CLASSICAL FORCE FIELDS, EXPLICIT SOLVENT MODELS AND CONTINUUM SOLVENT MODELS
Empirical force field-based methods- application of theoretical approaches to investigate structure–activity relationships in biological systems, used for investigating the structure–activity relationships of biological macromolecules at an atomic level of detail.
Force field-The combination of the potential energy function with the parameters used in that function, quality of force fields and their proper implementation is considered the most important determinant of the accuracy of empirical methods.
Potential Energy Function- The core of any force field used to describe the relationship of the structure to the energy of the system of interest.
Parameters- The terms used in the potential energy function that represent the actual force field.
Molecular dynamics (MD) simulations- A type of empirical force field-based method used to investigate structure–activity relationships in biological systems.
Target Data- The data used as the basis for parameter optimization during force field development, often comprised of Quantum Mechanical (QM) results for model compounds representative of the biomolecules of interest.
Empirical or knowledge-based optimization approach- optimization approach where parameters are adjusted to systematically "soften" the energy surface, often leading to poorer reproduction of QM energy surfaces but yielding improved agreement with survey data (e.g., from the Nucleic Acid Database, NDB)
Parameter correlation problem- phenomenon where the LJ parameters and partial atomic charges are highly correlated, and internal parameters are dependent on nonbond parameters, making it difficult to accurately determine the "correct" parameters based on reproduction of condensed phase properties alone.
Knowledge-based or free energy force fields- Force fields that are parameterized to directly yield free energies, in contrast to force fields defined by a potential energy function that yield potential energies from which thermodynamic quantities are obtained via statistical mechanics.
Potential Energy Functions and Interactions
Class I additive potential energy function- The functional form commonly used in biomolecular force fields, comprising a collection of simple functions where **bonds, angles, and improper
Harmonic terms- The simple functions used in Class I force fields to treat bonds, angles, and out-of-plane distortions (improper dihedral angles).
Lennard–Jones (LJ) 6–12 term- The term used to describe the atom–atom repulsion and dispersion interactions between atoms.
Coulombic term- The term used to treat electrostatics via partial atomic charges divided by the distance
Internal or intramolecular parameters- Parameters that describe the covalent structure, including the bond force constant and equilibrium distance the valence angle force constant and equilibrium angle, the dihedral force constant, multiplicity and phase angle , and the improper force constant and equilibrium improper angle
Nonbonded parameters (Interaction or external parameters) - Parameters used to treat interactions between atoms i and j, including the partial atomic charges , and the LJ well-depth and minimum interaction radius used for van der Waals (vdW) interactions.
1,4 Nonbonded interactions- Interactions involving atoms separated by three covalent bonds, which have nonbonded contributions and influence conformational energies. Different biomolecular force fields treat these differently, often using scale factors (e.g., 1.0, 0.5, or 0.83).
Combining rules – Used to take the Lennard-Jones (LJ) parameters for individual atoms and combine them to yield the atom i–atom j LJ interactions (epsilon and Rmin).
Lorentz–Berthelodt rules – A specific set of combining rules where the combined epsilon values are obtained via the geometric mean and Rmin via the arithmetic mean.
Electrostatics and Polarization
Additive force field – A model that uses static, partial atomic charges (via the Coulombic term) and LJ terms; this model generally does not explicitly treat electronic polarizability. Instead, polarizability is included implicitly by choosing partial atomic charges that overestimate molecular dipoles.
Induced dipole models – A method to explicitly treat polarizability, generally applying an isotropic dipole moment to each atom.
Fluctuating charge models (or electronegativity equalization model) – A method to explicitly treat polarizability where partial atomic charges are allowed to redistribute to yield equivalent electronegativity on each atom, changing the overall molecular dipole moment.
Classical Drude oscillator (or Shell model) – A method to explicitly treat polarizability where an additional “Drude” particle is attached to the nucleus of each atom, and polarization is attained by allowing the positions of the Drude particles to relax in the external electrostatic field.
Extended Lagrangian methods – Methods used in molecular dynamics simulations that treat the polarizability as a dynamic variable.
