Efficiencies of a molecular motor: a generic hybrid model applied to the F₁-ATPase

The one-dimensional model we will use to describe a molecular motor with an attached probe particle consists of two degrees of freedom, representing the motor protein at position n(t) and the probe at position x(t), respectively, see figure 1. For a rotary motor like the F₁-ATPase, the rotary motion is mapped to a linear one for simplicity. Both constituents are linked via a harmonic potential

$$\begin{equation} V(n,x)=\frac{\kappa}{2}(n-x)^2 \end{equation} \tag{ 1 }$$

with spring constant κ, where we have included a possible rest length of the linker in the definition of x.

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The motion of the probe particle is described by an overdamped Langevin equation with friction coefficient γ and constant external force f_ex,

$$\begin{equation} \dot{x}(t) =\left(-\frac{\partial V}{\partial x}-f_{\mathrm{ex}}\right)\Bigg/\gamma+\zeta(t), \end{equation}\noindent \tag{ 2 }$$

including the random force γζ(t) that the solvent exerts on the probe. The thermal fluctuations are assumed to be Gaussian white noise with zero mean and correlations 〈ζ(t₁)ζ(t₂)〉 = 2k_BTδ(t₁ − t₂)/γ, where k_B is Boltzmann's constant and T is the temperature of the solvent.

The motor protein jumps at times τ_j from n_j to n_j ± d with transition rates k^±(n_j,x), hydrolyzing (or synthesizing) one ATP molecule per jump, which corresponds to tight mechanochemical coupling. In this minimal model, we take into account only one chemical state. The transition rates have to fulfill a local detailed balance condition of the form

$$\begin{equation} \frac{k^{+}(n,x)}{k^{-}(n+d,x)}=\exp[(\Delta\mu-V(n+d,x)+V(n,x))/k_{\mathrm{B}}T], \end{equation}\noindent \tag{ 3 }$$

where we assume that the jumps of the motor protein take place instantaneously. The free energy change of the solvent

$$\begin{equation} \Delta\mu\equiv\mu_{\mathrm{ATP}}-\mu_{\mathrm{ADP}}-\mu_{ \mathrm{P}_{\mathrm{i}}} \end{equation} \tag{ 4 }$$

is associated with ATP turnover. Implementing mass action law kinetics and the concept of a barrier in the potential of mean force for the unresolved chemical steps, the individual rates become

$$\begin{eqnarray} &&\fl k^{+}(n,x) = k^{\mathrm{eq}} \exp [\Delta\mu_{\mathrm{ATP}}/k_{\mathrm{B}}T] \exp\left[\frac{1}{k_{\mathrm{B}}T}\int\nolimits_{n}^{n+d\theta_{+}} -\frac{\partial V(n,x)}{\partial n}\, \mathrm{d}n\right ] \nonumber \\ &&=k^{\mathrm{eq}} \exp [\Delta\mu_{\mathrm{ATP}}/k_{\mathrm{B}}T] \exp[(-\kappa d^2\theta_{+}^2/2 - \kappa(n-x)d\theta_{+})/k_{\mathrm{B}}T] \end{eqnarray}\noindent \tag{ 5 }$$

and

$$\begin{eqnarray} \fl k^{-}(n,x) &=& k^{\mathrm{eq}} \exp [(\Delta\mu_{\mathrm{ADP}}+\Delta\mu_{ \mathrm{P}_{\mathrm{i}}})/k_{\mathrm{B}}T] \exp\left[\frac{1}{k_{\mathrm{B}}T}\int\nolimits_{n}^{n-{\rm d}\theta_{-}} -\frac{\partial V(n,x)}{\partial n}\, \mathrm{d}n \right] \nonumber \\ \fl &=& k^{\mathrm{eq}} \exp [(\Delta\mu_{\mathrm{ADP}}+\Delta\mu_{ \mathrm{P}_{\mathrm{i}}})/k_{\mathrm{B}}T] \exp[(-\kappa d^2\theta_{-}^2/2 + \kappa(n-x)d\theta_{-})/k_{\mathrm{B}}T] \end{eqnarray}\noindent \tag{ 6 }$$

with

$$\begin{equation} \Delta\mu_i\equiv\mu_i-\mu_i^{\mathrm{eq}}=k_{\rm B}T\ln(c_i/c_i^{\mathrm{eq}}) \end{equation}\noindent \tag{ 7 }$$

and c_i the concentrations of the nucleotides (i = ATP, ADP, P_i). Here, the transition rate k^eq applies to equilibrium concentrations of nucleotides. The load sharing factors θ₊ and θ₋, with θ₊ + θ₋ = 1, depend on the unresolved shape of the free-energy landscape of the motor protein [5, 45].

