Introduction

Proton production and consumption processes play a pivotal role for bioenergetics across all organisms1. Most of these processes involve proton diffusion at the cellular membrane/water interface2, 3. For instance, the synthesis of adenosine triphosphate (ATP), the free energy carrier in living systems, relies on two types of membrane-bound enzymes: proton pumps creating the transmembrane proton gradient and ATP synthases consuming this transmembrane potential to drive ATP synthesis4, 5. Strikingly, protons move extremely fast along lipid membranes6, 7: their lateral proton diffusivity is almost as large as in bulk water7, 8. This fast proton migration establishes an efficient link between these proton release and consumption sites3, 9, 10.

The long interfacial travel distance observed for protons implies that a substantial free energy barrier, ΔG ‡ r, for proton release prevents the surface proton from readily equilibrating with its bulk counterparts7, 11. It allows placing regulatory proteins (uncoupling protein 4) at some distance from both ATP synthases and proton pumps on the inner mitochondrial membrane12. Due to the spatial separation the uncoupling protein cannot uncouple phosphorylation from proton pumping. However, the large ΔG ‡ r routes excessive protons along the membrane surface to the distant proton leak. In turn, the production of reactive oxygen species is decreased.

ΔG ‡ r values can be estimated from experiments in which protons are photo-released on a membrane area at distance x (tens of micrometers) from the observation patch6, 7. To explain the results, two different models have been proposed (Fig. 1): The first model assumes that proton uptake by the interface is not in equilibrium with proton surface-to-bulk release13, whereas the second assumes quasi-equilibrium between interfacial and bulk protons14.

Figure 1
figure 1

Schematic of the two different models for proton migration along the membrane surface. The non-equilibrium model (a) envisions proton diffusion within the confinement of the membrane hydration layers without the involvement of titratable residues on the surface. Proton surface-to-bulk release is thought to be irreversible (equation (1)). The quasi-equilibrium model (b) treats proton surface diffusion as a succession of jumps between titratable residues. The proton uptake and release reactions are in equilibrium. In the absence of real proton acceptors and donors, fictional moieties are assumed to take their place. Accordingly, their pK a value is obtained as a fitting parameter of the model (equations (3–6)).

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The non-equilibrium model describes the proton concentration, σ, in the water layers adjacent to the membrane at time t as a function of both the interfacial (lateral, two dimensional) proton diffusion constant, D l, and the release rate coefficient, k off, from the membrane surface13:

$$\sigma (x,t)={\sigma }_{0}+\frac{{A}_{neq}}{4\pi {D}_{{\rm{l}}}t}\exp (-\frac{{x}^{2}}{4{D}_{{\rm{l}}}t})\exp (-t\,{k}_{{\rm{off}}})\,$$
(1)

Here A neq is a measure of proton concentration increase (of the non-equilibrium model) and σ 0 denotes the pre-existing proton concentration adjacent to the surface. Previously measured k off values13 of about 0.5 s−1 can be used to estimate ΔG ‡ r ≈ 30 k B T:

$${k}_{off}={\nu }_{0}\,\exp (-{\rm{\Delta }}{{\rm{G}}}_{{\rm{r}}}^{\ddagger}/{k}_{B}T),$$
(2)

where T, k B, and ν 0 ≈ 1013 s−1 are the absolute temperature, the Boltzmann constant, and the universal transition state theory attempt frequency for rate processes at surfaces15.

The second, quasi-equilibrium model assumes that ΔG ‡ r can be computed from the bulk proton concentration, [H+]bulk, and σ as16: ΔG ‡ r = k B T ln(σ/[H+]bulk). Surface and bulk protons are thought to be coupled over distance L 0 (Fig. 1):

$$\sigma (x,t)={\sigma }_{0}+\frac{{A}_{eq}}{4\pi {D}_{{\rm{s}}}t}\exp (-\frac{{x}^{2}}{4{D}_{{\rm{s}}}t}){(1+{(\frac{\sqrt{\pi {D}_{{\rm{s}}}t}}{{L}_{0}})}^{\alpha })}^{-1}.$$
(3)

where A eq is a measure of proton concentration increase (of the quasi-equilibrium model)14. Equation (3) is a much simplified version of the original model17. It has been obtained by assuming that the proton surface diffusion coefficient, D s, and the proton bulk diffusion coefficient, D b, are equal to each other. The dimensionality α = 1 for transversal proton motion holds for an ideal infinite plane, i.e. when the interfacial water layer width, d ~ 1 nm, for surface proton diffusion is much smaller than L 0:

$${L}_{0}=d\,\exp ({{\rm{\Delta }}G}_{{\rm{r}}}^{\ddagger}/{k}_{B}T).$$
(4)

Assuming L 0 = 170 µm18 yields ΔG ‡ r ≈ 12 k B T.

