How Well Does a Restrained Electrostatic Potential (RESP) Model Perform in Calculating Conformational Energies of Organic and Biological Molecules? JUNMEI WANG, PIOTR CIEPLAK, * PETER A. KOLLMAN Department of Pharmaceutical Chemistry, University of California at San Francisco, San Francisco, California 94143-0446 Received 15 September 1999; accepted 21 March 2000 ABSTRACT: In this study, we present conformational energies for a molecular mechanical model (Parm99) developed for organic and biological molecules using the restrained electrostatic potential (RESP) approach to derive the partial charges. This approach uses the simple “generic” force field model (Parm94), and attempts to add a minimal number of extra Fourier components to the torsional energies, but doing so only when there is a physical justification. The results are quite encouraging, not only for the 34-molecule set that has been studied by both the highest level ab initio model (GVB/LMP2) and experiment, but also for the 55-molecule set for which high-quality experimental data are available. Considering the 55 molecules studied by all the force field models for which there are experimental data, the average absolute errors (AAEs) are 0.28 (this model), 0.52 (MM3), 0.57 (CHARMm [MSI]), and 0.43 kcal/mol (MMFF). For the 34-molecule set, the AAEs of this model versus experiment and ab initio are 0.28 and 0.27 kcal/mol, respectively. This is a lower error than found with MM3 and CHARMm, and is comparable to that found with MMFF (0.31 and 0.22 kcal/mol). We also present two examples of how well the torsional parameters are transferred from the training set to the test set. The absolute errors of molecules in the test set are only slightly larger than in the training set (differences of <0.1 kcal/mol). Therefore, it can be concluded that a simple “generic” force field with a limited number of specific torsional parameters can describe intra- and intermolecular interactions, although all comparison molecules were selected from our 82-compound training set. We also show how this effective two-body * Permanent affiliation: Department of Chemistry, University of Warsaw, Warsaw, Poland Correspondence to: P. A. Kollman; e-mail: [email protected]; web site: http://www.amber.ucsf.edu/amber/ Contract/grant sponsor: National Institutes of Health; con- tract/grant numbers: IH GM-29072; NIH CA-25644, NIH GM- 56609 Journal of Computational Chemistry, Vol. 21, No. 12, 1049–1074 (2000) c 2000 John Wiley & Sons, Inc.
Transcript
How Well Does a Restrained ElectrostaticPotential (RESP) Model Perform inCalculating Conformational Energies ofOrganic and Biological Molecules?
JUNMEI WANG, PIOTR CIEPLAK,∗ PETER A. KOLLMANDepartment of Pharmaceutical Chemistry, University of California at San Francisco, San Francisco,California 94143-0446
Received 15 September 1999; accepted 21 March 2000
ABSTRACT: In this study, we present conformational energies for a molecularmechanical model (Parm99) developed for organic and biological moleculesusing the restrained electrostatic potential (RESP) approach to derive the partialcharges. This approach uses the simple “generic” force field model (Parm94), andattempts to add a minimal number of extra Fourier components to the torsionalenergies, but doing so only when there is a physical justification. The results arequite encouraging, not only for the 34-molecule set that has been studied by boththe highest level ab initio model (GVB/LMP2) and experiment, but also for the55-molecule set for which high-quality experimental data are available.Considering the 55 molecules studied by all the force field models for which thereare experimental data, the average absolute errors (AAEs) are 0.28 (this model),0.52 (MM3), 0.57 (CHARMm [MSI]), and 0.43 kcal/mol (MMFF). For the34-molecule set, the AAEs of this model versus experiment and ab initio are 0.28and 0.27 kcal/mol, respectively. This is a lower error than found with MM3 andCHARMm, and is comparable to that found with MMFF (0.31 and 0.22 kcal/mol).We also present two examples of how well the torsional parameters aretransferred from the training set to the test set. The absolute errors of moleculesin the test set are only slightly larger than in the training set (differences of<0.1 kcal/mol). Therefore, it can be concluded that a simple “generic” force fieldwith a limited number of specific torsional parameters can describe intra- andintermolecular interactions, although all comparison molecules were selectedfrom our 82-compound training set. We also show how this effective two-body
∗Permanent affiliation: Department of Chemistry, Universityof Warsaw, Warsaw, Poland
Correspondence to: P. A. Kollman; e-mail: [email protected];web site: http://www.amber.ucsf.edu/amber/
Contract/grant sponsor: National Institutes of Health; con-tract/grant numbers: IH GM-29072; NIH CA-25644, NIH GM-56609
Keywords: additive force field; nonadditive force field; restrained electrostaticpotential (RESP); torsional angle parameterization
Introduction
M olecular mechanics models are useful forsimulations of conformational energies and
noncovalent interactions of complex molecular sys-tems. Nevertheless, they do suffer from not havingas firm a physical underpinning as quantum-me-chanically-based approaches. The use of empiricalparameters enables them to fit experimental databetter than all but the high-level and most expensivequantum-mechanical approaches. However, there isa wide variety of functional forms, as well as manydifferent approaches for derivation of parametersfor molecular mechanical methods1 – 19; therefore, itis very difficult to establish unequivocally which ap-proach and functional form is best.
In fact, the answer to what is “best” is clearly de-pendent on what properties one wishes to calculate.A further consideration is the desirability to havethe simplest, most extendable model consistent witha satisfactory performance. By simplest and mostextendable, one means, of course, fewer atom typesand fewer empirical parameters.
We hypothesize that if one’s research goal is theaccurate description of structures and nonbondedenergies for organic and bioorganic systems, a sim-ple functional form, such as eq. (1), is adequate.The lack of anharmonic and cross-terms is criti-cal only when accurately fitting energies for highlystrained molecules and for fitting vibrational fre-
quencies. Our second hypothesis is that, in order toaccurately fit conformational and nonbonded ener-gies, one should use restrained electrostatic poten-tial (RESP) charges20 – 23 for the partial charge, qi,and that choice should lead to the need for fewertorsional potentials, Vi, than in models that have anempirical scheme for derivation of qi:
Epair =∑
bonds
Kr(r− req)2 +∑
angles
Kθ (θ − θ eq)2
+∑
dihedrals
Vn
2
[1+ cos(nφ − γ )
]+∑i<j
[Aij
R12ij
− Bij
R6ij
+ qiqj
εRij
](1)
Epol = −12
∑i
µiE0i (2)
µi = αiEi (3)
Ei = E0i +
∑j 6= i
Tijµj (4)
E0i =
∑j 6= i
qjrij
r2ij
(5)
Tij = 1r3
ij
(3rij
rij
r2ij
− 1)
(6)
Etotal = Epair + Epol (7)
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In what follows, we demonstrate that these twohypotheses seem justified in our molecular mechan-ical model, which reproduces experimental confor-mational energies for a limited set of moleculesbetter than previously derived models, albeit onlymarginally better than MMFF. However, given thatour model is simpler in functional form than theMMFF model, it is encouraging that it performs aswell.
We also show how one can extend the additiveforce field described by eq. (1), which includes thenonadditive effect implicitly in its use of 6-31G∗RESP charges, to a nonadditive model. The non-additive force field applies a high-level ab initiomodel to derive the partial charges, qi. The po-tential energy is the sum of the pairwise additiveenergy and the polarization potential energy de-scribed by eqs. (2)–(7), where αi is the polarizabilityof atom I; rij is the vector for atom I, and qj is thecharge of atom j. Eqs. (3) and (4) are iterated toself-consistency during the minimization and dy-namics simulations. It is our assumption that theelectrostatic energy is not tightly coupled with thetorsional energy; therefore, the nonadditive model
can share the same set of torsional parameters de-veloped for the additive model. Naturally, furtheradjustments of the Vi torsional potential will makethe nonadditive model more accurate. In this way,our RESP-based molecular mechanical model al-lows for natural extension to nonadditive models.
Methods
QM METHOD
The ab initio relative energies of equilibrium con-formers listed in Table I have been published inrefs. 19 and 24 – 26. For the 34-molecule set (nos. 1–4, 15–18, 20–21, 25–26, 36–39, 44–47, 49, 55–60, 72–74,76, 78–80), both the MP4/TZP and GVB-LMP2 ener-gies are available; for 1,3-dioxane and its derivatives(nos. 63–71), the conformational energies have beencalculated at a lower level (MP2/6-31G∗).
