Introduction

One of the principal tools in the theoretical study of biological molecules is the method of molecular dynamics simulations (MD). This computational method calculates the time dependent behavior of a molecular system.  MD simulations have provided detailed information on the fluctuations and conformational changes of proteins and nucleic acids.   These methods are now routinely used to investigate the structure, dynamics and thermodynamics of biological molecules and their complexes. They are also used in the determination of structures from x-ray crystallography and from NMR experiments.

Biological molecules exhibit a wide range of time scales over which specific processes occur; for example

Local Motions (0.01 to 5 Å, 10-15 to 10-1 s) o Atomic fluctuations

o Sidechain Motions

o Loop Motions

Rigid Body Motions (1 to 10Å, 10-9 to 1s)

o Helix Motions

o Domain Motions (hinge bending)

o Subunit motions

Large-Scale Motions (> 5Å, 10-7 to 104 s)

o Helix coil transitions

o Dissociation/Association

o Folding and Unfolding

An overview of the theoretical foundations of classical molecular dynamics simulations, to discuss some practical aspects of the method and to provide several specific applications within the framework of the CHARMM program. Although the applications will be presented in the framework of the CHARMM program, the concepts are general and applied by a number of different molecular dynamics simulation programs. The CHARMM

program is a research program developed at Harvard University for the energy minimization and dynamics simulation of proteins, nucleic acids and lipids in vacuum, solution or crystal environments (Harvard CHARMM Web Page http://yuri.harvard.edu/).

Section I of this course will focus on the fundamental theory followed by a brief discussion of classical mechanics.  In section II, the potential energy function and some related topics will be presented. Section III will discuss some practical aspects of molecular dynamics simulations and some basic analysis. The remaining sections will present the CHARMM program and provide some tutorials to introduce the user to the program. This course will concentrate on the classical simulation methods (i.e., the most common) that have contributed significantly to our understanding of biological systems.

Molecular dynamics simulations permit the study of complex, dynamic processes that occur in biological systems. These include, for example,

Protein stability Conformational changes

Protein folding

Molecular recognition: proteins, DNA, membranes, complexes

Ion transport in biological systems

and provide the mean to carry out the following studies,

Drug Design

Structure determination: X-ray and NMR

 

Historical Background

The molecular dynamics method was first introduced by Alder and Wainwright in the late 1950's (Alder and Wainwright, 1957,1959) to study the interactions of hard spheres. Many important insights concerning the behavior of simple liquids emerged from their studies. The next major advance was in 1964, when Rahman carried out the first simulation using a realistic potential for liquid argon (Rahman, 1964). The first molecular dynamics simulation of a realistic system was done by Rahman and Stillinger in their simulation of liquid water in 1974 (Stillinger and Rahman, 1974). The

first protein simulations appeared in 1977 with the simulation of the bovine pancreatic trypsin inhibitor (BPTI) (McCammon, et al, 1977). Today in the literature, one routinely finds molecular dynamics simulations of solvated proteins, protein-DNA complexes as well as lipid systems addressing a variety of issues including the thermodynamics of ligand binding and the folding of small proteins. The number of simulation techniques has greatly expanded; there exist now many specialized techniques for particular problems, including mixed quantum mechanical - classical simulations, that are being employed to study enzymatic reactions in the context of the full protein. Molecular dynamics simulation techniques are widely used in experimental procedures such as X-ray crystallography and NMR structure determination.

 

References

Alder, B. J. and Wainwright, T. E. J. Chem. Phys. 27, 1208 (1957)

Alder, B. J. and Wainwright, T. E. J. Chem. Phys. 31, 459 (1959)

Rahman, A. Phys. Rev. A136, 405 (1964)

Stillinger, F. H. and Rahman, A. J. Chem. Phys. 60, 1545 (1974)

McCammon, J. A., Gelin, B. R., and Karplus, M. Nature (Lond.) 267, 585 (1977)

3. STATISTICAL MECHANICS

Molecular dynamics simulations generate information at the microscopic level, including atomic positions and velocities. The conversion of this microscopic information to macroscopic observables such as pressure, energy, heat capacities, etc., requires statistical mechanics. Statistical mechanics is fundamental to the study of biological systems by molecular dynamics simulation. In this section, we provide a brief overview of some main topics. For more detailed information, refer to the numerous excellent books available on the subject.

INTRODUCTION TO STATISTICAL MECHANICS:

In a molecular dynamics simulation, one often wishes to explore the macroscopic properties of a system through microscopic simulations, for example, to calculate changes in the binding free energy of a particular drug candidate, or to examine the energetics and mechanisms of conformational change. The connection between microscopic simulations and macroscopic properties is made via statistical mechanics which provides the rigorous mathematical expressions that relate macroscopic properties to the distribution and motion of the atoms and molecules of the N-body system; molecular dynamics simulations provide the means to solve the equation of motion of the particles and evaluate these mathematical formulas. With molecular dynamics simulations, one can study both thermodynamic properties and/or time dependent (kinetic) phenomenon.

Reference Textbooks on Statistical Mechanics

D. McQuarrie, Statistical Mechanics (Harper & Row, New York, 1976)

D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University Press, New York, 1987)

R. E. Wilde and S. Singh, Statistical Mechanics, Fundamentals and Modern Applications (John Wiley & Sons, Inc, New York, 1998)

 

 

Statistical mechanics is the branch of physical sciences that studies macroscopic systems from a molecular point of view. The goal is to understand and to predict macroscopic phenomena from the properties of individual molecules making up the system. The system could range from a collection of solvent molecules to a solvated protein-DNA complex. In order to connect the macroscopic system to the microscopic system, time independent statistical averages are often introduced. We start this discussion by introducing a few definitions.

