giving a view of the motion of the atoms. Because molecular systems generally consist of a vast number of particles, it is impossible to find the properties of such Complex Systems analytically; MD simulation circumvents this problem by using Numerical methods. It represents an interface between laboratory experiments and theory, and can be understood as a " Virtual experiment".
|
| |
de Laplace
|
| |
P S
|
| |
Oeuveres Completes de Laplace, Theorie Analytique des Probabilites
|
| |
1820
|
| |
Gauthier-Villars
|
| |
Paris, France
|
| |
French
|
|
|
_l r_{ij}^{-n_j} + ...
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.
''
Ab-initio ''
Quantum-mechanical Formula to calculate the
Potential Energy of a system of atoms or molecules. Compared to classical potential function, which is represented by empirical functions, the properties of the system in ''ab-intio'' calculations are calculating the wave-functions for electrons moving around the nucleus of atoms. 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. Of course, the computational price paid is high. A significant advantage of using ''ab-initio'' methods is the ability to study reactions that involved breakage or formation of covalent bonds, this would correspond to multiple electronic states. Classical molecular dynamics is unable to simulate breakage and formation of covalent bonds, However, in recent years techniques such as thermodynamic integration and ghost particles have been introduced to overcome these limitations. The success however remains limited.
A popular package for ''ab-initio'' molecular dynamics is the
Car-Parrinello Molecular Dynamics (CPMD) package based on the
Density Functional Theory .
QM (quantum-mechanical) methods are very powerful however they are computationally expensive, while the MM (classical or molecular mechanics) methods are fast but suffer from several limitations (require extensive parameterization; energy estimates obtained are not very accurate; cannot be used to simulate reactions where covalent bonds are broken/formed; and are limited in their abilities for providing accurate details regarding the chemical environment). A new class of method has emerged that combines the good points of QM (accuracy) and MM (speed) calculations. These methods are known as mixed or hybrid quantum-mechanical and molecular mechanics methods (hybrid QM/MM). The methodology for such techniques was introduced by Warshel and coworkers. In the recent years have been pioneered by several groups including: Arieh Warshel (
University Of Southern California ), Weitao Yang (
Duke University ), Sharon Hammes-Schiffer (
The Pennsylvania State University ), Donald Truhlar and Jiali Gao (
University Of Minnesota ) and Kenneth Merz (
University Of Florida ).
The most important advantage of hybrid QM/MM methods is the speed. The cost of doing classical molecular dynamics (MM) in the most straight forward case scales O(n
2), where N is the number of atoms in the system. This is mainly due to electrostatic interactions term (every particle interacts with everything else). However, use of cutoff radius, periodic pair-list updates and more recently the variations of the particle-mesh Ewald's (PME) method has reduced this between O(N) to O(n
2). In other words, if a system with twice many atoms is simulated then it would take between twice to four times as much computing power. On the other hand the simplest ''ab-initio'' calculations typically scale O(n
3) or worse (Restricted
Hartree-Fock calculations have been suggested to scale ~O(n
2.7)). To overcome the limitation, a small part of the system is treated quantum-mechanically (typically active-site of an enzyme) and the remaining system is treated classically.
At the other end of the detail scale are coarse-grained and lattice models. Instead of explicitly representing every atom of the system, one uses "pseudo-atoms" to represent groups of atoms. 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 prohibitively 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.
The parameterization of these very coarse-grained models must be done empirically, by matching the behavior of the model to appropriate experimental data or all-atom simulations. Ideally, these parameters should account for both enthalpic and entropic contributions to free energy in an implicit way. 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.
Examples of applications of coarse-graining in biophysics:
- Protein Folding studies are often carried out using a single (or a few) pseudo-atoms per amino acid;
- DNA Supercoiling has been investigated using 1-3 pseudo-atoms per basepair, and at even lower resolution;
- Packaging of Double-helical DNA into Bacteriophage has been investigated with models where one pseudo-atom represents one turn (about 10 basepairs) of the double helix;
- RNA structure in the Ribosome and other large systems has been modeled with one pseudo-atom per nucleotide.
The simplest form of coarse-graining is the "united atom" (sometimes called "extended atom") and was used in most early MD simulations of proteins, lipids and nucleic acids. For example, instead of treating all four atoms of a CH3 methyl group explicitly (or all three atoms of CH2 methylene group), one represents the whole group with a single pseudo-atom. This pseudo-atom 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 pseudo-atom 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"). An example of this is the
Charmm 19 force-field.
The polar hydrogens are usually retained in the model, because proper treatment of hydrogen bonds requires a reasonably accurate description of the directionality 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 pseudo-atom. 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.
Molecular dynamics is used in many fields of science.
