The thermodynamic H⁺/ATP ratios of the H⁺-ATPsynthases from chloroplasts and Escherichia coli

Abstract

The H⁺/ATP ratio is an important parameter for the energy balance of all cells and for the coupling mechanism between proton transport and ATP synthesis. A straightforward interpretation of rotational catalysis predicts that the H⁺/ATP coincides with the ratio of the c-subunits to β-subunits, implying that, for the chloroplast and Escherichia coli ATPsynthases, numbers of 4.7 and 3.3 are expected. Here, the energetics described by the chemiosmotic theory was used to determine the H⁺/ATP ratio for the two enzymes. The isolated complexes were reconstituted into liposomes, and parallel measurements were performed under identical conditions. The internal phase of the liposomes was equilibrated with the acidic medium during reconstitution, allowing to measure the internal pH with a glass electrode. An acid–base transition was carried out and the initial rates of ATP synthesis or ATP hydrolysis were measured with luciferin/luciferase as a function of ΔpH at constant Q = [ATP]/([ADP][P_i]). From the shift of the equilibrium ΔpH as a function of Q the standard Gibbs free energy for phosphorylation, ΔG_p⁰′; and the H⁺/ATP ratio were determined. It resulted ΔG_p⁰′ = 38 ± 3 kJ·mol⁻¹ and H⁺/ATP = 4.0 ± 0.2 for the chloroplast and H⁺/ATP = 4.0 ± 0.3 for the E. coli enzyme, indicating that the thermodynamic H⁺/ATP ratio is the same for both enzymes and that it is different from the subunit stoichiometric ratio.

Membrane-bound H⁺-ATPsynthases play a central role in the energy metabolism of all cells: They catalyze the synthesis of ATP from ADP and inorganic phosphate (P_i) in bacteria, chloroplasts, and mitochondria (1–4). In these organelles, electron-transport generates a transmembrane electrochemical potential difference of protons, Δμ̃_H⁺, and the H⁺-ATPsynthase couples the Δμ̃_H⁺-driven proton efflux with ATP synthesis (5, 6).

All H⁺-ATPsynthases have a similar structure: the hydrophilic F₁ part with subunits α₃β₃γδε contains the nucleotide and P_i binding sites, and the membrane integrated F₀ part, with subunits ab₂c₁₀ in Escherichia coli (I, II, III₁₄, and IV in chloroplasts), contains the proton binding sites. The kinetics of the enzyme was described by the binding change theory, in which the cooperativity of the three catalytic binding sites is accomplished by γ-subunit rotation (7). The high-resolution x-ray structure of F₁ corroborated this theory showing that the centrally located γ-subunit interacts differently with the three β-subunits and that the catalytic site on each β-subunit has a different conformation (8). Rotation of the γ-subunit was observed with different experimental techniques (9–11).

Current models of the coupling of proton translocation to ATP synthesis assume that proton flow through the F₀ part drives a rotation of the ring of c-subunits against the a- and b-stator subunits; the connection of the c-subunits to the γ- and ε-subunits leads to rotation of the latter within the α₃β₃ barrel, which brings about the conformational changes at the catalytic sites required for ATP synthesis. To accomplish rotation of the c-ring, protons are assumed to enter the F₀ part through an access channel on one side of the membrane, bind to the essential glutamate (or aspartate) on subunit c, and leave it through an asymmetrically placed exit channel on the other side (12–15). Cross-linking data (16–18), single molecule visualization (19–21), polarization data (22), and FRET measurements (23) indicate that the ring of c-subunits in F₀ rotates together with the γ- and ε-subunit in F₁.

This mechanism appears to directly predict the H⁺/ATP ratio: It should be equal to the number of c-subunits, divided by the number of β-subunits. Whereas all known F₀F₁-ATPsynthases contain three β-subunits, the number of c-subunits seems to be species-dependent. It has been reported to be 10 in mitochondria (24), 14 in chloroplasts (25), 10 in E. coli (26) and PS3 (27), 11 in the Na⁺-ATP synthase from Ilyobacter tartaricus (28), and 13 in Bacillus sp. strain TA2A1 (29). The unexpected β–c stoichiometric ratio mismatch (pointing to a nonintegral value of the H⁺/ATP ratio) and species specificity have stimulated anew much interest in this number (see, e.g., ref. 30).

