J. Phys. Chem. B 2010, 114, 6567–6578
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Energy Compensation Mechanism for Charge-Separated Protonation States in Aspartate-Histidine Amino Acid Residue Pairs Katsumasa Kamiya,*,†,‡ Mauro Boero,‡,§ Kenji Shiraishi,‡,|,⊥ Atsushi Oshiyama,†,‡,# and Yasuteru Shigeta†,‡ Picobiology Institute, Graduate School of Life Science, UniVersity of Hyogo, 3-2-1 Koto, Kamigori, Ako, Hyogo, 678-1297, Japan, CREST, Japan Science and Technology Agency, Sanban-cho, Tokyo 102-0075, Japan, Institut de Physique et Chimie des Mate´riaux de Strasbourg, UMR 7504 CNRS and UniVersity of Strasbourg, 23, rue du Loess, F-67034 Strasbourg 2, France, Graduate School of Pure and Applied Sciences, UniVersity of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8571, Japan, Center for Computational Sciences, UniVersity of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8577, Japan, and Department of Applied Physics, School of Engineering, The UniVersity of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ReceiVed: June 30, 2009; ReVised Manuscript ReceiVed: February 15, 2010
The initial stage of proton propagation in the D-path channel of bovine cytochrome c oxidase, consisting of the approach of an H+ to the entrance of this specific pathway, is inspected via first-principles calculations. Our model, extracted from the X-ray crystallographic structure, includes the amino acid residue pair aspartate (Asp91) and histidine (His503) as protonatable sites. Our calculations show that an additional proton, corresponding to the H+ uptake by the enzyme from the inner bulk water, is transferred to either Asp91 or His503, leading to the formation of a neutral or a charge-separated protonation state. The relative stability between the two states amounts to a total energy difference of about 5 kcal/mol; this indicates that both Asp91 and His503 are involved in the proton supply to the D-path, playing the role of proton acceptors. The hydrogen-bond environment around Asp91 and His503 has an important influence on both the energetics and the electronic structure of the system; for instance, it compensates the Coulomb-energy cost in the chargeseparated protonation state. An energy partitioning analysis shows that the compensatory effect is mainly due to local electrostatic interactions among the charged Asp91 and His503 side chains and the surrounding polar residues. The energy compensation mechanism we found in this work balances the energetics of Asp-His pairs, hence permitting an efficient and selective regulation of the protonatable amino acid residues, where several protonation states are accessible within energy differences of the order of a few H-bonds. 1. Introduction Aspartate (Asp) and histidine (His) are two protonatable amino acid residues generally present in proteins. Because of this peculiarity, they are essential constituents of protein active sites, participating in the catalytic reactions by taking (protonation) or releasing (deprotonation) protons, transferred from side chains. For instance, His and Asp are the main components of proton transfer paths both in proton channels and protonpumping enzymes.1,2 Furthermore, histidine and aspartate have also been discovered as a catalytic dyad His · · · Asp in the active sites of proteases,3 ribonuclease A,4 and in a wealth of enzymes5–8 and as an Asp-His-Fe triad in peroxidases.9 It is well-known that the side chains of these protonatable amino acid residues generally assume the most favorable protonation state, depending on their surrounding environments. Relationships between the suitable protonation states and physiological environments have been investigated on the basis of the relative positions of these residues in proteins, which are structurally well characterized by X-ray diffraction measurements. However, * To whom correspondence should be addressed. E-mail: kkamiya@ sci.u-hyogo.ac.jp. † University of Hyogo. ‡ Japan Science and Technology Agency. § UMR 7504 CNRS and University of Strasbourg. | Graduate School of Pure and Applied Sciences, University of Tsukuba. ⊥ Center for Computational Sciences, University of Tsukuba. # The University of Tokyo.
as far as the protonation state is concerned, these experiments can provide direct information only in the particular case where the side chains are completely exposed to the bulk solvating water. Otherwise, since protons cannot be detected by X-ray techniques, an a posteriori analysis of the X-ray structure is needed. This is exactly the situation encountered in cytochrome c oxidase (CcO), whose X-ray structures have been recently determined at unprecedented high resolution in membrane proteins.10–14 The CcO, the terminal oxidase of cellular respiration, reduces dioxygen to water, and this reaction is coupled with the protonpump activity. This enzyme has been the subject of extensive investigations targeting the relationship between structure and function, due to the high-resolution X-ray structure available.10–14 Recently, the 1.8/1.9 Å X-ray structures of the bovine CcO11 have provided interesting structural details about the entrance of incoming protons into one of three proposed proton transfer pathways: the D-path. This specific pathway is considered to promote the O2 reduction, and is schematically shown in panel a of Figure 1. The X-ray structure of CcO has evidenced that an amino acid residue pair, made of one aspartate and one histidine residue, Asp91 and His503, is present at the entrance of the D-path. These two amino acid residues are connected to each other by hydrogen bonds (H-bonds) through an oxygen atom identified as a crystallographic water molecule and labeled as H2O207 (Figure 1b). An important feature that must be mentioned is the fact that Asp91 is not exposed to a bulk
10.1021/jp906148m 2010 American Chemical Society Published on Web 04/22/2010
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Figure 1. Schematic representation of (a) the whole X-ray structure of the fully reduced bovine CcO,11 (b) the entrance of the D-path, and (c) the Asp91-His503 model system used in this work. In panel a, the redox-active metal sites are shown as either black balls or black sticks. The D-path is indicated in red, while the other two alternative proton transfer paths are shown in blue and green. In panel b, two atoms satisfying the H-bond criteria, i.e., a distance of 3.4 Å and an angle of 120°, are connected by dashed lines. In panel c, the atoms composing the model system are shown as thick sticks. Here and in the following figures, the color code for the atoms is yellow for C, white for H, red for O, blue for N, and tan for S, respectively. The fixed capping H atoms and terminal C atoms are shown as balls. The other H atoms are not represented for the sake of clarity.
