fig. 8D, 1 ↔ 2 transitions). But the center-of-mass displacement in 1 ↔ 2 and 3 ↔ 4 transitions occurs without involving any leg dissociation or binding. For a bipedal walker, only the dissociation and binding of its legs contribute to the walker's directionality, as shown by the D–entropy relation (for a rigorous derivation of the relation, see ref. 24).

The neat D–entropy relation identifies the energy cost for the direction D as the heat production along the two D-pertinent pathways, i.e., ΔQ_{D} = T(ΔS_{2,3} + ΔS_{4,1}), in an isothermal environment of temperature T. Following the D–entropy relation, the maximization of D for a fixed amount of heat yields an equal-entropy condition ΔS_{2,3} = ΔS_{4,1} = ΔQ_{D}/2T. The equal entropies are a necessary condition for producing the highest possible D at a certain price of heat or for producing a certain D with the lowest heat.

The neat D–entropy relation, which is a basis for this study, holds not only for the simplified four-state diagram but also for the complex transition schemes that involve intermediate states between the four doorway states and even more doorway states entering or exiting a station or bridge. As found in ref. 24, any motor-relevant kinetic schemes, however complex they may be, must support cyclic fuel consumption/inter-station motion, hence allowing their many transitions to be grouped into two D-pertinent pathways unambiguously. Two effective entropy productions can be defined accordingly to preserve the neat D–entropy relation. Therefore, the conclusions of this study are not limited by the four-state diagram. This kinetic scheme is suitable for studying efficient motors in optimal operation, which is the purpose of this study. Including more states into the kinetic scheme is necessary to account for spurious pathways of motors under less optimal conditions or in mutated versions, as has been done by many previous studies. Besides, the D–entropy relation is independent of step size, and thus is applicable to motors with fixed steps or variable ones (e.g., cytoplasmic dynein).

Eqn (2) follows from eqn (1)via the relation Δμ_{D} = ΔG_{min}(D_{0}) and the energy partition γ_{e} = (Δμ − Δμ_{D})/Δμ_{D}. Eqn (3) is the inversed F_{max}(D) relation for an optimal motor generating at the maximal force (i.e., F = F_{max}). For such a motor, the utilized energy ΔG_{η}(F) = ΔG_{min}(D(F)) + Fd yields eqn (4) through the F_{max}(D) relation plus the step partition d = d_{D} + d_{in}.

(5) |

(6) |

In eqn (5) and (6), w_{in} = exp(ΔQ_{in}/k_{B}T) with ΔQ_{in} = T(ΔS_{1,2} + ΔS_{3,4}) as the heat production along the D-uncoupled pathways; w_{D} = exp(ΔQ_{D}/2k_{B}T). Based on the energy conservation (i.e., Δμ = Fd + ΔQ_{in} + ΔQ_{D}), the heat production ΔQ_{D} is related to ΔQ_{in} as ΔQ_{D} = (1 − η)Δμ − ΔQ_{in}. k_{tot} is the joint rate for the cycle of transitions producing a full forward step (i.e., state 1 → 2 → 3 → 4 → 1). Hence 1/k_{tot} = 1/k_{1,2} + 1/k_{2,3} + 1/k_{3,4} + 1/k_{4,1}. For motor proteins with a tight chemomechanical coupling, the four forward transitions for the forward-stepping cycle k_{1,2}, k_{2,3}, k_{3,4}, and k_{4,1} correspond to the four processes of the fuel turnover cycle, namely ATP binding, hydrolysis, phosphate release and ADP dissociation. Hence k_{tot} is the overall ATP turnover rate for a motor of tight chemomechanical coupling. This joint forward-step rate can be re-organized as 1/k_{tot} = 1/k_{c} + 1/k_{d} with k_{c} as the joint rate for the forward transitions along the two D-uncoupled pathways (i.e., 1/k_{c} = 1/k_{1,2} + 1/k_{3,4}) and k_{d} as the joint rate for forward transitions along the D-pertinent pathways (1/k_{d} = 1/k_{2,3} + 1/k_{4,1}). a_{1} and a_{2} are two rate ratios a_{1} = k_{c}/k_{3,4} and a_{2} = k_{c}/k_{d}.

For the F_{1} rotor, its ATP binding rate is k_{3,4}, hydrolysis rate is k_{1,2}, phosphate release rate is k_{2,3}, and ADP dissociation rate is k_{4,1} according to the consensus chemomechanical coupling scheme shown in Fig. 8C. The load may affect each of the rates, but the load dependences are likely cancelled in the rate ratios. It is a good approximation to assume load-independent ratios a_{1} and a_{2}. It is also reasonable to assume a load-independent ΔQ_{in} for F_{1}, because this energy dissipation occurs in the transition pathways of ATP binding and hydrolysis that are not coupled to any displacement (see Fig. 8C). A previous study^{7} shows that a ΔQ_{in} value of ∼0.75k_{B}T explains the measured zero-load speed^{34} of F_{1} within a large rate uncertainty. Estimating the absolute power also requires the load dependence of the ATP turnover rate k_{tot}, which is unclear for this motor. As an approximation, we assume F_{1}'s k_{tot} to follow a load-dependent function previously adopted for the studies^{13,16,17,36} of kinesin-1 (also a motor known for tight chemomechanical coupling^{59,60}), namely k_{tot}(F) = k_{tot}(F = 0)/(p + qe^{Fδ/kBT}), in which δ is a coupling length and p is a branch ratio with p + q = 1. The absolute power in Fig. 6B and C is estimated using k_{tot}(F = 0), δ and p as fitting parameters.

