Charge transport along proton wires
© Karahka and Kreuzer; licensee Springer. 2013
Received: 20 April 2013
Accepted: 7 June 2013
Published: 20 June 2013
Using density functional theory we look at the quantum mechanics of charge transport along water wires both with free ends and donor/acceptor terminated. With the intermediate geometries in the DFT iterations we can follow the charge transfer mechanism and also construct the energy landscape explicitly. It shows activation barriers when a proton is transferred from one water molecule to the next. This, together with snapshots of intermediate geometries, leads to a justification and further elucidation of the Grotthuss mechanism and the Bjerrum effect. The charge transfer times and the conductivity of the proton wire are obtained in agreement with experimental results.
82.39 Jn, 31.15 E
In 1978 Nagle and Morowitz  suggested that linear water structures in membrane channels may act as carriers for proton transfer through the membrane wall. They also coined the term proton wires for these water whiskers. These proton wires extend through membrane channels and connect the enzyme active site with the liquid phase outside the cell with enzyme functional groups inside the cell. We list a few examples. In the transmembrane channel formed by gramicidin A proton wires are found within its helical structure [2, 3]. Proton wires are also found or suspected in bacteriorhodopsin [4–7], photosynthetic centers , in enzymes  and even in viruses . In most of these systems amino acid residues (His, Glu, Asp, Ser) act as proton donors and acceptors at the respective ends of the water whisker. A comprehensive review, both experimental and theoretical, was given by Cukierman .
As an early model of charge transfer in proton wires Nagle and Morowitz  adapted classical ideas by Grotthuss who, as early as 1806 suggested that in diffusion of protons in bulk water an ‘excess’ proton propagates through the hydrogen-bonded network by repeated cleavage and re-formation of hydrogen bonds . Theoretical efforts since then have focused mainly on mixed classical/quantum methods such as a polarization model with discretized Feynman path integral-molecular dynamics (FPI-MD) as used by Pomès and Roux [13–15], an ab-initioFPI Car-Parrinello MD used by Klein et al., a Born-Oppenheimer local-spin-density MD used by Sadeghi and Cheng , a multiconfigurational MD with quantum transitions used by Hammes-Schiffer et al.[18–20], a MD with a multi-state empirical valence band (EVB) model used by Voth et al., a quantum/molecular mechanical method used by Nemukhin et al., a “real time” MD used by de Groot and Grubmüller , and an EVB model with Langevin dynamics used by Warshel et al.. For the most part they are concerned with the stabilization of the proton wires by the walls of the membrane channel and also on simple kinetic models to describe the proton hopping process. Several groups looked at the effects of weak electrostatic fields [18, 20, 24, 25]. Various groups have also employed fully quantum mechanical approaches such as the multiconfiguration time-dependent Hartree method used by Vendrell and Meyer , and density functional theory [27–30].
Neutral and protonated water whiskers form helical structures.
Protonated water whiskers with more than four water molecules are only stable for fields below 0.3 V/Å.
Two electronic effects of field stabilization have been identified; (i) polarization and field alignment at small fields (< 0.1 V/Å), and (ii) charge transfer and re-hybridization due to field modification of the molecular level structure at high fields (> 0.1 V/Å) that ultimately lead to field fragmentation at the point when the HOMO-LUMO gap closes and the whisker becomes conducting and field expulsion from the region of the whisker occurs. The latter effects can be understood on the basis of field-induced chemistry: applying an electric field implies that in a cluster of atoms an atom further down the field by a distance d from a reference atom changes its energy levels relative to the reference atom by e F d. This changes the character of their joint molecular orbital either weakening or strengthening its bonding character .
In the classical Anway model it is assumed that the oxygen atoms lie in a straight line with the two hydrogen atoms oriented at some angle to facilitate energy gain through the field but maintain some possibility for hydrogen bonding as well. This picture is acceptable at small fields if it is modified by the fact that a lower energy state can be achieved by a staggered arrangement. Such an arrangement is also not planar but forms a helical conformer much akin to polymers. In this sense one might be tempted to call this field-induced polymerization. In addition, a simple classical dipole model does not allow for the transfer of charge down the field. In other words, a classical model does not allow field-dissociation.
The presence of extra protons in the whisker has important consequences in that it destabilizes the whiskers earlier i.e. at lower field strengths, simply because the additional proton moves up the field leading also to a re-arrangement of the electrons.
The most important result of this study is a stability diagram which delineates, for protonated water clusters of different sizes n, the minimum electric field needed to form linear whiskers and the maximum field at which these whiskers disintegrate. For instance, for a protonated tetramer and hexamer the maximum fields are 0.3 V/Å and 0.2 V/Å, respectively. The presence of the proton creates a field of approximately 0.1 V/Å. Noteworthy for the present discussion is the fact that cell membrane fields are of the order of 5-50 mV/ Å[1, 36] so the combination of these fields will not be sufficient to destabilize a proton water wire. This is not to say that the walls of a membrane channel do not provide a confinement potential to stabilize the linear structures even more.
