Proc. Nati. Acad. Sci. USA
Vol. 85, pp. 5350-5354, August 1988 Chemistry
Derivation of force fields for molecular mechanics and dynamics from ab initio energy surfaces JON R. MAPLE, URI DINURt, AND ARNOLD T. HAGLER Biosym Technologies, Inc., 9605 Scranton Road, No. 101, San Diego, CA 92121
Communicated by Cyrus Levinthal, March 21, 1988
experimental properties and followed the Lifson consistentforce-field approach (3). In the crystal studies, ab initio calculations were used to obtain information about patterns of charge distribution (11), while the values of the partial charges were determined from experimental properties. In the studies of the energetics of highly strained molecules, ab initio calculations were used to provide information on the functional form. The coupling between angle bending and bond stretching was shown and the need for an analytical term in the force field to represent this physical interaction was stressed. In this paper we present a technique fitting an ab initio molecular orbital energy surface and for deriving potential constants from ab initio data. This technique has been developed with several objectives in mind. First, it provides an efficient method for obtaining a set of potential constants for an arbitrary functional group of interest from an ab initio surface. Thus, it fills the need of providing a well-defined and well-characterized set of approximate force constants for functional groups for which no force field is available. More importantly, it may allow us to address two of the most important issues outstanding in the derivation of force fields for biomolecular and organic systems: (i) the validity of alternative analytical functional forms for representing the energy surface of these molecules and (ii) the transferability of both functional forms and potential constants from model compounds to the molecules of interest. Our approach deviates from existing methods in two respects. First, instead of fitting a grid of the potential energy surface (22) or numerically calculating second or higher order energy derivatives at the equilibrium geometry (23-43), we use an analytical evaluation of the first and second derivatives (44). By fitting the first and second derivatives of the energies (for a given analytical function) to the corresponding values obtained from ab initio calculations on a small number of distorted configurations, we are able to obtain a sufficient amount of information about the potential energy surface in the sampled region, including its anharmonic features, to determine the corresponding force constants. Second, we are less interested in a rigorous Taylor expansion of the potential function as used in classical vibrational analysis (33-35, 43, 45, 46)-although we do demonstrate the applicability of the method to this approach as well-than in a transferable force field including electrostatic and nonbonded interactions, and our fit of the potential surface proceeds accordingly. The degree to which the surface is fit allows us to judge the adequacy of a particular functional form and to compare a variety of analytical representations. The method is described in Section 2. Section 3 briefly reviews the force fields used in this work, while Section 4 presents an application of the method to formate anion. Finally, Section 5 discusses the merit of this approach.
ABSTRACT We present a technique for addressing the problem of deriving potential energy functions for the simulation of organic, polymeric, and biopolymeric systems, as well as for modeling vibrational spectroscopic properties. This method is designed to address three major objectives: deriving and comparing optimal functional forms for describing the energies of molecular deformations and interactions, developing a technique to rapidly and objectively determine reasonable force constants for intermolecular and intramolecular interactions, and determining the transferability of these potential forms and constants. The first two of these objectives are addressed in this paper, while the latter problem will be treated elsewhere. The technique uses ab initio molecular energy surfaces, which are described by the energy and its first and second derivatives with respect to coordinates. As an example, application to a small model compound (i.e., the formate anion) is given. A variety of analytical forms for the potential are tested against the data, to find which forms are best. The importance of anharmonicity and cross terms in accounting for structure and energy, as well as for dynamics, is demonstrated and a more accurate representation of the out-of-plane deformation for a trigonal center is derived from the energy surfaces.
