FORCE FIELD (CHEMISTRY)
In the context of molecular modeling, a force field refers to the form and parameters of mathematical functions used to describe the potential energy of a system of particles (typically molecules andatoms). Force field functions and parameter sets are derived from both experimental work and high-level quantum mechanical calculations. "All-atom" force fields provide parameters for every type of atom in a system, including hydrogen, while "united-atom" force fields treat the hydrogen and carbon atoms in each terminal methyl and each methylene bridge as a single interaction center. "Coarse-grained" force fields, which are frequently used in long-time simulations of proteins, provide even more crude representations for increased computational efficiency.
The usage of the term "force field" in chemistry and computational biology differs from the standard usage in physics. In chemistry it is a system of potential energy functions rather than the gradientof a scalar potential, as defined in physics.
CONTENTS
- Force fields and interatomic potentials 1
- Functional form 2
- Parameterization 3
- Deficiencies 4
- Future perspectives 5
- Popular force fields 6
- Classical force fields 6.1
- Second-generation force fields 6.2
- Polarizable force fields based on electronic structural theory 6.3
- Polarizable force field based on induced dipole 6.4
- Polarizable force fields based on point charges 6.5
- Polarizable force fields based on distributed multipoles 6.6
- Polarizable force fields based on density 6.7
- Polarizable force fields based on Bond Polarization Theory (BPT) 6.8
- Reactive force fields 6.9
- Coarse-grained force fields 6.10
- Water models 6.11
- Post-translational Modifications and Unnatural Amino Acids 6.12
- Other 6.13
- See also 7
- References 8
- Further reading 9
FORCE FIELDS AND INTERATOMIC POTENTIALS
In materials physics and materials science, the term interatomic potential or analytical potential is usually used instead of the term force field to describe interatomic interaction models. While force fields usually do not allow for the breaking of chemical bonds, most interatomic potentials are developed in a way in which all bonds can break and reform dynamically.[1] [2] .[3]Interatomic potentials include e.g. the bond order potentials commonly used for covalently bonded materials and theembedded-atom method (EAM) potentials widely used for metals.[2]
FUNCTIONAL FORM
The basic functional form of a force field encapsulates both bonded terms relating to atoms that are linked by covalent bonds, and nonbonded (also called "noncovalent") terms describing the long-range electrostatic and van der Waals forces. The specific decomposition of the terms depends on the force field, but a general form for the total energy in an additive force field can be written as E_{ ext{total}} = E_{ ext{bonded}} + E_{ ext{nonbonded}} where the components of the covalent and noncovalent contributions are given by the following summations:
E_{ ext{bonded}} = E_{ ext{bond}} + E_{ ext{angle}} + E_{ ext{dihedral}}
E_{ ext{nonbonded}} = E_{ ext{electrostatic}} + E_{ ext{van der Waals}}
The bond and angle terms are usually modeled as harmonic oscillators in force fields that do not allow bond breaking. A more realistic description of a covalent bond at higher stretching is provided by the more expensive Morse potential. Just as the potential energy can be written as a quadratic form in the internal coordinates, so it can also be written in terms of generalized forces. The resulting coefficients are termed compliance constants.
The functional form for the rest of the bonded terms is highly variable. Proper dihedral potentials are usually included. Additionally, "improper torsional" terms may be added to enforce the planarity of aromatic rings and other conjugated systems, and "cross-terms" that describe coupling of different internal variables, such as angles and bond lengths. Some force fields also include explicit terms for hydrogen bonds.
The nonbonded terms are most computationally intensive because they include many more interactions per atom. A popular choice is to limit interactions to pairwise energies. The van der Waals term is usually computed with a Lennard-Jones potential and the electrostatic term with Coulomb's law, although both can be buffered or scaled by a constant factor to account for electronicpolarizability and produce better agreement with experimental observations.