The quasi-equilibrium model17 takes into account all titratable groups at the membrane surface. Each of them is thought to occupy surface area A tg and to be characterized by proton binding and dissociation coefficients k on and k d, respectively18:

$${L}_{0}=\frac{{k}_{{\rm{on}}}}{{k}_{{\rm{d}}}{N}_{{\rm{A}}}{A}_{tg}},$$
(5)

where N A is Avogadro’s number. Setting the dwell time t 0 of the proton within the interfacial water layer equal to be d 2/D s times a Boltzmann factor depicting the delay in proton surface-to-bulk release, allows transforming equation (4)14:

$${t}_{0}=\frac{{d}^{2}\,}{{D}_{s}}\exp (\frac{{\rm{\Delta }}{G}_{r}^{\ddagger}}{{k}_{{\rm{B}}}T})=\frac{{L}_{0}d}{{D}_{{\rm{s}}}}$$
(6)

Thus with k d = 1/t 0, equations (5) and (6) imply that \({k}_{on}={N}_{{\rm{A}}}{A}_{tg}{D}_{s}/d\), namely a diffusion controlled association (k on proportional to D s). For phosphatidylethanolamine bilayers equations (5) and (6) yield t 0 ≈ 1 s and L 0 ≈ 13 m, respectively, since k on/k d and Atg are equal to 109.6 M−1 and 51 Å2, respectively19, 20. Phosphatidylcholine bilayers provide another example: L 0 must be in the order of one micrometer and t 0 in the order of 100 ns since k d is roughly 7 orders of magnitude larger. However, neither one of the extremes has been observed experimentally: dwell time and diffusion span were in the order of one second and 100 micrometers, respectively, for both bilayers7. In consequence, the quasi-equilibrium model now rests on fictitious moieties (equation (5))14. Thus, L 0, D s, and α are fitting parameters in equation (3).

One goal of the present work is to differentiate between the two theoretical models (equation (1) versus (3–6)) by means of measuring ΔG ‡ r. The second goal is to establish whether this quantity contains a large entropic contribution, which could explain the affinity of the proton to the hydration water in the absence of a potent proton acceptor. In order to address these issues, we monitored the temperature dependence of proton diffusion kinetics along the surface of lipid bilayers.

Results

We released the protons from a membrane bound caged compound6, by exposing a 10 × 10 µm2 membrane area to a UV flash (wavelength < 400 nm) (red square in Fig. 2a). We then recorded the time-dependent intensity changes in fluorescence of a lipid-anchored pH-sensitive dye (N-(fluorescein-5-thiocarbamoyl)-1,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine) at distance x from the release area that was excited by illumination at 488 nm (green square in Fig. 2a)6, 7. The resulting fluorescence decreases as protons reach the observation spot and then increases as they diffuse further away. The time between proton release and arrival increases with increasing x while the number of protons reaching the destination decreases (Fig. 2b). See Materials and Methods for additional detail.

Figure 2
figure 2

Monitoring proton surface diffusion. (a) The membrane bound caged protons were released by a UV flash from the area in the red square (10 × 10 µm2) and their arrival was observed as a change in fluorescence intensity in the green square (10 × 10 µm2). The light emitted by the lipid-anchored pH sensor fluorescein was collected using a 40x water immersion objective and a 515 nm high-pass filter. (b) The proton concentration σ adjacent to the membrane is monitored as a function of the time that elapsed after the flash at the indicated distances x (in µm) from the observation site (19 °C). σ has been calculated from the fluorescence intensity of membrane anchored fluorescein according to a calibration curve (Supplementary Fig. S1). It reaches it maximum at time t max, which according to equation (1) obeys: t max  = x 2/4D − k off t max 2. When the last term is small, t max depends linearly on x 2, as shown in the inset. The colored traces are averaged data from at least 10 individual uncaging reactions each. The black lines represent a global fit to average traces at four distances of the non-equilibrium model. Therefore equation (1) was modified to take into account the finite sizes of release and detection zones (equation (S3), Supplementary Fig. S2). The global fit parameters, D l = 5.1 × 10−5 cm2 s−1 and k off = 2.3 s−1, are common to all curves, whereas the amplitude A neq was allowed to vary (±15%).

