J. Mol. Biol. (1996) 264, 199–210

Hydrogen Bonding and Molecular Surface ShapeComplementarity as a Basis for Protein Docking

Michael Meyer*, Peter Wilson and Dietmar Schomburg

A geometric docking algorithm based upon correlation analysis forGBF (Gesellschaft furquantification of geometric complementarity between protein molecularBiotechnologische Forschung)

Abt., Molekulare surfaces in close interfacial contact has been developed by a detailedStrukturforschung optimization of the conformational search of the algorithm. In order to

reduce the entire conformation space search required by the method aMascheroder Weg 1, D-38124physico-chemical pre-filter of conformation space has been developedBraunschweig, Germanybased upon the a priori assumption that two or more intermolecularhydrogen bonds are intrinsic to the mechanism of binding within proteincomplexes. Donor sites are defined spatially and directionally by thepositions of explicitly calculated donor hydrogen atoms, and the vectorspace within a defined range about the donor atom-hydrogen atom bondvector. Acceptor sites are represented spatially and directionally by the vander Waals molecular surface points having normal vectors within apredefined range of vector space about the acceptor atom covalent bondvector(s). Geometric conditions necessary for the simultaneous hydrogenbonding interaction between both sites of functionally congruent hydrogenbonding site pairs, located on the individual proteins, are then tested onthe basis of a transformation invariant parameterization of the site pairspatial and directional properties. Sterically acceptable conformationsdefined by interaction of functionally, spatially, and directionallycompatable site pairs are then refined to a maximum contact ofcomplementary contact surfaces using the simplex method for the angularsearch and correlation techniques for the translational search.

The utility of the spatial and directional properties of hydrogen bondingdonor and acceptor sites for the identification of candidate dockingconformations is demonstrated by the reliable preliminary reduction ofconformation space, the improved geometric ranking of the minimumRMS conformations of some complexes and the overall reduction of CPUtime obtained.

7 1996 Academic Press Limited

Keywords: protein docking; molecular recognition; hydrogen bond;surface complementarity; molecular modelling*Corresponding author

Introduction

Specific recognition is the requirement for anunambiguous biochemical function of proteins. Theformation of enzyme complexes with a substrate oran inhibitor, or antibody-antigen complexes areexamples for such a function. The molecular com-plementarity model of molecular recognition has

several related interactions: shape (steric), hydro-phobicity, hydrogen bonding, electrostatic, van derWaals interactions. For the molecular recognitionprocesses involved in protein-protein interactiongeometric complementarity is of primary import-ance. Geometrically complementary molecularsurfaces are potentially capable of close approach,significantly reducing the molecular surface areasof the two proteins exposed to the medium, gainingstability for the complex from hydrophobic effects,facilitating the formation of hydrogen bonds andother electrostatic and van der Waals interactions,without unfavourable steric interaction.

Abbreviations used: D, donor atom; A, acceptoratom; AA, atom covalently bonded to A; FFT, fastFourier transformation; 3D, three-dimensional; MS,molecular surface.

0022–2836/96/460199–12 $25.00/0 7 1996 Academic Press Limited

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Considering geometric (steric) match betweenmolecular surfaces as a fundamental condition forthe formation of a complex, near crystal confor-mations of protein complexes have been repro-duced (Jiang & Kim, 1991; Shiochet & Kuntz, 1991;Wang 1991; Walls & Sternberg, 1992; Katchalski-Katzir et al., 1992). All practical methods invoke arigid body model of protein structure reducing thedegrees of freedom to three relative rotations andthree translations. However, the conformationalspace of two rigid body proteins is still so large, thatdocking algorithms based on an entire search ofconformation space for geometric molecular surfacecomplementarity are at present impractically slowfor routine use in multiple docking simulations.Only a few of the docking methods reported to dateaddress directly the intrinsic combinatorial prob-lems of protein docking, in order to achievepractical computation times.

The docking algorithms of Lenhoff (1995), Fischeret al. (1995), Lin et al. (1994) and Norel et al. (1994,1995) implement reduced molecular surface rep-resentations by identifying discrete points on themolecular surfaces of two proteins having specificlocal shape features, upon which the geometricmolecular recognition processes are based. Theconformational space search is constrained toconformations defined by alignment of shapecongruent points with matching spatial/directionalcharacteristics, implementing efficient geometrichashing techniques.