Definitions

The thermodynamic state of a system is usually defined by a small set of parameters, for example, the temperature, T, the pressure, P, and the number of particles, N. Other thermodynamic properties may be derived from the equations of state and other fundamental thermodynamic equations.

The mechanical or microscopic state of a system is defined by the atomic positions, q, and momenta, p; these can also be considered as coordinates in a multidimensional space called phase space. For a system of N particles, this space has 6N dimensions. A single point in phase space, denoted by , describes the state of the system. An ensemble is a collection of points in phase space satisfying the conditions of a particular thermodynamic state. A molecular dynamics simulations generates a sequence of points in phase space as a function of time; these points belong to the same ensemble, and they correspond to the different conformations of the system and their respective momenta. Several different ensembles are described below.

An ensemble is a collection of all possible systems which have

different microscopic states but have an identical macroscopic or thermodynamic state.

There exist different ensembles with different characteristics.

Microcanonical ensemble (NVE) : The thermodynamic state characterized by a fixed number of atoms, N, a fixed volume, V, and a fixed energy, E. This corresponds to an isolated system.

Canonical Ensemble (NVT): This is a collection of all systems whose thermodynamic state is characterized by a fixed number of atoms, N, a fixed volume, V, and a fixed temperature, T.

Isobaric-Isothermal Ensemble (NPT): This ensemble is characterized by a fixed number of atoms, N, a fixed pressure, P, and a fixed temperature, T.

Grand canonical Ensemble (VT): The thermodynamic state for this ensemble is characterized by a fixed chemical potential, , a fixed volume, V, and a fixed temperature, T.

CALCULATING AVERAGES FROM A MOLECULAR DYNAMICS SIMULATION

An experiment is usually made on a macroscopic sample that contains an extremely large number of atoms or molecules sampling an enormous number of conformations. In statistical mechanics, averages corresponding to experimental observables are defined in terms of ensemble averages; one justification for this is that there has been good agreement with experiment. An ensemble average is average taken over a large number of replicas of the system considered simultaneously.

 

In statistical mechanics, average values are defined as ensemble averages.

The ensemble average is given by

where

is the observable of interest and it is expressed as a function of the momenta, p, and the positions, r, of the system. The integration is over all possible variables of r and p.

The probability density of the ensemble is given by

where H is the Hamiltonian, T is the temperature, kB is Boltzmann’s constant and Q is the partition function

 

This integral is generally extremely difficult to calculate because one must calculate all possible states of the system. In a molecular dynamics simulation, the points in the ensemble are calculated sequentially in time, so to calculate an ensemble average, the molecular dynamics simulations must pass through all possible states corresponding to the particular thermodynamic constraints.

Another way, as done in an MD simulation, is to determine a time average of A, which is expressed as

where is the simulation time, M is the number of time steps in the simulation and A(pN,rN) is the instantaneous value of A.

The dilemma appears to be that one can calculate time averages by molecular dynamics simulation, but the experimental observables are assumed to be ensemble averages. Resolving this leads us to one of the most fundamental axioms of statistical mechanics, the ergodic hypothesis, which states that the time average equals the ensemble average.

The Ergodic hypothesis states

Ensemble average = Time average

The basic idea is that if one allows the system to evolve in time indefinitely, that system will eventually pass through all possible states. One goal, therefore, of a molecular dynamics simulation is to generate enough representative conformations such that this equality is satisfied. If this is the case, experimentally relevant information concerning structural, dynamic and thermodynamic properties may then be calculated using a feasible amount of computer resources. Because the simulations are of fixed duration, one must be certain to sample a sufficient amount of phase space.

Some examples of time averages:

AVERAGE POTENTIAL ENERGY

where M is the number of configurations in the molecular dynamics trajectory and Vi is the potential energy of each configuration.

AVERAGE KINETIC ENERGY

where M is the number of configurations in the simulation, N is the number of atoms in the system, mi is the mass of the particle i and vi is the velocity of particle i.

A molecular dynamics simulation must be sufficiently long so that enough representative conformations have been sampled.

 

4. CLASSICAL MECHANICS

The molecular dynamics simulation method is based on Newton’s second law or the equation of motion, F=ma, where F is the force exerted on the particle, m is its mass and a is its acceleration. From a knowledge of the force on each atom, it is possible to determine the acceleration of each atom in the system. Integration of the equations of motion then yields a trajectory that describes the positions, velocities and accelerations of the particles as they vary with time. From this trajectory, the average values of properties can be determined. The method is deterministic; once the positions and velocities of each atom are known, the state of the system can be predicted at any time in the future or the past. Molecular dynamics simulations can be time consuming and computationally expensive. However, computers are getting faster and cheaper. Simulations of solvated proteins are calculated up to the nanosecond time scale, however, simulations into the millisecond

regime have been reported.

 

Newton’s equation of motion is given by

where Fi is the force exerted on particle i, mi is the mass of particle i and ai is the acceleration of particle i. The force can also be expressed as the gradient of the potential energy,

Combining these two equations yields

 

where V is the potential energy of the system. Newton’s equation of motion can then relate the derivative of the potential energy to the changes in position as a function of time.

 

NEWTON’S SECOND LAW OF MOTION: A SIMPLE APPLICATION

Taking the simple case where the acceleration is constant,

we obtain an expression for the velocity after integration

and since

we can once again integrate to obtain

Combining this equation with the expression for the velocity, we obtain the following relation which gives the value of x at time t as a function of the acceleration, a, the initial position, x0 , and the initial velocity, v0..

 

The acceleration is given as the derivative of the potential energy with respect to the position, r,

 

Therefore, to calculate a trajectory, one only needs the initial positions of the atoms, an initial distribution of velocities and the acceleration, which is determined by the gradient of the potential energy function. The equations of motion are deterministic, e.g., the positions and the velocities at time zero determine the positions and velocities at all other times, t. The initial positions can be obtained from experimental structures, such as the x-ray crystal structure of the protein or the solution structure determined by NMR spectroscopy.