aqueous phase but is located at the interior of the protein. An independent confirmation comes from mutagenesis studies,15 which have shown that Asp91 is the entrance amino acid residue of the D-path and is involved in proton supply to this pathway. On the other hand, His503, known to be an omnipresent amino acid residue both in bacterial and bovine CcO enzymes,11–14 is located at the molecular surface; such a surface is defined by rolling a spherical probe of radius 1.4 Å on a superposition of spheres, centered on each atom composing the molecule, having the van der Waals radius. Recent X-ray structural and functional analyses, performed on the bovine CcO by using Zn2+ and Cd2+ as probes, have shown a partial inhibition in the O2 reduction process, due to the binding of Zn2+ and Cd2+ divalent cations to His503.11 This provides a further indication that His503 is involved in proton supply to the D-path. Despite these experimental findings, the role of the Asp91-His503 pair at the entrance of the D-path is still far from being unraveled. On the theoretical side, first-principle calculations have become a versatile and reliable theoretical tool to complement the experimental outcome and to inspect details not directly accessible to experimental probes. Specifically, density functional theory (DFT)16,17 provides a computationally efficient approach, successfully applied to a wide variety of chemical and biological systems.18–26 In particular, it has been shown to be particularly suited to the study of H-bond systems and proton transfer processes27–32 and applied successfully to investigate protonation states and underlying proton transfer in proteins.33–39 The scope of the present work is to provide insight into the relationships between the protonation states of the Asp91-His503 pair and their surrounding environment and to inspect the role of this amino acid pair at the entrance stage of the D-path in the bovine CcO. In particular, we analyze different possible protonation states of the Asp91-His503 system during the stage in which an incoming proton is supplied to the D-path. Furthermore,we investigate the effects of the surrounding H-bonding environment around both Asp91 and His503 on the energetics and relative stabilities of the protonation states. A comparison done with a different Asp-His catalytic dyad provides further insight into the role of the H-bond network and allows for a better understanding of our results. The outcome of the present set of calculations elucidates the peculiar role of both Asp91 and His503 amino acid residues and shows the importance of the surrounding H-bond connectivity at the entrance of the D-path in promoting the proton transfer in the initial stage of this pathway.
2. Computational Details The Asp91-His503 model system used in the present calculations (Figure 1c) has been extracted from the X-ray structure of the fully reduced bovine CcO (RCSB Protein Data Bank, accession code 2eij11). The model consists of Thr10, Asn11, His12, Lys13, Asp14, Ala89, Pro90, Asp91, Met92, Pro501, Tyr502, His503, Thr504, and Phe505 of subunit I, Met10, Val11, and Asn12 of subunit III, and 13 crystal water molecules. To reduce the computational costs, six amino acid residues, whose side chains are far from the active site, have been replaced with Ala: Met92Ala, Pro501Ala, Tyr502Ala, and Phe505Ala in subunit I and Val11Ala and Asn12Ala in subunit III (Figure 1c). Both the Asp91 and His503 amino acid residues are assumed to have deprotonated side chains in the initial stage of the process. The protonation states of the three residues His12, Lys13, and Asp14 were set to neutral (deprotonated Nε2), protonated, and deprotonated, respectively, as suggested by direct inspection of their local H-bonding environments. An additional H+, corresponding to the uptake of a proton by the enzyme from water bulk, has been attached to the O atom of either H2O4 or H2O207, two of the crystallographic water molecules which are close to the Asp91-His503 D-path entrance pair. The resultant Asp91-His503 model amounts to 255 atoms and has a net charge of Q ) 0. This ensures that no spurious Coulomb interactions exist with periodically repeated images of the simulation cell, since the system is placed in an orthorhombic supercell of sizes a ) 22.90 Å, b ) 23.60 Å, and c ) 23.10 Å with periodic boundary conditions (PBC). Such a size guarantees that our model system is separated from its periodic image by a minimum distance of 6 Å. We checked this by releasing the PBCs40,41 and by performing a few test calculations on isolated systems; the results show energy differences below 1 kcal/mol (Table S1 in the Supporting Information). Hence, due to the lower computational cost, PBC were adopted in all of the calculations presented in the ongoing discussion. The present DFT scheme makes use of gradient corrections on the exchange and correlation functionals according to the Hamprecht, Cohen, Tozer, and Handy (HCTH)42–44 prescription. We confirmed that using different exchange-correlation functionals (the Perdew-Burke-Ernzerhof (PBE)45 and the BeckeLee-Yang-Parr (BLYP)46,47 functionals) provides nearly the same results for the energetics of the Asp-His pair (Table S2 in the Supporting Information). The interaction between core
Energy Compensation Mechanism and valence electrons is described by Troullier-Martins normconserving pseudopotentials.48 Valence orbitals are expanded in a plane-wave basis set with an energy cutoff of 70 Ry. The conjugate gradient minimization procedure is used for both the electronic-structure calculations and the geometry optimizations. In the geometry optimizations, almost all atoms (226 atoms) are relaxed until the residual forces per atom are less than 2.3 kcal/mol/Å. During the optimization, the positions of the remaining 29 atoms, representing the terminal ones at the boundaries of our model system, are kept fixed at the X-ray crystallographic positions (Figure 1c). All of the calculations presented have been performed with the CPMD code.49 3. Results and Discussion 3.1. Two Possible Protonation States of the Asp91-His503 System. We start with the assessment of possible protonation states at the entrance of the D-path. The X-ray structure of bovine CcO indicates that His503 is located in proximity of Asp91, the entrance amino acid residue of the D-path (Figure 1b).11 The side chains of these two residues are separated by one water molecule (H2O207), which is located at the intersection of two arrays of crystallographic H2O molecules (from H2O207 to H2O4 and H2O201) extending to the solventaccessible surface of CcO. A supplied proton from the bulk to the D-path is thus expected to reach H2O207 through one of the two H-bond connected water arrays via an ordinary proton wire mechanism, and a hydronium ion (H3O+) can be formed. The two H-bond networks could easily undergo rearrangements at room temperature, since the water molecules form H-bonds with the nearby amino acid residues and another crystal water molecule (Figure 1b). It is thus possible for H2O207 to be in a different orientation when it receives a supplied proton from the H-bond networks. Given this situation, we verified whether or not the protonated form of H2O207 (H3O+) behaves like a proton donor and if the His503 side chain can act as a proton acceptor during the proton supply stage to the D-path. We added an additional proton on the O atom of H2O207 to construct a (tentative) protonated Asp91-H2O207-His503 system. Considering the possibility of taking different orientations of H3O+207, we examined as an initial state all of the possible configurations where a H3O+ initially forms H-bonds with at least two H-bonded acceptors, selected from five possible H-bonded ones around H2O207 (Figure 1b).50 Geometry optimizations of the Asp91-H3O+-His503 models have shown that an additional proton is readily transferred either to the carboxyl group of the Asp91 side chain or to the imidazole ring of the His503 side chain (details are shown in Table S3 of the Supporting Information). This leads to the formation of neutral and charge-separated protonation states of the Asp91His503 system. The former consists of a dyad in which a protonated carboxyl group of Asp91 coexists with a deprotonated imidazole ring of His503 ([Asp91-CγOOH and His503Im-Nδ1]). The latter, instead, is a deprotonated carboxyl group and a protonated imidazole ring ([Asp91-CγOO- and His503Im-Nδ1H+]). Our calculations have demonstrated that the chargeseparated state is energetically favored, since its total energy is 5.4 kcal/mol (on average) lower than the neutral one. This provides a clear indication that His503 is involved in the proton supply to the D-path and, at the same time, shows that Asp91 acts as a proton acceptor and takes active part in the process. The calculations have also shown that all of the optimized protonated systems are more stable than the corresponding initial H3O+ configurations by 59.1 kcal/mol on average. This indicates
