Framework model for single proton conduction through gramicidin

Synthesis of Hidden Subgroup Quantum Algorithms and Quantum Chemical Dynamics

We describe a general formalism for quantum dynamics and show how this formalism subsumes several quantum algorithms, including the Deutsch, Deutsch-Jozsa, Bernstein-Vazirani, Simon, and Shor algorithms as well as the conventional approach to quantum dynamics based on tensor networks. The common framework exposes similarities among quantum algorithms and natural quantum phenomena: we illustrate this connection by showing how the correlated behavior of protons in water wire systems that are common in many biological and materials systems parallels the structure of Shor's algorithm.

Discovering Reaction Pathways, Slow Variables, and Committor Probabilities with Machine Learning

A significant challenge faced by atomistic simulations is the difficulty, and often impossibility, to sample the transitions between metastable states of the free-energy landscape associated with slow molecular processes. Importance-sampling schemes represent an appealing option to accelerate the underlying dynamics by smoothing out the relevant free-energy barriers, but require the definition of suitable reaction-coordinate (RC) models expressed in terms of compact low-dimensional sets of collective variables (CVs). While most computational studies of slow molecular processes have traditionally relied on educated guesses based on human intuition to reduce the dimensionality of the problem at hand, a variety of machine-learning (ML) algorithms have recently emerged as powerful alternatives to discover meaningful CVs capable of capturing the dynamics of the slowest degrees of freedom. Considering a simple paradigmatic situation in which the long-time dynamics is dominated by the transition between two known metastable states, we compare two variational data-driven ML methods based on Siamese neural networks aimed at discovering a meaningful RC model─the slowest decorrelating CV of the molecular process, and the committor probability to first reach one of the two metastable states. One method is the state-free reversible variational approach for Markov processes networks (VAMPnets), or SRVs─the other, inspired by the transition path theory framework, is the variational committor-based neural networks, or VCNs. The relationship and the ability of these methodologies to discover the relevant descriptors of the slow molecular process of interest are illustrated with a series of simple model systems. We also show that both strategies are amenable to importance-sampling schemes through an appropriate reweighting algorithm that approximates the kinetic properties of the transition.

Theory of single-file multiparticle diffusion in narrow pores

Single-file diffusion of multiple strongly interacting particles in a one-dimensional pore is described within a general analytical framework. The theory accounts for nonequilibrium conditions, explicit particle-particle interactions, external potential acting on the particles and the fluctuations of the number of particles due to their exchange with external equilibrium reservoirs. It is shown that the problem can be reduced to a closed hierarchical set of partial differential equations of increasing dimensionality, which can be solved numerically. Our framework allows computing any macroscopic characteristic of multiparticle diffusion in the pore. It is shown that the pore occupancy probabilities and the current are rational functions of external concentrations in the steady state. The theory is tested on a simplified model of the narrow rigid pore inspired by the selectivity filter of biological ion channel. Perspectives and limitations of the theory are discussed.

Mechanisms of proton transfer in proteins: localized charge transfer versus delocalized soliton transfer

Proton translocation coupled to redox chemistry is ubiquitous for membrane enzymes involved in energy generation in cells. In such enzymes, proton transport occurs in special proton conducting channels, which consist of a series of protonatable groups of the protein connected by chains of mobile water molecules. Here we discuss two possible mechanisms of proton transport along such structures: diffusion of a localized charge and delocalized soliton transitions, in which several protons are collectively shifted along a chain of hydrogen bonds.

The role of pacemaker currents in neuropathic pain

Hyperpolarization-activated cation nonselective cyclic nucleotide-gated (HCN) channels mediate pacemaker currents that control basic rhythmic processes including heartbeat. Alterations in HCN channel expression or function have been described in both epilepsy and cardiac arrhythmias. Recent evidence suggests that pacemaker currents may also play an important role in ectopic neuronal activity that manifests as neuropathic pain. Pacemaker currents are subject to endogenous regulation by cyclic nucleotides, pH and perhaps phosphorylation. In addition, a number of neuromodulators with known roles in pain affect current density and kinetics. The pharmacology of a number of drugs that are commonly used to treat neuropathic pain includes effects on pacemaker currents. Altered pacemaker currents in injured tissues may be an important mechanism underlying neuropathic pain, and drugs that modulate these currents may offer new therapeutic options.