The initial distribution of velocities are usually determined from a random distribution with the magnitudes conforming to the required temperature and corrected so there is no overall momentum, i.e.,

The velocities, vi, are often chosen randomly from a Maxwell-Boltzmann or Gaussian distribution at a given temperature, which gives the probability that an atom i has a velocity vx in the x direction at a temperature T.

The temperature can be calculated from the velocities using the relation

where N is the number of atoms in the system.

 

INTEGRATION ALGORITHMS

The potential energy is a function of the atomic positions (3N) of all the atoms in the system. Due to the complicated nature of this function, there is no analytical solution to the equations of motion; they must be solved numerically.

Numerous numerical algorithms have been developed for integrating the equations of motion. We list several here.

Verlet algorithm Leap-frog algorithm

Velocity Verlet

Beeman’s algorithm

Important: In choosing which algorithm to use, one should consider the following criteria:

The algorithm should conserve energy and momentum. It should be computationally efficient

It should permit a long time step for integration.

INTEGRATION ALGORITHMS

All the integration algorithms assume the positions, velocities and accelerations can be approximated by a Taylor series expansion:

Where r is the position, v is the velocity (the first derivative with respect to time), a is the acceleration (the second derivative with respect to time), etc.

To derive the Verlet algorithm one can write

Summing these two equations, one obtains

The Verlet algorithm uses positions and accelerations at time t and the positions from time t-t to calculate new positions at time t+t. The Verlet algorithm uses no explicit velocities. The advantages of the Verlet algorithm are, i) it is straightforward, and ii) the storage requirements are modest . The disadvantage is that the algorithm is of moderate precision.

THE LEAP-FROG ALGORITHM

 

 

In this algorithm, the velocities are first calculated at time t+1/2t; these are used to calculate the positions, r, at time t+t. In this way, the velocities leap over the positions, then the positions leap over the velocities. The advantage of this algorithm is that the velocities are explicitly calculated, however, the disadvantage is that they are not calculated at the same time as the positions. The velocities at time t can be approximated by the relationship:

 

THE VELOCITY VERLET ALGORITHM

This algorithm yields positions, velocities and accelerations at time t. There is no compromise on precision.

  

BEEMAN’S ALGORITHM

This algorithm is closely related to the Verlet algorithm

 

 

The advantage of this algorithm is that it provides a more accurate expression for the velocities and better energy conservation. The disadvantage is that the more complex expressions make the calculation more expensive. 

 

Use  of Molecular Dynamics Simulation 

Kinetics and  irreversible processes

chemical reaction kinetics (with QM) conformational changes, allosteric mechanisms

Protein folding

Equilibrium ensemble  sampling

Flexibility thermodynamics (free energy changes, binding)

Modeling tool

structure prediction / modeling solvent effects

NMR/crystallography (refinement)

Electron microscopy (flexible fitting)

    

Why  use molecular dynamics? 

MD is a sampling  method.  But there are other sampling  methods like MonteCarlo (MC). So why use  MD? MD gives you DYNAMICS. Other methods can give you the ensemble (smeared picture), but MD gives you a movie.

Dynamics are important because Biological systems are compartmentalized and are FAR FROM EQUILIBRIUM.

From a small molecule’s standpoint, it doesn’t matter what the list of potential structures of a protein are. Instead the molecule cares about the protein’s

structures over the time it can diffusively sample (about a nsec). And can the molecule influence the dynamics during its contact time?

Consider highway traffic at rush hour, midday and at 2 in the morning. The average (ensemble) picture of the two doesn’t help the poor frog trying to get across the highway. 

   

Atomic  Detail Computer Simulation 

Model System 

Molecular Mechanics  Potential 

Energy  Surface

Exploration by Simulation.. 

© Jeremy Smith

    

Bonded  Interactions: Stretching 

Estr represents the energy required to stretch or compress a covalent bond: 

A bond can be thought  of as a spring having its own equilibrium  length, ro, and the energy required to stretch or compress it can be approximated by the Hookean potential for an ideal spring: 

      Estr = ½ ks,ij ( rij - ro )2  

    

Bonded  Interactions: Bending 

Ebend is the energy required to bend a bond from its equilibrium angle, o: 

Again this system can  be modeled by a spring, and the energy 

is given by the Hookean potential with  respect to angle: 

      Ebend = ½ kb,ijk (ijk - o )2  

    

Bonded  Interactions: Improper Torsion 

Eimproper is the energy required to deform a planar group of atoms from its equilibrium angle, o, usually equal to zero: 

Again this system can be modeled by a spring, and the energy is given by the Hookean potential with respect to planar angle: 

      Eimproper = ½ ko,ijkl ( ijkl - o )2    

© Thomas W. Shattuck 

 

i 

j 

l 

k 

   

Bonded  Interactions: Torsion 

Etor is the energy of torsion needed to rotate about bonds: 

   

      

© Thomas W. Shattuck 

Torsional interactions are  modeled by the potential:  

Etor = ½ ktor,1 (1 - cos ) + ½ ktor,2 (1 - cos 2 ) + ½ ktor,3 ( 1 - cos 3 )     

asymmetry (butane) 2-fold groups e.g. COO- standard tetrahedral torsions

    

Non-Bonded  Interactions: van der Waals 

EvdW is the steric exclusion and long-range attraction energy (QM origins): 

© Thomas W. Shattuck 

Two frequently used formulas:  

        

E 

E 

   

Non-Bonded  Interactions: Coulomb 

Eqq is the Coulomb potential function for electrostatic interactions of charges: 

© Thomas W. Shattuck 

   

Formula:  

The Qi and Qj are the partial atomic charges for atoms i and j separated by a distance rij. is the relative dielectric constant. For gas phase calculations is normally set to 1. Larger values of are used to approximate the dielectric effect of intervening solute (60-80) or solvent atoms in solution. k is a units conversion constant; for kcal/mol, k=2086.4.

