Case study: F 1 -ATPase

4. Efficiencies of molecular motors with probe particles

of water (η ' 0.001Ns/m²). The frictional torque N = Γ ˙ϕ acting on the probe with angular velocity ˙ϕcorresponds to a frictional force

f_fr = Γ

r²x˙ =γx˙ (4.32)

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

γ = Γ

r² = 4π²Γ

9d² (4.33)

leading to γ = 0.407s/d².

Following the mass action law assumption, the equilibrium transition ratew^eq is supposed to depend linearly on the concentrations of nucleotides in the solvent. For low ATP concentrations (c_ATP '10⁻⁶M), 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 w^eq to be the experimentally determined rate of ATP binding w^eq ' 3×10⁷M⁻¹s⁻¹c^eq_ATP[74]. For known nonequilibrium concentrations of nucleotides like in the experiments, the structure of the transition rates (3.3) and (3.4) leaves the choice of the equilibrium concentrations arbitrary as long as they obey

c^eq_ATP

c^eq_ADPc^eq_Pi '4.89×10⁻⁶ 1

M (4.34)

for pH 7 (and T = 23^◦C) [41]. For given w^eq and ∆µ, one possible choice of the nonequilibrium concentrations of nucleotides isc_ADP =c^eq_ADP,c_Pi =c^eq_Pi and c_ATP = c^eq_ATPexp[∆µ] 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 [15] and the histogram of the angular position of the probe at a jump [148]. While both data sets depend on both parameters, the velocity, especially for large w^eq, is more sensitive to κ whereas the peak position of the histogram mainly depends onθ⁺. 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 [148] with the left peak position of the corresponding histograms obtained by our simulation as the ones shown in Fig. 4.2.

As a result, we obtain κ = 40 ±5d⁻² and a value of θ⁺ in the range 0.1 . θ⁺ .0.3. In Fig. 4.4, 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. Experiments on the ATP binding and hydrolysis rates as functions of

54

4.5. Case study: F₁-ATPase

the rotary angle indicate that 9% of the ATP-binding step is supported by thermal fluctuations which agrees very well with θ⁺ = 0.1 [138]. For later purposes, we also include data for θ⁺ = 0.01.

4.5.2. Comparison of efficiencies with experimental data

Pseudo efficiency

Experimentally, the heat flow of the probe is determined using the Harada-Sasa relation [156]. In the appendix, we show that this heat flow is equal to ˙Q_Pas defined in (4.9). In Fig. 4.5, 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 = 40d⁻² and θ⁺ = 0.1 and compare it with the experimental results [15]. We find 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) also the simulated mean velocities coincide well with the experimental values with a maximum deviation of 10% as shown in Fig. 4.4. For illustrative purposes, we also plot Q_Pplus W forθ⁺= 0.01 which shows better agreement with the experimental data (but is not consistent with the range of θ⁺ obtained in section 4.5.1).

Discrepancies between our theory and the experiment are visible in Figs. 4.4 and 4.5 where for parameter set V both the average velocity and the pseudo efficiency deviate significantly from the experimental values for κ = 40d⁻² and θ⁺ = 0.1.

For ∆µ= 28.12(k_BT) corresponding to the data set V in Fig. 4.5, the probe just reaches the potential minimum between consecutive jumps of the motor protein.

Therefore on average at most 20(k_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 caused by the simplicity of our model which does not capture the structural complexity of the motor and the linker.

The confinement of θ⁺ to the range 0.1 . θ⁺ . 0.3 implies on the one hand that the potential of 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 [157]. If θ⁺ was larger, η_Q would decrease and the experimentally determined values ofη_Q would not be reached in the simulation. If θ⁺ was 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 Fig. 4.2 would then no longer coincide with the experimentally observed distribution (see [148]).

