Water at hydrophobic interfaces delays proton surface-to-bulk transfer and provides a pathway for lateral proton diffusion Chao Zhanga,1, Denis G. Knyazevb,1, Yana A. Vereshagaa,1, Emiliano Ippolitia, Trung Hai Nguyena, Paolo Carlonia,c,2, and Peter Pohlb a Computational Biophysics, German Research School for Simulation Sciences, 52425 Jülich, Germany; bInstitute of Biophysics, Johannes Kepler University Linz, 4040 Linz, Austria; and cComputational Biomedicine, Institute for Advanced Simulations, Forschungszentrum Jülich, 52425 Jülich, Germany
Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved May 1, 2012 (received for review December 23, 2011)
Fast lateral proton migration along membranes is of vital importance for cellular energy homeostasis and various proton-coupled transport processes. It can only occur if attractive forces keep the proton at the interface. How to reconcile this high affinity to the membrane surface with high proton mobility is unclear. Here, we tested whether a minimalistic model interface between an apolar hydrophobic phase (n-decane) and an aqueous phase mimics the biological pathway for lateral proton migration. The observed diffusion span, on the order of tens of micrometers, and the high proton mobility were both similar to the values previously reported for lipid bilayers. Extensive ab initio simulations on the same water∕n-decane interface reproduced the experimentally derived free energy barrier for the excess proton. The free energy profile GH þ adopts the shape of a well at the interface, having a width of two water molecules and a depth of 6 2RT . The hydroniums in direct contact with n-decane have a reduced mobility. However, the hydroniums in the second layer of water molecules are mobile. Their in silico diffusion coefficient matches that derived from our in vitro experiments, ð5.7 0.7Þ × 10 −5 cm 2 s −1 . Conceivably, these are the protons that allow for fast diffusion along biological membranes. hydrophobic liquid ∣ hydrophilic liquid interface ∣ surface acidity ∣ free energy calculations
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roton transfer is only fairly understood in bulk water (1, 2) but has not been comprehended along biological interfaces, even though the process is of vital importance for cellular bioenergetics and membrane transport processes, as has been postulated a long time ago (3). Convincing evidence revealing the membrane surface as a major pathway for proton transport was obtained by the observation that protons that were released on one side of a purple membrane fragment appeared first at the opposing interface and only afterwards in the corresponding bulk solution (4). Unfortunately, it remained unclear whether proton mobility is reduced at the membrane surface. The reported lateral diffusion coefficients D varied between 9.6 × 10 −7 cm 2 s −1 (4) and 3 × 10 −5 cm 2 s −1 (5). It was believed that the protons moved by jumping along the membrane surface between titrable groups. Their proton attraction was deemed responsible for the delayed proton surfaceto-bulk transfer (6). In line with these considerations, measurements of the protonation kinetics served to predict the proton surface diffusion coefficient (7). According to such a calculation, D was more than an order of magnitude smaller than the diffusion coefficient of a proton carrier in bulk. In contrast, direct measurements on simple planar bilayers devoid of proton accepting proteinaceous residues revealed a D that was 10 times higher and proton residence times at the interface on the order of hundreds of milliseconds (8). Moreover, lateral diffusion along those bilayers persisted upon removal of titrable lipid groups. In the presence of such groups, D appeared to depend only weakly on their pK (9). The observed size of the isotope effect strongly sup9744–9749 ∣ PNAS ∣ June 19, 2012 ∣ vol. 109 ∣ no. 25
ported the conclusion that the protons may travel along surface bound water (9). Proton movement along hydrogen-bonded water molecules is a scenario known from narrow membrane channels, where these waters bridge the distance between titrable amino acids (10). The mechanism of proton hopping between protonation sites on the membrane surface may be quite similar (11, 12). However, the protons on the membrane surface lack the geometrical confinement of a narrow pore, which raises the question of why the protons remain for hundreds of milliseconds on the surface instead of being released into the bulk. Molecular dynamics simulations using the multistate empirical valence bond (MS-EVB) model suggested that the protons may be trapped by charged groups such as the phosphate moieties of phospholipids. In this case, they would move together with the lipids (13, 14). The mobile (untrapped) fraction of surface protons faces a shallow energy barrier. Accordingly, they are released from the interfacial region in less than one nanosecond (13, 14). According to simulations carried out with nonbiological models (15–21), well-defined binding sites may be not required for proton attraction to interfaces between high and low dielectric regions. For example, for the water/vapor interface, ab initio and MS-EVB simulations revealed the acidification of the top surface water layer as compared to bulk water (15–18). Free energy minima about 3 Å wide from the interface were also found in MS-EVB simulations at water/carbon nanotubes (19) and water∕ CCl4 interfaces (20). However, the affinity of the hydronium to a hydrophobic interface came at the cost of reduced mobility as has been observed in ab initio simulations for the hydronium close to the water-graphene interface (21). How to reconcile the high affinity to the membrane-water interface with high proton mobility is, thus, totally unclear. To explain both the high surface diffusion coefficient and the long lateral diffusion span (5, 8, 22), the membrane-water interface must possess a feature that has thus far escaped notice. The simplest explanation would be that, in the absence of titrable groups (9), polar groups might stabilize the proton close to the interface. Since their attraction is weaker than the one of titrable groups, the proton retains high mobility. To test this hypothesis, we developed a minimalistic model that lacks the polar groups. Our experimental observation of long-range proton diffusion between release and detection spots on a water-n-decane interface sugAuthor contributions: P.C. and P.P. designed research; C.Z., D.G.K., Y.A.V., and T.H.N. performed research; C.Z., D.G.K., Y.A.V., and E.I. analyzed data; and C.Z., D.G.K., E.I., P.C., and P.P. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. 1
These authors contributed equally to this work.
