Advancing beyond the atom-centered model in additive and nonadditive molecular mechanics

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Advancing Beyond the Atom-CenteredModel in Additive and NonadditiveMolecular Mechanics

RICHARD W. DIXON, PETER A. KOLLMANDepartment of Pharmaceutical Chemistry, University of California, San Francisco, San Francisco,California 94143

Received 28 March 1996; revised 1 April 1997; accepted 2 April 1997

ABSTRACT: A computational approach to the inclusion of off-center chargesin both additive and nonadditive molecular mechanics calculations is presented.The additional sites in the molecular skeleton are placed in the approximatelocations of the chemically intuitive electron lone pair, and are treated as formalparticles throughout the calculation. The increase in the number of charge sitesresults in overall improvement in the energy associated with the angulardependence of hydrogen bonds and improved statistical accuracy of theelectrostatic potential derived charges. The addition of lone pairs also results inimproved accuracy in relative solvation free energy calculation for the pyridineto benzene and methanol to methane mutations. Because the number of atomsthat require lone pairs is small, the extra accuracy can be achieved with littlecomputational overhead. Q 1997 John Wiley & Sons, Inc. J Comput Chem 18:1632]1646, 1997

Keywords: force field; electrostatics; hydrogen bonding

Correspondence to: R. W. DixonContractrgrant sponsor: National Institutes of Health; con-

tractrgrant numbers: P41-RR01081, GM-29072Contractrgrant sponsor: National Science Foundation; con-

tractrgrant number: CHED-94-17458This article includes Supplementary Material available from

the authors upon request or via the Internet at ftp.wiley.comrpublicrjournalsrjccrsuppmatr18r1632 or http:rrjour-nals.wiley.comrjccr

( )Journal of Computational Chemistry, Vol. 18, No. 13, 1632]1646 1997Q 1997 John Wiley & Sons, Inc. CCC 0192-8651 / 97 / 131632-15

BEYOND THE ATOM-CENTERED MODEL

Introduction

he electrostatic energy is often the major con-T tribution to molecular interactions in chemi-cal and biochemical systems.1 It is crucial to theunderstanding of reactivity, and molecular pro-cesses in general, that electrostatic forces be de-scribed as accurately as possible.2 In the case ofmolecular mechanicsrdynamics studies of systemsof organic and biochemical interest, the total en-ergy is usually described by a set of simpleequations, the molecular mechanical force field,empirically parameterized from experimental orquantum mechanical data. The electrostatic por-tion of the force field is usually of the simple form:

n n1 q qi j Ž .E s 1Ý Ýel ec 2 « Ri ji-j j

where q and q are the formal charges associatedi jwith atomic centers i and j, R is the distancei jbetween them, and « is the dielectric constant. Ananalogous equation is implemented in MM23 andMM34 for the interaction between point dipoles.

� 4How can one derive the charges q or theianalogous point dipoles? There are two mainmethods: first, one can derive them empirically,based on experiment andror quantum mechanical

Ž 3 ] 6.calculations e.g., gas phase hydrogen bonds .Alternatively, one can derive them using an analy-sis of the quantum mechanical charge distribution,either through distributed multipole analysisŽ .DMA , or by statistically fitting to reproduce thequantum mechanically calculated electrostatic po-tential outside of the van der Waals envelope ofthe molecule. Although there are advantages anddisadvantages to each approach, as discussed else-where,7 it is clear that use of electronic structurecalculations has the potential to give the moreaccurate charge distribution. The DMA approachis promising; however, it has not yet reached alevel of robustness necessary for use in the devel-

� 4opment of q for general force field calculations,iwhereas electrostatic potential derived chargeshave. It is clear that such electrostatic potentialcharges can be particularly powerful because theyfit very well not only the first nonvanishing multi-pole of the molecule, but higher order moments aswell. The pioneering work in this area9,10 was

� 4followed up by the derivation of q for use in aiprotein and nucleic acid force field.11,12 By means

of these potential derived charges, force field cal-culations have been successful in reproducing in-teraction and conformational energies, liquid prop-erties, and relative free energies of hydration andbinding.7,12 ] 15

The organicrbiochemical force field derived us-ing 6-31G*16 electrostatic potential charges7 is im-proved over that presented in Ref. 12, probably

� 4because its q is more balanced with respect toiŽ 17 18.models of water TIP3P and SPCrE used with

it. It is our opinion that the most important re-maining inaccuracies vis-a-vis the ability of the`force field to simulate structures and energies ofmolecules of organic and biochemical interest are:Ž .1 the use of an effective two-body model to

Ž .represent many-body effects; and 2 the use ofcharges centered only on atoms. This latter approx-imation was chosen for consistency with the abovewater models. Given the crucial nature of the elec-trostatic contribution to the total energy, these twoinaccuracies appear to be a logical place to attemptto improve the quality of the force field. One of thedrawbacks of forcing the charges to be located atthe atomic centers is, in general, lower accuracyw x19 , and, in particular, the lower accuracy of suchmodels in representing directionality of hydrogen

w xbonds 20 .In our earlier force field, drawing on the results

of our ab initio calculations on H S ??? HF,21 we2did include lone pairs on sulfur. These were notincluded in our later force field only because sul-fur is so rarely a hydrogen bond acceptor in pro-teins.22 Although force fields often accuratelyreproduce the lowest energy conformation of hy-drogen bonded systems, suboptimal structures canbe poorly represented. Average properties such assolvation, liquid structure, and energetics are oftenhandled well, but detailed interactions may suffer.These interactions will be crucial in any problemdependent upon specific atomic interactions, suchas protein]ligand design. Take for example theconformational change represented in Figure 1. Ascan be seen, the force field significantly underesti-mates the energy cost of this conformationalchange. To improve the molecular mechanics de-scription of this type of system, it is likely to benecessary to impart some nonspherical character tothe hydrogen bond acceptor.

Several alternative methods designed to accom-plish this present themselves, some of which sub-stantially increase the computational burden of theunderlying model. Such approaches includeadding, in a general fashion, higher order electriceffects on all atoms.19,24 Although this methodol-

JOURNAL OF COMPUTATIONAL CHEMISTRY 1633

DIXON AND KOLLMAN

FIGURE 1. Conformational change involving pyridine ??? HOH hydrogen bond.