    

Newton’s  Law 

Newton’s Law: 

Esteric energy = Estr + Ebend + Eimproper + Etor + EvdW + Eqq   

   

Verlet’s  Numeric Integration Method 

Taylor expansion: 

Verlet’s Method 

   

Timescale  Limitations 

Protein Folding  - milliseconds/seconds (10-3-1s) Ligand Binding - micro/milliseconds (10-

6-10-3 s)

Enzyme catalysis - micro/milliseconds (10-6-10-3 s)

Conformational transitions - pico/nanoseconds (10-12-10-9 s)

Collective vibrations -

1 picosecond (10-12 s)

Bond vibrations -

1 femtosecond (10-15 s) 

   

Timescale  Limitations 

Molecular  dynamics:

Integration timestep  - 1 fs, set by fastest varying force. 

Accessible timescale:  about 10 nanoseconds. 

   

Cutting  Corners 

SHAKE,  Schlick 12.5 

MTS,  Schlick 13.3 

PME,  Schlick 9.4

    

Input  files for MD simulation 

A starting  structure (.pdb) A description of structural connections and atom types (.psf)

A force field (.par) for the atom types (charmm, gromacs, amber, etc)

An input script for the MD program (.conf) 

   

Input  Files: PDB 

 atomic structures 

   

Input  Files: PDB 

We will use  ubiquitin (1UBQ) as an example.  The  pdb structure is available in the protein data bank. 

First, take a  look at the file. 

Then view it  using vmd. 

   

Input  Files: Topology Files 

blueprints for building  a PSF file 

   

Input  Files: Topology Files 

The topology  file represents residues in internal coordinates 

   

Input  Files: Topology Files 

Take a look  at the charmm/xplor topology file top_all27_prot_lipid.inp

    

Input  Files: PSF 

Use information in  topology file to “fill in the gaps”  of the pdb structure:

Patch residues Add hydrogens

Add waters 

NAMD/VMD uses the  utility psfgen to construct the psf file. 

   

Input  Files: PSF 

   

Input  Files: PSF 

Look at the  Tcl script, ubq.pgn, that uses psfgen  to render the psf file.  

   

Input  Files: Parameter Files 

defining the MM energy  terms 

   

Input  Files: Parameter Files 

defining the MM energy  terms 

Take a look  at the charmm force field par_all27_prot_lipid.inp 

   

Production  Run Protocol 

Different  protocols:

One can do constrained dynamics

at the  final temp

Or

Do free dynamics with a temp ramp-up. 

   

Input  Files: MD run script 

defining the MM energy  terms 

NAMD input script ubq_wb_eq.conf

Output is a  trajectory (mcd) in binary format.  Depending  on input script, mcd may contain trajectory  coordinates and trajectory velocities. 

Can use vmd  to visualize the trajectory.

    

Data  reduction and analysis 

MD runs, by virtue  of temperature coupling are stochastic. 

One can calculate  average trajectories by summing over multiple  runs to get an AVERAGE TRAJECTORY. This  gives information on non-equilibrium systems. 

More commonly, one  calculates average properties within a single  run. IFF the MD run has sampled the  entire configurational phase space ( is  ergotic), then the time average = ensemble  average.  One recovers an ENSEMBLE AVERAGE.  

Ensemble properties of interest include: thermodynamic  state functions: free energy (partition function),  entropy, enthalpy 

   

Data  reduction and analysis 

Averaging is done by  constructing a correlation matrix, where 

Off-diagonal terms  give correlations between atoms. 

And  

Diagonal elements  give an average property of an atom.  (ex: the RMSD of a residue).  RMSD  from MD can be compared to B-factors  from X-ray chrystallography, H/D protection 

factors from NMR data, and order parameters  from NMR, EPR, or fluorescence. 

   

RMSD 

   

MD  - energies 

energies: kinetic and  potential 

   

MD  - sampling a local basin 

exploring conformations 

   

Biased  MD - jumping barriers 

exploring conformations 

   

Biased  MD 

exploring conformations 

MD simulations are  generally bad at sampling phase space. 

To cover configurational  phase space, one needs to be able  to do two things well: explore canyons  and jump over energy barriers.  Biased  MD algorithms have been devised to overcome  these deficiencies. 

Explore canyons  - accelerated collective motions (ACM) T4 lysozyme example.

http://cmm.info.nih.gov/intro_simulation 

MOLECULAR DYNAMICS

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Molecular dynamics (MD) is a form of computer simulation in which atoms and molecules are allowed to interact for a period of time by approximations of known physics, giving a view of the motion of the particles. Molecular dynamics simulation is frequently used in the study of proteins and biomolecules, as well as in materials science. It is tempting, though not entirely accurate, to describe the technique as a "virtual microscope" with high temporal and spatial resolution. Whereas it is possible to take "still snapshots" of crystal structures and probe features of the motion of molecules through NMR, no current experimental technique allows access to all the time scales of motion with atomic resolution. Richard Feynman once said that "If we were to name the most powerful assumption of all, which leads one on and on in an attempt to understand life, it is that all things are made of atoms, and that everything that living things do can be understood in terms of the jigglings and wigglings of atoms." Molecular dynamics lets scientists peer into the motion of individual atoms in a way which is not possible in laboratory experiments.