Alanine dipeptide and tetrapeptide have beenwell studied by different level ab initio methods,27 – 30
from simple MP2/6-31G∗ to MP4-BSSE/cc-pVTZ(-f). The relative energies with high-level electron
TABLE I.Conformational Energies (kcal/mol) Found in Experiment, Ab Initio Calculations, and Different Force Fields.
Comp. Compound name and MP4/ GVB- MSIno. conformational comparisona Expt.b TZP LMP2 AMBER MMFF 94 MM3 CHARMm
a For brevity, the conformational abbreviations a, g, t, and c are sometimes used for anti, gauche, trans, and cis, respectively. Thesedesignations correspond to torsional angles of 180◦ , 60◦ (or −60◦), 180◦ , and 0◦. For the skew conformation, the desired torsionalangle is 120◦ .b The experimental data are mainly from refs. 1, 19, and 24. The special sources of some data are as the following: no. 1 (ref. 42),no. 10 (refs. 43, 44), no. 11 (ref. 45), no. 12 (ref. 46), nos. 13–14 (ref. 47), no. 19 (ref. 48), no. 48 (ref. 17), nos. 63–68 (refs. 25 and 26).In total, there are six kinds of data according to the measurements: (I) 1H, gas phase; (II) 1E, gas phase; (III) 1G, gas phase;(IV) 1H, solution; (V) 1G, solution; (VI) 1G, solution, low temperature. Nos. 1, 4, 15–16, 19, 26, 48, 50–54, and 78 belong to class I;nos. 9, 18, 20, 22, 25, 27–30, 33, 36, 38, and 57 belong to class II; nos. 2–3, 5–6, 17, 23, 31, 34–35, 44–45, 49, 55–56, 59, 63–68, and79 belong to class III; nos. 39, 46–47, 72–73, and 76–77 belong to class IV; nos. 8, 24, 37, 40–43, belong to class V; Nos. 7, 32, and60 belong to class VI. For literature citation of special cases, see the Supplementary Material of ref. 19.c Conf1 is the all-staggered conformation; Conf2 has the methylene group to eclipse one methyl group; Conf3 has the methylenegroup to eclipse both methyl groups.d Compounds 1–3, 4a–4e, and 5 are defined in ref. 25.e The ab initio method is MP2/6-31G∗.f The ab initio method is B3LYP/6-311+G(2d,p).g Torsional restraint was applied.
correlation were obtained by single-point calcula-tion for the minimized structures at the HF/6-31G∗or the MP2/6-31G∗ level. The data (Table II) used inthis work were carefully selected from refs. 27 – 30.
Five nucleosides were studied in this work. Foreach, geometry optimization was first performed atthe HF/6-31G∗ level and then the electron correla-tion energy was evaluated with the same basis setat the MP2 level.
In order to develop the generic torsional para-meters X—C—OS—X and X—C—OH—X (X can beany atom type), which are necessary for the stud-ies of organic acids and esters, quantum-mechanicalcalculations were carried out for two model mole-cules, acetic acid and methylformate. Potential en-ergy surface scans were performed using four dif-ferent ab initio methods, HF/6-31G∗, MP2/6-31G∗,MP2/6-311+G (2d,p), and B3LYP/6-311+G (2d,p).
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TABLE II.Conformational Energies (kcal/mol) of Alkanes for Different Torsional Parameter Sets.a
Compound AMBER AMBER AMBER
no. Compound name and conformational comparison Expt.b (set I) (set II) (set III)
a Set I: Parm99; set II: Parm94; set III: Parm99; however, for CT—CT—CT—CT, only V3 is included, and the optimized force constantis 0.28.b For sources and measure types of data, see Table I.
The relative energies of three stationary points onthe energy surfaces were selected to derive the cor-responding torsional parameters.
RESP CHARGES
The atom-centered point monopole charges usedfor the molecular-mechanical calculations were de-rived from the electrostatic potential. First, quan-tum-mechanical optimizations were performed forall the compounds using the 6-31G∗ basis set withthe GAUSSIAN-94 software package.31 For the AFFmodel, electrostatic potentials were then calculatedat the same level for the minimized geometry. How-ever, for the NAFF model, electrostatic potentialsused for RESP charge fitting were calculated at ahigher level (B3LYP/cc-pVTZ). For each compound,RESP charges were derived using only the electro-static potential of the lower energy conformer.
MOLECULAR MECHANICS METHOD
All minimization calculations reported for thiswork were carried out using the AMBER-5 pack-age.32 Scale factors of 1/1.2 and 1/2 were appliedto the 1–4 electrostatic and van der Waals interac-tions, respectively. The dielectric constant was setto 1.0 and no cutoff was used for the nonbondedinteractions. When the minimized structure devi-
ated far away from the reference one, torsional anglerestraints were applied. The torsional restraint weemployed is a well with a square bottom and twoparabolic sides out to a defined distance, with linearsides beyond that. If R was the value of the torsionalangle restraint in question, two force constants (K1
and K2) and four sequential torsional angles (R1 <
R2 < R3 < R4) were used to define this flexible re-straint. If R was between R2 and R3, no restraint wasapplied; if R was between R1 and R2, or betweenR3 and R4, restraint was exerted with a force con-stant of K1; if R was smaller than R1, or larger thanR4, restraint was applied with another force con-stant of K2. In our cases, K1 and K2 were set to 50kcal/mol·rad2 and 500 kcal/mol·rad2, respectively.The reference torsional angle was at the midpointbetween R2 and R3, and the distances between R1
and R2, R2 and R3, and R3 and R4 were set to 5◦. Ta-bles I and II list the compounds and conformationsfor which torsional restraint was applied during theminimization.
MOLECULAR MECHANICAL ENERGIES OFOTHER FORCE FIELDS
The energy differences of the other force fieldswere selected from refs. 19 and 24 – 26.
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MOLECULAR DYNAMICS SIMULATIONS
Molecular dynamics simulations of ubiquitinwere carried out using the SANDER module ofthe AMBER-5 program32 for both the Cornell et al.model and the new model. The simulations werecarried out at 300 K with a time step of 2.0 fs. Thenonbonded cutoff was set to 9.0 Å and the particle-mesh Ewald (PME) method33 was used. A periodicbox of water was added using the TIP3P potential34
and approximately 10,000 waters were added. Thedimensions of the box were 61.0, 62.0, and 54.6 Å.SHAKE35 was applied to bonds involving hydrogenatoms.
Molecular dynamics simulations were alsoperformed for the Dickerson dodecamer,d(CGCGAATTCGCG)2,36 in aqueous solution.A similar protocol was used for molecular dynam-ics simulations as that for the ubiquitin molecule;that is: a 9.0-Å nonbonded cutoff; the PME methodfor electrostatic interactions; the SHAKE procedurefor bonds involving hydrogen atoms; a 2.0-fs timestep; and the periodic box of water containedapproximately 3800 TIP3P water molecules. Thetotal time of the molecular dynamics simulationfor this system was 1 ns. Two trajectories weregenerated. One started from the crystallographicB-form36 and the other was initiated from thecanonical A-form. The second trajectory wasused to estimate the A-to-B transition time inaqueous solution for comparison with previousdata obtained using the Cornell et al.1 force field.