J. Phys. Chem. B, Vol. 114, No. 19, 2010 6569 that the initially protonated water is a highly unstable H3O+ unable to form an Eigen (or Zundel) metastable state29,30 and thus H3O+ behaves like an acid,32 donating a proton to either Asp91 or His503. We also checked that a bulk proton entering the H-bond network could reach H2O207, and eventually transferred to Asp91 or His503. As shown in Figure 1b, H2O4 is one of the terminal water molecules that are completely exposed to the bulk water phase and thus H-bond connected with the rest of the solvent. Under these conditions, this water monomer is able to uptake protons from the bulk to transfer it into its participating H-bond network. For this reason, we decided to protonate H2O4, changing it into H3O+. Using this starting configuration, we performed geometry optimizations. We found that an additional proton is readily transferred from H2O4 to H2O207 through the H-bond network via the proton wire mechanism (Figure 2 and Table S4, Supporting Information). Eventually, a proton reaches the carboxyl group of the Asp91 side chain for this specific initial configuration. Considering all of the results shown in this section, it is suggested that H2O207 is temporarily protonated once a bulk proton enters the H-bond networks, finally passing a proton to either Asp91 or His503 according to its tentative molecular orientation. 3.2. Compensatory Effects of H-Bonding Environment on the Energy Difference between Two Protonation States. It must be remarked that the charge-separated and neutral protonation states of the Asp91-His503 system are still close in energy, being comparable with the typical value of H-bond strength, ∼5 kcal/mol.51 As stated above, protonatable amino acid residues assume suitable protonation states, depending on the physiological environments. In the case of the present Asp91-His503 system, the side chains of Asp91 and His503 are tightly H-bonded by their surrounding amino acid residues and water molecules (Figure 1b). We have thus investigated effects of the H-bonding environment on the energetics of the charge-separated and neutral protonation states of the Asp91His503 system. For this purpose, we have constructed five models, shown in the insets of panels a-e in Figure 3. These systems, of grown complexity, possess several types of Hbonding environments selected on the basis of the full Asp91-His503 system (inset f in Figure 3). The chargeseparated state of the full system corresponds to the one obtained in our previous geometry optimizations. On the basis of this configuration, the corresponding neutral one has been constructed. We extracted from these full systems smaller models that include the nearby amino acid residues and water molecules in the vicinity of both Asp91 and His503. The calculations performed on these models have evidenced compensatory effects of the H-bonding environment on the energy difference between the charge-separated and neutral protonation states in the CcO Asp91-His503 system (Figure 3 and Table S5, Supporting Information). As shown in Figure 3a, the charge-separated state of the model consisting only of Asp91 and His503, hereafter referred to as the simplest Asp-His system, is highly unstable; in fact, the energy difference between the (locally) charge-separated system and the neutral charge state is rather large, namely, ∆Esimplest ) Echarge-separated - Eneutral ) 50.8 kcal/mol. This difference decreases upon insertion of one H2O molecule between Asp91 and His503 side chains (Figure 3b). A further significant reduction is achieved by including the H-bonding environments around Asp91 (Figure 3c) and His503 (Figure 3d). Eventually, a more extended model including all of the nearby residues (Figure 3e) gives an energy difference that matches the full Asp91-His503 system (Figure
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Figure 2. Total energies during geometry optimizations of the Asp91-H3O+4-His503 system. Energies are measured from that of the optimized structure. Asp91, His503, H2O4, H2O28, H2O260, and H2O207 are shown as balls and sticks.
Figure 3. Total energy differences between the charge-separated and the neutral protonation states: (a) the simplest Asp-His system; (b) the H2O207 insertion model; (c) the Asp91 H-bonding environment addition model; (d) the His503 with its H-bond environment; (e) the compensatory H-bond environment model; (f) the full Asp91-His503 system. In each case, only the charge-separated form [Asp91-CγOO- and His503-ImNδ1H+] is shown for the sake of clarity.
3f). These results provide a clear indication that the H-bonding environment around Asp91 and His503 cannot be neglected, since it is the driving factor for the stabilization of the system in the different protonation states and shows how the local environment compensates for the large energy difference between the charge-separated and neutral protonation states (by ∼51 kcal/mol) affecting smaller models. A first conclusion that can be drawn is then that results obtained on minimal models are not representative of the real system and cannot provide reliable insights into the proton transfer mechanism at the entrance stage of the D-path. Instead, more extended models shown in panels e and f of Figure 3 give converged results for the energetics of the protonation states, suggesting that these models have all of the features needed to correctly reproduce the compensatory H-bond environment around Asp91 and His503. Such a compensatory effect in the Asp91-His503 system can be explained by local electrostatic interactions among the
charged Asp91 and His503 side chains and the surrounding polar residues. To make this issue clearer, we have constructed a model (Figure 4), where all the polar amino acid residues are replaced with water molecules to create a compensatory H-bond environment depicted by the model reported in Figure 3e. Geometry optimizations of the substituting water molecules show that a similar compensation effect on the energy difference between the charge-separated and neutral states is kept also in this model including almost exclusively water molecules in the H-bond network connecting Asp91 to His503. In fact, the energy difference, ∆Ereplacing-wat, is ∼1 kcal/mol, leading to an energy gain of about ∆∆E ) ∆Ereplacing-wat - ∆Esimplest ) -50 kcal/ mol. Figure 4 also shows that the number of H-bonds is almost the same in both of the protonation states: 14 H-bonds in the charge-separated state and 13 in the neutral state. By simple H-bond counting, one would infer that both states should be characterized by a similar stability. However, this might not be the case, since the strength of the H-bonds in the two cases is
Energy Compensation Mechanism
J. Phys. Chem. B, Vol. 114, No. 19, 2010 6571 The calculated Ei values for each water molecule are summarized in Table 1. We found that using an effective charge q ) 0.7 au comparable values are obtained for all of the ∆∆Ei contributions. This q value is also comparable with the net difference in the Mulliken charge of the Asp or His side chain between the charge-separated and the neutral protonation states: ∆Q ) -0.5 au (Asp) and ∆Q ) +0.6 au (His). The sum of Ei is related to the energy gain ∆∆E by the relationship n
∆∆E ) ∆Ereplacing-wat - ∆Esimplest ∼
∑ Ei + ∆E1HB i)1
since n
∆Ereplacing-wat ∼
∑ Ei + ∆E1HB + ∆Esimplest i)1
Figure 4. Schematic representation of the charge-separated and neutral protonation states in the replacing water model. The colored sticks show the charge-separated state, while the gray ones represent the neutral state. Two atoms satisfying the same H-bond criteria used in Figure 1 are connected by black dashed lines for the charge-separated state and gray dotted lines for the neutral state, respectively. The local dipole moments of each water molecule are shown by a cyan arrow.
