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Department of Biomathematics and the Institute for Pure and Applied Mathematics, Los Angeles, California
Correspondence: Address reprint requests to Tom Chou, Dept. of Biomathematics and IPAM, Los Angeles, CA 90095-1766. Tel.: 310-206-2787; E-mail: tomchou{at}ucla.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMODEL AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX: NONINTERACTING MEAN...ACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMODEL AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX: NONINTERACTING MEAN...ACKNOWLEDGEMENTSREFERENCES |
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; Grabe and Oster, 2001). Proton transport in confined geometries is also relevant for ATP synthesis (Boyer, 1997) and light transduction by bacteriorhodopsin (Lanyi, 1995). In this article, we develop a lattice model for describing proton transport in one-dimensional environments. This study is motivated by numerous measurements of proton conduction across lipid membrane channels (Akeson and Deamer, 1991; Busath et al., 1998; Cotten et al., 1999; Cukierman et al., 1997; Deamer, 1987; Eisenman et al., 1980). Experiments are typically performed using membrane-spanning gramicidin channels that are only a few Ångstroms in diameter. This geometric constraint imposes a single-file structure on the interior water molecules (Hille and Schwarz, 1978; Hladky and Haydon, 1972).
Under equal electrochemical potential gradients, conduction of protons across ion channels occurs at a rate typically an order-of-magnitude higher than that of other small ions. This supports a "water-wire" mechanism (Akeson and Deamer, 1991; Nagle and Morowitz, 1978; Nagle and Tristram-Nagle, 1983; Nagle, 1987), first proposed by Grotthuss (Agmon, 1995; Grotthuss, 1806). Across a water-wire, protons are shuttled across lone-pairs of water oxygens as they successively protonate the waters along the single-file chain. Since the hydrogens are indistinguishable, any one of the hydrogens in a water cluster (e.g., any of the three hydrogens on a hydronium) can hop forward along the chain to protonate the next water molecule or cluster of water molecules (compare to Fig. 1). This mechanism naturally allows much faster overall conduction of protons compared to other small ions which have to wait for the entire chain of water molecules ahead of it to fluctuate across the pore to traverse the channel.
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were carried out in symmetric solutions in the 1–3 pH range, and the results were recently reproduced by Busath et al. (1998) and Rokitskaya et al. (2002). These experiments were performed using simple, relatively featureless gramicidin A channels. One leading hypothesis is that the nonlinear proton current-voltage relationships arise from the intrinsic proton dynamics within such simple channels. Specifically, multiple proton occupancy and repulsion among protons within the channel may give rise to the observed nonlinearity (Hille and Schwarz, 1978; Phillips et al., 1999; Schumaker et al., 2001).
There have been a number of recent theoretical studies of water-wire proton conduction. Extensive simulations on the quantum dynamics of proton exchange across small water clusters in vacuum have been used to predict microscopic hopping rates between water clusters (Bala et al., 1994; Sadeghi and Cheng, 1999; Marx et al., 1999; Mavri and Berendsen, 1995; Mei et al., 1998; Sagnella et al., 1996; Schmitt and Voth, 1999). Pomès and Roux (1996) have performed classical molecular dynamics (MD) simulations on water-channel interactions, proton hopping, and water reorientation. They derive effective potentials of mean force describing the energy barriers encountered by a single proton within the pore. Since MD simulations are presently limited to only processes that occur over a few nanoseconds, none of these computational methods are efficient at probing very long-time, steady-state transport behavior. On a more macroscopic, phenomenological level, Sagnella and Voth (1996) and Schumaker et al. (2000, 2001) have considered the long-time behavior of a single proton and dipole defect diffusing in a single-file channel. The parameters used in these studies, including effective energy profiles and kinetic rates, were derived from MD simulations. Although the basic underlying structure assumed by most of these transport models qualitatively resembles the Grotthuss mechanism, they have not addressed multiple proton occupancy. One exception are fully dynamical models that treat proton transfer in ordered water structures in the context of soliton dynamics (Bazeia et al., 2001; Pang and Feng, 2003; Pnevmatikos, 1988).