2
To whom correspondence should be addressed. E-mail:
[email protected].
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gests that a free energy barrier ΔGH þ is intrinsic to any interface separating hydrophobic and hydrophilic phases. This supports the opinions from previous simulations (15–21). Proton retention close to the surface decreases interfacial pH at the first two water layers by about 2–3 units below bulk pH, as indicated by our extensive 75 ps-long ab initio metadynamics-based free energy simulations, including 1,707 atoms. At least in the case of a liquid hydrophobic phase, proton mobility in the second interfacial hydration shell is not impaired. Its high mobility coincides with the observed fast surface proton diffusion in the experiment. Results Long-Range Proton Diffusion on Hydrophobic Liquid-Water Interfaces.
σðx; tÞ ¼
A x2 exp − expð−k1 tÞ: 4πDt 4Dt
[1]
Here, parameter k1 is a mere fitting parameter. It combines several phenomena that affect proton escape. Fitting Eq. 1 to the
Fig. 1. Experimental scheme for fluorometrical detection of lateral proton diffusion. On top of a water droplet (140 μl) containing 0.1 or 1 mM Mes we added n-decane (280 μl) containing 0.7 μM of fluorescent dye Oregon Green 488 1,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine, which accumulated at the interface. The protons were injected at a distance 20 Å apart from each other (assuming exclusive interfacial localization). The dye responded to decreased pH with a decrease in fluorescence intensity F. The observed fluorescence signal is FðtÞ ¼ Fðt ¼ 0Þ − Bσðx; tÞ, where σðx; tÞ is the proton surface density, x is the distance to the sink, and B is a proportionality factor. σðx; tÞ reaches a maximum at time t ¼ τmax . We tested proton transport by (i) microinjecting protons on a spot at the water∕n-decane interface and (ii) measuring the kinetics of their arrival at a distant position on the same interface. Assuming that surface diffusion is decoupled from bulk, we fitted FðtÞ for σðx; tÞ by the solution of a twodimensional diffusion equation (23):
Substituting H2 O for D2 O (i) increased τmax twofold and (ii) halved Dapp (Fig. 3), as does an increased buffer capacity. This difference is too big to be explained by an increased viscosity of D2 O. The increased strength of hydrogen bonds in D2 O conceivably slowed down proton hopping. Alternatively, some deuterons may have reached the measurement spot via bulk diffusion. Both the buffer and the isotope effect rule out the possibility that microinjecting protons caused convection, since either changing the isotope or the buffer would not significantly alter drift velocity (22). We conclude that the water∕n-decane interface captures key features of proton migration along the water/ membrane interface, and thus the contribution of dipolar lipid moieties to ΔGH þ is not required. To test whether the lateral migration of protons depends on their recombination with molecules known to adsorb at hydrophobic boundaries (24), we varied the chloride concentrations in the buffer solution (Fig. S1). We found that lateral surface proton diffusion persisted when chloride was not at all present. This observation indicates that lateral proton migration is intrinsic to the boundary of hydrophobic and hydrophilic liquids. For an experimental evaluation of ΔGH þ , we measured k1 at temperatures ranging from 287 K to 316 K. The resulting Arrhenius plot of ln k1 versus 1∕T is nonlinear, similar to what was shown in previous work (25). The plot is composed of two temperature regimes (Fig. 4). Assuming that proton delivery to membrane channels occurs by surface migration, the activation energy for transmembrane proton transport must show similar temperature dependence. Indeed, concave Arrhenius plots have already been observed for proton transport through membrane channels at comparable temperatures (26, 27). In the temperature regime around 310 K, for which simulations have been carried out (see the next section), we find that ΔGH þ ≃ 8.7 RT.