FITTING PROCEDURE
Parmscan, an automatic force field parameter op-timization program developed by our group wasused to derive new torsional parameters. Unlikeother automated parameterization programs,37 – 39
parmscan primarily attempts to find the best Fourierseries and force constants so as to reproduce pre-cisely the energy differences of the training set. Themain purpose of parmscan is to change the force con-stants systematically, with a certain step, to find theoptimum torsional parameters that give the smallestabsolute error of molecular mechanical (MM) en-ergy differences when compared with experimentalor ab initio data. For each step, the file of modifiedforce field parameters (frcmod) was first regener-ated; molecular mechanics minimization was thenperformed for both of the two conformers of thecompounds that share the same parameters. Eq. (8)or eq. (9) was applied to evaluate the score for eachstep, where Econf1 is the MM energy of the first con-former, Econf2 is that of the second conformer, and n
is the number of conformational pairs. In this work,the scores are represented by the average absoluteerror as determined by eq. (8). The least-squaresgeometry fitting procedure was also necessary toensure that the minimized structure did not deviatetoo far from the starting geometry. The root meansquare of distance (RMSD) should be smaller thana certain criterion (we used 0.2 Å). If the RMSDwas larger than the criterion, reoptimization witha torsional constraint from the initial structure wastriggered automatically:
Score =∑
abs(Econf1 − Econf2)/n (8)
Score = sqrt(∑
(Econf1 − Econf2)2/n)
(9)
Results
The 82 conformational pairs examined herein arelisted in Table I. They are further grouped into tenclasses: alkanes (14); alkenes (3); conjugated com-pounds (2); fluorides (5); chlorides (7); bromides (4);amide and amino compounds (10); alcohols (9);ethers (17); and compounds with a carbonyl func-tional group (11). The numerals in parentheses referto the number of conformational pairs for each class.Two special molecular sets were extracted from thewhole set for comparison purposes. For the 34-molecule set (nos. 1–4, 15–18, 20–21, 25–26, 36–39,44–47, 49, 55–60, 72–74, 76, 78–80), the experimen-tal relative energies, high-level ab initio energies(GVB/LMP2 and MP4/TZP), as well as the con-formational energy differences of AMBER, MMFF,MM3, and CHARMm (MSI), are all available. For the55-molecule set (nos. 1–9, 15–18, 20–47, 49, 55–60,72–74, 76, 78–80), both the experimental data andthe energy differences of the four mentioned forcefields are available.
TORSIONAL PARAMETER DEVELOPMENT
The following paragraphs describe the proce-dures and results of parameter development basedon the functional groups. A torsional parameter isusually depicted as: Vn(c1, c2), where Vn signifiesthe n-fold Fourier component, and c1 and c2 arethe force constants (in kcal/mol) and phase angles(in degrees), respectively. For each compound men-tioned in what follows, its two conformers and theexperimental relative energy are given, with thesecond conformer, in all cases, having the more fa-vorable energy than the first.
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ALKANES
Key studies for understanding the use ofquantum-mechanical electronic structure calcula-tions for analysis of molecular torsional preferencesincluded those by Radom et al.40 and Brunck andWeinhold.41 As shown in ref. 41, the single mostimportant feature in the conformational preferencesin single-bonded molecules, X—Y—Z—W, is thestrength of the bond–antibond interaction, which islargest in a trans conformation, with cis next, andsmallest in a skew orientation. The greater the dif-ference in electronegativity between X and W, thestronger the stabilizing bond–antibond interactions.The bond–antibond interaction is caused in part bythe electronic delocalization from the bond involv-ing the atom with the lower electronegativity to thebond involving the atom with the higher electroneg-ativity. This simple analysis can rationalize most ofthe rotational barrier in ethane (where only ∼0.3kcal/mol is due to nonbond H. . .H interactions),because staggered ethane has three trans H—C—C—H interactions and eclipsed ethane has only cisand skew H—C—C—H interactions. It can also ra-tionalize the fact that 1,2-difluoroethane prefers thegauche to trans conformation, because the formerhas two trans H—C—C—F interactions and the lat-ter has trans F—C—C—F and H—C—C—H interac-tions. The conceptual basis underlying the Cornellet al. force field is that one should only add extratorsional potential in the case of specific electrondelocalization effects. For example, for aliphatic hy-drocarbons, given the similar electronegativity of Hand C, Cornell et al. showed that a single V3 (X—C—C—X), where X = C or H, could accuratelyreproduce the conformational energies of ethane,propane, and butane, and the assumption was thatthis result would extrapolate to other hydrocarbons.
With the single V3 parameter, the Cornell et al.model shows a reasonable performance in mostof the cases. The average absolute error for 14compounds is 0.63 kcal/mol. However, this simplemodel does not accurately predict the conforma-tional preferences for cyclohexane (twist boat vs.chair, 1Etwist boat–chair = 5.5 kcal/mol) and cyclooc-tane (D4d vs. Cs boat chair, 1ED4d–Cs boat chair =1.9 kcal/mol) as well as cyclononane (C2 vs. D3,1EC2–D3 = 1.0 kcal/mol). The absolute errors are2.03, 1.77, and 0.99 kcal/mol, respectively. There-fore, a more complex model is necessary to improvethe energies.
There are two ways to extend this simple model.The first approach is to break a generic torsional pa-rameter into several parameters, each with a specific
force constant. The second approach is to includeother Fourier components in addition to the V3
term. In the case of hydrocarbons, both approachesare necessary to achieve the best performance.
First, we broke the generic torsional parame-ter (X—Csp3—Csp3—X) into three specific ones:Csp3—Csp3—Csp3—Csp3; Csp3—Csp3—Csp3—H; andH—Csp3—Csp3—H. Considering ethane (eclipsedvs. staggered), which involves only H—Csp3—Csp3—H, the force constant was set directly to0.15 kcal/mol (V3), which can predict the energydifference (2.93 kcal/mol) as essentially the same asthat of experiment (2.88 kcal/mol). Then the forceconstant of Csp3—Csp3—Csp3—H was determinedby fitting the energy differences of propane (Conf1vs. Conf2 and Conf2 vs. Conf3, with the definitionsof Conf1, Conf2, and Conf3 described in Table I),which involves only Csp3—Csp3—Csp3—H and H—Csp3—Csp3—H torsions. The final parameter was setto 0.16 kcal/mol (V3), which can reproduce the twoenergy differences very well. Finally, the other 11molecules that involve all of the three parameterswere used to fit the force constant of Csp3—Csp3—Csp3—Csp3 with parmscan. We found that the forceconstant of 0.25 kcal/mol (V3) gave the minimumaverage absolute error, which is 0.54 kcal/mol. Al-though this model is a great improvement overthe Cornell et al. model, it cannot predict the en-ergy differences of the three aforementioned casesvery well. Inspired by MM3, which uses all ofV1, V2, and V3 for Csp3—Csp3—Csp3—Csp3, we per-formed a systematic search for the 11 compoundsto find the best force constants of the three Fouriercomponents. Parmscan suggested that the force con-stants be 0.20, 0.25, and 0.18 kcal/mol for V1, V2,and V3, respectively. Using this model, the AAE isreduced to 0.38 kcal/mol and the RMS deviation isonly 0.47 kcal/mol. Moreover, it leads to a muchbetter representation of the energies of the threecycloparaffins. The absolute errors are 1.08, 0.31,and 0.20 kcal/mol for cyclohexane, cyclooctane, andcyclononane, respectively. Although we made ourmodel slightly more complex, it is encouraging thatthe V3 term for all three torsional types are nearlyidentical, and the V1 and V2 terms for Csp3—Csp3—Csp3—Csp3 are also rather small. Table II lists theconformational energies for each model discussedearlier.
ALKENES
Three simple compounds were studied in thiswork: (1) propene (H—Csp3—Csp2—Csp2 trans vs.cis, 1Etrans–cis = 2.00 kcal/mol); (2) 1-butene (cis
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PERFORMANCE OF RESP
vs. skew, 1Ecis–skew = 0.22 kcal/mol); and (3) 2-butene (cis vs. trans, 1Ecis–trans = 1.20 kcal/mol).The energy differences in the Cornell et al. modelare −0.32, 1.87, and 0.49 kcal/mol, respectively. It iswell known that, in propene, the Csp3—H prefers toeclipse the Csp2—Csp2 bond rather than the Csp2—Hbond. As for 1—butene (cis vs. skew), the relativeconformational energy is a compromise betweenCsp3—H and Csp3—Csp3, eclipsing the C=C. Wewere able to fit these two compounds with twospecific Fourier terms for H—Csp3—Csp2—Csp2 (V1
[1.15, 0.0] and V3 [0.38, 180.0]). When we consider2-butene (cis vs. trans), a V1 (1.90, 180.0) term is re-quired to correctly represent this preference. Withthis complex model, very encouraging results areobtained and the energy differences are 1.96, 0.18,and 1.25 kcal/mol for the three compounds men-tioned earlier, respectively.