Section 1. Introduction
Theoretical techniques are currently being applied to problems such as ligand binding to receptors, simulation of the structure and conformational fluctuations of flexible molecules, and the design of drugs. These techniques include molecular mechanics and dynamics simulations, Monte Carlo simulations, and vibrational normal-mode analysis (1). Such techniques ultimately will offer the possibility of understanding the complex behavior of organic molecules, polymers, biomolecules, and biomolecular complexes in terms of fundamental intermolecular and intramolecular physical forces, thus allowing an understanding of the molecular behavior of these systems at a level that is inaccessible to experimental techniques alone. However, the results of these simulations depend critically on the set of potential energy functions (i.e., force field) used. Although a great deal ofeffort has gone into the derivation offorce fields now in use (2-21), there remain many more organic and biomolecular functional groups for which adequate parameters have yet to be derived. Moreover, even the functional form required to describe molecular energy surfaces is still a subject of research (1). The derivation of the force constants and the appropriate functional forms to be used in describing the energy surfaces for biomolecular systems have been addressed previously (6, 7, 9-12). These studies relied for the most part on the fit of The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
tPermanent address: Dept. of Chemistry, Ben Gurion University of the Negev, Beer Sheva, Israel. 5350
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Proc. Natl. Acad. Sci. USA 85 (1988)
Section 2. Method of Calculation
Section 3. The Potential Energy Function
The basic objective is to find an analytical function (i.e., force field) that best describes a given ab initio potential energy surface. This implies both the form of the function and the constants for each interaction term. The information we invoke comprises the quantum chemical potential energy and its first and second Cartesian derivatives evaluated at certain points in the vicinity of the equilibrium geometry. For a molecule with n atoms there are 3n first derivatives and 3n(3n + 1)/2 second derivatives or, excluding rotation and translations, (4.5)(n - 2)(n - 1) total internal first and second derivatives for each molecular configuration. This constitutes a sizable amount of data, and a relatively small number of properly chosen configurations can provide ample information about the potential surface surrounding the equilibrium geometry. For example, five configurations of a sixatom molecule result in 454 data points (including the four relative energies), and this should suffice for determining a good quality force field with 50 parameters. As noted by Pople et al. (44), the evaluation of both first and second derivatives requires four to five times more computation time than the evaluation of the gradients and seven times more than for a simple Hartree-Fock calculation. It follows that with the same amount of computation time the number of data points that can be obtained for a six-atom molecule is 13 times greater when first and second derivatives are used than when simply calculating the energy at various points. This ratio increases for larger molecules and has a qualitative effect on the ability to derive and characterize force fields. The distorted configurations are generated by random displacements of the internal coordinates about their equilibrium values or by displacing them along the molecular normal modes, if the latter are available. In some applications-e.g., study of rotational barriers-the coordinates of interest are systematically varied, while the other coordinates are distorted at random. Once the configurations are determined, a set of ab initio calculations is carried out to determine the energy and its first and second (Cartesian) derivatives for each of them. Then, the task is to fit these derivatives with an empirical potential function. Let us denote the various configurations by Greek subscripts and the Cartesian coordinates by lowercase letter subscripts. The ab initio energy of structure, a, relative to some reference value (which is the energy of one of the configurations at hand), will be denoted accordingly by AE', its gradient aE'/axi will be denoted by g°, and its second derivatives a2E'/axiaxj by H°,ij. Let the corresponding quantities derived from the empirical potential function be denoted by AEaE, goji and Heyij. These will depend on the parameters that appear in the empirical function. One now forms the following sum of squared differences:
The potential energy, V, can be expressed as a function of the internal degrees of freedom and the nonbonded distances through various analytical representations. One example is given by
where tog and cn are appropriate weighting factors. Minimizing S with respect to the empirical parameters yields the least-squares fit to the potential energy surface. It should be noted that the differences between the empirical and the ab initio relative energies were not included in the sum of squares defined above. We have found it useful to exclude the energies from the fitting procedure and use them instead as a check on the goodness of the fit. As we show below, if the fit to the derivatives is good, then the empirical relative energies are close to the ab initio ones. Of course, they, as well as other observables (e.g., the dipole moment components), could be included with appropriate weights, if desired.
