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A force field for virtual atom molecular mechanics of proteins

Contributed by Wayne A. Hendrickson, July 13, 2009 (received for review March 4, 2009)
Abstract
Activities of many biological macromolecules involve large conformational transitions for which crystallography can specify atomic details of alternative end states, but the course of transitions is often beyond the reach of computations based on fullatomic potential functions. We have developed a coarsegrained force field for molecular mechanics calculations based on the virtual interactions of Cα atoms in protein molecules. This force field is parameterized based on the statistical distribution of the energy terms extracted from crystallographic data, and it is formulated to capture features dependent on secondary structure and on residuespecific contact information. The resulting force field is applied to energy minimization and normal mode analysis of several proteins. We find robust convergence in minimizations to low energies and energy gradients with low degrees of structural distortion, and atomic fluctuations calculated from the normal mode analyses correlate well with the experimental Bfactors obtained from highresolution crystal structures. These findings suggest that the virtual atom force field is a suitable tool for various molecular mechanics applications on large macromolecular systems undergoing large conformational changes.
Accurate understanding of the dynamic properties of proteins has been a major challenge in biophysics (1, 2). With advances in macromolecular crystallography, structural information has been obtained on very large complexes such as ribosome particles (3), chaperone complexes (4), virus particles (5), and RNA polymerases (6) as well as on thousands of individual proteins. In addition, snapshots of protein structures in different states of activity demonstrate the existence of very large conformational changes (7, 8). Computational analysis of the dynamics of such systems is extremely difficult, if not impossible, when using fullatomic computational approaches, due not only to computational limitations but also to the complexity of the resulting information. Thus, coarsegrained approaches have gained importance for addressing large systems and large conformational changes (9–11).
Coarse graining reduces computational complexity by greatly decreasing degrees of freedom of a molecular system with appropriate assumptions to achieve simplification without compromise of essential features (12). For reducing complexity in proteins, Cαonly models have been the most popular, but other coarsegraining approaches have also been taken, such as the inclusion of sidechain centroids (SCs) (13). An important aspect of coarsegrained analysis is the use of an appropriate pseudoforce field to model the forces and constraints exerted on the molecular system.
The use of simple harmonic potentials to model the CαtoCα interactions in proteins as flexible springs has been most popular (14). Such simple harmonic potentials are very effective in defining the nearnativestate fluctuations of proteins calculated by elastic network models, but these models fail to represent the specific restraints that true interatomic potentials impose on virtual bond angles and dihedrals. Unrealistic distortions can result, especially for calculations that aim to model properties far from the native state. A few potential functions have been designed to account for true molecular forces.
The first major approach to a realistic coarsegrained force field is the united residue (UNRES) potential developed by Scheraga and coworkers (13, 15). It has been used mainly in ab initio protein prediction by means of global conformational searches with energy evaluations. With UNRES, the polypeptide chain is represented by Cα and SC positions. Various energy terms such as virtual Cα–Cα bond terms, virtual dihedral and bondangle constraints, electrostatic interactions, Cα–SC interactions, SC rotamer energies and local correlation energies are taken into consideration. Several other knowledgebased coarsegrained functions have also been designed for structure prediction and design (16). One of particular interest here is the Cα, knowledgebased, virtual potential OPUSCA (17), which includes solvent and hydrogenbonding energies in addition to virtual local and packing energy terms. Interestingly, this potential models the curvilinear terms in a simple secondarystructurespecific approach based on three structure types (helix, sheet, and loop). Unlike the UNRES potential, but as for many related functions, OPUSCA is not formulated for first or second derivatization as is needed for molecular mechanics.
Other coarsegrained potentials have been devised for applications in dynamic simulation. In one model (18), the SCs are used to calculate only the linear energy terms (i.e., virtual bonds and nonbonding interactions). Forces here are formulated in analogy to the fullatomic CHARMM force field; thus, the nonbonded interactions, virtual Cα–Cα dihedrals, and bond angles are similar in character to CHARMM curvilinear terms. This model has been used in molecular dynamics simulations of complex biological systems. In a different approach, atomic features are mapped into a reduced representation for a coarsegrained potential that accounts for the doublewell character of virtual dihedral and bond angle potentials (19). This potential, parameterized by statistical information solely from HIV protease, has been used in Brownian dynamics simulations of HIV1 protease (20). None of these “nonharmonic” force fields has been verified systematically against experimental data except for comparisons of UNRESgenerated ab initio structures to crystal structures.
