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{{Otheruses|Force field (disambiguation)}}

[[Image:Bond stretching energy.png|thumb|right|A [[Force field (chemistry)|force field]] is used to minimize the bond stretching energy of this ethane molecule.]]

In the context of [[molecular mechanics]], a '''force field''' (also called a '''forcefield''') refers to the [[potential function|functional form]] and [[parameter]] sets used to describe the [[potential energy]] of a system of particles (typically but not necessarily [[atom]]s). Force field functions and parameter sets are derived from both experimental work and high-level [[quantum mechanics|quantum mechanical]] calculations. "All-atom" force fields provide parameters for every atom in a system, including [[hydrogen]], while "united-atom" force fields treat the hydrogen and [[carbon]] atoms in [[methyl]] and [[methylene]] groups as a single interaction center. "Coarse-grained" force fields, which are frequently used in long-time simulations of [[protein]]s, provide even more abstracted 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 usage a force field is defined as a potential function, while the term is [[force field (physics)|used in physics]] to denote the negative [[gradient]] of a [[scalar potential]].

==Functional form==
{{further|[[Molecular mechanics]]}}
[[Image:MM PEF.png|thumb|right|Molecular mechanics potential energy function with continuum 
solvent.]]
The basic functional form of a force field encapsulates both [[chemical bond|bonded]] terms relating to atoms that are linked by [[covalent bond]]s, and nonbonded (also called "noncovalent") terms describing the long-range [[electrostatic]] and [[van der Waals force|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 
<math>\ E_{total} = E_{bonded} + E_{nonbonded} </math> 
where the components of the covalent and noncovalent contributions are given by the following summations:

<math>\ E_{bonded} = E_{bond} + E_{angle} + E_{dihedral}</math>

<math>\ E_{nonbonded} = E_{electrostatic} + E_{van der Waals} </math>

The bond and angle terms are usually modeled as [[harmonic oscillator]]s 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]]. 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 system]]s, 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 bond]]s.

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.

==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 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 length]]s, bond angles, and [[dihedral angle]]s for pairs, triplets, and quandruplets 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 [[electrostatic charge|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 particle]]s, or massless, charge-carrying virtual sites attached by a springlike [[harmonic oscillator|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 [[macromolecule]]s such as [[protein]]s, [[DNA]], and [[RNA]], the parameters for given atom types are generally derived from observations on small [[organic compound|organic]] molecules that are more tractable for experimental studies and quantum calculations. Different force fields can be derived from dissimilar types of experimental data, such as [[enthalpy]] of [[vaporization]] ([[OPLS]]), [[enthalpy]] of [[sublimation]] ([[CFF]]), [[dipole moment]]s, or various spectroscopic parameters (CFF). 

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 usually 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” <ref name="Ponder">Ponder JW and Case DA. (2003) Force fields for protein simulations. ''Adv. Prot. Chem.'' 66: 27-85.</ref> <ref name="warshel">Warshel A, Sharma PK, Kato M and Parson WW (2006) Modeling Electrostatic Effects in Proteins. Biochim. Biophys. Acta 1764:1647-1676.</ref> 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 <ref name="Shultz">Schutz CN. and Warshel A. 2001. What are the dielectric "constants" of proteins and how to validate electrostatic models? Proteins 44: 400-417. </ref>.