likely to be different. In fact, in the case of the charge-separated state, the local dipole moment of the system increases because of the charge separation, and the strength of an H-bond is known to be affected by the induced dipole. The charge-dipole interaction between the charged Asp and His side chains and the surrounding H-bonds is then expected to be different in the two cases despite a similar amount of H-bonds. To quantify this qualitative idea, we have first computed the energy difference ∆∆E for the case in which water molecules are inserted into the simplest Asp-His pair shown in Figure 3a. In this case, the energy gain is given by the sum
where ∆E1HB (∼ -5 kcal/mol) represents the difference in the number of H-bonds between the charge-separated and neutral states. The value estimated from the sum of Ei is thus -42 5 ) -47 kcal/mol, indicating that the charge-dipole interaction energy represents a large amount of the energy gain ∆∆E. The importance of the local dipole moments can be understood by a direct inspection of the specific configuration of the polar residues around Asp-His pairs. We can formally divide the polar residues in the Asp-His system into two groups: the ones near the Asp-His pair and the others. The polar residues which are likely to be essential for making a compensatory H-bond environment should be specifically oriented with respect to the Asp-His pair in order to give rise to favorable chargedipole interactions. These residues form H-bonds with the Asp-His pair in most cases. The number of local H-bonds is thus proportional to the number of such polar residues; hence, n
n
∆∆E )
n
∑ (∆Ewat i - ∆Esimplest) ) ∑ ∆∆Ei i)1
n
∑ ∑ qI i)1
∑ Ei ∼ N · Eave i)1
i)1
As it can be deduced from the results of Table 1, the sum of the various contributions ∆∆Ei amounts to ∼51 kcal/mol, which is identical to the previous value of ∆∆E. For comparison, we have also estimated the charge-dipole interaction energies by using a schematic model consisting of a dipole moment attributed to each water molecule near two point charges representing the (charged) Asp91 and His503. The charge-dipole interaction energies in this model are written as
E)
∆∆E ∼
I
pi · (rI - ri) |rI - ri | 3
n
)
∑ Ei i)1
where the point charges, qI, refer to the charged Asp91-CγOOand His503-Im-Nδ1H+, located at each geometrical center rI. The dipole moments, pi, represent the surrounding n water molecules at positions ri. All of the dipole values are set to 1.85 D, corresponding to the experimental gas-phase water molecule. An auxiliary calculation done by using a set of the water dipole values obtained within the same first-principles approach used in this work did not significantly alter the result.
where N is the number of H-bonds around the Asp-His pair. Eave represents the average charge-dipole interaction energy, i.e., 4-5 kcal/mol as inferred from the previous case (Table 1). Further discussion on this issue will be given in the next subsection. On the other hand, a large number of polar residues that are far from the Asp-His pair could be oriented in several different ways with respect to the Asp-His pair. Furthermore, charge-dipole interactions decrease as rapidly as ∼1/r2; thus, their overall contribution to a compensatory H-bond environment is likely to be small. The compensatory effects are reflected in the electronic structure of the Asp91-His503 system, particularly near the lower level region (Figure 5). We found that O 2s-like and N 2s-like Kohn-Sham (KS) states appearing in the region are the ones that are mostly affected by the H-bond environment. The spatial distributions of the O 2s-like and N 2s-like KS states are characterized by large amplitudes on top of the carboxyl and imidazole groups of Asp91 and His503, respectively. Both of these states have ssσ bonding and ssσ* antibonding character. As shown in Figure 5a, the energy levels of these two states in the simplest Asp-His system are significantly shifted in the charge-separated protonation state (right panel), in comparison with the neutral one (left panel). The levels of the O 2s-like
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TABLE 1: Energy Gain (∆∆Ei) by the Insertion of Each Water Molecule into the Simplest Asp-His Pair and the Charge-Dipole Interaction Energies (Ei) by Estimating in a Model Consisting of a Dipole Moment on Each Water Molecule near Two Point Chargesa kcal/mol ∆∆Ei Ei a
wat 1
wat 2
wat 3
wat 4
wat 5
wat 6
wat 7
wat 8
wat 9
wat10
wat 11
total
ave
-4 -3
-1 -1
-2 -3
-5 -6
-11 -10
-3 -2
-6 -4
-6 -4
-2 -1
-8 -5
-2 -2
-51 -42
-5 -4
The corresponding geometry of each water molecule is depicted in Figure 4.
Figure 5. Energy levels of the KS states near the lower level region of (a) the simplest Asp-His system, (b) the compensatory H-bond environment model, and (c) the full Asp91-His503 system. The energy levels in the neutral protonation states are shown in the left panel for each case, whereas the levels of the charge-separated states are reported in the right panel. The KS levels of the O 2s-like and N 2s-like states, having large amplitudes on the carboxyl and imidazole groups of Asp91 and His503, respectively, are highlighted as black lines, while other ones are shown in gray. The energy is measured from the level of the lowest, O 2s-like bonding KS state in the neutral protonation state of each model. The spatial distributions of the corresponding KS orbitals are represented in the figure. All isosurfaces are at +0.19 (pink) and -0.19 (ice blue) (e/Å3)1/2.