DFT-MD calculations are well suited for the study of proton transfer in complex systems (29, 30). Their accuracy, however, dramatically depends on the choice of the exchange-correlation functional. Here we use the Becke–Lee–Yang–Parr (BLYP) functional, along with Grimme’s correction for the dispersion forces (31), which turned out to be reasonably accurate yet computationally affordable for our system, which is very large. The BLYP functional consists of 25 n-decane molecules, 302 water molecules in slab geometry, and one excess proton, making 1,707 atoms in total (see Materials and Methods). The free energy profile GH þ is calculated by 75 ps-long ab initio simulations based on metadynamics (32) at 310 K. GH þ is a function of the distance L from the excess proton to the water∕ndecane instantaneous interface (33) (SI Text and Fig. S2). A wide minimum of GH þ is located within 6 Å from the hydrophobic surface with a depth of 6 RT (Fig. 5) and an estimated statistical error of approximately 2 RT (Figs. S3 and S4). This is in reasonable agreement with the experimental estimate of GH þ described before (approximately 8.7 RT). Within this minimum, two populations of the excess proton I and II (Fig. 5) are observed. In population I, the hydronium ion is in direct contact with the n-decane surface. Its electron lone pair is oriented towards n-decane (Fig. 5, Inset), consistent with the fact that it is a weak Hbond acceptor (Fig. S5) (15, 21). The hydronium ion forms three H-bonds with surrounding water molecules [the so-called ‘Eigen structure’ (34)]. Similar asymmetric amphiphile-like solvation patterns through both hydrophilic and hydrophobic interactions have previously been found for the excess proton in methanolwater solutions (35, 36). Its electronic density is significantly polarized by the first and second solvation shells (Fig. 5, Inset).
Mechanistic Insights. We performed density functional theory (DFT)-based molecular dynamics (MD) simulations (28) to investigate the molecular details of proton diffusion at the surface.
Fig. 4. Experimental estimation of the free energy barrier for proton surface to bulk escape. (A) Kinetics of surface proton concentration change in a spot at distance x ¼ 45 μm from the proton release area. The left and right panels were recorded at T ¼ 287.5 K and 316 K, respectively. The buffer contained 1 mM Mes (pH 6.3) and 100 mM NaCl. Fitting of Eq. 1 (main text) to the data resulted in the red lines. The parameters were D ¼ 4.3 × 10 −5 cm 2 s −1 , k1 ¼ 0.2 s −1 and D ¼ 9.5 × 10 −5 cm 2 s −1 , k1 ¼ 31.9 s −1 , for the left and right panels, respectively. (B) Experiments similar to those shown in (A) were performed at temperatures between 287 and 316 K. Subsequently we used the fitting parameter k1 to build the Arrhenius plot. In order to extract ΔGH þ , we treated the data at low and high temperature regimes separately. The slope of the linear regression at 310 K corresponds to ΔGH þ ¼ 8.7 RT.
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Fig. 5. The free energy profile of the excess proton GH þ and density profiles of water and n-decane molecules as a function of the distance L to the water∕n-decane interface (see SI Text for the definition of L). In the background, water molecules are shown in CPK representation and n-decane molecules are represented as green rods. The blue vertical stripe indicates the instantaneous interface between water and n-decane. The system fully treated with DFT consists of 25 n-decane molecules, 302 water molecules, and 1 excess proton (Fig. S8). Free energy simulations were 75 ps-long and performed with a thermostat at T ¼ 310 K. See SI Text and Figs. S3 and S4 for technical discussion. (Inset) Electron density differences of the two excess proton populations at the surface (I and II), Δρ ¼ ρsystem − ρH3 O þ − ρrest . System refers to the whole ab initio simulation system; Rest refers to the rest of the whole simulation system without H3 O þ . Purple refers to positive Δρ and yellow is for the negative Δρ.