CONJUGATED COMPOUNDS
Two conjugated compounds, 1,3-butadiene(gauche vs. trans, 1Egauche–trans = 2.89 kcal/mol)and 2-methyl-1,3-butadiene (gauche vs. trans,1Egauche–trans = 2.65 kcal/mol), were studied inthis work. Although glyoxylic acid (no. 73 inTable I) and acrolein (no. 80 in Table I) are alsoconjugated systems, they are discussed with othercarbonyl-containing compounds. Considering thatthe inner two carbons of butadiene are not pure sp2
carbons, and the bond lengths between them areslightly longer than a pure single Csp2—Csp2, a newatomic type (CD) can be introduced. The van derWaals parameter is the same as that of sp2 carbonand the equilibrium bond length of CD—CD is setto 1.40 Å, similar to that of Csp2—Csp2 bonds inpure benzene. A new generic torsional parameter,X—CD—CD—X, was optimized by parmscan and,finally, the force constant was set to 4.0 kcal/molwith a phase angle of 180.0◦. The average absoluteerror for the two compounds is 0.36 kcal/mol.
FLUORIDES, CHLORIDES, AND BROMIDES
Five fluorides, seven chlorides, and four bro-mides were studied for halides. The Cornell et al.model gives average absolute errors of 1.32, 1.11,and 1.86 kcal/mol for the three kinds of com-pounds, respectively. Given the fact that halogensin 1,2-difluoroethane prefer to be trans to hydro-gen atoms, rather than to each other, two kindsof torsional parameters, H—Csp3—Csp3—X and X—Csp3—Csp3—X, were introduced. Instead of usingonly the default threefold Fourier component, we
applied the onefold Fourier component for thesetwo torsions with phase angles of 0.0◦ and 180.0◦,respectively. For the former, the force constantschange from 0.19, to 0.25, and then to 0.55 kcal/molfor fluorides, chlorides, and bromides, respectively.For the latter torsion parameter, the sequence or-der of force constants is still qualitatively consis-tent with the periodic table, changing from 1.20 to0.45 and then to 0.0 kcal/mol for fluorides, chlo-rides, and bromides, respectively. With the newmodel, the mean absolute errors are 0.24, 0.30,and 0.41 kcal/mol for fluorides, chlorides, and bro-mides, respectively. We also tried to apply twofoldFourier components to improve the model, whichoccurs to some degree. However, the model ap-plying the onefold Fourier terms shows a betterperformance.
AMIDES AND AMINO COMPOUNDS
Two amides and eight amino molecules were in-vestigated in this study. A new atomic type, NT,that is, the neutral sp3 nitrogen, was generated.This atom type shares the same van der Waalsparameter as other nitrogen atomic types. Bondstretching and bending parameters are also sharedwith the N3 atom type, which is the charged am-monium nitrogen in the Cornell et al. force field.For the torsional parameter of X—Csp3—Nsp3—X(X can be any atomic type), a generic threefoldFourier component was applied, just as that inthe Cornell et al. model for X—CT—N3—X (V3
[1.80, 0.0]). The average absolute error of this sim-ple model is 0.58 kcal/mol. Encouraged by thegreat improvement made for alkanes after apply-ing all of the first three Fourier components forCsp3—Csp3—Csp3—Csp3, we added a small twofoldcomponent for Csp3—Csp3—Nsp3—Csp3 (V3 [0.3, 0.0],V2 [0.48, 180.0]). We found that the average absoluteerror is further reduced to 0.20 kcal/mol and theRMS deviation is only 0.23 kcal/mol.
ALCOHOLS
For alcohols, nine energy conformational pairswere considered for optimizing the torsional para-meters. For the first four conformational pairs (nos.46–49 in Table I), experimental data are available,whereas only high-level ab initio (MP4SDQ/TZP)data are available for the last five conformationalpairs (nos. 50–54 in Table I).
The average absolute error for the Cornell et al.model is 0.74 kcal/mol, and therefore it is neces-sary to add specific torsional parameters to further
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reduce the error. Similar to the H—Csp3—Csp3—X(X = F, Cl, or Br) torsional parameters noted ear-lier, we included the onefold Fourier component forH—Csp3—Csp3—O V1 (0.25, 0.0). The force constantfor this torsion is the same as that of H—Csp3—Csp3—Cl, which is consistent with the similar elec-tronegativities of oxygen and chlorine. Another spe-cific torsional angle, Csp3—Csp3—O—H, was alsointroduced to regenerate the relative energies tothe extent possible. After optimization by parmscan,both onefold component V1 (0.25, 0.0) and threefoldcomponent V3 (0.16, 0.0) were included for this tor-sion. This new model reduces the average absoluteerror to 0.48 kcal/mol. Although the AAE is stillslightly larger than those of other organic species,we still believe that the new model works quitewell because only MP4SDQ/TZP data are availablefor the last five conformational pairs. It is knownthat this ab initio method has an RMS deviation of∼0.4 kcal/mol, similar to the experimental data.
ETHERS
Seventeen ethers, including nine 1,3-dioxanederivatives (nos. 63–71 in Table I), were investigatedin this work. For the last three 1,3-dioxane deriva-tives (nos. 69–71 in Table I), only MP2/6-31G∗ ener-gies are available. The Cornell et al. model appliesonly a generic torsional parameter for X—Csp3—O—X (X can be any atom type), and it works wellfor the most ethers except those with OCO units,such as 1,3-dioxane and its derivatives. The AAE is0.44 kcal/mol without considering 1,3-dioxane andits derivatives. However, the AAE is 2.08 kcal/molfor 1,3-dioxane and its eight derivatives. Howard etal. found that a onefold Fourier component (V1) in-stead of a threefold one (V3) for Csp3—O—Csp3—Oreproduces the relative energies very well. More-over, the relative energies of their model show verygood correlation with those of ab initio MP2/6-31G∗.25 Although the AAE of this model is only∼0.5 kcal/mol compared with the MP2/6-31G∗ en-ergies, it is still significantly larger, when comparedwith the experimental data. In our opinion, addingV2 and V3 torsional parameters was necessary to de-pict the energies more accurately.
For our new model, V1 was introduced first andthen V2 and V3 were added in sequence. It is usu-ally thought that bond stretching and bond bendingmake little contribution to the relative conforma-tional energy. However, we found that the bondangle parameter, O—Csp3—O, plays an importantrole in the successful modeling of 1,3-dioxane andits derivatives. The equilibrium value of this bond
angle and force constant were first set to 109.5◦and 50.0 kcal/mol·Å2, respectively, as shown byHoward et al. However, we found that MM3 uses asignificantly smaller equilibrium bond angle, 103.1◦.We therefore sought to optimize the equilibriumbond angle as well as the force constant of O—Csp3—O using parmscan. The equilibrium bond an-gle and force constant were refined at 101◦ and 160kcal/mol·Å2, respectively. It is of interest to pointout that, in phosphates, the pendant oxygen groupalso applies a large force constant for O2—P—O2(140.0 kcal/mol·Å2), although it is much smaller forthe ester oxygen OS—P—OS. Encouragingly, oursomewhat more complex model not only repro-duces the ab initio data better than Howard et al.’smodel but also reproduces the experimental dataquite well. The AAE is only 0.34 kcal/mol for all17 ethers. Table III lists the conformational energiesand the AAEs of the different models. From this ta-ble, one can estimate the contributions of differentparameters.
CARBONYL COMPOUNDS
The 11 compounds with carbonyl groups werestudied. First, acetic acid and methylformate wereselected as model molecules to derive the genericparameters of X—C—O—H and X—C—O—X,where X can be any atomic type. Potential en-ergy surface scanning and minimization were per-formed for O—C—O—H of acetic acid and O—C—O—C of methylformate. With the aim to investigatethe effects of basis sets and different ab initio ap-proaches, the following four methods were appliedfor the ab initio calculations: HF/6-31G∗; MP2/6-31G∗; MP2/6-311+G (2d,p); and B3LYP/6-311+G(2d,p). The curves of two potential energy surfacesare very similar in shape: both of them are sym-metrical around 180◦, and have global minima at0.0◦, a local minima at 180◦ and two other station-ary points around±90◦. Because the profiles are notthe standard cosine curves, a single twofold Fouriercomponent cannot depict the whole curve very well.With the intention of making our model as simpleas possible, we tried only to reproduce the relativeenergies between the stationary points. The follow-ing are the relative energies of different methodsfor acetic acid. For the local minimum at 180◦, therelative energies for the global minimum confor-mation are 7.19, 6.99, 5.64, and 5.50 kcal/mol forthe four methods in sequence; for the maxima at90◦ and −90◦, the relative energies for the globalminimum conformation are 13.62, 14.47, 13.01, and12.81 kcal/mol, respectively. As to methylformate,
1058 VOL. 21, NO. 12
PERFORMANCE OF RESP
TABLE III.Conformational Energies of Ethers for Different Parameter Sets.a
Compound AMBER AMBER AMBER AMBER
no. Compound name and conformational comparison Expt.b (set I) (set II) (set III) (set IV)
a Set I: Parm99; set II: Parm94; set III: Parm99; however, for OS—CT—OS—CT only V1 is included and the optimized force constantis 1.80; set IV: Parm99; however, the reference bond angle of OS—CT—OS is set to 109.0◦ and the force constant is optimized to160 kcal/mol (conformational energies in kcal/mol).b For the sources and measure types of data, see Table I.c Compounds 1–3, 4a–4e, and 5 are defined in ref. 25.