Complementarity of molecular surface physico-chemical properties such as electrostatic potential(Nakamura et al., 1985; Chau & Dean, 1994),hydrophobicity (Young et al., 1994; Korn & Burnett,1991) and hydrogen bond donor/acceptor sites(Janin & Chothia, 1990; Krystek et al., 1993;McPhalen & James, 1988) have been demonstratedto exist between the contact docking surfaces ofprotein complexes. Several protein docking algor-ithms implementing physico-chemical and geo-metric parameters have been reported. Shoichet &Kuntz (1993) commented on the combinatoricadvantages of incorporation of chemical filters intogeometric docking schemes applicable to smallmolecule ligands, enabling early discrimination ofpotential conformations that are sterically accept-able but have unfavourable physico-chemicalinteraction, reducing the number of conformationsthat need be considered for subsequent energeticrefinement. Walls & Sternberg (1992) have im-plemented such an approach based on electrostaticpotential surface complementarity into a geometricprotein docking algorithm. A docking schemebased on a semantic net type representation ofprotein molecular surfaces, segmenting the entiremolecular surface on the basis of combinedgeometric and physico-chemical properties hasrecently been reported, giving a detailed unifiedfunctional representation of protein molecularsurfaces (Ackermann et al., 1995). Kasinos et al.(1992) have described a functional chemical graphtheoretic approach for protein docking. However,

with the exception of the algorithm of Kasinos et al.(1992), physico-chemical parameters do not appearto have been implemented in any reportedalgorithms for the preliminary reduction ofconformation space prior to geometric docking.

The correlation algorithm of Katchalski-Katziret al. (1992), has been further developed within ourgroup by a detailed optimization of the confor-mational space search involved in the molecularsurface correlation. The method fulfils the necess-ary criteria of generality, accuracy, precision andreliability required of any practical dockingscheme, but has the inherent combinatorial disad-vantages of requiring a full conformational spacesearch. In order to constrain the global searchrequired by the algorithm, we have devised aphysico-chemical conformational space pre-filterbased on the intrinsic contribution of hydrogenbonding to both the stability and specificity ofbinding between proteins, and the a prioriassumption that only conformations potentiallyforming several hydrogens bonds need be con-sidered.

Hydrogen bonding

The importance of hydrogen bonding for thestructure and function of biomacromolecules hasbeen demonstrated by extensive statistical, exper-imental and theoretical studies (Legon & Miller,1987; Buckingham et al., 1988; Baker & Hubbard,1984; Ippolito et al., 1990; Mitchell & Price, 1989;Jeffrey & Saenger, 1991; Stickle et al., 1992; Thorntonet al., 1993; McDonald & Thornton, 1994). Thedetailed attribution of binding free energy hasdemonstrated the intrinsic importance of inter-molecular hydrogen bonding to the detailedmechanisms of binding specificity and stabilizationof antibody-antigen and protease-inhibitor com-plexes in solution (Fersht et al., 1985; Fersht &Serrano, 1993; Smith-Gill et al., 1982; Mariuzza et al.,1987; Novotny et al., 1989). Although the resolutionof the X-ray data is probably not sufficient toestablish the complete interfacial hydrogen bondsystems of protein complexes with absolutecertainty (Morris et al., 1992), a significant numberof interfacial hydrogen bonds with good distanceand angular geometry for interaction, can reason-ably be assumed to be formed in the crystalstructures of the protease-inhibitor and antibody-antigen complexes within the PDB forming 8 to 13and an average of ten hydrogen bonds between thedocking surfaces. Moreover, only a few of thedonor/acceptor atoms involved in these inter-molecular interactions are capable of formingintramolecular hydrogen bonds suggesting theirprincipal role is one of functional recognition. Bycomparison however, the subunit interfaces ofoligomeric proteins form far fewer hydrogen bondsin proportion to the docking surface areas, someexamples forming none. The majority of interfacialhydrogen bonds between oligomeric subunitsinvolve charged donor or acceptor groups, which

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should make a significant contribution to thesubunit interactions (Janin et al., 1988).

The energy of hydrogen bonding interactiondepends critically upon the donor-acceptor spatialseparation and line of approach of the donorhydrogen to the acceptor lone pairs. Other factorssuch as specific environments of individualhydrogen bonds and specific arrangement of aninterfacial hydrogen bond system have also beendemonstrated to be major determinants in theoverall hydrogen bond stabilization of proteincomplexes (Smith-Gill et al., 1982; Fersht et al.,1985). On the basis of the physical nature ofthe hydrogen bond, the interacting donor andacceptor sites of a protein complex intermolecularhydrogen bonding system are required to have acertain degree of spatial and directional comple-mentarity.

Given the intrinsic importance of hydrogenbonding to the specificity and stabilization ofbinding within protease-inhibitor and antibody-antigen complexes that are involved in activefunctional recognition, a method for the recognitionof functionally, spatially and directionally comp-lementary hydrogen bonding donor and acceptorsites existing on the surfaces of two proteins ispotentially very useful for implementation intodocking processes. The combination of spatial/directional constraints and the chemical filterimplicit to such an approach, has the abilityto greatly reduce the number of conformationsthat need be considered for a ‘‘secondary’’ geo-metric refinement. Molecular recognition analysismethods primarily directed by hydrogen bondinginteractions have been implemented within pro-tein-small molecule docking schemes (Yamada &Itai, 1993; Leach & Kuntz, 1992), template, design ofpharmacophores (Danziger & Dean, 1989; Bohm,1992) and functional molecular similarity analysisof small ligands (Danziger & Dean, 1985). However,protein docking schemes reported in the literatureto date, appear to have confined hydrogen bondinginteractions to the energetic refinement of geometri-cally acceptable solutions.