Ž . Ž . Ž . Ž .Model E tot, 1808 E int, 1808 E tot, 908 E int, 90* DE

6-31G*rr6-31G* y322.7155701 y5.65 y322.7102914 y2.34 I3.31MP2r6-31G*rrG-31G* y323.6884628 y7.61 y323.6821843 y3.67 I3.94AMBER 4rTIP3P y9.0951 y5.25 y8.6571 y4.81 I0.44

ogy may be useful in some cases, what is proposedhere is a much simpler approach which retainsalmost all of the power and efficiency of the origi-nal force field model, while improving its applica-bility. In particular, we will delineate the effect ofincluding formal lone pairs in the atomic descrip-tion of the chemical systems on force field perfor-mance. In addition to the simplicity of the pro-posed model, the added flexibility should allowthe use of more realistic charge distributions; thatis, ones in which calculated electric moments moreclosely reproduce experimental measurements,with both additive and nonadditive force fields.

As noted previously, in two-body additive forcefield development, one seeks a charge distributionthat implicitly includes polarization effects. This isthe logic behind the choice of the 6-31G* basis setfor electrostatic potential generation,7 in spite of itsconsistent overemphasis of dipole moments. How-ever, as force fields go beyond the two-body ap-proach, and include polarization, it will be essen-tial to have a more precise charge distribution thatreproduces more than just the first nonvanishingmultipole moment. For example, an atom-centeredmodel is inherently incapable of reproducing boththe dipole and quadrupole moment of the watermolecule.25 If the restriction is made that calcu-lated electric moments must be accurate, lone pairswill be one of the few ways in which interactionenergies and hydrogen bond directionality can befine tuned in a force field, particularly a nonaddi-tive one. The results presented will more clearlyillustrate this point.

It should be pointed out that lone pairs havebeen included in many treatments of electrostaticsprior to this investigation.26 ] 29 Several investiga-tors have explored the effect of off-atom chargesites on the accuracy of charge-fitting proce-dures,19,29 and have concluded that adding someatomic anisotropy can noticeably improve thecharge description. Lone pair containing models of

the water molecule for use in liquid simulationshave also been proposed, notably ST2,30 whichcontains two tetrahedrally oriented lone pairs, andTIP4P,17 which includes a single off-atom chargesite on the hydrogen side of the molecule. Weearlier showed that in a model of water with lonepairs, if the position of the lone pairs is optimized,the lone pairs invert to a TIP4P-like orientation,31

which was also found by Popkie et al.32 The watermodels with lone pairs have proven useful, but arenot dramatically more effective than water modelswithout lone pairs.17 As noted previously, an earlyversion of the AMBER force field12 included lonepairs on sulfur atoms, which have not been re-tained in the current version.7 The MM23 forcefield included lone pairs on sp3-hybridized oxygenand nitrogen, which were abandoned in the MM34

force field in an effort to accurately reproducevibrational frequencies. In some of the above cases,the inclusion of lone pairs was found to be useful,but was eventually superseded by other improve-ments in force field design, such as larger trainingsets, availability of higher level ab initio chargesfor more substantial molecules and molecular frag-ments, and longer simulation times for model test-ing and validation. In many ways, some of theearlier force fields had not yet reached the limit ofaccuracy inherent in the chosen functional form.Thus, investigators were not in a position to assessthe importance of inclusion of off-center charges,because the atom-centered model had not beenmade as accurate as possible. Recent trends inforce field development efforts7 seem to suggestthat this limit is being approached. It is in thisspirit that the current investigation is undertaken.We have only analyzed the effect of lone pairs onthe Cornell et al.7 force field, because for forcefields involving empirically derived charges,3 ] 6 itwould not be possible to separate the effect of‘‘adjusting’’ atom-centered charges from addinglone pairs. In this context, we will also evaluate

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the effect of explicitly including polarization ef-fects as well as considering nonatomic-centeredcharges.

As noted, the inclusion of lone pairs shouldprovide some immediate benefit independent offorce field performance. Because the charges usedare derived by means of a statistical fit to thequantum mechanical molecular electrostatic poten-tial, the addition of degrees of freedom shouldresult in a better fit. As far as force field perfor-mance is concerned, it is hoped that treating spe-cific atom types as multicentered, multichargedobjects will result in significant improvement inmodel performance. The assessment of this im-provement will proceed first by surveying a groupof small molecule, hydrogen bonded systems. Fromthese studies a preliminary set of parameters willbe derived and will be tested in more realisticsituations. In particular, the relative solvation freeenergies of the pyridinerbenzene and the metha-nolrmethane systems will be obtained.

Theory and Methods

The first issue that must be resolved is thenumber and location of additional charge sites forthe various molecules in the test suite. Using thepyridine ??? HOH system of Figure 1 as the pri-mary test case, many different atomicrextra siteconfigurations were examined. Both one and twoextra site models were considered, and it wasfound that a single extra site, in the plane of, andfacing ‘‘out’’ from, the C—N—C angle was mosteffective in reproducing the quantum mechanicalenergy change. Having determined the number ofextra sites and their approximate location, only theN]LP distance needs to be determined. At first, a

˚value of 0.5 A was obtained using two indepen-dent methods, although all values in the range of

˚0.3 and 0.7 A gave good fits. The first was anempirical fit to the differential quantum mechani-cal energy in Figure 1, and the second was anoptimization of the extra site position with respectto the statistical accuracy of reproduction of the

˚electrostatic potential. The 0.5-A extra site bonddistance was initially chosen for all second rowelements regardless of hybridization. In the case ofsulfur, a better fit to the potential was found with

˚this distance increased to 0.75 A. Several of themolecules in the test suite were studied with thesestandard values, with promising success. Beforeproceeding with a more thorough survey, an anal-ysis of the charge density 33 of many of the

molecules in the test suite was performed to moreaccurately ascertain the ‘‘best’’ general values forthe X]LP distances to be used. These studies sug-gested that X]LP distance should be slightlyshorter than the values noted previously. The

˚O]LP and N]LP distances were reset to 0.35 A˚and the S]LP distance to 0.7 A. All of the distances

for a particular element were very similar, regard-less of hybridization, suggesting that the lone pairdistance could be a transferable parameter. Forthis study, this has been taken as an assumption,with all lone pairs being treated in the same fash-ion.