for the local minimum at 180◦, the relative ener-gies for the global minimum conformation are 6.26,6.36, 5.45, and 4.76 kcal/mol for the four meth-ods in sequence, whereas the experimental value is4.75 kcal/mol (no. 74 in Table I); for the maximaat 90◦, and −90◦, the energies relative to the globalminimum conformation are 12.79, 14.44, 13.88, and13.45 kcal/mol, respectively. Given that the first twomethods are not as accurate (the AAE for the ex-periment was estimated to be 0.8 kcal/mol19), wefocused on the latter two methods. One may believethat the two methods have similar accuracy, but ourchoice is the B3LYP/6-311+G (2d,p) model, whichrequires less CPU time than the MP2 method withthe same basis set. In this work, the B3LYP/6-311+G(2d,p) energies were chosen as references to derivethe two generic torsional parameters. With parm-scan, we can easily obtain the best force constantsto reproduce the ab initio relative energies, and theforce constants are 4.6 and 5.4 kcal/mol for X—C—O—H and X—C—O—X, respectively.
Besides the two generic torsional parameters,the following specific ones were included to fur-
ther minimize the average absolute error: H—Csp3—C=O (V1 [0.80, 0.0], V3 [0.08, 180.0]); H—O—C=O (V1 [1.90, 0.0], V2 [2.30, 180.0]); C=C—C=O(V2 [2.175, 180.0], V3 [0.30, 0.0]); and Csp3—O—C=O(V1 [1.40, 180.0], V2 [2.70, 180.0]). Again, parmscanwas used to choose the Fourier components aswell as the force constants. First, only the compo-nent making the largest contribution was includedin our model and other components were addedonly when necessary, because we prefer to keep ourmodel simple. The final model reproduced the rela-tive energies very well for all the 11 molecules. TheAAE was 0.4 kcal/mol and the RMS deviation was0.59 kcal/mol.
ADDITIVE FORCE FIELD (AFF) MODEL FORORGANIC MOLECULES
We studied a total of 82 conformational pairs us-ing the simple pairwise additive model presented ineq. (1). The average absolute error is 0.33 kcal/moland the RMS deviation is 0.47 kcal/mol. To com-pare with other widely available force fields, weextracted two subsets from the 82 conformational
JOURNAL OF COMPUTATIONAL CHEMISTRY 1059
WANG, CIEPLAK, AND KOLLMAN
TABLE IV.Conformational Energies (kcal/mol) of Additive Force Field (AFF) and Nonadditive Force Field (NAFF).a
Compound no. Compound name and conformational comparison Expt.b AFF NAFF (I) NAFF (II) NAFF (III)
a Three schemes of NAFF are: I—polarizable model without lone pairs; II—polarizable model with lone pairs having distances 0.2 Åfrom heavy atoms; III—polarizable model with lone pairs having distances 0.35 Å from heavy atoms. The polarizability parametersfor NAFF models were developed by Applequist et al.48
b For the sources and measure types of data, see Table I. For nos. 40–44, no experimental data were available, so ab initio energiesat the MP4SDQ/TZP level were used.
pairs, the 34-molecule set and 55-molecule set, forcomparison purposes. We compared our resultswith MMFF, MM3, and CHARMm, which have beenused widely and have shown good performancein characterizing the geometries and relative con-
formational energies. If the force field value differsfrom the experiment or ab initio by>1.0 kcal/mol, itis regarded as an outlier. Both the average absoluteerror and RMSD are used to evaluate the perfor-mance of different force fields.
TABLE V.Summary of AMBER Conformational Energies (kcal/mol) versus Experiment by Compound Classes.
a Conformational energies in kcal/mol.b With the cc-pVTZ (-f) basis set.
Table VI lists the results of comparisons for the34-molecule set. Compared with the experimen-tal data, AMBER shows the smallest AAE (0.28kcal/mol), slightly better than MMFF, which showsan AAE of 0.31 kcal/mol. MM3 and CHARMmshould be regarded at the same level, having AAEsof 0.53 and 0.58 kcal/mol, respectively. When com-pared with MP4/TZP and GVB/LMP2, MMFFshows the smallest AAE and RMS deviation, bothmarginally better than AMBER. It is understandablethat MMFF gives the best performance for the com-parisons to the ab initio data, given that these datawere used extensively in the MMFF parameteriza-tion.
Table VII gives a summary of comparison resultsfor the 55-molecule set. Both the AAE and RMSdeviation for each compound class are listed forthe four force fields. The AAEs are 0.28, 0.43, 0.52,and 0.57 kcal/mol for AMBER, MMFF, MM3, andCHARMm, respectively. As for RMS, the same trendcan be seen. It is of interest to point out that MMFF,MM3, and CHARMm perform better than AMBERif only alcohols are considered. For compounds inother classes, AMBER always has the smallest AAEand RMS deviations. The performance of each forcefield is illustrated in Figures 1–3. From these, we canquickly find the number of outliers and the numberof wrong conformers favored (wrong sign). For the
TABLE VII.Summary Comparisons of Difference Force Fields for 55-Molecule Set.a
AMBER MMFF MM3 CHARMm
Average of RMS Average of RMS Average of RMS Average of RMSNumber absolute error dev. absolute error dev. absolute error dev. absolute error dev.
FIGURE 1. The performance of AMBER (Parm99) forcefield for reproducing the relative experimentalconformational energies. Three parallel lines are definedas y = x + 1.0, y = x, and y = x − 1.0 from top to bottom.The greater density of dots close to the middle lineindicates better performance. The dots above y = x+ 1.0or below y = x − 1.0 are defined as outliers. A dot belowthe line that is parallel with the X axis predicts the wrongsign (favoring the wrong conformation). For Parm99,there are two outliers and three wrong signs.
55-molecule set, AMBER has two outliers, whereasMMFF, MM3, and CHARMm have four, five, andnine outliers, respectively. As for wrong sign, AM-BER has three, whereas MMFF, MM3, and CHARMmhave five, five, and seven, respectively.
Considering that all the molecules for compar-isons are selected from our 82-molecule training set,we should note that the comparison with MM3 and
FIGURE 2. The performance of the MMFF force fieldfor reproducing the relative experimental conformationalenergies. Notation is the same as that in Figure 1. ForMMFF, there are four outliers and five wrong signs.
FIGURE 3. The performance of the MM3 force field forreproducing the relative experimental conformationalenergies. Notation is the same as that in Figure 1. ForMM3, there are six outliers and five wrong signs.
CHARMm might be biased considering that theirforce fields may have not included all of the mole-cules for parameterization. We thus investigatedhow well the parameters transfer from training setto test set by considering two examples for furtherstudy. For the first example, 11 hydrocarbons thatinvolve the CT—CT—CT—CT torsional parameterare grouped into the training set (nos. 2, 4, 5, 7, 9,and 11 in Table VIII) and the test set (nos. 1, 3, 6,8, and 10 in Table VIII). For the second example,seven chlorides, three (nos. 12, 14, and 17 in Ta-ble VIII) in the training set and four (nos. 13, 15, 16,and 18 in Table VIII) in the test set, involve the tor-sional parameter Cl—CT—CT—HC. Parameter set I
FIGURE 4. The performance of the CHARMm force fieldfor reproducing the relative experimental conformationalenergies. Notation is the same as in Figure 1. ForCHARMm, there are nine outliers and eight wrong signs.
JOURNAL OF COMPUTATIONAL CHEMISTRY 1063
WANG, CIEPLAK, AND KOLLMAN
TABLE VIII.Conformational Energies of 11 Hydrocarbons and 7 Chlorides with Parameters Derived Using Training Set andFull Molecular Set (Parm99).