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An increase in temperature enhances the probability of proton surface-to-bulk release, so that fewer protons arrive at the observation spot. At the same time, proton mobility increases, so that σ reaches its maximum earlier (Fig. 3). We obtain D l and k off from global fits to equation (S3) (black lines). Equation (S3) differs from equation (1) by taking into account the exact sizes of the proton release and the observation areas (see Supplement). An Arrhenius plot (inset, Fig. 3)

$${D}_{{\rm{l}}}={A}_{{\rm{l}}}\exp (\frac{-{\rm{\Delta }}{H}_{{\rm{l}}}^{\ddagger}}{{k}_{B}T}),$$
(7)

permits calculation of the activation enthalpy,\(\,{\rm{\Delta }}{H}_{{\rm{l}}}^{\ddagger}\) = 5.9 ± 1.1 k B T. It is roughly 20% larger than the experimental activation enthalpy of 4.3 k B T for bulk proton mobility21. The pre-exponential factor A l allows assessment of ν 0 = 17 × 1013 s−1 using the Einstein relation (in two dimensions):

$${v}_{0}={A}_{{\rm{l}}}\frac{4}{{l}^{2}}\,,$$
(8)

where l = 2.8 Å is the O-O distance in liquid water across which the proton hops. D l indicates that interfacial proton mobility is very close to its value in unbuffered bulk water22. ν 0 is about 10 times larger than its commonly accepted value. Such an increment is sometimes taken as indicative for proton tunneling23. Indeed, the previously observed isotope effect for proton surface diffusion (about 8)7 is larger than for proton mobility in bulk water (ca. 1.5).

Figure 3
figure 3

Kinetics of the proton concentration adjacent to the membrane surface at 80 µm from the release spot for different temperatures. The colored traces are averaged data from at least 10 individual release events each. At least three such averaged traces had been obtained at every temperature for four distances: 80, 90, 100 and 110 µm. The global fits of the non-equilibrium model (equation (S3)) to all traces at a given temperature is depicted as solid black lines. Inset: Temperature dependencies of the rate coefficient for proton surface-to-bulk release (k off, in units of s−1) and for the lateral diffusion constant (D l, in units µm2 s−1). The slopes correspond to ΔH ‡ l ≈ 5.9 ± 1.1 k B T and ΔH ‡ r ≈ 5.7 ± 0.7 k B T, whereas the intercepts with the y-axis are A l = (3.3 ± 0.5) × 106 µm2 s−1 and A r = (8.1 ± 0.9) × 102 s−1, respectively (compare equations (7) and (9)).

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With this encouraging result, we proceed to analyze the proton release reaction:

$${k}_{off}={A}_{r}\,\exp (-\frac{{\rm{\Delta }}{H}_{{\rm{r}}}^{\ddagger}}{{k}_{B}T})$$
(9)

The Arrhenius plot (inset Fig. 3) gives ΔH ‡ r = 5.7 ± 0.7 k B T. Interestingly, the activation enthalpies for proton motion perpendicular and parallel to the membrane are nearly identical. Thus, the small preexponential, A r, is responsible for the long retention time of protons on the membrane. From transition state theory we anticipate that A r = ν 0 exp(ΔS ‡ r/k B), where ΔS ‡ r is the entropy of activation for proton release:

$$T{\rm{\Delta }}{{\rm{S}}}_{{\rm{r}}}^{\ddagger}={k}_{B}T\,\mathrm{ln}({A}_{r}/{\nu }_{0})=-26.1{k}_{B}T$$
(10)

where we have used ν 0 = 17 × 1013 s−1 (equation (8)). This implies ΔG ‡ r = ΔH ‡ r −TΔS ‡ r = 31.8 k B T. The equation ΔG ‡ r = −k B T ln(k off/ν 0) yields a similar value (for 24 °C) attesting to the consistency of this analysis. It is therefore predominantly an entropy effect which opposes proton release from surface-to-bulk.