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The first peak of the water density profile lies behind the position of population I at about 1.5 Å, as shown in Fig. 5. Therefore, the hydronium ion is closer to the n-decane surface than the water molecules. In population II, the ion forms solvent-separated contacts with the n-decane surface. It is located in the second layer of water molecules without a specific orientation. In our 10 ps-long unbiased DFT-MD simulations, the excess proton diffuses and remains in the interfacial region (Fig. S6A). The calculated proton lateral diffusion coefficient is ð8 2Þ × 10 −5 cm 2 s −1 (Fig. S6B). This is in agreement with our experimental value ð5.7 0.7Þ × 10 −5 cm 2 s −1 . Very importantly, population II represents the main species that diffuses quickly along the interface (Fig. 6), because its adjacent water molecules form more H-bonds, on average, than the water molecules adjacent to population I do (Fig. S5).
Free Energy Minimum at the Interface. To understand the origin of the free energy well involving two water layers, we consider key contributions to GH þ as functions of the proton distance L from the interface (Fig. 7D and SI Text). The first stabilizing contribution Gbind is the excess binding free energy of a hydronium ion to the n-decane surface. An approximate estimate for Gbind by considering only the enthalpy can be tabulated by ab initio cal-
Fig. 6. A lateral diffusion path of the excess proton at the water∕n-decane interface extracted from a 10 ps-long NVE ab initio MD simulation. (Inset) Simulation snapshots of the excess proton and its coordinating water molecules at the interface. The n-decane molecules and remaining water molecules are not plotted for clarity. The blue area represents the instantaneous water∕n-decane interface, the same as in Fig. 5. The background color is set to green.
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Fig. 7. (A) and (B) Calculated binding energies of a hydronium ion and a water molecule to the n-decane slab as a function of the distance to the interface, respectively. The n-decane slab was extracted by removing the water molecules from the equilibrated water∕n-decane system. No geometric optimization has been performed. (C) Calculated average electrostatic potential Φ of the water∕n-decane system as a function of the distance to the RRR interface from a 10 ps-long NVT ab initio MD simulation. ΦðLÞ ¼ 1 dxdydzhΦðrÞδðl − LÞi, where ΦðrÞ is the electrostatic potential of the sysA tem at the position r, and A is the area of the interface. ΦðrÞ ¼ ∑B jRZB B−rj − 0 ∫ dr 0 jrρðr0 −rjÞ , in which Z B is the charge on nucleus B located at RB , and ρðrÞ is the electron density of the system at the position r in the real space grid. (D) Gbind , GΦ , GBorn-image and their sum Gsum , comparing with GH þ as a function of the distance to the interface. Gbind stems from the excess short-range attractive interactions of a hydronium ion with the n-decane surface. The GΦ is the electrostatic energy of the proton due to the surface electrostatic potential. GBorn-image is due to Born energy of the excess proton and the interaction with its image in continuum electrostatics. The GH þ is the profile generated from DFT-MD free energy simulations.
culations. The binding energy of the hydronium ion to the n-decane surface amounts to approximately −25 RT (Fig. 7A), which exceeds that of a water molecule (approximately −4 RT) sixfold (Fig. 7B). This leads to the value of Gbind approximately −21 RT at the minimum (SI Text). The next stabilizing contribution Gstrain is due to the water molecules solvating the excess proton. The latter strains the hydrogen-bond pattern of the surrounding liquid water, as discussed already for water/vapor (17), water/carbon nanotubes (19), or water∕CCl4 interfaces (20). The strain is much smaller at the boundary because of the reduced number of hydrogen bonds (Fig. S5). At present, no simple way to quantify Gstrain has been found. Then, we consider the destabilizing contribution of Born free energy and the free energy costs for polarization of the interfaces (image energy), i.e. GBorn-image . This contribution arises from moving a proton from the hydrophilic environment of bulk water (ε approximately 80) to the more hydrophobic environment adjacent to the interface (ε approximately 10–20). We use continuum electrostatics for the calculation of this contribution (SI Text). Finally, we consider the destabilizing contribution GΦ of surface electrostatic potential Φ across two dielectric phases (SI Text). Φ is analogous to the so-called dipole potential of biological membranes and lipid bilayers, which is positive inside the membrane (37, 38). Although somewhat larger in size, Φ has the same orientation at the water∕n-decane interface. Our ab initio simulations provide an estimation, showing that the potential PNAS ∣
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Discussion Simulations and experiments provide a consistent picture indicating that (i) the hydronium ion forms attractive interactions at the water∕n-decane interface, and that (ii) the hydronium ion retains high mobility that exceeds bulk mobility of proton carriers. Moreover, both the in vitro and in silico results indicate that the interfacial energy barrier adopts the form of a well. If the proton would face a barrier upon approaching the interface from the bulk water, proton microinjection (Fig. 1) would not result in surface diffusion spans and mobilities that are comparable to those obtained upon proton photo-release from the hydrophobic interface (8, 9).