In this article, we will explore the intrinsically nonlinear proton dynamics along a single-file water-wire. We formulate a stochastic lattice model that defines the discrete structural states of the water-wire to approximate the continuous molecular orientations. Although the lattice model provides a different approach from MD simulations, it is more amenable to analysis at longer timescales, yet is connected to the microphysics inherent in MD simulations provided a consistent correspondence between the parameters is made. Rather than enumerating all possible molecular configurations, our lattice approach resembles that developed for molecular motors (Fisher and Kolomeisky, 1999), mRNA translation (MacDonald and Gibbs, 1969; Chou, 2003), traffic flow (Karimipour, 1999; Schreckenberg et al., 1995), and ion and water transport in single-file channels (Chou, 1998, 1999; Chou and Lohse, 1999). Here, the proton occupancy along the water-wire will be self-consistently determined by the prescribed lattice dynamics. The parameters used in our model are transition rates among discrete states that, in principle, can be independently computed from relatively short-time MD simulations (Dècornez et al., 1999). The approximations inherent in our discrete model qualitatively take into account the effects of proton-proton repulsion and water-water dipole interactions.
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MODEL AND METHODS |
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TOPABSTRACTINTRODUCTIONMODEL AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX: NONINTERACTING MEAN...ACKNOWLEDGEMENTSREFERENCES |
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3–4 Å and can only accommodate water in a single-file chain. Although the number of water molecules in this chain is a fluctuating quantity, their dynamics in and out of the channel will be assumed to be much slower than that of their orientational rearrangements and proton hopping (Hummer et al., 2001; Kalra et al., 2003). We thus treat the water-wire as containing a fixed, average number of water molecules. There are N 
8–26 single-file waters within a typical transmembrane channel (Levitt et al., 1978; Wu and Voth, 2003).
Fig. 1 A shows a schematic of our model. We first assume that each site along the pore is occupied by a single oxygen atom which may either be part of neutral water (H2O), or a hydronium (H3O+) ion. Although protonated oxygens in bulk are often associated with larger complexes such as 
(Zundel cation), or 
(Eigen cation), in confined geometries, the formation of the larger complexes is suppressed (Lynden-Bell and Rasaiah, 1996). Furthermore, the model depicted in Fig. 1 A can incorporate the dynamics of reactive proton transfer among transient clusters by an appropriate redefinition of a lattice site to contain the entire cluster.
Neutral waters have permanent dipole moments and electron lone-pair orientations that can rotate thermally. For simplicity, we bin all water dipoles (hydrogens) that point toward the right as "+" particles, whereas those pointing more or less to the left are denoted "–" particles. The singly-protonated species H3O+ is hybridized to a nearly planar molecule. Therefore, we will assume that hydronium ions are symmetric with respect to transferring a proton forward or backward, provided the adjacent waters are in the proper orientation and there are no external driving forces (electric fields). Each lattice site can exist in only one of three states: 0, +, or –, corresponding to protonated, right, or left states, respectively. Labeling the occupancy configurations 
i = {–1, 0, +1}, allows for fast integer computation in simulations.
In addition to proton exclusion, the transition rules are constrained by the orientation of the waters at each site and are defined in Fig. 1 B. A proton can enter the first site (i = 1) from the left reservoir and protonate the first water molecule with rate 
only if the hydrogens of the first water are pointing to the right (such that its lone-pair electrons are left-pointing, ready to accept a proton). Conversely, if a proton exits from the first site back into the left reservoir (with rate 
), it leaves the remaining hydrogens right-pointing. In the pore interiors, a proton at site i can hop to the right(left) with rate p+(p–) only if the adjacent particle is a right- (left-) pointing, unprotonated water molecule. If such a transition is made, the water molecule left at site i will be left- (right-) pointing. Physically, as a proton moves to the right, it leaves a wake of – particles to its left. A left-moving proton leaves a trail of + particles to its right. These trails of – or + particles are unable to accept another proton from the same direction. Protons can follow each other successively only if water molecules can reorient such that these trails of +'s or –apos;s are thermally washed out. Water reorientation rates are denoted k± (compare to Fig. 1 B and Fig. 2). Protons at the rightmost end of the water-wire (at site i = N), exit with rate ß, which is different from p+ inasmuch as the local microenvironment (e.g., typical distance to acceptor electrons) of the bulk waters that accept this last proton is different from that in the pore interior. From the right reservoir, protons can hop back into the water-wire with rate 
if a water in the "–" configuration is at site i = N. The entrance rates and are functions of at least the proton concentration in the respective reservoirs. Fig. 2 shows a representative time series of the evolution of a specific configuration. The rate-limiting steps in steady-state proton transfer across biological water channels are thought to be associated with water flipping (Pomès and Roux, 1998).