Molecular dynamics is a specialized discipline of molecular modeling and computer simulation based on statistical mechanics; the main justification of the MD method is that statistical ensemble averages are equal to time averages of the system, known as the ergodic hypothesis. MD has also been termed "statistical mechanics by numbers" and "Laplace's vision of Newtonian mechanics" of predicting the future by animating nature's forces[1][2] and allowing insight into molecular motion on an atomic scale. However, long MD simulations are mathematically ill-conditioned, generating cumulative errors in numerical integration that can be minimized with proper selection of algorithms and parameters, but not eliminated entirely. Furthermore, current potential energy functions (also called force-fields) are, in many cases, not sufficiently accurate to reproduce the dynamics of molecular systems, so the much more computationally demanding Ab Initio Molecular Dynamics method must be used. Nevertheless, molecular dynamics techniques allow detailed time and space resolution into representative behavior in phase space for carefully selected systems.

Before it became possible to simulate molecular dynamics with computers, some undertook the hard work of trying it with physical models such as macroscopic spheres.

The idea was to arrange them to replicate the properties of a liquid. J.D. Bernal said, in 1962: "... I took a number of rubber balls and stuck them together with rods of a selection of different lengths ranging from 2.75 to 4 inches. I tried to do this in the first place as casually as possible, working in my own office, being interrupted every five minutes or so and not remembering what I had done before the interruption."[3] Fortunately, now computers keep track of bonds during a simulation.

Because molecular systems generally consist of a vast number of particles, it is in general impossible to find the properties of such complex systems analytically. When the number of particles interacting is higher than two, the result is chaotic motion (see n-body problem). MD simulation circumvents the analytical intractability by using numerical methods. It represents an interface between laboratory experiments and theory, and can be understood as a "virtual experiment". MD probes the relationship between molecular structure, movement and function. Molecular dynamics is a multidisciplinary method. Its laws and theories stem from mathematics, physics, and chemistry, and it employs algorithms from computer science and information theory. It was originally conceived within theoretical physics in the late 1950s[4] and early 1960s [5], but is applied today mostly in materials science and the modeling of biomolecules.

Example of a molecular dynamics simulation in a simple system: deposition of a single Cu atom on a Cu (001) surface. Each circle illustrates the position of a single atom; note that the actual atomic interactions used in current simulations are more complex than those of 2-dimensional hard spheres.

Highly simplified description of the molecular dynamics simulation algorithm. The simulation proceeds iteratively by alternatively calculating forces and solving the equations of motion based on the accelerations obtained from the new forces. In practise, almost all MD codes use much more complicated versions of the algorithm, including two steps (predictor and corrector) in solving the equations of motion and many additional steps for e.g. temperature and pressure control, analysis and output.

CONTENTS

[hide]

1 Areas of Application 2 Design Constraints

o 2.1 Microcanonical ensemble (NVE)

o 2.2 Canonical ensemble (NVT)

o 2.3 Isothermal-Isobaric (NPT) ensemble

o 2.4 Generalized ensembles

3 Potentials in MD simulations

o 3.1 Empirical potentials

o 3.2 Pair potentials vs. many-body potentials

o 3.3 Semi-empirical potentials

o 3.4 Polarizable potentials

o 3.5 Ab-initio methods

o 3.6 Hybrid QM/MM

o 3.7 Coarse-graining and reduced representations

4 Examples of applications

5 Molecular dynamics algorithms

o 5.1 Integrators

o 5.2 Short-range interaction algorithms

o 5.3 Long-range interaction algorithms

o 5.4 Parallelization strategies

6 Major software for MD simulations

7 Related software

8 Specialized hardware for MD simulations

9 See also

10 References

o 10.1 General references

11 External links

[EDIT ] AREAS OF APPLICATION

There is a significant 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 and biophysics, the interaction between the particles is either described by a "force field" (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 and called the potential energy, or just the "potential".

Beginning in theoretical physics, the method of MD gained popularity in materials science and since the 1970s also in biochemistry and biophysics. In chemistry, MD serves as an important tool in protein structure determination and refinement using experimental tools such as X-ray crystallography and NMR. It has also been applied with limited success as a method of refining protein structure predictions. 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 nanotechnological devices that have not or cannot yet be created.

In applied mathematics and theoretical physics, molecular dynamics is a part of the research realm of dynamical systems, ergodic theory and statistical mechanics in general. The concepts of energy conservation and molecular entropy come from thermodynamics. Some techniques to calculate conformational entropy such as principal components analysis come from information theory. Mathematical techniques such as the transfer operator become applicable when MD is seen as a Markov chain. Also, there is a large

community of mathematicians working on volume preserving, symplectic integrators for more computationally efficient MD simulations.

MD can also be seen as a special case of the discrete element method (DEM) in which the particles have spherical shape (e.g. with the size of their van der Waals radii.) Some authors in the DEM community employ the term MD rather loosely, even when their simulations do not model actual molecules.

[EDIT ] DESIGN CONSTRAINTS

Design of a molecular dynamics simulation should account for the available computational power. Simulation size (n=number of particles), timestep and total time duration must be selected so that the calculation can finish within a reasonable time period. However, the simulations should be long enough to be relevant to the time scales of the natural processes being studied. To make statistically valid conclusions from the simulations, the time span simulated should match the kinetics of the natural process. Otherwise, it is analogous to making conclusions about how a human walks from less than one footstep. Most scientific publications about the dynamics of proteins and DNA use data from simulations spanning nanoseconds (1E-9 s) to microseconds (1E-6 s). To obtain these simulations, several CPU-days to CPU-years are needed. Parallel algorithms allow the load to be distributed among CPUs; an example is the spatial or force decomposition algorithm [1].

During a classical MD simulation, the most CPU intensive task is the evaluation of the potential (force field) as a function of the particles' internal coordinates. Within that energy evaluation, the most expensive one is the non-bonded or non-covalent part. In Big O notation, common molecular dynamics simulations scale by O(n2) if all pair-wise electrostatic and van der Waals interactions must be accounted for explicitly. This computational cost can be reduced by employing electrostatics methods such as Particle Mesh Ewald ( O(nlog(n)) ), P3M or good spherical cutoff techniques ( O(n) ).

