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Molecular Motors: Power Strokes Outperform Brownian Ratchets Jason A. Wagoner, and Ken A Dill J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b02776 • Publication Date (Web): 02 May 2016 Downloaded from http://pubs.acs.org on May 3, 2016

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The Journal of Physical Chemistry

Molecular Motors: Power Strokes Outperform Brownian Ratchets Jason A. Wagoner† and Ken A. Dill∗,‡ †Laufer Center for Physical and Quantitative Biology, Stony Brook University, Stony Brook, New York 11794, USA ‡Laufer Center for Physical and Quantitative Biology, and Departments of Physics and Astronomy and Chemistry, Stony Brook University, Stony Brook, New York 11794, USA E-mail: [email protected]

Abstract Molecular motors convert chemical energy (typically from ATP hydrolysis) to directed motion and mechanical work. Their actions are often described in terms of ‘Power Stroke’ (PS) and ‘Brownian Ratchet’ (BR) mechanisms. Here, we use a transition state model and stochastic thermodynamics to describe a range of mechanisms ranging from PS to BR. We incorporate this model into Hill’s diagrammatic method to develop a comprehensive model of motor processivity that is simple but sufficiently general to capture the full range of behavior observed for molecular motors. We demonstrate that, under all conditions, PS motors are faster, more powerful, and more efficient at constant velocity than BR motors. We show that these differences are very large for simple motors but become inconsequential for complex motors with additional kinetic barrier steps. We dedicate this paper to Bill Gelbart, a long-time dear friend of KD. Bill’s enthusiasm for statistical physics in chemistry and biology, and his great depth of engagement in any discussion, have long been a great inspiration to us.

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Introduction Molecular motors drive force, movement, and other actions in living systems. These protein complexes transduce a difference in chemical potential of reactants (usually ATP) and products (ADP and inorganic phosphate) to perform directed motion or work against a gradient. 1–6 Motors act in muscle contraction, 7,8 cellular cargo transport, 9,10 DNA unwinding and repair, 11,12 balancing pH in the cell, 13,14 and the beating of cilia and flagella. 15,16 Details of motor operation have been gained from advanced techniques in structural biology, single molecule force spectroscopy, 10,17–26 biomolecular simulation, 27–35 and modeling using stochastic thermodynamics 36–51 –the statistical description of small systems operating out of equilibrium 52,53 Molecular motors are often understood in terms of the Power Stroke (PS) and the Brownian Ratchet (BR) (also called the Feynmann ratchet or thermal ratchet) mechanisms. 2,54,55 For a simple summary, we can consider a molecular motor like Kinesin with two motor domains that ‘walk’ along a microtubule track. 9,10 The BR model, which can be traced back to the work of Feynmann 56 and Huxley, 57 assumes that the mechanical displacement step occurs through the thermal diffusion of one motor relative to a stator (the other, fixed motor domain). The PS model 58–61 alternatively postulates that the displacement step occurs through a conformational change (a ‘pre-wound’ spring) that pulls on the motor relative to the stator. Though it is instructive to speak of power strokes and Brownian ratchets as separate mechanisms, it is clear that the true mechanism of motor operation is likely some combination of these two limiting regimes. Much attention has been given to the mechanistic differences between power strokes and Brownian ratchets. 2,34,54,60–66 Here, we are interested in the performance characteristics of these two models. It has been shown that the PS/BR distinction has no impact on the thermodynamic properties of net directionality, stepping ratio, and efficiency. 67 However, it is well known that working against an external load, a power stroke motor will be faster than a Brownian ratchet, 54,68 though the quantitative extent of this difference has only been shown 2


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for a few simple models. 44,46,68 It follows that other kinetic quantities, like power output or the efficiency at a given speed, will be higher for power stroke mechanisms. 44,46 Here, we are interested in the extent of these differences in kinetic performance over a range of models that encompass the full complexity of biomolecular motors and other cellular processes. Our goal is to better understand the biological mechanisms of and evolutionary design principles for biomolecular motors.