QM electrostatic potential (ESP) – A method for charge determination based on the optimization of charges to reproduce a quantum mechanically determined electrostatic potential mapped onto a grid surrounding a model compound.
Restrained ESP (RESP) fitting – An adaptation of ESP fitting where restraints are used during the fitting process to overcome the issue of ambiguity and underdetermination of charges on buried atoms.
Supramolecular approach – A charge determination method where charges are optimized to reproduce quantum mechanically determined interaction energies and geometries of the model compound, typically with individual water molecules or dimers. This approach implicitly includes local electronic polarization.
Solvation and Boundaries
Solvation models – Models used to accurately treat the condensed aqueous environment, categorized as either explicit or implicit.
Explicit water models – A microscopically complete method for treating solvation, where the solvent (water) is represented by specific models like TIP3P, TIP4P, SPC, extended SPC/E, and F3C.
Implicit solvation models – Approaches that offer significant computational savings while providing an accurate treatment of solvation; they are typically used when extensive sampling of conformational space is required.
Poisson–Boltzmann (PB) model – An implicit solvation model where contributions from solvent polarization along with the asymmetric shapes of biological molecules are taken into account.
Generalized-Born (GB)-based solvation models – Effective implicit solvation models that have significantly enhanced computational speeds.
Ewald summation method – A method that treats long-range electrostatic interactions by taking advantage of crystal symmetry combined with reciprocal space.
Particle mesh Ewald (PME) method – A specific implementation of the Ewald summation method that has been widely implemented in simulation packages.
Periodic boundary conditions – Considered the most rigorous approach for setting up a simulation system, typically of cubic or orthorhombic symmetries, allowing Ewald methods to be used.
Stochastic boundary approach – Alternatives used when the macromolecular size disallows a periodic system; a spherical system is created that is surrounded by a potential (including a reaction field) that maintains the density of the system.
Atom Types
All-atom force field – A force field that explicitly treats all the atoms in the molecule.
United atom protein backbone in MD simulations – Refers to models where nonpolar hydrogens are not explicitly represented, simplifying the backbone representation.
Ramachandran map – The map defined by the phi and psi dihedral angles.
Alanine dipeptide (Ace-Ala-Nme) – The quintessential model compound used for optimization of protein backbone parameters.
2D dihedral energy correction map (CMAP) approach – An extension of the potential energy function allowing for any two-dimensional dihedral surface (for example, a quantum mechanical phi/psi surface) to be reproduced nearly exactly by the force field.
Glycosyl bonds – The variety of chemical connectivities found in polysaccharides (for example, 1,1; 1,2; 1,3; 1,4; 1,6; 2,2 types).
Hyperconjugation – The delocalization of oxygen lone pairs into antibonding orbitals in carbohydrates, which influences their conformational energies.
Anomeric effect – A delocalization effect associated with hyperconjugation that influences the conformational energies of carbohydrates.
Exoanomeric effect – A delocalization effect associated with hyperconjugation that influences the conformational energies of carbohydrates.
—------
Next Paper:
Force Field Concepts and Terminology
Force Field – Expressions for the total energy, typically used in a classical framework for computer modeling methods like Molecular Dynamics (MD) and Monte Carlo (MC). Most widespread force fields include harmonic bond stretching and angle bending, Fourier series for torsional energetics, and Coulomb plus Lennard-Jones terms for nonbonded interactions.
OPLS All-Atom (OPLS-AA) Force Field – The specific force field described and tested in the paper, parameterized for organic molecules and peptides. It uses parameters for torsional and nonbonded energetics derived from ab initio calculations and liquid simulations, while bond stretching and angle bending parameters are mostly adopted from the AMBER all-atom force field.
OPLS Partially United-Atom (OPLS-UA) Model – The original OPLS potential functions. In this model, sites for nonbonded interactions are placed on all non-hydrogen atoms and on hydrogens attached to heteroatoms or carbons in aromatic rings, meaning hydrogens attached to aliphatic carbons are implicit. This model is computationally attractive because it uses fewer interaction sites.