Quantum mechanical calculations were carriedout using the GAUSSIAN9234 system of programs,and the density functional program DeFT.35 Geom-etry optimizations were performed at the 6-31G*16

level of theory. Single-point energies were ob-tained at the MP236 level of theory for the 6-31G*-optimized structures. Counterpoise correctionswere not applied to improve the quality of theabsolute interaction energies because, at this stage,the main concern is with energy differences. Elec-trostatic potential data were generated at the 6-31G* level for use with the standard, additiveforce field model, and with a triple-z plus polar-ization DFT treatment,37 for use with a polarizableforce field model.

Point charges for all sites were determined bymeans of the RESP14 methodology using electro-static potential data generated at the levels oftheory noted previously. The quality of the de-rived charges was measured by means of the stan-dard error:

2xŽ .S s 2)Nes p

where x 2 is the merit function from the least-squares fit, and N is the number of grid pointses pat which the electrostatic potential was derived.

To incorporate the additional charge sites intothe molecular dynamics calculations, we treatedthem as formal particles. To ensure that the lonepairs would remain relatively fixed, the force con-stants for bond stretching and angle bending wereapproximately doubled from those of standard hy-drogen atoms types. However, to ensure that mo-tions involving hydrogen atoms remain the high-est frequency motions present, the lone pair wasgiven a mass of three. This should ensure thatconventional wisdom with regard to protocol se-

Ž 38 .lection i.e., the use of SHAKE , choice of stepsize, length of simulation, etc., should apply un-


DIXON AND KOLLMAN

changed to molecular dynamics simulations in-volving lone pair systems. Although lone pairs canbe incorporated into models by forcing them tohave a fixed position relative to the heteroatom,39

for testing the model we adopted the simpler ap-proach of assigning mass and force constants tothe lone pair. The complete list of parameters forthe lone pair atom type are shown in Table I. Thismay effect dynamics but not equilibrium proper-ties. Several approaches deliberately vary theatomic masses in the molecular system to improvesampling behavior.40,41 Thus, our use of this ap-proach in solvation free energy studies of pyridineand methanol is justifiable.

Solvation free energies and molecular mechan-ics energies were obtained by means of the AM-BER molecular dynamics program,42 employingthe Cornell et al. force field.7 The free energysimulations were carried out over a 200-ps timescale with a 2-fs time step. SHAKE was applied toall bonds, electrostatic decoupling was employed,and the pmf correction was included to account forbond shrinkage. Initial solvated structures wereequilibrated for 40 ps prior to the free energysimulations. To assess the error inherent in thesecalculations, the simulations were run in reverse,preceded by 20 ps of equilibration at the finalstructure from the forward simulation. Errors re-ported are the difference between the forward andbackward simulations.

Results and Discussion

A set of 21 molecules was chosen as an initialtraining set to test the effect of lone pairs on force

TABLE I.( )Parameters Used for Lone Pair LP Atom Type.

Parameter Value

Mass 3.0 amu˚( )r X]LP , X g N, O 0.35 Aeq

˚( )r X]LP , X g S 0.70 Aeq( )K X]LP , X g N, O, S 600.0r

2( )u LP]X]Y , X g sp N, O, S 120.08eq( )u LP]X]Y , pyridine 120.08eq( )u LP]X]Y , imidazole, furan, oxazole 126.08eq

3( )u LP]X]Y , X g sp N, O 109.58eq3( )u LP]X]Y , X g sp S 90.08eq

( )K LP]X]Y , X g N, O, S 150.0u

r* 0.00e 0.00

field performance. There are examples of moleculescontaining sp nitrogen and sp2, sp3, and aromaticnitrogen, oxygen, and sulfur atoms. The first issueregarding the effect of lone pairs on these systemsis the quality of the charge derivation. In Table II,data are presented that address this question. Foreach molecule considered, electrostatic potentialfield data have been generated using two quantummechanical methods. The first is a Hartree-Fockscheme and the second is a density functionalmethod. The Hartree]Fock calculation involves thenow standard 6-31G* basis set, whereas the den-sity functional calculation employs triple-z pluspolarization basis set on heavy atoms and a dou-ble-z plus polarization on hydrogen. This basis sethas been shown to well reproduce experimentalgas-phase dipole moments.37 For both of thesepotentials, RESP point charges are derived for eachmolecule both with and without lone pairs. In eachof these cases, the lone pairs were placed in thechemically intuitive positions. The molecules con-sidered, along with a schematic representation ofthe explicit lone pairs included for each case, andthe charge values obtained are available as Supple-mentary Material. The resulting dipole momentsand standard errors from these calculations arereported, as well as the quantum mechanical andexperimental dipole moments. As can be seen fromthese data, in almost every case, inclusion of lonepair sites improves the quality of the charge fit.The amines and dimethylether show a consistentworsening of the charge fit, whereas ammonia andacetamide show a slight decrease in accuracy forthe 6-31G* potential. In all of these cases, however,this loss of accuracy was due to the extra con-straints associated with the RESP charge-fittingprocess, either the harmonic restraint or the pro-cess of equivalencing interchangeable atoms. Uponremoval of all constraints, every molecule showedimprovement in the charge-fitting accuracy due tothe inclusion of lone pairs. This is consistent withpurely statistical considerations. It has been noted43

that chemical lone pairs are often not the mosteffective means of improving the quality of de-rived charges. Although this may be true from apurely statistical point of view, the most importantcriterion to evaluate is force field performance. Itseems more reasonable, therefore, to stay with aphysically justifiable picture of molecular struc-ture. However, many nonstandard charge distribu-tions were evaluated with the result that thechemically intuitive lone pair structures alwaysperformed better in force field calculations. This isundoubtedly due to the fact that optimal hydrogen

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TABLE II.Charge Fitting Dataa.

b( )Molecule Par 6-31G* HF HF, LP DFT tz DFT DFT, LP Expt.

