Average and extreme multiatom Van der Waals interactions: Strong coupling of multiatom Van der Waals interactions with covalent bonding
 Alexei V Finkelstein^{1}Email author
DOI: 10.1186/1752153X121
© Finkelstein et al 2007
Received: 30 March 2007
Accepted: 30 July 2007
Published: 30 July 2007
Abstract
Background
The prediction of ligand binding or protein structure requires very accurate force field potentials – even small errors in force field potentials can make a 'wrong' structure (from the billions possible) more stable than the single, 'correct' one. However, despite huge efforts to optimize them, currentlyused allatom force fields are still not able, in a vast majority of cases, even to keep a protein molecule in its native conformation in the course of molecular dynamics simulations or to bring an approximate, homologybased model of protein structure closer to its native conformation.
Results
A strict analysis shows that a specific coupling of multiatom Van der Waals interactions with covalent bonding can, in extreme cases, increase (or decrease) the interaction energy by about 20–40% at certain angles between the direction of interaction and the covalent bond. It is also shown that on average multibody effects decrease the total Van der Waals energy in proportion to the square root of the electronic component of dielectric permittivity corresponding to dipoledipole interactions at small distances, where Van der Waals interactions take place.
Conclusion
The study shows that currentlyignored multiatom Van der Waals interactions can, in certain instances, lead to significant energy effects, comparable to those caused by the replacement of atoms (for instance, C by N) in conventional pairwise Van der Waals interactions.
Background
Van der Waals (VdW) forces, which are very important for the structure and interactions of biological molecules, are usually treated as a simple sum of pairwise interatomic interactions even in dense systems like proteins [1–5]. However, multiatom VdW interactions are usually ignored. This seems to follow the AxilrodTeller theory [6] which predicts a drastic (stronger than for pairwise interactions) decrease of threeatom interactions with distance; and indeed, detailed computations of singleatom liquids [7] and solids [8, 9] show that MB (multibody) effects amount to only ~5% of the total energy. However, this work shows that multiatom VdW interactions can become quite large in the presence of covalent bonds. This finding, which equally concerns atomic interactions in biological molecules and solvents, implies a necessity to revise the allatom force fields currently used.
Results and Discussion
Theory
here θ_{ ij }/r_{ ij }^{3} = (δ  3n_{ ij }⊗ n_{ ij })/r_{ ij }^{3} is the usual dipoledipole interaction tensor (where δ is the 3D unit matrix, n_{ ij }⊗ n_{ ij }the tensor product of vectors n_{ ij }= r_{ ij }/r_{ ij }, and r_{ ij }= r_{ i } r_{ j }; i ≠ j). As is usually done (see Refs. [6, 10–12]), relatively week quadrupledipole and so on interactions of oscillators are ignored in addition to possible inharmoniousness of the oscillators.
Energy of multibody VdW interactions
In general, one can compute all 3n eigenvalues (Ω_{ i })^{2} of 3n × 3n matrix B(and thus eigenvalues Ω_{ i }of B^{1/2} and their sum Sp [B^{1/2}]) in a time proportional to (3n)^{3}. Computationally, this solves a problem of exact calculation of VdW dispersion forces for any system of polarizable dipoles.
However, to get a physical understanding of the main terms contributing to these forces, one has to consider the main terms of Sp[B^{1/2}].
where matrix B_{0} corresponds to uncoupled oscillators and ΔB to weak coupling of the oscillators. Now let us consider an auxiliary matrix B λ = B_{0} + λΔB (where λ is a small multiplier), and present its square root, ${B}_{\lambda}^{1/2}$, as a series ${B}_{\lambda}^{1/2}={B}_{0}^{1/2}+\lambda {Z}_{1}+{\lambda}^{2}{Z}_{2}+\mathrm{...}$, where ${B}_{0}^{1/2}$ is a diagonal matrix with 3D blocks (${B}_{0}^{1/2}$)_{ ij }= ω_{ i }δ if i = j, and (${B}_{0}^{1/2}$)_{ ij }= 0 if i≠j, and matrices Z_{1}, Z_{2},... have to be calculated.
This system can be solved recursively: equation MZ + ZM = Y (where matrices consist of equalsize blocks (M)_{ ij }, (Z)_{ ij }, (Y)_{ ij }, i, j = 1, ..., n) unambiguously determines Z for any Y when M = ω_{ i }δ_{ ij }δ is a positively determined diagonal matrix: ${(MZ+ZM)}_{ij}={\displaystyle \sum _{p}\left[{\omega}_{i}{\delta}_{ip}\delta \cdot {(Z)}_{pj}+{(Z)}_{ip}\cdot {\omega}_{p}{\delta}_{pj}\delta \right]}=\text{[}{\omega}_{i}+{\omega}_{j}]{(Z)}_{ij}$, and from [ω_{ i }+ ω_{ j }](Z)_{ ij }= (Y)_{ ij }= (Y)_{ ij }we have (Z)_{ ij }= (Y)_{ ij }/(ω_{ i }+ ω_{ j }).
can be presented as a sum of pairwise, triple, quadruple, etc. interactions (the term $\text{Sp[}{Z}_{1}]={\displaystyle \sum _{i}\text{Sp[}{h}_{ii}]}$ is equal to 0, since all diagonal blocks h_{ ii }≡ 0).
is a convenient dimensionless parameter for the interaction of oscillators i and k, and α_{ i }= e^{2}/(m_{ i }${\omega}_{i}^{2}$) is the electronic polarizability of atom i.
Thus, a triple interaction can be both repulsive and attractive: attractive, when atoms i, k, p stay nearly along a line; repulsive, when these atoms form an acute or rightangled triangle [6].
where ${\Gamma}_{ikpq}=\frac{\hslash}{\text{16}}\left({\omega}_{i}+{\omega}_{k}+{\omega}_{p}+{\omega}_{q}+\frac{{\omega}_{k}{\omega}_{q}}{{\omega}_{i}+{\omega}_{p}}+\frac{{\omega}_{i}{\omega}_{p}}{{\omega}_{k}+{\omega}_{q}}\right)\cdot {\gamma}_{ik}{\gamma}_{kp}{\gamma}_{pq}{\gamma}_{qi}\cdot \left(\text{Sp[}{\theta}_{ik}{\theta}_{kp}{\theta}_{pq}{\theta}_{qi}\text{]}\right)$.
In twoatom quadruple terms Γ_{ ikik }, Sp[θ_{ ik }θ_{ ki }θ_{ ik }θ_{ ki }] = 18. In threeatom quadruple terms like Γ_{ ikip },
Sp[θ_{ ik }θ_{ ki }θ_{ ik }θ_{ pi }] = 9  9cos^{2}φ_{ i }
is always negative (Fig. 1b), i.e., neighbor k of atom i increases attraction of i to p, and neighbor p of i increases attraction of i to k. In fouratom quadruple terms Γ_{ ikpq }, the value of Sp[θ_{ ik }θ_{ kp }θ_{ kp }θ_{ qi }] varies from 18 to +9 depending on the mutual arrangement all four atoms.
Microscopic dielectric permittivity in a system of oscillators
when the "medium" consists of oscillators described at a microscopic level.
Defined thus, the scalar ε_{ ab }coincides with the conventional permittivity in a uniform macroscopic medium. When the "medium" consists of oscillators described at a microscopic level, ε_{ ab }corresponds to the dielectric permittivity for the interaction of dipoles a and b via the polarizable environment.
If the sum is taken over the oscillating electrons only (while the nuclei positions are fixed), ε_{ ab }is an electronic component of dielectric permittivity.
Multiatom VdW interactions are important in the presence of covalent bonds
Equations [10] – [14] show that energies of pairwise, triple and quadruple interactions are proportional to ~$\hslash $ω·γ_{ ik }γ_{ ki }, ~$\hslash $ω·γ_{ ik }γ_{ kp }γ_{ pi }and ~$\hslash $ω·γ_{ ik }γ_{ kp }γ_{ pq }γ_{ qi }, respectively, where $\hslash $ω is an oscillator excitation energy. The γ values are small for nonbonded contacts. Indeed, $\frac{\sqrt{{\omega}_{i}{\omega}_{k}}}{{\omega}_{i}+{\omega}_{k}}in{\gamma}_{ik}=\frac{\sqrt{{\omega}_{i}{\omega}_{k}}}{{\omega}_{i}+{\omega}_{k}}\cdot \left(\frac{\sqrt{{\alpha}_{i}{\alpha}_{k}}}{{\text{r}}_{ik}{}^{3}}\right)$ is normally close to 1/2, because ω_{ i }and ω_{ k }are of equal order of magnitude [13], and the α values are about 1 Å^{3} for atoms typical in proteins (0.40 Å^{3} for H atoms, 0.69 – 0.85 Å^{3} for O atoms, 0.90–0.97 Å^{3} for N atoms, 1.03 – 1.32 Å^{3} for C atoms [13]). Thus, the dimension less value γ is about 0.02 for the closest nonbonded contact of atoms (r ≈ 2.2–3.4 Å for atoms H, O, N, C [13]).
The very small value of γ at "nonbonded" distances means that triple, quadruple, etc., nonbonded atomatom contacts are very weak compared to those that are pairwise.
However, the situation changes when the γ factor involves two atoms connected by a covalent bond. The atomatom distance in this case is at least two times smaller than that for the closest nonbonded contact. Thus, the γ factor for covalently bonded atoms is about tenfold (see Eq. [11]) larger than the γ factor for the closest nonbonded contact of atoms. In this case, the energy of highorder interactions can approach that of a pairwise interaction.
In the case of the collinear arrangement of atoms and bonds the attraction can grow by 70% (compare values in columns "configuration and its energy" and "pairwise" in Fig. 2), while in the case of the orthogonal arrangement of atoms and bonds, the attraction can decrease by 6%. The decrease is less than the increase because, as mentioned above, the quadruple interaction is mainly attractive. The main part of this attraction is due to the threeatom quadruple interaction. This can be divided in two parts: Equation [15] can be presented as Sp[θ_{ ik }θ_{ ki }θ_{ ip }θ_{ pi }] = 12 + (9cos^{2}φ_{ i } 3), where the first (larger) term is an orientationindependent constant (actually, it simply increases the VdW attraction of any atom involved in covalent bonding to all other atoms, see Fig. 1b; in available force fields [1–5] this term is implicitly taken into account by ascribing and fitting different pairwise attraction forces to atoms in different covalent states). The second, the orientationdependent part equals zero if averaged in 3D space over all possible atomtobond orientations (see examples in Fig. 2); this part is rather similar to that for the interaction of a separate atom with a remote covalent bond (cf. Fig.1b to Fig. 1a).
The orientationdependent terms, which mainly arise from triple interactions, can increase or decrease VdW energy by ~±20–40% at extremes. This energy effect is significant, being comparable to that caused by the replacement of an N atom (α_{ N }≈ 0.95 Å^{3}) by a C (α_{ C }≈ 1.2 Å^{3}) or O (α_{ O }≈ 0.75 Å^{3}) atom in a pairwise VdW interaction (see equations [10,11] at ω_{ N }≈ω_{ C }≈ω_{ O }). Moreover, the orientation effect is much stronger than the effect of replacing the pairwise energies of two CN nonbonded contacts with the sum of pairwise energies of CC and NN contacts (α_{ C }α_{ C }+ α_{ N }α_{ N } 2α_{ C }α_{ N }≈ 5% × α_{ C }α_{ N }). Note that the contact replacements are the main VdW effects currently used to distinguish ''correct'' protein folds from those that are ''wrong''.
It is necessary to note that at short distances, which are the most important for VdW interactions, one can expect effects of inharmonicity, higher multipole interactions, etc. [11, 12]. These effects are avoided in the "harmonic oscillator" model used in this work. Nevertheless, they exist in reality, and their existence makes the expressions obtained for manybody interactions approximate to the same extent as the conventional London's expression is for pairwise interactions. In particular, the value of γ for each covalent bond may be treated as a parameter, the value of which should be obtained from a fit of the experimental data in the same way as the London's pairwise interaction energy [11, 13, 14]. The necessity of fitting the experimental data is underlined by a relatively slow convergence of both the orientationdependent and orientationindependent series in Figure 2.
"VdW permittivity"
VdW forces and the electronic part of dielectric permittivity are evidently connected: both originate from dipoledipole interactions. The relationship between VdW forces and the macroscopic dielectric permittivity of the medium is well studied in the continuum approximation [11, 14, 16]. However, this approach is useful and relatively simple when applied to large uniform bodies, like drops or layers. This work concerns interactions of small fragments composed of various atoms, and the frequencydependent dielectric permeability, taken as a macroscopic characteristic, seems not to be an appropriate tool to investigate this case. Here we will consider an electronic part of the microscopic dielectric permittivity created by and acting at interacting harmonic oscillators, with the goal of demonstrating that multiatom interaction creates a kind of "VdW permittivity" for pairwise VdW forces in addition to elucidating a connection between VdW forces and the electronic part of microscopic dielectric permittivity.
The role of dielectric permittivity in VdW forces is ambiguous. On the one hand, the VdW interaction of two atoms is proportional to the square of the electrostatic interaction between their fluctuating electronic polarizations. Thus, one might expect the pairwise VdW interaction to be inversely proportional to the square of the electronic part of the dielectric permittivity. On the other hand, the medium's electrons (which create the electronic component of dielectric permittivity) are involved in VdW interactions of "their own" atoms, and it is not clear if they are "free" enough to influence the VdW interactions of the other atoms in such a strong manner.
Thus, on average, MB effects decrease the pairwise VdW energy roughly in proportion to the square root of the electronic part of the microscopic dielectric permittivity for the dipoledipole interaction via a polarizable atomic environment.
The main effect is caused by the electronic permittivity, which is pertinent to small, atomic distances, where the main VdW interaction takes place. One can expect that the electronic component of the microscopic dielectric permittivity for interacting dipoles at these distances is significantly smaller than the electronic dielectric permittivity at macroscopic distances: the decrease in VdW energy caused by threeatom interaction in liquid argon [7] is approximately fivefold less than that expected the from its macroscopic permittivity.
Conclusion
Ligand binding and protein structure prediction require especially accurate force field potentials, because one "correct" structure struggles against billions of "wrong" ones [17]. Despite huge efforts to optimize them, the force fields currently used are still not able, in a vast majority of cases, to bring an approximate, homologybased model of protein structure (whose atoms usually deviate by only 1.5 – 3 Å from their native positions) closer to its native conformation [18–22]. At present, even the most successful methods (one or two of the five dozen methods used) show some improvement of homology models by a force fieldbased refinement in only a half of cases [21–23]. This shows that a "significant difficulty still exists in both sampling and force field accuracy" [23]. One of the force field problems is that which concerns the "not well captured" balance between intramolecular and solvent dispersion (i.e., VdW) interactions [23]. On the other hand, a partial success of modern molecular mechanics methods in the prediction of detailed protein structure shows that (i) modern, improved methods of sampling of protein chain conformations seem to work (possibly, at the cost of averaging that decreases the effect of energy errors [17]) in some, but not all, cases (especially when a nearnative conformation basin is reached from homology modeling), and (ii) current force field quality is sufficient in some, but not all, cases, so that an increase in the accuracy of the force field seems to be crucial for the final success of protein structure prediction methods.
This study shows that currentlyignored multiatom VdW interactions can be important for the further development of force fields. The coupling of covalent bonding with multiatom VdW interactions is not the only important MB effect. Some of those, like MB electrostatic interactions, are taken into account by the existing force fields (see [24, 25]). Here the effect is caused by the interaction of the permanent charges or permanent molecular multipoles with particles of polarizable media. The other interactions, which include interaction of permanent multipoles with induced ones [26], are not yet taken into account by existing force fields.
The discovered specific coupling of covalent bonding with multiatom VdW interactions does not need permanent charges or multipoles, and therefore involves the multitude of covalent bonds in molecules. Multiatom interaction with any of the bonds can lead to a significant energy effect, which is comparable to that caused by the replacement of atoms in conventional pairwise VdW interactions. As a result, the effect described can be rather significant in choosing the molecular shape.
Abbreviations
 3D:

threedimensional
 VdW:

Van der Waals
 MB:

multibody
 CASP:

Critical Assessment of Techniques for Protein Structure Prediction meeting
Declarations
Acknowledgements
I am grateful to D. Reifsnyder for assistance in the preparation of this paper, to A.G. Donchev for assistance in computations and to A.M. Dykhne, A.A. Vedenov, R.V. Polozov, M. Levitt and G. Vriend for seminal discussions. This work was supported in part by the Russian Foundation for Basic Research and Russian programs "Molecular and Chemical Biology" and "Leading Scientific Schools", INTAS, IHES and an International Research Scholar's Award from the Howard Hughes Medical Institute.
Authors’ Affiliations
References
 Levitt M, Hirshberg M, Sharon R, Dagget V: Potential energy function and parameters for simulations of the molecular dynamics of proteins and nucleic acids in solution. Comp Phys Commun. 1995, 91: 215231. 10.1016/00104655(95)00049L.View ArticleGoogle Scholar
 MacKerell A, WiorkiewiczKuczera J, Karplus M: An allatom empirical energy function for the simulation of nucleic acids. J Am Chem Soc. 1995, 117: 1194611975. 10.1021/ja00153a017.View ArticleGoogle Scholar
 Jorgensen WL, Maxwell DS, TiradoRives J: Development and testing of the opls allatom force field on conformational energetics and properties of organic liquids. J Am Chem Soc. 1996, 118: 1122511236. 10.1021/ja9621760.View ArticleGoogle Scholar
 Halgren TA: Merck molecular force field. I. Basis, form, scope, parameterization and performance of MMFF94. J Comp Chem. 1995, 17: 490519. 10.1002/(SICI)1096987X(199604)17:5/6<490::AIDJCC1>3.0.CO;2P.View ArticleGoogle Scholar
 Wang J, Wolf RM, Caldwell JW, Kollman PA, Case DA: Development and testing of a general amber force field. J Comp Chem. 2004, 25: 11571174. 10.1002/jcc.20035.View ArticleGoogle Scholar
 Axilrod BM, Teller E: Interaction of the van der Waals' type between three atoms. J Chem Phys. 1943, 11: 299300. 10.1063/1.1723844.View ArticleGoogle Scholar
 Sadus RJ: Exact calculation of the effect of threebody AxilrodTeller interactions on vapourliquid phase coexistence. Fluid Phase Equilibria. 1998, 144: 351360. 10.1016/S03783812(97)002793.View ArticleGoogle Scholar
 MacRury TB, Linder B: Manybody aspects of physical adsorption. J Chem Phys. 1971, 54: 20562966. 10.1063/1.1675136.View ArticleGoogle Scholar
 Donchev AG: Manybody effects of dispersion interaction. J Chem Phys. 2006, 125: 07471310.1063/1.2337283.View ArticleGoogle Scholar
 London F: The general theory of molecular forces. Trans Faraday Soc. 1937, 33: 826. 10.1039/tf937330008b.View ArticleGoogle Scholar
 Barash YS: Van der Waals Forces. 1988, Moscow: Nauka, chapters 2, 5:Google Scholar
 Landau LD, Lifshitz EM: Quantum Mechanics. 1987, New York: Pergamon, §23:Google Scholar
 Pauling L: General chemistry. 1970, New York: W.H.Freeman & Co, chapters 6, 11:Google Scholar
 Parsegian AV: VdW Forces. 2005, Cambridge: Cambridge Univ. Press, pp.429, 214227, 241260Google Scholar
 Basharov MA, Vol'kenshtein MV, Golovanov IB, Ermakov GL, Nauchitel' VV, Sobolev VM: Bondbond interactions. I. Simple relationship for estimating the energy of a bondbond interaction. J Struct Chem (Moscow). 1984, 25: 2635. 10.1007/BF00808546.View ArticleGoogle Scholar
 Dzyaloshinskii IE, Lifshitz EM, Pitaevskii LP: General theory of Van der Waals' forces. Sov Phys Uspekhi. 1961, 4: 153176. 10.1070/PU1961v004n02ABEH003330.View ArticleGoogle Scholar
 Finkelstein AV, Gutin AM, Badretdinov AY: Perfect temperature for protein structure prediction and folding. Proteins. 1995, 23: 151162. 10.1002/prot.340230205.View ArticleGoogle Scholar
 Krieger E, Darden T, Nabuurs SB, Finkelstein A, Vriend G: Making optimal use of empirical energy functions: force field parameterization in crystal space. Proteins. 2004, 57: 678683. 10.1002/prot.20251.View ArticleGoogle Scholar
 Vincent JJ, Tai CH, Sathyanarayana BK, Lee B: Assessment of CASP6 predictions for new and nearly new fold targets. Proteins. 2005, 6783. 10.1002/prot.20722. Suppl 7
 Clarke N: Free modeling assessment. 2006, [http://predictioncenter.gc.ucdavis.edu/casp7/meeting/presentations/Presentations_assessors/CASP7_FM_Clarke.pdf]Google Scholar
 Read RJ: Assessing high accuracy models. 2006, [http://predictioncenter.gc.ucdavis.edu/casp7/meeting/presentations/Presentations_assessors/CASP7_HA_RJRead.pdf]Google Scholar
 Schwede T: Assessment of template based modeling (TMB). 2006, [http://predictioncenter.gc.ucdavis.edu/casp7/meeting/presentations/Presentations_assessors/CASP7_TBM_Schwede.pdf]Google Scholar
 Chen J, Brooks CL: Can molecular dynamics simulations provide highresolution refinement of protein structure?. Proteins. 2007, 67: 922930. 10.1002/prot.21345.View ArticleGoogle Scholar
 Honig B, Nicholls A: Classical electrostatics in biology and chemistry. Science. 1995, 268: 11441149. 10.1126/science.7761829.View ArticleGoogle Scholar
 Simonson T, Archontis G, Karplus M: Free energy simulations come of age: proteinligand recognition. Acc Chem Res. 2002, 35: 430437. 10.1021/ar010030m.View ArticleGoogle Scholar
 Ernesti A, Hutson JM: Nonadditive intermolecular forces from spectroscopy of van der Waals trimers: A theoretical study of Ar_{2}HF. Phys Rev A. 1995, 51: 239250. 10.1103/PhysRevA.51.239.View ArticleGoogle Scholar
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