- ea-(b-b0)]2 -
out of plane
E H1(1 - s cos nT)
(b - bo) (6 >Fbb,(b - bo)(b' - bl) + >F7,O bO
>F.0(6 - 60) (6' -00) 00'
XX + Z
COS T (6- 00)(I'
Basically, the terms in Eq. 2 represent the energies required to deform the internal coordinates (which have the values b, 0, T, X) from their unstrained standard values, denoted by the subscript 0, and the nonbonded interaction energy. These terms represent the energies for (in the order used in Eq. 2) bond stretching or compression; angle bending; out-of-plane deformation of a planar system; torsion angle twisting; coupling between bond deformations; coupling between a bond deformation and an angle bending; coupling between angle bendings; coupling between out-of-plane deformations; coupling between a torsion angle twisting and the two associated angle bendings; and repulsive, dispersive coulombic interactions between nonbonded atoms. The parameters Db, H,1,HTIH., and Fij are the force constants for the corresponding intramolecular deformations, r* and E characterize the size of the atoms and the strength of the van der Waals interaction between them, and qj are the partial charges carried by each atom. Eq. 2 constitutes the Discover force field, which is based upon the VFF approach (2-7, 15). Other force fields exist [e.g., MM2 (8), AMBER (16, 17, 19, 21), and CHARMM (18, 20)] and mainly vary with respect to the inclusion or omission of cross terms and the degree of anharmonicity of the bonds and bending angles. These analytical force fields are to be distinguished from standard ab initio and vibrational force fields (22-43, 45, 46), sometimes termed GVFF (general valence force field) or GHFF (general harmonic force field). The latter are straightforward and rigorous Taylor expansions of the potential about the equilibrium configuration in terms of a well-determined system of coordinates. Their disadvantage with regard to molecular mechanics is they are not transferable from one molecular to another. The analytical force fields, on the other hand, use a redundant set of coordinates (47, 48) but attain transferability by explicitly modeling physical interactions that occur in the molecule (e.g., Coulomb and Lennard-Jones interactions, Morse potential for the bonds). Section 4. Probing the Energy Surface of the Formate Ion In this section we demonstrate the method by fitting the ab initio surface of the formate anion (HCOO -), determining
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Proc. Natl. Acad. Sci. USA 85 (1988) 0
trum*. The energies of the distorted configurations spanned
lcC 114.40 1 1 4.4'
0.000 0.000 kcal/mole
ul) CD Cn
0.000 12.617 kcal/mole X
X 0.000 14.197 kcal/mole
X 0.023 14.602 kcal/mole
X = -0.343 49.031 kcal/mole
FIG. 1. Configurations of the formate anion used in the ab initio calculations. The equilibrium configuration (E = 0.0 kcal/mol) is shown at the upper left.
optimal potential constants for various functional forms, and testing the accuracy of a variety of analytical representations (force field forms). In this case, eight different configurations of this molecular ion were generated by a random walk around a guessed reference point estimated to be near the (a priori unknown) equilibrium geometry (Fig. 1). Ab initio 4-31G* Hartree-Fock calculations of the energies and its first and second derivatives were then performed for all eight structures, yielding 720 Cartesian first and second derivatives and seven relative energies. The ab initio energy was minimized later to obtain the equilibrium geometry of the molecule, its energy, second derivatives, and the harmonic spec-
a range of 49 kcal/mol (1 cal = 4.18J) above the equilibrium point, much of which was stored in the out-of-plane mode. From all these data only the energy derivatives of the eight distorted configurations were used to derive the potential constants. The rest of the data-i.e., relative energies, equilibrium geometries, and harmonic spectra-were used to test how well the optimized force fields could predict observables outside the domain of the fitting. The details of the fitting procedure were described in Section 2. Weighting factors of 100 and 1 were used for the first and second derivatives, respectively. In this work we are interested in testing the ability of the method to fit the ab initio energy surface and identify those functional forms that are more appropriate than others in representing the molecular potential surface. Accordingly, we have performed a series of calculations in which the ab initio data were fitted with a hierarchy of potential functions that conform with the general structure of Eq. 2 but differ with respect to the degree of elaboration as well as the system of coordinates that are being used. The results are presented in Table 1. The system of coordinates in this case consists of three bond deformations (C-H and two C-0 bonds), three angle deformations (0-C--C and two 0-C-H), and an out-ofplane deformation. The latter is not uniquely defined, and several possibilities exist in the literature. For the following force fields we have defined this coordinate as the distance of the central atom from the plane formed by the other three. We address the question of which functional form best represents the energy of this deformation as a function of coordinates below. Six force fields were tested with this system ofcoordinates. The first one was a simple diagonal force field in which the molecule was represented by seven noninteracting harmonic springs. Each spring was characterized by a coordinate s (bond, angle, or out-of-plane coordinate) and two parameters, a force constant k and a "strain free" reference value so, Vspring = k(s - so)2. Basically, this simple diagonal form is the force field used in most protein simulations (16-21). The second force field was equally diagonal but included some anharmonicities in the form of cubic terms for all coordinates
tFor this minimization we used as an initial geometry the equilibrium structure calculated with the analytical form (Eq. 2). The analytical representation was parameterized with the distorted configuration data. In general, this procedure significantly reduced the computation time required for ab initio geometry optimization. This is particularly valuable for larger molecules.