KS states are displaced, along the energy axis, toward higher values by about 4.0 eV, while the levels of the N 2s-like KS ones shift downward by ∼3.1 eV. These two level shifts, however, are balanced by the compensatory H-bonding environment (Figure 5b and c). The energy differences of the shifts of the O 2s-like and N 2s-like KS states are eventually stabilized around ∼2.5 and ∼2.2 eV, respectively, in the case of the full Asp91-His503 model (Figure 5c). 3.3. Digression about the Energy Compensation Mechanism. The results discussed in the previous section indicate that the peculiar H-bond environment in the Asp91-His503 system might be responsible for the compensation of the large level shifts observed in the two protonation states. This idea is not entirely new. Indeed, the effects of the local H-bond network in the stabilization of a charge-separated protonation state in a proton donor (AH) and proton acceptor (B) pair have already been reported in the literature.52,53 In general, in a strong enough acid-base pair, the proton transfer occurs from A to B, converting the pair into a charge-separated protonation state (also referred to as “ion pair”, “salt bridge”, or “zwitterion” if A and B are intramolecular moieties):
AH · · · B f A- · · · H+B Although it has been pointed out earlier that the charge-separated state could be stabilized by polar environments, necessary and sufficient conditions for the conversion from a neutral state to a charge-separated one are not obvious, and their investigation has been the target of intense efforts.52–66 For instance, studies of hydrogen halide-amine pairs, such as HCl and NH3, have indicated that the addition of one to two H2O molecules to the
HCl-NH3 pair is sufficient to confer stability to the chargeseparated protonation state.54–56 Analogously, experimental and theoretical studies57–59 on the simplest amino acid, glycine, have suggested that a minimum number of four or five H2O molecules is needed for the energetic balance of the neutral and chargeseparated (zwitterion) forms. Further studies on other amino acids, alanine and arginine,60–63 as well as chemical systems not related to amino acids64–66 led also to similar conclusions, namely, that water molecules are responsible and necessary for the stabilization of charge-separated protonation states. In the case of proteins, polar amino acid residues and water molecules are generally H-bonded to a catalytic pair of amino acid residues, thus forming the surrounding polar H-bond environment at active sites. Related stabilization effects of charge-separated protonation states have been extensively investigated,67–74 and recently reviewed.75 These studies have shown rather clearly that the H-bond environment around the catalytic pairs can stabilize charge-separated protonation states appearing as ground and/or transition states in enzymatic reactions. Furthermore, recent studies73–75 have pointed out that active sites of proteins tend to form a preorganized polar environment suitable to stabilize ion pairs (charge-separated states or zwitterions), at least at the transition state stage, and this concept seems to hold in general for a large class of charge separation reactions. However, to date, not much has been reported on the effects of H-bond environments as a tool for compensation of energy differences between a neutral and a charge-separated protonation state of catalytic pairs. Nonetheless, whether or not such H-bond environments are necessary and sufficient for the conversion from a neutral state to a charge-
Energy Compensation Mechanism
Figure 6. Total energy differences between the charge-separated protonation state (left inset) and neutral one (right inset) as a function of the distance d for the simplest Asp-His system. Calculated values are indicated by a solid line, while the 1/d fitting curve is shown as a dashed line. The vertical lines crossing the curve at the abscissa values of 2.5, 3.2, and 4.0 Å define (i) strong, (ii) moderate, and (iii) weak H-bond regions, according to Jeffrey’s classification of H-bonds.76 The vertical line at 5.4 Å corresponds to the experimental Asp-His distance in the bovine CcO.
separated one is a fundamental issue in assessing the fine regulation of catalytic pairs. In this respect, our calculations highlight compensatory effects of the local H-bond environment on the energetics of neutral/ charge-separated protonation states in the case of an Asp-His pair in the bovine CcO. In an attempt at a further understanding of these effects, we first analyze the energetics of the simplest Asp-His system sketched in Figure 3a. Figure 6 shows the total energy difference between the charge-separated and neutral protonation states for the system as a function of the distance d between the carbonyl Oδ2 of Asp and the imidazole Nδ1 of His. We find that for any distance the charge-separated protonation state is always located at higher energy with respect to the neutral state in the Asp-His pair, consistently with previous studies in the gas phase.52 The energy difference increases in a Coulomb-like way as q2/d (dashed line), where the parameter q corresponds to an effective charge q ) (0.6 au on each side chain. This q value is comparable with the net difference in the Mulliken charge, of either the Asp or the His side chain, between the charge-separated and the neutral protonation states, namely, ∆Q ) -0.5 (Asp) and ∆Q ) +0.6 (His). Hence, the large energy difference between the two protonation states is mostly due to the effective Coulomb interaction between the residues. A hint that can be inferred from the analysis summarized in Figure 6 is that different H-bond environments are needed for the energy compensation in Asp-His pairs, and this might depend on the separation between Asp and His. According to Jeffrey’s classification of H-bonds,76 compensation energies should fall in the ranges 15-20, 20-30, and 30-40 kcal/mol for Asp-His distances typical of strong H-bond lengths (2.2-2.5 Å), moderate H-bond lengths (2.5-3.2 Å), and weak H-bond lengths (3.2-4.0 Å), respectively. On the other hand, a larger energy is required to compensate the energy difference between charge-separated and neutral states when Asp and His are separated by distances larger than the maximum H-bond distance (4.0 Å). In particular, an Asp-His separation of ∼5.4 Å, corresponding to the case of Asp91-His503 in the bovine CcO, needs a compensation energy of about 50 kcal/mol. Interestingly, such a compensation energy corresponds to the energy gain due to the formation of about
