Force field (chemistry)

A force field is used to minimize the bond stretching energy of this ethane molecule.

In the context of molecular modeling, a force field (a special case of energy functions or interatomic potentials; not to be confused with force field in classical physics) refers to the functional form and parameter sets used to calculate the potential energy of a system of atoms or coarse-grained particles in molecular mechanics and molecular dynamics simulations. The parameters of the energy functions may be derived from experiments in physics or chemistry, calculations in quantum mechanics, or both.

All-atom force fields 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 methyl group (terminal methyl) and each methylene bridge as one interaction center. Coarse-grained potentials, which are often used in long-time simulations of macromolecules such as proteins, nucleic acids, and multi-component complexes, provide even cruder representations for higher computing efficiency.

Functional form

Molecular mechanics potential energy function with continuum solvent.

The basic functional form of potential energy in molecular mechanics includes bonded terms for interactions of atoms that are linked by covalent bonds, and nonbonded (also termed noncovalent) terms that describe 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 by quadratic energy functions 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 dihedral energy is highly variable. Additional, "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. 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.

bond stretching

As it is rare for bonds to deviate significantly from their reference values the Morse potential is seldom employed for molecular mechanics due to it not being efficient to compute. The most simplistic approaches utilize a Hooke's law formula:

Where is the force constant, is the bond length and is the value for the bond length when all other terms in the force field are set to 0. The term is oft referred to as the equilibrium bond length which may cause confusion. The equilibrium bond length would be the value adopted in a minimum energy structure with all other terms contributing.

In most cases the effects on accuracy of modelling a bond as a harmonic oscillator are small however a real bond stretching potential is not harmonic and modelling it as such may lead to inaccuracies in predictions of bond lengths to the thousandth of an angstrom. The strong nature of the interactions between atoms yields large values for the force constant (the stronger the bond the higher the value of the force constant).

Though the Hooke's law formula provides a reasonable level of accuracy at bond lengths near the equilibrium distance it is less accurate as one moves away. In order to model the Morse curve better one can employ cubic and higher powers.[1]

Parameterization

In addition to the functional form of the potentials, force fields define a set of parameters for different types of atoms, chemical bonds, dihedral angles and so on. The parameter sets are usually empirical. 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 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 parameters use a fixed-charge model by which each atom is assigned one 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, massless, charge-carrying virtual sites attached by a springlike harmonic oscillator 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 force field 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 together with another interatomic potential function.

Deficiencies

All interatomic potentials are based on many approximations and derived from different types of experimental data. Thus, they are termed empirical. Some existing energy functions 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[2][3] or using macroscopic dielectric constant. However, application of one 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.[4]

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).[5] 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.[6] 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 fully repulsive, as observed for liquid helium.[5] 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".[5] Such effects are unaccounted in standard molecular mechanics.

Another round of criticism came from practical applications, such as protein structure refinement. It was noted that Critical Assessment of protein Structure Prediction (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".[7] The force fields have been applied successfully for protein structure refinement in different X-ray crystallography and NMR spectroscopy applications, especially using program XPLOR.[8] However, such refinement is driven mainly 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[9] (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.[10] Such situation gave rise to development of alternative empirical scoring functions specifically for ligand docking,[11] protein folding,[12][13][14] homology model refinement,[15] computational protein design,[16][17][18] and modeling of proteins in membranes.[19]

There is also an opinion that molecular mechanics may operate with energy which is irrelevant to protein folding or ligand binding.[20] 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.[21][22][23] Thus, 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.[5] 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,[24][25] but the same energies estimated from sublimation enthalpy of molecular crystals were -4 to -6 kcal/mol.[26] 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.[20]

Future perspectives

The use of interatomic potentials in chemistry was first introduced in 1949, apparently independently by Hill and by Westheimer,[27] applied mainly 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. Further, 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.[28][29] 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.[28][29]

Different force fields are designed for different purposes. All are implemented in various computer software.

MM2 was developed by Norman Allinger mainly 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).[30][31][32][33][34]

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.[35]

AMBER, CHARMM, and GROMOS have been developed mainly for molecular dynamics of macromolecules, although they are also commonly used for energy minimizing. Thus, the coordinates of all atoms are considered as free variables.