The H⁺/ATP ratio is important not only for understanding the coupling mechanism of the enzyme, but also for the energy balance of cells. Values of H⁺/ATP between 2 and 4 have been reported and also variable ones (reviewed in ref. 31). The most recent data give H⁺/ATP = 4 for the chloroplast enzyme (32–34), whereas H⁺/ATP = 3 was reported for the mitochondrial enzyme (reviewed in ref. 35). Few data on the H⁺/ATP ratio of the E. coli enzyme are found in the literature (31), pointing to values between 2 and 3.6.

Results

To determine the H⁺/ATP ratio in the absence of other interfering reactions taking place in energy transducing membranes, we set up a minimal chemiosmotic system containing only the H⁺-ATPsynthase and a lipid membrane. Liposomes from phosphatidylcholine/phosphatidic acid were prepared, and either CF₀F₁ or EF₀F₁ was reconstituted into these liposomes with a stoichiometric ratio of ≈1 CF₀F₁ or 3 EF₀F₁ per liposome [see supporting information (SI) Fig. 4]. These liposomes catalyze high rates of ATP synthesis (36, 37) when subjected to an acid-base transition (38).

The chemiosmotic theory describes the thermodynamics of the coupling between proton transport and ATP synthesis. For the chemical reaction,

the Gibbs free energy of ATP synthesis is given by

where ΔG_p⁰′ is the Gibbs free energy in the biochemical standard state, Q is the stoichiometric product, and c⁰ is the standard concentration (equal to 1 M).

Proton transport from the internal to the external phase is described by the following equation:

with n = H⁺/ATP. Using the definitions ΔpH = pH_out − pH_in and Δϕ = ϕ_in − ϕ_out, the transmembrane electrochemical potential difference of protons (Gibbs free energy of proton transport) is given by

The Gibbs free energy of the coupled reaction is the sum of Eqs. 2 and 4, i.e.,

At the point of equilibrium of the coupled reaction (ΔG′ = 0) and with Δϕ = 0, we obtain from Eq. 5:

With experimentally determined Q and ΔpH(eq), the unknown parameters in Eq. 6 are the H⁺/ATP ratio, n, and the phosphorylation standard free energy, −ΔG_p⁰′.

The different parameters in Eqs. 5 and 6 were determined as follows.

• Δϕ. An acid-to-base transition gives rise to a transmembrane ionic imbalance, which automatically generates a diffusion potential, Δϕ. We have short-circuited it by using the same high K⁺ concentration (50 mM) on both sides of the membrane in the presence of 10 μM valinomycin. Under these conditions, the contribution of all other ions to the Goldmann diffusion potential was negligible, and Δϕ = 0.

• ΔpH. For reconstitution of the enzyme, the liposome membrane was permeabilized by addition of detergent. We assume that, during this process (2 h), a full equilibration of all ion concentrations between the internal and external phases has taken place, so that the composition of the internal phase is known. Correspondingly, the pH_in value was taken to be identical to the pH of the reconstitution buffer, which was measured with a glass electrode after the reconstitution. The pH_out value resulted from the mixing of the acidic reconstitution buffer containing the proteoliposomes with the basic medium and was measured with the same glass electrode. The error of the absolute pH was considered to result from the limited precision of the calibration buffers, with a standard deviation of ±0.02 pH units. We have varied the pH_in values at constant pH_out, i.e., the error associated with ΔpH was ± 0.03 pH units.

• ΔpH(eq). For each given Q value, the rates of ATP synthesis and ATP hydrolysis were measured as a function of ΔpH, and the ΔpH(eq) was interpolated as the value at which the rate was zero.

• Q. The stoichiometric product Q was established by setting the ADP, ATP, and P_i concentrations to the desired values. Because we measured initial rates of ATP synthesis and ATP hydrolysis (slope at t = 0), substrate and product concentrations remained at their preestablished values. Therefore, the error in lgQ was very small and was neglected in our analysis.

To measure the initial rates of ATP synthesis and ATP hydrolysis, an ATP-monitoring luciferin/luciferase mixture was added to the basic medium in a luminometer cuvette. Additionally, the baseline of luminescence was recorded indicating the ATP concentration in the medium at the preestablished Q value. At reaction time t = 0, the acidic buffer with proteolipomes was injected with a syringe, and the luminescence was measured for ≈3 min (see SI Fig. 5). The composition of the external phase resulted from the mixing of the basic medium with the acidic reconstitution buffer containing the proteoliposomes (see SI Table 1). The CF₀F₁ and EF₀F₁ concentrations in the reaction medium were 7.6 nM and 23 nM, respectively.