In this paper we intend to elucidate the process of charge transfer along the water wire further. We will present results from density functional calculations on the stability of proton wires in weak electrostatic fields of less than 0.1 V/Å as they occur across a cell membrane. To simulate the membrane environment we terminate the water whisker with N H 3 at one end and with a cation Z n ++(N H 3)3 as recently used by Isaev [28, 29]. Employing a procedure developed in our previous paper  we will use the interim geometries in the iterations towards the geometry-optimized final result as snapshots to track the motion of one positive charge from the cation cluster up the water whisker to the ammonia group. By following the energy of the intermediate geometries and the position of the center of mass of the charge cloud as a function of the iteration steps we can plot the energy as a function of the center of mass position. This curve of course decreases to lower energies as the energy minimization proceeds but it also has local maxima at certain positions which are interpreted as the barriers that the charge cloud, i.e. the proton, must overcome as it moves up the wire. We take this result as the justification of the classical proton hopping model. We will also show that proton “hopping” is actually the exchange of a proton from one water molecule to the next by moving the electronic charge cloud in the opposite direction. Having the energy barriers we can get the transit times across the “membrane” and estimate the conductivity of the proton wire quantitatively. In addition, our quantum mechanical approach leads to a justification and further elucidation of the Grotthuss mechanism of charge transfer along water wires .
We will base our calculations on density functional theory with a large basis set as used in previous studies of water and appropriate gradient-corrected exchange-correlation functionals as implemented in the GAUSSIAN’09 software package . As shown elsewhere [38–41] a reliable choice for an exchange/correlation potential and basis set for water is B3LYP/6-311++G ∗∗. It should be noted that electric field effects are quite large producing relative changes in molecular energy levels of the order of eV. Thus more sophisticated models, such as for bulk water or van der Waals interactions, are not needed in this case because we only look at linear structures.
In the absence of a field or a confining membrane channel a water cluster will be compact with the well known coordination of three to four hydrogen bonds. However, putting this cluster into a weak external field will turn it into a quasi-linear water whisker in its converged ground state geometry. The field must be weak enough to ensure stability because for stronger fields the whisker dissociates into smaller clusters. From our previous study we know this happens e.g. for a tetramer or an hexamer in a field range of 0.01 to 0.1 V/Å i.e. in the range of typical membrane fields.
Geometry optimizations in the GAUSSIAN’09 software package are implemented using the Berny algorithm using GEDIIS . Because we will make extensive use of the intermediate steps in the self-consistency scheme of DFT, we recall that the optimization iterations proceed as follows: the geometry optimization begins at the molecular structure specified in the input and then steps along the potential energy surface. The energy and gradient are calculated at that point and then this determines which direction and how far to take the next step. Energy and geometry are now available. This process is repeated automatically until the convergence criteria are less than the standard cutoff values of 0.00045 Hartrees/Bohr for the maximum force component, 0.0003 Hartrees/Bohr for the root-mean square force, 0.0018 Å for the maximum step component, and 0.0012 Å for the root-mean-square step. When the convergence criteria have been satisfied, the global minimum has been reached. Further details of the optimization algorithm used can be found in the Gaussian 09 Users Reference .
3 Results and discussion
3.1 Terminated tetramer
As indicated in the introduction we will take [Z n ++(N H 3)3[H 2 O]4 N H 3 as the prototype of a short proton wire in a membrane although we have also done calculations with more water molecules which however, do not reveal new insights. This aggregate is not linear but in its lowest energy state is globular. However, if we apply a small field of 0.005 V/Å it will remain straight albeit helical. As the DFT iterations proceed a positive charge will be transferred up to the ammonia group at the other end with the lowest energy state being a linear cluster [Z n +(N H 3)3][H 2 O]4[N H 3]+; this had been the starting configuration in Isaev’s work . It should be recognized that a doubly charged ion generates a field F = 2 × 14.4/r 2[V/Å], i.e. about 1 V/Å at the other end of the complex, compared to which the field applied in the present work is minimal but still strong enough to ensure a linear structure.
The diffficulty with implementing such an idea is the fact the local atomic charges within an assembly of atoms such as molecules are NOT quantum mechanical observables, i.e cannot be defined or calculated rigorously. This point is amply demonstrated in the present case by the fact that we cannot assign a proton that is halfway between two oxygen atoms to either one. Having this ambiguity we simply put forward as a criterion that we assign a proton to a given water molecule if it is within half the distance to the next oxygen i.e. a distance of about 1.2 Å. In the upper curve of Figure 2 we show the local charges on the Zn-cluster (including the lowest water molecule, the remaining three water molecules and the ammonia group at the other end) as a function of the iterations. Anticipating that one charge remains on the Zn cluster we have only plotted the one charge that is moving up. As the proton moves across the halfway distance between two oxygen atoms the charge drops abruptly on one water molecule and is picked up on the next one up the ladder. Noteworthy is the fact that there is never a charge of 1.0e on one of the subclusters but approximately 0.8e because the other subclusters continue to carry some charge. In other words, the total charge cloud ρ (n)(r) is rather diffuse. Yet the picture that a “charge” or proton moves up the wire is quite convincing.