An Elementary Model of Motor Processivity We consider a motor that is driven by hydrolyzing ATP into products ADP and inorganic phosphate, ∆µ = µATP − µADP − µPi , and that performs work against a constant external force, ∆w = F d > 0, for a motor step size of d. When ∆µ > ∆w, the motor acts processively forward. A simple model for motor processivity is the single-state unicyclic network shown in Figure 1. For this elementary model, one complete forward transition always includes both an ATP hydrolysis event and a forward step, while one reverse transition always includes a backward step and the formation of ATP from its constituent molecules ADP and Pi . Here, ‘step’ specifically refers to the physical output step of a motor. Here and throughout this work, there are no irreversible transitions; reverse transitions are explicitly represented, no matter how unlikely.

Figure 1: An elementary model of a processive motor. Xi denotes the physical position of a motor, like kinesin, on it’s track. Each forward step is catalyzed by ATP hydrolysis. The net work performed on a motor over one complete forward transition, ∆wnet = 3



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reverse rates:

kf = k0 exp [β (∆µ − δ∆w)] , kr = k0 exp [β(1 − δ)∆w] ,

(2)

where k0 is the equilibrium rate for forward and reverse transitions of the motor (kf = kr = k0 at equilibrium). An applied force is split between forward and reverse rates according to the location of the transition state, given by the load sharing factor times the step size, δd. The value of δ corresponds to the different possible motor mechanisms. For the pure power stroke, δ = 0, while for the pure Brownian ratchet, δ = 1. We note that, while we are bounding δ ∈ [0, 1], Fisher and Kolomeisky originally discussed the possibility of a load sharing factor that falls outside of this range. 43,44 This transition state model also fits with early illustrations of motor function in a ‘two-landscape’ model, where forward motion can be viewed as biased diffusion through alternating landscapes that are switched upon binding or release of ATP. Both the power stroke and Brownian ratchet mechanisms can be depicted using such a model if the transition state is given in the appropriate location for the motor step. 2 Real biomolecular motors operate through a sequence of transitions that is more complicated than the simple model described in this section and shown in Figure 1. In the following section, we derive a more general and comprehensive network of motor processivity. We then combine the transition state model with this more comprehensive network to analyze the kinetic performance of motors across the spectrum of power stroke and Brownian ratchet mechanisms.

A more comprehensive model of motor processivity Each forward step in the elementary model of Figure 1 entails an ATP hydrolysis event and each reverse step entails the formation of ATP from ADP and Pi . While this elementary 5



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model is a useful reference point and gives analytical insights into the kinetic performance of motors (shown in the results section), most real biological systems are more complex. For example, kinesin is known to have both ATP-driven backsteps 10,70 and futile cycles (ATP hydrolysis events that lead to neither a forward nor a backward step); 10 RNA Polymerase may have an off-path force-dependent waiting state; 17 and F0F1-ATPase shows a degree of cooperativity of motor speed on its three ATP binding sites. 71–73 There are many other possible complications and, beyond molecular motors, there are other nonequilibrium cyclic processes in the cell that are too complex to be captured by the unicycle network of Figure 1.

Figure 3: A more comprehensive model of motor processivity. Each Xi is a mesostate that itself may contain any number of microstates and transitions between them. There are three ways for the motor to take a mechanical step: (E1) A forward step is driven by ATP hydrolysis; or (E2) A backstep is driven by ATP hydrolysis; or (E3) a forward or reverse step that progresses with no ATP hydrolysis. The transition along edge E1 highlighted in red (ATP-driven forward motion) is the primary origin of forward motion. The net flux of this model is given in equations 3-5. Figure 3 shows a more comprehensive model, which allows for futile cycles of ATP hydrolysis, ATP-hydrolysis driven backsteps, etc. We explicitly consider motor transitions that fall into one of three classes: (1) An ATP-driven forward step, in which one forward transition includes ATP hydrolysis coupled to a forward step of the motor (E1 in Figure 3); (2) an ATP-driven back step, in which one forward transition includes ATP hydrolysis coupled to a reverse step of the motor (E2); or (3) an uncoupled cycle, in which the motor makes a forward or reverse step with no ATP hydrolysis (E3). The ATP-driven forward step E1 is the origin of forward processivity. The ATP-driven backstep E2 is negligible or nonexistent for many systems, though it has been observed for some motors like kinesin. 10,70 The uncoupled step of E3 includes any possible motion of the motor independent of ATP hydrolysis. We 6