We have developed a restrained coarsegrained force field for protein molecules. Our potential function is based on virtual atoms at Cα atomic positions, and it is constituted to preserve accurate geometry in computations on the structure and dynamics of biological macromolecules. We therefore call this the virtual atom molecular mechanics (VAMM) potential. VAMM includes both linear and curvilinear terms, parameterized against crystalstructure data by the Boltzmann conversion method (21) and also local restraints that ensure computational stability. Energy minimizations with VAMM converge to energy gradients of order 10^{−6} kcal/molÅ without significant distortions. Normal mode calculations with VAMM yield excellent fits to experimentally obtained Bfactors. The current VAMM force field is useful in various computational approaches to protein dynamics, even for the largest systems, and it has flexibility for extensions to include other types of molecules, such as nucleic acids and lipids, and other protein features, such as SCs.
Theoretical Formulation
CoarseGrained Force Field Parameterization.
VAMM is a coarsegrained model of polypeptide chains based on Cα atoms, and it defines the restraints and pseudoforces acting upon these atoms (Fig. 1). The potential function is formulated as where V_{bonded}, V_{angle}, V_{dihedral}, V_{nonbonded}, and V_{local} are the virtual bond, angle bending, dihedral, nonbonded, and local restraint potentials, respectively. All of the energy terms correspond to the virtual interactions between the Cα atoms.
To parameterize the functional forms for each term, statistical distributions of the properties are calculated from the evaluation (EVA) database (22), which contains more than 2,600 unique structures of proteins. The resulting probability distributions are used to calculate a potential of mean force by the Boltzmann conversion method, where k_{B} is the Boltzmann constant, T is the temperature, P_{i} is the probability of a property at value i, and P_{0} is the referencestate probability.
Virtual Cα–Cα Bonds.
The virtual interaction between the bonded Cα atoms are described by a harmonic potential, such that Three types of virtual bonded restraints exist in polypeptide chains. The first two are the cis and trans peptide bonds on the main chain, and the third includes the disulfide bridges that link the cysteine residues. Each type is explained by the same harmonic potential but with different parameters, specified here by the superscript BT for bond type.
The statistical analysis shows a narrow distribution of Cα–Cα distances centered on 3.78 Å (σ = 0.0093 Å) for trans peptides (Fig. 2A). Hence, a minimum at 3.8 Å can be adopted for the trans peptides or, alternatively, crystalstructure distances can be set as the minima point. Cis peptide bonds, most common before proline residues, adopt a different conformation where the Cα–Cα distance is 2.97 Å on average. To avoid deficiencies around the cis–peptide bonds—or a more complicated functional form to account for the cis conformation as well—the same functional form is preserved, whereas the crystalstructure distances given for each Cα pair are taken as the minima points in a fashion similar to those seen in harmonic network models (14). Harmonic potential fitting to the energy profile obtained from Boltzmann conversion of the probability distribution from trans peptides (Fig. 2A) yields a spring constant of ≈70 kcal/molÅ^{2}. The same force constant is also adopted for cis peptides, because bonding interactions are similar in cis and trans peptides.
Disulfide bridges are also modeled by a harmonic potential as for the Cα–Cα virtual bonds. A virtual bond is assumed to exist between the Cα atoms of the Cys residues that form the bridges. In contrast to Cα–Cα interactions, however, there is a fairly large heterogeneity in the distribution of distances for disulfide bridges. In this circumstance, the equilibrium distances in the harmonic potentials for these virtual bonds are assigned directly as the crystalstructure values. Due to the relative heterogeneity of the Cα atoms related to the disulfide bridges, an empirical force constant value of 20 kcal/molÅ^{2} is adopted. This value permits relative flexibility, but it also preserves the disulfide–bridge restraints.
Virtual Dihedral Restraints.