Next we tested whether the quasi-equilibrium model, equation (3), satisfactorily describes the data. α = 1 did not fit the data, although we accounted for the exact sizes of the proton release and measurement spots (equation (S4)). Augmenting α to 2 also resulted in an unsatisfactory fit. Solely upon setting α = 3 were we able to fit the quasi-equilibrium model to the data (Fig. 4).

Figure 4
figure 4

The quasi-equilibrium model (compare equation (3)) does not fit the data for d ≪ L 0 (dashed line, for parameters see Supplementary Table S1) if the surface-to-bulk release reaction is assumed to be one dimensional (α = 1). As in the case of the non-equilibrium model we took the finite sizes of proton release and detection areas into account (equation (S4)). The experimental data (colored lines) were taken from Fig. 3. The assumption α = 3 (solid black lines) yielded a satisfactory fit. Inset: The global fit of the quasi-equilibrium model (α = 3) to the traces measured at 80, 90, 100, and 110 µm produced a temperature independent L 0 of roughly 100 µm. The corresponding Arrhenius plot of D s (in units of µm2 s−1) revealed ΔH ‡ l = 5.8 ± 2.0 k B T and a pre-exponential factor A l  = (3.3 ± 0.5) × 106 µm2 s−1. These values are essentially identical to those obtained from the non-equilibrium model in Fig. 3.

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That is, we globally fitted equation (S4) to several σ(t) profiles that were measured at distances x = 80, 90, 100, and 110 µm (Fig. 2). We repeated the procedure at four different temperatures and thereby extracted both D s and L 0 from the experiments (Supplementary Table S1). L 0 showed little variation with temperature being always roughly equal to ~100 µm (Fig. 4, inset). We find ΔG ‡ r ≈ 0 and d = L 0 ≈ 100 µm according to equation (4) (Fig. 4, inset). The temperature-independent L 0 means that 1/t 0 of equation (6) has the same temperature dependence (Supplementary Fig. S3) as D s.

The large d value is in stark contrast to the originally assumed value of only 1 nm, contrasting with the requirement that d ≪ L 0. d ≈ L 0 would be consistent with a three dimensional proton escape reaction (α = 3) (Fig. 4). However, augmenting α above 1 renders the mathematical description of the quasi-equilibrium model questionable. For α = 3, the dimensionality of the membrane degenerates to 0. As we show in our Supplement, for α = 2 equation (3) represents an approximation of equation (1), i.e. it no longer describes quasi-equilibrium (Supplementary Fig. S4). It is thus not surprising that the fitting parameters D s and 1/t 0 of the non-equilibrium model are close to those of the equilibrium model, D l and k off, respectively (Supplementary Table S1). Accordingly, also the pre-exponents of the corresponding Arrhenius plots are close to each other (Figs 3, 4 and S3). We calculated them by substituting D l for D s in equation (7).

The fitting result ΔG ‡ r ≈ 0 (Fig. 4, inset) suggests the highly unrealistic scenario of a barrier-free proton escape. Proton hopping requires breaking of hydrogen bonds, so that ΔH ‡ r must be (i) at least as large as its counterpart in bulk water, i.e. ≈ 4.3 k B T 21 or (ii) equal to the corresponding value for hopping along the membrane water interface, i.e. ≈ 5.8 k B T (Fig. 4). If so, TΔS ‡ r must adopt a similar value. A positive entropic energy term suggests that protons prefer the bulk solution over the more hydrophobic water/membrane interface. This is difficult to reconcile with the observation that an excess proton disturbs the tetrahedral bulk water structure and thus preferentially accommodates close to hydrophobic interfaces24.

Since the simplified version of the quasi-equilibrium model (equations (3–6)) did not describe the experiment, we abandoned the assumption of D s and D b equality. Instead, we repeated our analysis with the original model17. However, the corresponding analysis (equation (S5)) only revealed a satisfactory accuracy of the global fit (with three independent parameters: L 0, D s, D b) to the experimental data for negligibly small D s (i.e. D s ≪ D b) (Supplementary Fig. S5). To compensate for the lack of proton surface diffusion, the model returned large D b values (Supplementary Table S1). However, buffer molecules must have carried the majority of the bulk protons in our system, so that much smaller values were expected. This is evident from the fact that the buffer capacity (~50 µM) in our experiments exceeded the bulk proton concentration (1 nM) by more than four orders of magnitude. Thus D b ~ 500 µm² s−1 was expected at room temperature instead of the calculated value of 12500 µm² s−1. Together with the temperature independent L 0 (Supplementary Fig. S5), this observation strongly argues against the quasi-equilibrium model.