Another factor that impacts total CPU time required by a simulation is the size of the integration timestep. This is the time length between evaluations of the potential. The timestep must be chosen small enough to avoid discretization errors (i.e. smaller than the fastest vibrational frequency in the system). Typical timesteps for classical MD are in the order of 1 femtosecond (1E-15 s). This value may be extended by using algorithms such as SHAKE, which fix the vibrations of the fastest atoms (e.g. hydrogens) into place. Multiple time scale methods have also been developed, which allow for extended times between updates of slower long-range forces.[6][7][8]

For simulating molecules in a solvent, a choice should be made between explicit solvent and implicit solvent. Explicit solvent particles (such as the TIP3P, SPC/E and SPC-f water models) must be calculated expensively by the force field, while implicit solvents use a mean-field approach. Using an explicit solvent is computationally expensive, requiring inclusion of roughly ten times more particles in the simulation. But the granularity and

viscosity of explicit solvent is essential to reproduce certain properties of the solute molecules. This is especially important to reproduce kinetics.

In all kinds of molecular dynamics simulations, the simulation box size must be large enough to avoid boundary condition artifacts. Boundary conditions are often treated by choosing fixed values at the edges (which may cause artifacts), or by employing periodic boundary conditions in which one side of the simulation loops back to the opposite side, mimicking a bulk phase.

[EDIT ] MICROCANONICAL ENSEMBLE (NVE)

In the microcanonical, or NVE ensemble, the system is isolated from changes in moles (N), volume (V) and energy (E). It corresponds to an adiabatic process with no heat exchange. A microcanonical molecular dynamics trajectory may be seen as an exchange of potential and kinetic energy, with total energy being conserved. For a system of N particles with coordinates X and velocities V, the following pair of first order differential equations may be written in Newton's notation as

The potential energy function U(X) of the system is a function of the particle coordinates X. It is referred to simply as the "potential" in Physics, or the "force field" in Chemistry. The first equation comes from Newton's laws; the force F acting on each particle in the system can be calculated as the negative gradient of U(X).

For every timestep, each particle's position X and velocity V may be integrated with a symplectic method such as Verlet. The time evolution of X and V is called a trajectory. Given the initial positions (e.g. from theoretical knowledge) and velocities (e.g. randomized Gaussian), we can calculate all future (or past) positions and velocities.

One frequent source of confusion is the meaning of temperature in MD. Commonly we have experience with macroscopic temperatures, which involve a huge number of particles. But temperature is a statistical quantity. If there is 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 nkBT/2 where n is the number of degrees of freedom of the system.

A temperature-related phenomenon 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 (1010 or more) with no big change in temperature. When there are only 500 atoms, however, the substrate is almost immediately vaporized by the

deposition. Something similar happens in biophysical simulations. The temperature of the system in NVE is naturally raised when macromolecules such as proteins undergo exothermic conformational changes and binding.

[EDIT ] CANONICAL ENSEMBLE (NVT)

In the canonical ensemble, moles (N), volume (V) and temperature (T) are conserved. It is also sometimes called constant temperature molecular dynamics (CTMD). In NVT, the energy of endothermic and exothermic processes is exchanged with a thermostat.

A variety of thermostat methods is available to add and remove energy from the boundaries of an MD system in a more or less realistic way, approximating the canonical ensemble. Popular techniques to control temperature include velocity rescaling, the Nosé-Hoover thermostat, Nosé-Hoover chains, the Berendsen thermostat and Langevin dynamics. Note that the Berendsen thermostat might introduce the flying ice cube effect, which leads to unphysical translations and rotations of the simulated system.

It is not trivial to obtain a canonical distribution of conformations and velocities using these algorithms. How this depends on system size, thermostat choice, thermostat parameters, time step and integrator is the subject of many articles in the field.

[EDIT ] ISOTHERMAL-ISOBARIC (NPT) ENSEMBLE

In the isothermal-isobaric ensemble, moles (N), pressure (P) and temperature (T) are conserved. In addition to a thermostat, a barostat is needed. It corresponds most closely to laboratory conditions with a flask open to ambient temperature and pressure.

In the simulation of biological membranes, isotropic pressure control is not appropriate. For lipid bilayers, pressure control occurs under constant membrane area (NPAT) or constant surface tension "gamma" (NPγT).

[EDIT ] GENERALIZED ENSEMBLES

The replica exchange method is a generalized ensemble. It was originally created to deal with the slow dynamics of disordered spin systems. It is also called parallel tempering. The replica exchange MD (REMD) formulation [9] tries to overcome the multiple-minima problem by exchanging the temperature of non-interacting replicas of the system running at several temperatures.

[EDIT ] POTENTIALS IN MD SIMULATIONS

Main articles: Force field and Force field implementation

A molecular dynamics simulation requires the definition of a potential function, or a description of the terms by which the particles in the simulation will interact. In chemistry and biology this is usually referred to as a force field. Potentials may be defined at many

levels of physical accuracy; those most commonly used in chemistry are based on molecular mechanics and embody a classical treatment of particle-particle interactions that can reproduce structural and conformational changes but usually cannot reproduce chemical reactions.

The reduction from a fully quantum description to a classical potential entails two main approximations. The first one is the Born-Oppenheimer approximation, which states that the dynamics of electrons is so fast that they can be considered to react instantaneously to the motion of their nuclei. As a consequence, they may be treated separately. The second one treats the nuclei, which are much heavier than electrons, as point particles that follow classical Newtonian dynamics. In classical molecular dynamics the effect of the electrons is approximated as a single potential energy surface, usually representing the ground state.

When finer levels of detail are required, potentials based on quantum mechanics are used; some techniques attempt to create hybrid classical/quantum potentials where the bulk of the system is treated classically but a small region is treated as a quantum system, usually undergoing a chemical transformation.

