Molecular dynamics

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Molecular dynamics (MD) simulation is a special discipline of molecular modelling. It addresses numerical solutions of Newton's equations of motion on an atomistic or similar model of a molecular system to obtain information about its time-dependent properties.

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Applications

Beginning in theoretical physics, the method of MD gained popularity in material science and since the 1970s also in biochemistry and biophysics. In chemistry, MD serves as an important tool in protein structure determination and refinement (see also crystallography, NMR). In physics, MD is used to examine the dynamics of atomic-level phenomena that cannot be observed directly, such as thin film growth and ion-subplantation. It is also used to examine the physical properties of nanotechnology devices that have not or cannot yet be created.

Note that there is a large difference between the focus and methods used by chemists and physicists, and this is reflected in differences in the jargon used by the different fields.

In Chemistry, the interaction between the objects is either described by a force field (chemistry) (classical MD), a quantum chemical model, or a mix between the two. These terms are not used in Physics, where the interactions are usually described by the name of the theory or approximation being used.

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Design Constraints

Design of a molecular dynamics simulation can often encounter limits of computational power. Simulation sizes and time duration must be selected so that the calculation can finish within a useful time period.

The simulation's time duration is dependent on the time length of each timestep, between which forces are recalculated. The timestep must be chosen small enough to avoid discretization errors, and the number of timesteps, and thus simulation time, must be chosen large enough to capture the effect being modeled without taking an extraordinary period of time.

The simulation size must be large enough to express the effect without the boundary conditions disrupting the behavior. Boundary constraints are often treated by choosing fixed conditions at the boundary, or by choosing periodic boundary conditions in which one side of the simulation loops back to the opposite side. The scalability of the simulation with respect to the number of molecules is usually a significant factor in the range of simulation sizes which can be simulated in a reasonable time period. In Big O notation, common molecular dynamics simulations usually scale by either <math>O(n \log(n))</math>, or with good use of neighbour tables, <math>O(n)</math>, with <math>n</math> as the number of molecules.

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Physical Principles

All of the information needed to calculate the dynamics of a system can be found from the potential energy function U (referred to simply as the potential in Physics, or the force field in Chemistry) of the system. From Newtown's laws, the force on atom i in the system can then be determined from the equation:

<math>F_i = -\nabla_i U</math>

An integrator (such as the Verlet Integrator) is then used to calculate the trajectories of the atoms from the forces.

One of the difficulties in MD is calculating the temperature of the system, because temperature is a statistical quantity. However, if there were a large enough number of atoms, statistical temperature can be estimated from the instantaneous temperature, which is found by equating the kinetic energy of the system to 3kT/2.

Another temperature-related problem arises due to the small number of atoms that are used in MD simulations. For example, consider simulating the growth of a copper film starting with a substrate containing 500 atoms and a deposition energy of 100 eV. In the real world, the 100 eV from the deposited atom would rapidly be transported through and shared among a large number of atoms (<math>10^{10}</math> or more) with no real change in temperature. When there are only 500 atoms, however, the substrate is almost immediately vaporized by the deposition. To overcome this problem, a variety of thermostat methods are required to remove energy from the boundaries of an MD system in a realistic way.

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Types of MD Systems

Systems studied via MD are typically divided into the following categories:

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Empirical Potentials

Empirical potentials either ignore quantum mechanical effects, or attempt to capture quantum effects in a limited way through entirely empirical equations. Parameters in the potential are fitted against known physical properties of the atoms being simulated, such as elastic constants and lattice parameters.

Empirical potentials can further be subcategorized into pair potentials, and many-body potentials.

For pair potentials, the total potential energy of a system can be calculated from the sum of energy contributions from pairs of atoms. One example of a pair potential is the Lennard-Jones potential (also known as the 6-12 potential).

In many-body potentials, the potential energy cannot be found by a sum over pairs of atoms. For example, the Tersoff Potential, which is used to simulate Silicon and Germanium, involves a sum over groups of three atoms, with the angles between the atoms being an important factor in the potential. Another example is the Tight-Binding Second Moment Approximation (TBSMA), where the electron density of states in the region of an atom is calculated from a sum of contributions from surrounding atoms, and the potential energy contribution is then a function of this sum.

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Semi-Empirical Potentials

Semi-empirical potentials make use of the matrix representation from quantum mechanics. However, the values of the matrix elements are found through empirical formulae that estimate the degree of overlap of specific atomic orbitals. The matrix is then diagonalized to determine the occupancy of the different atomic orbitals, and empirical formulae are used once again to determine the energy contributions of the orbitals.