Gramicidin A Channel in a Matrix from a Semifluorinated Surfactant Monolayer

Gramicidin A, a polypeptide antibiotic forming transmembrane ion channels, has been incorporated into a Langmuir monolayer formed by a semifluorinated alkane (SFA). In this work, partially fluorinated tetracosane, perfluorohexyloctadecane (F6H18), has been applied, aiming at finding a suitable matrix for gramicidin A to be transferred onto solid support for a biosensor design. For this purpose, the physiological conditions were of special interest (mixed monolayers containing low gramicidin proportion and the surface pressure of 30 mN/m). Mixed monolayers of gramicidin and SFA were found to be miscible within the whole range of mole fractions. A very significant increase of the stability of SFA monolayer has been found in the presence of gramicidin, even at such a low proportion as X(gramicidin) = 0.1, which is reflected in a 3.5-fold increase of the collapse pressure value of mixed monolayer as compared to the film from pure SFA. This interesting phenomenon has been interpreted as being due to the existence of a strong dipole-dipole interaction between both film-forming molecules. Opposite sign of the measured electric surface potential for gramicidin and SFA, resulting from different directions of the dipole moment vectors in both film molecules, implies that the ordered, antiparallel orientation of the dipole moments in the mixed gramicidin/SFA system can be responsible for its extremely high stability.

Framework models of ion permeation through membrane channels and the generalized King-Altman method

A modern approach to studying the detailed dynamics of biomolecules is to simulate them on computers. Framework models have been developed to incorporate information from these simulations in order to calculate properties of the biomolecules on much longer time scales than can be achieved by the simulations. They also provide a simple way to think about the simulated dynamics. This article develops a method for the solution of framework models, which generalizes the King-Altman method of enzyme kinetics. The generalized method is used to construct solutions of two framework models which have been introduced previously, the single-particle and Grotthuss (proton conduction) models. The solution of the Grotthuss model is greatly simplified in comparison with direct integration. In addition, a new framework model is introduced, generalizing the shaking stack model of ion conduction through the potassium channel.

Water alignment, dipolar interactions, and multiple proton occupancy during water-wire proton transport

A discrete multistate kinetic model for water-wire proton transport is constructed and analyzed using Monte Carlo simulations. In the model, each water molecule can be in one of three states: oxygen lone-pairs pointing leftward, pointing rightward, or protonated (H(3)O(+)). Specific rules for transitions among these states are defined as protons hop across successive water oxygens. Our model also includes water-channel interactions that preferentially align the water dipoles, nearest-neighbor dipolar coupling interactions, and Coulombic repulsion. Extensive Monte Carlo simulations were performed and the observed qualitative physical behaviors discussed. We find the parameters that allow the model to exhibit superlinear and sublinear current-voltage relationships, and show why alignment fields, whether generated by interactions with the pore interior or by membrane potentials, always decrease the proton current. The simulations also reveal a "lubrication" mechanism that suppresses water dipole interactions when the channel is multiply occupied by protons. This effect can account for an observed sublinear-to-superlinear transition in the current-voltage relationship.

Structural Determinants of Proton Blockage in Aquaporins

Aquaporins are an important class of membrane channels selective for water and linear polyols but impermeable to ions, including protons. Recent computational studies have revealed that the relay of protons through the water-conduction pathway of aquaporin channels is opposed by a substantial free energy barrier peaking at the signature NPA motifs. Here, free-energy simulations and continuum electrostatic calculations are combined to examine the nature and the magnitude of the contribution of specific structural elements to proton blockage in the bacterial glycerol uptake facilitator, GlpF. Potential of mean-force profiles for both hop and turn steps of structural diffusion in the narrow pore are obtained for artificial variants of the GlpF channel in which coulombic interactions between the pore contents and conserved residues Asn68 and Asn203 at the NPA signature motifs, Arg206 at the selectivity filter, and the peptidic backbone of the two half-helices M3 and M7, which are arranged in head-to-head fashion around the NPA motifs, are turned off selectively. A comparison of these results with electrostatic energy profiles for the translocation of a probe cation throughout the water permeation pathway indicates that the free-energy profile for proton movement inside the narrow pore is dominated by static effects arising from the distribution of charged and polar groups of the channel, whereas dielectric effects contribute primarily to opposing the access of H+ to the pore mouths (desolvation penalty). The single most effective way to abolish the free-energy gradients opposing the movement of H+ around the NPA motif is to turn off the dipole moments of helices M3 and M7. Mutation of either of the two NPA Asn residues to Asp compensates for charge-dipole and dipole-dipole effects opposing the hop and turn steps of structural diffusion, respectively, and dramatically reduces the free energy barrier of proton translocation, suggesting that these single mutants could leak protons.