The virtual dihedral angle defined by four consecutive Cα atoms is directly related to the secondary structure of the residues involved. Based on this main observation, the dihedral angle distributions are calculated for every secondary structure combination possible for each quartet of residues. For this purpose, Dictionary of Protein Secondary Structure (DSSP) (23) assignments are made to all proteins represented in the EVA database. DSSP assigns eight different secondary structure types to residues (i.e., αhelix [H], 3_{10} helix [G], πhelix [I], βsheet [E], turn [T], βbulge [B], bend [S], and loop [L]), but we combine βbulge with the βsheet assignment [E] because these types have very similar dihedral and bending angle propensities. Hence, a series of secondarystructurebased classes of dihedral angle quartets are defined, such as HHHH, HHHL, EEEE, etc. Among such quartets, the dihedrals that are not represented more than 1,000 times in the EVA database are combined into more general classes. For example, XGLX, which groups all of the virtual dihedrals whose central residues are a 3_{10} helix and loop, are grouped in the same quartet. These groupings yielded a total of 126 quartet types having specific dihedral energy parameters.
Having classified the Cα quartets, the probability distribution for dihedral angles in each group is calculated from the EVA database. The probability distribution for the HHHH quartet is given in Fig. 2B as an example. Each probability distribution is used to calculate the corresponding potential of mean force (PMF) by using the Boltzmann conversion method. In the final step, PMF is fit to the Fourier series as where θ is the virtual dihedral angle, ω is the multiplicity, and k_{n} and k_{n}′ values are the force constants. The superscript SS denotes that the virtual dihedral potential is secondarystructure specific. V_{o} is the base energy for each SS dihedral type. The fitting to the PMF curve for the HHHH quartet is given in Fig. 2B.
Virtual Bond Angle.
PMFs for virtual bond angles are calculated in a way similar to dihedral angles. For the bond angles, quartets of dihedrals are replaced by the residue triplets that define the bond angle. This procedure has yielded a total of 79 triplet types having specific anglebending parameters. Different from the dihedral distribution, the virtualbondangle distributions indicate that only a limited number of angle values are populated in the Protein Data Bank (PDB), giving rise to localized, sharp peaks. This finding leads to a PMF defined only in a subset of the angle space (i.e., 0 to π), and the PMF cannot be described outside of these boundaries of definition. An example is given for the HHH (helix–helix–helix) triplet in Fig. 2C. The sharp peak is centered on roughly π/2, and the probability values are 0 outside the boundary [0.3 π, 0.75 π]. To define the energy outside the boundaries and avoid overfitting, potential energy is described by a Fourier series around the populated regions (i.e., in the boundary) and fitting a harmonic potential outside the boundaries. The resulting potential is described as follows: The k values describe the force constants, ω is the multiplicity, and τ is the angle. SS again denotes the particular secondarystructure specification, and τ_{0} is the base value of the angle for a given secondarystructure state. The effective part of the anglebending potential is specified by the Boolean operator a(τ), whereby the angle is within the boundary of the peak of probability distribution for a(τ) = 1, and it is outside for a(τ) = 0.
Nonbonded Interactions.
The interaction between nonbonded Cα atoms can be modeled as springs linking the atoms in a similar fashion to the network models (14). However, the radial distribution of Cα atoms within a 12Å sphere indicates the existence of two interaction shells, as noted before (24). The generic radial distribution function is given in Fig. 2D. For the generic distribution, the shortrange interaction shell peaks at ≈5.5 Å, whereas the long range shell peaks at ≈9.9 Å. Residuespecific radial distributions differ from one another, but a similar bipartite character is preserved.
