Molecular mechanics

Aforce fieldis used to minimize the bond stretching energy of this ethane molecule.

Molecular mechanicsusesclassical mechanicsto modelmolecularsystems. TheBorn–Oppenheimerapproximation is assumed valid and the potential energy of all systems is calculated as a function of the nuclear coordinates usingforce fields.Molecular mechanics can be used to study molecule systems ranging in size and complexity from small to large biological systems or material assemblies with many thousands to millions of atoms.

All-atomistic molecular mechanics methods have the following properties:

• Each atom is simulated as one particle
• Each particle is assigned a radius (typically thevan der Waals radius), polarizability, and a constant net charge (generally derived from quantum calculations and/or experiment)
• Bonded interactions are treated asspringswith an equilibrium distance equal to the experimental or calculated bond length

Variants on this theme are possible. For example, many simulations have historically used aunited-atomrepresentation in which each terminalmethyl groupor intermediatemethylene unitwas considered one particle, and large protein systems are commonly simulated using abeadmodel that assigns two to four particles peramino acid.

Functional form

Molecular mechanics potential energy function with continuum solvent.

The following functional abstraction, termed aninteratomic potentialfunction orforce fieldin chemistry, calculates the molecular system's potential energy (E) in a given conformation as a sum of individual energy terms.

$\ E=E_{\text{covalent}}+E_{\text{noncovalent}}\,$

where the components of the covalent and noncovalent contributions are given by the following summations:

$\ E_{\text{covalent}}=E_{\text{bond}}+E_{\text{angle}}+E_{\text{dihedral}}$

$\ E_{\text{noncovalent}}=E_{\text{electrostatic}}+E_{\text{van der Waals}}$

Theexact functional form of the potential function,or force field, depends on the particular simulation program being used. Generally the bond and angle terms are modeled asharmonic potentialscentered around equilibrium bond-length values derived from experiment or theoretical calculations of electronic structure performed with software which doesab-initiotype calculations such asGaussian.For accurate reproduction of vibrational spectra, theMorse potentialcan be used instead, at computational cost. The dihedral or torsional terms typically have multiple minima and thus cannot be modeled as harmonic oscillators, though their specific functional form varies with the implementation. This class of terms may includeimproperdihedral terms, which function as correction factors for out-of-plane deviations (for example, they can be used to keepbenzenerings planar, or correct geometry and chirality of tetrahedral atoms in a united-atom representation).

The non-bonded terms are much more computationally costly to calculate in full, since a typical atom is bonded to only a few of its neighbors, but interacts with every other atom in the molecule. Fortunately thevan der Waalsterm falls off rapidly. It is typically modeled using a6–12Lennard-Jones potential,which means that attractive forces fall off with distance asr⁻⁶and repulsive forces asr⁻¹²,where r represents the distance between two atoms. The repulsive partr⁻¹²is however unphysical, because repulsion increases exponentially. Description of van der Waals forces by the Lennard-Jones 6–12 potential introduces inaccuracies, which become significant at short distances.Generally a cutoff radius is used to speed up the calculation so that atom pairs which distances are greater than the cutoff have a van der Waals interaction energy of zero.

The electrostatic terms are notoriously difficult to calculate well because they do not fall off rapidly with distance, and long-range electrostatic interactions are often important features of the system under study (especially forproteins). The basic functional form is theCoulomb potential,which only falls off asr⁻¹.A variety of methods are used to address this problem, the simplest being a cutoff radius similar to that used for the van der Waals terms. However, this introduces a sharp discontinuity between atoms inside and atoms outside the radius. Switching or scaling functions that modulate the apparent electrostatic energy are somewhat more accurate methods that multiply the calculated energy by a smoothly varying scaling factor from 0 to 1 at the outer and inner cutoff radii. Other more sophisticated but computationally intensive methods areparticle mesh Ewald(PME) and themultipole algorithm.

In addition to the functional form of each energy term, a useful energy function must be assigned parameters for force constants, van der Waals multipliers, and other constant terms. These terms, together with the equilibrium bond, angle, and dihedral values, partial charge values, atomic masses and radii, and energy function definitions, are collectively termed aforce field.Parameterization is typically done through agreement with experimental values and theoretical calculations results.Norman L. Allinger's force field in the last MM4 version calculate for hydrocarbons heats of formation with aRMS errorof 0.35 kcal/mol, vibrational spectra with a RMS error of 24 cm⁻¹,rotational barriers with a RMS error of 2.2°,C−Cbond lengths within 0.004 Å andC−C−Cangles within 1°.Later MM4 versions cover also compounds with heteroatoms such as aliphatic amines.

Each force field is parameterized to be internally consistent, but the parameters are generally not transferable from one force field to another.

Areas of application

The main use of molecular mechanics is in the field ofmolecular dynamics.This uses theforce fieldto calculate the forces acting on each particle and a suitable integrator to model the dynamics of the particles and predict trajectories. Given enough sampling and subject to theergodic hypothesis,molecular dynamics trajectories can be used to estimate thermodynamic parameters of a system or probe kinetic properties, such as reaction rates and mechanisms.

