AddRemove: A New Link Model for Usein QM/MM Studies

MARCEL SWARTTheoretische Chemie (MSC), Rijksuniversiteit Groningen, Nijenborgh 4, 9747 AG Groningen,The Netherlands

Received 8 January 2001; accepted 30 May 2002

DOI 10.1002/qua.10463

ABSTRACT: The division of a system under study in a quantum mechanical (QM)and a classical system in QM/MM molecular mechanical calculations is sometimes verynatural, but a problem arises in the case of bonds crossing the QM/MM boundary. Anew link model that uses a capping (link) atom to satisfy the valences of the quantumchemical system is presented, with the position of the capping atom depending on thepositions of the real atoms involved in the link bond. Using this method no degrees offreedom for the capping atom are introduced. Moreover, the introduction of thisartificial atom is corrected for by subtracting the classical MM interactions with the realQM system it would have if it were a classical atom. That is, the capping atoms areadded and removed. The new model has been tested on three amino acid residues andshows a clear improvement over other link models (as represented by the IntegratedMolecular Orbital and Molecular Model (IMOMM)/ADF implementation). The averageabsolute deviation for the C�–C� bond distance, as obtained when comparing the fullQM and QM/MM results, is around 0.75 pm. The IMOMM model predicts distances forthe C�–Cbackbone and C�–Nbackbone bonds, with an average absolute deviation of 2.3–2.8and 5.3–5.5 pm, respectively; this is an increase by a factor of 3.1–4.0 and 7.1–7.3compared with the C�–C� bond. For the new AddRemove model, the average absolutedeviations are 1.0–1.2 and 0.6–0.9 pm, respectively, for the C�–Cbackbone andC�–Nbackbone bonds; compared with the C�–C� bond, this indicates only a slight change,with a factor of 1.3–1.6 and 0.8–1.2, respectively. The new AddRemove model thereforeperforms much better and is shown to be a substantial improvement over the IMOMMmodel. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem 91: 177–183, 2003

Key words: hybrid QM/MM; link atoms; molecular mechanics; quantum mechanics

Correspondence to: M. Swart; e-mail: m.swart@chem.rug.nl.Contract grant sponsor: The Netherlands Organization for

Scientific Research.Contract grant sponsor: Unilever Research Vlaardingen.

International Journal of Quantum Chemistry, Vol 91, 177–183 (2003)© 2002 Wiley Periodicals, Inc.

Introduction

H ybrid quantum mechanical/molecular me-chanical (QM/MM) methods [1] split up a

system under study into two parts: the electroni-cally more important one is treated with quantummechanics (real QM system), whereas the remain-ing part is treated on a classical level (MM system).The division into the two subsystems is sometimestrivial, as in the case of solvation of an organiccompound where one puts the organic compoundin the QM system and the solvent in the MM sys-tem. However, in biochemical systems the divisionoften results in bonds that cross the QM/MMboundary. Several approaches have been used tohandle this situation, for instance, localized orbitals[2], pseudopotentials [3–6], density matrix, or elec-tron density partitioning (see Ref. 1 for an over-view), but the easiest and most commonly chosensolution is to use link atoms [7, 8]. In the latterconcept one introduces a (artificial) capping atom inthe quantum system (comprising the capped QMsystem), which is usually taken to be a hydrogenatom. Singh and Kollman [7] (who originally pro-posed the concept) refrained from putting any con-straints on the position of the capping atom, therebyadding three degrees of freedom to the system. Toavoid the addition of these unwanted and artificialdegrees of freedom, many link models, developedsubsequently, impose the capping atom LC, the quan-tum atom LQ and the classical atom LM to have amutual coordinate dependence (for instance, to lie ona straight line), thereby removing again the additionaldegrees of freedom (see Fig. 1).

Although there are many link models in the re-cent literature [9–13], the newly developed modelas described in the current work (AddRemove), iscompared primarily with the currently imple-mented model in the ADF [14] program, as thismodel is more or less representative of the othermodels. After introducing the new model, a com-parison is made also with other (well-known) mod-els.