Based on these observations, a separate Morse potential with parameters defined for longrange and shortrange interactions is fitted to the PMF for each of the 210 types of residue pairs. The boundary position between the two shells, r_{boundary}, is defined as the distance where the fitted functions for longrange and shortrange interactions cross (Fig. 2D). An artificially high potential energy is obtained for the atoms within +/−1 Å from r_{boundary} due to uncertainty about belonging to short vs. longrange interaction shells. Thus, we smooth the potential in this interval to the average of nonbonded energies at r_{boundary}−1Å and r_{boundary}+1Å. This boundarysmoothening dramatically improves VAMM energy minimizations, permitting convergence to very low energy gradients without significant structural distortions (see Applications section below). A generic form for the nonbonded interactions is represented in Fig. 2D for simplicity, but unlike the Morse potential in ref 19. our function has residuespecific parameters. The potential is formulated as where RR denotes the particular residue pair, r_{0} values are at energyminimum distances, and ε_{1} and ε_{2} are residuespecific energy parameters. The r_{0}, ε_{1}, and ε_{2} values are fitted separately for the long and shortrange interactions of each residue pair. Because the radial distribution function is normalized according to the 12Å cutoff distance, the energy assumes the value of ≈0 at the cutoff distance without any shifting or switching of function. Nonbonded interactions are defined only for the atom pairs that are more than five residues away from each other on the polypeptide chain.
Local Restraints.
It is well known that Cα atoms generally enjoy less flexibility compared with the sidechain atoms in proteins. Most fullatomic force fields actually account for this difference inherently. When a protein system is represented only by its Cα atoms, however, proper restrictions on flexibility are not imposed by typical pseudoforces. We note that experimental distributions of Cα–Cα distances for closely linked residues on the polypeptide chain (2 ≤ j−i ≤ 5) are similar to, albeit broader than, those for virtually bonded pairs (j−i = 1), whereas the distributions for truly nonbonded pairs (j−i > 5) have two peaks (Fig. 2D). Computational distortions can arise because, in the absence of a fullatomic model, strict restraints can be maintained for dihedral and angular terms while permitting deviations from statistical values for other local features. To prevent such unrealistic distortions, local harmonic restraints are incorporated into the force field. The local harmonic restraint potential is defined as where r_{ij} is an interCα distance, r_{0} is the corresponding equilibrium value taken from the crystal structure, and k_{local} is the pseudoforce constant. Local restraints are defined between Cα atoms within a contiguous span of five residues, i.e., 2 ≤ j−i ≤ 5. A statistical analysis of local residue pairs yields slightly different force constants depending on separations along the peptide chain, but we use the average value of 5 kcal/molÅ^{2} for k_{local} in our applications. This value suffices to maintain structural integrity during energy minimizations and does not exert excessive constraints during coarsegrained normal mode analysis.
Applications
Energy Minimization with Truncated Newton Method.
Extensive energy minimization of a protein crystal structure can cause distortions of the protein from its native state if computational procedures are inappropriate. An accurate minimization algorithm and force field should not yield significantly high distortions from an accurate starting model, such as a highresolution crystal structure (25). Structure preservation upon energy minimization of highresolution crystal structures is a test to assure that the force field will be useful to define macromolecular structures around their known native states and to evolve meaningful characteristics during further computational analysis.
To detect the behavior of proteins under the VAMM restraints and prepare them for further analysis, energy minimizations are performed. For this purpose, the TN algorithm (TNPACK) provided by Schlick and colleagues (26) is adapted to the coarsegrained systems and VAMM. The TN method (27) is a secondorder minimization algorithm that can minimize large systems to very low energy gradients in few steps. Derivatives are calculated by using the chain rule and the mathematical formulation given in refs. 28 and 29.
Results from energy minimizations on a test set of highresolution structures, performed to reach gradient values down to the order of 10^{−6} kcal/molÅ, are given in Table 1. These VAMM minimizations caused an average rootmeansquareddeviation (RMSD) of 0.72 Å, whereas minimizations of the same set with the CHARMM19 force field and an adopted basis Newton–Raphson (ABNR) algorithm left deviations (RMSD = 1.74 Å) much higher than those with VAMM. Distortions observed for the minimizations with VAMM are in the same range as those observed for minimization with other fullatomic force fields such as AMBER (RMSD = 0.41 Å), OPLS (RMSD = 0.92 Å), and GROMOS96 (RMSD = 1.36 Å), which performed on a different set of proteins (25).
An examplerun VAMMbased minimization is given in Fig. 3 for the prokumamolisin proteinase (PDB ID 1T1E). Among all 15 tested cases, only one exceeded 1 Å in residual RMSD between experimental and minimized structures. Initial gradient values (Fig. 3B) all fall into the range from 1.4 to 2.0 kcal/molÅ. These results verify that VAMM defines native structures in a robust and accurate way, giving minimized states in accordance with protein structure space as defined by crystal structures.