A framework model based on the Smoluchowski equation in two reaction coordinates

The general form of the Smoluchowski equation in two reaction coordinates is obtained as the diffusion limit of a random walk on an infinite square grid using transition probabilities that satisfy detailed balance at thermodynamic equilibrium. The diffusion limit is then used to construct a generalization of the single-particle model to two reaction coordinates. The state space includes a square on which diffusion takes place and an isolated empty state. Boundary conditions on opposite sides of the square correspond to transitions between the empty state and the square. The two-dimensional (2D) model can be reduced to a 1D single-particle model by adiabatic elimination. A finite element solution of the 2D boundary value problem is described. The method used to construct the 2D model can be adapted to state spaces that have been constructed by other authors to model K+ conduction through gramicidin, proton conduction through dioxolane-linked gramicidin, and chloride conduction through the bacterial H(+)-Cl- antiporter.

Glycerol conductance and physical asymmetry of the Escherichia coli glycerol facilitator GlpF

The aquaglyceroporin GlpF is a transmembrane channel of Escherichia coli that facilitates the uptake of glycerol by the cell. Its high glycerol uptake rate is crucial for the cell to survive in very low glycerol concentrations. Although GlpF allows both influx and outflux of glycerol, its structure, similar to the structure of maltoporin, exhibits a significant degree of asymmetry. The potential of mean force characterizing glycerol in the channel shows a corresponding asymmetry with an attractive vestibule only at the periplasmic side. In this study, we analyze the potential of mean force, showing that a simplified six-step model captures the kinetics and yields a glycerol conduction rate that agrees well with observation. The vestibule improves the conduction rate by 40% and 75% at 10- micro M and 10-mM periplasmic glycerol concentrations, respectively. In addition, neither the conduction rate nor the conduction probability for a single glycerol (efficiency) depends on the orientation of GlpF. GlpF appears to conduct equally well in both directions under physiological conditions.

Energetics of ion conduction through the gramicidin channel

The free energy governing K(+) conduction through gramicidin A channels is characterized by using over 0.1 micros of all-atom molecular dynamics simulations with explicit solvent and membrane. The results provide encouraging agreement with experiments and insights into the permeation mechanism. The free energy surface of K(+), as a function of both axial and radial coordinates, is calculated. Correcting for simulation artifacts due to periodicity and the lack of hydrocarbon polarizability, the calculated single-channel conductance for K(+) ions is 0.8 pS, closer to experiment than any previous calculation. In addition, the estimated single ion dissociation constants are within the range of experimental determinations. The relatively small free energy barrier to ion translocation arises from a balance of large opposing contributions from protein, single-file water, bulk electrolyte, and membrane. Mean force decomposition reveals a remarkable ability of the single-file water molecules to stabilize K(+) by -40 kcal/mol, roughly half the bulk solvation free energy. The importance of the single-file water confirms the conjecture of Mackay et al. [Mackay, D. H. J., Berens, P. H., Wilson, K. R. & Hagler, A. T. (1984) Biophys. J. 46, 229-248]. Ion association with the channel involves gradual dehydration from approximately six to seven water molecules in the first shell, to just two inside the narrow pore. Ion permeation is influenced by the orientation of the single-file water column, which can present a barrier to conduction and give rise to long-range coupling of ions on either side of the pore. Small changes in the potential function, including contributions from electronic polarization, are likely to be sufficient to obtain quantitative agreement with experiments.