Pyridine m 2.3201 2.2981 2.3163 2.2041 2.1702 2.1838 2.19y3std. err = 10 — 1.82 0.85 — 2.24 1.51 —

Imidazole m 3.8625 3.8096 3.8595 3.6728 3.5976 3.6421 3.87y3std. err = 10 — 2.86 1.84 — 2.98 1.98 —

Oxazole m 1.5833 1.5546 1.5814 1.6194 1.6016 1.6107 —y3std. err = 10 — 3.40 1.87 — 3.37 1.88 —

Methanimine m 2.2435 2.2210 2.2180 1.9760 1.9232 1.9422 1.971y3std. err = 10 — 1.89 1.75 — 2.24 1.74 —

Methyliminomethane m 1.7372 1.8135 1.9469 1.4847 1.6413 1.7746 1.53y3std. err = 10 — 2.83 2.15 — 3.40 2.80 —

Hydrogen cyanide m 3.1693 3.2015 2.8962 2.9315 2.95y3std. err = 10 — 1.12 0.42 — 1.41 0.73 —

Ammonia m 1.9189 1.9552 1.9610 1.5304 1.6074 1.5958 1.47y3std. err = 10 — 2.66 2.74 — 4.07 3.91 —

Methylamine m 1.5337 1.7659 1.9222 1.3120 1.6165 1.7357 1.30y3std. err = 10 — 3.85 3.97 — 4.32 4.63 —

Dimethylamine m 1.1421 1.4540 1.7053 0.9506 1.3600 1.6692 1.03y3std. err = 10 — 3.58 4.03 — 4.16 5.06 —

Trimethylamine m 0.7440 0.9859 1.2375 0.4179 0.7646 1.2088 0.63y3std. err = 10 — 2.56 3.49 — 3.15 5.31 —

Methanol m 1.8674 1.9664 2.0038 1.6056 1.6917 1.7315 1.662y3std. err = 10 — 2.82 2.62 — 3.31 3.08 —

Dimethylether m 1.4777 1.6240 1.7030 1.1434 1.3089 1.4247 1.30y3std. err = 10 — 2.12 2.31 — 2.58 2.95 —

Formaldehyde m 2.6642 2.6622 2.6823 2.2003 2.1941 2.2110 2.33y3std. err = 10 — 1.30 0.82 — 1.77 1.05 —

cFuran m 0.7717 0.7447 0.7831 — — — 0.66y3std. err = 10 — 2.72 1.91 — — — —

Formamide m 4.1010 4.0844 4.1145 3.8142 3.7856 3.8129 3.73y3std. err = 10 — 1.87 1.39 — 1.79 1.38 —

Acetamide m 4.0586 4.0543 4.1019 3.7640 3.8033 3.7943 3.76y3std. err = 10 — 1.21 1.33 — 1.28 1.25 —

N-methylacetamide m 4.0424 4.1065 4.1495 3.7563 3.8193 3.8486 3.73std. err = 10y3 — 1.72 1.64 — 1.97 1.82 —

dMethanethiol m 1.7875 2.0319 1.9422 — — — 1.52y3std. err = 10 — 4.56 3.51 — — — —

dDimethylthioether m 1.7969 1.5510 1.7398 — — — 1.50y3std. err = 10 — 6.05 4.98 — — — —

dMethanethial m 2.2325 2.2788 2.3074 — — — 1.647y3std. err = 10 — 3.76 3.60 — — — —

dThiophene m 0.8973 0.8869 0.9122 — — — 0.55y3std. err = 10 — 2.30 1.86 — — — —

a All dipole moments in debyes and standard errors in ey/ bohr3.b From Ref. 55.c DeFT calculation failed to converge.d DeFT basis set not available for sulfur.

bonding configurations almost always correspondto maximized donorrlone pair interactions.44 Aswill be presented in what follows, inclusion of lonepairs will not always dramatically improve forcefield performance, but in no case was an atom-

centered or counterintuitive lone pair model supe-rior to the chemically intuitive lone pair model.

As noted previously, there have been otherstudies that have attempted to include off-atomcharge sites in statistical fits of molecular electro-


DIXON AND KOLLMAN

static potentials. Some of these address issues per-tinent to the current line of investigation and de-serve some comment, particularly in view of thefact that our approach yields somewhat differentresults. The investigation of optimal lone pair sitesin a series of azabenzene molecules, undertaken byWilliams and Weller,23,24 indicated that lone pairswere essential to describe accurately the crystalstructures of these compounds with molecular

Ž .packing analysis MPA . However, the optimalN]LP distances were considerably shorter than the

˚0.35-A proposed here. In that study, mostmolecules had optimal off-atom charge sites within

˚0.1 A of the nitrogen atom. We have performed thesame analysis of the six azabenzene moleculespresented in that study with respect to our RESPcharge-fitting procedure. In our case, we find thatoptimal N]LP distances, with respect to the rmsfit to the electrostatic potential, are in the range of

˚0.40 to 0.50 A. We can only conclude that choice ofbasis set, external point selection criteria, and theparticular features of the RESP algorithm accountfor the difference. It should be noted that, for thepractical application of crystal packing calculationsundertaken by Williams and Weller, The N]LP

˚distance was uniformly lengthened to 0.25 A, inspite of indications of the charge fit results. Themodest increase in the observed rms fit error thischoice imposed was justified by the quality of thecrystal packing calculations and the more intu-itively reasonable charge values obtained. We alsowish to use the performance of the model in forcefield calculations as our main criterion for qualityassessment. This is not to say that we are notconcerned with the most effective way to repro-duce the quantum mechanically calculated molec-ular electrostatic potential—quite the contrary.However, a highly accurate potential model whichperforms poorly in force field calculations is oflittle value for our current investigations.