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), we defined each pair of waters as occupying a single lattice site, k±, as an effective reorientation time for the following pair of waters, and p± as the hopping rate to an adjacent oxygen lone-pair. The Grotthuss water-wire mechanism is qualitatively preserved as long as the proper identification with the microphysics is made.
All eight parameters used in our model (the rates p±, k±, , ß, , and ), can be related to measured bulk quantities or derived from short-time MD simulations. They are a minimal set and are equivalent to the numerous bulk parameters used in other models (Schumaker et al., 2001), such as the bulk proton diffusion constant, water orientational diffusion constants, etc. Using similar MD approaches then, one should be able to approximately fix the parameters used in our model. For example, variations in the potential of mean force along the pore (resulting from interactions of the different species with the constituents of the pore interior) are embodied by site-dependent transition rates p± and k±. Thus, MD-derived potentials of mean force used in previous models can also be implemented within our lattice framework. Such effects of local inhomogeneities in the hopping rates have been studied analytically and with MC simulations in related models (Kolomeisky, 1998).
The basic model described above has been studied analytically in certain limits where exact asymptotic results for the steady-state proton current J were derived (Chou, 2002). However, this study did not explicitly include any interactions other than proton exclusion and proton transfer onto properly aligned water dipoles. Effects arising from forces such as repulsion between protons in close proximity, interactions between water dipoles and external electric fields, and dipolar coupling between neighboring waters need to be considered.
In Fig. 3 A, a proton moves down the electric potential reducing the total enthalpy by V, and a right-pointing dipole is converted into a left-pointing dipole at an energy cost of H. Since both initial and final states have adjacent, repelling protons, the repulsion energy R does not enter in the overall energy change. In Fig. 3 B, a proton moves down the potential (–V), a "+" water is converted to a "–" (+H), a dipole domain wall is removed (–K), and the repulsive energy between adjacent charged protons is relieved (–R). The representation of these nearest-neighbor effects can be succinctly written in terms of the energy of a specific configuration,
 | (1) |
i}] are all in units of kBT, and represent
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iN (Edwards et al., 2002; Jordan, 1984; Partenskii and Jordan, 1992; Syganow and von Kitzing, 1999). As a charge moves from the dielectric 
80 water phase through the 2 lipid bilayer, the polarization energy varies. This smooth (on length scales over a typical water-water separation, or lattice site) energy variation ultimately gives rise to a smooth variation in the internal hopping rates, p±. In this study, we neglect this variation and assume constant V and p± across the entire lattice.
More complicated interactions, such as non nearest-neighbor repulsion and interactions between protons and water dipole orientation changes have also been considered (Dellago et al., 2003). Longer-ranged electrostatic repulsion can be easily incorporated by assigning an energy for, say, next-nearest-neighbor protons. We neglect these more complicated contributions to the free energy of the system and focus on the qualitative effects of Coulomb repulsion by only considering nearest-neighbor interactions.
To connect the quantities H, K, R, and V to the rates , ß, , , p±, and k±, we will assume the transitions occur over thermal barriers. Although barriers to proton hopping may be small, we employ the Arrhenius forms to obtain a simple relationship so that qualitative aspects of the effects of H, K, R, and V can be illustrated. Activation energy-based treatments for conduction across gramicidin channels have been previously studied (Chernyshev and Cukierman, 2002). When the more complicated interactions and external potentials are turned on, the effective transition rates 
{, ß, , , p±, k±} on which we base our Metropolis Monte Carlo become
 | (2) |
0 {0, ß0, 0, 0, p0, k0} are rate prefactors when H, K, R, V, and 
E are zero. In defining Eq. 2, we have assumed that the energy barrier due to the difference E = E[{'i}] – E[{i}] (where {'i} and {i} are the final and initial state configurations, respectively) is evenly split between the barrier energies in the forward and backward directions. We use the convention that p+ = p– = p0 and k+ = k– = k0 when V = 0 and H = 0, respectively. The constraints and the state-dependent transition rates determined by Eqs. 1 and 2 completely define a nonequilibrium dynamical model which we study using MC simulations. Note that in the original model (Fig. 2) we do not assume transition barriers, but rather only that the dynamics are Markovian.