Some of the rate measurements with CF₀F₁ are shown in Fig. 1Left. An increase of luminescence indicates ATP synthesis; a decrease indicates ATP hydrolysis. The arrows indicate the time point of injection of the proteoliposomes into the basic medium, i.e., the start of the reaction. Each panel shows the data obtained at the same constant stoichiometric product. The Q values and the corresponding concentrations of ADP, ATP, and P_i are collected in SI Table 2. The different curves within each panel refer to different ΔpH values, and the slope of each curve at t = 0 gives the initial rate in units of ATP per enzyme per second. In each panel, the initial rate switches from ATP synthesis at the highest ΔpH to ATP hydrolysis at the lowest ΔpH. No change in ATP concentration was observed when the ΔpH was dissipated by 10 μM nigericin, indicating both the absence of adenylate kinase activity and the absence of ATP hydrolysis when the enzyme was not activated by a ΔpH.

Fig. 1.

ATP synthesis and ATP hydrolysis after generation of a transmembrane ΔpH. The basic medium (900 μl) with the preestablished Q value and with luciferin/luciferase was placed into a cuvette in the luminometer, and the baseline was registered. The acid-base transitions were carried out by injection of 100 μl of proteoliposomes, incubated in the acidic reconstitution medium at different pH_in. The final pH_out was 8.47 ± 0.02, and the ΔpH values are given next to each trace. The addition gives rise to an immediate jump of the baseline. For clarity, the baseline was shifted back to the original luminescence level. The luminescence increase indicates ATP synthesis, the decrease indicates ATP hydrolysis. The different panels refer to different stoichiometric ratios Q as indicated. The luminescence was calibrated by addition of standard ATP. The solid lines show a biexponential fit of the complete time trace. The initial rates and their errors are given in units of 10⁻³ s⁻¹; they were calculated from the monoexponential fit of the first part of the traces, as detailed in SI Fig. 6. (Left) CF₀F₁ proteoliposomes. The final CF₀F₁ concentration was 7.6 nM. (Right) EF₀F₁ proteoliposomes. The final EF₀F₁ concentration was 23 nM.

Similar measurements were carried out with EF₀F₁, using the same method. For both enzymes, reconstitutions were carried out in parallel, using the same buffers, mixing ratios, pH electrode, etc. The only difference was that the EF₀F₁ proteoliposomes contained a 3-fold higher average number of ATPsynthase molecules per liposome. Despite this higher enzyme concentration, the signal-to-noise ratio in the EF₀F₁ traces was significantly lower. Lower absolute rates in EF₀F₁ were expected, because the maximal ATP synthesis turnover of EF₀F₁ was 80 s⁻¹ compared with 280 s⁻¹ for CF₀F₁. Moreover, in the present measurements, the Δϕ was clamped to 0 and the EF₀F₁ turnover decreases much stronger with Δϕ than that of CF₀F₁ (37).

Fig. 1 Right shows rate measurements with EF₀F₁ from Q = 12 to Q = 152. For each stoichiometric product, ATP synthesis is observed at the highest ΔpH and ATP hydrolysis at the lowest. In all ATP synthesis traces, the initial rate decreases and switches to ATP hydrolysis after 5–20 s. This reflects the decrease of the initial ΔpH due to proton efflux. The data in the presence of 10 μM nigericin indicate the absence of ATP hydrolysis and adenylate kinase activities; at Q = 152, a constant ATP concentration is initially observed, followed by a slow decrease of ATP, i.e., the enzyme is initially inactive and is activated by binding and hydrolysis of ATP.

For both enzymes, the initial rates and their errors were evaluated by monoexponential fitting of the first part of the traces (see details in SI Fig. 6) and were plotted versus ΔpH (see Fig. 2). At the thermodynamic equilibrium, catalysis switches from ATP synthesis to ATP hydrolysis, and this point, ΔpH(eq), was determined by interpolation.

Fig. 2.

Rate of ATP synthesis and hydrolysis as a function of the transmembrane ΔpH. The initial rates and their errors obtained from Fig. 1, and additional measurements, were plotted as a function of ΔpH. Some error bars are not visible, because they are smaller than the size of the symbols. Each curve represents catalysis rates measured at a constant stoichiometric product, Q. The data were fitted, using their errors as weight, by a function that was the product of a linear function (describing the rate-force relation near equilibrium) and of a sigmoidal function (describing the ΔpH dependence of activation). The fitting functions were used to interpolate the point at which the rate is zero [ΔpH(eq)]. (Left) CF₀F₁ proteoliposomes. (Right) EF₀F₁ proteoliposomes.