The pioneering ideas of Danziger & Dean (1989)introduced the concept of a hydrogen bondingmolecular surface of a protein, mapping potentialdonor and acceptor regions on to the accessiblemolecular surface for the template design of smallpharmacophoric ligands on the basis of hydrogenbonding complementarity. In principle, the searchfor hydrogen bonding complementarity could bebased on a continuous optimization of the proteinhydrogen bonding molecular surface interaction,but would involve some form of combinatorialsearch. A method for the discrete pattern recog-nition of the similarity of hydrogen bonding sitesexisting between two molecules, based on thequantitative description of the spatial distributionof donor and acceptor atoms has been reported(Danziger & Dean, 1985, 1989). The method couldalso in principle be adapted to the case of hydrogenbonding system complementarity recognition be-

tween two proteins. Application of such a methodto global comparison of molecules as large asproteins would however be combinatorially com-plex, due to the scale of such a system andthe complicating factors of atom bifunctionalityand donor/acceptor site null correspondenceconditions.

A graph theoretic method has been reported forthe search of functionally and spatially complemen-tary sets of small ligand-protein hydrogen bondingsites based upon graph clique finding (Smellie et al.,1991). However, the computational time requiredfor clique finding between two large proteinswould be prohibitively long.

Based on detailed investigations made intopotential methods for the quantification of spatialand directional complementarity between donor/acceptor sites of interfacial hydrogen bondingsystems, we have devised a physico-chemicalprefilter of conformation space, constraining thesearch to conformations between proteins that arepotentially capable of intermolecular hydrogenbonding, on the basis of molecular surfacedonor/acceptor site spatial and directional proper-ties. This provides a reliable, sufficiently preciseand rapid reduction of conformation space toenable secondary geometric screening withinpractically realistic CPU times. A minimum of twopotential intermolecular hydrogen bonds in thecomplex is required for the prediction of relativeorientations of the components. The methoddefines a reduced molecular representation, basedupon the molecular van der Waals surface pointnormal vectors of acceptor atoms, and the atomiccentres of explicitly calculated donor hydrogenatoms and donor atom-hydrogen atom bondvectors, gives a detailed representation of both thespatial and directional properties of donor andacceptor atom sites. Conformations that arepotentially capable of intermolecular hydrogenbonding are identified by testing functionallycongruent donor-donor, donor-acceptor, and ac-ceptor-acceptor pairs between the proteins fornecessary conditions of spatial and directionalcomplementarity on the basis of a transformationinvariant geometric parameterization (Fischeret al., 1995; Norel et al., 1995) of the spatialand directional properties of the site pairs,represented by the site points and vectors. Eulerangles defining the candidate conformations arethen calculated by alignment of the compatible sitepairs, which are then refined by the geometricdocking algorithm.

Algorithm

Figure 1 gives a summary of the dockingprocedure. For a complex of two proteins we startwith an analysis of potential intermolecularhydrogen bonds and generate a small list ofEuler angles for the correlation analysis. Other-wise a global rotational search has to be carriedout.

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Figure 1. Scheme of the docking algorithm.

Donor and acceptor atom types includedin the definition of hydrogen bondDonor/acceptor sites

Hydrogen bonding donor and acceptor groups atphysiological pH are listed in Table 1 by theirrespective Brookhaven PDB name code (Bernsteinet al., 1977). Hydrogen bonds involving alphacarbon atoms (Derewenda et al., 1995), aromaticring acceptors and NH donors (Levitt & Perutz,1988) and sulphur atoms, although possible, arerelatively weak and are not included. Two possibleorientations of the amide groups of Asn and Glnand the imidazole group of His are used whendefining donor and acceptor sites. The initialstructural survey of interfacial hydrogen bonds

between proteins used the criteria of Baker &Hubbard (1984), The distances H. . .A and D. . .Amust be less than 2.5 A and 3.9 A, respectively, andthe angles {D-H. . .A, {D. . .A-AA, and {H. . .A-AA must exceed 90° where AA symbolises otheratoms bonded to the acceptor. The hydrogencoordinates and bond angles of side-chains werederived from ab initio bond lengths and smallmolecule models, respectively. For example, 4-methyl-imidazole was used as a model of theside-chain of histidine (Meyer, 1994). Free rotationin amine and primary hydroxyl groups wasmodelled by location of the hydrogen atoms atthree equidistant positions about the circle describ-ing the trajectory of the possible loci of thehydrogen. Similarly, for the primary hydroxyl

Table 1. Hydrogen bond donors and acceptorsDonors(1) N (main-chain N-H)(2) Asn OD1, His NE2, His ND1, His CD2a, His CE1a, Lys NZ, Asn ND2, Gln NE2 Arg NE, Arg NH1, Arg NH2, Ser OG, Thr OG1,

Tyr OH, Trp NE1

Acceptors(1) O (main-chain C = O)(2) Asp OD1, Asp OD2, Glu OE1, Glu OE2, His ND1, His CD2a, His CE1a, Asn OD1, Gln OE1, Gln NE2, Asn ND2, Ser OG, Thr

OG1, Tyr OHa We include also the carbon atoms CE1 and CD2 in the prediction of Euler angles from hydrogen bonds to take an ambigous

orientation of the His imidazole into account.