The other major body of work investigatinginclusion of off-atom charge sites in moleculescenters around the water molecule. As noted pre-viously, optimization of lone pair positions in thewater molecule lead to a collapse of off-atom sitesinto a single site, usually located along the C2 vaxis on the hydrogen side of the molecule. Greatersuccess in developing improved solvent modelsfor water that include off-atom sites has been notedfor those models which place a single site withinthe H—O—H angle rather than two arranged tet-rahedrally. We have explicitly investigated thislone pair arrangement in the sp3 oxygen systemsincluded in our study, methanol and dimethyl

ether, and found that the tetrahedral arrangementis at least as good as, if not superior to, the stan-dard single-site model in force field calculations.We have attempted to optimize the lone pair posi-tion with respect to the fit the electrostatic poten-tial, and did not observe the lone pair collapsewhich is usual in the water calculation. The opti-mal arrangement from this criterion was tetrahe-

˚dral, with an O]LP distance of 0.45 to 0.50 A,although a fairly broad range of distances resultedin only minor increases in the observed fit error. Asingle-site model for sp3 oxygen would probablywork for the molecules considered; however, thereis no compelling reason from our initial investiga-tions to suggest that this is the preferred model forany molecule other than water. Indeed, it may bethat water is an unique model subject, and that therequirements of a good solvent model for molecu-lar simulations lead to the improved performancefor single off-site models. In our investigations,which do not include water or any neat solventcalculations, we have chosen to stay with thechemically intuitive lone pair orientation for allmolecules considered, including those containingsp3-hybridized oxygen.

As noted previously, the most important ques-tion to be answered is how the inclusion of lonepairs affects force field performance. As a firstsimple step toward this end, all of the smallmolecules considered here were optimized in thegas phase, both with and without lone pairs. Theresults of these calculations are available as Sup-plementary Material. In addition to the standardmolecular mechanics model, a methodology incor-porating atomic polarizabilities45 was also em-ployed. The atomic polarizabilities employed inthis model are taken from Applequist et al.,46 anddo not include a value for sulfur. A value of 1.25˚3A was determined by noticing that changing oxy-

Ž .gen to sulfur e.g., H O to H S increases the2 2molecular polarizability by a factor in the 2.5 to 3.0range.47 Therefore, the oxygen atomic polarizabil-ity presented by Applequist et al., approximately0.47, increased by the aforementioned factor, yields

˚3 ˚3a sulfur polarizability between 1.18 A and 1.41 A .The final value was chosen as a compromise be-tween these two values.

The point charges used for the standard modelwere derived from the 6-31G* potential, whereasthe polarizable model charges were obtained fromthe DFT-generated field. The exceptions to thisrule include furan, for which the DFT calculationfailed to converge, and the sulfur compounds, forwhich there is no basis set available. In these cases,

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the 6-31G* potential charges were scaled by 0.88for use in the polarizable model.48,49 The standardreasoning behind this distinction in charge modelsis that 6-31G* already contains some ‘‘implicit’’polarization as demonstrated by its consistent

Ž .overestimation of dipole moments see Table II .This can be beneficial in aqueous solution calcula-tions which do not explicitly include a polarizationterm, particularly in view of the fact that the TIP3Pand SPC water models include such ‘‘implicit’’polarization. In the case of the polarizable model,however, this overestimation of the dipole mo-ment leads to error if the resulting charges areused unchanged. This is the reasoning behind thechoice of an unusually large basis set for the DFTcalculation performed; that is, the calculated dipolemoments are much more consistent with gas phaseexperimental values.

For each of the molecules considered in thisstudy, energies associated with both optimal andsuboptimal hydrogen bonded systems were ob-tained. In every case, both the quantum mechani-cal and molecular mechanical models had nearly

linear X ??? H—O angles. The X ??? O distances weresimilar among all the models with the molecularmechanics treatments resulting in slightly shorter

˚distances, on the order of 0.2 A, than the quantummechanical results. The particular conformationsconsidered are presented in Table VI. In all cases,the hydrogen bond donor was a water molecule,specifically at TIP3P17 water molecule for the ad-ditive force field calculations, and a POL348 watermolecule for the polarizable force field calcula-tions. In addition, the formamide dimer was con-sidered as a representative of the type of hydrogenbonding found in protein structures. The results ofthese calculations are summarized in Table III andreported in full in the Supplementary Material.The results can best be summarized by consider-ing atom types and hybridizations of the acceptoratoms as families. Comparison with ab initio ener-gies refers to the MP2r6-31G*rr6-31G* valuesand, in the case where three conformations havebeen reported, the best and worst are compared.Even though comparisons are being made against

TABLE III.Relative Conformational Energy Data.a

b b b b bSystem 6-31G* MP2 AMBER AMBER POL AMBER LP AMBER POL]LP

Pyridine ??? HOH y3.31 y3.94 y0.44 y0.64 y3.51 y5.44Imidazole ??? HOH y2.81 y3.03 q0.22 q0.58 y3.52 y4.72

( )Oxazole ??? HOH N-acceptor y2.97 y3.18 y0.85 y0.44 y3.34 y5.13( )Oxazole ??? HOH O-acceptor y1.75 y1.77 y1.28 y0.79 y2.83 y3.22

Methanimine ??? HOH y2.92 y3.53 y1.48 y1.55 y3.83 y3.75Methyliminomethane ??? HOH y3.26 y3.81 y0.53 y0.71 y3.27 y4.61Hydrogen cyanide ??? HOH y1.24 y1.53 q0.19 q0.04 y1.11 y1.50Ammonia ??? HOH y3.53 y4.06 y2.30 y2.16 y4.87 y3.67Methylamine ??? HOH y4.07 y4.90 y2.98 y3.31 y5.05 y4.35Dimethylamine ??? HOH y5.55 y7.13 y5.10 y4.07 y6.60 y6.54Trimethylamine ??? HOH y5.50 y7.49 y3.40 y2.88 y4.38 y4.13Methanol ??? HOH y1.29 y0.76 y1.40 y1.10 y1.52 y1.43Dimethylether ??? HOH y1.41 y0.79 y0.87 y0.64 y1.01 y0.91Formaldehyde ??? HOH y2.46 y3.12 y1.03 y0.99 y1.97 y2.16FutAn ??? HOH y0.95 y0.82 q0.63 q1.14 y1.43 y0.82