J. Phys. Chem. B, Vol. 114, No. 19, 2010 6573 10 H-bonds, and can be provided solely by the local H-bond environment around the Asp91-His503 pair. In fact, a direct inspection of the X-ray structure (Figure 1b) shows that several water molecules and polar amino acid residues exist around Asp91 and His503, forming a hydration shell structure. The H2O207 water molecule inserted between the Asp91 and His503 side chains leads to the formation of the Asp91H2O207-His503 moiety, which, in turn, involves several surrounding polar residues in its local H-bond environment, thus amounting to about 10 H-bonds. Given this scenario, it seems that compensatory effects due to the H-bond environment are indeed present, as one can infer by looking at Figure 3e and f, where just the most relevant amino acid residues and water molecules are shown (panel e); these are likely to be the necessary and sufficient polar residues responsible for the compensatory local H-bond environment for the Asp91-His503 pair. Moreover, the local character of the H-bond system around this pair is consistent with the indications reported in the previous subsection, namely, the fact that compensation effects are mainly due to charge-dipole interactions between the charged Asp91 and His503 side chains and the nearby polar residues. A crucial feature of the bovine CcO Asp91-His503 pair is worthy of note. As mentioned, the Asp-His pair is a rather common catalytic dyad present in active sites of several enzymes, such as serine proteases, esterases, and so on. At variance with other Asp-His dyads, in the Asp91-His503 pair considered here, the crystallographic water molecule H2O207 is inserted between Asp91 and His503 (Figure 1b). From the present study, we can infer that the H2O207 water molecule plays at least three roles in the CcO enzymatic activity. First, as shown in subsection 3.1, the H2O207 water molecule could donate a proton to either Asp91 or His503 during the proton supply stage to the D-path. Moreover, this water molecule could serve as a bridge for a proton transfer between these two residues. Second, as stated in subsection 3.2, the insertion of the H2O207 water molecule into the Asp-His pair contributes to the stabilization of the charge-separated protonation state. Last, H2O207 is essential to involve several surrounding polar residues in the compensatory H-bond environment around the Asp-His pair. An indirect support to the catalytic role of a water molecule bridging Asp and His amino acid residues in the stabilization of charge-separated protonation states comes also from analogous conclusions drawn for other enzymes.77 A clear indication that emerges from this work is the fact that the presence of a compensatory local H-bond network is a condition necessary and sufficient to balance the energetics of the two competing states, neutral and charge separated, at the entrance of the D-path in the bovine CcO. This is shown by the analysis summarized in Figures 3, 4, and 6, where the energy gain is due to the subsequent increase in the number of H-bonds around the Asp-His pair, as discussed in the previous subsection. This can offer also a guideline to understand, at least at a qualitative level, whether or not a local H-bond network around a general Asp-His pair moiety is suited to stabilize a chargeseparated state. To gain further insight into this concept, we examined the X-ray crystallographic structures of several other enzymes presenting Asp-His or Glu-His dyads as catalytic pairs. The results are summarized in Figures 7 and 8 and in Table 2. Specifically, Figure 7 shows a set of X-ray structures of hydroxynitrile lyase from HeVea brasiliensis (Hb-HNL). Recent NMR spectroscopy measurements and Poisson-Boltzmann calculations performed on this enzyme have suggested that neutral and charge-separated protonation states of the Asp207-
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Figure 7. Schematic representations of active sites in hydroxynitrile lyase from HeVea brasiliensis. The eight X-ray structures examined in ref 72 are reported. Only the surrounding amino acid residues and crystallographic water molecules around Asp207-His235 pairs are shown for the sake of clarity. The corresponding PDB IDs are given in Table 2. Two atoms satisfying the same H-bond criteria used in Figure 1 are connected by dashed lines. The Cl atoms are shown in purple.
His235 dyad are energetically close to each other by less than 5 kcal/mol (on average) for eight available X-ray structures.72 Although the main reason for this energetic balance has not been pointed out in these studies, we remark that the local H-bond environment around the Asp207-His235 dyad seems to be already in a compensatory configuration, energetically speaking, on the basis of our analysis (Figures 6 and 7). As shown in Figure 7, the Hb-HNL X-ray structure shows that the separation between Asp and His amounts to about 2.79 Å and the number of H-bonds to the dyad is about five on average. By looking at Figure 6, a compensation energy of about 25 kcal/ mol can be estimated, corresponding to energy gain due to the formation of five H-bonds. Therefore, the energetics of neutral/ charge-separated protonation states of Asp207-His235 is likely to be balanced. Two of the better studied enzyme systems having Asp (Glu)-His catalytic dyads are the protease and esterase families. The panels from a to h in Figure 8 show some representative configurations of the serine and cysteine protease family, listed according to types of catalytic triad: (a-c) His-Asp-Ser, (d) Ser-Asp-His, (f, g) His-Asp (Glu)-Cys, and (h) Glu-Cys-His. From the figure, it can be deduced that the number of H-bonds to the catalytic dyads ranges from three to five, whereas the Asp-His or Glu-His distances range from 2.64 to 2.90 Å. On the basis of former analysis, sketched in Figure 6, the compensation energy for the neutral/charge-separated protonation states of the dyads is expected to be ∼25 kcal/mol, which is comparable with energy gain arising from the formation of five H-bonds. Hence, it can be inferred that the energy differences
between the neutral and charge-separated states of the Asp-His or Glu-His dyads would be largely compensated by the surrounding local H-bonding environments. The situation is similar in the case of esterases (Figure 8i and j). Although further calculations are clearly needed for the determination of accurate energetics, the present idea provides a tool for a posteriori analysis of X-ray structures of proteins. 3.4. Interconversion between Two Protonation States in the Bovine CcO Asp-His System. This section focuses on possible transition mechanisms between the charge-separated [Asp91-CγOO- and His503-Im-Nδ1H+] and the neutral protonation state [Asp91-CγOOH and His503-Im-Nδ1] in the bovine CcO Asp-His system. As mentioned, the large total energy difference (∼51 kcal/mol) between these states in the simplest Asp91-His503 system is reduced to only ∼5 kcal/mol via the energy compensation mechanism. This suggests that an incoming proton could be in a sort of bistable state characterized by two shallow minima, i.e., the Asp91 and His503 protonation sites, separated by an activation barrier of the order of ∼5-10 kcal/mol. Hence, changes of the order of one or two H-bonds could be sufficient to trigger local proton transfers between Asp91 and His503, which is an elementary step for the transition between the charge-separated and neutral protonation states. On the basis of the bovine CcO X-ray structure, there are at least two candidates suitable for such a trigger. The first one can be identified in the thermal fluctuations that the crystallographic H2O4 water molecule, H-bonded to the imidazole Nε2 of His503 (Scheme 1), can undergo. As stated above, the H2O4 water molecule is completely exposed to the bulk but is involved
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Figure 8. Schematic representations of active sites in (a-d) serine proteases, (f-h) cysteine proteases, and (i, j) esterases. Only the main surrounding amino acid residues and crystallographic water molecules around Asp (Glu)-His pairs are shown for the sake of clarity. The corresponding PDB IDs are given in Table 2. Two atoms satisfying the same H-bond criteria used in Figure 1 are connected by dashed lines.