The transition rates, as well as the force the motor protein exerts on the probe, depend on the distance

$$\begin{equation} y(t)\equiv n(t)-x(t) \end{equation}\noindent \tag{ 8 }$$

between the motor and the probe. The corresponding probability density p(y,t) obeys the Fokker–Planck-type equation

$$\begin{eqnarray} \fl \partial_t p(y,t) & =& k^{+}(y-d)\, p(y-d,t)+k^{-}(y+d)\, p(y+d,t)-(k^{+}(y)+k^{-}(y))\, p(y,t) \nonumber \\ \fl &&+\partial_y ((\kappa y-f_{\mathrm{ex}})\, p(y,t)+ k_{\mathrm{B}}T\partial_y \, p(y,t))/\gamma, \end{eqnarray}\noindent \tag{ 9 }$$

which contains in the first line the contributions of the transitions of the motor protein and in the second line the drift and diffusion of the probe particle.

For large t and constant nucleotide concentrations, the system reaches a stationary state with time-independent p^s(y) and constant mean velocity $\langle \dot {n} \rangle =\langle \dot {x} \rangle \equiv v$ with

$$\begin{equation} \langle \dot{x} \rangle=(\kappa\langle y \rangle-f_{\mathrm{ex}})/\gamma \end{equation}\noindent \tag{ 10 }$$

and

$$\begin{equation} \langle \dot{n} \rangle=d\langle k^{+}(y)-k^{-}(y) \rangle, \end{equation}\noindent \tag{ 11 }$$

where 〈...〉 denotes the average using the stationary distribution p^s(y), which, however, cannot be determined analytically.

Following the concept of stochastic thermodynamics [46, 47], one can assign a first law at the level of a single trajectory. If the probe moves a small distance Δx, the first law becomes

$$\begin{equation} \Delta q_{\mathrm{P}}=\left(-\frac{\partial V}{\partial x}-f_{\mathrm{ex}}\right)\Delta x=(\kappa y-f_{\mathrm{ex}})\Delta x, \end{equation} \tag{ 12 }$$

where Δq_P is the heat dissipated by the probe, f_exΔx is the work against the external force and (∂_xV )Δx is the change in the internal energy of the spring due to the motion of only the probe. A jump of the motor protein gives rise to a first law in the form of [48]

$$\begin{eqnarray} 0&=&\Delta V + \Delta E_{\mathrm{Sol}}+\Delta q_{\mathrm{M}} \nonumber \\ &=&\Delta V - \Delta\mu + T\Delta S_{\mathrm{Sol}}+\Delta q_{\mathrm{M}} \end{eqnarray} \tag{ 13 }$$

without a contribution of the internal energy of the motor, as its internal energy does not change in the one-state model. The change in the internal energy of the spring is given by

$$\begin{equation} \Delta V \equiv V(n\pm d,x)-V(n,x), \end{equation} \tag{ 14 }$$

where the sign depends on the direction of the jump. Due to ATP turnover, the internal energy of the solution changes by ΔE_Sol = −Δμ + TΔS_Sol, where ΔS_Sol is the change in the entropy of the solution. The heat dissipated by the motor protein in this transition is denoted by Δq_M.