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do not assume these transitions proceed through a single rate; there may be any number of intermediate states along the edges of Figure 3. In addition, the nodes of Figure 3 are mesostates that themselves may contain microstates and intermediate transitions that couple to the input ∆µ or output ∆w. These mesostates may contain, for example, cycles of futile ATP hydrolysis or a force-dependent waiting state. 17 We impose only the following requirements on this model: 1. All transition rates obey local detailed balance/microscopic reversibility. That is, around any complete cycle, the forward and reverse rates are equivalent at equilibrium (when ∆µ = ∆w = 0) and otherwise satisfy an analog of equation 1 where any variation from equilibrium is dictated by the work done on or performed by the system around that cycle. 2. The net flux of the motor (number of forward steps per unit time) is positive when ∆µ ≥ 0; 0 ≤ ∆w ≤ ∆µ. 3. By definition, the parameter δ acts identically on all steps along E1, E2, and E3 in Figure 3 according to the transitions state model (equations 2) and does not act on any other rates of the network. These conditions are quite general. Conditions 1 and 2 constrain the mathematics of the network shown in Figure 3 to correspond to real motors moving forward. Condition 3 defines the effect of the load sharing factor δ. As discussed above, there may be other force-dependent transitions within the mesostates of Figure 3. The rates associated with these transitions are not perturbed by the load sharing factor. As our analyses are focused on the mechanisms of motor stepping, we have defined δ to correspond to the transition states of motor steps (edges E1, E2, and E3 in Figure 3) only. The net flux through an arbitrary network can be calculated using various methods. 74–78 Here, we adapt the diagrammatic method of Terrell Hill 74,75 to derive a general equation for 7



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the net flux (number of mechanical steps per unit time) through this comprehensive network. Below is a brief description; more details are given in the Supporting Information (SI) and in Ref. 74. Because the linear network of Figure 3 is periodic, the net flux can be modeled as a cyclic network. 38–40 The three possible transitions for mechanical steps (E1, E2, E3) are subcycles of this network. The net flux through this network (the number of steps taken by the motor per unit time) is J = Π/F

(3)

where

Π = a1 eβ(∆µ−δ∆w) − eβ(1−δ)∆w F1 (∆µ, ∆w) +a2 e−βδ∆w − eβ∆µ+(1−δ)∆w F2 (∆µ, ∆w) +a3 e−βδ∆w − eβ(1−δ)∆w F3 (∆µ, ∆w) ,

(4)

F = G0 (∆µ, ∆w) + G1 (∆µ, ∆w) e−βδ∆w +G2 (∆µ, ∆w) eβ(1−δ)∆w + G3 (∆µ, ∆w) eβ(∆µ−δ∆w) +G4 (∆µ, ∆w) eβ(∆µ+(1−δ))∆w .

(5)

In equation 4, the flux has been partitioned into subcycles 75 containing ATP-driven forward steps (E1 in Figure 3), ATP-driven backsteps (E2), and uncoupled steps (E3). The nonnegative constant ai is the product of equilibrium transition rates around subcycle i (labeled as Ei in Figure 3). The function Fi () is a nonnegative algebraic combination of a subset of rate constants in the network but outside of subcycle i; following the discussion above, these functions may depend on ∆µ and ∆w but will be independent of δ since that parameter perturbs the mechanical step only. The denominator F is a positive complex algebraic combination of all rate constants in the network. Of relevance here is that this polynomial 8