in the H-bonding environment of His503. The Debye-Waller factors (B-factors) deduced from the X-ray data indicate a relatively high mobility of H2O4 at room temperature. The thermal fluctuations of H2O4 including water reorientations and translations could change its H-bond partners, thus possibly changing the His503 H-bonding environment. Indeed, a water molecule in the bulk can rotate in concert to reform a H-bond with the nearby water molecule, thus lowering the energy cost associated with the H-bond breaking.80,81 This molecular jump mechanism of water reorientation suggests that a H-bonding environment composed of the water molecules exposed to the bulk could easily undergo rearrangements of the number of H-bonds due to thermal effects. As illustrated in panels c and d of Figure 3, breaking the H-bond between H2O4 and His503 shifts the energy difference between the charge-separated and neutral protonation states by ∼4 kcal/mol. These suggest that the thermal fluctuation of H2O4 could be large enough to change the H-bonding environment of His503.
The second possibility for the triggering of proton transfer between Asp91 and His503 is the redox-coupled movement of His503. A comparison of the fully oxidized bovine CcO X-ray structure (Figure 1) with the fully reduced one (PDB code 2DYR12) shows that the imidazole plane of His503 rotates by 180°.11 This redox-coupled movement could break some Hbonds around the histidine, leading to a change of the nearby H-bonding environment and inducing compensatory effects on the energetics of the Asp91-His503 system. Further firstprinciples dynamical calculations are underway to elucidate these local proton transfer mechanisms and the effects of thermal fluctuation/redox-coupled His503 rotation. Changes in the local electrostatic potential caused by capture/ release of electrons at the redox-active metal sites could be another trigger for the proton transfer in the Asp91-His503 system. The nearest metal sites to Asp91-His503 are heme a and heme a3. In each case, the central Fe atom in each heme is located at a distance of ∼35 and ∼40 Å from Oδ2 of Asp91
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TABLE 2: Representatives of Enzymes Having Asp (Glu)-His Dyads as Catalytic Pairsa enzymes
PDB ID
resolution (Å)
d(His-Asp/Glu) (Å)
number of H-bonds
1QJ4 1YAS 2YAS 3YAS 4YAS 5YAS 6YAS 7YAS
1.10 1.90 1.72 1.85 2.00 2.20 2.20 1.75
average 2.79 2.67 2.84 2.68 2.85 2.84 2.84 2.78 2.79
5/6b 5 5 5 5 5 5 5b
2AGE 2AGI 2TGA 1BCS 1GCI 2H6M 1NLN 1AUG
1.15 1.14 1.80 2.08 0.78 1.40 1.60 2.00
2.75 2.75 2.90 2.83 2.64 2.66 2.63 2.79
5 4 3 5 4 3 5 3
1VXR 1MC2
2.20 0.85
2.64 2.77
3 4
78
HeVea brasiliensis hydroxynitrile lyases (a) (b) (c) (d) (e) (f) (g) (h) proteases79 (a) Bos taurus succinyl-AAPR-trypsin acyl-enzyme (b) Bos taurus leupeptin-trypsin covalent complex (c) Bos taurus trypsinogen (d) Triticum aestiVum serine carboxypeptidase II (e) Bacillus lentus subtilisin (f) Hepatitis A viral 3C proteinase (g) Human adenovirus 2 proteinase (h) Bacillus amyloliquefaciens pyroglutamyl peptidase I esterases (i) Torpedo californica acetylcholinesterase (j) Deinagkistrodon acutus phospholipase A2
a The corresponding X-ray structures are shown in Figures 7 and 8. b The corresponding PDB file has a multiconformer around the Asp-His dyad.
SCHEME 1
also in other systems from proteins90,91 to DNA.24,92 Work is now in progress to include both quantum effects and redox coupled proton transport mechanism in CcO. 4. Conclusions
and Nδ1 of His503, respectively. If we estimate the difference in the electrostatic potential between Asp91 and His503 by placing a point charge Q ) -1e at the iron site, the energy change turns out to be ∼1.2 kcal/mol per heme, by assuming a dielectric constant of ε ) 1. Using ε ) 1 provides an upper limit of the estimated value. However, the value is weak, comparable to a H-bond strength. These estimations suggest that it is necessary for a system to be energetically well-balanced in order to use such weak electrostatic potential difference as a trigger, as in the case of the CcO. Another important issue in proton transfer reactions is the importance of quantum effects, namely, proton tunneling and zero-point vibration corrections, with important isotope effect implication in the case of protonated and deuterated compounds.82 Quantum effects make the distribution of the proton broader in comparison with a classical treatment of the nuclei.29 Recently, Hammes-Schiffer has investigated the quantum nature of the proton transfer reactions in enzymes83 by means of molecular-orbital-based methods.84–87 The tunneling ratio exhibits ordinary and extraordinary isotope effects that depend strongly on the specific system considered. Recently, one of us performed an analysis of quantum effects on the multiple proton transfer reactions in DNA base pairs using cumulant-based methods.88,89 The results show that the zero-point vibration energy correction for the equilibrium geometries lifts the energy levels along the energy axis and that, at the same time, the energy barrier decreases. These effects seem to be rather important and coupled with the redox reaction. Thus, the global mechanism undergoing at the D-pathway entrance in CcO can be categorized as a proton-coupled electron transfer (PCET) and/ or electron-coupled proton transfer (ECPT) reactions, as reported