The activity of CF₀F₁ is regulated by a disulfide bridge in the γ-subunit and by ΔpH (39–42). When CF₀F₁ is in its reduced state—achieved in this work by incubation with DTT—the rate of coupled ATP hydrolysis is controlled by two opposing effects. With decreasing ΔpH, the back pressure of protons is decreased; thus, the hydrolysis rate should increase, but the fraction of active enzyme also decreases, thus decreasing this rate. Both effects can be seen in Fig. 1 Left. When the ΔpH is collapsed by nigericin, i.e., the enzyme is not activated, no hydrolysis is detected. An increase of the hydrolysis rate with increasing reaction time, i.e., with decreasing ΔpH, can also be observed (see, e.g., trace at Q = 58, ΔpH = 1.96).

Only active enzymes catalyze proton transport coupled ATP synthesis/hydrolysis. Whereas the rates depend on the number of active enzymes, the position of the equilibrium described by Eq. 6 does not. The rates were measured in this work close to equilibrium, so that the activation should not influence the interpolated ΔpH(eq). Nevertheless, we corrected for the effect of activation as described in ref. 34 by taking into account the sigmoidal ΔpH dependency measured in thylakoid membranes (40) (SI Fig. 7 shows the corrected rates as a function of ΔpH). The interpolated ΔpH(eq) values were, within error limits, the same for the original and the corrected data (see SI Table 1).

Regulatory phenomena dependent on Δμ̃_H⁺ have been shown in EF₀F₁ as well (43), although a quantitative ΔpH dependency is not available. The EF₀F₁ traces of Fig. 1 Right are consistent with a kinetic control exerted by ΔpH: similar to CF₀F₁, no hydrolysis is detected when the ΔpH is collapsed by nigericin, and an increase of the hydrolysis rate with increasing reaction time (i.e., with decreasing ΔpH) can be observed in several traces.

The ΔpH(eq) determined in Fig. 2 for both enzymes and the corresponding Q values were then plotted as 2.3RTlgQ versus 2.3RTΔpH(eq) in Fig. 3Upper. According to Eq. 6, such a plot should yield a straight line with slope n (the thermodynamic H⁺/ATP ratio) and axis intercept −ΔG_p⁰′. The fitting by linear regression of the CF₀F₁ data (Fig. 3 Upper Left) gave values of n = 4.0 ± 0.2 and ΔG_p⁰′ = (38 ± 3) kJ·mol⁻¹. The fitting by linear regression of the corresponding EF₀F₁ data (Fig. 3, Upper Right) gave values of n and −ΔG_p⁰′, which were loaded with a much higher uncertainty, mainly due to the limited range of Q values and ΔpH(eq) obtained for EF₀F₁. Therefore, the CF₀F₁ dataset was used as a reference, and the −ΔG_p⁰′ resulting from their fit was included in the EF₀F₁ dataset. The linear regression of these data yielded a value of n = 4.0 ± 0.3, i.e., very similar to the value obtained for CF₀F₁. The dashed line has been drawn by imposing n = 3.3 (with ΔG_p⁰′ = 38 kJ·mol⁻¹), and the dotted lines indicate that the expected ΔpH(eq) values in this case should have been ≈0.5 ΔpH units higher.

Fig. 3.

Determination of the thermodynamic H⁺/ATP ratio. ΔpH(eq) was plotted versus the stoichiometric product Q according to Eq. 6 (data from Fig. 2). (Upper Left) CF₀F₁ data (from Fig. 2 Left). The error associated with ΔpH(eq) has been taken to coincide with the error estimated for ΔpH determination by the pH electrode (±0.03). The data were fitted by a straight line, using their errors as weight. The slope of the linear regression gives the H⁺/ATP ratio. The y intercept gives the standard Gibbs free energy −ΔG_p⁰′ at pH_out = 8.47. The arrow marked as R&S indicates the value of −ΔG_p⁰′ published in ref. 43. (Upper Right) EF₀F₁ data (from Fig. 2 Right). The −ΔG_p⁰′ value (diamond) is the best-fitting parameter out of the CF₀F₁ data (Left) and has been included in the linear regression of the EF₀F₁ data. The data were fitted by a straight line, using their errors as weight. The slope of the linear regression gives the H⁺/ATP ratio. To account for the error on the y axis of the ΔG_p⁰′ value, two additional linear fits were calculated, using its two extreme values of 35 and 41; the resulting H⁺/ATP values were 3.7 and 4.3 respectively, whose difference from 4.0, being higher than the error resulting from each linear regression, has been taken as the error in the determination of this parameter. (Lower) The CF₀F₁ (circles) and EF₀F₁ (triangles) data from Upper are plotted together for better comparison; the squares are CF₀F₁ data from ref. 34, in which the Δϕ component has been converted in ΔpH units (15 mV, corresponding to 0.26 ΔpH units). The straight line is the regression line of Upper Left.