On average, the chemical energy gained from ATP consumption that involves changes in the entropy of the solvent will be dissipated as heat Q in the environment and/or is delivered as work against the external force. In the stationary state, the internal energy of the spring is constant on average. Taking the average rates of (12) and (13) and summing the two contributions, this first-law condition can be expressed as

$$\begin{equation} \dot{\Delta\mu}=\dot{Q}_{\mathrm{P}}+\dot{Q}_{\mathrm{M}}+T\dot{S}_{\mathrm{Sol}}+f_{\mathrm{ex}} v, \end{equation} \tag{ 15 }$$

where the dot denotes a rate and

$$\begin{equation} \dot{\Delta\mu}\equiv-\langle \dot{F}_{\mathrm{Sol}} \rangle=\Delta\mu v/d \end{equation} \tag{ 16 }$$

is the rate of free energy consumption. The rate of dissipated heat $\dot {Q}=\dot {Q}_{\mathrm {P}}+\dot {Q}_{\mathrm {M}}$ has two contributions. First, the heat flow through the motor protein is given by

$$\begin{equation} \dot{Q}_{\mathrm{M}}\equiv\langle \dot{q}_{\mathrm{M}} \rangle=\dot{\Delta\mu}-\dot{V}_n-T\dot{S}_{\mathrm{Sol}}, \end{equation} \tag{ 17 }$$

representing the fact that while jumping, the motor protein uses free energy from the hydrolysis to load the spring which corresponds to a change of the internal energy of the spring $\dot {V}_n$ with

$$\begin{equation} \dot{V}_n \equiv \int_{-\infty}^{\infty}p^{\mathrm{s}}(y)[k^{+}(y)(V(y+d)-V(y))+k^{-}(y)(V(y-d)-V(y))] \, \mathrm{d}y \end{equation} \tag{ 18 }$$

$$\begin{equation} \hspace{1.2pc}= \frac{\kappa d^2}{2}\langle k^{+}(y)+k^{-}(y) \rangle+\kappa d\langle y(k^{+}(y)-k^{-}(y)) \rangle. \end{equation} \tag{ 19 }$$

The energy thus stored in the spring is then dissipated by the probe whose heat flow is given by

$$\begin{equation} \dot{Q}_{\mathrm{P}}\equiv\langle \dot{q}_{\mathrm{P}} \rangle=\langle (\kappa y-f_{\mathrm{ex}})\nu(y) \rangle \end{equation} \tag{ 20 }$$

where

$$\begin{equation} \nu(y)\equiv((\kappa y-f_{\mathrm{ex}})+ k_{\mathrm{B}}T\partial_y \ln p^{\mathrm{s}}(y))/\gamma \end{equation}\noindent \tag{ 21 }$$

is the local mean velocity of the probe for a given y [49, 50] which corresponds to the current arising from the motion of only the probe in (9).

We will now focus on three different definitions of efficiency that have been proposed for motor proteins.

In the absence of an external force (f_ex = 0), one can compare the energy that the motor protein transfers to the spring, $\dot {V}_n$, with its available chemical energy $\dot {\Delta \mu }$. From (15) and (17) it follows that $\dot {V}_n=\dot {Q}_{\mathrm {P}}$. The ratio of the on average dissipated heat through the probe and the available free energy

$$\begin{equation} \eta_{\mathrm{Q}}\equiv\frac{\dot{Q}_{\mathrm{P}}}{\dot{\Delta\mu}}=\frac{d\kappa\langle y\nu \rangle}{v\Delta\mu} \end{equation}\noindent \tag{ 22 }$$

was proposed as the definition of efficiency [33]. We will see below that η_Q is not bounded by 1, as has been anticipated earlier [16, 51], and therefore we will call it a pseudo efficiency. A second type of efficiency is the Stokes efficiency,

$$\begin{equation} \eta_{\mathrm{S}}\equiv\frac{\gamma v^2}{\dot{\Delta\mu}}=\frac{d\kappa\langle y \rangle}{\Delta\mu}, \end{equation}\noindent \tag{ 23 }$$

which compares the mean drag force γv the probe feels with the available chemical force. In contrast to η_Q, η_S is bounded by 1 [16]. If the motor protein exerted a constant force on the probe, the Stokes efficiency would be equal to the pseudo efficiency η_Q, because in this case the average heat dissipated by the probe is the mean drag force times d.