PARAMETERIZATION
In addition to the functional form of the potentials, a force field defines a set of parameters for each type of atom. For example, a force field would include distinct parameters for an oxygen atom in acarbonyl functional group and in a hydroxyl group. The typical parameter set includes values for atomic mass, van der Waals radius, and partial charge for individual atoms, and equilibrium values ofbond lengths, bond angles, and dihedral angles for pairs, triplets, and quadruplets of bonded atoms, and values corresponding to the effective spring constant for each potential. Most current force fields use a "fixed-charge" model by which each atom is assigned a single value for the atomic charge that is not affected by the local electrostatic environment; proposed developments in next-generation force fields incorporate models for polarizability, in which a particle's charge is influenced by electrostatic interactions with its neighbors. For example, polarizability can be approximated by the introduction of induced dipoles; it can also be represented by Drude particles, or massless, charge-carrying virtual sites attached by a springlike harmonic potential to each polarizable atom. The introduction of polarizability into force fields in common use has been inhibited by the high computational expense associated with calculating the local electrostatic field.
Although many molecular simulations involve biological enthalpy of vaporization (OPLS), enthalpy of sublimation, dipole moments, or various spectroscopic parameters.
Parameter sets and functional forms are defined by force field developers to be self-consistent. Because the functional forms of the potential terms vary extensively between even closely related force fields (or successive versions of the same force field), the parameters from one force field should never be used in conjunction with the potential from another.
DEFICIENCIES
All force fields are based on numerous approximations and derived from different types of experimental data. Therefore they are called empirical. Some existing force fields do not account for electronic polarization of the environment, an effect that can significantly reduce electrostatic interactions of partial atomic charges. This problem was addressed by developing "polarizable force fields" [4][5] or using macroscopic dielectric constant. However, application of a single value of dielectric constant is questionable in the highly heterogeneous environments of proteins or biological membranes, and the nature of the dielectric depends on the model used.[6]
All types of van der Waals forces are also strongly environment-dependent, because these forces originate from interactions of induced and "instantaneous" dipoles (see Intermolecular force). The original Fritz London theory of these forces can only be applied in vacuum. A more general theory of van der Waals forces in condensed media was developed by A. D. McLachlan in 1963 (this theory includes the original London's approach as a special case).[7] The McLachlan theory predicts that van der Waals attractions in media are weaker than in vacuum and follow the "like dissolves like" rule, which means that different types of atoms interact more weakly than identical types of atoms.[8] This is in contrast to "combinatorial rules" or Slater-Kirkwood equation applied for development of the classical force fields. The "combinatorial rules" state that interaction energy of two dissimilar atoms (e.g. C…N) is an average of the interaction energies of corresponding identical atom pairs (i.e. C…C and N…N). According to McLachlan theory, the interactions of particles in a media can even be completely repulsive, as observed for liquid helium.[7] The conclusions of McLachlan theory are supported by direct measurements of attraction forces between different materials (Hamaker constant), as explained by Jacob Israelachvili in his book "Intermolecular and surface forces". It was concluded that "the interaction between hydrocarbons across water is about 10% of that across vacuum".[7] Such effects are unaccounted in the standard molecular mechanics.
Another round of criticism came from practical applications, such as protein structure refinement. It was noted that CASP participants did not try to refine their models to avoid "a central embarrassment of molecular mechanics, namely that energy minimization or molecular dynamics generally leads to a model that is less like the experimental structure".[9] Actually, the force fields have been successfully applied for protein structure refinement in different X-ray crystallography and NMR spectroscopy applications, especially using program XPLOR.[10] However, such refinement is driven primarily by a set of experimental constraints, whereas the force fields serve merely to remove interatomic hindrances. The results of calculations are practically the same with rigid sphere potentials implemented in program DYANA [11] (calculations from NMR data), or with programs for crystallographic refinement that do not use any energy functions. The deficiencies of the force fields remain a major bottleneck in homology modeling of proteins.[12] Such situation gave rise to development of alternative empirical scoring functions specifically for ligand docking,[13] protein folding,[14][15][16] homology model refinement,[17] computational protein design,[18][19][20] and modeling of proteins in membranes.[21]