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is order 1 for any particular rate constant, so we may write F as it is shown in equation 5, where the function Gi () is an algebraic combination of rates that may depend on ∆w and ∆µ but are independent of δ. Equations 3, 4, and 5 define our comprehensive model. This model explicitly allows for any number of intermediate states and for the many complex transitions that have been observed for molecular motors, described above. The results we develop in the following section are quite general and do not depend on the specific form of the constants and functions in these equations (ai , Fk , etc.). These constants and functions depend on a specific network model, we leave them undetermined here. In the SI, we use a 7-state network to derive these terms explicitly, as an illustrative example. What is relevant here is that (1) these terms are nonnegative, and (2) the dependence of the net flux on ∆w and δ is well understood. There are a number of possible transitions that are not explicitly considered in this comprehensive model, such as cooperative action that would depend on multiple ATP binding sites 71–73 or an output mechanical step that is variable in size or broken into smaller substeps. We exclude these possibilities to maintain simplicity of notation: including one of these transitions would require an additional transition edge in Figure 3 and an additional subcycle in equation 4, and would not affect our conclusions in the following section. The transition state model used here assumes that an output load is applied through a one-dimensional vector and that the location δ of the transition state stays fixed. These approximations are limited. 79,80 A more advanced model for load distribution would alter our comprehensive model, but we do not expect these changes to impact our findings for PS and BR mechanisms given in the following section. In the results section below, we show how the motor efficiency η, power P , and maximum velocity Vmax depend on the parameter δ, which controls the range of model behavior from power-stroke to Brownian ratchet for the comprehensive class of molecular motors represented by equations 3-5.

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Results Power stroke motors are faster, more powerful, and more efficient than molecular ratchets Using the comprehensive model of Figure 3 and equations 3-5, we show below that ∂η ≤ 0, ∂δ J,∆µ ∂P ≤ 0, ∂δ ∆µ,∆w ∂Vmax ≤ 0. ∂δ ∆µ,∆w

(6)

That is, the efficiency η at constant velocity, output power P , and maximum velocity of a molecular motor all increase with a decreasing δ and are maximal for a pure power stroke mechanism, δ = 0. Equations 6 are valid for all motors that can be represented by the comprehensive model of the previous section and are the central results of the present work. We use the flux J (the number of steps per unit time) and the velocity V (distance per unit time) interchangeably, V = Jd

(7)

where d is the motor step size. To obtain the results in equations 6, we first take the derivative of Eq 3 to get ∂J Π dF 1 dΠ − 2 . = ∂δ ∆µ,∆w F dδ F dδ 1 dF = J −β∆w − F dδ ≤ 0.

10


(8)

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The last line follows from 1 dF F dδ

=

−β∆w G1 e−βδ∆w + G2 eβ(1−δ)∆w F

(9)

+G3 eβ(∆µ−δ∆w) + G4 eβ(∆µ+(1−δ))∆w . Noting again that ai , Fi , and Gi of equations 3-5 are nonnegative, we have

0≥

1 dF ≥ −β∆w. F dδ

(10)

Inserting eq. 10 into the middle line of equation 8 shows that dJ/dδ is strictly negative. Next, we note that for any real physical motor, increasing its load must slow it down. For the comprehensive model (equations 3-5), this is a consequence of constraints 1 and 2 given at the beginning of the last section. So, we have ∂J ≤ 0. ∂∆w ∆µ,δ

(11)

We now use equations 8 and 11 to derive equations 6. First, we consider the motor efficiency, η=

∆w . ∆µ

(12)

Note that if we take η as a purely thermodynamic property– a function of the input ∆µ and the output work ∆w–then η must be independent of δ, as noted by Astumian. 67,81 However, we are interested in the efficiencies of power strokes and ratchet motors operating at the same velocity, relevant to a biological system that must accomplish some task in a given period of time. We take J and ∆µ as the independent variables of η. Differentiating with

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respect to δ while holding J and ∆µ fixed: ∂η = ∂δ J,∆µ

1 ∂∆w ∆µ ∂δ J,∆µ

= −

dJ/dδ dJ/d∆w

≤ 0,

(13)

where we have used equations 8 and 11 and implicit differentiation in the middle step. Equation 13 shows that a molecular motor operating under the chemical potential difference ∆µ will be able to move at some speed J while producing more work if it is a power stroke than if it is a ratchet. We also consider how the power output of a molecular motor,

P = J∆w.