cFormamide ??? formamide y3.42 y2.24 y2.04 y0.90 y2.78 y1.76Formamide ??? HOH y2.84 y3.54 y1.15 q0.06 y3.50 y2.03Acetamide ??? HOH y2.37 y2.69 y1.53 y0.51 y3.15 y1.86N-methylacetamide??? HOH y1.96 y2.36 y0.86 y0.77 y2.94 y3.08Methanethiol ??? HOH q1.01 q1.31 q0.44 q0.34 q1.29 q1.58Dimethylthioether ??? HOH q1.05 q1.41 q0.28 q0.33 q1.59 q1.54Methanethial ??? HOH y1.86 y2.32 y0.86 y0.69 y0.98 y0.78Thiophene ??? HOH q0.49 q0.48 q0.92 q1.07 q0.90 q0.99

a All energies in kilocalories per mole representing difference between best and worst conformers in Table VI.b Hartree]Fock energies obtained at the level of theory noted with 6-31G* structures. AMBER calculations performed using theforce field in Ref. 7.c Structure constrained to single H-bond conformation.


DIXON AND KOLLMAN

TABLE IV.Overall Errors in Model Performancea

b c( ) ( )E interaction, optimal DE conf.

MM model MP2 6-31G* MP2 6-31G*

Average absolute errorAMBER 4 / TIP3P 1.313 0.843 1.711 1.305AMBER 4 POL / POL3 1.958 0.940 1.985 1.602AMBER 4-LP / TIP3P 0.469 1.541 0.654 0.617AMBER 4 POL-LP / POL3 1.068 1.057 1.014 1.017

Root mean square errorAMBER 4 / TIP3P 1.579 1.632 2.021 1.538AMBER 4 POL / POL3 2.232 1.361 2.286 1.826AMBER 4-LP / TIP3P 0.758 1.745 0.921 0.700AMBER 4 POL-LP / POL3 1.428 1.511 1.383 1.264

a All energies in kilocalories per mole.b Absolute interaction energy of optimal configuration.c ( )Relative conformational energy between best and worst conformers see text .

absolute interaction energies, it is clear that thesevalues are not the most reliable. Absolute interac-tion energies can be difficult to calculate accu-rately.50 For this reason, errors are reported inTable IV against both MP2 and 6-31G* values.From these data, one can get an overall sense ofthe improvement in model performance due to theinclusion of lone pairs. Because of the fact that theab initio calculations at the 6-31G* and 6-31G*rMP2 levels of theory exaggerate the hydro-gen bond energies, particularly when the counter-poise correction is not applied, our goal is not toreproduce these values. We do consider it signifi-cant, however, that the models with lone pairs dosignificantly improve the relative conformationalenergies which do more accurately reflect hydro-gen bond directionality. Overall improvement isabout a factor of two for predicting hydrogen bonddirectionality. These results will be examined inmore detail.

NITROGEN ACCEPTORS

For aromatic nitrogen, specifically pyridine imi-dazole, and oxazole, the inclusion of a single lonepair dramatically improved discrimination be-tween the hydrogen bonded conformations stud-ied. In these cases, the optimal hydrogen bond wasalong the N]LP vector whereas the suboptimalconfiguration was perpendicular to the plane ofthe molecule. The sp2 nitrogen-containing system,methanimine and methyliminomethane, also ex-hibited a dramatic improvement through the addi-tion of lone pairs. The structures considered inthese instances were the same as the aromaticexample. The optimal hydrogen bond was alongthe N]LP axis and the suboptimal structure wasperpendicular to the plane of the molecule. For theone sp nitrogen compound considered, hydrogencyanide, the addition of a single lone pair also hada dramatic effect. The standard models incorrectly

TABLE V.Free Energy Results for Pyridine and Methanol Simulations.a

b( ) ( )System DDG calc. DDG expt. Diff.

( )Pyridine lp ª benzene 3.95 " 0.05 3.90 0.05Pyridine ª benzene 2.28 " 0.02 3.90 1.62

( )Methanol lp ª methane 7.14 " 0.01 7.11 0.03cMethanol ª methane 6.86 " 0.01 7.11 0.25

a All energies in kilocalories per mole.b From Ref. 56.c From Ref. 57.

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TABLE VI.Hydrogen Bond Conformations.


DIXON AND KOLLMAN

predict the minimum energy structure, whereasthose with the lone pair correctly mimic the abinitio data.

The case of sp3 nitrogen presents a set of mixedresults. Methylamine and dimethylamine showdramatic improvement in both the absolute inter-action energy of the optimal conformation as wellas the conformation energy difference. Perfor-mance of the model for ammonia and trimeth-ylamine was not as positive. In the case of ammo-nia, the charge on the lone pair was positive, andled to a smaller hydrogen bond directionality thanfound quantum mechanically. This may be analo-gous to TIP4P water in which the extra charge islocated ‘‘inside’’ the HOH angle; that is, simplymoving the charge off of the heteroatom is moreeffective than adding more total charge sites. No-ticeably improved results in this case are obtainedby forcing the charge on the nitrogen atom to bezero for the lone pair models. This yields muchbetter interaction and relative conformational ener-gies, and represent the results presented in TableIII and the Supplementary Material. The case oftrimethylamine does not permit such a simplesolution. Overall interaction energies and confor-mational energies are too low for both the additiveand polarizable models. The source of the diffi-culty seems to be in the charge fitting procedure.The standard model employed imposes extra con-straints on methyl and methylene groups. In amolecule such as trimethylamine, there is verylittle flexibility left in the fitting procedure. In thiscase, the limitation results in a charge on the lonepair which is small compared with that found inmethylamine and dimethylamine, y0.45 ratherthan y0.6 electrons. Lengthening of the N]LPbond results in improved model performance, butreproduction of the ab initio data requires imprac-tical bond distances. The model performance canalso be improved by fixing the nitrogen and lonepair atomic charges at the values obtained for theother amines. The interaction energies and relativeconformational energies still fall short of the abinitio values, however, and it was decided that themodel would not be fine-tuned further to treat thismolecule more accurately. It should be pointedout, however, that the inclusion of lone pairs doesimprove model performance; it is quantitativeagreement with quantum mechanical data whichis currently at issue. Some of these difficulties indeveloping amine charge models may be relatedto the difficulties in reproducing the free energy ofsolvation of amines.51,52