Classical force fields

  • Assisted Model Building and Energy Refinement (AMBER) – widely used for proteins and DNA.
  • Chemistry at HARvard Molecular Mechanics (CHARMM) – originally developed at Harvard, widely used for both small molecules and macromolecules
  • CVFF – also used broadly for small molecules and macromolecules.
  • COSMOS-NMR – hybrid QM/MM force field 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.[36]
  • GROningen MOlecular Simulation (GROMOS) – a force field that comes as part of the GROMOS software, a general-purpose molecular dynamics computer simulation package for the study of biomolecular systems.[37] GROMOS force field A-version has been developed for application to aqueous or apolar solutions of proteins, nucleotides, and sugars. A B-version to simulate gas phase isolated molecules is also available.
  • Optimized Potential for Liquid Simulations (OPLS, variants include OPLS-AA, OPLS-UA, OPLS-2001, OPLS-2005) – developed by William L. Jorgensen at the Yale University Department of Chemistry.
  • ECEPP[38] – first force field for polypeptide molecules - developed by F.A. Momany, H.A. Scheraga and colleagues.[39][40]
  • QCFF/PI – A general force fields for conjugated molecules.[41][42]
  • Universal Force Field (UFF) – A general force field with parameters for the full periodic table up to and including the actinoids, developed at Colorado State University.[43]
  • Consistent Force Field (CFF) – a family of forcefields adapted to a broad variety of organic compounds, includes force fields for polymers, metals, etc.
  • Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies (COMPASS) – developed by H. Sun at Molecular Simulations Inc., parameterized for a variety of molecules in the condensed phase, now available through Accelrys.[44]
  • Merck Molecular Force Field (MMFF) – 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).
  • Transferable Potentials for Phase Equilibria (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 force fields

  • X-Pol: the Explicit Polarization Theory[45][28][29] – 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 method (HF), semiempirical molecular orbital theory, correlated wave function theory, or Kohn-Sham (KS) density functional theory (DFT). It can perform over 3,200 steps (3.2 ps) of MD simulations of a fully solvated protein in water with periodic boundary conditions, consisting of about 15,000 atoms and 30,000 basis functions on one processor in 24 hours in 2008, with a full quantum mechanical representation of the whole 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]
  • CFF/ind and ENZYMIX – The first polarizable force field[48] which has subsequently been used in many applications to biological systems.[3]
  • DRF90 developed by P. Th. van Duijnen and coworkers.[49]
  • 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.[50][51]
  • Polarizable Force Field (PFF) – developed by Richard A. Friesner and coworkers.
  • SP-basis Chemical Potential Equalization (CPE) – approach developed by R. Chelli and P. Procacci.[52]
  • PHAST - polarizable potential developed by Chris Cioce and coworkers.[53]
  • CHARMM – polarizable force field developed by S. Patel (University of Delaware) and C. L. Brooks III (University of Michigan).[54][55]
  • 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).[56][57]
  • Sum of Interactions Between Fragments Ab initio computed (SIBFA) – force field[58][59] 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.
  • Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) – force field developed by Pengyu Ren (University of Texas at Austin) and Jay W. Ponder (Washington University).[60]
  • ORIENT – procedure developed by Anthony J. Stone (Cambridge University) and coworkers.[61]
  • Non-Empirical Molecular Orbital (NEMO) – procedure developed by Gunnar Karlström and coworkers at Lund University (Sweden)[62]
  • 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 at Paris VI University.[59][63][64]
  • Polarizable procedure based on the Kim-Gordon approach developed by Jürg Hutter and coworkers (University of Zürich)
  • Computer Simulation of Molecular Structure (COSMOS-NMR) – 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.[65] Atomic charge fluctuation is possible in each molecular dynamics step.
  • Atomistic Polarizable Potential for Liquids, Electrolytes, and Polymers(APPLE&P), developed by Oleg Borogin, Dmitry Bedrov and coworkers, which is distributed by Wasatch Molecular Incorporated.

Reactive force fields

  • 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.[66] Parallelized ReaxFF allows reactive simulations on >>1,000,000 atoms.
  • Empirical valence bond (EVB) – this reactive force field, introduced by Warshel and coworkers, is probably the most reliable and physically consistent way to use force fields in modeling chemical reactions in different environments. The EVB facilitates calculating 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 simuling bond formation-breaking of water and acids.[67]

Coarse-grained force fields

  • Virtual atom molecular mechanics (VAMM) – 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.[68]
  • MARTINI – a coarse-grained potential developed by Marrink and coworkers at the University of Groningen, initially developed for molecular dynamics simulations of lipids,[69] 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.
  • SIRAH – a coarse-grained force field developed by Pantano and coworkers of the Biomolecular Simulations Group, Institut Pasteur of Montevideo, Uruguay; developed for molecular dynamics of water, DNA and proteins. Free available for AMBER and GROMACS packages.