4. Efficiencies of molecular motors with probe particles

0 5 10 15 20 25 30 35

I II III IV V

Figure 4.5.: Average heat QP released through the probe (green and cyan) and work W ≡ f_exd against the external force (blue) compared to the available free energy per step ∆µ (red line). The dissipated heat of the probe is split into two contributions QS and QV according to the two terms of the Harada–Sasa relation (B.3). The contribution from the linear motion with constant mean velocity, Q_S (cyan), appears in the numerator of the Stokes efficiency while Q_V (green) is the contribution due to the non–uniform jumping motion of the motor protein. In each of the five parameter sets labeled by I-V, the left and the central bar represent results from the simulation for θ⁺ = 0.1 and θ⁺ = 0.01, respectively, while the right bar shows the experimental results and error bars from [15]. The following parameters were used in the five cases: (I) c_T = 0.4µM, c_D = 0.4µM, c_P = 1mM, i.e, w^eq = 5.87×10⁻⁸s⁻¹ and ∆µ= 19.14; (II) c_T = 2µM, c_D = 2µM, c_P = 1mM, i.e, w^eq = 2.93×10⁻⁷s⁻¹ and ∆µ = 19.14; (III) cT = 100µM, cD = 100µM, c_P = 1mM, i.e,w^eq = 1.47×10⁻⁵s⁻¹ and ∆µ= 19.14; (IV)c_T = 2µM,c_D= 2µM, c_P = 1mM, f_ex = 9.27d⁻¹, i.e, w^eq = 2.93 × 10⁻⁷s⁻¹ and ∆µ = 19.14; (V) cT = 2µM, cD= 0.5µM,cP= 0.5µM, i.e, w^eq = 3.67×10⁻¹¹s⁻¹ and ∆µ= 28.12.

Thermodynamic efficiency

Information about the thermodynamic efficiency of the motor protein can be ob-tained by applying an external force to the probe. In Fig. 4.6 we show the ex-tracted/delivered work as well as the dissipated heat and their sum obtained from our simulations in comparison with experimental results from [118, 158]. The data is obtained for a fixed value of ∆µ by increasing the external force from 0 far beyond the stall force.

56

4.5. Case study: F₁-ATPase

−50

−40

−30

−20

−10 0 10 20 30

0 10 20 30 40 50

Figure 4.6.: Comparison of the simulation results (left) with experimental data from [118] (right). The concentrations used in the simulation are the same as in the experiment: c_T =c_D = 10µM, c_P = 1mM,θ⁺ = 0.1.

The extracted work increases up tof_exd= ∆µat the stall force, which is found to be f_ex^st = ∆µ/d also in the experiment [77, 158]. Thus, the motor is able to convert the full ∆µ into extractable work without any dissipation. Both in the experiment and in the simulation, the dissipated heat through the probe decreases linearly from almost ∆µat fex = 0 to zero at the stall force.

Beyond stall conditions, the external force has to be considered as energy supply to run the F1-ATPase in reverse to synthesize ATP. Producing one ATP molecule, the motor uses (−)∆µ from the available (−)f_exd per step. The excess energy delivered by the external force is then dissipated through the motor and the probe as heat. We find that the dissipated heat increases linearly withf_ex which implies that the excess energy is mainly dissipated through the probe.

The sum of both contributions, extracted/delivered work and dissipated heat through the probe, is almost equal to ±∆µ for external forces both smaller and larger than the stall force. This corresponds to the almost dissipation-free energy conversion in the motor. Note that even if the energy conversion in the motor is dissipation-free, any v 6= 0 involves dissipation at the probe. The maximum extracted work with ˙Q_P = 0, ˙Q_M = 0 can be attained only at v = 0. For the parameters used in the experiment, we find a very good agreement between our simulation and the experimental data also for external forces far beyond the stall force.

One has to keep in mind that the parameters used in this illustrative example

4. Efficiencies of molecular motors with probe particles

yield a pseudo efficiency of almost 1 in the absence of external forces. Using parameters which yield smaller or larger pseudo efficiencies forf_ex = 0, one should also expect deviations of η_Q'1 in the presence of external forces.

It is important to note that observingf_exd=∆µat stall conditions does not imply that the motor does not have idle cycles. That condition just states that the full

∆µ obtained from one hydrolysis event can be converted into mechanical power.

However, as long as one does not measure the rate of ATP turnover, there can be many additional idle cycles wasting ∆µwithout any displacement (see detailed discussion in section 6.5). Experiments controlling also the ATP concentrations indicate that the F1-ATPase does not have idle cycles [86].