All-Atom (AA) Representation/Model – A force field representation where sites for nonbonded interactions are placed on all atoms. This model allows for more flexibility in charge distributions and torsional energetics compared to united-atom models.
Nonbonded Interactions – Interactions between molecules represented by the sum of Coulomb and Lennard-Jones terms. The same expression is used for intramolecular nonbonded interactions between all pairs of atoms separated by three or more bonds.
Lennard-Jones Terms – Components of the nonbonded interaction expression used alongside Coulomb terms. Standard combining rules are used such that the distance parameter is averaged and the energy parameter is based on the geometric mean.
Intramolecular 1,4-interactions – Nonbonded interactions between atom pairs separated by three or more bonds. For OPLS-AA, scaling factors of one-half for both the Coulombic and Lennard-Jones interactions were used for these specific interactions.
Bond Stretching Energy – The energy related to the stretching of bonds, represented by a harmonic term.
Angle Bending Energy – The energy related to the bending of angles, represented by a harmonic term.
Torsional Energy – The energy associated with rotation about a bond, which is expressed as a Fourier series for each dihedral angle. Torsional parameters are determined by fitting to rotational energy profiles from ab initio calculations.
Potential Energy Change for Rotating about a Bond – The total energy change during bond rotation, broken down into components: torsional energy, bond stretching, angle bending, and nonbonded interactions.
Empirical Charges – The charges used for the OPLS force fields, obtained largely from fitting to reproduce properties of organic liquids rather than directly from ab initio calculations.
Computational Methods and Properties
Molecular Dynamics (MD) – A principal computational method used in a classical framework for modeling fluid systems.
Monte Carlo Statistical Mechanics (MC) – A principal computational method used in a classical framework for modeling fluid systems. Extensively used in the development and testing of OPLS parameters by simulating thermodynamic and structural properties of pure organic liquids.
Heats of Vaporization – A key thermodynamic property used to test the force field. It is calculated as the difference between enthalpy in the gas phase and enthalpy in the liquid phase.
Density – A thermodynamic property of pure organic liquids used for testing and validating the force field.
Free Energies of Hydration – A property tested to give confidence in the description of nonbonded interactions, including hydrogen bonding, and in the size of molecules.
NPT Ensemble – The isothermal, isobaric ensemble in which all Monte Carlo liquid simulations were carried out (at a pressure of 1 atm).
Eintra(g) – The total potential energy of a single molecule in the gas phase, obtained from gas-phase Monte Carlo simulations.
Eintra(l) – The average internal energy of a molecule determined in the liquid simulations.
Einter(l) – The intermolecular energy in the liquid.
Cp(inter) – The intermolecular component of the liquid’s heat capacity at constant pressure, calculated from fluctuations in the total intermolecular energy.
Isothermal Compressibility – A property calculated from fluctuations in the volume.
Radial Distribution Functions (rdfs) – Functions used to visualize the structure of liquids, such as the O–H radial distribution functions for alcohols reflecting hydrogen bonding.
—---------------
Analytical Potential Energy Functions (or Force Fields)
Analytical Potential Energy Functions (or Force Fields) – Functions used within classical mechanics in computer-based molecular mechanics models to study biochemical and organic molecules. These functions use energy minimization, molecular dynamics, and Monte Carlo methods to explore potential energy surfaces.
Force Field (Molecular Mechanical Force Field) – A potential function used for simulating structures, conformational energies, and interaction energies of proteins, nucleic acids, and organic molecules in condensed phases. The force field presented is an effective two-body force field.
Effective Two-Body Force Field – A force field that simulates the structures, conformational energies, and interaction energies of proteins, nucleic acids, and organic molecules in condensed phases.
Molecular Mechanics – The application of computer-based models using analytical potential energy functions within classical mechanics to study biochemical and organic molecules.
Diagonal Potential Function – A feature of the Weiner et al. force field, representing bond and angle energies with a simple diagonal harmonic expression.
Electrostatic Potential Fit Atom Centered Charges (ESP charges) – Charges determined by fitting to the electrostatic potential using a basis set (such as STO-3G or 6-31G*). Standard ESP charges can vary by conformation and may assign unrealistic values to buried atoms.