There is also an opinion that molecular mechanics may operate with energy which is irrelevant to protein folding or ligand binding.[22] The parameters of typical force fields reproduce enthalpy ofsublimation, i.e. energy of evaporation of molecular crystals. However, it was recognized that protein folding and ligand binding are thermodynamically very similar to crystallization, or liquid-solid transitions, because all these processes represent "freezing" of mobile molecules in condensed media.[23][24][25] Therefore, free energy changes during protein folding or ligand binding are expected to represent a combination of an energy similar to heat of fusion (energy absorbed during melting of molecular crystals), a conformational entropy contribution, and solvation free energy. The heat of fusion is significantly smaller than enthalpy of sublimation.[7] Hence, the potentials describing protein folding or ligand binding must be weaker than potentials in molecular mechanics. Indeed, the energies of H-bonds in proteins are ~ -1.5 kcal/mol when estimated from protein engineering or alpha helix to coil transition data,[26][27] but the same energies estimated from sublimation enthalpy of molecular crystals were -4 to -6 kcal/mol.[28] The depths of modified Lennard-Jones potentials derived from protein engineering data were also smaller than in typical force fields and followed the "like dissolves like" rule, as predicted by McLachlan theory.[22]
FUTURE PERSPECTIVES
Molecular mechanics or force field was first introduced apparently independently by Hill and by Westheimer in 1949,[29] primarily applied to organic chemistry to estimate properties such as strain energies among others. The functional form of the force field, focused in this article applied to biological systems, was established by Lifson in the 1960s. For over a half century, force fields have served us well, providing useful insights into and interpretation of biomolecular structure and function. Undoubtedly, it will continue to be widely used, thanks to its computational efficiency, while its reliability will continue to be improved. Yet, there are many well-known deficiencies as noted above. In addition, the number of energy terms used in a given force field cannot be uniquely determined and a highly redundant number of degrees of freedom are typically used. Consequently, the "parameters" in different force fields can be vastly different. Of course, the emphasis to incorporate polarization into the standard pair-wise potentials can be very useful; however, there is no unique way of treating polarization in molecular mechanics because it is of quantum mechanical origin.[29][30] Furthermore, often we are more interested in the properties derived from the dynamic dependence of the force field itself on molecular fluctuations.
One possibility is that the future development of force field ought to move beyond the current molecular mechanics approach, by using quantum mechanics explicitly to construct the force field. A number of the "polarizable force fields" listed below, such as density fitting and bond-polarization, already included some of the key ingredients towards this goal. The explicit polarization (X-Pol) method appears to have established the fundamental theoretical framework for a quantal force field; the next step is to develop the necessary parameters to achieve more accurate results than classical mechanics can offer.[29][30]
POPULAR FORCE FIELDS
Different force fields are designed for different purposes.
MM2 was developed by
- Israelachvili, J. N. (1992). Intermolecular and surface forces. San Diego: Academic Press.
- Schlick, T. (2002). Molecular Modeling and Simulation: An Interdisciplinary Guide. Interdisciplinary Applied Mathematics: Mathematical Biology. New York: Springer-Verlag.
- Warshel, A. (1991). Computer Modeling of Chemical Reactions in Enzymes and Solutions. New York: John Wiley & Sons.
FURTHER READING
SEE ALSO
- VALBOND - a function for angle bending that is based on valence bond theory and works for large angular distortions, hypervalent molecules, and transition metal complexes. It can be incorporated into other force fields such as CHARMM and UFF.
OTHER
- Forcefield_PTM - An AMBER-based forcefield and webtool for modeling common post-translational modifications of amino acids in proteins developed by Chris Floudas and coworkers. It utilizes the ff03 charge model and has several side-chain torsion corrections parameterized to match the quantum chemical rotational surface.[64]
- Forcefield_NCAA - An AMBER-based forcefield and webtool for modeling common non-natural amino acids in proteins in condensed-phase simulations using the ff03 charge model.[65] The charges have been reported to be correlated with hydration free energies of corresponding side-chain analogs.[66]
POST-TRANSLATIONAL MODIFICATIONS AND UNNATURAL AMINO ACIDS
The set of parameters used to model water or aqueous solutions (basically a force field for water) is called a water model. Water has attracted a great deal of attention due to its unusual properties and its importance as a solvent. Many water models have been proposed; some examples are TIP3P, TIP4P, SPC, Flexible SPC, and ST2.
WATER MODELS
- MARTINI - a coarse-grained force field developed by Marrink and coworkers at the University of Groningen, initially developed for molecular dynamics simulations of lipids,[63] later extended to various other molecules. The force field applies a mapping of four heavy atoms to one CG interaction site and is parameterized with the aim of reproducing thermodynamic properties.