Compound Compound name and Parameter derivedno. conformational comparisona Expt. Parm99 with training set
a For the sources and measure types of data, see Table I.
is defined as the torsional parameter derived us-ing all the molecules (Parm99); set II is the sameparameter optimized using only the training set.Table VIII lists the relative energies of these two tor-sional parameter sets. For CT—CT—CT—CT, set I(V1 [0.18, 0.0], V2 [0.25, 180.0], V3 [0.20, 180.0]) andset II (V1 [0.18, 0.0], V2 [0.29, 180.0], V3 [0.24, 180.0])are very similar, and the AAEs of the 11 hydrocar-bons are almost the same for the two sets (0.355 vs.0.359 kcal/mol). It is understandable that set II givesthe smaller AAE for the training set (0.35) than thatof set I (0.39); on the contrary, for the test set, theAAE of set II (0.37) is a little larger than that of set I(0.31). For the second example, parameter set II hasa marginally smaller force constant (V1 [0.23, 0.0])than that of set I (V1 [0.25, 0.0]). The AAEs of theseven chlorides are 0.30 and 0.33 kcal/mol for sets Iand II, respectively. For the training set, set II alsogives smaller AAE (0.297) than that of set I (0.313),whereas, for the test set, set I gives a smaller AAE(0.30) than that of set II (0.36).
From the two examples just presented, we be-lieve that the torsional parameters of Parm99 can
be reliably transferred from the training set tothe test set. It is our estimate that, for the or-ganic compounds outside the training set, in casesof no parameter missing, Parm99 can reproducethe relative energies of experiment with an AAEof <0.40 kcal/mol. We conclude that a well-para-meterized, simple additive model with RESP chargecan describe the structure and the intramolecularenergies for organic systems very well.
NONADDITIVE FORCE FIELD (NAFF) MODELFOR ORGANIC MOLECULES
For the nonadditive model described by eqs. (2)–(7), we worked out three schemes. The first is asimple polarizable model without lone pairs [NAFF(I)], whereas the other two models include lonepairs for oxygen and nitrogen. The distances fromthe lone pairs to the center atoms were set to 0.2 Å[NAFF (II)] and 0.35 Å [NAFF (III)] for the lattertwo models, respectively. Table IV lists the confor-mational energies of the three models. Sixty con-formational pairs were studied. For the first 26
1064 VOL. 21, NO. 12
PERFORMANCE OF RESP
TABLE IX.Summary Comparisons of Additive Force Field (AFF) and Nonadditive Force Fields (NAFF) for 60 Compounds.a
a Three projects of NAFF are studied: I—polarizable model without lone pairs; II—polarizable model with lone pairs having distances0.2 Å from heavy atoms; III—polarizable model with lone pairs having distances 0.35 Å from heavy atoms (conformational energiesin kcal/mol).b Nos. 27–60 in Table IV.c For 34-molecule subset in Table IV (nos. 27–60).
conformational pairs that are isolated hydrocarbonsand halides, only the standard nonadditive model(NAFF (I)] was used, because the electrostatic andpolarization effects are negligible for hydrocarbonsand no lone pairs are used for halogens in ourmodel. For the remaining 34 conformational pairs,all of the three NAFF models were investigated.
All of the NAFF models applied the same tor-sional parameters as those developed for the AFFmodel. Table IX lists the results of comparison ofeach model with the experimental data or ab ini-tio energy in cases unavailable experimental values.For the 60 conformational pairs, the AFF modelachieves better performance (AAE and RMS devi-ation 0.31 and 0.44 kcal/mol, respectively) than allof the three polarizable models. As for NAFF (I), theAAE is 0.44 and the RMS deviation is 0.55 kcal/mol.The AAEs are almost the same for NAFF (II) andNAFF (III), which are 0.46 and 0.48 kcal/mol, re-spectively. The RMS deviations of NAFF (II) andNAFF (III) are also similar, 0.56 kcal/mol for NAFF(II) and 0.62 kcal/mol for NAFF (III). It is encourag-ing that our polarizable models perform reasonablywell when applying the same parameters devel-oped for the additive model. All three nonaddi-
tive models have performance levels comparable toMMFF, which has an AAE of 0.43 kcal/mol (55-molecule set).
It is likely that our nonadditive model can beimproved significantly with the logical adjustmentof some torsional parameters. In this case, “a log-ical adjustment” means that both the additive andnonadditive models use different force constants,but they should use the same phase angles. Ta-ble X presents some results of comparing a polar-izable model with and without logical adjustmentsof the torsional parameters. From Table X we con-clude that, in most cases, the polarizable models doachieve comparable or better performance than theadditive model, after making a logical adjustment oftorsional parameters.
PEPTIDES
The development of an accurate model to re-produce precisely the relative energies of peptideconformations is crucial in successfully modelingbiological systems. With the great progress madein computer performance, one can undertake high-level ab initio calculations for large systems suchas peptides. The reconsideration of organic mole-
JOURNAL OF COMPUTATIONAL CHEMISTRY 1065
WANG, CIEPLAK, AND KOLLMAN
TAB
LEX
.C
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0.0,
1)(0
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(1.5
0,0.
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(0.3
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0,3)
F—C
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sp3—
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HB
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Csp
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ol).
1066 VOL. 21, NO. 12
PERFORMANCE OF RESP
TABLE XI.MM Results for Alanine Dipeptides and Tetrapeptides Using Standard Charges.a
Parm 99 Parm 94
Conformers Ab initiob Additive model Polarizable model Additive model Polarizable model
a Conformational energies in kcal/mol. Torsional restraints were applied to all conformers except C7eq, C7ax, C5 of alanine dipeptide,and Conf1, Conf3 of alanine tetrapeptide. The mean RMS of distances between the minimized and the ab initio structures are 0.20,0.21, 0.20, and 0.22 Å for the four models from left to right, respectively.b Ab initio energies of C7ax, C5 and AlphaR of alanine dipeptide are from ref. 28 and other conformational data of the dipeptide arefrom ref. 27. All conformational data of alanine tetrapeptide are from ref. 27 except the AlphaR conformation, which is obtained fromprivate correspondence.
cules in this study provides an opportunity to revisitthe parameterizations of the 9 , 8 torsional anglesretaining the same bond, angle, and nonbonded pa-rameters as those of the Cornell et al. model. For ex-ample, the parameter H—Csp3—C=O for carbonylcompounds will change peptide energies unless oneuses different atom types for ketone C=O other thanamide C=O, which we choose not to do. In addition,the availability of more accurate ab initio energiesfor di- and tetrapeptides than when the Cornell etal. model was derived provides further motivationto rederive the 9 , 8 torsional parameters. We thusattempted to improve our model for peptides andproteins by optimizing the 9 , 8 torsional parame-ters to reproduce the relative ab initio energies. Onceagain, the parameter optimization engine parmscanwas used to optimize the torsional parameters ofpeptides, which are N—C—C—N (V1 [1.70, 180.0],V2 [2.00, 180.0]) and C—N—C—C (V1 [0.80, 0.0], V2
[0.85, 180.0]). Our main aim was to reproduce tothe extent possible the relative energies of alaninedipeptides and tetrapeptides: In the present case we
could not able to give a clear “physical picture” ofevery Fourier component of the torsional angles.
The six alanine dipeptide conformational ener-gies were selected as a training set to optimize thetorsional parameters, and 11 alanine tetrapeptideconformational energies were used to test the pa-rameters. Table XI lists the relative conformationalenergies of our new model and that of the Cornell etal. model. Both the additive model and the nonad-ditive model without lone pairs were investigated.Table XII summarizes the results of Table XI. Forthe alanine dipeptides (the training set), the aver-age absolute errors are 0.21 and 0.58 kcal/mol forthis model and the Cornell model, respectively. Forthe alanine tetrapeptides (the test set), the averageabsolute errors are 1.21 and 2.58 kcal/mol, respec-tively. When the polarizable energy is included, theperformance of the Cornell et al. model is improvedslightly, which now has an average absolute errorof 2.20 kcal/mol. The new model with polariza-tion is slightly worse than the additive one, but wehave not reoptimized the torsional parameters forthis case. It must be emphasized that a torsional
JOURNAL OF COMPUTATIONAL CHEMISTRY 1067
WANG, CIEPLAK, AND KOLLMAN
TABLE XII.Comparison of Performance of Parm99 and Parm94 for Alanine Peptides.a
constraint was added if the final structure devi-ated greatly from the optimized structure found inthe ab initio calculations. This makes sense becausemany of the ab initio structures were optimized onlyat HF/6-31G∗, which does not include any disper-sion energies. Table XI also shows which conforma-tion was minimized with the torsional constraint.Least-square fittings were performed for the min-imized and the ab initio-optimized structures andthe average RMSDs were found to be approximately0.2 Å for the aforementioned models. One shouldnote that, for those conformations minimized with-out adding torsional constraints, our new model hasa somewhat larger RMSD than that of the Cornell etal. model. For example, for the C7eq conformationof the dipeptide, the RMSD in our model is 0.34 Åcompared with 0.29 Å in the Cornell et al. model.