We first gain insight into the dynamics by considering numerical solutions to the full master equation for a short three-site (N = 3) channel. If we explicitly enumerate all 27 = 33 states of the three site model, the master equation for the 27-component-state vector 
is
 | (3) |
. In steady state, the Pi are solved by inverting M with the constraint 
The steady-state currents are found from the appropriate elements in Pi times the proper rate constants in the model. For example, if the probability that the three-site chain is in the configuration (+ –0) is denoted P12, then the transition rate to state P13 (+ – –) (corresponding to the ejection of a proton from the last site into the right reservoir) is ßeV–H–K and the steady-state current J = ß
'iPi (where the sum 'i runs over all configurations that contain a proton at the last site), will contain the term ßeV–H–KP12.
Monte Carlo (MC) simulations were implemented for relatively small (N = 10) systems by randomly choosing a site, and making an allowed transition with the probability exp(Ei – Ef)/rmax, where rmax is the maximum possible transition rate of the entire system. In the next time step, a particle is again chosen at random and its possible moves are evaluated. The currents were computed after the system reached steady state by counting the net transfer of protons across all interfaces (which separate adjacent sites and the reservoirs) and dividing by N + 1. Physical values of J are recovered by multiplying by rmax. Particle occupation statistics within the chain were tracked by using the definitions of +, 0, and – particle densities at each site i: 
and 
respectively. However, for our subsequent discussion, it will suffice to analyze simply the chain-averaged proton concentration 
All MC results were checked and compared with the exact numerical results from the three-site, 27-state master equation.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMODEL AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX: NONINTERACTING MEAN...ACKNOWLEDGEMENTSREFERENCES |
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; Mavri and Berendsen, 1995; Mei et al., 1998; Schmitt and Voth, 1999). Moreover, a direct simulation of proton movement in carbon nanotube water-wires yields a proton diffusion constant of 0.1 nm2/s (Dellago et al., 2003). With a typical interwater spacing of 0.242 nm, this diffusion constant corresponds to a hopping rate of p0 4 ps–1. The resulting steady-state proton currents under realistic driving forces are on the order of 10 ns–1, consistent with that observed in gramicidin channels.
One of the main features we wish to explore is the effect of multiple proton occupancy on current-voltage relationships. To understand what values of transition rates would permit multiple proton occupancy, consider water at pH = 7, which has 10–7 M protons and hydroxyls. This concentration corresponds to 60 H3O+ and 60 OH– species per cubic micron. Even at pH 4, one would only have 60,000 hydroniums per µm3, corresponding to a typical distance between hydroniums of 25 nm. Since there are only 10–20 waters within a single-file channel, and at pH 4, only 1 in 500,000 waters are protonated in bulk, multiple protons in a single channel can occur only if protonated species within the channel are highly stabilized by interactions with the chemical subgroups comprising the pore interior. This stabilizing effect is modeled by small escape rates ß0, 0, and assumed to be distributed equally such that p0 remains constant across all sites within the lattice. Although from a concentration point of view, small entrance rates 0, 0 arise from infrequent protons that wander into the first site of the channel, their exit rates ß0, 0 can be suppressed even more by their stabilization once inside the channel. Multiple ion occupancy has also been observed in related pore systems such as the potassium channel containing three sites for K+ ions (Morais-Cabral et al., 2001; Bernèche and Roux, 2001). Despite low bulk ion concentrations, the channel interior stabilizes the ions such that exit rates 0, ß0 are small enough for appreciable simultaneous multiion occupancy. In all of our simulations, we will assume proton stabilization is moderately strong and limit ourselves to the rates ß0, 0 < 0, 0. The values we use give steady-state proton occupancies across the whole range of values from 
1 to N.