In the current implementation in ADF [14](IMOMM/ADF [15–17]), the degrees of freedom of

the real classical atom LM are removed; it “follows”the capping atom LC with its position dependingon the positions of the LQ and LC atoms as:

r�LM � �r�LC � �1 � ��r�LQ (1)

In this equation, � is an arbitrary constant param-eter, but its value is difficult to generalize or choose.The energy depends on the value chosen for it andone can compare energies between different mole-cules only if the �s are chosen equal for both.

The capping atom interacts only within thecapped QM system and has no interactions with theMM environment. Moreover, the real classical LMatom does not interact with the real QM system; itsinteractions with the QM system are “replaced” bythe QM interactions of the capping atom LC. Thereal classical LM atom, of course, does have MMinteractions with the rest of the MM system, butthey are not used for updating its position (becauseit is constrained to follow the LC atom).

Before doing the QM calculation, the MM systemis optimized. In this part, the atoms in the real QMsystem are kept fixed at their positions and interactwith atoms in the MM system through classical MMinteractions. In the IMOMM/ADF implementation[15–17], also the classical LM atoms are kept fixedin this part. A schematic example for the optimiza-tion scheme is given on the left in Figure 2. After theMM optimization has ended, the QM calculation isFIGURE 1. Schematic representation of link bond.

FIGURE 2. Schematic representation of QM/MM opti-mization.

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performed and the QM energy, gradients, andcharges calculated. Then the MM system is opti-mized again before starting a new QM cycle, and soon, until the QM system is fully optimized.

One of the advantages of the IMOMM andAddRemove model is that the interactions betweenthe QM and MM system are treated at the samelevel of theory as are the interactions within theMM system itself. Then, if one uses for those inter-actions the AMBER95 or GROMOS96 force fields [18,19] (which are designed for treating the interactionswithin proteins and are known to perform well forthem), one is certain that the interactions (in theMM system itself as well as between the QM andMM systems) are treated properly.

AddRemove Methodology

In the AddRemove model, the position depen-dence is reversed: it is the capping atom LC thatfollows the classical atom LM, and not vice versa:

r�LC � r�LQ � Req,LQ–LCu� LQ–LM (2)

Here, u�LQ–LM is a unit vector pointing from atomLQ to atom LM (see Fig. 1 for atom labels), andReq,LQ–LC is the equilibrium distance for the LQ–LCbond it would have in the classical force field that isused for the MM system. This has the major advan-tage that one does not remove the degrees of free-dom of the real classical atom LM (as is the case inthe IMOMM/ADF model [15–17]). Instead, the ar-tificial degrees of freedom of the capping atom LCare removed. Also, unlike the IMOMM/ADF im-plementation [15–17], the energy does not dependanymore on some arbitrary parameter �. Further-more, the real classical LM atoms interact normallywith the real QM system and are allowed to movein the optimization of the MM system (see also theright hand side of Fig. 2). Then, before the QMcalculation starts, the positions of the capping at-oms are updated with Eq. (2) from the currentpositions of the LM atoms. After the QM energyand gradients have been obtained, they are cor-rected for by removing the interactions of the cap-ping atoms: the MM interactions, which the cap-ping atoms would have with the atoms in the realQM system if it were a classical atom, are sub-tracted. That is, the capping atoms are added andremoved. Schematically, this would result in the

following energy expression for the interaction be-tween the QM and MM systems:

UQM/MM � �j�MM∧j�LM

i�realQM

UijMM � �

n�1

Nlinks �i�realQM

Ui�LC,nQM

� Ui�LC,nMM � Ui�LM,n

MM (3)

The first summation runs over all classical atomsthat are not involved in the link bond, which couplenormally with the real QM system through stan-dard MM interactions. The second part is specificfor the atoms involved in the link bonds. Here, the“quantum” interactions of the capping atoms LCwith the real QM system (Ui�LC,n

QM ) are corrected forby subtracting its MM counterparts (Ui�LC,n

MM ), andreplaced by the MM interactions of the classical LMatom with the real QM system (Ui�LM,n

MM ). Of course,it is not possible to project out the Ui�LC,n

QM interac-tion, but schematically this is justified. Moreover,after summation and combination with the “QMenergy” of the real QM atoms, simply the total QMenergy (of the capped QM system) is recovered.