Restrained Electrostatic Potential (RESP) Fitting – A method for deriving partial charges by least-squares fitting to the electrostatic potential with restraints to reduce excessive charges on buried or nonpolar atoms. Tailored for condensed phase simulations.
6-31G Basis Set –* A basis set suggested for deriving ESP-fit charges for condensed phases because it uniformly overestimates molecular polarity.
STO-3G Basis Set – A basis set used by Weiner et al. for computational efficiency. It produces dipole moments similar to or smaller than the gas-phase values but tends to underestimate quadrupole moments.
Van der Waals (VDW) Parameters – Parameters optimized for reproducing liquid properties. They include hydrogen parameters to account for nearby electronegative atoms and are represented by a 6-12 potential.
10-12 Hydrogen Bonding Parameters – Parameters used in the Weiner et al. force field for hydrogen bonds; made unnecessary in newer force fields with improved charge and VDW models.
Bonded Parameters – Parameters represented by a diagonal harmonic expression, modified as needed to reproduce vibrational frequencies and molecular structures.
Dihedral Parameters (Torsional Parameters) – Parameters representing dihedral energies, often through Fourier expansions. Complex phi and psi parameters were developed for peptide backbones to reproduce low-energy conformations.
1-4 Interactions – Nonbonded interactions separated by exactly three bonds, reduced by a scaling factor.
Coulombic Interaction – Electrostatic interactions modeled with atom-centered point charges.
Simple Diagonal Force Fields – A category of force fields (Weiner et al., GROMOS, CHARMM, OPLS/AMBER) using harmonic diagonal representations for bond and angle terms.
MM2 and MM3 – Force fields developed by Allinger. MM3 is the advanced version for gas-phase organic molecules, using more complex functional forms, higher-order and cross-terms, and a 6-exponential nonbonded potential.
OPLS (Optimized Potentials for Liquid Simulations) – A model developed by Jorgensen focusing on reproducing liquid enthalpies and densities. Its nonbonded parameters were combined with Weiner et al. parameters to create OPLS/Amber.
OPLS/Amber Force Field – A diagonal force field combining OPLS nonbonded parameters with Weiner et al. bond, angle, and dihedral parameters.
CHARMm – A diagonal force field. Early versions derived electrostatics from quantum mechanical dimers and VDW parameters from crystal data.
GROMOS – A diagonal force field using empirical data for electrostatics and VDW parameters from crystal data.
ECEPP Force Field – A force field developed by Scheraga with rigid internal geometries, improving efficiency but sometimes overestimating barriers.
SYBYL Force Field – A force field for calculating internal geometries and conformational energies. Lacks an electrostatic term, making it unsuitable for condensed-phase studies.
YETI Force Field – A modification of the Weiner et al. force field, featuring damped electrostatics and an added angular-dependent hydrogen bond/metal ligation potential.
Class II Force Field – A more complex force field under development by Hagler, similar in complexity to MM2/MM3, extensively calibrated with quantum mechanical data.
Merck Molecular Force Field (MMFF) – A complex force field by Halgren designed for pharmaceutical applications. It uses a buffered 7-14 potential for nonbonded terms and an empirical bond dipole model for charges.
Gauche Effect – A physical effect where electronegative substituents stabilize the gauche conformation over trans (e.g., in X–C–C–X bonds).
Fourier Expansion – A method to represent dihedral energy using periodic components (two-fold, three-fold, etc.).
Free Energies of Solvation – A condensed-phase intermolecular property that the force field is designed to reproduce accurately.
Energy Minimization – A molecular mechanics method for finding the lowest energy structure.
Molecular Dynamics (MD) – A method used in molecular mechanics for simulating molecular motion.
Monte Carlo Methods (MC) – A method in molecular mechanics using random sampling to explore conformational space.
Particle Mesh Ewald Method – A method in MD simulations for treating long-range electrostatics in periodic systems.
TIP3P Water Model – A widely used water model for condensed-phase simulations.