The following two biophysical examples are not run-of-the-mill MD simulations. They illustrate almost heroic efforts to produce simulations of a system of very large size (a complete virus) and very long simulation times (500 microseconds):
- MD simulation of the complete ) This virus is a small, icosahedral plant virus which worsens the symptoms of infection by Tobacco Mosaic Virus (TMV). Molecular dynamics simulations were used to probe the mechanisms of Viral Assembly . The entire STMV particle consists of 60 identical copies of a single protein that make up the viral Capsid (coating), and a 1063 nucleotide single stranded RNA Genome . One key finding is that the capsid is very unstable when there is no RNA inside. The simulation would take a single 2006 desktop computer around 35 years to complete. It was thus done in many processors in parallel with continuous communication between them. Molecular dynamics simulation of the Satellite Tobacco Mosaic Virus (STMV) Peter Freddolino, Anton Arkhipov, Steven B. Larson, Alexander McPherson, Klaus Schulten. Theoretical and Computational Biophysics Group, University of Illinois at Urbana Champaign
- Folding Simulations of the program installed, a large-scale distributed computing effort coordinated by Vijay Pande at Stanford University. The kinetic properties of the Villin Headpiece protein were probed by using many independent, short trajectories run by CPU's without continuous real-time communication. One technique employed was the Pfold value analysis, which measures the probability of folding before unfolding of a specific starting conformation. Pfold gives information about Transition State structures and an ordering of conformations along the Folding Pathway . Each trajectory in a Pfold calculation can be relatively short, but many independent trajectories are needed. The Folding@Home Project and recent papers published using trajectories from it. Vijay Pande Group. Stanford University
- VMD - MD simulation trajectories can be loaded and visualized
- PyMol - Molecular Modelling software written in python
- Packmol Package for building starting configurations for MD in an automated fashion
- Sirius - Molecular modeling, analysis and visualization of MD trajectories
- AGM Build - Molecular builder and conformational editor with proper partial charges and MM atom types association.
- esra - Lightweight molecular modeling and analysis library (Java/Jython/Mathematica).
- M. P. Allen, D. J. Tildesley (1989) Computer simulation of liquids. Oxford University Press. ISBN 0-19-855645-4.
- J. A. McCammon, S. C. Harvey (1987) Dynamics of Proteins and Nucleic Acids. Cambridge University Press. ISBN 0521307503 (hardback).
- D. C. Rapaport (1996) The Art of Molecular Dynamics Simulation. ISBN 0-521-44561-2.
- Daan Frenkel, Berend Smit (2001) Understanding Molecular Simulation. Academic Press. ISBN 0-12-267351-4.
- J. M. Haile (2001) Molecular Dynamics Simulation: Elementary Methods. ISBN 0-471-18439-X
- R. J. Sadus, ''Molecular Simulation of Fluids: Theory, Algorithms and Object-Orientation'', 2002 , ISBN 0-444-51082-6
- Oren M. Becker, Alexander D. Mackerell Jr, Benoît Roux, Masakatsu Watanabe (2001) Computational Biochemistry and Biophysics. Marcel Dekker. ISBN 0-8247-0455-X.
- Andrew Leach (2001) Molecular Modelling: Principles and Applications. (2nd Edition) Prentice Hall. ISBN 978-0582382107.
- Tamar Schlick (2002) Molecular Modeling and Simulation. Springer. ISBN 0-387-95404-X.
- The Blue Gene Project ( IBM )
- BioSimGrid , a database for biomolecular simulations
- Towards the Dynome : Adding a 4th Dimension to the Protein Database by Terascale Simulation
- MolMovDB : A database of molecular motions
- VigyaanCD : A free electronic workbench for bioinformatics, computational biology and computational chemistry.
- Molecular Workbench : A free educational tool based on molecular dynamics simulations
- Molecular Physics
- Statistical mechanics of Nonequilibrium Liquids Lecture Notes on non-equilibrium MD
- [http://www.fz-juelich.de/nic-series/volume10/sutmann.pdf Introductory Lecture on Classical Molecular Dynamics] by Dr. Godehard Sutmann, NIC, Forschungszentrum Jülich, Germany
- [http://www.fz-juelich.de/nic-series/volume10/tuckerman2.pdf Introductory Lecture on Ab Initio Molecular Dynamics and Ab Initio Path Integrals ] by Mark E. Tuckerman, New York University, USA
- [http://www.fz-juelich.de/nic-series/Volume1/marx.pdf Introductory Lecture on Ab initio molecular dynamics: Theory and Implementation] by Dominik Marx, Ruhr-Universität Bochum and Jürg Hutter, Universität Zürich