In comparing the performance of the additiveand the polarizable force field models overall, onecan see that, generally, smaller interaction ener-gies, and often the relative conformational energiesas well, are obtained with the polarizable model.However, for the aromatic nitrogen acceptors, therelative conformational energies are noticeablylarger with the polarizable model. As in the case oftrimethylamine, this anomaly can be traced to thecharge-fitting procedure. The magnitude of thecharges in these molecules is rather large com-pared with both the 6-31G* RESP charges and theDFT RESP charges in the absence of lone pairs.These large charges destabilize the suboptimalconformations too significantly. The cause of thisseems to be that the lone pair is too close to thenitrogen atom for these molecules and this basisset. The N]LP distance was determined by analyz-ing the 6-31G** charge density, not the triple-zDFT charge density. In addition, recall that prelim-inary studies of pyridine suggested an N]LP dis-tance longer than the one used here. Lengtheningof this bond does indeed result in improved modelperformance in these cases. Improved performancecan also be obtained by fixing one charge value;for instance, the charge on the nitrogen, to somereasonable value. The resulting charges will thenbe more consistent with the expectation of theforce field, specifically, that charges used for thepolarizable model should be ‘‘smaller’’ than those

Žused for the additive model cf. the previous dis-.cussion of basis set and charge derivation . As in

the case of trimethylamine, results have been re-ported for the standard model, rather than thoseassociated with any individual adaptation. Thestandard model does an excellent job on the abso-lute interaction energies of the optimal confor-mations, and should prove useful in further cal-culations. Knowledge of potentially significantrefinements, however, may prove useful if a morespecific force field treatment is required.

OXYGEN ACCEPTORS

Like nitrogen hydrogen bond acceptors, oxygenacceptors play important roles in biochemicalstructure and function. Two examples of aromaticoxygen have been included in the test suite, furanand oxazole. In both cases, the lowest energy con-figuration places the hydrogen bond along theO]LP axis. The addition of a single, lone pair tothe oxygen atom results in noticeable model per-

VOL. 18, NO. 131642


formance improvement. In particular, the standardforce field approach incorrectly predicts the lowestenergy conformation in the case of furan. Theaddition of a lone pair reverses this prediction andmuch more closely mimics ab initio results. Asmentioned previously with respect to the oxazolenitrogen acting as an acceptor, the charges onoxazole can be unusually large, particularly thosederived from the DFT electrostatic potential. As inthe case of nitrogen, fixing the acceptor atomiccharge to some reasonable value leads to muchimproved results. As has been the practice up tothis point, the results of the standard lone pairmodel are reported in Table III SupplementaryMaterial, and individual solutions to particularmodel shortcomings are treated separately.

The one example of pure sp2 oxygen consid-ered, formaldehyde, reflects marked improvementwith the inclusion of lone pairs. In this case, therelative conformational energy is approximatelydoubled with respect to the results without lonepairs. This energy is still lower than expected, butmost of the error has been removed. Further im-provement can be achieved with a lengthening ofthe O]LP bond.

As was the case with nitrogen, sp3-hybridizedoxygen acceptors show mixed results. As was alsothe case previously, these model shortcomings canbe traced to the quality of the charge fit. Formethanol, the addition of lone pairs improves theinteraction energy associated with the lowest en-ergy conformer. The relative conformational ener-gies are already larger than the ab initio valueswithout the lone pairs. Adding these into the sys-tem increases the relative energies further. Theerror is still less than 1 kcal; however, it is clearthat addition of the lone pairs has pushed thisenergy in the wrong direction. The case of dimeth-ylether exhibits almost no effect due to the addi-tion of lone pairs to the system. The optimalconfiguration interaction energy is still low andalmost identical to the value without lone pairs.This is a situation similar to trimethylamine dis-cussed previously; that is, there is little flexibilityin the charge fitting procedure. In this case, themagnitude of the resulting charges is particularlysmall. Simply scaling the charges up results inbetter model performance overall, at the expenseof the calculated dipole moment. As a special case,this seems like a reasonable solution; however, itwould be more satisfying for charges among atomtypes to be more consistent. The results presentedin Table III and Supplementary Material were gen-erated using unscaled charges. Lone pairs can be

useful for oxygen sp3 systems, but, at this point,they must be applied with care. It should be notedthat hydrogen bonding interactions with carbonyland ether oxygen atoms have ‘‘soft’’ bending po-tentials.53 This can make selection of optimal con-figurations challenging from the standpoint ofmolecular mechanics, particularly in light of therestriction that lone pair orientations are fixed.

The amide functionality is of crucial importancein biochemical systems, so it is of particular inter-est to assess the model’s performance in thesecases. In all of the amide ??? HOH systems consid-ered, significant benefit accrued from the inclusionof lone pairs. Both the interaction energies andrelative conformational energies are much moreconsistent with quantum mechanical calculations.In addition, the performance of the model on theformamide ??? formamide system improved notice-ably with lone pairs. In this case, the global mini-mum structure is a cyclic structure involving twohydrogen bonds. This is also the case in the for-mamide and acetamide ??? HOH systems. How-ever, we wished to limit consideration, at thisstage, to singly hydrogen bonded species. There-fore, the optimal structure for the formamide dimerwas restrained to the single hydrogen bond struc-ture illustrated in Figure 2. The suboptimal struc-ture used here fixes the 0 ??? H—N angle at 908.The success of this model on the amide systemssuggests that lone pairs could contribute signifi-cantly to the accuracy of protein modeling studies.

SULFUR ACCEPTORS

The cases involving sulfur atoms seem to followthe trend exemplified by the oxygen systems. The

FIGURE 2. Single hydrogen bonded configuration ofthe formamide dimer.