[EDIT ] EMPIRICAL POTENTIALS

Empirical potentials used in chemistry are frequently called force fields, while those used in materials physics are called just empirical or analytical potentials.

Most force fields in chemistry are empirical and consist of a summation of bonded forces associated with chemical bonds, bond angles, and bond dihedrals, and non-bonded forces associated with van der Waals forces and electrostatic charge. Empirical potentials represent quantum-mechanical effects in a limited way through ad-hoc functional approximations. These potentials contain free parameters such as atomic charge, van der Waals parameters reflecting estimates of atomic radius, and equilibrium bond length, angle, and dihedral; these are obtained by fitting against detailed electronic calculations (quantum chemical simulations) or experimental physical properties such as elastic constants, lattice parameters and spectroscopic measurements.

Because of the non-local nature of non-bonded interactions, they involve at least weak interactions between all particles in the system. Its calculation is normally the bottleneck in the speed of MD simulations. To lower the computational cost, force fields employ numerical approximations such as shifted cutoff radii, reaction field algorithms, particle mesh Ewald summation, or the newer Particle-Particle Particle Mesh (P3M).

Chemistry force fields commonly employ preset bonding arrangements (an exception being ab-initio dynamics), and thus are unable to model the process of chemical bond breaking and reactions explicitly. On the other hand, many of the potentials used in physics, such as those based on the bond order formalism can describe several different coordinations of a system and bond breaking. Examples of such potentials include the Brenner potential [10] for hydrocarbons and its further developments for the C-Si-H and C-O-

H systems. The ReaxFF potential[11] can be considered a fully reactive hybrid between bond order potentials and chemistry force fields.

[EDIT ] PAIR POTENTIALS VS. MANY-BODY POTENTIALS

The potential functions representing the non-bonded energy are formulated as a sum over interactions between the particles of the system. The simplest choice, employed in many popular force fields, is the "pair potential", in which the total potential energy can be calculated from the sum of energy contributions between pairs of atoms. An example of such a pair potential is the non-bonded Lennard-Jones potential (also known as the 6-12 potential), used for calculating van der Waals forces.

Another example is the Born (ionic) model of the ionic lattice. The first term in the next equation is Coulomb's law for a pair of ions, the second term is the short-range repulsion explained by Pauli's exclusion principle and the final term is the dispersion interaction term. Usually, a simulation only includes the dipolar term, although sometimes the quadrupolar term is included as well.

In many-body potentials, the potential energy includes the effects of three or more particles interacting with each other. In simulations with pairwise potentials, global interactions in the system also exist, but they occur only through pairwise terms. In many-body potentials, the potential energy cannot be found by a sum over pairs of atoms, as these interactions are calculated explicitly as a combination of higher-order terms. In the statistical view, the dependency between the variables cannot in general be expressed using only pairwise products of the degrees of freedom. For example, the Tersoff potential [12] , which was originally used to simulate carbon, silicon and germanium and has since been used for a wide range of other materials, involves a sum over groups of three atoms, with the angles between the atoms being an important factor in the potential. Other examples are the embedded-atom method (EAM)[13] and the Tight-Binding Second Moment Approximation (TBSMA) potentials[14], 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.

[EDIT ] 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.

[EDIT ] POLARIZABLE POTENTIALSMain article: Force field

Most classical force fields implicitly include the effect of polarizability, e.g. by scaling up the partial charges obtained from quantum chemical calculations. These partial charges are stationary with respect to the mass of the atom. But molecular dynamics simulations can explicitly model polarizability with the introduction of induced dipoles through different methods, such as Drude particles or fluctuating charges. This allows for a dynamic redistribution of charge between atoms which responds to the local chemical environment.

For many years, polarizable MD simulations have been touted as the next generation. For homogenous liquids such as water, increased accuracy has been achieved through the inclusion of polarizability.[15] Some promising results have also been achieved for proteins.[16] However, it is still uncertain how to best approximate polarizability in a simulation. [citation

needed]

[EDIT ] AB-INITIO METHODS

In classical molecular dynamics, a single potential energy surface (usually the ground state) is represented in the force field. This is a consequence of the Born-Oppenheimer approximation. In excited states, chemical reactions or a more accurate representation is needed, electronic behavior can be obtained from first principles by using a quantum mechanical method, such as Density Functional Theory. This is known as Ab Initio Molecular Dynamics (AIMD). Due to the cost of treating the electronic degrees of freedom, the computational cost of this simulations is much higher than classical molecular dynamics. This implies that AIMD is limited to smaller systems and shorter periods of time.

Ab-initio quantum-mechanical methods may be used to calculate the potential energy of a system on the fly, as needed for conformations in a trajectory. This calculation is usually made 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 calculations produce a vast amount of information that is not available from empirical methods, such as density of electronic states or other electronic properties. A significant advantage of using ab-initio methods is the ability to study reactions that involve breaking or formation of covalent bonds, which correspond to multiple electronic states.

A popular software for ab-initio molecular dynamics is the Car-Parrinello Molecular Dynamics (CPMD) package based on the density functional theory.

[EDIT ] HYBRID QM/MM

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 straightforward case scales O(n2), where N is the number of atoms in the system. This is mainly due to electrostatic interactions term (every particle interacts with every other particle). 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(n2). In other words, if a system with twice as many atoms is simulated then it would take between two to four times as much computing power. On the other hand the simplest ab-initio calculations typically scale O(n3) or worse (Restricted Hartree-Fock calculations have been suggested to scale ~O(n2.7)). 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.