(14)

depends on the load sharing parameter δ. For a fixed efficiency (fixed ∆µ and ∆w), we take the derivative with respect to δ, to get ∂J ∂P = ∆w ∂δ ∆µ,∆w ∂δ ∆µ,∆w ≤ 0,

(15)

where we have used equation 8. Equation 15 shows that a power stroke motor operating at fixed efficiency η generates more power than an analogous ratchet. Finally, we consider how the maximum velocity Vmax depends on the load sharing factor δ. From equation 7: Vmax = Jd

∆µ→∞

(16)

and, from equation 8: ∂Vmax ≤ 0. ∂δ ∆µ,∆w 12


(17)

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Hence, we have derived equations 6 for a broad class of models. Power strokes are more advantageous than ratchet motors for measures of kinetic performance: they have a higher efficiency at a fixed velocity, greater power output, and greater maximum velocity. In the following section, we show analytical results for simple models of a power stroke and ratchet. Subsequently, we show that, despite the validity of the above results under general conditions, the distinction between power stroke and ratchet mechanisms becomes negligible when there are other rate-limiting steps in the cycle.

Illustrating the difference between PS and BR mechanisms in an elementary model Here, we consider the simplest motor: a single-state unicycle. This is the elementary model shown in Figure 1. This model assumes that the motor operates processively through a single step which contains both ATP hydrolysis and an output step against ∆w. There are no intermediate states and each complete step includes only a single transition. For this model, a simple analytical treatment shows the extent of the differences in performance for power stroke and Brownian ratchet mechanisms. Similar simple models have been analyzed by others. 44,68,82 The flux for this model is the difference of equations 2:

J1-state = k0 (exp [β (∆µ − δ∆w)] − exp [β(1 − δ)∆w]) ,

(18)

We set k0 = 1.0 ms−1 and ∆µ = 10 kT, which is less than the expected driving force of ATP hydrolysis (∼25kT). 3 We also use the relationships for force F = ∆w/d and velocity V = Jd, with d = 8 nm for kinesin. The parameters used here and in the following section are for illustrative purposes and generate velocities that are higher than would be expected for real motors. Figure 4 shows that the power stroke operates at much higher speeds than a corresponding ratchet that has the same input ∆µ and the same given load ∆w. 13




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where w(z) is the solution to the Lambert-W function z = w(z)ew(z) . Interestingly, Figure 5b shows that the efficiency of the power stroke motor increases for increasing motor velocity, as has been previously shown by Schmiedl and Seifert. 82 Figure 5c shows that power-stroke machines deliver more power at high speeds, as much as ∼20x more than the corresponding ratchet mechanism for the same input ∆µ. This section has demonstrated, using a single-state unicycle, the extent to which a power stroke operates more efficiently and can generate more power than a corresponding Brownian ratchet. Below, we examine these differences for a more complex model that includes intermediate transitions.

Figure 6: A four-state model of motor processivity. Here, ATP hydrolysis occurs on edge E1 while the output step occurs on edge E3 (the motor takes one physical step forward for every C → D transition).

The role of ‘cycle friction’ and network organization in a 4-state model Consider the 4-state unicycle of motor processivity in Figure 6. As before, we assume every full forward cycle includes one ATP hydrolysis event and one forward step (there are no ATP-driven backsteps or futile hydrolysis cycles). But, we now assume that there are four 16


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load; its Vmax drops markedly with an external load ∆w. The calculations in this figure are done in the limit of low friction (E2 and E4 are fast steps; they instantly equilibrate). Next, Figure 8 shows the effects of cycle friction (additional slow steps within the motor cycle). Compare Figure 8a (no cycle friction) to Figure 8b (higher cycle friction; E4 is slow). As in the elementary model, Figure 8a shows that, in a low friction cycle, a power stroke is much faster when operating against a large load. However, when edge E4 is slow, as in Figure 8b, the PS and BR models operate identically in response to a load. Here is the explanation. As described above, the difference between a power stroke and Brownian ratchet comes down to whether or not a load ∆w decreases the forward rate or increases the reverse rate (see equations 2). In the limit of high friction, this difference is inconsequential: the entire cycle is limited by the intermediate, high-friction, step. In that limit, the kinetics of the motor is determined only by the stepping ratio, equation 1. This would be the case for a molecular motor if there is an additional, rate-limiting transition of conformational change that does not correspond to ATP hydrolysis or motor stepping. This effect of cycle friction on velocity extends to power output, efficiency, and Vmax for these motors. In the SI, we show that PS and BR motors operate at an indistinguishable efficiency in a high friction cycle. Next, we compare these two models (Figures 8a and 8b) to Figure 8c (edge E2 is slow). When E2 is slow, the PS motor is still much faster than a BR motor at high load, just as it was for the low friction cycle. It is only when we increase the friction along E2 even further (∼ 1000× decrease in rates along E2) that the PS and BR motors give identical results (data not shown). This shows that the effect of increased friction is different for edges 2 and 4, demonstrating that network organization plays a substantial role for even this simple cycle. We don’t explore this in any more detail here, but it is important an important detail for more complicated models corresponding to the general network of Figure 3, which may include other transitions, futile cycles, cooperativity, etc. Thus we expect not just the rates, but also the organization of these other processes in the network–the order in which conformational