Molecular mechanics is also used withinQM/MM,which allows study of proteins and enzyme kinetics. The system is divided into two regions—one of which is treated withquantum mechanics(QM) allowing breaking and formation of bonds and the rest of the protein is modeled using molecular mechanics (MM). MM alone does not allow the study of mechanisms of enzymes, which QM allows. QM also produces more exact energy calculation of the system although it is much more computationally expensive.

Another application of molecular mechanics is energy minimization, whereby theforce fieldis used as anoptimizationcriterion. This method uses an appropriate algorithm (e.g.steepest descent) to find the molecular structure of a local energy minimum. These minima correspond to stable conformers of the molecule (in the chosen force field) and molecular motion can be modelled as vibrations around and interconversions between these stable conformers. It is thus common to find local energy minimization methods combined with global energy optimization, to find the global energy minimum (and other low energy states). At finite temperature, the molecule spends most of its time in these low-lying states, which thus dominate the molecular properties. Global optimization can be accomplished usingsimulated annealing,theMetropolis algorithmand otherMonte Carlo methods,or using different deterministic methods of discrete or continuous optimization. While the force field represents only theenthalpiccomponent offree energy(and only this component is included during energy minimization), it is possible to include theentropiccomponent through the use of additional methods, such asnormal modeanalysis.

Molecular mechanics potential energy functions have been used to calculate binding constants,protein folding kinetics,protonation equilibria,active site coordinates,and todesign binding sites.

Environment and solvation

In molecular mechanics, several ways exist to define the environment surrounding a molecule or molecules of interest. A system can be simulated in vacuum (termed a gas-phase simulation) with no surrounding environment, but this is usually undesirable because it introduces artifacts in the molecular geometry, especially in charged molecules. Surface charges that would ordinarily interact with solvent molecules instead interact with each other, producing molecular conformations that are unlikely to be present in any other environment. The most accurate way to solvate a system is to place explicit water molecules in the simulation box with the molecules of interest and treat the water molecules as interacting particles like those in the other molecule(s). A variety ofwater modelsexist with increasing levels of complexity, representing water as a simple hard sphere (a united-atom model), as three separate particles with fixed bond angle, or even as four or five separate interaction centers to account for unpaired electrons on the oxygen atom. As water models grow more complex, related simulations grow more computationally intensive. A compromise method has been found inimplicit solvation,which replaces the explicitly represented water molecules with a mathematical expression that reproduces the average behavior of water molecules (or other solvents such as lipids). This method is useful to prevent artifacts that arise from vacuum simulations and reproduces bulk solvent properties well, but cannot reproduce situations in which individual water molecules create specific interactions with a solute that are not well captured by the solvent model, such as water molecules that are part of the hydrogen bond network within a protein.

Software packages

Main article:Comparison of software for molecular mechanics modeling

This is a limited list; many more packages are available.

• Abalone
• ACEMD - GPU MD
• AMBER
• Ascalaph Designer
• BOSS
• CHARMM
• COSMOS
• CP2K
• Ghemical
• GROMACS
• GROMOS
• Internal Coordinate Mechanics(ICM)
• LAMMPS
• MacroModel
• MDynaMix
• Molecular Operating Environment(MOE)
• NAMD
• Q
• Q-Chem
• Spartan
• StruMM3D (STR3DI32)
• Tinker
• X-PLOR
• Yasara
• Zodiac

See also

• Molecular graphics
• Molecular dynamics
• Molecule editor
• Force field (chemistry)
• Comparison of force field implementations
• Molecular design software
• Molecular modeling on GPU
• Comparison of software for molecular mechanics modeling
• List of software for Monte Carlo molecular modeling

Literature

• Allinger NL, Burkert U (1982).Molecular Mechanics.American Chemical Society.ISBN978-0-8412-0885-8.OCLC987878077.
• Box VG (March 1997). "The Molecular Mechanics of Quantized Valence Bonds".J Mol Model.3(3): 124–41.doi:10.1007/s008940050026.S2CID93821090.
• Box VG (12 November 1998)."The anomeric effect of monosaccharides and their derivatives. Insights from the new QVBMM molecular mechanics force field".Heterocycles.48(11): 2389–417.doi:10.3987/REV-98-504.
• Box VG (2004). "Stereo-electronic effects in polynucleotides and their double helices".J. Mol. Struct.689(1–2): 33–41.Bibcode:2004JMoSt.689...33B.doi:10.1016/j.molstruc.2003.10.019.
• Becker OM (2001).Computational biochemistry and biophysics.Marcel Dekker.ISBN978-0-8247-0455-1.OCLC54078278.
• Mackerell AD (October 2004). "Empirical force fields for biological macromolecules: overview and issues".J Comput Chem.25(13): 1584–1604.doi:10.1002/jcc.20082.PMID15264253.S2CID9162620.
• Schlick T (2010).Molecular modeling and simulation: an interdisciplinary guide(2nd ed.). Springer.doi:10.1007/978-1-4419-6351-2.ISBN978-0-387-95404-2.
• Namboori, Krishnan; Ramachandran, K.S.; Gopakumar, Deepa (2008).Computational Chemistry and Molecular Modeling: Principles and Applications.Springer.doi:10.1007/978-3-540-77304-7.ISBN978-3-540-77302-3.

External links

• Molecular dynamics simulation methods revised
• Molecular mechanics - it is simple