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group, three dummy hydrogen atoms are placed atequidistant positions about the circle describingthe possible loci of the hydrogen. This model ofthe amine and primary hydroxyl donor sitesallows for all recognised interactions within thegeometric parameterization of the group spatialand directional properties (see later). For thehydroxyl of tyrosine, two in plane configurationsare allowed.

Generation of orientations with potentialhydrogen bonding interactions

The algorithm assumes that only defined donorand acceptor atoms within a specified distance ofthe solvent accessible molecular surface (Richards,1977) are capable of intermolecular interaction. Thesolvent accessible surfaces of the two proteins arefirst calculated using the MS-dot routine ofConnolly (1983) with a density of five points/A2

and a solvent probe radius of 1.4 A, The molecularsurfaces are then projected onto a grid with a gridspacing of 0.25 A. The van der Waals surfaces andnormals of the proteins are calculated by theMS-dot routine with a probe radius of zero.Excluding acceptor atoms not within 3.0 A of thesolvent accessible surface, the acceptor sites arethen initially defined by the van der Waalsmolecular surface points of recognised acceptoratoms having associated normal vectors within 60°of the A-AA bond vector. These points are thenrefined further by testing points between 0.1 A and1.8 A along the normal vectors for van der Waalsoverlap with other atoms, points where the vectorhas van der Waals overlap are rejected as the linesof interaction defined by the vector are blocked.Atoms having less than 30 of such neighbouringpoints were excluded from the analysis. Thehydrogen positions of donors within 3.0 A of thesolvent accessible surface boundary are thencalculated according to standard geometries usingab initio bond lengths and angles derived fromsmall molecule models of side-chain geometry(Momany et al., 1975; Meyer, 1994). Calculatedatomic positions of hydrogen atoms and the vectorspace within 60° of the donor atom-hydrogen atombond vector, define the donor sites, implicitlyallowing for all recognized interactions of freelyrotating hydroxyl and amine groups. Effects of thesteric hinderance on the possible restriction offreely rotating donor hydrogen site positions areneglected.

The seemingly complicated approach of calculat-ing the solvent accessible surface and the van derWaals molecular surface individually, then exclud-ing atoms not within a specified distance of thesolvent accessible surface is adopted in order toaccurately identify donors and acceptors that areable to interact, while also retaining an accuraterepresentation of acceptor site directional proper-ties. Inclusion of all donors and acceptors at themolecular van der Waals surface would give aninaccurate representation of the donor and acceptor

Figure 2. Illustration of the the necessary angular anddistance conditions for simultaneous interaction betweentwo sites. (a) The angles b1, b2, b3 and b4 are defined byvectors with an origin at hydrogen or acceptor sitesand the line segments joining them on both proteins.The distance and angular criteria are given in the text.The mutual angles between the vectors at different sitesof the same protein are a1 and a2. (b) Dihedral angle ofthe planes between two normal vectors and the joiningline for two donor or acceptor sites located at the sameprotein.

atoms that could interact. Due to the nature of thevan der Waals surface, atoms are capable of havinga component in the molecular van der Waalssurface at positions located within the proteinsolvent accessible surface boundary that could notpossibly participate in any form of intermolecularinteraction. Using only atoms that have a solventaccessible surface in the definition of donor andacceptor sites may exclude atoms that are justbeneath an area of re-entrant solvent acces-sible surface and may be capable of significantinteraction.

Donor/donor, acceptor/acceptor, and donor/acceptor site pairs on the individual proteinsseparated by a minimum distance of 2 A anda maximum of 8.0 A for the protease-inhibitor orantibody-antigen complexes and 15 A foroligomeric subunits are then listed, this includesmulticentre hydrogen bonds implicitly in theanalysis. Two functionally congruent site pairslocated on two proteins are considered to fulfil thenecessary (but not sufficient) spatial and directionalrequirements for simultaneous hydrogen bondingbetween the two sites, if the normals on both thesites of the pairs are capable of having anti-parallelorientation within threshold values, and thedifferences in Euclidean distance between the sitepoints of the pairs are within the definedthresholds, modelling the condition of simul-taneous approach of the two hydrogen atoms tothe acceptor lone pairs within the spatial anddirectional constraints stipulated for recognizedhydrogen bonds (Figure 2). This condition is testedfor all congruent site pairs between the two

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Figure 3. Illustration of the selection of acceptor sitepoints having a normal vector within 60° of the C = Obond vector. The donor sites are defined by the explicitlycalculated hydrogen atom coordinates. The vectorswithin the 60° range of the donor hydrogen bond vectordescribe the constrained line of an approach of theacceptor atom. The line segment between the hydrogenatoms and acceptor atoms illustrate the distance criterion.

from grid points within spatial domains about thesite points and the joining line segment arecalculated. The moment of inertia Iij may be writtenin tensor notation as

Iij = Sm(r2dij − rirj ) (1)

where dij is the Kronecker symbol, ri is the positionvector with the components (xi , yi , zi ) of a point andthe sum runs over all points within the defineddomains. The mass m of this second rank tensor isset to one. After a principal axis transformation thecorresponding axes of the domains are aligned andthe Euler angles u1, u2, u3 derived from the rotationmatrix.