In more sophisticated implementations, QM/MM methods exist to treat both light nuclei susceptible to quantum effects (such as hydrogens) and electronic states. This allows generation of hydrogen wave-functions (similar to electronic wave-functions). This methodology has been useful in investigating phenomena such as hydrogen tunneling. One example where QM/MM methods have provided new discoveries is the calculation of hydride transfer in the enzyme liver alcohol dehydrogenase. In this case, tunneling is important for the hydrogen, as it determines the reaction rate.[17]

[EDIT ] COARSE-GRAINING AND REDUCED REPRESENTATIONS

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.

Examples for coarse graining (CG) methods are discontinuous molecular dynamics (CG-DMD)[18][19] and Go-models[20]. Coarse-graining is done sometimes taking larger pseudo-atoms. Such united atom approximations have been used in MD simulations of biological membranes. The aliphatic tails of lipids are represented by a few pseudo-atoms by gathering 2 to 4 methylene groups into each pseudo-atom.

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.

[EDIT ] EXAMPLES OF APPLICATIONS

Molecular dynamics is used in many fields of science.

First macromolecular MD simulation published (1977, Size: 500 atoms, Simulation Time: 9.2 ps=0.0092 ns, Program: CHARMM precursor) Protein: Bovine Pancreatic Trypsine Inhibitor. This is one of the best studied proteins in terms of folding and kinetics. Its simulation published in Nature magazine paved the way for understanding protein motion as essential in function and not just accessory.[21]

MD is the standard method to treat collision cascades in the heat spike regime, i.e. the effects that energetic neutron and ion irradiation have on solids an solid surfaces.[22][23]

The following two biophysical examples are not run-of-the-mill MD simulations. They illustrate notable 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 satellite tobacco mosaic virus (STMV) (2006, Size: 1 million atoms, Simulation time: 50 ns, program: NAMD) 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.[24]

Folding Simulations of the Villin Headpiece in All-Atom Detail (2006, Size: 20,000 atoms; Simulation time: 500 µs = 500,000 ns, Program: folding@home) This simulation was run in 200,000 CPU's of participating personal computers around the world. These computers had the folding@home 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.[25]

[EDIT ] MOLECULAR DYNAMICS ALGORITHMS

[EDIT ] INTEGRATORS Verlet-Stoermer integration Runge-Kutta integration

Beeman's algorithm

Gear predictor - corrector

Constraint algorithms (for constrained systems)

Symplectic integrator

[EDIT ] SHORT-RANGE INTERACTION ALGORITHMS Cell lists Verlet list

Bonded interactions

[EDIT ] LONG-RANGE INTERACTION ALGORITHMS Ewald summation Particle Mesh Ewald (PME)

Particle-Particle Particle Mesh P3M

Reaction Field Method

[EDIT ] PARALLELIZATION STRATEGIES Domain decomposition method (Distribution of system data for parallel computing) Molecular Dynamics - Parallel Algorithms

[EDIT ] MAJOR SOFTWARE FOR MD SIMULATIONS

Main article: List of software for molecular mechanics modeling

AutoDock suite of automated docking tools, Autodock Vina improved local search algorithm, suite of automated docking tools,

Abalone (classical, implicit water)

ABINIT (DFT)

ACEMD (running on NVIDIA GPUs: heavily optimized with CUDA)

ADUN (classical, P2P database for simulations)

AMBER (classical)

Ascalaph (classical, GPU accelerated)

CASTEP (DFT)

CPMD (DFT)

CP2K (DFT)

CHARMM (classical, the pioneer in MD simulation, extensive analysis tools)

COSMOS (classical and hybrid QM/MM, quantum-mechanical atomic charges with BPT)

Desmond (classical, parallelization with up to thousands of CPU's)

Culgi (classical, OPLS-AA, Dreiding, Nerd, and TraPPE-UA force fields)

DL_POLY (classical)

ESPResSo (classical, coarse-grained, parallel, extensible)

Fireball (tight-binding DFT)

GROMACS (classical)

GROMOS (classical)

GULP (classical)

Hippo (classical)

HOOMD-Blue (classical, accelerated by NVIDIA GPUs, heavily optimized with CUDA)

Kalypso MD simulation of atomic collisions in solids

LAMMPS (classical, large-scale with spatial-decomposition of simulation domain for parallelism)

LPMD Las Palmeras Molecular Dynamics: flexible an modular MD.

MacroModel (classical)

MDynaMix (classical, parallel)

MOLDY (classical, parallel) latest release

Materials Studio (Forcite MD using COMPASS, Dreiding, Universal, cvff and pcff forcefields in serial or parallel, QMERA (QM+MD), ONESTEP (DFT), etc.)

MOSCITO (classical)

NAMD (classical, parallelization with up to thousands of CPU's)

nano-Material Simulation Toolkit

NEWTON-X (ab initio, surface-hopping dynamics)

ORAC (classical)

ProtoMol (classical, extensible, includes multigrid electrostatics)

PWscf (DFT)

RedMD (coarse-grained simulations package on GNU licence)

S/PHI/nX (DFT)

SIESTA (DFT)

VASP (DFT)

TINKER (classical)

YASARA (classical)

XMD (classical)

[EDIT ] RELATED SOFTWARE

Avizo - 3d visualization and analysis software. BOSS - MC in OPLS

esra - Lightweight molecular modeling and analysis library (Java/Jython/Mathematica).

Molecular Workbench - Interactive molecular simulations on your desktop.

Packmol Package for building starting configurations for MD in an automated fashion.

Punto is a freely available visualisation tool for particle simulations.

PyMol - Molecular Visualization software written in python.

Sirius - Molecular modeling, analysis and visualization of MD trajectories.

VMD - MD simulation trajectories can be visualized and analyzed.

[EDIT ] SPECIALIZED HARDWARE FOR MD SIMULATIONS

Anton - A specialized, massively parallel supercomputer designed to execute MD simulations. MDGRAPE - A special purpose system built for molecular dynamics simulations,

especially protein structure prediction.