First consider symmetric solutions and featureless, uniform pores where 0 = 0, ß0 = 0. The only possible driving force is an external voltage V. In Fig. 4, we plot the current-voltage relationship for various flipping rates k0. We initially ignore interaction effects and set H = K = R = 0. Currents for sufficiently small V are always nearly linear. However, for sufficiently large V, the rate-limiting step eventually becomes the water-flipping rate k0. Further increases in V do not increase the overall steady-state current, and the current-voltage curve becomes sublinear before saturating. The crossover to sublinear (water-flipping rate-limited) behavior depends on the value of k0, with sublinear onset occurring at higher voltages V for larger k0. In the noninteracting case, for most reasonable values of rate constants, any possible superlinear regime does not arise as it is washed out by the sublinear, water-flip rate-limited saturation. The only instance found where noticeable superlinear behavior in the steady-state proton current arises is in the limit of large k0 and when 
Superlinear relationships can occur via other mechanisms not inherent in our model. For example, the transmembrane potential may compress the bilayer and mechanically increasing the effective diameter of the channel, and increasing the mean number of waters in the pore. A small decrease in the interwater spacing could dramatically increase the internal hopping rate p0, leading to a superlinear J–V relationship.
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Fig. 5 displays the effects of a fixed, external, dipole-orienting field H 
0. All other interactions and fields, except the external driving voltage V, are turned off. The convention used in the energy Eq. 1 favors a "+" state for H > 0. This asymmetry leads to an asymmetry in the J–V relationship (Fig. 5 A). After an initial proton has traversed the channel, flipping of the "–" waters left in its wake is suppressed for H > 0, thereby preventing further net proton movement. The persistent blockade induced by increasing H is evident in Fig. 5 B where the proton density decreases for increasing H.
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The effects of proton-proton repulsion (R > 0) are considered in Figs. 7 and 8. These simulations are consistent with the hypothesis that proton-proton repulsions can give rise to superlinear current (Hille and Schwarz, 1978). Fig. 7 A shows a slight preference for superlinear behavior as repulsion R is increased. Not surprisingly, Fig. 7 B shows that the overall density of protons within the pore decreases with increasing repulsion.
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0 = 0 is increased. Measurements, however, also show rather modest superlinear behavior (Eisenman et al., 1980; Phillips et al., 1999; Rokitskaya et al., 2002). The occupancy also increases with decreasing pH, enhancing the effect of proton-proton repulsion. These behaviors are consistent with experimental findings (Eisenman et al., 1980) and those in the simulations depicted in Fig. 7, where increased repulsion exhibited superlinear J–V curves.
Finally, we consider the effects of dipole coupling K 0 between adjacent water molecules. This interaction is analogous to a nearest-neighbor ferromagnetic coupling in e.g., Ising models. Fig. 9 A shows that for sufficiently large 0 = 0, a superlinear behavior arises (for small enough V and large enough k0 such that saturation has not yet occurred). Notice that as 0 = 0 is increased, the J–V relationship can become more sublinear before turning superlinear. Here we have used a higher value of k0 to suppress sublinear behavior to larger V, but the qualitative shift from sublinear to slightly superlinear behavior exists for small k0. Moreover, recent comparisons between gramicidin A and gramicidin M channels suggest that water reorientation is not rate-limiting (Gowen et al., 2002). The nature of the superlinear behavior can be deduced from Fig. 9 B, where the mean proton density is shown to increase with 0 = 0. Waters that neighbor a proton are relieved of their dipolar coupling and can more readily flip to a configuration that would allow acceptance of another proton. For example, the transition ...0 – 0...
...0 + 0... will occur faster than ...– –0......–+0..... This lubrication effect arises only when the proton density is high and K 0.
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SUMMARY AND CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMODEL AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX: NONINTERACTING MEAN...ACKNOWLEDGEMENTSREFERENCES |
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, 2001) to include multiple proton occupancy and the memory effects of protons that have recently traversed the water-wire. Monte Carlo simulations of the lattice model were performed to test conjectures on a number of observed qualitative features in proton transport across water-wires. Four interaction energies that modify the kinetic rates are considered: A dipole-orienting field, which tends to align the water molecules; a ferromagnetic dipole-dipole interaction between neighboring water molecules; a penalty from the repulsion between neighboring protons; and an external electric field (transmembrane potential) that biases the hops of the charged protons. We find current-voltage relationships that can be both superlinear and sublinear depending on the voltage V. For large enough voltages, the proton-hopping step is no longer rate-limiting. Water-flipping rates limit proton transfer and further increases in V do little to increase the steady-state proton current J. This observation suggests that the observed transition from sublinear to superlinear behavior can be effected by varying an effective water-flipping rate, although we find that, indeed, proton-proton repulsion can lead to slightly superlinear J–V characteristics—particularly for large repulsions and proton injection rates (low pH).