increases by approximately 1.2 V from the water phase to the ndecane phase (Fig. 7C). In the absence of analytical expression for Gstrain , we calculate Gsum ¼ Gbind þ GBorn-image þ GΦ . Gsum reproduces GH þ surprisingly well (Fig. 7D), although Gbind represents the only stabilizing contribution. This observation suggests that (i) Gstrain might not be very large; (ii) the minimum of the free energy spans over two water layers (Fig. 5) because, beyond that, Gbind ¼ 0 (Fig. 7A); (iii) since our calculated value of the free energy barrier ΔGH þ ¼ 6 2 RT is not very different from those calculated for water/ vapor (17, 18), water/carbon nanotube (19), and water∕CCl4 (20) interfaces, some of these findings may be general to dielectric mismatched interfaces. Fast Diffusion at the Interface. The free energy barrier provided by our in vitro and in silico studies on the n-decane interface (ΔGH þ of about 6 to 8 RT) is similar to that of MS-EVB calculations on the membrane interface (ΔGH þ of about 6.7 to 8.3 RT) (14). In that case, these protons localize in the deep interface region of a phospholipid membrane and move essentially with the lipid (14). Our simulations allow us to suggest a different picture for the n-decane interface. Here, the protons in the second interfacial hydration layer retain their high mobility, although they are subjected to attractive forces large enough to prevent their fast release into the bulk. This observation is in perfect agreement with the experiment, where the highly mobile excess proton stays in the interfacial region for tens and hundreds of milliseconds (Figs. 2 and 3). Our in silico system contains neither ions nor buffer molecules. This omission is unlikely to distort the principal molecular picture that emerged from the simulations because varying the concentration of both kinds of molecules in the experimental system did not induce dramatic effects on either lateral diffusion span or on its speed. Although chloride ions may adsorb at the interface (24), proton binding to interfacial Cl − is not essential for lateral proton migration (Fig. S1). An increase in mobile buffer concentration seems to inhibit surface diffusion, most probably by a recombination of the mobile buffer with the excess proton (Fig. 3). We may thus be confident that the in vitro and in silico studies have captured the same process. Moreover, we have found a proper explanation for how the requirements for proton attraction and high proton transport rate, which at first glance appear conflicting, may actually both be realized at the same time. The simulations show that the hydroniums in direct contact with the hydrophobic liquid are indeed immobile. However, the hydroniums in the second layer of water molecules are capable of migrating along the surface given that they encounter a weaker binding force. We expect proton attraction to be an intrinsic property of all water/hydrophobic interfaces and that, at least for hydrophobic liquid phase, fast surface proton diffusion is retained. 1. Agmon N (1995) The Grotthuss mechanism. Chem Phys Lett 244:456–462. 2. Knight C, Voth GA (2012) The curious case of the hydrated proton. Acc Chem Res 45:101–109. 3. Williams RJP (1988) Proton circuits in biological energy interconversions. Annu Rev Biophys Biophys Chem 17:71–97. 4. Heberle J, Riesle J, Thiedemann G, Oesterhelt D, Dencher NA (1994) Proton migration along the membrane surface and retarted surface to bulk transfer. Nature 370:379–382. 5. Alexiev U, Mollaaghababa R, Scherrer P, Khorana HG, Heyn MP (1995) Rapid longrange proton diffusion along the surface of the purple membrane and delayed proton transfer into the bulk. Proc Natl Acad Sci USA 92:372–376. 6. Cherepanov DA, Junge W, Mulkidjanian AY (2004) Proton transfer dynamics at the membrane/water interface: Dependence on the fixed and mobile pH buffers, on the size and form of membrane particles, and on the interfacial potential barrier. Biophys J 86:665–680. 7. Brändén M, Sandén T, Brzezinski P, Widengren J (2006) Localized proton microcircuits at the biological membrane-water interface. Proc Natl Acad Sci USA 103:19766–19770. 8. Serowy S, et al. (2003) Structural proton diffusion along lipid bilayers. Biophys J 84:1031–1037.