Finally, in the presence of an external force acting on the probe, the thermodynamic efficiency of the system is the ratio between the mechanical work delivered to the external force and the available free energy [10]

$$\begin{equation} \eta_{\mathrm{T}}\equiv \frac{f_{\mathrm{ex}} v}{\dot{\Delta\mu}} =\frac{f_{\mathrm{ex}} d}{\Delta\mu}. \end{equation}\noindent \tag{ 24 }$$

For f_ex ≠ 0, the pseudo efficiency η_Q can be defined as

$$\begin{equation} \eta_{\mathrm{Q}}=\frac{\dot{Q}_{\mathrm{P}}+f_{\mathrm{ex}} v}{\dot{\Delta\mu}}. \end{equation} \tag{ 25 }$$

For a comparison with the simulations and in order to gain more analytical insights, it will be convenient to have a simple approximation for the stationary distribution p^s(y). For a Gaussian probability distribution

$$\begin{equation} p^{\mathrm{G}}(y)\equiv\frac{1}{\sqrt{2\pi}\sigma}\exp\!\left[-\frac{(y-\bar{y})^2}{2\sigma^2}\right] \end{equation} \tag{ 26 }$$

the free parameters $\bar {y}$ for the mean and σ² for the variance can be determined by requiring that the time derivative of these quantities as calculated with the Fokker–Planck equation (9) vanishes in the steady state. These conditions result in the following two equations for $\bar {y}$ and σ²:

$$\begin{equation} (\kappa\bar{y}-f_{\mathrm{ex}})/\gamma=d(\bar{k}^{+}-\bar{k}^{-}) \end{equation} \tag{ 27 }$$

and

$$\begin{eqnarray} &&\fl (\kappa\sigma^2+\kappa\bar{y}^2-f_{\mathrm{ex}}\bar{y}-k_{\rm B}T)/\gamma=d[(\bar{y}-k \,d \theta_{+}\sigma^2/k_{\mathrm{B}}T)\bar{k}^{+}-(\bar{y}+k \,d \theta_{-}\sigma^2/k_{\mathrm{B}}T)\bar{k}^{-}]\nonumber \\ &&\hspace{6.9pc}+\,d^2(\bar{k}^{+}+\bar{k}^{-})/2, \end{eqnarray} \tag{ 28 }$$

where we have introduced the average jump rates

$$\begin{eqnarray} \fl \bar{k}^{+}&\equiv& \int_{-\infty}^{\infty}k^{+}(y) p^{\mathrm{G}}(y)\,\mathrm{d}y \nonumber \nonumber\\ \fl &=&k^{\mathrm{eq}}\exp[\Delta\mu_{\mathrm{ATP}}/k_{\mathrm{B}}T-\kappa d^2\theta_{+}^2(k_{\mathrm{B}}T-\kappa\sigma^2)/2(k_{\mathrm{B}}T)^2-\kappa \,d\theta_{+}\bar{y}/k_{\mathrm{B}}T], \end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray} \fl \bar{k}^{-}&\equiv& \int_{-\infty}^{\infty}k^{-}(y) p^G(y)\,\mathrm{d}y \nonumber \\ \fl &=&k^{\mathrm{eq}}\exp[(\Delta\mu_{\mathrm{ADP}}+\Delta\mu_{ \mathrm{P}_{\mathrm{i}}})/k_{\mathrm{B}}T-\kappa d^2 \theta_{-}^2(k_{\mathrm{B}}T-\kappa\sigma^2)/2(k_{\mathrm{B}}T)^2+\kappa \,d \theta_{-}\bar{y}/k_{\mathrm{B}}T] . \end{eqnarray} \tag{ 30 }$$

These equations can easily be solved numerically.

Close to chemical equilibrium, i.e. Δμ = 0, and for f_ex = 0, we expand $\bar {y}$ and κσ² − k_BT up to first order in Δμ and find that

$$\begin{equation} \bar{y}\approx A\Delta\mu+\tilde{A}(\theta_{+}-\theta_{-})^2\Delta\mu \end{equation} \tag{ 31 }$$

and

$$\begin{equation} \kappa\sigma^2-k_{\mathrm{B}}T\approx B(\theta_{+}-\theta_{-})\Delta\mu. \end{equation} \tag{ 32 }$$

The coefficients A, $\tilde {A}$ and B obtained by solving the first order of (27) and (28) are too long to be shown here.