For the forces working on the atoms, the sameprocedure is followed. That is, the following ex-pression is valid for the gradients of the interactionenergy:

gQM/MM � �j�MM∧j�LM

i�realQM

gijMM � �

n�1

Nlinks �i�realQM

gi�LC,nQM

� gi�LC,nMM � gi�LM,n

MM (4)

This expression follows directly from using Eq. 3for the energy expression. Note that this expressionis one of the things where the AddRemove modeldiffers from other link models (such as ONIOM[10–12]); this topic is discussed below in more de-tail.

For convenience, the gradient of the artificial LCatoms is set to zero, because any resulting nonzerogradient would be the result of the inability of theMM force field to represent QM interactions; thiseffect is ignored also for the MM system. Moreover,the position of the capping atom is not a degree offreedom, and leaving the gradients of the cappingatoms nonzero would affect only the convergenceof the optimization procedure of the QM systemand not the positions of the atoms.

The electrostatic interactions between the realQM and MM systems are obtained in a classical

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manner but can be handled in an iterative way(updated electrostatic coupling): during the optimi-zation procedure, the charges of the QM system areupdated after each QM optimization cycle and usedin the next MM optimization cycle. The charges forthe QM system are obtained from a recently devel-oped charge analysis, which is derived from anatomic multipole expansion (the multipole derivedcharge analysis) [20]. In the ADF program [14], theCoulomb potential can be obtained from a set ofatomic multipoles [21] (together with a short-rangefunction). These atomic multipoles are reproducedexactly by the charges and therefore the molecularmultipoles as well as the Coulomb potential. In thisapproach, we take as much data from the QM cal-culations as possible.

Comparison with Other Link Models

The expression for getting the interaction energyseems similar to ones used by other link models, forinstance, the ONIOM model [10–12], when the lat-ter is used with two layers. However, this similarityis present only in the energy expression. The spe-cific implementation of this expression differs to agreat extent.

Unlike the ONIOM model [10–12], the degreesof freedom of all real atoms are completely free,that is, all real atoms are completely free to move.This is in contrast to the ONIOM method, wherenot only the bond between the real quantum atomLQ and the capping atom LC but also the bondbetween the real atoms LQ and LM is frozen at acertain predefined distance. This is similar to theintroduction of the IMOMM/ADF [15–17] model�-parameter, upon which the energy and geometrydepends. In fact, it is found that depending on thespecific value given for this parameter (g parameterin ONIOM terminology), the bond between the realLQ and LC atoms ranges from 1.46 to 1.67 Å for aCC bond when the parameter � is varied from 0.5 to1.0 [10].

More importantly, the gradient expressions donot follow naturally from the energy expressions;unlike our implementation, where not only the en-ergy but also the gradient is corrected (see Eqs. 3and 4), in the ONIOM method the gradient of thecapping atoms is projected onto the LQ and LCatoms. This not only introduces the coordinates ofthe capping atoms as degrees of freedom but, moreimportantly, introduces inconsistencies in the totalpicture of the system. Finally, unlike the ONIOM

method, which needs several calculations for sev-eral systems on a different level that are coupled ina manner that is not always well defined, theAddRemove model can be used directly within aQM scheme. The only modifications needed are theoptimizations of the MM system before every QMstep, followed by adding the QM/MM interactionenergies and forces to the QM and MM forces to getthe total energies and forces working on the atoms.In this respect, the AddRemove model is more sim-ilar to the QMPot model by Sauer and Sierks [13].However, unlike their work, the interactions be-tween the QM and MM systems are simply takenfrom a classical force field (AMBER95 [19]) that isknown to perform well for treating interactionswithin proteins; therefore, no additional assump-tions have to be made.