Normal Mode Analysis.
Normal mode analysis (NMA) has been one of the most popular applications of the coarsegrained systems (14). NMA methods, such as network models, when applied to coarsegrained systems generally give accurate and useful information on the functional/collective motions of the protein (12). The atomic fluctuations calculated by the NMA are usually compared with the experimental Bfactors of crystal structures to verify the NMA results (30). Interestingly, simple harmonic potentials have proven to be the strongest method in terms of reproducing the experimental Bfactors. In a recent study, five different NMA procedures are compared with one another in terms of reproducing the Bfactors of ultrahighresolution crystal structures (31). The analysis shows that the Elastic Network Model (ENM) and the similar ElNemo networkmodeling server can reproduce the isotropic Bfactors better than the other approaches, whereas the block normal mode analysis (BNM) approach (32) performs slightly better to reproduce anisotropic Bfactors of ultrahighresolution structures.
In light of previous results, we have performed NMA with VAMM and compared its performance with those of ENM and BNM. For VAMM and BNM, the protein structures are briefly minimized to gradient values of 0.1 kcal/molÅ by the TN method and 10^{−3} kcal/molÅ by the ABNR algorithm, respectively. No further energy minimization was required with the coarsegrained representation as no imaginary normal mode is observed. We justify our NMA approach in the SI Appendix (33). The Hessian matrix is computed by standard methods. Second derivatives of dihedral and angular terms for the VAMMbased NMA are calculated by the chain rule (28, 29). The Hessian is diagonalized to obtain normal mode vectors and frequencies with singular value decomposition. The resulting eigenvectors are used to compute atomic fluctuations, which are compared with experimental Bfactors (Table 1).
VAMM yields an average correlation coefficient of 0.66, whereas this value is 0.59 when ENM or BNM is used. The statistical significance of these difference are tested with a t test showing P = 0.04 for differences related to both ENM and BNM. Thus, VAMM performs better than both other methods in reproducing the experimental Bfactors of highresolution structures. Specific examples comparing VAMM with network models are shown in Fig. 4. At some positions with ENM, Cα atoms show unrealistically high fluctuations; VAMM prevents such improper behavior by incorporating local restraint terms (Eq. 4). Ma and colleagues have introduced alternative approaches to mitigate this “tip effect” problem by using a modified elastic potential function in ENM calculations (34) and a pairwise Hessian diagonalization scheme in BNM calculations (35). In practice, tip effects may not introduce a very serious problem because the important slow modes do not usually display this abnormality; however, it causes defects in analysis of the overall normal modes. The problem most probably arises because restraints in simple models derive solely from packing density within a protein, and systems become unstable where packing density is extremely low.
We note that the utility of Bfactor comparisons as a test of normal mode calculations (30) is limited by crystal lattice effects. An analysis of 43 T4 lysozyme molecules in 25 crystal lattices showed an average of 0.55 for the correlation coefficient of main chain Bfactors, each against the average of all others (36), and Bfactor distributions also vary significantly among four myoglobin structures (37). Calculated fluctuations are the same independent of crystal lattice, and we find that VAMMNMA fluctuations correlate with the average Bfactors from T4 lysozyme structures at the level of 0.66 and with the myoglobin averages at 0.75. Thus, the average of 0.66 from VAMM for the 15structure test set (Table 1) may be near the limit of intrinsic variation.
These results indicate that VAMM provides excellent accordance with experimental Bfactors and controls against abnormally high local fluctuations at illdefined Cα positions. This is the second verification of VAMM against experiment, following energy minimizations. It thus implies that VAMM can be used to calculate the normal modes of biological macromolecules to model protein dynamics, and it opens the way for incorporating this force field into other simulation methods.
Transition Pathway Calculations.
