Force field (chemistry)
In the context of molecular modeling, a force field refers to both an interatomic potential functional form and its relative parameters. Interatomic potentials are mathematical functions used to describe the potential energy of a statistical mechanical model formed by a system of particles (typically molecules and atoms).
This usage is in contrast to the definition in physics, where the term refers to a vector field representing forces. The simulation methods used in molecular modeling where the interatomic potential applies, are physical methods. This means that the theoretical background on which such methods of molecular modeling are based, belong to physics. An interatomic potential is defined in physics as a scalar, discrete quantity that can be estimated by a function for each particle of the system. For this reason, the usage of term force field to define an interatomic potential is indeed wrong. Nevertheless, the usage of the term "force field" to mean "interatomic potential" or its parameters is widely used and generally accepted as technical slang.
To provide an example of the relationship between force and potential energy, the following equation describes the total force acting on each particle i of a one-component system of N particles interacting via a spherically symmetric i,j-pair potential where, and is a 3D vector representing the particle position in space:
Potential functions and their parameters (when present) are in principle arbitrary, because the consistency of a mechanical model is dictated by its Hamiltonian form and not by its potential energy term. However, in chemistry both the potential function and its parameters are often derived from both experimental work and high-level quantum mechanical calculations. "All-atom" interatomic potentials provide parameters for every type of atom in a system, including hydrogen, while "united-atom" interatomic potentials treat the hydrogen and carbon atoms in each terminal methyl and each methylene bridge as a single interaction center. "Coarse-grained" potentials, which are frequently used in long-time simulations of macromolecules such as proteins, nucleic acids, and multi-component complexes, provide even more crude representations for increased computational efficiency.
- 1 Interatomic potentials
- 2 Functional form
- 3 Parametrization
- 4 Deficiencies
- 5 Future perspectives
- 6 Popular interatomic potentials
- 6.1 Classical interatomic potentials
- 6.2 Second-generation interatomics potentials
- 6.3 Polarizable interatomic potentials based on electronic structural theory
- 6.4 Polarizable interatomic potential based on induced dipole
- 6.5 Polarizable interatomic potentials based on point charges
- 6.6 Polarizable interatomic potentials based on distributed multipoles
- 6.7 Polarizable interatomic potentials based on density
- 6.8 Polarizable interatomic potentials based on Bond Polarization Theory (BPT)
- 6.9 Reactive interatomic potentials
- 6.10 Coarse-grained potentials
- 6.11 Water models
- 6.12 Post-translational Modifications and Unnatural Amino Acids
- 6.13 Other
- 7 See also
- 8 References
- 9 Further reading
While interatomic potentials sometimes do not allow for the breaking of covalent chemical bonds, many such potentials are developed such that all bonds can break and reform dynamically.  . Interatomic potentials include e.g. the bond order potentials commonly used for covalently bonded materials and the embedded-atom method (EAM) potentials widely used for metals.
The basic functional form of an interatomic potential 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 where the components of the covalent and noncovalent contributions are given by the following summations:
The bond and angle terms are usually modeled as harmonic oscillators in interatomic potentials 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 energies. 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 interatomic potentials 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 electronic polarizability and produce better agreement with experimental observations.
In addition to the functional form of the potentials, just to increase the overall confusion, force field supposedly defines also a set of parameters for each type of atom. Most statistical mechanical models potentials are parametric. An interatomic potential would include distinct parameters for an oxygen atom in a carbonyl 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 of bond 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 interatomic potential parameters 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 macromolecules such as proteins, DNA, and RNA, the parameters for given atom types are generally derived from observations on small organic molecules that are more tractable for experimental studies and quantum calculations. Different interatomic potential parameters can be derived from dissimilar types of experimental data, such as enthalpy of vaporization (OPLS), enthalpy of sublimation, dipole moments, or various spectroscopic parameters.
Parameter sets and functional forms are defined by interatomic potentials developers to be self-consistent. Because the functional forms of the potential terms vary extensively between even closely related interatomic potentials (or successive versions of the same interatomic potential), the parameters from one interatomic potential function should clearly never be used in conjunction with another interatomic potential function.
All interatomic potentials are based on numerous approximations and derived from different types of experimental data. Therefore they are called empirical. Some existing interatomic potentials 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 interatomic potentials"  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.