In the limit Δμ → ∞ and f_ex = 0, we obtain for $\bar {y}$ and κσ² − k_BT

$$\begin{equation} \bar{y}\approx \frac{\Delta\mu}{\kappa d}-C\Delta\mu\exp[-\Delta\mu\theta_{-}/k_{\mathrm{B}}T] \end{equation} \tag{ 33 }$$

and

$$\begin{equation} \kappa\sigma^2-k_{\mathrm{B}}T\approx D\Delta\mu\exp[-\Delta\mu\theta_{-}/k_{\mathrm{B}}T] \end{equation} \tag{ 34 }$$

as long as θ₋ > 0. The coefficients

$$\begin{equation} C=\frac{1+\kappa d^2(\theta_{+}-\theta_{-})^2/4}{\gamma \kappa d^3k^{\mathrm{eq}}\exp[(\Delta\mu_{\mathrm{ADP}}+\Delta\mu_{ \mathrm{P}_{\mathrm{i}}})/k_{\mathrm{B}}T]} \end{equation} \tag{ 35 }$$

and

$$\begin{equation} D=-\frac{\theta_{+}-\theta_{-}}{2\gamma d^2k^{\mathrm{eq}}\exp[(\Delta\mu_{\mathrm{ADP}}+\Delta\mu_{ \mathrm{P}_{\mathrm{i}}})/k_{\mathrm{B}}T]} \end{equation} \tag{ 36 }$$

are obtained by solving (27) and (28) to first and second order in Δμ.

Within this Gaussian approximation, the average heat flow through the probe as given by (20) is calculated using the local mean velocity (21)

$$\begin{equation} \nu(y)=(\kappa y-f_{\mathrm{ex}})/\gamma-k_{\mathrm{B}}T(y-\bar{y})/(\gamma\sigma^2). \end{equation} \tag{ 37 }$$

The average over y can now be performed leading to

$$\begin{equation} \dot{Q}_{\mathrm{P}} =(\kappa^2\sigma^2+\kappa^2\bar{y}^2-k_{\mathrm{B}}T\kappa-2\kappa f_{\mathrm{ex}}\bar{y}+f_{\mathrm{ex}}^2)/\gamma. \end{equation} \tag{ 38 }$$

This expression is used to determine η_Q in this approximation as

$$\begin{equation} \eta_{\mathrm{Q}}= d\kappa\frac{\kappa\sigma^2-k_{\mathrm{B}}T+\kappa\bar{y}^2-f_{\mathrm{ex}}\bar{y}}{\Delta\mu(\kappa\bar{y}-f_{\mathrm{ex}})} \end{equation} \tag{ 39 }$$

with $\bar {y}$ and σ² being the solution of (27) and (28) for given Δμ and k^eq.

For small Δμ, using (31) and (32), η_Q takes the form

$$\begin{equation} \eta_{\mathrm{Q}}\approx\frac{dB(\theta_{+}-\theta_{-})}{(A+\tilde{A}(\theta_{+}-\theta_{-})^2)\Delta\mu}+\kappa \,dA+\kappa \,d\tilde{A}(\theta_{+}-\theta_{-})^2. \end{equation} \tag{ 40 }$$

If θ₊ ≠ θ₋, η_Q diverges for vanishing Δμ. For θ₊ > θ₋, η_Q can become negative due to those jumps of the motor protein that occur when the previous diffusion of the probe has resulted in y < −0.5d. Then, the energy stored in the spring is dissipated by the motor protein during jumping.

In the limit of large Δμ, we use (34) and (35) to obtain

$$\begin{equation} \fl \eta_{\mathrm{Q}}\approx 1-\kappa\, dC\exp[-\Delta\mu\theta_{-}/k_{\mathrm{B}}T]+\frac{dD\exp[-\Delta\mu\theta_{-}/k_{\mathrm{B}}T]}{\Delta\mu/(\kappa d)-C\Delta\mu\exp[-\Delta\mu\theta_{-}/k_{\mathrm{B}}T]}, \end{equation} \tag{ 41 }$$

which approaches 1.