NUCLEIC ACID BASES
Given that we have added some parametersto our molecular mechanical models (e.g., H—Csp3—Csp3—O) and changed others (e.g., Csp3—Csp3—Csp3—Csp3) that will affect the molecular me-chanical energies for nucleic acids, it was also
necessary to test how this model reproduces theproperties of nucleosides. Recently, we modifiedour torsional parameters for χ (OS—CT—N∗—CK,OS—CT—N∗—CM) and CT—OS—CT—N∗, andalso OS—CT—CT—OS, in order to better repro-duce the sugar-puckering properties and χ valueof nucleosides to derive Parm98.52 Despite this,one of the less desirable properties of Parm98when compared with Parm94 was the increasedbarrier between C2′ endo and C3′ endo sugars.Thus, this study enabled us to revisit this para-meterization and to achieve nucleoside torsionalenergies better than or comparable to Parm98.We achieved this with a significantly smaller V2(1.15, 0.0) instead of (1.5, 0.0) for OCCO and V2
(0.65, 0.0) instead of (1.0, 0.0) for CT—OS—CT—N∗(Table XIII). Furthermore, the new parameters leadto a significantly lower barrier (1EO1′endo–C2′endo =2.15 kcal/mol for Parm99), comparable to that ofParm94 (2.61 kcal/mol) and significantly less thanthat of Parm98 (3.27 kcal/mol). Table XIV comparesthe structural parameters of the minimized struc-tures of our new model and those of MP2/6-31G∗.It can be seen that MP2/6-31G∗ and Parm99 obtainquite similar structures.
TABLE XIII.Conformational Energies Difference for DNA and RNA Bases.a
Ab initio Ab initioBase name and puckers (HF/6-31G∗) (MP2/6-31G∗) Parm99 Parm98 Parm94
a q and W are defined in ref. 50; χ is defined in ref. 51; angles in degrees; conformational energies in kcal/mol.
MOLECULAR DYNAMICS SIMULATION ONUBIQUITIN AND d(CGCGAATTCGCG)2
Molecular dynamics simulations were performedfor the protein ubiquitin and DNA dodecamer de-oxynucleic acid duplex using both our new modeland the Cornell et al. model (and in the case ofthe DNA, Parm98). Figure 5a shows that the RMSdeviation from the crystal structure of ubiquitin iscomparable for the two models, consistent with aprevious analysis by Fox and Kollman.53 For DNA,Parm98 and Parm99 give comparable RMSDs forthe crystal structure, both of which are lower thanin Parm94 (Fig. 5b). For DNA, Parm99 shows an ADNA to B DNA transition at the same time scale asfor Parm9454 (Fig. 5c).
Discussion
Molecular mechanics methods have now becomea standard tool for chemists to study the structureand conformational energy as well as noncovalentmolecular interactions. To calculate these proper-ties as accurately as possible, one should choose agood force field model. Which factors are respon-sible for a successful force field model? First, accu-rate treatment of electrostatic interactions is crucialfor correctly describing the intermolecular energies,which is also an important factor for differentiatingthe force fields that have similar functional forms.Second, a successful force field should have ac-curate and enough specific parameters—especiallytorsional parameters, which are important for de-scribing the geometry and conformational energy.
Good parameter transferability is another crucial el-ement for a successful molecular mechanical model.In our opinion, the minimal functional form ineq. (1) should be adequate to study geometry, in-termolecular interactions, and conformational en-ergies accurately for all but the most restrainedmolecules. Among the several widely used forcefields, MM2/MM3 and MMFF apply more com-plex functional forms, including anharmonic stretchand bend and cross terms, whereas AMBER, OPLS,DREIDING, and other harmonic force fields usethe simpler functional forms of eq. (1). MM3 andMMFF usually represent conformational energiesbetter than the simpler force fields [eq. (1)], but oneshould remember that MM3 and MMFF are alsotwo of the most extensively parameterized forcefields. Many torsional parameters are specified and,of these, many have three Fourier terms. The re-sults presented suggest that it is not the functionalform that is crucial in representing conformationalenergies but rather the parameterization. We haveshown that a well-parameterized model, althoughsimple, is capable of studying tasks for both organicand bioorganic systems.
Different force fields have different methods forhandling electrostatic interactions. Some force fieldscalculate atomic charges with heuristic algorithmsthat are based on electronegativities (e.g., MMFF).Other force fields, such as the OPLS,6 derive chargeempirically, based on fitting to liquid properties(structure, vaporization, and sublimation enthalpy).Both the Weiner et al.2 and the Cornell et al.1 ver-sions of AMBER force fields use electrostatic po-tential charges based on quantum-mechanical cal-culation of fragments (RESP in the case of the
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FIGURE 5. (a) RMS deviation (Å) of backbone atoms (only amino acids 1–72 are taken into account as noted inref. 50) between the crystal structure of ubiquitin and the structure along the MD trajectory as given by the Cornell et al.model (dotted line) and our new model (solid line). Average RMS deviations are 0.95 and 0.86 Å for the Cornell et al.model and the new model, respectively. (b) All-atom RMS deviation (Å) as a function of time with respect to initialstructure for the Dickerson dodecamer d(CGCGAATTCGCG)2 MD simulation in aqueous solution. Average RMSs overthe 1-ns trajectory are 2.46, 2.33, and 3.92 Å for Parm99, Parm98, and Parm94, respectively. (c) All-atom RMS deviation(Å) as a function of time for the Dickerson dodecamer d(CGCGAATTCGCG)2 MD simulation initiated from the A-form inaqueous solution. Solid and dotted lines are RMSs with respect to initial A-form structure and minimizedcrystallographic B-form structure, respectively.
Cornell et al.1) to describe the electrostatic interac-tions. Although the previous electrostatic potentialmodels (Weiner et al.2 and Cornell et al.1) have notbeen widely parameterized to reproduce the rela-tive conformational energy of a large number oforganic systems, they have done a reasonable job indescribing intermolecular interactions. Gundertofiesuggested that it is the treatment of charges thatmade AMBER significantly better than other har-monic force fields with regard to such a test (see
ref. 24). Hobza et al. also showed that the Cornellet al. model represented nucleic acid–base interac-tions better than any other MM and many quantum-mechanical models relative to high-level ab initiocalculations.55
As to which ab initio model is appropriate forthe derivation of RESP charges, it makes sense tochoose the 6-31G∗ basis set for an effective two-bodymodel [eq. (1)] for simulations in polar media. It hasbeen suggested that the 6-31G∗-based ESP-fit charge
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model is capable of providing an excellent repro-duction of condensed-phase intermolecular proper-ties such as liquid enthalpies and densities as wellas free energies of solvation.21 Moreover, Kuyper etal. suggested that the logical choice of basis set forESP-fit charges for use in condensed phases is the6-31G∗ basis set,56 which uniformly overestimatesmolecular polarity; that is, it is well balanced withthe commonly available water models (SPC/E,57
TIP3P,34 TIP4P34), which have dipole moments thatare about 20% higher than the gas phase valuefor water. Unfortunately, the electrostatic potential(EP)-based charge methods are less than ideal fortwo reasons. First, one must carry out quantum-mechanical calculations to obtain the electrostaticpotentials before performing the least-squares fit.Thus, it cannot be applied for deriving charges asefficiently as the empirical methods, which may bea bottleneck for database screening studies. Second,charges generated using different conformations ofa molecule are usually not identical. The variationcan be ≥10% for those charges that have large ab-solute values (>0.1). We should point out that thisdeficiency is much more serious for standard ESPcharges. Although it is partially mitigated by theRESP charge-fitting scheme, it may remain unclearas to how to determine which conformations shouldbe chosen for deriving the RESP charges.