A listing of the number of hydrogen bondspotentially formed within the candidate confor-mations is made by representation of the entireEulerian orientation space as a 3D 72 × 36 × 72array, each index corresponding to 5° in eachEuler angle. The incrementing of each arraypoint for each candidate conformation fallingwithin the corresponding conformational spaceallows an approximate hydrogen bond numberranking to be established for domains of confor-mation space.

Surface representation and correlation

The protein structures are digitally representedon a 3D grid g(p)

klm (p = 1 or 2 for the first or secondprotein) with a typical sample rate of 1.5 A for a fastinitial search and on a finer grid for the refinementsteps. We apply an all atom model for proteinstructures with explicit hydrogen atoms (NMR-structures) and a united atom model for proteinstructures without hydrogen atom coordinates. Inthe latter case we use extended radii for thoseatoms which are bonded to hydrogen (Richards,1974). These radii have been generalized forstandard amino acids, nucleotides, some co-factorsand common small molecules.

Similarly to Katchalski-Katzir et al. (1992), weassign negative numbers g(1)

klm = − 5 to all grid pointsinside of the larger of both proteins, which wekeep in a fixed orientation. This values has beenoptimized to reproduce a series of complexstructures accurately. The grid points outside of theproteins are zero, except for a 2 A surface layer withpositive grid points g(1)

klm = 1. The smaller protein isdigitized in such a way that all grid points insidethe protein are positive (g(2)

klm = 1) and the otherpoints are set to zero. Then we calculate correlationCabc of both proteins, which corresponds to ascoring of the contacts of both proteins resultingfrom different relative translations. The optimaltranslation for a given rotation is the one with thehighest correlation corresponding to a contact oflarge complementary surface layers with a mini-mum of overlaps between both proteins. A negativecorrelation corresponds to strong overlaps, and ifthe proteins do not touch each other, a zerocorrelation is computed.

proteins on the basis of a transformation invariantparameterization of the site pair spatial anddirectional characteristics:

(1) The difference of distances between two pairsof complementary donor or acceptor points locatedat the different proteins must not exceed athreshold of 2.5 A (Figure 2a).

(2) The mutual angle between vectors at thetwo points of a site pair must agree within 60°(Figure 2a).

(3) For angles formed between vectors of the twosites and the joining line segment the sums of theangles b1 + b3 and b2 + b4 must be within 60°threshold of 180° (Figure 2a).

(4) The dihedral angle t must not exceed 260°.This angle is formed between the planes defined bythe normal vectors of two donor or acceptor sites P1

and P2 at the same protein and the line segmentjoining them (Figure 2b)

The parameters are calculated and compared forcombinations of vectors between two site pairs.Allowing for a 60° threshold in the donoratom-hydrogen bond vectors, the two site pairsshould have coincident ranges of normal vectorangles (Figure 2a) and dihedral angles (Figure 2b).The threshold in the angle about the donor-hydrogen bond vector to 60°, and the restrictionof acceptor points to those having normals within60° of the A-AA bond vector (Figure 3), con-strains the potential intermolecular hydrogenbonds recognised to those capable of havinggood angular interaction geometry with a D-H. .A angle of 120° or greater, a D. .A-AA angle ofgreater than 120° and a H. . .A distance of 2.5 A orless.

For the generation of the starting Euler angles forthe rotational search, the moment of inertia tensors

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Gpuvw = s

N − 1

k = 0

sN − 1

l = 0

sN − 1

m = 0

gpklm exp[2pi(ku + lv + mw)/N]

C'uvw = G1*uvwG2

uvw

Cabc = 1N3 s

N − 1

u = 0

sN − 1

v = 0

sN − 1

w = 0

C'uvw

× exp[−2pi(ua + vb + wc)/N] (2)

According to the correlation theorem the gridrepresentations of both proteins p are transformedto give the complex Fourier coefficients G(p)

uvw and thefinal correlation Cabc is obtained from an inversetransformation of the intermediate product C'uvw

(see equation (2)). The 3D-FFT of the real data canbe computed via 1D-FFTs with respect to eachdimension using vectorized routines for multiplereal sequences (Sweet et al., 1990) or with theefficient Green Mountain Software Real FFTpackage (1995).

Rotations

For the determination of optimal Eulerian angleswe rotate the smaller of the proteins, sample itagain on a grid and recompute the correlation.Finally we sort the coordinate transformationsaccording to the magnitude of the correlation todetermine the best relative orientation.

d = zDu2+ cos2(u2/2) + Du2

2 + Du2− sin2(u2/2) (3)

The systematic rotational search is carried outusing u2 and the linear combinations u+ = u1 + u3 andu− = u1 − u3 of Eulerian angles u1, u2, u3. For a givenu2 the angular steps Du+, Du2 and Du− are adjustedto keep constant and equal each term of equation(3) representing the distance d of two nearbyorientations (Lattman, 1972).