Functional Dynamics of Ion Channels: Modulation of Proton Movement by Conformational Switches

Detailed comparative studies of proton relay in native and chemically modified gramicidin channels provide a unique opportunity to uncover the structural basis of biological proton transport. The function of ion channels hinges on their ability to provide surrogate solvation in narrow pore filters so as to overcome the dielectric barrier presented by biological membranes. In the potassium channel KcsA and in the cation channel gramicidin, permeant selectivity and mobility are determined by the proteinaceous matrix via hydrogen bonding, charge-dipole, and dipole-dipole interactions. In particular, main-chain carbonyl groups in these pore interiors play an essential role in the solvation of alkali ions and of protons. In this study, molecular dynamics simulations reveal how the translocation of H(+) is controlled by nanosecond conformational transitions exchanging distorted states of the peptidic backbone in the single-file region of a dioxolane-linked analogue of the gramicidin dimer. These results underline the functional role of channel dynamics and provide a mechanism for the modulation of proton currents by fluctuating dipoles.

A microscopic view of ion conduction through the K+ channel

Recent results from x-ray crystallography and molecular dynamics free-energy simulations have revealed the existence of a number of specific cation-binding sites disposed along the narrow pore of the K+ channel from Streptomyces lividans (KcsA), suggesting that K+ ions might literally "hop" in single file from one binding site to the next as permeation proceeds. In support of this view, it was found that the ion configurations correspond to energy wells of similar depth and that ion translocation is opposed only by small energy barriers. Although such features of the multiion potential energy surface are certainly essential for achieving a high throughput rate, diffusional and dissipative dynamical factors must also be taken into consideration to understand how rapid conduction of K+ is possible. To elucidate the mechanism of ion conduction, we established a framework theory enabling the direct simulation of nonequilibrium fluxes by extending the results of molecular dynamics over macroscopically long times. In good accord with experimental measurements, the simulated maximum conductance of the channel at saturating concentration is on the order of 550 and 360 pS for outward and inward ions flux, respectively, with a unidirectional flux-ratio exponent of 3. Analysis of the ion-conduction process reveals a lack of equivalence between the cation-binding sites in the selectivity filter.

Membrane peptides and their role in protobiological evolution

How simple membrane peptides performed such essential protocellular functions as transport of ions and organic matter across membranes separating the interior of the cell from the environment, capture and utilization of energy, and transduction of environmental signals, is a key question in protobiological evolution. On the basis of detailed, molecular-level computer simulations we explain how these peptides fold at water-membrane interfaces, insert into membranes, self-assemble into higher-order structures and acquire functions. We have investigated the interfacial behavior and folding of several peptides built of leucine and glutamine residues and have demonstrated that many of them tend to adopt ordered structures. Further, we have studied the insertion of an alpha-helical peptide containing leucine (L) and serine (S) of the form (LSLLLSL)3 into a model membrane. The transmembrane state is metastable, and approximately 15 kcal mol(-1) is required to insert the peptide into the membrane. Investigations of dimers formed by (LSLLLSL)3 and glycophorin A demonstrate how the favorable free energy of helix association can offset the unfavorable free energy of insertion, leading to self-assembly of peptide helices in the membrane. An example of a self-assembled structure is the tetrameric transmembrane pore of the influenza virus M2 protein, which is an efficient and selective voltage-gated proton channel. Our simulations explain the gating mechanism and provide guidelines how to re-engineer the channel to act as a simple proton pump. In general, emergence of integral membrane proteins appears to be quite feasible and may be easier to envision than the emergence of water-soluble proteins.

Structural proton diffusion along lipid bilayers

For H(+) transport between protein pumps, lateral diffusion along membrane surfaces represents the most efficient pathway. Along lipid bilayers, we measured a diffusion coefficient of 5.8 x 10(-5) cm(2) s(-1). It is too large to be accounted for by vehicle diffusion, considering proton transport by acid carriers. Such a speed of migration is accomplished only by the Grotthuss mechanism involving the chemical exchange of hydrogen nuclei between hydrogen-bonded water molecules on the membrane surface, and the subsequent reorganization of the hydrogen-bonded network. Reconstitution of H(+)-binding sites on the membrane surface decreased the velocity of H(+) diffusion. In the absence of immobile buffers, structural (Grotthuss) diffusion occurred over a distance of 100 micro m as shown by microelectrode aided measurements of the spatial proton distribution in the immediate membrane vicinity and spatially resolved fluorescence measurements of interfacial pH. The efficiency of the anomalously fast lateral diffusion decreased gradually with an increase in mobile buffer concentration suggesting that structural diffusion is physiologically important for distances of approximately 10 nm.