Table 1. Format anion calculations
Force field ° Arms AE., Aors Avl,avg Diagonal 1. Quadratic (9) 2.18 0.17 0.298 0.27 155.6 2. Cubic (13) 0.137 70.1 0.010 0.41 186.6 3. Quartic (18) 0.125 2.05 0.009 0.62 172.3 With cross terms 4. Full second order (14) 0.278 1.00 0.010 0.15 88.3 5. Full third order (35) 0.032 0.13 0.002 0.42 31.1 6. Full third order + 0.017 0.17 0.001 0.16 7.2 quartic diagonals (40) Six with various definitions of the o.o.p. coordinate 0.017 0.17 0.001 0.16 7.2 Pyramid height 0.032 0.45 0.53 0.001 13.4 Improper torsion 0.018 0.21 0.11 Wilson et al. 0.000 11.8 Energy is in kcal/mol, distances are in A, angles are in degrees, as frequency is in cm-l. Numbers in parentheses are numbers of parameters. o.o.p., out of plane.
Chemistry: Maple et al. except the out-of-plane, which is bound by symmetry to include only even terms. The third force field was also diagonal, but the anharmonicity was now represented by quartic terms (for all springs) in addition to the cubic ones. The next three force fields allowed the coordinates to interact. Force field 4 involved the addition of quadratic cross terms to the diagonal harmonic force field (i.e., 1). Except for the lack of anharmonicity in the bonds, this is essentially the form of the force field used by Lifson for alkanes and Hagler and coworkers (1-7, 9-12, 15) for peptides and proteins. Force field 5 had quadratic and cubic cross terms added to the cubic force field (i.e., 2) and force field 6 resulted similarly from the addition of quadratic and cubic cross terms to the fully anharmonic quartic force field (i.e., 3). The number of parameters for these three force fields was greatly reduced by excluding coupling terms that are symmetry forbidden or that are redundant because of the redundancy in the coordinates (22, 47, 48). Importance of Cross Terms. The relative rms deviation J71 between the ab initio derivatives of the eight distorted configurations and the corresponding derivatives that were obtained from the optimized empirical force fields are given in Table 1. As expected, more elaborate force fields fit the ab initio data progressively better, but some modifications are more effective. The two most significant modifications are the addition of cubic terms to the harmonic diagonal force field, which leads to a reduction of more than a factor of 4 in the sum of squares of residuals, and the further addition of cross terms to the cubic diagonal force field, which reduces the sum of squares by an additional factor of 20. Other modifications have a much smaller effect on the sum of squares. In particular, there seems to be an intrinsic limit to the fit that can be obtained with a purely diagonal force field (compare force fields 2 and 3). Although the force fields in this work are derived by fitting energy derivatives, their performance, with respect to other observables such as structure, energy, and dynamics as reflected in vibrational frequencies, is of more direct interest. The deviations of the empirical calculations from the ab initio values with respect to the relative energies of the eight distorted configurations, the equilibrium geometry, and the harmonic spectrum at the equilibrium geometry are given in Table 1. None of these observables was included in the fitting procedure and, consequently, the magnitude of the deviations directly demonstrates the predictive power of the force fields. Inspecting the predicted energies first, one sees that inclusion of anharmonicity (through cubic and quartic terms) in a diagonal force field, while improving the fit to the derivatives, does not necessarily yield more accurate relative energies. The effect on equilibrium geometry varies: the anharmonic diagonal force fields are not significantly better than the diagonal harmonic force field with respect to the geometry but are clearly worse with respect to the harmonic frequencies. The latter finding merely indicates that anharmonicity cannot replace the cross terms. This is further manifested in the fact that the results are substantially and uniformly improved when cross terms are included in the force field (see force fields 4-6). Thus, for example, on going from a diagonal to a full quadratic force field the deviation of the calculated energies from the ab initio values is reduced from 2.18 to 1.0 kcal/mol, and the deviation with respect to the harmonic spectrum is reduced from 156 to 88 cm- . The deviations with respect to the bond lengths and angles are also reduced, although these observations are reasonably calculated even at the diagonal level. This observation with respect to the cross terms is somewhat contrary to the common belief that such terms are important for evaluating the harmonic spectrum but not for the energetics and geometry. At the same time it is clear that to obtain a reasonable fit for the harmonic spectrum (devia-
Proc. Natl. Acad. Sci. USA 85 (1988)