The Stokes efficiency in the Gaussian approximation without an external force is simply given by

$$\begin{equation} \eta_{\mathrm{S}}=\frac{d\kappa\bar{y}}{\Delta\mu}. \end{equation} \tag{ 42 }$$

For κσ² > k_BT, which is the case for θ₊ < 0.5, the Stokes efficiency is always smaller than η_Q. For vanishing Δμ, η_S approaches a finite value, $\eta _{\mathrm {S}}\approx d\kappa A+d\kappa \tilde {A}(\theta _{+}-\theta _{-})^2$, while for Δμ → ∞ it also converges to 1.

We will first investigate the pseudo efficiency η_Q as a function of Δμ, k^eq and θ₊. We extract $\dot {Q}_{\mathrm {P}}$ from the numerical data by averaging over one sufficiently long trajectory. The results are shown in figure 3. The most striking fact regarding these data is the observation that η_Q is larger than 1 for small enough Δμ and θ₊, which shows up in the Gaussian approximation as well. This effect can be understood as follows. In a jump, the motor protein can take heat from the solution in order to change the internal energy of the spring by an amount larger than Δμ. If, subsequently, the probe dissipates this internal energy of the spring as heat back into the environment, $\dot {Q}_{\mathrm {P}}$ can indeed become larger than $\dot {\Delta \mu }$ without any violation of the second law. Using the obtained parameter for the spring constant κ, the motor protein transfers 20 k_BT to the spring if it starts the jump from the minimum of the harmonic potential. For small values of θ₊, the forward jump rate of the motor protein depends only weakly on the current position of the probe as shown in figure 4. Therefore, jumps will occur even if the associated change in internal energy of the spring, ΔV , is larger than Δμ. For a rather small k^eq, backward jumps are rare and the probe relaxes to the potential minimum between successive forward jumps (see data set I in figure 2). This leads to $\dot {V}_n>\dot {\Delta \mu }$ on average and hence to η_Q > 1 for Δμ considerably smaller than 20k_BT as shown in figure 3. As the value of θ₊ increases, η_Q decreases because the forward jumps of the motor protein are suppressed. On average, in this case the motor protein jumps only if the probe has diffused forward and exerts a pulling force on the motor through the spring.

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Increasing the absolute concentrations of the nucleotides, i.e. increasing k^eq, results in more forward but also more backward jumps, which can be seen for data set III in figure 2. For small Δμ, the occasional backward jumps follow especially those forward jumps for which the change in internal energy of the spring has been larger than Δμ, leading to a smaller η_Q.

In the limit of large Δμ, the motor protein jumps even when the spring is previously stretched, which can result in changes of the internal energy of the spring by an amount larger than 20k_BT. The coupling between the motor protein and the probe induces a balancing effect between the forward motion of the motor protein and the drag of the probe maintaining a typical $\dot {V}$ that turns out to be approximately $\dot {\Delta \mu }$, leading to η_Q ≃ 1.

We also obtain the Stokes efficiency (23) from the simulated trajectories and the Gaussian approximation as shown in figure 5. Characteristically, starting close to 0 for small Δμ, η_S monotonically increases with Δμ reaching 1 for Δμ → ∞. For small Δμ, the trajectory of the probe shows a staircase form with small average velocity leading to small values of the Stokes efficiency in contrast to values of the pseudo efficiency η_Q > 1. For large Δμ, the probe does not relax to the potential minimum between consecutive jumps, resulting in a more linear trajectory of the probe as if it was exposed to an almost constant force. In this limit of an almost linear motion of the probe, the pseudo efficiency becomes the Stokes efficiency. As η_S is bounded by 1, η_Q cannot reach values larger than 1 in this limit either.

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Increasing the load sharing factor θ₊ results in decreasing average velocities. Therefore, the Stokes efficiency also decreases, which can be seen in figures 5(a) and (c). With increasing absolute concentrations of nucleotides, i.e. with increasing k^eq, the average velocity and therefore also the Stokes efficiency at fixed Δμ increases as shown in figures 5(b) and (d).