A final refinement search can be carried out witha combination of the simplex method andsimulated annealing (Press et al., 1992) if accuratestructures are required. For a given simplexrepresenting four sets of linear combinations ofEulerian angles, the highest correlation is computedand with a series of reflections, expansions andcontractions a higher correlation with respect to theangles is searched. The simplex criterion for theacceptance of a new set of Eulerian anglesC(u+, u2, u−)new > C(u+, u2, u−)old can be modified togive C(u+, u2, u−)new − T log Znew > C(u+, u2, u−)old +T log zold for a simulated annealing optimization. Ata temperature T = 0 both criteria are identical, butfor T > 0 the acceptance depends on randomnumbers 0 < zE1. A step leading to a lowercorrelation can be accepted, if the randomcontributions from the logarithmic terms aresufficiently large to enable the method to escapefrom local maxima. The method can be used for arefinement in a small angular range starting withlow temperatures or for a global search with a high

initial temperature. However, it is necessary to takeinto account the periodicity of the Eulerian anglesto prevent the program from increasing the stepsizes successively and rotating the protein severaltimes once a large angular step is successful.

Results

Fourty five complexes have been selected fromthe PDB to test the method. This sample, whichincludes test examples given by Fischer et al. (1995)in Table 2 of their work and additional examples,consists of enzyme inhibitor complexes, dimers,antibody-antigen complexes and some complexeswith nucleotides or small molecules. The watermolecules of these complexes have been removedand one of the components has been subjected to arandom rotation and the docking method has beenemployed to generate the correct structures. For thenon protein-protein complexes a complete ro-tational search has been carried out first, whereasfor the protein-protein complexes an analysis of thehydrogen bonds has been carried out prior to thecorrelation search (Figure 1). Subsequently wecomputed potential contact regions and generatedinitial values for the rotational search. Between 40and 50 different angular sets have been generatedby these initial steps for each complex; the typicalnumber of sets was between 50 and 150.

For these sets we carried out a refinement in threesteps. First we computed the correlation at a gridsize of approximately 1.5 A. Then we incrementedthe Eulerian angle u2 with a step of 10° and −10°,adjusted the corresponding steps for u+ and u−

according to equation (3) and computed thecorrelations at a grid size below 1.2 A for thoseangular sets with a relative correlation of at least75% relative to the highest one in the precedingstep. For the best sets of Eulerian angles with arelative correlation of at least 90% in the secondstep we started the simulated annealing with a lowrelative temperature adjusted to 0.05 times thecorrelation at the corresponding angles. Wereduced the temperature in ten steps to zerocarrying out a maximum of 50 simplex search stepsat each temperature. When T reached 20% of theinitial value we reinitialized the simplex par-ameters with the best ones that have been found sofar. The results are given in Table 2. The typicalRMS errors are in the order of 1 A and the highestcorrelation corresponds to predicted structuresclose to the experimental one. For the antibody-antigen complexes 1fdl, 2hfl and 3hfm the bestpredicted structures correspond to the first threeranked solutions in each case.

Discussion

Reduction of conformational search spacebased upon hydrogen bond interaction

The utility of protein surface hydrogen bondingdonor and acceptor site complementarity as a

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Table 2. Results from the docking calculationsa

Complex Chain Atoms RMS/A

2est e,i 1822 31 0.583dfr 1342 33 0.615sga e,p 1259 35 0.504mbn 1217 44 0.971rga 792 44 0.561sgc 1259 45 1.124hvp ab,i 1516 54 1.523apr e,i 2417 57 0.442igf lh,p 3378 58 0.612cpk e,i 2665 75 0.465hvp ab,c 1540 90 0.554phv ab,i 1520 92 0.901cpk e,i 2665 157 0.994cpa 2443 285 1.204sgb e,i 1310 380 0.511cho e,i 1750 400 0.851tgs z,i 1646 416 0.312kai ab,i 1799 438 0.892ptc e,i 1629 454 1.581tpa e,i 1628 454 0.632tgp z,i 1629 454 0.664tpi e,i 1629 471 0.791lcc a,bc 497 492 0.951gat a,b 949 510 0.792sni e,i 1940 513 1.221acb e,i 1769 522 0.981tec e,i 1881 522 1.881cse e,i 1920 522 1.202sec e,i 1923 530 1.182utg a,b 548 548 0.571aar a,b 602 601 0.544hvp a,b 811 758 0.634phv a,b 760 760 0.672sic e,i 1938 764 1.952pab a,b 872 872 1.412rsp a,b 891 881 0.742ccy a,b 976 976 0.653hfm lh,y 2114 1001 0.822hfl lh,y 3227 1001 2.561fdl lh,y 3308 1001 2.012mhb a,b 1178 1113 0.682mcp h,l 1720 1692 0.274fab h,l 1700 1695 0.366adh h,y 2885 2784 0.794cts a,b 3453 3453 0.61

a The predicted structures correspond to the highest corre-lation except for antibody-antigen complexes (see the text).