All types of [[van der Waals force]]s 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 force]]s in condensed media was developed by A. D. McLachlan in 1963 (this theory includes the original London’s approach as a special case) <ref name="Israelachvili">Israelachvili, J.N. 1992. ''Intermolecular and surface forces.'' Academic Press, San Diego.</ref>.  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.  <ref name="Leckband"> Leckband, D. and Israelachvili, J. (2001) Intermolecular forces in biology. ''Quart. Rev. Biophys.'' 34: 105-267.</ref>.  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]] <ref name="Israelachvili" />.  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''" <ref name="Israelachvili" />.  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''". <ref name="Koehl">Koehl P. and Levitt M. (1999) A brighter future for protein structure prediction.  ''Nature Struct. Biol.'' 6: 108-111.</ref>  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 <ref name="Brunger">Brunger AT and Adams PD. (2002) Molecular dynamics applied to X-ray structure refinement. ''Acc. Chem. Res.'' 35: 404-412.</ref>.  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 <ref name="Guntert"> Guntert P. (1998) Structure calculation of biological macromolecules from NMR data. ''Quart. Rev.  Biophys.'' 31: 145-237.</ref> (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 <ref name="Tramontano"> Tramontano A. and Morea V. 2003. Assessment of homology-based predictions in CASP5. ''Proteins.'' 53: 352-368.</ref>.  Such situation gave rise to development of alternative empirical scoring functions specifically for [[ligand docking]] <ref name="Gohlke">Gohlke H. and Klebe G. (2002) Approaches to the description and prediction of the binding affinity of small-molecule ligands to macromolecular receptors. ''Angew. Chem. Internat. Ed.'' 41: 2644-2676. </ref>, [[protein folding]] <ref name="Edgcomb">Edgcomb SP. and Murphy KP. (2000) Structural energetics of protein folding and binding. ''Current Op. Biotechnol.'' 11: 62-66.</ref> <ref name="Lazaridis">Lazaridis T. and Karplus (2000) Effective energy functions for protein structure prediction. ''Curr. Op. Struct. Biol.'' 10: 139-145</ref>  <ref name="awml">Levitt M. and Warshel A. (1975) Computer Simulations of Protein Folding, Nature 253: 694-698</ref>, computational [[protein design]] <ref name="Gordon">Gordon DB, Marshall SA, and Mayo SL (1999) Energy functions for protein design. ''Curr. Op. Struct. Biol.'' 9: 509-513.</ref> <ref name="Mendes"> Mendes J., Guerois R, and Serrano L (2002) Energy estimation in protein design. ''Curr. Op. Struct. Biol.'' 12: 441-446. </ref> <ref name="Rohl">Rohl CA, Strauss CEM, Misura KMS, and Baker D. (2004) Protein structure prediction using Rosetta. ''Meth. Enz.'' 383: 66-93. </ref>, and modeling of proteins in membranes <ref name="Lomize1">Lomize AL, Pogozheva ID, Lomize MA, Mosberg HI (2006) Positioning of proteins in membranes: A computational approach. ''Protein Sci.'' 15, 1318-1333.</ref>.

There is also an opinion that molecular mechanics may operate with energy which is irrelevant to protein folding or ligand binding <ref name="Lomize2">Lomize A.L., Reibarkh M.Y. and Pogozheva I.D. (2002) Interatomic potentials and solvation parameters from protein engineering data for buried residues. ''Protein Sci.'', 11:1984-2000.</ref>. The parameters of typical force fields reproduce [[enthalpy]] of [[sublimation (physics)|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 <ref name="Murphy"> Murphy K.P. and Gill S.J. 1991. Solid model compounds and the thermodynamics of protein unfolding. ''J. Mol. Biol.'', 222: 699-709.</ref> <ref name="Shakhnovich"> Shakhnovich, E.I. and Finkelstein, A.V. (1989) Theory of cooperative transitions in protein molecules.  I. Why denaturation of globular proteins is a first-order phase transition. ''Biopolymers'' 28: 1667-1680. </ref> <ref name="Graziano"> Graziano, G., Catanzano, F., Del Vecchio, P., Giancola, C., and Barone, G. (1996) Thermodynamic stability of globular proteins: a reliable model from small molecule studies. ''Gazetta Chim. Italiana'' 126: 559-567.</ref>.  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 <ref name="Israelachvili" />. Hence, the potentials  describing protein folding or ligand binding must be weaker than potentials in molecular mechanics.  Indeed, the energies of [[H-bond]]s in proteins are ~ -1.5 kcal/mol when estimated from [[protein engineering]] or [[alpha helix]] to [[Random coil|coil]] transition data <ref name="Myers">Myers J.K. and Pace C.N. (1996) Hydrogen bonding stabilizes globular proteins, ''Biophys. J.'' 71: 2033-2039. </ref> <ref name="Scholtz"> Scholtz J.M., Marqusee S., Baldwin R.L., York E.J., Stewart J.M., Santoro M., and Bolen D.W. (1991) Calorimetric determination of the enthalpy change for the alpha-helix to coil transition of an alanine peptide in water. ''Proc. Natl. Acad. Sci. USA'' 88: 2854-2858. </ref>, but the same energies estimated from [[sublimation (physics)|sublimation]] [[enthalpy]] of molecular [[crystal]]s were -4 to -6 kcal/mol <ref name="Gavezotti"> Gavezotti A. and Filippini G. (1994) Geometry of intermolecular X-H...Y (X,Y=N,O) hydrogen bond and the calibration of empirical hydrogen-bond potentials. ''J. Phys. Chem.'' 98: 4831-4837.</ref>.  The depths of modified [[Lennard-Jones potential]]s 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 <ref name="Lomize2" />.