Noncontact dipole effects on channel permeation. VI. 5F- and 6F-Trp gramicidin channel currents

Fluorination of peptide side chains has been shown to perturb gramicidin channel conductance without significantly changing the average side chain structure, which, it is hoped, will allow detailed analysis of electrostatic modulation of current flow. Here we report a 1312-point potassium current-voltage-concentration data set for homodimeric channels formed from gramicidin A (gA) or any of eight fluorinated Trp analogs in both lecithin and monoglyceride bilayers. We fit the data with a three-barrier, two-site, two-ion (3B2S) kinetic model. The fluorination-induced changes in the rate constants were constrained by the same factor in both lipids. The rate constant changes were converted to transition-state free-energy differences for comparison with previous electrostatic potential energy differences based on an ab initio force field. The model allowed a reasonably good fit (chi = 8.29 with 1271 degrees of freedom). The measured changes were subtle. Nevertheless, the fitted energy perturbations agree well with electrostatic predictions for five of the eight peptides. For the other three analogs, the fitted changes suggested a reduced translocation barrier rather than the reduced exit barrier as predicted by electrostatics.

Molecular dynamics simulation of proton transport through the influenza A virus M2 channel

The structural and dynamical properties of a solvated proton in the influenza A virus M2 channel are studied using a molecular dynamics (MD) simulation technique. The second-generation multi-state empirical valence bond (MS-EVB2) model was used to describe the interaction between the excess proton and the channel environment. Solvation structures of the excess proton and its mobility characteristics along the channel were determined. It was found that the excess proton is capable of crossing the channel gate formed by the ring of four histidine residues even though the gate was only partially open. Although the hydronium ion itself did not cross the channel gate by traditional diffusion, the excess proton was able to transport through the ring of histidine residues by hopping between two water molecules located at the opposite sides of the gate. Our data also indicate that the proton diffusion through the channel may be correlated with the changes in channel conformations. To validate this observation, a separate simulation of the proton in a "frozen" channel has been conducted, which showed that the proton mobility becomes inhibited.

The role of Trp side chains in tuning single proton conduction through gramicidin channels

We present an extensive set of measurements of proton conduction through gramicidin A (gA), B (gB), and M (gM) homodimer channels which have 4, 3, or 0 Trp residues at each end of the channel, respectively. In gA we find a shoulder separating two domains of conductance increasing with concentration, confirming the results of Eisenman, G., B. Enos, J. Hagglund, and J. Sandblom. 1980. Ann. NY. Acad. Sci. 339:8-20. In gB, the shoulder is shifted by approximately 1/2 pH unit to higher H(+) concentrations and is very sharply defined. No shoulder appears in the gM data, but an associated transition from sublinear to superlinear I-V values occurs at a 100-fold higher [H(+)] in gM than in gA. The data in the low concentration domain are analyzed using a configuration space model of single-proton conduction, assuming that the difference in the proton potential of mean force (PMF) between gA and its analogs is constant, similar to the results of Anderson, D., R. B. Shirts, T. A. Cross, and D. D. Busath. 2001. Biophys. J. 81:1255-1264. Our results suggest that the average amplitudes of the calculated proton PMFs are nearly correct, but that the water reorientation barrier calculated for gA by molecular dynamics using the PM6 water model (Pomès, R., and B. Roux. 1997. Biophys. J. 72:246a) must be reduced in amplitude by 1.5 kcal/mol or more, and is not rate-limiting for gA.

Computational studies of the gramicidin channel

Ion channels are highly specific membrane-spanning protein structures which serve to facilitate the passage of selected ions across the lipid barrier. In the past decade, molecular dynamics simulations based on atomic models and realistic microscopic interactions with explicit solvent and membrane lipids have been used to gain insight into the function of these complex systems. These calculations have considerably expanded our view of ion permeation at the microscopic level. This Account will mainly focus on computational studies of the gramicidin A channel, one of the simplest and best characterized molecular pore.