Discussion

We observed proton migration along the membrane-water interface between two membrane patches. The rise and the decay of the fluorescence signal agree with a simple model, in which protons diffuse laterally on the two-dimensional surface of the membrane (diffusion coefficient D l), detaching from it slowly and irreversibly (rate coefficient k off) (equation (1)). D l and k off were in excellent agreement with previous studies where protons have similarly been released from a membrane adsorbed caged compound6, 7. The incompatibility of the large D l value with proton jumps along titratable interfacial moieties7 raised the question concerning the origin of the high proton affinity to membranes. How are protons retained on the surface for so long (seconds) if not by attraction to titratable groups? The small value of k off suggests a large free-energy barrier (ΔG ‡ r > 30 k B T) for proton release, which is unphysically large if most of it is enthalpic. To help decipher this enigma, we have conducted a detailed temperature-dependent study of the membrane proton transport process.

The temperature dependence of both D l and k off (equation (1)) is a major result of our work that sheds light on the mechanism of proton conduction along membrane-water interfaces. Their Arrhenius plots reveal essentially identical enthalpies of activation for protons moving parallel and perpendicular to the membrane. Moreover, this enthalpy cost is not large – approximately the strength of a single water-water hydrogen bond, as in the Grotthuss mechanism for proton mobility25. Consequently, high energy binding forces are not involved in keeping the proton at the surface.

The intercepts of the Arrhenius plots show that the major contribution to ΔG ‡ r is entropic, and not enthalpic. The proton enjoys considerably larger entropy at the interface than in the bulk24. For example, there may be an enhanced probability for the H5O2 + cation at the interface, with the proton delocalizing between two water oxygens. Such a possibility is suggested e.g. from the infrared spectrum of H5O2 + attached to a benzene molecule26 or the distribution of heteropolyanions at the air-water interface27. In contrast, the dominant species in the bulk is the hydronium ion, H3O+, which forms exceedingly strong hydrogen-bonds in its first solvation shell28, restructuring the water network around it. Thus, the entropy of the water solvent will reduce once a proton moves from the interface to the bulk.

Proton binding to aqueous buffer molecules should also be considered7. However, it is important to note that buffer molecules do not contribute to D l but only to D b 6. This follows from the simple considerations that (i) proton release from buffer molecules is much slower than H3O+ dissociation due to the higher pKa value of buffer molecules and (ii) the bulkier buffer molecules perform diffusion in all three dimensions. The hopping along hydrogen bonded surface water molecules also explains the observation that D l is very close to the diffusion coefficient of protons in pure water.

Conceivably, water structuring at the membrane interface also contributes to the entropic nature of proton membrane affinity. Evidence for the non-random orientation of interfacial water molecules has been obtained by (a) phase-sensitive vibrational sum frequency generation spectroscopy29 and (b) measurements of membrane dipole potential30,31,32. Aligned water molecules are thought to contribute roughly 50% to the total value of membrane dipole potential of about 220–250 mV30, 32. Bilayers from (i) phospholipids with or without conventional headgroups (like ethanolamine, choline, or glycerol headgroups) or (ii) lipids that do not contain the anionic phosphate moiety, have similar dipole potentials32 suggesting that charged moieties are not required to orient water dipoles. This notion is supported by reports about a net orientation of water molecules at the interface to alkene33 or to other hydrophobic interfaces34. It is also in line with the finding that titratable residues are not required for interfacial proton migration7.

Mechanistic insight about how a preferential alignment of water dipoles normal to the membrane surface promotes proton diffusion parallel to the surface awaits discovery by molecular dynamics simulations. One possibility would be that water dipole orientation toward the interface11, 35 electrostatically favors proton movement in that direction while disfavoring surface-to-bulk release. Another explanation may be that hydrated excess protons create their own water wires parallel to the membrane boundary. Such an effect has been observed in silico for proton transport through a hydrophobic nanotube36. It could explain why ΔG ‡ r is so much larger than the previously calculated free energy difference ΔG for passing from close proximity of the phosphate moieties to the bulk. Multistate empirical valence bond (MS-EVB) calculations37 and classical molecular dynamic calculations using a HYDYN protocol35 resulted in ΔG ~ 8 k B T and ΔG = 5 k B T, respectively. They are in quantitative agreement with equilibrium experiments on DOPC38, which suggested a 100 fold increase of proton concentration at the surface, i.e ΔG ~ 4–5 k B T. In contrast to ΔG, ΔG ‡ r does not allow conclusions about the difference between surface and bulk pH.

