Molecular dynamics (MD) is a form of computer simulation in which atoms and molecules are allowed to interact for a period of time under known laws of physics, giving a view of the motion of the atoms. In physics the term dynamics customarily refers to the time evolution of physical processes A computer simulation, a computer model or a computational model is a Computer program, or network of computers that attempts to simulate an 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. This article describes complex systems as field of Science. For other meanings see Complex system. Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) It represents an interface between laboratory experiments and theory, and can be understood as a "virtual experiment". Virtual reality ( VR) is a technology which allows a user to interact with a Computer-simulated environment be it a real or imagined one 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. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Information theory is a branch of Applied mathematics and Electrical engineering involving the quantification of Information. It was originally conceived within theoretical physics in the late 1950's[1], but is applied today mostly in materials science and biomolecules. Materials Science or Materials Engineering is an interdisciplinary field involving the properties of matter and its applications to various areas of Science and A biomolecule is any organic Molecule that is produced by living Organisms including large Polymeric molecules such as Proteins

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: ". John Desmond Bernal FRS (born 10 May 1901 died 15 September 1971 was an Irish-born scientist known for pioneering X-ray crystallography. . . 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. "[2] Fortunately, now computers keep track of bonds during a simulation.

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. Molecular modelling is a collective term that refers to theoretical methods and computational techniques to model or mimic the behaviour of Molecules The techniques A computer simulation, a computer model or a computational model is a Computer program, or network of computers that attempts to simulate an Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Mathematical physics, especially as introduced into Statistical mechanics and Thermodynamics by J In Physics and Thermodynamics, the ergodic hypothesis says that over long periods of time the time spent by a particle in some region of the Phase space 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[3][4] and allowing insight into molecular motion on an atomic scale. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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. In Numerical analysis, the condition number associated with a problem is a measure of that problem's amenability to digital computation that is hownumerically well-conditioned In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension