Kernel energy method applied to vesicular stomatitisvirus nucleoproteinLulu Huanga, Lou Massab, and Jerome Karlea,1

aNaval Research Laboratory, 4555 Overlook Avenue, S.W., Washington, DC 20375; and bDepartment of Chemistry, Hunter College and the Graduate Center,City University of New York, New York, NY 10065

Contributed by Jerome Karle, November 24, 2008 (sent for review October 10, 2008)

The kernel energy method (KEM) is applied to the vesicular sto-matitis virus (VSV) nucleoprotein (PDB ID code 2QVJ). The calcula-tions employ atomic coordinates from the crystal structure at 2.8-Åresolution, except for the hydrogen atoms, whose positions weremodeled by using the computer program HYPERCHEM. The calcu-lated KEM ab initio limited basis Hartree-Fock energy for the full33,175 atom molecule (including hydrogen atoms) is obtained. Inthe KEM, a full biological molecule is represented by smaller‘‘kernels’’ of atoms, greatly simplifying the calculations. Collectionsof kernels are well suited for parallel computation. VSV consists offive similar chains, and we obtain the energy of each chain.Interchain hydrogen bonds contribute to the interaction energybetween the chains. These hydrogen bond energies are calculatedin Hartree-Fock (HF) and Møller-Plesset perturbation theory tosecond order (MP2) approximations by using 6–31G** basis orbit-als. The correlation energy, included in MP2, is a significant factorin the interchain hydrogen bond energies.

Hartree-Fock � KEM � Møller-Plesset � quantum mechanics

The kernel energy method (KEM) combines structural crys-tallographic information with quantum-mechanical theory,

and is of practical use in the calculation of molecular interactionenergies, which is otherwise a challenging problem for largemolecular targets such as the vesicular stomatitis virus (VSV)nucleoprotein. This article focuses on calculations of total energyfor the entire molecule and the hydrogen bond interactionenergies between the chains of VSV which make up the fullnucleoprotein, whose PDB ID code is 2QVJ (1).

The KEM here used determines the quantum mechanicalmolecular energy by the use of parts of a whole molecule, whichare referred to as kernels, as in supporting information (SI) Fig.S1. Because the kernels are much smaller than a full biologicalmolecule, the calculations of kernels and double kernels arepracticable. Subsequently, kernel contributions are summed in amanner affording an estimate of the energy for the wholemolecule. Thus, the task of obtaining a quantum mechanicalenergy is simplified for large biological molecules. The compu-tational time is much reduced by using the KEM, and theaccuracy obtained appears to be quite satisfactory, as shown inprevious work (2–8).

As the crystal structure is known for 2QVJ under study, themolecule may be mathematically broken into the tractable piecescalled kernels. The kernels are chosen such that each atomoccurs in only one kernel. Only kernels and double kernels areused for all quantum calculations. The total molecular energy isreconstructed there from, by summation over the contributionsof the double kernels reduced by those of any single kernels thathave been over counted.

If all double kernels are included, the total energy is,

Entotal � �

1�i�j�n

Eij � �n � 2� �1�i�n

Ei [1]

where Eij � energy of a double kernel of name ij, Ei � energyof a single kernel of name i, i, j � running indices, and n �number of single kernels.

In this article we obtain the total molecular energy of 2QVJand all of its interchain hydrogen-bond interaction energies.Such fundamental information can be used to understand thebasis of the VSV nucleoprotein stability which is essential to itsstructural contribution in the encapsidation of the viral RNAwhich it envelopes in vivo.

The definition of the interaction energy between any pair ofkernels is:

Iij � Eij � Ei � Ej [2]

where, the subscript indices name the pair of kernels in question,Iij is the pair interaction energy, Eij is the energy of a doublekernel, and Ei and Ej are the energies of a single kernel. The signof the interaction energy, Iij, indicates whether the kernels i andj attract (negative I) or repel (positive I).

Knowledge of the list of the hydrogen bond interactionenergies is an aid to understanding their contribution to the VSVnucleoprotein structural stability. Moreover, it would be ex-tremely difficult to obtain by experimental methods the hydro-gen-bond interaction energies that flow naturally from imple-mentation of the KEM to the problem.

The crystal structure of 2QVJ is known (1). The original refer-ence 1 contains the crystal structure analysis, and gives the 2.80-Åresolution, the R factor magnitude as Rcryst � 0.243, root meansquare deviations of bonds (Å) equal to 0.006, and angles (°) equalto 0.95. As indicated in the reference 1, in-place model rebuildingwas carried out followed by additional manual model building andreal-space refinement. The crystal structure strongly suggests thatintermolecular contacts among the chains are critical for encapsu-lation of viral RNA, but the energetic magnitude of these interac-tions does not follow directly from the crystallography. However,the hydrogen bond energies between the chains do follow from theKEM calculations applied to 2QVJ using the crystal’s atomiccoordinates.

ResultsTotal Energy Calculation. The 2QVJ molecule, Fig. S2 A is com-posed of five chains (A–E), one of which (chain A) is shown asa ball and stick representation in Fig. S2B. Each chain has 421residues, 6,635 atoms, and carries a charge of �3. The entiremolecule contains 33,175 atoms. Each of the five chains isdivided into 66 kernels, each of whose amino acid sequence isshown in Fig. S3; we have altogether 66 � 5 � 330 kernels,contained in the entire molecule composed of five chains. In Fig.S3 each of the amino acids is enumerated from 2 to 422. The

Author contributions: L.H., L.M., and J.K. designed research; L.H., L.M., and J.K. performedresearch; and L.H., L.M., and J.K. wrote the paper.

The authors declare no conflict of interest.

1To whom correspondence should be addressed at: Naval Research Laboratory, Code 6030,4555 Overlook Avenue, S.W., Washington DC 20375. E-mail: jerome.karle@nrl.navy.mil.

This article contains supporting information online at www.pnas.org/cgi/content/full/0811959106/DCSupplemental.

© 2009 by The National Academy of Sciences of the USA

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demarcations between kernels are indicated as vertical blacklines.

In Table S1, each of the 66 kernels associated with a singlechain is enumerated, and for each kernel there is given thenumber of its residues, the numerical identity of the amino acidsthat define the kernel, and the amino acid sequence associatedwith the kernel.

To calculate the energies of the title molecule of this paper andeach of its five subchains, we have used the atomic coordinatesof the crystal structure which has been solved for the 2QVJmolecule. However, the crystal structure does not deliver thehydrogen atom coordinates. These have been added automati-cally to the non-H atoms of the crystal structure using theprocedures of the computer program HYPERCHEM. Theamino acid sequence, which defines each of five chains that makeup the full molecule, are exactly the same. But the relativepositions occupied by each of the chains in the full molecule willdiffer, and this in turn will affect slightly the automatic place-ment of hydrogen atoms from chain to chain. As a result theenergy of the chains will differ slightly, from one to another. InTable 1 we list the limited basis (STO-3G) (9) Hartree-Fock (10,11) energies, calculated using Eq. 1, for each of the five chains(A–E) that make up the full molecule.

Eq. 1 is also applied to calculate the total energy of the fullmolecule 2QVJ, and that result is �825,954.57 [a.u.].

Hydrogen Bond Calculations. The interface statistics betweenchains, as shown in Fig. S4, are important in studies of bio-molecules, and among the most interesting interchain interac-tions are those associated with the hydrogen bonds. All of the

interface statistics between chains for 2QVJ molecule weredetermined by use of the computer program HBAT 1.0 (12).Table S2 lists the number of hydrogen bonds and the number ofnon-bonded contacts between chains. The third column in thetable indicates the residue pairs including all hydrogen bondswhich are the focus of these calculations. We determined all suchhydrogen bond interactions between the chains and calculatedtheir magnitudes by using Eq. 2. The figures below give arepresentation of each of the hydrogen bonds between residues,and the tables below deliver the energy magnitudes whichcorrespond to the pictures. The table entries and the figures maybe correlated according to the uniform interaction nomenclatureused in both. The letters used name a particular chain, and thenumbers indicate a residue in the chain. For example, thedesignation A7-B256 refers to residue of number 7 attached tochain A, and residue of number 256 attached to chain B. Insubsequent sections we calculate all of the hydrogen bondinteraction energies which exist, based on distance criteria,between the various chain pairs. All of the hydrogen bond

Fig. 1. Hydrogen bonds between residues of chain A and chain B.

Table 1. Energies for individual chains

Chain A B C D E

Energies in �a.u. �165,191.04 �165,192.19 �165,189.29 �165,193.27 �165,189.20

Table 2. The interaction energies of residue pairs between chainA and chain B

Residue no., residue name Figure EMP2 �a.u. EHF �a.u.

A7-B256, ARG-ASP 1A �13.9218 �12.9576

A17-B262, LYS-MET 1B �6.9953 �0.0996

A17-B264, LYS-PRO 1C �11.1624 �9.6337

A250-B345, LEU-ALA 1D �6.7089 �2.8166

A326-B343, SER-ASP 1E �1.1593 3.0751

A326-B373, SER-ARG 1F �0.0104 2.1006

A383-B354, GLU-LYS 1G �17.1379 �14.6983

A383-B356, GLU-THR 1H �6.9593 �3.4514

A387-B342, ARG-ALA 1 I �13.5646 �12.6138

A387-B341, ARG-SER 1 J �11.9558 �8.6722

A387-B371, ARG-GLN 1K �11.3254 �10.0584

A418-B403, SER-SER 1L �8.2930 �5.9281

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calculations are reported by using 6–31G** (14–17) basis func-tions in both the Hartree-Fock (11, 12) and MP2 approximations(18–20).

Hydrogen Bonds Between A and B Chains. In Table 2 we reportresults for 13 hydrogen bonds which occur between 12 residuepairs. In Fig. 1 A–L are the hydrogen bond donor-acceptorgeometries which correspond to the energy calculations ofTable 2.

Hydrogen Bonds Between A and E Chains. There are 13 interchainhydrogen bonds associated with chains A and E. The calculatedinteraction energies of 11 residue pairs, are listed in Table 3. Thepictures of the hydrogen bonds which correspond to Table 3 areshown in Fig. 2 A–K.

Hydrogen Bonds Between B and C Chains. There are 13 hydrogenbonds between chains B and C. The calculated interaction

energies of 12 residue pairs are shown in Table 4. All of thehydrogen bond geometries which correspond to the energies ofTable 4 are shown in Fig. 3 A–L.

Hydrogen Bonds Between C and D Chains. There are 14 hydrogenbonds between chains C and D. The calculated interactionenergies between 12 residue pairs are listed in Table 5. Thegeometries that correlate to the energies in Table 5 are shown inFig. 4 A–L.

Hydrogen Bonds Between A and C Chains; B and D Chains; and B andE Chains. There are 2 sets of hydrogen bonds between the A andC chains. In Table 6 second and third columns we report resultsfor the hydrogen bonds which occur. In Fig. 5A and B are thehydrogen bond donor-acceptor geometries that correspond tothe energy calculations of Table 6, second and third columns.There are three hydrogen bonds between chains B and D. The

Table 3. The interaction energies of residue pairs between chainA and chain E

Residue no., residue name Figure EMP2 �a.u. EHF �a.u.

A233-E321, HIS-ASP 2A �2.4865 2.1043

A256-E7, ASP-ARG 2B �10.3089 �8.7526

A262-E17, MET-LYS 2C �10.8980 �2.4873

A309-E419, ARG-GLU 2D �12.6716 �9.9359

A311-E320, THR-ASP 2E �0.9871 �0.1436

A311-E410, THR-LYS 2F 9.9610 14.8662

A342-E326, ALA-SER 2G �2.1579 �0.4368

A354-E383, LYS-GLU 2H �10.0289 �7.3687

A371-E387, GLN-ARG 2I �0.8196 4.3493

A399-E422, ARG-LYS 2J �8.5428 0.6294

A403-E418, SER-SER 2K �5.2246 �2.6152

Fig. 2. Hydrogen bonds between residues of chain A and chain E.

Table 4. The interaction energies of residue pairs between chainB and chain C

Residue no., residue name Figure EMP2 �a.u. EHF �a.u.

B17-C262, LYS-MET 3A �7.3387 1.5067B17-C263, LYS-LEU 3B �10.0313 �9.5654B26-C207, GLU-LYS 3C �16.4127 �16.0597B247-C348, THR-PHE 3D �0.5421 4.0847B250-C345, LEU-ALA 3E �6.6727 �3.1411B285-C207, SER-LYS 3F 5.7492 8.3305B321-C233, ASP-HIS 3G �3.5113 0.36168B326-C342, SER-ALA 3H �4.2342 �1.2526B387-C371, ARG-GLN 3I �10.5116 �6.1860B387-C343, ARG-ASP 3J �8.9174 �4.3528B418-C403, SER-SER 3K �5.6354 �3.1167B419-C309, GLU-ARG 3L �2.3255 1.4410

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calculated interaction energies in two residue pairs are listed inTable 6, fourth and fifth columns. The corresponding hydrogenbond geometries are shown in Fig. 5 C and D. There are threehydrogen bonds between B and E chains. The calculated inter-action energies of two pairs of residues are shown in Table 6,sixth and seven columns. The geometries of the correspondinghydrogen bonds are displayed in Fig. 5 E and F.

Discussion and ConclusionsWithin RNA viruses the viral genome RNA is completelyenwrapped by a nucleoprotein. Vesicular stomatitis virus is sucha case. The nucleoprotein in VSV is a 10 member ‘‘cylindrical’’oligomer, half of which is the five member oligomer 2QVJ thatretains a ‘‘half cylinder shape’’, has been crystallized and is thesubject of study in this paper. As suggested by the authors of thecrystal structure reference (1) the intermolecular interactionsamong the chains that make up the nucleoprotein play a criticalrole in providing the structural stability it acquires before

encapsulation of the viral RNA. This is a conclusion whichfollows from the crystal structure study. Knowledge of thecrystal structure alone does not dictate the actual magnitude ofthe interchain interaction energies. However, given the crystalstructure, it becomes possible to extract data from the hydrogen-bond donors and acceptors, and with that information, tocalculate the interchain hydrogen bond interaction energies.That has been accomplished in this paper.

To begin, we calculated the total energy of the entire 2QVJmolecule by using the basic ideas of the KEM. The coordinatesof the atoms used were obtained from the crystal structure at2.8-Å resolution, except for the hydrogen atoms. The position ofthe hydrogen atoms were modeled by a subroutine of thecomputer program HyperChem. Because of the fairly largenumber of atoms in the molecule as a whole (33,175), wecalculated the energy in the Hartree-Fock approximation only,by using a limited basis of Gaussian orbitals. We considered eachof the molecule’s five chains separately, breaking each chain into66 kernels. A total of 330 kernels make up the whole molecule.Each kernel was chosen to contain approximately 100 atoms,which is of practicable size. In this way, using Eq. 1 the data ofTable S2 were able to be obtained. And, Eq. 1 also delivers atotal energy for the full protein equal to �825,954.57 a.u.

It is likely that the interchain hydrogen bonds are among themost important contributors to the stability of the interchainstructure of the whole molecule. All of the many interchainhydrogen bonds have been considered, their geometries dis-played, and their corresponding energies have been calculated.This is the information illustrated in Figs. 1–5, and listed inTables 2–6, respectively. The figures illustrate a variety ofinteresting hydrogen bond geometries. In addition to the usualtwo point geometry of a hydrogen bond as in Fig. 1I, see also 3-,4-, and 5-point geometry in for example, Figs. 1 A and G and 4L,respectively. The tabulated hydrogen-bond energies indicate theimportance of correlation energy in representing the hydrogen-

Fig. 3. Hydrogen bonds between residues of chain B and chain C.

Table 5. The interaction energies of residue pairs between chainC and chain D

Residue no., residue name Figure EMP2 �a.u. EHF �a.u.

C7-D256, ARG-ASP 4A �12.5683 �10.0383

C17-D262, LYS-MET 4B �13.5815 �6.6309

C26-D207, GLU-LYS 4C �18.1539 �17.4105

C321-D233, ASP-HIS 4D �8.6284 �5.9868

C321-D312, ASP-ARG 4E �9.1469 �7.4286

C323-D373, GLU-ARG 4F �15.2136 �15.1048

C383-D354, GLU-LYS 4G �13.5921 �12.5601

C383-D356, GLU-THR 4H �2.0119 0.3896

C387-D356, ARG-THR 4I �3.9021 �2.3766

C410-D311, LYS-THR 4J 5.5745 11.5802

C419-D309, GLU-ARG 4K �15.1070 �13.0429

C422-D399, LYS-ARG 4L �24.0087 �17.0815

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bond interactions. Not only are the MP2 energies quite a bitlower than the Hartree-Fock values, in some cases the Hartree-Fock results indicate repulsion (positive sign) instead of attrac-tion (negative sign), as for example in Table 2 (1e), (1f), Table3 (2a), Table 4 (3a), (3d), (3g), (3l), etc.

In summary, the quantum calculations of VSV complementthe crystal structure determination of the molecule by deliveringthe energetics that follow from knowledge of the atomic coor-dinates. One obtains by the KEM an approximation to the totalenergy of the whole molecule, and the individual chains whichmake it up. Principal contributors to the chain interactions arethe hydrogen bonds between them. All of these hydrogen-bondinteractions have been calculated in both the Hartree-Fock andMP2 approximations.

Quantum calculations of large biological molecules such asthose of this paper should find increasing application in thefundamental study of medical problems.

Methods of CalculationThe identification of all hydrogen bonds between chains was determined

by use of the computer program HBAT 1.0 (12), which receives as input the

protein databank (PDB) file with the atomic coordinates of each pair of chains,and analyzes them for all possible hydrogen bond donors and acceptorswithin a given distance criterion. Once all of the hydrogen-bond donors andacceptors have been identified, other graphics software is used to producetheir images, including their dashed line representations and the indicationsof relevant distances.

We calculated the quantum mechanical interaction energies representingthe hydrogen bonds in two different ways. First by the Hartree-Fock (HF) (10,

Fig. 4. Hydrogen bonds between residues of chain C and chain D.

Table 6. The interaction energies of residue pairs between chainA and chain C; chain B and chain D, and chain B and chain E

Residue no., residue name Figure EMP2 �a.u. EHF �a.u.

A and C chains

A6-C349, LYS-CYS 5A �2.9026 2.375

A8-C347, ILE-GLN 5B �4.6134 �0.7500

B and D chains

B6-D349, LYS-CYS 5C �6.5468 �3.2822

B8-D347, ILE-GLN 5D �4.0897 �0.6056

B and E chains

B347-E8, GLN-ILE 5E �5.3435 �1.4382

B349-E6, CYS-LYS 5F �6.7108 �0.1835 Fig. 5. Hydrogen bonds between residues of chain A and chain C (A and B);B chain and D chain (C and D), and B chain and E chain (E and F).

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11) self-consistent field (SCF) method. The calculations here were imple-mented by using the analytical basis functions of type 6–31G** (13–16). Theenergy error that is inherent to the independent particular Hartree-Fockequations is called the correlation energy error, which in absolute terms isquite small although it is nonetheless important.

An interaction energy calculation which is significantly more accurate thanthose of the Hartree-Fock results may be obtained by using Møller-Plessetperturbation theory (17–19). Møller-Plesset perturbation theory is a usefulway to go beyond the Hartree-Fock model, and thus include correlationeffects in the calculation of molecular energy. We have used MP2 and

6–31G** basis functions to calculate the interaction energies that representthe hydrogen bonds between chains of the VSV nucleoprotein. All of theenergy calculations of this paper were obtained using the standard proce-dures of the computer program Gaussian 03.

ACKNOWLEDGMENTS. We thank the Office of Naval Research for supportingthe work at the Naval Research Laboratory (NRL). L.M. thanks the U.S. NavySummer Faculty Research Program administered by the American Society ofEngineering Education for the opportunity to spend summers at NRL. Thiswork was funded by the National Institutes of Health Grant RR-03037 (to L.M.)and the National Center for Research Resources.

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