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The term molecular mechanics refers to the use of Newtonian mechanics to model molecular systems. Molecular mechanics approaches are widely applied in molecular structure refinement, molecular dynamics simulations, Monte Carlo simulations and ligand docking simulations. Molecular mechanics can be used to study small molecules as well as 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 a single hard spherical particle
- Each such particle is assigned a radius (typically the van der Waals radius) and a constant net charge (generally derived from high-level quantum calculations and/or experiment)
- Bonded interactions are treated as "springs" with an equilibrium distance equal to the experimental or calculated bond length
Variations on this theme are possible; for example, many simulations have historically used a "united-atom" representation in which methyl and methylene groups were represented as a single particle, and large protein systems are commonly simulated using a "bead" model that assigns two to four particles per amino acid.
Functional Form
The following functional abstraction, known as a potential function, calculates the system's potential energy in a given conformation as a sum of individual energy terms.
The exact functional form of the potential function depends on the particular simulation program being used. Generally the bond and angle terms are modeled as harmonic potentials centered around equilibrium values derived from experiment or theoretical calculations of electronic structure performed with software like Gaussian. For accurate reproduction of vibrational spectra, the Morse potential can 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 include "improper" dihedral terms, which function as correction factors for out-of-plane deviations (for example, they can be used to keep benzene rings planar).
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 simulation. Fortunately the van der Waals term falls off rapidly - it is typically modeled using a "6-12 Lennard-Jones potential", which means that attractive forces fall off with distance as r6 and repulsive forces as r12, where r represents the distance between two atoms. Generally a cutoff radius is used so that atom pairs whose distances is 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 for proteins). The basic functional form is the Coulomb potential, which only falls off as r2. 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 are known as particle mesh Ewald (PME) and the multipole algorithm.
In addition to the functional forms of each energy term, a useful energy function must also include 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 known as a force field. Parameterization is typically done through consultation with experimental values and theoretical calculation results. Each force field is parameterized to be internally consistent, but the parameters are generally not transferable from one force field to another.
Integration and Propagation
Required initialization information includes spatial coordinates and starting velocities for each atom or particle in the system. Since velocities are generally not known for an initial conformation, they are assigned at random from a Boltzmann distribution at a user-specified temperature. The user must also specify a timestep - that is, the assumed amount of time between frames of the trajectory that is produced when Newton's equations of motion are integrated. This timestep, which is usually 1-2 femtoseconds, is used as input to the method used for numerical integration. Various types of Verlet integrators are most commonly used in molecular mechanics due to their improved stability over the more natural choice, the Euler integration methods. Especially at large timesteps, a poor integration method can introduce significant errors, resulting in rapid divergence of two trajectories started from virtually identical initial conditions, accompanied by a failure of energy conservation.
Environment and Solvation
There are several ways of defining the environment surrounding the molecule or molecules of interest in molecular mechanics. A system can be simulated in vacuum (known as a gas-phase simulation) with no surrounding environment at all, but this is usually not desirable 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 "best" 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 molecule. A variety of water models exist with increasing levels of complexity, representing water as a simple hard sphere (a united-atom approach), as three separate particles with fixed bond angles, or even as four or five separate interaction centers to account for unpaired electrons on the oxygen atom. Unsurprisingly, the more complex the water model, the more computationally intensive the simulation. A compromise approach has been found in "implicit 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 for preventing artifacts that arise from vacuum simulations and reproduces bulk solvent properties well, but cannot reproduce situations in which individual water molecules have interesting interactions with the molecules under study.
Software Packages
Limited list; many more are available
References
- U. Burkert and N.L. Allinger, Molecular Mechanics, 1982, ISBN 0841208859
- Vernon G. S. Box, The Molecular Mechanics of Quantized Valence Bonds, J. Mol. Model., 3, 124, 1997
- Vernon G. S. Box, The anomeric effect of monosaccharides and their derivatives. Insights from the new QVBMM molecular mechanics force field, Heterocycles, 48, 2389 1998
- Vernon G. S. Box, Stereo-electronic effects in polynucleotides and their double helices, J. Mol. Struct., 689, 33-41 2004
- O. Becker, A.D. MacKerell, Jr., B. Roux and M. Watanabe, Editors, Computational Biochemistry and Biophysics, Marcel Dekker Inc., New York, 2001, ISBN 082470455X
- MacKerell, A.D., Jr., Empirical Force Fields for Biological Macromolecules: Overview and Issues, Journal of Computational Chemistry, 25: 1584-1604, 2004
- Schlick, T. Molecular Modeling and Simulation: An Interdisciplinary Guide. Springer-Verlag, New York, NY: 2002. ISBN 038795404X.
Molecular physics | Computational chemistry
Mécanique moléculaire | Mekanika molekul | moleculaire mechanica
"Molecular mechanics"