Dipole-dipole interactions between neighboring waters are also incorporated. Previous single-proton theories (Schumaker et al., 2000, 2001) have considered the propagation of a single defect back and forth in the pore. In our model, the number of protons and defects are dynamical variables that depend on the injection rates and the dipole-dipole coupling, respectively. For large coupling K, we expect very few defects, and effective water-flipping rates will be low. However, when injection rates and proton occupancy in the pore is high, some dipole-dipole couplings are broken up by the intervening protons. Thus, protons can "lubricate" their neighboring dipoles, allowing them to flip faster than if they were neighboring a dipole pointed in the same direction. Using simulations, we showed that this lubrication effect can give rise to a superlinear J–V relationship.
The parameters used in our analyses can be estimated from shorter time MD simulations, or other continuum approximations (Dècornez et al., 1999); Edwards et al., 2002; Partenskii and Jordan, 1992). More complicated local interactions with membrane lipid dipoles (Rokitskaya et al., 2002) and internal pore constituents (such as Trp side groups; Dorigo et al., 1999; Gowen et al., 2002) can be incorporated by allowing H, K, p0, and/or k0 to reflect the local molecular environment by varying along the lattice site (position) within the channel (Kolomeisky, 1998).
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APPENDIX: NONINTERACTING MEAN-FIELD RESULTS |
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TOPABSTRACTINTRODUCTIONMODEL AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX: NONINTERACTING MEAN...ACKNOWLEDGEMENTSREFERENCES |
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).
If V = 0 ( = 0), only pH differences between the two reservoirs can induce a nonzero steady-state proton current. The proton concentration difference is reflected by a difference between the entry rates from the two reservoirs 0 0, and the steady-state current can be expanded in powers of 1/N: 
In the long chain limit, we found (Chou, 2002)
 | (A1) |
0; and Eq. A1 can be further simplified by expanding in powers of k– – k+,
 | (A2) |
and = 0 limit,
 | (A3) |
> , ß > , and p+ > p–, a finite current persists in the N 
limit. We can use mean-field approximations familiar in the totally asymmetric simple exclusion process (TASEP) (Derrida, 1998; Schütz and Domany, 1993) to conjecture that three current regimes exist. If the both proton entry and exit is fast, and the rate-limiting steps involve water flipping, or interior protons hops with rate p+, we expect that a maximal current regime exists and that the densities of the three states along the interior of a long chain are spatially uniform. Mean-field analysis from previous work (Chou, 2002) yields
 | (A4) |
 | (A5) |
limit.
A similar approach is taken when the currents are entry- or exit-limited. From the mean-field approximation of the steady-state equation for 
± near the channel entry,
 | (A6) |
= 0. Upon using normalization – + 0 + + = 1, and the expressions in Eq. A6, we find the mean densities near the left boundary,
 | (A7) |
 | (A8) |
; Chou, 2003), except for the factor k–/( + k– + k+), representing the fraction of time the first site is in the "+" state, and able to accept a proton from the left reservoir.
When the rate ß is rate-limiting, we consider the mean-field equations near the exit of the channel
 | (A9) |
 | (A10) |
 | (A11) |
). Only in this limit, where the memory of a previously passing proton is quickly erased, are the mean-field results quantitatively accurate (Chou, 2002). Nonetheless, the mean-field calculations of the simplified system (H = K = R = 0) yields qualitatively correct results for the steady-state current, provides a connection with well-known results of the TASEP, and gives an explicit qualitative description of the mechanisms at play. |
ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMODEL AND METHODSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSAPPENDIX: NONINTERACTING MEAN...ACKNOWLEDGEMENTSREFERENCES |
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This work was performed with the support of the National Science Foundation through grant DMS-0206733.
Submitted on September 22, 2003; accepted for publication January 28, 2004.
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–) for a nonzero steady-state current to exist.