== Popular force fields ==
Different force fields are designed for different purposes.  

[[MM2]] was developed primarily for conformational analysis of 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 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 <ref name="Schaumann">Schaumann, T., Braun, W. and Wutrich, K. (1990) The program FANTOM for energy refinement of polypeptides and proteins using a Newton-Raphson minimizer in torsion angle space. ''Biopolymers'' 29: 679-694.</ref>.

[[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 force fields===
*[[AMBER]] (Assisted Model Building and Energy Refinement) - widely used for proteins and DNA
*[[CHARMM]] - 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
*[[GROMACS]] - The force field optimized for the package of the same name
*[[GROMOS]] - A force field that comes as part of the [http://www.igc.ethz.ch/gromos/ 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-AA, OPLS-UA, OPLS-2001, OPLS-2005 - Members of the [[OPLS]] family of force fields developed by [http://zarbi.chem.yale.edu/ 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 [http://futura.usc.edu/programs/index.html#molaris MOLARIS] package.
*[http://msdlocal.ebi.ac.uk/docs/mmrefs.html#ecepp ECEPP/2] - First force field for polypeptide molecules - developed by F.A.Momany, H.A.Scheraga and colleagues.
*QCFF/PI – A general force field for conjugated molecules. <ref name="warshel2"> Warshel A (1973). Quantum Mechanical Consistent Force Field (QCFF/PI) Method: Calculations of Energies, Conformations and Vibronic Interactions of Ground and Excited States of Conjugated Molecules, Israel J. Chem. 11: 709.</ref>. <ref name='warshel-levitt"> Warshel A and Levitt M (1974). QCFF/PI: A Program for the Consistent Force Field Evaluation of Equilibrium Geometries and Vibrational Frequencies of Molecules, QCPE 247, Quantum Chemistry Program Exchange, Indiana University.</ref>

===Second-generation force fields===
*[[CFF]] - a family of forcefields adapted to a broad variety of organic compounds, includes forcefields for polymers, metals, etc.
*[[Merck Molecular Force Field|MMFF]] - developed at Merck, for a broad range of chemicals
*[[MM2]], [[MM3]], [[MM4]] - developed by Norman L. Allinger, for a broad range of chemicals

===Polarizable force field based on induced dipole===
*-CFF/ind and ENZYMIX – The first polarizable force field <ref name="awml3">Warshel A. and Levitt M. (1976) Theoretical Studies of Enzymatic Reactions: Dielectric Electrostatic and Steric Stabilization of the Carbonium Ion in the Reaction of Lysozyme, J. Mol. Biol. 103: 227-249</ref> which has subsequently been used in many applications to biological systems.<ref name="warshel"> Warshel A, Sharma PK, Kato M and Parson WW (2006) Modeling Electrostatic Effects in Proteins. Biochim. Biophys. Acta 1764:1647-1676.</ref>.