Unlike the non-equilibrium model, the quasi-equilibrium model does not properly describe the temperature dependence of the proton release reaction. Its simplified mathematical version14 returns an incredibly large thickness of the near-membrane water layers of ~100 µm. The mathematically more involved, original version17 nullifies D s and renders D b incredibly large (see Supplementary Table S1). Both versions nullify ΔG ‡ r, while producing ΔS ‡ r with the wrong (positive) sign, which is quite implausible. The quasi-equilibrium fails because it is rooted on the assumption of an enthalpic attraction of the surface proton to the membrane surface. The disguise by cosmetic adaptations - as represented by abandoning the pK a values of real titratable moieties and substituting them for pK a values of fictitious moieties14 - cannot repair the principal misconception: the protons are not held at the membrane surface by covalent bonds, but they are captured by an entropic trap. The trap is provided by interfacial water, along which the proton migrates. This strips titratable moieties from the position to govern interfacial proton mobility by proton uptake or release reactions7. Occasionally a proton may be lost to these moieties or born by them, but since there are so many protons released at the surface in our experiments, the overall proton mobility is little affected by their presence. The conclusion holds for micrometer-sized objects, like the planar bilayer studied here, as well as for the much smaller lipid vesicles or nanodiscs. It is equally valid for the pump-probe approach used in the current study, as well for equilibrium experiments. In any case, the residence time of a proton that is covalently bound to a titratable moiety does not reflect the mobility of all the other interfacial protons and hence, it cannot be used to calculate D s. It is thus not surprising that attempts to calculate D s from equilibrium protonation kinetics of fluorescent surface dyes38, 39 severely underestimated the mobility of the surface proton6, 7, 40 (compare also Fig. 1).

We conclude that the low proton acceptability of water does not exclude interfacial water wires from acting as “proton railways”. This mechanism markedly differs from the concept of titratable lipid moieties acting as proton collecting antennae18. By dissecting ΔG ‡ r into entropic and enthalpic contributions, we were able to show that only a minor part of proton’s surface affinity is due to energetic attraction to the interface. Both proton movement along and perpendicular to the membrane requires breaking of hydrogen bonds. The energy associated with that process does not depend on the directionality of proton movement, while the entropy increases substantially as the proton moves from the interface to the bulk, and this now appears to be a key factor in membrane energetics.

Materials and Methods

The Experimental setup has been described previously6, 7. In brief, horizontal planar lipid bilayers were formed from a solution of 20 mg 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC, Avanti Polar Lipids, Alabama) in 1 ml n-decane (Sigma-Aldrich, Missouri) in a 200–300 µm wide aperture of a Teflon septum. The solution contained ~1 mol % of the pH-sensor fluorescein that was covalently linked to N-(fluorescein-5-thiocarbamoyl)-1,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine (Fluorescein DHPE (FPE), ThermoFisher, Massachusetts), and a caged-proton compound, 6,7-dimethoxycoumarin-4-yl)methyl diethyl phosphate, synthesized as previously described41. Protons were released by UV light pulse (<400 nm) emitted by a xenon flash lamp that was focused onto a 10 × 10 µm2 membrane patch. The FPE fluorescence was excited by illuminating another 10 × 10 µm2 large membrane area using 488 nm radiation from a second xenon lamp (150 W). The emitted light passed a 515 nm high-pass filter and was measured by a photomultiplier. Proton surface concentrations were calculated from the fluorescence intensities with the help of a calibration curve that depicted the fluorescence intensity (normalized to peak fluorescence intensity) as a function of bulk pH. We measured that curve in equilibrium. The procedure ignores any pH difference that may have existed between bulk and the bilayer surface. The buffer consisted of 10 mM KCl and 0.1 mM Capso. pH was adjusted to 9.0.