We have investigated the aspartate (Asp91) and histidine (His503) pair positioned at the entrance of the D-pathway in the bovine CcO by using a density-functional approach. We have found that an additional proton reaching the crystal water molecule (H2O207), which links Asp91 and His503, is readily transferred to either Asp91 or His503. This leads to the formation of either a neutral protonation state [Asp91-CγOOH and His503-Im-Nδ1] or a charge-separated protonation state [Asp91-CγOO- and His503-Im-Nδ1H+]. The energy difference between the two states is comparable with the typical value of H-bond strength, ∼5 kcal/mol, indicating that His503 is actively involved in the proton supply to the D-path, as well as Asp91, and acts as a proton acceptor. The H-bond environment around Asp91 and His503 has effects on both the energetics and the electronic structure of the system and can account for the reduction in the Coulomb-energy difference between these two protonation states with respect to minimal models neglecting the local H-bond environment. A dependence on the Asp91His503 distance has also been evidenced and linked to the X-ray structure of the bovine CcO. The energy compensation mechanism that we have found in this study balances the energetics of Asp-His pairs, allowing for a fine regulation of the protonatable amino acid residues. The present studies indicate that the entrance part of the D-path is an energetically wellbalanced system, where protonatable amino acid residues are allowed to take several protonation states just by changes of a few H-bonds. Acknowledgment. We gratefully acknowledge fruitful discussions with T. Tanaka, K. Muramoto, S. Nakashima, Y. Takano, H. Shimada, T. Tsukihara, and S. Yoshikawa. Computations were performed on the computer facilities at Institute for Solid-State Physics, University of Tokyo, at Research Center for Computational Science, Okazaki Research Facilities, National Institutes of Natural Sciences, and at Picobiology Institute,
Energy Compensation Mechanism Graduate School of Life Science, University of Hyogo. This work was supported by the subsidy for scientific research, University of Hyogo. This research is also supported by a Grantin-Aid for Core Research for Evolutional Science and Technology from the Japan Science and Technology Agency, by a Grant-in-Aid for the scientific research on priority area “Molecular theory for real systems” (No. 20038008), and Grant-inAid for Young Scientists (B) (No. 20750004). Supporting Information Available: Tables of total energy differences in the Asp91-His503 systems between the chargeseparated and neutral protonation states calculated by different boundary conditions and exchange-correlation functionals, total energies and optimized geometries (Cartesian coordinates) of Asp91-H3O+-His503 system, and coordinates and total energy differences between the charged-separated and neutral protonation states of different H-bonding environment models. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Decoursey, T. E. Physiol. ReV. 2003, 83, 476–579. (2) Agmon, N. Chem. Phys. Lett. 1995, 244, 456–462. (3) Voet, D.; Voet, J. G. Biochemistry, 3rd ed.; John Wiley and Sons: Hoboken, NJ, 2004. (4) Raines, R. T. Chem. ReV. 1998, 98, 1045–1065. (5) Zhao, L.; Liao, H.; Tsai, M. D. J. Biol. Chem. 2004, 279, 31995– 32000. (6) Li, Y.; Tsai, M. D. J. Am. Chem. Soc. 1993, 115, 8523–8526. (7) Cosgrove, M. S.; Gover, S.; Naylor, C. E.; Vandeputte-Rutten, L.; Adams, M. J.; Levy, H. R. Biochemistry 2000, 39, 15002–15011. (8) Laio, D. I.; Breddam, K.; Sweet, R. M.; Bullock, T.; Remington, S. J. Biochemistry 1992, 31, 9796–9812. (9) Goodin, D. B.; McRee, D. E. Biochemistry 1993, 32, 3313–3324. (10) Tsukihara, T.; Shimokata, K.; Katayama, Y.; Shimada, H.; Muramoto, K.; Aoyama, H.; Mochizuki, M.; Shinzawa-Itoh, K.; Yamashita, E.; Yao, M.; Ishimura, Y.; Yoshikawa, S. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 15304–15309. (11) (a) Muramoto, K.; Hirata, K.; Shinzawa-Itoh, K.; Yoko-o, S.; Yamashita, E.; Aoyama, H.; Tsukihara, T.; Yoshikawa, S. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 7881–7886. (b) The RCSB Protein Data Bank (PDB: http://www.pdb.org/), PDB ID 2eij. (12) Shinzawa-Itoh, K.; Aoyama, H.; Muramoto, K.; Terada, H.; Kurauchi, T.; Tadehara, Y.; Yamasaki, A.; Sugimura, T.; Kurono, S.; Tsujimoto, K.; Mizushima, T.; Yamashita, E.; Tsukihara, T.; Yoshikawa, S. EMBO J. 2007, 26, 1713–1725. (13) Ostermeir, C.; Harrenga, A.; Ermler, U.; Michel, H. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 10547–10553. (14) Svensson-Ek, M.; Abramson, J.; Larsson, G.; To¨rnroth, S.; Brzezinski, P.; Iwata, S. J. Mol. Biol. 2002, 321, 329–339. (15) Gennis, R. B. Biochim. Biophys. Acta 1998, 1365, 241–248. (16) Hohenberg, P.; Kohn, W. Phys. ReV. 1964, 136, B864-B871. (17) Kohn, W.; Sham, L. J. Phys. ReV. 1965, 140, A1133-A1138. (18) Gervasio, F. L.; Carloni, P.; Parrinello, M. Phys. ReV. Lett. 2002, 89, 108102. (19) Piana, S.; Sebasitani, D.; Carloni, P.; Parrinello, M. J. Am. Chem. Soc. 2001, 123, 8730–8737. (20) von Lilienfeld, O. A.; Tavernelli, I.; Ro¨thlisberger, U.; Sebastiani, D. J. Chem. Phys. 2005, 122, 014133. (21) Boero, M.; Terakura, K.; Tateno, M. J. Am. Chem. Soc. 2002, 124, 8949–8957. (22) Carloni, P.; Ro¨thlisberger, U.; Parrinello, M. Acc. Chem. Res. 2002, 35, 455–464. (23) Boero, M.; Tateno, M.; Terakura, K.; Oshiyama, A. J. Chem. Theory Comput. 2005, 1, 925–934. (24) Gervasio, F. L.; Boero, M.; Parrinello, M. Angew. Chem., Int. Ed. 2006, 45, 5606–5609. (25) Boero, M.; Ikeda, T.; Ito, E.; Terakura, K. J. Am. Chem. Soc. 2006, 128, 16798–16807. (26) Dal Peraro, M.; Ruggerone, P.; Raugei, S.; Gervasio, F. L.; Carloni, P. Curr. Opin. Struct. Biol. 2007, 17, 149–156. (27) Grossman, J. C.; Schwegler, E.; Draeger, E. W.; Gygi, F.; Galli, G. J. Chem. Phys. 2004, 120, 300–311. (28) Ferna´ndez-Serra, M. V.; Artacho, E. Phys. ReV. Lett. 2006, 96, 016404. (29) Marx, D. ChemPhysChem 2006, 7, 1848–1870.
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