Molecular mechanism of H+ conduction in the single-file water chain of the gramicidin channel

The conduction of protons in the hydrogen-bonded chain of water molecules (or "proton wire") embedded in the lumen of gramicidin A is studied with molecular dynamics free energy simulations. The process may be described as a "hop-and-turn" or Grotthuss mechanism involving the chemical exchange (hop) of hydrogen nuclei between hydrogen-bonded water molecules arranged in single file in the lumen of the pore, and the subsequent reorganization (turn) of the hydrogen-bonded network. Accordingly, the conduction cycle is modeled by two complementary steps corresponding respectively to the translocation 1) of an ionic defect (H+) and 2) of a bonding defect along the hydrogen-bonded chain of water molecules in the pore interior. The molecular mechanism and the potential of mean force are analyzed for each of these two translocation steps. It is found that the mobility of protons in gramicidin A is essentially determined by the fine structure and the dynamic fluctuations of the hydrogen-bonded network. The translocation of H+ is mediated by spontaneous (thermal) fluctuations in the relative positions of oxygen atoms in the wire. In this diffusive mechanism, a shallow free-energy well slightly favors the presence of the excess proton near the middle of the channel. In the absence of H+, the water chain adopts either one of two polarized configurations, each of which corresponds to an oriented donor-acceptor hydrogen-bond pattern along the channel axis. Interconversion between these two conformations is an activated process that occurs through the sequential and directional reorientation of water molecules of the wire. The effect of hydrogen-bonding interactions between channel and water on proton translocation is analyzed from a comparison to the results obtained previously in a study of model nonpolar channels, in which such interactions were missing. Hydrogen-bond donation from water to the backbone carbonyl oxygen atoms lining the pore interior has a dual effect: it provides a coordination of water molecules well suited both to proton hydration and to high proton mobility, and it facilitates the slower reorientation or turn step of the Grotthuss mechanism by stabilizing intermediate configurations of the hydrogen-bonded network in which water molecules are in the process of flipping between their two preferred, polarized states. This mechanism offers a detailed molecular model for the rapid transport of protons in channels, in energy-transducing membrane proteins, and in enzymes.

Molecular dynamics simulation of proton transport near the surface of a phospholipid membrane

The structural and dynamical properties of a hydrated proton near the surface of DMPC membrane were studied using a molecular dynamics simulation. The proton transport between water molecules was modeled using the second generation multistate empirical valence bond model. The proton diffusion was found to be inhibited at the membrane surface. The potential of mean force for the proton adsorption to the membrane surface and its release back into the bulk water was also determined, yielding a small barrier in each direction. An efficient algorithm for Ewald summation calculations for the multistate empirical valence bond model is also introduced.

Membrane dipole potential modulates proton conductance through gramicidin channel: movement of negative ionic defects inside the channel

The effect of membrane dipole potential on gramicidin channel activity in bilayer lipid membranes (BLMs) was studied. Remarkably, it appeared that proton conductance of gramicidin A (gA) channels responded to modulation of the dipole potential oppositely as compared with gA alkali metal cation conductance. In particular, the addition of phloretin, known to reduce the membrane dipole potential, resulted in a decrease in gA proton conductance, on one hand, and an increase in gA alkali metal conductance, on the other hand, whereas 6-ketocholestanol, the agent raising the membrane dipole potential, provoked an increase in gA proton conductance as opposed to a decrease in the alkali metal cation conductance. The peculiarity of the 6-ketocholestanol effect consisted in its dependence on the H(+) concentration. The experiments with the impermeant dipolar compound, phloridzin, showed that the response of proton transport through gramicidin channels to varying the membrane dipole potential did not change qualitatively if the dipole potential of only one monolayer or both monolayers of the BLM was altered. In contrast to gA proton conductance, the single-channel lifetime changed similarly with varying the membrane dipole potential, regardless of the kind of permeant cations (protons or potassium ions). The results of this study could be tentatively accounted for by an assumption that one of the rate-limiting steps of proton conduction through gramicidin channels represents, in fact, movement of negatively charged species (negative ionic defects) across a membrane.

Gramicidin derivatives as membrane-based pH sensors

Ion channels provide a means for sensitive pH measurement at membrane interfaces. Detailed knowledge of the structure and function of gramicidin channels permits the engineering of pH-sensitive derivatives. Two derivatives, gramicidin-ethylenediamine and gramicidin-histamine, are shown to exhibit pH-dependent single-channel behaviour over the pH ranges 9-11 and 6.5-8.5, respectively. Thermal isomerization of a carbamate group at the entrance of the channels leads to a pattern of steps in single-channel recordings. The size of the steps depends on the time-averaged degree of protonation of the appended group (ethylenediamine or histamine). Measurement of the size of the steps thus permits single-molecule pH sensing under symmetrical pH conditions or in the presence of a pH gradient.