===Polarizable Force Fields based on point charges===
*- PFF (Polarizable Force Field) developed by Richard A. Friesner and coworkers
*- [http://iqc.udg.es/~marcel/manuals/drf90/ DRF90] developed by P. Th. van Duijnen and coworkers.
*- SP-basis [[Chemical Potential Equalization]] (CPE) approach developed by R. Chelli and P. Procacci 
*- CHARMM polarizable force field developed by B. Brooks and coworkers.
*- AMBER polarizable force field developed by Jim Caldwell and coworkers.

===Polarizable Force Fields based on distributed multipoles===
*- The [http://www.lct.jussieu.fr/pagesperso/jpp/ SIBFA (Sum of Interactions Between Fragments Ab initio computed)] force field <ref name="SIBGEM">N. Gresh, G. A. Cisneros, T. A. Darden and J-P Piquemal(2007) Anisotropic, polarizable molecular mechanics studies of inter-, intra-molecular interactions, and ligand-macromolecule complexes. A bottom-up strategy, J. Chem. Theory. Comput. 3: 1960 </ref> 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.
*- [http://dasher.wustl.edu/tinker/ AMOEBA force field.] developed by Pengyu Ren (University of Texas at Austin) and Jay W. Ponder (Washington University).
*- [http://www-stone.ch.cam.ac.uk/programs.html 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 Force Fields based on density===
*- Gaussian Electrostatic Model (GEM)<ref name="SIBGEM">N. Gresh, G. A. Cisneros, T. A. Darden and J-P Piquemal(2007) Anisotropic, polarizable molecular mechanics studies of inter-, intra-molecular interactions, and ligand-macromolecule complexes. A bottom-up strategy, J. Chem. Theory. Comput. 3: 1960 </ref><ref name="GEM0">J.-P. Piquemal, G. A. Cisneros, P. Reinhardt, N. Gresh and T. A. Darden (2006), Towards a a Force Field based on Density Fitting.,
J. Chem. Phys. 124: 104101</ref><ref name="GEM2">G. A. Cisneros, J-P. Piquemal and T. A. Darden (2006), Generalization of the Gaussian Electrostatic Model: extension to arbitrary angular momentum, distributed multipoles and speedup with reciprocal space methods, J. Chem. Phys. 125:184101</ref>, 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 Zurick)

===Reactive 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.
*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.

===Other===
*[[VALBOND]] - a function for angle bending that is based on [[valence bond theory]] and works for large angular distortions, [[hypervalent molecule]]s, and [[complex (chemistry)|transition metal complexes]]. It can be incorporated into other force fields such as CHARMM and UFF.

===Water Models===
''Main article: [[water model]]

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, and ST2.

== See also ==

* [http://msdlocal.ebi.ac.uk/docs/mmrefs.html#comp Comparisons and Evaluations of various forcefields]
* [[Molecular dynamics]]
* [[Molecular modelling]]
* [http://www.agilemolecule.com/Manual/Construction.html Molecular model construction with proper Force field association.] [http://www.agilemolecule.com/AGM_Build.html (AGM Build)]

==References==
<div class="references-small">
<references />
</div>

==Further reading==
# Schlick T. (2000). ''Molecular Modeling and Simulation: An Interdisciplinary Guide'' Interdisciplinary Applied Mathematics: Mathematical Biology. Springer-Verlag New York, NY.
# Israelachvili, J.N. (1992) ''Intermolecular and surface forces.'' Academic Press, San Diego.
#Warshel A (1991). "Computer Modeling of Chemical Reactions in Enzymes and Solutions" John Wiley & Sons New York.

[[Category:Force fields| ]]
[[Category:Intermolecular forces]]
[[Category:Molecular physics]]

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