Water models

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 simple point charge water model (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 uses the ff03 charge model and has several side-chain torsion corrections parameterized to match the quantum chemical rotational surface.[70]
  • 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.[71] The charges have been reported to be correlated with hydration free energies of corresponding side-chain analogs.[72]

Other

Servers for force fields generation

Parameter assignment (mostly rule-based)

Parameter optimization

See also

References

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  2. ↑ Ponder JW and Case DA. (2003). Interatomic potentials and their relative parameters for protein simulations. Adv. Prot. Chem. 66 27-85.
  3. 1 2 Warshel A, Sharma PK, Kato M and Parson WW (2006). "Modeling Electrostatic Effects in Proteins." Biochim. Biophys. Acta 1764 1647-1676.
  4. ↑ Schutz CN. and Warshel A. (2001). "What are the dielectric "constants" of proteins and how to validate electrostatic models?". Proteins 44 400-417.
  5. 1 2 3 4 Israelachvili, J.N. (1992). Intermolecular and surface forces. Academic Press, San Diego.
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  57. ↑ Yu, H., Whitfield, T.W., Harder, E., Lamoureux, G., Vorobyov, I., Anisimov, V. M., MacKerell, A.D., Jr., and Roux, B. (2010) "Simulating Monovalent and Divalent Ions in Aqueous Solution Using a Drude Polarizable Force Field". J. Chem. Theory Comput. 6 774–786.
  58. ↑ Sum of Interactions Between Fragments Ab initio computed (SIBFA)
  59. 1 2 Gresh, N., Cisneros, G. A., Darden, T. A. and Piquemal, J-P. (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.
  60. ↑ Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA)
  61. ↑ ORIENT procedure
  62. ↑ O. Engkvist, P.-O. Åstrand, and G. Karlström. "Accurate intermolecular potentials obtained from molecular wave functions: Bridging the gap between quantum chemistry and molecular simulations". Chem. Rev., 100:4087-4108, 2000.
  63. ↑ Piquemal, J-P, Cisneros, G.A., Reinhardt, P., Gresh, N. and Darden, T.A. (2006). "Towards a Force Field based on Density Fitting". J. Chem. Phys. 124 104101.
  64. ↑ Cisneros, G.A., Piquemal, J-P and Darden, T.A. (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.
  65. ↑ Sternberg, Ulrich, Koch, Frank-Thomas and Möllhoff, Margit (1994). "New Approach to the Semiempirical Calculation of Atomic Charges for Polypeptides and Large Molecular Systems" J. Comp. Chem. 15 524-531.
  66. ↑ van Duin, Adri C. T.; Dasgupta, Siddharth; Lorant, Francois; Goddard, William A. (2001). "ReaxFF: A Reactive Force Field for Hydrocarbons" (PDF). The Journal of Physical Chemistry A. 105 (41): 9396–9409. Bibcode:2001JPCA..105.9396V. doi:10.1021/jp004368u.
  67. ↑ Hofmann, Detlef W. M.; Kuleshova, Liudmila; d'Aguanno, Bruno (November 2007). "A new reactive potential for the molecular dynamics simulation of liquid water". Chemical Physics Letters. 448 (1–3): 138–143. Bibcode:2007CPL...448..138H. doi:10.1016/j.cplett.2007.09.063.
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  69. ↑ S.J. Marrink, H.J. Risselada, S. Yefimov,D.P. Tieleman,A.H. de Vries (2007). "The MARTINI force field: coarse grained model for biomolecular simulations." J. Phys. Chem. B. 111 7812-7824.
  70. ↑ Khoury, G. A.; Thompson, J. P.; Smadbeck, J.; Kieslich, C. A.; Floudas, C. A. (2013). "Forcefield_PTM: Ab Initio Charge and AMBER Forcefield Parameters for Frequently Occurring Post-Translational Modifications". Journal of Chemical Theory and Computation: 131122135653005. doi:10.1021/ct400556v. PMC 3904396.
  71. ↑ Khoury, G. A.; Smadbeck, J.; Tamamis, P.; Vandris, A. C.; Kieslich, C. A.; Floudas, C. A. (2014). "Forcefield_NCAA: Ab Initio Charge Parameters to Aid in the Discovery and Design of Therapeutic Proteins and Peptides with Unnatural Amino Acids and Their Application to Complement Inhibitors of the Compstatin Family". ACS Synthetic Biology: 140106144011006. doi:10.1021/sb400168u.
  72. ↑ Khoury, G. A.; Bhatia, N.; Floudas, C. A. (2014). "Hydration free energies calculated using the AMBER ff03 charge model for natural and unnatural amino acids and multiple water models". Computers & Chemical Engineering. doi:10.1016/j.compchemeng.2014.07.017.

Further reading

  • Israelachvili, J. N. (1992). Intermolecular and surface forces. San Diego: Academic Press. ISBN 0-12-375181-0.
  • Schlick, T. (2002). Molecular Modeling and Simulation: An Interdisciplinary Guide. Interdisciplinary Applied Mathematics: Mathematical Biology. New York: Springer-Verlag. ISBN 0-387-95404-X.
  • Warshel, A. (1991). Computer Modeling of Chemical Reactions in Enzymes and Solutions. New York: John Wiley & Sons. ISBN 0-471-53395-5.
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