The thermodynamic efficiency of the system can be studied only if an external force is applied to the probe. As an illustrative example, for fixed Δμ = 19k_BT, we examine the thermodynamic efficiency in the presence of external forces smaller than the stall force as shown in figure 6. The thermodynamic efficiency increases linearly with f_ex and reaches 1 at the stall force which is possible only due to the tight mechanochemical coupling in this model. In figure 6 we also show the pseudo efficiency η_Q as defined in (25) in the presence of external forces. While η_Q is almost independent of the external force, the contribution from the dissipated heat, $\dot {Q}_{\mathrm {P}}/\dot {\Delta \mu }$, decreases linearly with f_ex and reaches zero at the stall force.

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In stall conditions, the work corresponding to the stall force refers to the maximum work the motor protein can convert on average. In the simulation, we find that applying f_ex = Δμ/d generates a diffusive motion of the motor protein with v ≃ 0 and $\dot {Q}_{\mathrm {P}}\simeq 0$ for various values of θ₊ and Δμ, including the ones with η_Q > 1 and η_Q < 1. This observation implies that the motor protein seems, in principle, to be able to convert the full Δμ into extractable work. In our model, where the motor interacts with the external force only via the spring, this result is not trivial, as it would have been if we had applied f_ex directly to the motor. Under these conditions, p^s(y) is Gaussian with 〈y〉 = Δμ/(dk) and the same variance as the Boltzmann distribution in equilibrium, σ² = k_BT/κ.

Within the Gaussian approximation we can insert f_ex = Δμ/d into (27) and (28). With κσ² = k_BT, $\bar {y}=\Delta \mu /(d\kappa )$, i.e. v = 0, is a solution for f_ex = Δμ/d, implying that also in the Gaussian approximation the motor protein is able to convert the full Δμ into extractable work for any values of the load sharing factors given that θ₊ + θ₋ = 1.

For a quantitative comparison, we have to map the rotary motion of the F₁-ATPase to our linear model and determine the model parameters. In our model, one jump of the motor protein covering a distance d corresponds to a rotation of the γ shaft of 120°. Using large probe particles such as polystyrene beads or actin filaments, the substeps in one 120° rotation are not resolved experimentally. Therefore, we will omit the substeps here, too. We assume that the temperature of the solution is T ≃ 24 °C and that the probe consists of two beads of diameter 287 nm [33]. The friction coefficient of the probe can be calculated using the formula for the rotational frictional coefficient Γ from [36, 37] with the viscosity of water (η ≃ 0.001 N s m⁻²). The frictional torque $N=\Gamma \dot {\varphi }$ acting on the probe with angular velocity $\dot {\varphi }$ corresponds to a frictional force

$$\begin{equation} f_{\mathrm{fr}}=\frac{\Gamma}{r^2}\dot{x}=\gamma\dot{x}, \end{equation} \tag{ 43 }$$

acting on the probe at distance r from the γ shaft. Within one 120° rotation, the probe at distance r covers d = 2πr/3. For the linear model, the friction coefficient γ can be calculated as

$$\begin{equation} \gamma=\frac{\Gamma}{r^2}=\frac{4\pi^2\Gamma}{9d^2}, \end{equation} \tag{ 44 }$$

leading to γ = 0.407k_BT s/d².

Following the mass action law assumption, the equilibrium transition rate k^eq is supposed to depend linearly on the concentrations of nucleotides in the solvent. For low ATP concentrations (c_ATP ≃ 10⁻⁶ M), the mean velocity of the motor protein is dominated by the rate of ATP binding. In the one-step model this feature holds for all concentrations. Therefore, we choose k^eq to be the experimentally determined rate of ATP binding k^eq ≃ 3 × 10⁷ M⁻¹ s⁻¹ c^eq_ATP [42]. For known non-equilibrium concentrations of nucleotides like in the experiments, the structure of the transition rates (5) and (6) leaves the choice of the equilibrium concentrations arbitrary as long as they obey