There are a wide variety of semi-empirical potentials, known as tight-binding potentials, which vary according to the atoms being modeled.

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Ab-initio (First Principles) Methods

Ab-initio (first principles) methods use full quantum-mechanical formula to calculate the potential energy of a system of atoms or molecules. This calculation is usually made "locally", i.e., for nuclei in the close neighborhood of the reaction coordinate. Although various approximations may be used, these are based on theoretical considerations, not on empirical fitting. Ab-Initio produce a large amount of information that is not available from the empirical methods, such as density of states information.

Perhaps the most famous ab-initio package is the Car-Parrinello Molecular Dynamics (CPMD) package based on the density functional theory.

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Coarse-Graining: "Reduced Representations"

MD simulations on very large systems may require such large computer resources that they cannot easily be studied by traditional all-atom methods. Similarly, simulations of processes on long timescales (beyond about 1 microsecond) are prohibitely expensive, because they require so many timesteps. In these cases, one can sometimes tackle the problem by using reduced representations, which are also called coarse-grained models. Instead of explicitly representing every atom of the system, one uses pseudoatoms to represent groups of atoms.

The earliest MD simulations on proteins, lipids and nucleic acids generally used "united atoms", which represents the lowest level of coarse graining. For example, instead of treating all four atoms of a methyl group explicitly (or all three atoms of a methylene group), one represents the whole group with a single pseudoatom. This pseudoatom must, of course, be properly parameterized so that its van der Waals interactions with other groups have the proper distance-dependence. Similar considerations apply to the bonds, angles, and torsions in which the pseudoatom participates. In this kind of united atom representation, one typically eliminates all explicit hydrogen atoms except those that have the capability to participate in hydrogen bonds ("polar hydrogens"). The polar hydrogens are usually retained in the model, because proper treatment of hydrogen bonds requires a reasonably accurate description of the geometry and the electrostatic interactions between the donor and acceptor groups. A hydroxyl group, for example, can be both a hydrogen bond donor and a hydrogen bond acceptor, and it would be impossible to treat this with a single OH pseudoatom. Note that about half the atoms in a protein or nucleic acid are nonpolar hydrogens, so the use of united atoms can provide a substantial savings in computer time.

Similar united atom approximations have been used in MD simulations on biological membranes. In these, the aliphatic tails of lipids are represented by a few pseudoatoms by gathering 2-4 methylene groups into each pseudoatom.

When coarse-graining is done at higher levels, the accuracy of the dynamic description may be less reliable. But very coarse-grained models have been used successfully to examine a wide range of questions in structural biology. Following are just a few examples.

  • protein folding studies are often carried out using a single (or a few) pseudoatoms per amino acid;
  • DNA supercoiling has been investigated using 1-3 pseudoatoms per basepair, and at even lower resolution;
  • Packaging of double-helical DNA into bacteriophage has been investigated with models where one pseudoatom represents one turn (about 10 basepairs) of the double helix;
  • RNA structure in the ribosome and other large systems has been modeled with one pseudoatom per nucleotide.

The parameterization of these very coarse-grained models must be done empirically, by matching the behavior of the model to appropriate experimental data.

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Major software for MD simulations

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Related software

  • VMD - MD simulation trajectories can be loaded and visualized
  • atomeye - MD simulation trajectories can be loaded and visualized
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See also

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References

  • M. P. Allen, D. J. Tildesley (1989) Computer simulation of liquids. Oxford University Press. ISBN 0198556454.
  • J. A. McCammon, S. C. Harvey (1987) Dynamics of Proteins and Nucleic Acids. Cambridge University Press. ISBN 0-52-135652-0 (paperback); ISBN 0-52-130750 (hardback).
  • D. C. Rapaport (1996) The Art of Molecular Dynamics Simulation. ISBN 0521445612.
  • Daan Frenkel, Berend Smit (2001) Understanding Molecular Simulation. Academic Press. ISBN 0122673514.
  • J. M. Haile (2001) Molecular Dynamics Simulation: Elementary Methods. ISBN 047118439X
  • Oren M. Becker, Alexander D. Mackerell Jr, Benoît Roux, Masakatsu Watanabe (2001) Computational Biochemistry and Biophysics. Marcel Dekker. ISBN 082470455X.
  • Tamar Schlick (2002) Molecular Modeling and Simulation. Springer. ISBN 038795404X.
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External links

fr:Dynamique moléculaire ja:分子動力学法 nl:Moleculaire dynamica pl:Dynamika molekularna ru:Молекулярная динамика zh:分子动力学

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