To investigate how this drawback can affect themolecular mechanical energies, we designed threecharge schemes to calculate the relative energies forfive fluorides in Table I. Charge sets I and II wereRESP charges derived from the electrostatic poten-tials of the lower energy conformation and thoseof the higher energy conformation, respectively.Charge set III was generated by multiple molecularfitting procedures using the electrostatic potentialsof both conformers. The torsional parameters usedfor all three schemes were derived based on chargeset I, and it is not surprising that the best resultwas obtained for this charge set. The AAE comparedwith the experiment is 0.24 kcal/mol. For chargeset II, the AAE was 0.58 kcal/mol, and for chargeset III it was 0.35 kcal/mol, which is slightly worsethan charge set I. We conclude that the multiple-molecule RESP charge scheme performs adequatelyand can be used effectively for complicated mole-cules in which the global minimum conformation isnot obvious. A similar tendency was found in theCornell et al.1 model (Parm94). We suggest one mayuse the RESP charges derived from the electrostaticpotentials of the more stable conformers, or use themultiple-molecule RESP charges.
Another key characteristic of a successful forcefield is parameter transferability. One can consider“transferability” on two different levels. First, eachparameter should not only work well for the modelmolecules in the training set, but also for the otherrelated molecules. Second, all the parameters of theforce field should be consistent. For instance, as tothe torsional parameters of HC—CT—CT—X (X canbe F, Cl, or Br), not only should the Fourier seriesand phase angles be the same, but the force con-stants should also follow the periodic table.
To improve transferability one can use parmscanor another heuristic search program37 – 39 to derivethe parameters for all representative test cases avail-able. The more high-quality the experimental orhigh-level the ab initio data, the more reliably theparameters can be transferred to the test sets. Asto how to collect high-quality reference data, thereare several possibilities. First, the molecules mustbe chosen to reflect the chemistry area for whichthe force fields are developed. Second, one shouldunderstand that the relative energies of molecu-lar mechanics are steric energies (1E), which canusually be substituted for the enthalpy difference(1H). However, these data cannot be used to pre-dict the free energy differences (1G) unless thereis good reason to suspect that the entropy differ-ence (1S) is near zero. Although the reference datashould include the experimentally determined en-thalpy differences and the energy differences fromthe potential energy curves in the gas phase, suchvalues are relatively scarce, and, to obtain a reason-able scope for the test set, a few instances of1G mayneed to be included.
It is the parameter developer that is responsiblefor the second level of transferability—the para-meter consistency of the whole force field. Thereare some encouraging examples of parameter con-sistency with our new model. As an example oftorsional parameters of halides, both H—Csp3—Csp3—X and X—Csp3—Csp3—X (X can be F, Cl, orBr) apply the same Fourier components with thesame phase angles for all three kinds of halides.Moreover, the torsional force constants follow theperiodic table. For H—Csp3—Csp3—X, the force con-stant changes from 0.19, 0.25, to 0.55 kcal/mol forfluorides, chlorides, and bromides, respectively; forX—Csp3—Csp3—X, the force constant changes from1.20, to 0.45, to 0.0 kcal/mol for fluorides, chlo-rides, and bromides, respectively. Furthermore, thetorsional parameter of H—Csp3—Csp3—O also ap-plies the onefold Fourier component with the phaseangle of 0.0◦. The force constant (0.25 kcal/mol)is the same as that of chloride. Another example
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is that, for both Csp3—Csp3—Csp3—Csp3 and Csp3—Csp3—Nsp3—Csp3, a onefold as well as a threefoldterm can improve the performance of reproduc-tion of the relative energies. The third example isthat all the following parameters apply a twofoldFourier component with a phase angle of 180.0◦to represent planar systems: X—CA—CA—X (CAis the carbon in pure benzene); X—Csp2—Nsp2—X,X—CD—CD—X (CD is the middle carbon in bu-tadiene); and X—Csp2—CT—X. The force constantschange from 14.5 to 10.0 to 4.0 to 0.0 kcal/mol. Thetendency responses the strength order of the doublebonds formed between the two middle atoms.
The more complex intramolecular force field in-volves anharmonic terms and coupling terms suchas “stretch–bend” in addition to the harmonic termsof eq. (1). This kind of model may be necessaryto study other molecular properties (e.g., vibra-tional frequencies) besides structure and inter- andintramolecular interactions. Including polarizationand three-body exchange repulsion can also extenda simple additive force field.58 Polarization energycan be described using eqs. (2)–(7) and it can besolved for self-consistently. In most cases, it is verydifficult to extend a simple additive model to a non-additive one while keeping most of the parametersunchanged. We are encouraged that our new modelperforms reasonably well in reproducing the rela-tive energies for the three polarizable models withthe same torsional parameters developed for theadditive model. Furthermore, we find that perfor-mance can be improved significantly after slightadjustment of some of the torsional parameters(Table X). Of course, using the high-level (B3LYP-pVTZ) ab initio RESP-based charges with polariza-tion should enable more accurate intermolecular in-teraction in various different solvent environments.In the future, we plan to develop a complete non-additive force field for proteins, nucleic acids, andorganic molecules.
Conclusion
A new version of the AMBER force field has beendeveloped using 82 conformational pairs with RESPcharges. For the pairwise additive model, very en-couraging results have been obtained upon addinga limited set of torsional parameters. The newmodel gives the smallest AAE and RMS deviationcompared with experimental data for both 34- and55-molecule sets of MMFF, MM3, and CHARMm. Asfor comparison to high-level ab initio methods, thepresent model also provides excellent results, and
is comparable in performance to MMFF. Consider-ing that all the molecules chosen for comparisonsare selected from our “training set” and MM3 andCHARMm may have not included all moleculesfor parameterization, the comparison with the twoforce fields might be biased. To make comparisonsmore valid and to investigate the transferability ofthe torsional parameters, we took 11 hydrocarbonsand 7 chlorides for further study by dividing theminto a training set and a test set. The absolute errorsof molecules in the test set are only slightly largerthan in the training set (differences<0.1 kcal/mol).In summary, we conclude that a well-parameterizedharmonic force field with a reliable charge methodcan describe the structure and intramolecular ener-gies for organic systems very well.
It is very encouraging to point out that nonaddi-tive models also give reasonable results using thesame parameters derived for the additive model,although additional torsional parameterization isrequired to achieve the same high level of accuracyas that found using the additive model.
The torsional angle parameters involved in pep-tides have been optimized by the new programparmscan to reproduce the relative energies of ala-nine dipeptide and tetrapeptide conformers calcu-lated by high-level ab initio methods. Although,for those conformers, which are fully flexible dur-ing minimization, the new model predicts geome-try slightly less accurately than the Cornell et al.model, it reproduces the relative energies much bet-ter than the Cornell et al. force field. The averageabsolute errors for alanine dipeptide and tetrapep-tide are 0.21 and 1.21 kcal/mol, respectively. Thisis noteworthy because the absolute accuracy ofthe LMP2/cc-pVTZ (-f) relative energies for thetetrapeptide has been estimated to be 1.0 kcal/mol.Moreover, two conformations, Conf8 and Conf9,of alanine tetrapeptide are of rather high energyand are not often found; if we neglect them, theAAE of the test set is reduced to 0.92 kcal/mol,which is within the absolute accuracy of the ab initiomethod.
Parmscan has also been applied to optimize sometorsional parameters involved in nucleic acids andthe new model achieves DNA structures closer tothe crystal structures than Parm94 and comparableto those found with Parm98. In the process of modi-fying Parm98 to “reach” Parm99, we have been ableto lower the sugar-pucker barrier of Parm98 to alevel comparable to that of Parm94, which shouldenable it to sample conformational space as effi-ciently as Parm94.
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Acknowledgment
P.A.K. is grateful to acknowledge research sup-ported from NIH GM-29072 and NIH CA-25644(P. Kollman, P. I.) and NIH GM-56609 (E. Arnold,P. I.)
Supplementary Material
The parameter file (Parm99) and topology filesfor the organic molecules in our test sets are avail-able as Supplementary Material. All the atom typesof Parm99 are the same as those in the Cornellet al. model (Parm94), with the exception of thesp3 neutral nitrogen (NT) and the inner sp2 carbonof dienes (CD). Detailed information on the non-bond, bond, and angle parameters of the new atomtypes are available in the “Results” section of thiswork.
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