For clarification: the jaggedness within the activation barriers is a reflection that the hopping proton will make several attempts to get to the next water molecule, i.e. it is a reflection of local fluctuations. This can be taken as evidence of the Bjerrum effect  as recently discussed again by Pavlenko .
At the height of the green barrier the hydrogen bonds to the second oxygen are stretched which costs energy, i.e. results in a barrier for further migration. One of the two bonds is re-formed at the end of the green barrier re-gaining this energy and at the beginning of the orange barrier the second hydrogen bond is re-formed.
3.2 Protonated hexamer
The most remarkable feature of this graph is the large reduction in the barriers to 0.001 (0.029), 0.66 (15.19), and 0.09 (1.98) eV (kcal/mol), (apart for the one really high barrier around 3 Å) as compared to the terminated whisker. The main reason for this is the fact that a whisker without heavy donor and acceptor groups at the ends has greater rotational freedom i.e the two water molecules involved in a particular transfer event can more easily rotate at lower energy cost. This is obvious when one examines a series of snapshots like those in Figure 1. The exceptionally high barrier is associated with an attempt by the chain to twist as a whole.
The extra proton moves close to the lowest water molecule forming a H 3 O + which in turn rotates and loosens its bond to the hydrogen in the direction of the field. This proton in turn moves closer to the next water molecule leaving a “neutral” H 2 O behind and forming a new H 3 O + which again rotates to accommodate the extra charge which is by now located beyond the second water molecule. This process - motion of the extra proton up the wire, rotation of the two closest water molecules and eventual attachment to the new host molecule repeats itself until the proton has reached the fourth water molecule in the chain.
Charge transfer along a water wire is an atomic exchange mechanism in which a proton moves the short distance of less than the O − O distance leaving behind a quasi-neutral water molecule and forming a temporary diffuse hydronium ion H 3 O +.
What is then the overall picture of charge or proton transfer through a membrane channel? We start from the equilibrium configuration [Z n +(N H 3)3][H 2 O]4[N H 3]+. At some point in time a proton or hydronium ion, getting close to the Zn cluster will accept an electron getting neutralized and turning the Z n + into Z n 2+. If in a time interval before this charge transfer is reversed by another fluctuation, an electron is transferred from a water molecule on the other side of the membrane to the ammonia molecule a charge transfer has occurred across the membrane with the proton now on the other side. This results in the configuration [Z n ++(N H 3)3][H 2 O]4[N H 3] which will trigger the next round of charge transfer.
We stress again that a protonated water whisker without acceptor and donor groups has negligible activation barriers i.e. typically of the order of thermal energy at room temperature. This obviously casts doubt on the validity of classical Ising-type chain models in which one postulates well-defined and permanent minima along the chain for the extra proton which then hops over a barrier in the same way as an adsorbed particle hops from one adsorption site to the next in lattice gas models for surface diffusion. What emerges from our quantum mechanical calculations is a picture in which the transfer of a charge cloud from one water molecule to the next creates a local minimum and a barrier that disappears as soon as the charge has moved on.
where the barrier heights can be read off from left to right in Figure 2 to be 0.45 (10.39), 0.04 (0.89), 0.48 (11.07), 0.22 (5.02), and 0.33 (7.71) eV (kcal/mol) in agreement with the measured values for carbonic anhydrase of 0.043-0.11 eV (1.0-2.5 kcal/mol) for proton transfer and an overall energy barrier of 0.42-0.47 eV (9.8-11.0 kcal/mol) [45–47]. Because the transit time from one barrier to the next one a distance of a few angstroms away is less than a picosecond the proton arrives with some kinetic energy that has not been completely thermalized leading to a small reduction in the barrier heights. Taking this into account we get overall transit times of the order of microseconds, and possibly nanoseconds.
Here V is the membrane potential. For the two estimated transit times we get conductances of nS down to pS. We could not find measured values for isolated proton wires but note that for proton channels in Gramicidin A one finds 1.5 nS . One should note that in a Gramicidin A channel the transit time is on the order of nanoseconds in the range of our estimates.
Our final comments concern the relation of this work with previous studies.
In his quantum mechanical studies of proton wires Isaev [28, 29], using the same cluster and the same DFT framework apart from a somewhat smaller basis set rightly observes that without a field an extra proton will settle in the middle of the wire, i.e. at equal distances from the acceptor and donor which are both charged. He examines extensively the variation in hydrogen bond lengths along the cluster which he finds are within a few hundredth of an angstrom. This is also what we find but do not put any emphasis on it as such a small variation is easily within the thermal fluctuations of the hydrogen bond at room temperature.
In the classical Ising-type model of proton hopping Pavlenko  assumes that the hydrogen has four well-defined adsorption sites between any two oxygens along the chain. Such an assignment assumes that there are energy barriers on either side of these sites. We see some evidence for that, see Figure 3.
This work was supported by grants from NSERC and the Office of Naval Research, Washington DC.
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