tions in frequencies significantly lower than 100 cm-1), as well as for the energetics and the equilibrium geometry, both anharmonicities and cross terms ought to be included (force fields 5 and 6). Thus, our results suggest that the force field has to be balanced with respect to both types of interactions. Note also that it is not simply a matter of adding parameters that achieves a better fit. For example, the quadratic force field (i.e., 4) with cross terms, which has 14 parameters, has an overall better predictive ability (energies and equilibrium observables in Table 1) than the diagonal quartic force field, which has 18 parameters (i.e., four more parameters). The conclusions stated above regarding the importance of anharmonicity and cross terms remain valid for molecules larger than formate anion. These calculations will be reported in detail elsewhere. Out-of-Plane Representation, Using the ab Initio Energy Surface to Probe the Functional Form of Potential Function. We now turn to the question of how to define the out-of-plane coordinate (Fig. 2). Most force fields used in simulating proteins and other biomolecular systems define this coordinate as an improper torsion. This definition is somewhat artificial and two possible alternatives are the height of the pyramid formed by the four atoms (i.e., the distance coordinate discussed in the beginning of this section) and the angle between a bond and the plane formed by the other two bonds [as given by Wilson et al. (46)]. The results obtained by varying the three definitions in force field 6 are given in Table 1. (Note that the line corresponding to the pyramid height is identical to that for force field 6 given above.) As demonstrated, differences are not drastic but the improper torsion definition results in energy deviation that is 2-3 times larger than those obtained with the other two definitions. Equilibrium geometry and harmonic frequencies are also not as well fit, although the differences are slight. These differences may become more pronounced for larger molecules such as peptides that have many trigonal centers. This case is a particular example of the importance of the functional form used in the force field. All three force fields listed in the bottom of Table 1 are identical, except for the definition of this coordinate, and consequently have the same number of parameters. The differences in performance stem directly from differences in the quality of approximation of the out-of-plane coordinate. Thus, this demonstrates how the method proposed here is particularly suitable for resolving questions of the appropriate functional form to represent the energy of a molecule as a function of the coordinates. In other words, the procedure is highly useful for determinations of the "laws of force" for intramolecular deformations. Section 5. Conclusions The results presented above indicate that the derivatives of the molecular potential energy, calculated for a small number of distorted structures, can provide us with a powerful tool for investigating the functional form and parameters of force fields for molecular systems. This allows us to explore analytical and transferable representations of the energy for organic molecules, polymers, and biomolecular systems. In fact, the method is general and can be used to derive spectroscopy-quality force fields for small molecules of any 0
Wilson, Decius and Cross
FIG. 2. Various definitions of the out-of-plane coordinate.
Chemistry: Maple et al.
kind. The advantage of this approach is in the relative ease with which ab initio "observables" may be obtained and in the reduced computational effort required for obtaining the necessary information when derivatives are included. We have shown that force fields derived in this way can reproduce the potential energy surface up to a significant energy above the equilibrium point (49 kcal/mol for formate anion). In addition, the method has been used to evaluate the applicability of alternative functional forms such as different definitions of the out-of-plane and torsional coordinate. Finally, we have shown that inclusion of cross terms in the force field improves the performance with respect to reproducing molecular energetics, as well as dynamics (as reflected in the harmonic spectrum). We thank Dr. Tom Halgren (Merck Sharp & Dohme Research Laboratories) for reading the manuscript and for many helpful discussions. Ab initio calculations reported here were done at Cray Research and Abbott Laboratories. The latter calculations were carried out by Dr. Tetsuro Oie. The program GRADSCF, written by Dr. Andrew Komornicki (Polyatomic Institute), was used for the calculations at Cray Research while Gaussian 82 was used at Abbott Laboratories. This work was supported by the Consortium for Research and Development of Potential Energy Functions (including Abbott Laboratories, Battelle Pacific Northwest Laboratory, Biosym Technologies, Cray Research, E. I. Du Pont de Nemours & Co., ETA Systems, ICI, Merck Sharp & Dohme Research Laboratories, Monsanto Company, Rohm and Haas Company, and The Upjohn Company). A.T.H. is also affiliated with the Agouron Institute, 505 Coast Boulevard South, La Jolla, CA 92037. 1. Hagler, A. T. (1985) in The Peptides, eds. Hruby, V. J. & Meienhofer, J. (Academic, New York), Vol. 7, pp. 213-299. 2. Lifson, S. & Warshel, A. (1968) J. Chem. Phys. 49, 5116-5129. 3. Lifson, S. (1968) J. Chem. Phys. 65, 40-43. 4. Warshel, A. & Lifson, S. (1970) J. Chem. Phys. 53, 582-594. 5. Ermer, 0. & Lifson, S. (1973) J. Am. Chem. Soc. 95, 41214132. 6. Hagler, A. T., Huler, E. & Lifson, S. (1974) J. Am. Chem. Soc. 96, 5319-5327. 7. Hagler, A. T. & Lifson, S. (1974) Acta Crystallogr. Sect. B 30,
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