Below we compare results from using theAddRemove model with those found for theIMOMM/ADF model [15–17]; although there aredifferences between the IMOMM [15–17] andONIOM [10–12] energy expressions, both use thesame gradient expressions (by projecting the gradi-ent of the artificial link atoms onto its neighboringatoms). As the geometry obtained in a geometryoptimization depends only on the gradient and noton the energy, the IMOMM results are representa-tive also for a two-layered ONIOM model.

Results

In another work [22], we report the results ofusing a QM/MM approach for getting the active-site geometries of copper proteins, where theAddRemove model has been used to couple theQM and MM systems. In that article, the QM sys-tem consists of the complete active site that is sur-rounded by a MM system, which consists of the restof the protein as well as a layer of solvent moleculessurrounding the protein.

To check the validity of the AddRemove modelfor treating the interactions involved in cuttingsuch a biochemical system into a QM and MMsystem, a few amino acid residues were treatedeither by a full QM or a QM/MM description. Asthe main objective in the study on copper proteins[22] is the active-site geometry, in the current studyonly the geometry of these residues is discussed.Moreover, as it is the gradient expression (see Eq. 4)in which the AddRemove differs from other linkmodels (such as ONIOM [10–12]) and the geometryis determined completely by the gradient and not

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180 VOL. 91, NO. 2

by the energy, this enables a meaningful compari-son between AddRemove and other models.

For the QM/MM description, both theIMOMM/ADF [15–17] and the AddRemove modelhave been used. The residues studied are methio-nine (Met), cysteine (Cys), and tyrosine (Tyr). Theamino acid residues were described with a dipep-tide surroundings, where on both sides of the back-bone connection, the peptide bond is included ex-plicitly (i.e., a CHONHCHRCONH2 molecule whereR is the side chain of the residue). The residueswere studied with either the complete molecule inthe QM system (full QM), or with the QM systemcut off at the C-alpha position with hydrogensadded to complete the link bonds (QM/MM). In theQM/MM description, the electrostatic coupling ofthe QM and MM systems was done by either asimple or an update approach. In the latter ap-proach, the charges of the QM atoms are updatedafter every QM geometry optimization cycle,whereas in the simple approach the standard AM-BER95 charges were used [19].

The calculations were performed with the ADFprogram [14, 23], using the Becke [24]–Perdew [25]exchange-correlation potential (xc-potential) in atriple-zeta valence plus polarization (TZP) basis set.In another study [26], where a number of xc-poten-tials available in ADF were tested in several stan-dard basis sets, it was found that the Becke–Perdewxc-potential in that basis set predicts the bond dis-tances of a certain set of molecules with an averageabsolute deviation of 1.24 pm. This value gives anindication of the accuracy of the bond distances thatcould be reached.

The most relevant property to look at in thesecalculations is the distance between the real atomsinvolved in the link bonds; as the system is cut offat the C-alpha position, there are two distances permolecule, which result from the bonds of the C-alpha to the peptide bond on either side (C�–Nbackbone and C�–Cbackbone distances). For conve-nience, the distance between the C� and C� atomsare also given, to allow for a check of the perfor-mance of the methods in that respect. The distancesfor the full QM and IMOMM/AddRemove QMMMdescriptions are given in Table I. These resultsshow a rather poor performance for the IMOMM/ADF method [15–17] to describe the link bonds; theaverage absolute deviation in the C�–Cbackbone bondis 2.3 pm (simple electrostatic coupling) or 2.8 pm(updated electrostatic coupling), whereas it is evenhigher for the C�–Nbackbone bond: 5.5 (simple) and5.3 (updated) pm. The other two regular bonds ofthe C-alpha atoms are well described, with an av-erage absolute deviation of 0.7 pm (C�–C�) and 0.1pm (C�–H; not shown). Taking the C�–C� bond as areference for the accuracy that could be reached byputting part of the molecule in the MM systemresults in an increase by a factor of 3.1–4.0 for theC�–Cbackbone bond deviation and 7.1–7.3 for the C�–Nbackbone bond.