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transitions or other steps occur in relation to ATP hydrolysis and the mechanical step–to have an important impact on the kinetics of molecular motors. For many motors, the rate limiting transitions are ATP hydrolysis and mechanical stepping (corresponding to large conformational change). In these cases, there are significant differences in the performance of PS and BR mechanisms. For more complex motors, having other rate-limiting steps, the performance differences between PS and BR motors become negligible. Table 1: The load sharing factor δ as determined from previous models of molecular motors. References for the experimental data are given in the left hand column and references for the models are given in the right hand column. These models suggest that molecular motors operate close to the power stroke mechanism. Motor F0-F1 ATPase 20 Kinesin 24

δ 0.1-0.3 83 0.08 -.25 48 a .035-.135 42a Myosin 84 -0.01-0.045 41 a Dynein 18 0.3 85 b a These models contain multiple output steps and thus multiple values of δ that are not constrained to [0, 1]. Values of δ close to 0 are still indicative of a power-stroke mechanism. b In this model, the load sharing factor δ was not parameterized against experimental data. This value was shown to reproduce experimental stall forces. 85

The mechanisms of real biological motors Table 1 summarizes the values of δ that have previously been obtained from transition state models for experimental measurements on a range of biomolecular motors: F0 F1 -ATPase, kinesin, myosin, and dynein. These values of δ are relatively small (δ ∈ [−0.01, 0.3]). Thus, these models suggest that these biological motors have mechanisms that tend toward the power-stroke end of the spectrum. Studies from other approaches, including ODE models and mechanistic analyses, support either power stroke mechanisms, 47,49,84,86–95 Brownian ratchets, 96,97 or some intermediate combination. 8,9,25,26,34,65 Hence, while the results of this manuscript and of the models summarized in Table 1 give evidence that motors would operate 20


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as power strokes, it is not clear that biology has opted for one universal mechanism.

Conclusions We have used the diagrammatic method of Terrell Hill 74 to study a wide range of models of molecular motor processivity. Our comprehensive model allows for multiple kinetic steps, including on- and off-pathway intermediates, ATP-driven backsteps, futile hydrolysis cycles, load-dependent waiting states, etc. 10,17,70–73,85,98 Within this broad model framework, we find that the power stroke mechanism always operates more quickly, more powerfully, and with higher efficiency at a given velocity when compared to an analogous Brownian ratchet motor. However, complex systems with additional slow steps in their kinetic cycles can cause BR and PS models to behave identically over these performance metrics. And, interestingly, the performance characteristics of complex motors can depend on where the slow kinetic step is located in the kinetic cycle. This modeling may be useful for understanding how biology optimizes molecular motors and for designing synthetic motors. The transition state model used here and elsewhere 41–49 assumes that an output load is applied through a one-dimensional vector and that the location δ of the transition state stays fixed. It also assumes that the load degrees of freedom (cellular cargo or the probe particle of an in vitro assay) need not be explicitly considered. These approximations are limited. 79,80 Future studies of molecular motors may require more realistic models of load distribution. These more advanced models would alter our comprehensive model, but we hypothesize that equation 8, and thus our general conclusions on the performance of power stroke vs. Brownian ratchet mechanisms, will hold.

Acknowledgement The authors thank the Laufer Center for Physical and Quantitative Biology and NSF grant 21



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1205881 for support.

Supporting Information Available Supporting Information includes a supplementary theory for the comprehensive model and three additional figures. This material is available free of charge via the Internet at http://pubs.acs.org/.

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