An accurate and efficient description of largescale conformational transition pathways and intermediate states of proteins is an important challenge for computational biology. Coarsegrained pathwayanalysis algorithms based on network models and simple harmonic potentials, or potentials derived directly from the proteins of interest, have been applied by us (38) and others (39–42) to transitions between alternative protein conformations. Such computations typically generate intermediate structures that are highly distorted from allowed conformations of polypeptide chains; the computations are, in fact, so loaded with high strain energies that intermediate structures have been interpreted as undergoing “conformational transition through cracking” (40, 42). Moreover, no doubt because of such distortions, the pictured transition pathways do not converge efficiently (42). In contrast, the VAMM force field provides an accurate and comprehensive energy function, but one simple enough for calculating largescale conformational changes (38). VAMM also provides a basic framework for expansion to include sidechain positions without significant sacrifice from its simplicity.
Molecular Dynamics Simulations.
A potential and important application of the VAMM force field is its implementation into molecular dynamics simulations (43). The general applicability and accuracy of the VAMM force field suggests that such implementation is feasible and has the potential to dramatically expand the use of the VAMM force field for specific types of applications. Such coarsegrained molecular dynamics simulations may be used, for example, in undirected sampling of largescale conformational changes, for steered transformation pathway simulations, and in examination of conformational ensembles about NMAbased transition pathway calculations.
Methods
Protein Structures.
The protein structures are chosen randomly from a set of ultrahighresolution crystal structures with resolutions better than 2 Å. The crystal structure of HIV1 protease (PDB ID 1HHP) is an exception in this selection with a resolution of 2.7 Å (44); it is chosen due to its common use in analysis of protein dynamics. Secondarystructure assignments are made by using the DSSP software (23) and the Cα atom coordinates are extracted from PDB files for VAMMbased calculations.
Energy Minimization.
Energy minimizations with VAMM are performed by using the TN algorithm (26). The TNPACK numeric algorithm for Hessianvector multiplication is generally used; a userdefined analytical multiplication algorithm did not provide any significant improvement. A goldensection line search is adapted to increase minimization efficiency. Default settings of TNPACK are used for other options.
Energy minimizations with the Charmm19 force field are performed by the CHARMM simulation package (45). Nonbonded interaction parameters are set such that the electrostatic interaction is shifted to zero at 12 Å, and the van der Waals interaction is switched off from 8 Å to 12 Å. A distancedependent dielectric constant (ε = 4r) is adopted for minimizations in vacuum to mimic the effect of the solvent.
NMA.
Elastic Network Analysis is performed, similar to the case defined in ref. 14. A simple harmonic potential function is adapted to model all of the hypothetical springs connecting the Cα atoms of proteins. where V_{ij} is the pair wise energy between the atoms i and j, r_{ij} is the fluctuating distance, and r_{0} is the equilibrium distance given in the crystal structures. A cut of distance of 13 Å is adopted. A Hessian is constructed and diagonalized to calculate the eigenvectors and eigenvalues corresponding to normal mode fluctuation vectors and frequencies respectively. VAMMbased NMA differs only in the use of VAMM for Hessian construction instead of harmonic potentials.
BNM is performed by the vibran module of the CHARMM simulation package (45). The protein molecules are energyminimized to a gradient of 10^{−3} kcal/molÅ to avoid negative eigenvalues. The same nonbonded interactions are used for energy minimizations.
Atomic fluctuations (theoretical Bfactors) are calculated from the frequencyweighted linear sum of all normal modes as computed by any one of the methods.
A manageable set of 15 structures used for comparison of theoretical and experimental Bfactors was chosen from a large set of highresolution structures analyzed by Eyal et al. (37), such as to have virtually the same average correlation coefficient against ENM normal mode fluctuations for these 15 (0.59) as for all 176 (a range of 0.54–0.58 with varying ENM parameters).
Acknowledgments
We thank Barry Honig and Ogan Gurel for comments on the manuscript. This work was supported in part by National Institues of Health Grant GM56550 (to W. A. H.).
Footnotes
 ^{1}To whom correspondence should be addressed: wayne{at}convex.hhmi.columbia.edu

Author contributions: A.K. and W.A.H. designed research; A.K. performed research; A.K. and W.A.H. analyzed data; and A.K. and W.A.H. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/cgi/content/full/0907674106/DCSupplemental.

Freely available online through the PNAS open access option.
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