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). 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. 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. 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". 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". 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. However, such refinement is driven primarily by a set of experimental constraints, whereas the interatomic potentials serve merely to remove interatomic hindrances. The results of calculations are practically the same with rigid sphere potentials implemented in program DYANA  (calculations from NMR data), or with programs for crystallographic refinement that do not use any energy functions. The deficiencies of the interatomic potentials remain a major bottleneck in homology modeling of proteins. Such situation gave rise to development of alternative empirical scoring functions specifically for ligand docking, protein folding, homology model refinement, computational protein design, and modeling of proteins in membranes.
There is also an opinion that molecular mechanics may operate with energy which is irrelevant to protein folding or ligand binding. The parameters of typical force fields reproduce enthalpy of sublimation, 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. 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. 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, but the same energies estimated from sublimation enthalpy of molecular crystals were -4 to -6 kcal/mol. The depths of modified Lennard-Jones potentials derived from protein engineering data were also smaller than in typical potential parameters and followed the "like dissolves like" rule, as predicted by McLachlan theory.
The use of interatomic potentials in chemistry was first introduced apparently independently by Hill and by Westheimer in 1949, primarily applied to organic chemistry to estimate properties such as strain energies among others. The functional form of the interatomic potential, focused in this article applied to biological systems, was established by Lifson in the 1960s. For over a half century, interatomic potentials 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 interatomic potential cannot be uniquely determined and a highly redundant number of degrees of freedom are typically used. Consequently, the "parameters" in different interatomic potentials 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. Furthermore, often we are more interested in the properties derived from the dynamic dependence of the interatomic potential itself on molecular fluctuations.
One possibility is that the future development of interatomic potential ought to move beyond the current molecular mechanics approach, by using quantum mechanics explicitly to construct the interatomic potential. A number of the "polarizable interatomic potentials" 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.
Popular interatomic potentials
Different interatomic potentials are designed for different purposes.
MM2 was developed by Norman Allinger primarily for conformational analysis of hydrocarbons and other small organic molecules. It is designed to reproduce the equilibrium covalent geometry of molecules as precisely as possible. It implements a large set of parameters that is continuously refined and updated for many different classes of organic compounds (MM3 and MM4).
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.
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.
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.
Classical interatomic potentials
- 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.
- COSMOS-NMR - hybrid QM/MM interatomic potential adapted to a variety of inorganic compounds, organic compounds and biological macromolecules, including semi-empirical calculation of atomic charges and NMR properties. COSMOS-NMR is optimized for NMR based structure elucidation and implemented in COSMOS molecular modelling package.
- GROMOS - a interatomic potentials 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 interatomic potentials for modeling chemical reactions in biological molecules. This interatomic potentials 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 interatomic potentials for polypeptide molecules - developed by F.A. Momany, H.A. Scheraga and colleagues.
- QCFF/PI – A general interatomic potentials for conjugated molecules.
- UFF - A general force field with parameters for the full periodic table up to and including the actinoids - developed at Colorado State University.
- While it is common to refer to the "GROMACS interatomic potential", such an interatomic potential has never existed. For a period of time, the ffgmx variant of the standard GROMOS interatomic potential was referred to as the GROMACS interatomic potential, but this was only to distinguish it from the main GROMOS interatomic potential. This has been confirmed by David van der Spoel, one of GROMACS' chief authors.
Second-generation interatomics potentials
- 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.
- 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.
Polarizable interatomic potentials based on electronic structural theory
- X-Pol: the Explicit Polarization Theory - a fragment-based electronic structure method introduced by Jiali Gao  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. 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.
Polarizable interatomic potential based on induced dipole
- CFF/ind and ENZYMIX – The first polarizable force field  which has subsequently been used in many applications to biological systems.
- 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  at the University of Minnesota.
Polarizable interatomic potentials based on point charges
- 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).
- 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).
Polarizable interatomic potentials based on distributed multipoles
- The SIBFA (Sum of Interactions Between Fragments Ab initio computed) force field  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)
Polarizable interatomic potentials based on density
- Gaussian Electrostatic Model (GEM) - 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 interatomic potentials based on Bond Polarization Theory (BPT)
- 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. Atomic charge fluctuation is possible in each molecular dynamics step.
Reactive interatomic potentials
- ReaxFF - reactive force field (interatomic potential) 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. 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.[according to whom?] 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[quantify], reproduces the experimental data of neutron scattering accurately, and allows the simulation of bond formation/breaking of water and acids.
- 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.
- MARTINI - a coarse-grained potential developed by Marrink and coworkers at the University of Groningen, initially developed for molecular dynamics simulations of lipids, 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.
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.
Post-translational Modifications and Unnatural Amino Acids
- 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.
- 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. The charges have been reported to be correlated with hydration free energies of corresponding side-chain analogs.
- 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.
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