proportion of the total combined molecular surfaceareas and/or the docking areas have a relativelylow degree of absolute shape complementarity(Lawrence & Colman, 1993). Significant numbers ofconformations with larger contact areas givingbetter overall surface correlation exist, resulting ina low geometric ranking of the correct confor-mation. Therefore geometry-based immunoglobu-lin-lysozyme docking studies have been restrictedpreviously to the matching immunoglobulin epi-tope and lysozyme (Fischer et al., 1995), which isnot necessary when the hydrogen bonding confor-mation pre-filter is applied. It reduces the numberof incorrect conformations that are consideredsterically acceptable by the geometric algorithmprior to secondary geometric refinement, with thecorrect solution always being geometrically rankedwithin the top three for each of the complexesinvestigated. These ranks can stand up well to theresults of other methods (e.g. Fischer et al., 1995).The surface correlation analysis as a second step forthe geometric refinement of the hydrogen bondcandidate solutions has the distinct advantage ofnot limiting the conformations for assessment ofgeometric goodness of fit between the proteins tothose defined by alignment of fixed surface pointsand vectors, allowing the imprecisely definedcandidate conformations to be refined to minimumRMS values on the basis of entire contact molecularsurface complementarity (Fischer et al., 1995; Norelet al., 1995).

The a priori assumption of the existence of two ormore hydrogen bonds (Table 3) between theproteins in the native conformation upon which

Table 3. Intermolecular hydrogen bondsa

Complex Chain Hydrogen bonds

1aar a,b 24sgb e,i 36adh h,y 31cho e,i 41tec e,i 42utg a,b 41tgs z,i 52hf1 lh,y 52sni e,i 54hvp ab,i 55sga e,p 52mhb a,b 72sec e,i 72mcp l,h 72mhb a,b 71fdl lh,y 82cpk e,i 82igf lh,p 82kai b,i 82tgp z,i 82pab a,b 102ptc e,i 103hfm lh,y 104tpi z,i 104phv a,b 212rsp a,b 22

a Criteria of Baker & Hubbard (1984).

primary basis for protein-protein functional recog-nition is demonstrated by the ability of theapproach to provide a reliable and acceptablyprecise pre-filter of conformation space for ageometric docking algorithm requiring a globalsearch. The primary chemical filter implicit to themethod combined with the spatial/directionalconstraints identifies potential hydrogen bondswith good distance and interaction geometry,efficiently reducing the number of conformationsfor secondary geometric refinement to typically 40to 200 in the complexes tested, which is in mostcases at least one magnitude less than the numberof rotations required for a stepwise globalrotational search (see below). In the specific cases ofantibody-antigen complexes (PDB entries 1fdl, 2hfland 3hfm) and the protease inhibitor 3sgb thedocking sites account for a relatively small

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the entire approach depends, is readily justified forantibody-antigen and protease enzyme-inhibitorcomplexes and physically realistic cases ofoligomeric protein subunits (Janin et al., 1988).Native oligomeric subunit conformations, on theother hand, in which two hydrogen bonds couldnot reasonably be assumed to be formed, arestabilized primarily by hydrophobic interactionderived from large (>1000 A2) contact surface areaswith close geometric fit. On the basis of themolecular weights and accessible hydrophobicsurface areas of the individual subunits, the tertiarystructures of the subunits within the complexwould be thermodynamically unstable in isolationunder normal solution conditions.

The work of Danziger & Dean (1989) im-plements a simple valence model to identifydonor and acceptor groups at the molecularsurface that are capable of intermolecular inter-action. They assumed that acceptors and donorscapable of forming intramolecular hydrogenbonds with all lone pairs or hydrogen atoms neednot be considered for intermolecular interactionwith small ligands. However, in the context ofprotein-protein interaction such assumptions arenot necessarily valid. Additional factors such aslocal environment and specific arrangement of anintermolecular hydrogen bonding system, willalso determine the preference for intra- orintermolecular interaction where both interactionsare geometrically feasible. The algorithm thereforeincludes all donor and acceptor sites within aspecified distance of the solvent accessible surfacein the analysis. The simple valence model ofdonor and acceptor sites does however signifi-cantly reduce the number of sites to be con-sidered globally when implemented within thealgorithm, while sufficient sites remain (ca five tonine) from the active/epitopic sites of proteaseinhibitor and antibody-antigen complexes to en-able the proteins to be accurately and efficientlydocked. Work is currently being directed toassessing the reliability of using such donor andacceptor sites with unsatisfied hydrogen bondingpotential (McDonald & Thornton, 1994) alone asa basis for hydrogen bond functional recognition,as such sites appear to play a principal role inintermolecular hydrogen bonding interaction be-tween proteins.

The number of functionally congruent site pairsgenerated from the matching is sufficiently small toallow the site pair matchings to be completedwithin 20 minutes of CPU time for the complexesused in the present study.