- VAMM (Virtual atom molecular mechanics) - a coarse-grained force field developed by Korkut and Hendrickson for molecular mechanics calculations such as large scale conformational transitions based on the virtual interactions of C-alpha atoms. It is a knowledge based force field and formulated to capture features dependent on secondary structure and on residue-specific contact information in proteins.[62]
COARSE-GRAINED FORCE FIELDS
- ReaxFF - reactive force field developed by Adri van Duin, William Goddard and coworkers. It is fast, transferable and is the computational method of choice for atomistic-scale dynamical simulations of chemical reactions.[60]Parallelized ReaxFF allows reactive simulations on >>1000,000 atoms.
- EVB (empirical valence bond) - this reactive force field, introduced by Warshel and coworkers, is probably the most reliable and physically consistent way of using force fields in modeling chemical reactions in different environments. The EVB facilitates calculations of actual activation free energies in condensed phases and in enzymes.
- RWFF - reactive force field for water developed by Detlef W. M. Hofmann, Liudmila N. Kuleshova and Bruno D'Aguanno. It is very fast, reproduces the experimental data of neutron scattering accurately, and allows the simulation of bond formation/breaking of water and acids.[61]
REACTIVE FORCE FIELDS
- COSMOS-NMR (Computer Simulation of Molecular Structure) - developed by Ulrich Sternberg and coworkers. Hybrid QM/MM force field enables explicit quantum-mechanical calculation of electrostatic properties using localized bond orbitals with fast BPT formalism.[59] Atomic charge fluctuation is possible in each molecular dynamics step.
POLARIZABLE FORCE FIELDS BASED ON BOND POLARIZATION THEORY (BPT)
- Gaussian Electrostatic Model (GEM)[55][57][58] - a polarizable force field based on Density Fitting developed by Thomas A. Darden and G. Andrés Cisneros at NIEHS; and Jean-Philip Piquemal (Paris VI University).
- Polarizable procedure based on the Kim-Gordon approach developed by Jürg Hutter and coworkers (University of Zürich)
POLARIZABLE FORCE FIELDS BASED ON DENSITY
- The SIBFA (Sum of Interactions Between Fragments Ab initio computed) force field [55] for small molecules and flexible proteins, developed by Nohad Gresh (Paris V, René Descartes University) and Jean-Philip Piquemal (Paris VI, Pierre & Marie Curie University). SIBFA is a molecular mechanics procedure formulated and calibrated on the basis of ab initio supermolecule computations. Its purpose is to enable the simultaneous and reliable computations of both intermolecular and conformational energies governing the binding specificities of biologically and pharmacologically relevant molecules. This procedure enables an accurate treatment of transition metals. The inclusion of a ligand field contribution allows computations on "open-shell" metalloproteins.
- AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications) force field developed by Pengyu Ren (University of Texas at Austin) and Jay W. Ponder (Washington University).
- ORIENT procedure developed by Anthony J. Stone (Cambridge University) and coworkers.
- Non-Empirical Molecular Orbital (NEMO) procedure developed by Gunnar Karlström and coworkers at Lund University (Sweden)[56]
POLARIZABLE FORCE FIELDS BASED ON DISTRIBUTED MULTIPOLES
- PFF (Polarizable Force Field) developed by Richard A. Friesner and coworkers.
- SP-basis Chemical Potential Equalization (CPE) approach developed by R. Chelli and P. Procacci.
- CHARMM polarizable force field developed by S. Patel (University of Delaware) and C. L. Brooks III (University of Michigan).[51][52]
- AMBER polarizable force field developed by Jim Caldwell and coworkers.
- CHARMM polarizable force field based on the classical Drude oscillator developed by A. MacKerell (University of Maryland, Baltimore) and B. Roux (University of Chicago).[53][54]
POLARIZABLE FORCE FIELDS BASED ON POINT CHARGES
- CFF/ind and ENZYMIX – The first polarizable force field [48] which has subsequently been used in many applications to biological systems.[5]
- DRF90 developed by P. Th. van Duijnen and coworkers.