In Fig. 3 Lower, the CF₀F₁ (circles) and EF₀F₁ (triangles) data are plotted together for better comparison. Also plotted are the previously obtained CF₀F₁ data from (34) (squares); the latter data had been measured in the presence of a Δϕ component of 15 mV.

It is interesting that the data presented in Fig. 2 can also be analyzed in a reciprocal manner by plotting the rates obtained at constant ΔpH versus RTlnQ and determining RTlnQ(eq) by interpolation (SI Fig. 8). When the interpolated RTlnQ(eq) data were plotted versus 2.3RTΔpH, a straight line was obtained with a slope of n = 4.1 ± 0.2 and ΔG_p⁰′ = 39 ± 3 for CF₀F₁ and a slope of n = 4.0 ± 0.3 for EF₀F₁ (SI Fig. 9).

Discussion

The aim of this work was to determine the H⁺/ATP ratio from two H⁺-ATPsynthases, which show differences in the c/β stoichiometric ratio, in particular EF₀F₁ and CF₀F₁, for which the c/β stoichiometric ratios have been reported as 3.3 and 4.7, respectively. The thermodynamic parameters of proton transport coupled ATP synthesis/hydrolysis were measured under identical conditions for CF₀F₁ and EF₀F₁ and the H⁺/ATP ratio for both enzymes was obtained from the change of the equilibrium ΔpH with lnQ.

These identical experimental conditions allowed us to use, for evaluating the EF₀F₁ data, the ΔG_p⁰′ value, which was obtained with higher precision from the CF₀F₁ data, thereby lowering the error limits in the H⁺/ATP determination for EF₀F₁. In fact, because of the lower rate of EF₀F₁, these measurements could not be extended to the wide range of Q values required to determine with high precision both ΔG_p⁰′ and H⁺/ATP. The analysis of the CF₀F₁ data resulted in ΔG_p⁰′ = 38 ± 3 kJ·mol⁻¹ and H⁺/ATP = 4.0 ± 0.2. Using this ΔG_p⁰′ value, the EF₀F₁ data gave H⁺/ATP = 4.0 ± 0.3. It is evident that, similarly to CF₀F₁, the measured thermodynamic H⁺/ATP ratio in EF₀F₁ is not consistent with the reported subunit stoichiometric ratio (26).

The H⁺/ATP ratio of 4 obtained for CF₀F₁ is in accordance with earlier results (32–34), and ΔG_p⁰′ is in accordance with data in ref. 44. Values of 2–3.6 have been reported for H⁺/ATP in EF₀F₁; however, these measurements included use of indirect probes (45) and assumptions for estimating Δμ̃_H+ and ΔG_p⁰′ in whole cells (46, 47).

The reason for the surprising difference found here between the thermodynamic H⁺/ATP and the c/β stoichiometric ratio is not known yet. The thermodynamic H⁺/ATP ratio reflects the minimal number of H⁺ necessary for ATP synthesis (or hydrolysis) of one ATP at equilibrium. At high rates of ATP synthesis far from equilibrium, it is conceivable that additional protons have to be translocated through the enzyme for each synthesized ATP, e.g., slip protons, leading to what can be called a “kinetic” H⁺/ATP ratio. It might be that the c/β stoichiometric ratio is optimized for such high rates, which would explain the mismatch between that ratio and the thermodynamic H⁺/ATP ratio. However, earlier proton flux measurements carried out far from equilibrium with thylakoid membranes also gave H⁺/ATP = 4 (48), lending no support to this speculation.