DIXON AND KOLLMAN

aromatic sulfur acceptor considered, thiophene,shows improvement when lone pairs are added tothe system. The energy improvements are quitesmall, however, they are in the right direction. Inthe sp2-hybridized case, methanethial, modest im-provement in both interaction and relative confor-mational energies is seen. The relative conforma-tional energy is still low relative to the quantummechanical standard and can be improved by in-creasing the magnitude of the atomic charges. Likethe trimethylamine and dimethylether cases dis-cussed previously, methanethial charges seemrather small. Correcting this leads to further im-provement in model performance.

The sp3-hybridized systems, methanethiol anddimethylthioether, show improvement in the inter-action energy of the optimal conformer. However,the relative conformational energies do not im-prove, and the overall trend among the three con-formers is not reproduced. This is particularly pro-nounced in dimethylthioether where the differencebetween conformers I and III should be very closeto 0.0 kcalrmol; that is, the 908 conformer shouldbe of lowest energy, with the tetrahedral arrange-ment very close energetically. However, the en-ergy difference turns out to be significant. A solu-tion to this is to move the lone pairs from thetetrahedral position to perpendicular to the C]S]Cplane. Previous studies have found this arrange-ment of lone pairs to be superior to a tetrahedralarrangement as well.12,44 In this manner, the over-all trends exhibited in the quantum mechanicalcalculations are preserved in the force field calcu-lations, and the interaction and conformationalenergies are reproduced. In view of the clear su-periority of this arrangement of lone pairs forsp3-hybridized sulfur, these results are reported inTable III and Supplementary Material.

FREE ENERGY CALCULATIONS

Calculations on simple hydrogen bonded com-plexes are very important in the development andassessment of force field performance. However, itis essential that the method be applicable to con-densed phase systems as well. In particular, it isnecessary to insure that the changes due to theintroduction of lone pairs has not changed theeffectiveness of the original force field. One of themost important ways to assess force field simula-tions in condensed phases is to evaluate the abilityto calculate relative free energies. This type ofcalculation should provide an important test of thenew molecular description. The relative solvation

free energies of pyridine with respect to benzeneand methanol with respect to methane have beencalculated. The results of these simulations arereported in Table V. The addition of the lone pairon the nitrogen of pyridine results in a dramaticimprovement in the calculated solvation freeenergy. This is particularly satisfying in lightof the improvement shown in the specificpyridine ??? HOH interaction. The methanol simu-lation results are also encouraging. In this case,additional lone pairs provided only modest im-provement in the specific methanol ??? HOH inter-action energies, and the standard force field does avery good job describing the solvation energy.However, the simulation with the lone pairs pre-sent showed improvement over those without. Inspite of the excellent agreement with experimentshown by the calculations involving lone pairs, themost encouraging aspect of these calculations isthe relative changes due to lone pair inclusion. Inthe pyridine case, a large improvement was neededand was obtained. The methanol case requiredonly a small improvement, which was observed.These results seem to indicate that parametersderived from the small molecule gas phase calcu-lation can be effectively transferred to condensed-phase calculations. The quantitative agreementwith experiment should simply be considered for-tuitous. Indeed, it should be pointed out that muchof the error inherent in the pyridine calculationwithout the lone pair can be eliminated by usingatom centered ESP-fit charges, rather than RESPcharges, derived from a much finer field grid. Thisapproach yields much more accurate potential de-rived charges, with respect to reproducing quan-tum mechanical electric moments, which can im-prove the calculated solvation free energy.54 Thesecharges do not perform as well as the RESP chargeswith respect to interaction energies, however. Thispoint is made to indicate that there are othermethods by which particular shortcomings of theforce field may be addressed. It is our hope thatlone pairs provide a more general improvement tothe force field, applicable to a wide variety ofproblems. The free energy results presented hereare a positive endorsement for our choice to in-clude lone pairs.

Conclusion

In this study we have assessed the role of inclu-sion of lone pairs as well as explicit polarization onforce field performance. The effect of the addition

VOL. 18, NO. 131644


of lone pairs on force field performance has beenseen to be positive overall. The addition of chargesites to a molecular framework leads to improvedstatistical accuracy of the potential derived charges.The choice of chemically intuitive lone pair posi-tions for these additional charge sites is a verygood first step and has resulted in dramatic im-provement in discrimination between hydrogenbonded conformations. In one case, sp3-hybridizedsulfur, the lone pair position was refined to beperpendicular to the X]S]X plane. In the fewcases, where consistent agreement with referencequantum mechanical data was not observed, thedifferences are subject to eradication through achange in the parameters associated with the lonepair particle. In particular, careful attention to thevalues of the atomic charges often resulted innoticeable improvement over the ‘‘standard’’model performance. In no case did the addition oflone pairs result in an incorrect minimum energystructure and, in three cases, hydrogen cyanide,imidazole, and furan, the standard force field wascorrected. The success of the free energy simula-tions of pyridine and methanol suggests that thepower and applicability of the force field havebeen retained, with the lone pairs providing in-creased accuracy.

The positive results presented here suggest thatlone pairs, or some other method which incorpo-rates anisotropy into the charge distribution ofdonor centers, deserve to play a role in futureforce field developments. The dramatic improve-ment observed should prove critical in any molec-ular mechanics investigation focusing on specificatomic interactions. In addition, the improved flex-ibility imparted to the polarizable force field modelprovides an excellent avenue for consistent modelperformance improvement. Because the lone pairsneed to be added only at electron donor centers,the additional computational cost should be quitemodest for simulation of macromolecules.

Further work in this area will concentrate onrefining the methodology by which lone pairs areincorporated into the calculations. In addition,other problems of biochemical interest will also beaddressed.

Acknowledgments

R. W. D. thanks Christophe Chipot for the com-parative free energy of solvation data provided,and Paul Popelier for the charge density analysis.

Visualization facilities were provided by the UCSFŽ .Computer Graphics Lab T. Ferrin, PI supported

.by NIH Grant P41-RR01081 . We are also gratefulŽ .for support from the NSF Grant CHE-94-17458

Ž .and the NIH Grant GM-29072 .

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Advancing beyond the atom-centered model in additive and nonadditive molecular mechanics