- PIPF – The polarizable intermolecular potential for fluids is an induced point-dipole force field for organic liquids and biopolymers. The molecular polarization is based on Thole's interacting dipole (TID) model and was developed by Jiali Gao [3] at the University of Minnesota.[49][50]
POLARIZABLE FORCE FIELD BASED ON INDUCED DIPOLE
- X-Pol: the Explicit Polarization Theory[1][29][30] - a fragment-based electronic structure method introduced by Jiali Gao [2] at the University of Minnesota, which can be used at any level of theory—ab initio Hartree–Fock (HF), semiempirical molecular orbital theory, correlated wave function theory, or Kohn-Sham (KS) density functional theory (DFT). It is capable of performing more than 3200 steps (3.2 ps) of MD simulations of a fully solvated protein in water with periodic boundary conditions, consisting of about 15000 atoms and 30000 basis functions on a single processor in 24 hours in 2008, with a full quantum mechanical representation of the entire system.[46] Note that the first MD simulation of a protein by McCammon, Gelin, and Karplus in 1977 lasted 8.8 ps using a united-atom force field without solvent.[47]
POLARIZABLE FORCE FIELDS BASED ON ELECTRONIC STRUCTURAL THEORY
- CFF (Consistent Force Field) - a family of forcefields adapted to a broad variety of organic compounds, includes force fields for polymers, metals, etc.
- COMPASS (Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies) - developed by H. Sun at Molecular Simulations Inc., parameterized for a variety of molecules in the condensed phase, now available through Accelrys.[45]
- MMFF (Merck Molecular Force Field)- developed at Merck, for a broad range of molecules.
- MM2, MM3, MM4 - developed by Norman Allinger, parametrized for a broad range of molecules.
- QVBMM - developed by Vernon G. S. Box, parameterized for all biomolecules and a broad range of organic molecules, and implemented in StruMM3D (STR3DI32).
- TraPPE - a family of molecular mechanics force fields developed by the Siepmann group at the University of Minnesota for molecular simulations of complex chemical systems.
SECOND-GENERATION FORCE FIELDS
- While it is common to refer to the "GROMACS force field", such a force field has never existed. For a period of time, the ffgmx variant of the standard GROMOS forcefield was referred to as the GROMACS forcefield, but this was only to distinguish it from the main GROMOS forcefield.[43] This has been confirmed by David van der Spoel, one of GROMACS' chief authors.[44]
Note
- AMBER (Assisted Model Building and Energy Refinement) - widely used for proteins and DNA.
- CHARMM (Chemistry at HARvard Molecular Mechanics) - originally developed at Harvard, widely used for both small molecules and macromolecules
- CHARMm - commercial version of CHARMM, available through Accelrys.
- CVFF - also broadly used for small molecules and macromolecules.
- [37]
- GROMOS - a force field that comes as part of the GROMOS (GROningen MOlecular Simulation package), a general-purpose molecular dynamics computer simulation package for the study of biomolecular systems. GROMOS force field (A-version) has been developed for application to aqueous or apolar solutions of proteins, nucleotides and sugars. However, a gas phase version (B-version) for simulation of isolated molecules is also available.
- OPLS (Optimized Potential for Liquid Simulations) (variations include OPLS-AA, OPLS-UA, OPLS-2001, OPLS-2005) - developed by William L. Jorgensen at the Yale University Department of Chemistry.
- ENZYMIX – a general polarizable force field for modeling chemical reactions in biological molecules. This force field is implemented with the empirical valence bond (EVB) method and is also combined with the semimacroscopic PDLD approach in the program in the MOLARIS package.
- ECEPP - first force field for polypeptide molecules - developed by F.A. Momany, H.A. Scheraga and colleagues.[38][39]
- QCFF/PI – A general force field for conjugated molecules.[40][41]
- UFF - A general force field with parameters for the full periodic table up to and including the actinoids - developed at Colorado State University.[42]
CLASSICAL FORCE FIELDS
AMBER, CHARMM and GROMOS have been developed primarily for molecular dynamics of macromolecules, although they are also commonly applied for energy minimization. Therefore, the coordinates of all atoms are considered as free variables.
ECEPP was developed specifically for modeling of peptides and proteins. It uses fixed geometries of amino acid residues to simplify the potential energy surface. Thus, the energy minimization is conducted in the space of protein torsion angles. Both MM2 and ECEPP include potentials for H-bonds and torsion potentials for describing rotations around single bonds. ECEPP/3 was implemented (with some modifications) in Internal Coordinate Mechanics and FANTOM.[36]
CFF was developed by Arieh Warshel, Lifson and coworkers as a general method for unifying studies of energies, structures and vibration of general molecules and molecular crystals. The CFF program, developed by Levitt and Warshel, is based on the Cartesian representation of all the atoms, and it served as the basis for many subsequent simulation programs.
[35][34][33][32][31]
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