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Consequently, these findings may have wide implications beyond membrane transport (39). Materials and Methods The 1,707 atom system was set up with classical MD (SI Text, Figs. S7 and S8) and underwent ab initio Car-Parrinello MD simulations (28) in the NVT and NVE ensembles, where N is the number of particles, V is volume, and T is temperature. The quantum problem was solved within density functional theory (DFT), using the BLYP exchange-correlation functional and Grimme’s correction for the London dispersion interactions (31). The accuracy of the correction was investigated by comparing dispersion-corrected DFT calculations with MP2 calculations as well as with results from the highly accurate dispersion-corrected atom-centered potentials (DCACP) (40) (SI Text and Fig. S9). The electronic wave functions were expanded by plane wave basis sets within an energy cutoff of 70 Ry. A fictitious electron mass of 400 a.u. and a timestep of 4 a.u. (approximately 0.1 fs) were used. For the NVT simulations, the Nosé-Hoover chains thermostat (41) was used to keep the temperature at T ¼ 310 K. First, several simulated annealing steps and then a 3.0 ps-long ab initio MD equilibration in the NVT ensemble were carried out. Then, based on the last ab initio MD snapshot, three production simulations were performed. 1. We performed 75 ps-long ab initio multiple-walker metadynamics simulations (42) in the NVT ensemble to reconstruct the free energy profile as a function of an appropriate collective variable (CV). The method allows the system to escape the free energy minima and scales linearly on a massively parallel supercomputer (42). The CV identifies the distance from the excess proton to the interface (SI Text). The key parameters of the method (42, 43) are the width of Gaussian ‘hills’ added to the history-dependent biasing potential, the height of Gaussian hills, and the frequency at which the hills are added. These parameters were set to 0.25 Å, 0.2 RT, and 4.84 fs, respectively, similar to what was done in previous applications of the method (42, 43). 2. A 10 ps-long ab initio MD simulation in the NVE ensemble was performed. The lateral diffusion coefficient of the excess proton at the water∕n-decane interface was calculated. The averaged temperature of this simulation is 318 5 K. 3. A 10 ps-long ab initio MD simulation in the NVT ensemble, at T ¼ 310 K, was carried out. Electrostatic potential was calculated from 200 equally spaced snapshots. The real-space grid was 360 × 240 × 256. Density profiles of water and n-decane molecule as a function of the distance to the interface were also calculated from this simulation. All the ab initio simulations were performed by using the CPMD code (44). For metadynamics simulations, the PLUMED-CPMD interface was employed (45). ACKNOWLEDGMENTS. C.Z. and E.I. thank Fabio Pietrucci for providing the PLUMED-CPMD interface for free energy simulations and Giovanni Bussi for the discussions of recovering canonical probability from metadynamics simulations. We acknowledge that this work has been achieved using the PRACE (http://www.prace-project.eu/) research infrastructure resource JUGENE hosted by Forschungszentrum Jülich in Germany. 9. Springer A, Hagen V, Cherepanov DA, Antonenko YN, Pohl P (2011) Protons migrate along interfacial water without significant contributions from jumps between ionizable groups on the membrane surface. Proc Natl Acad Sci USA 108:14461–14466. 10. Freier E, Wolf S, Gerwert K (2011) Proton transfer via a transient linear water-molecule chain in a membrane protein. Proc Natl Acad Sci USA 108:11435–11439. 11. Agmon N (2010) Diffusion across proton collecting surfaces. Chem Phys 370:232–237. 12. Medvedev ES, Stuchebrukhov AA (2011) Proton diffusion along biological membranes. J Phys Condens Matter 23:234103. 13. Smondyrev AM, Voth GA (2002) Molecular dynamics simulation of proton transport near the surface of a phospholipid membrane. Biophys J 82:1460–1468. 14. Yamashita T, Voth GA (2010) Properties of hydrated excess protons near phospholipid bilayers. J Phys Chem B 114:592–603. 15. Petersen MK, Iyengar SS, Day TJF, Voth GA (2004) The hydrated proton at the water liquid/vapor interface. J Phys Chem B 108:14804–14806. 16. Buch V, Milet A, Vácha R, Jungwirth P, Devlin JP (2007) Water surface is acidic. Proc Natl Acad Sci USA 104:7342–7347. 17. Köfinger J, Dellago C (2008) Biasing the center of charge in molecular dynamics simulations with empirical valence bond models: Free energetics of an excess proton in a water droplet. J Phys Chem B 112:2349–2356. 18. Lee H-S, Tuckerman ME (2009) Ab initio molecular dynamics studies of the liquid-vapor interface of an HCl solution. J Phys Chem A 113:2144–2151.
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