The AddRemove model performs much better inthat respect, with average absolute deviations of 0.9(simple) or 0.6 (updated) pm for the C�–Nbackbonebond and 1.0 (simple) and 1.2 (updated) pm for theC�–Cbackbone bond. The regular bonds are equallywell described, as indicated by the average absolutedeviation of 0.8 pm for the C�–C� bond and �0.1

TABLE I ______________________________________________________________________________________________Link bond distances (pm) from full QM and QMMM studies.

Full QMIMOMM,

simple electrostaticsIMOMM,

updated electrostaticsAddRemove,

simple electrostaticsAddRemove,

dated electrostatics

CysteineC�–N 146.0 151.7 151.1 147.1 146.7C�–C 155.6 157.8 158.9 154.3 155.4C�–C� 153.3 153.6 153.5 152.5 152.6

MethionineC�–N 145.8 152.1 151.6 147.3 145.7C�–C 154.2 157.1 157.7 153.9 155.2C�–C� 154.7 154.3 154.4 153.4 153.2

TyrosineC�–N 146.9 151.5 151.8 147.0 147.8C�–C 155.9 157.7 157.6 154.4 153.4C�–C� 153.1 154.5 154.7 152.7 153.0

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pm for the C�–H bond (data not shown). By com-paring again the accuracies of the link bonds withthe one found for the C�–C� bond deviation, wefind factors of 1.3–1.6 for the C�–Cbackbone bond and0.8–1.2 for the C�–Nbackbone bond, which is a sub-stantial improvement over the results of theIMOMM model.

Conclusions

The division of a system under study in a quan-tum mechanical and a classical system in QM/MMcalculations is sometimes very natural (as in thecase of solvation of an organic compound, wherethe organic compound can be treated at the QMlevel and the solvent at the MM level), but a prob-lem arises in the case of bonds crossing theQM/MM boundary. A new link model that uses acapping (link) atom to satisfy the valences of thequantum chemical system is presented, with theposition of the artificial capping atom depending onthe positions of the real atoms involved in the linkbond. Therefore, the introduction of the cappingatom does not lead to additional degrees of free-dom; moreover, the degrees of freedom of all realatoms are kept (unlike other link models). Further-more, the introduction of this artificial atom is cor-rected for after every QM optimization cycle bysubtracting the classical molecular mechanics inter-actions with the real QM system it would have if itwere a classical atom. Charges of the real QM atomscan be updated after every QM geometry cycle andused in the electrostatic coupling between theQM/MM systems in the subsequent optimizationof the MM system.

The new model is compared with other (well-known) link models and tested in comparison withthe other link model implemented in the ADF pro-gram (IMOMM/ADF) on a few amino acid resi-dues. In these test calculations the residue is de-scribed either by a complete QM description (fullQM) or by a QM/MM description, where the QMsystem of the residue has been cut off at the C-alphaposition. For the QM/MM description, both theAddRemove and IMOMM models have been used,with either simple or updated electrostatic couplingbetween the QM and MM systems. The distancesbetween the C-alpha and the other two real atomsinvolved in the link bonds differ only marginallywhen using either one of these two electrostaticcouplings. This is a positive result, especially inview of the systems where the QM/MM method

will be applied to (metalloproteins). For those sys-tems, there are no predefined charges in the forcefield, and one has no other option than to use theupdated ones.

A large difference is found between the perfor-mance of the IMOMM/ADF and AddRemovemodels. Taking the deviation for predicting the dis-tance of the regular C�–C� bond (0.7 pm) as areference value when part of the system has beenput in the MM system, the IMOMM model gives arather poor performance: the deviation increases bya factor of 3.1–4.0 for the C�–Cbackbone bond and7.1–7.3 for the C�–Nbackbone bond. The AddRemovemodel gives a much better description; the devia-tion changes only slightly, by a factor of 1.3–1.6 forthe C�–Cbackbone bond and 0.8–1.2 for the C�–Nbackbone bond.

ACKNOWLEDGMENTS

The investigations were supported by the Neth-erlands Foundation for Chemical Research (NWO/CW), with financial aid from The Netherlands Or-ganization for Scientific Research (NWO) andUnilever Research Vlaardingen.

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