By making a priori assumptions about thenumber of the hydrogen bonds that acceptableconformations can form, or by considering onlydonors and acceptors located within areas aboutenzyme active sites or antibody epitopic regionscandidate proteins could be efficiently screenedfor active site inhibitor activity or immunogeniccross reaction on the basis of hydrogen bondcomplementarity. Further simplification may be

possible in specific cases, by making structuralgeneralizations about hydrogen bonding inter-actions within a given taxonomic protein groupsuch as the proteases. All protease-inhibitorsform a minimum of five interfacial hydrogenbonds between main-chain amidic nitrogendonors and carbonyl acceptors. A restrictedsearch using only main-chain COs and NHs andstipulating the requirement that three or morehydrogen bonds are formed, finds the minimumRMS solution in under ten minutes. Such ap-proaches limit the generality of the method, butwould be useful tools within a protein functionaldatabase.

The method is directly applicable to smallmolecule protein interactions, and could serve asa basis of a precise and accurate hydrogenbonding molecular similarity index for structuraldatabase screening. By matching donor hydrogenatom sites with donor atom sites and acceptoratom sites with acceptor atom sites betweenmolecules on a basis of spatial/directional simi-larity, then performing an alignment of matchingsite pairs, molecules with similar spatial anddirectional hydrogen bonding properties wouldhave comparable numbers of hydrogen bonddonor and acceptor sites within the same ‘‘align-ment’’ space. This simplifies the combinatorialproblems of discrete pattern recognition of simi-larity between hydrogen bond systems of differentmolecules.

Conformational search

The stepwise refinement enables a globalsearch at a small grid with fast evaluations ofthe correlation and leaves only a few sets ofEuler angles for the more time consumingcomputation at a fine grid. The use of linearcombinations of Euler angles for the systematicsearches is superior to the uneven samplingbased on the Eulerian angles themselves(Lattmann, 1972). Comparative calculations showthat the rotational search with linear combi-nations requires approximately 30% fewer angu-lar steps. Nevertheless, a complete rotationalsearch is still time consuming. For typicalenzyme-inhibitor complexes 1857 rotations arerequired and for large dimers 2578 or moresteps are necessary, because a small angularinaccuracy may lead to a large overlap for largemolecules. This number of rotations is somewhatsmaller than previously reported (Katchalski-Katzir et al., 1992) because the locally orthogonallinear combinations of Euler angles facilitate amore regular sampling. The generation of aninitial guess for the Euler angles from potentialintermolecular hydrogen bonds can reduce thefirst rotational search drastically and gives morereliable results. Less than 500 sets of Eulerianangles are generated in the hydrogen bondanalysis and the error of the best proposal forthe angles is usually somewhat below 10°.

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In principle, the systematic search for thecomplete rotational space can be replaced by asimulated annealing search. In the beginning ofsuch a search we carried out some initial searchsteps to determine the typical correlation for afew sets of Eulerian angles. Then we switch tothe simulated annealing with an initial tempera-ture to 0.75 times the mean value of thepreviously determined correlations. Our simu-lated annealing schedule consists of a linearreduction of the temperature to zero in 40 stepswith a re-initialization of the simplex with thebest angular set when the temperature has beenreduced by a factor of five. For each step amaximum of 50 simplex search steps is carriedout. Due to prior convergence typically 1800search steps are actually carried out. The methodis able to find the correct relative orientation forenzyme-inhibitor complexes and the efficiency iscomparable with the systematic search, but moresteps are required for large dimers. In favourablecases like 3dfr or 2utg the correct maximum canbe found with much fewer steps than with thesystematic search, which spends much time withcompletely irrelevant angles. But since we do notknow in advance, whether we have to deal witha favourable case or not, we cannot use aschedule with fewer steps.

We have carried out a detailed analysis ofdifferent types of complexes. The correlationmethod can be used for complexes containingsmall ligands with 30 to 40 atoms and it is aswell applicable for complexes consisting oflarge dimers (Table 2). The algorithm is ratherflexible. If high accuracy results are not requiredthe speed of the calculation can be increased,e.g. the grid distance can be increased or thesimplex search can be omitted. Near crystalstructures have been obtained using a three-stepgeometric refinement with typical RMS errorbelow 1.0 A. These results can compete withcurrent docking programs. High RMS errors haveonly been determined for the antibody-antigencomplexes which have a low degree of absoluteshape complementarity. A smaller grid spacingdoes not lead to more accurate results for thesecases.

Conclusions

The joint use of an analysis of potential hydrogenbonds and correlation techniques leads to fast andreliable solution for the protein docking problem.The hydrogen bond pre-filter reduces the requirednumber of rotational search steps substantially andremoves a high number of potentially incorrectstructural solutions. The correlation method leadsto accurate complex structures with a high degreeof molecular surface complementarity and incombination with the pre-filter the correct com-puted structure is within the top three rankedproposals.

AcknowledgementsWe thank R. A. Sweet and W. Bothe for helpful

suggestions, the Bundesministerim fur Bildung, Wis-senschaft, Forschung und Technologie for funds. Aportion of this work was supported by a European UnionAccess to Large Scales Facilities grant (ER-BCHGECT940062) to EMBL.

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Edited by R. Huber

(Received 7 July 1996; received in revised form 4 September 1996; accepted 11 September 1996)