$$\begin{equation} \frac{c_{\rm {ATP}}^{\mathrm{eq}}}{c^{\mathrm{eq}}_{\rm {ADP}}c^{\mathrm{eq}}_{\rm P_{\mathrm{i}}}}\simeq 4.89\times 10^{-6}\mathrm{M}^{-1} \end{equation} \tag{ 45 }$$

for T = 23 °C and pH 7 [37]. For given k^eq and Δμ, one possible choice of the non-equilibrium concentrations of nucleotides is c_ADP = c^eq_ADP, c_Pi = c^eq_Pi and $c_{\mathrm {ATP}}=c^{\mathrm {eq}}_{\mathrm {ATP}}\exp [\Delta \mu /k_{\mathrm {B}}T]$ which was used for the simulation and the Gaussian approximation.

In order to determine the spring constant κ and the load sharing factor θ₊, we use both the experimental data of the mean velocities [33] and the histogram of the angular position of the probe at a jump [35]. While both data sets depend on both parameters, the velocity, especially for large k^eq, is more sensitive to κ, whereas the peak position of the histogram mainly depends on θ₊. Therefore, we primarily use the velocity data to fit κ and determine the load sharing factor θ₊ by comparing the peak position of the experimental histogram [35] with the left peak position of the corresponding histograms obtained by our simulation as the ones shown in figure 4.

As a result, we obtain κ = 40 ± 5k_BT/d² and a value of θ₊ in the range 0.1 ≲ θ₊ ≲ 0.3. In figure 7, we show how for this value of κ changing the load sharing factor affects the mean velocity for which we get the best overall agreement for θ₊ = 0.1. For later purposes, we also include data for θ₊ = 0.01.

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Experimentally, the heat flow of the probe is determined using the Harada–Sasa relation [53]. In the appendix, we show that this heat flow is equal to $\dot {Q}_{\mathrm {P}}$ as defined in (20). In figure 8, we plot the average heat released through the probe per step, Q_P, plus the work against the external force, W, obtained by the simulation for κ = 40k_BT/d² and θ₊ = 0.1 and compare it with the experimental results [33]. We found quite good agreement between theory and experiment for the parameter sets I–IV where either the maximum deviation is 15% (II–IV) or our theoretical value is included in the experimental error range (I). As an aside, we note that for the parameter sets I–III (without external force) the simulated mean velocities also coincide well with the experimental values with a maximum deviation of 10% as shown in figure 7. For illustrative purposes, we also plot Q_P plus W for θ₊ = 0.01 which shows better agreement with the experimental data (but is not consistent with the range of θ₊ obtained in section 6.1).

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Discrepancies between our theory and the experiment are visible in figures 7 and 8 where for the parameter set V both the average velocity and the pseudo efficiency deviate significantly from the experimental values for κ = 40k_BT/d² and θ₊ = 0.1. For Δμ = 28.12k_BT corresponding to the data set V in figure 8, the probe just reaches the potential minimum between consecutive jumps of the motor protein. Therefore on average at most 20k_BT can be transferred to the spring leading to η_Q ≃ 0.7, which is less than the experimental value. This discrepancy is most likely due to the fact that we have omitted substeps in our model. In the simulation, the average velocity then does not show saturation as one would expect it to result from the hydrolysis step [42] which should be experimentally observable at the higher concentrations used in [33].

The confinement of θ₊ to the range 0.1 ≲ θ₊ ≲ 0.3 implies, on the one hand, that the potential of the mean force of the motor protein should be asymmetric and, on the other hand, that asymmetric potentials with a barrier state close to the initial state seem to enhance the ability of the motor protein to perform work on the spring, in accordance with [54]. If θ₊ were larger, η_Q would decrease and the experimentally determined values of η_Q would not be reached in the simulation. If θ₊ were smaller, η_Q would approach the experimental values better; however, the distribution of the position of the probe just before a jump as shown in figure 4 would then no longer coincide with the experimentally observed distribution (see [35]).

Information about the thermodynamic efficiency of the motor protein can be gained by applying an external force to the probe. In our simulation, the stall force is found to be f_ex = Δμ/d, implying that the motor protein is able to convert the full Δμ into extractable work without dissipation in accordance with the experiments performed in [34].