(7) |

Consider a reference value for the energy partition γ_{e0}, thereby a reference efficiency η_{max0} by eqn (2). The values of ΔQ_{in}, ΔQ_{D}, and the four forward rates at γ_{e0} decide the value of v_{max}(F = 0) at η_{max0}. When the energy partition changes from γ_{e0} to a higher value γ_{e}, ΔQ_{in} and ΔQ_{D} change as ΔQ_{D} = Δμ/(1 + γ_{e}) and ΔQ_{in} = Δμγ_{e}/(1 + γ_{e}), i.e., more energy is partitioned to the D-uncoupled pathways. A motor's speed at zero load is likely not limited by the rates for the D-coupled pathways (k_{2,3} and k_{4,1}), but by the rates for the D-uncoupled pathways (k_{1,2} and k_{3,4}), since the former pathways contribute to the motor's direction that is maximal at zero load. Hence a reasonable assumption is that the partition change affects only the rates k_{1,2} and k_{3,4}. Following the spirit of Bell's equation^{61} or the Arrhenius equation, the two rates are assumed to change as k_{1,2}(γ_{e})/k_{1,2}(γ_{e0}) = exp{[ΔS_{1,2}(γ_{e}) − ΔS_{1,2}(γ_{e0})]/k_{B}} and k_{3,4}(γ_{e})/k_{3,4}(γ_{e0}) = exp{[ΔS_{3,4}(γ_{e}) − ΔS_{3,4}(γ_{e0})]/k_{B}}. Here ΔS_{1,2}(γ_{e0}) and ΔS_{1,2}(γ_{e}) are entropy productions at the reference and new partition, which are calculated from the corresponding ΔQ_{in} values by eqn (6). However, calculating ΔS_{1,2}(γ_{e}) from eqn (6) requires the a_{1} value at γ_{e}, which in turn requires the values of k_{1,2} and k_{3,4} at γ_{e} in the first place. Re-organize the rates as k_{1,2}(γ_{e}) = ρ_{1,2}exp(ΔS_{1,2}/k_{B}), k_{3,4}(γ_{e}) = ρ_{3,4} exp(ΔS_{3,4}/k_{B}) with ρ_{1,2} = k_{1,2}(γ_{e0}) exp[−ΔS_{1,2}(γ_{e0})/k_{B}] and ρ_{3,4} = k_{3,4}(γ_{e0}) exp[−ΔS_{3,4}(γ_{e0})/k_{B}]. Applying the two rates to a_{1} = k_{1,2}/(k_{1,2} + k_{3,4}) and further using ΔQ_{in} = T(ΔS_{1,2} + ΔS_{3,4}) yields ΔS_{1,2} in terms of a_{1}. Plugging the obtained relation into the left-side relation in eqn (6) produces a closed equation for a_{1}, which yields the a_{1} value at γ_{e} as with ρ = ρ_{1,2}/ρ_{3,4}. The solution further yields ΔS_{1,2}(γ_{e}) and ΔS_{3,4}(γ_{e}), and thereby k_{1,2}(γ_{e}) and k_{3,4}(γ_{e}). Then the joint rate k_{tot} is obtained at γ_{e}, and the speed at the corresponding efficiency η_{max} is calculated from eqn (7).

For myosin V and other myosin motors, k_{1,2}, k_{2,3}, k_{3,4}, and k_{4,1} correspond to ADP dissociation, ATP binding, hydrolysis, and phosphate release, respectively (Fig. 8D). Hence the ADP dissociation, known as the rate-limiting process,^{41} is indeed in the D-uncoupled pathways. Fig. 7 shows the η_{max}–v_{max}(F = 0) relation predicted for myosin motors by the above procedures using myosin V as the reference, i.e., γ_{e0} = 1. The rates at this γ_{e0} value are taken from those of myosin V reported in ref. 41: k_{1,2} = 12 s^{−1}, k_{2,3} = 0.9 μM^{−1} s^{−1} × [ATP] = 2000 s^{−1} for saturating ATP concentration [ATP] = 2 M, k_{3,4}, = 250 s^{−1}, and k_{4,1} = 750 s^{−1}. The speed–efficiency correlation predicted using these rates (the dashed line in Fig. 7) has a speed limit because the rate-limiting process becomes the phosphate release (k_{4,1}) at low efficiency (i.e., high γ_{e}). Raising this rate to 2000 s^{−1} removes the speed limit (solid line in Fig. 7). The ADP dissociation rate is also adjusted to k_{1,2} = 6 s^{−1} for the solid line to show that a better agreement in magnitude with experimental data is readily available.

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