Alternatively, these data might indicate that the number of c-subunits in the holoenzyme CF₀F₁ is different from that found in the isolated ring of subunit III. As to the E. coli enzyme, it has been suggested that the number of c-subunits in this organism might change in response to the prevailing metabolic conditions (31), which would imply that the issue of the c-subunit number is not definitively settled in this organism either. The discrepancy presently found between the thermodynamic requirements (H⁺/ATP = 4 for both enzymes) and the subunit stoichiometric ratio (c/β = 3.3 for EF₀F₁ and c/β = 4.7 for CF₀F₁) raises interesting questions for future research.

Materials and Methods

CF₀F₁ from spinach (Spinacia oleracea) was isolated as described in ref. 34. CF₀F₁ was obtained in a buffer containing 1.25 M sucrose, 30 mM NaH₂PO₄/NaOH (pH 7.2), 2 mM MgCl₂, 0.5 mM Na₂EDTA, and 4 mM dodecyl maltoside, with a protein concentration of 4.2 mg/ml, and stored at −80°C. EF₀F₁ from pBWU13 E. coli strain DK8 (Δunc) was isolated as described in refs. 49 and 50. The EF₀F₁ complex was obtained in a buffer containing 0.9 M sucrose; 10 mM Mes; and 10 mM Tricine/NaOH (pH 7.0), 0.5 mM MgCl₂, 5 mM thioglycerol, and 10 g/liter octyl glucoside, with a protein concentration of 2.2 mg/ml, and stored in liquid nitrogen.

Liposomes from phosphatidylcholine and phosphatidic acid were prepared by dialysis and stored at −80°C (37). CF₀F₁ or EF₀F₁ was reconstituted into these preformed liposomes with 0.8% Triton X-100 and Biobeads (stored in 50 mM KCl), using buffers at 12 different pH values. These acidic buffers contained 25 mM Mops/NaOH, 25 mM Mes/NaOH, 50 mM KCl, 4.4 mM MgCl₂ (3.15 mM + 1.25 mM from the dialysis buffer), and 2.5 mM NaH₂PO₄ plus 5 mM Tricine, 0.1 mM EDTA, and 0.1 mM DTT from the dialysis buffer. Because of Triton X-100 treatment during reconstitution, all ion concentrations can be considered to have equilibrated between the bulk phase and the internal proteoliposome phase. The pH was measured with a glass electrode after reconstitution and is referred to as pH_in. The final enzyme concentrations were 80 nM for CF₀F₁ and 240 nM for EF₀F₁.

The ATP concentration was measured with luciferin/luciferase (34). All suspensions and solutions were equilibrated at room temperature (23°C). Valinomycin was added to proteoliposomes to a final concentration of 10 μM. A luminometer cuvette was filled with 20 μl of luciferin/luciferase kit; 349 μl of a solution containing 100 mM KCl and 9 mM MgCl₂; 1 μl of a solution containing 1 M DTT and 50 mM KCl; 200 μl of a solution at pH 8.60 (containing 0.5 M Tricine, 5 mM NaH₂PO₄, 4.5 mM MgCl₂, 50 mM KOH, and 330 mM NaOH); and 330 μl of a solution containing different concentrations of ADP and ATP (final volume 900 μl). The reaction was started by injection of 100 μl of proteoliposomes into the cuvette placed in the luminometer. The pH value measured after this mixing was the pH of the external phase during the ΔpH transition (pH_out). The resulting ion concentrations of the internal and external phases and the nucleotide concentrations are summarized in SI Tables 2 and 3. The final ATP and ADP concentrations were 25 ÷ 800 nM and 1.7 ÷ 16 μM, and the final P_i concentration was 1.25 mM. The luminescence signal was calibrated by three additions of 10 μl of an ATP standard (10⁻⁵ M) to each cuvette. The luminescence data were sampled in 55-ms intervals. In the graphical representation of the data in Fig. 1, ten 55-ms time intervals have been averaged to reduce the noise. To calculate the initial rates and their errors, the initial phases of the signals at the original time resolution were fitted with either a monoexponential or a linear function (using the software package Origin). SI Fig. 6 shows the luminescence time traces at the original time resolution, the fitted functions, and the residuals for one Q value (Q = 152).

Measurements of the rate of ATP synthesis under standard conditions (pH_in = 4.8, pH_out = 8.5, [K⁺_in] = 50 mM, [K⁺_out] = 180 mM) were carried out as described in ref. 37. Concentrations of ATP and ADP were measured as described in ref. 34. The ATP content in ADP solutions was 0.16%.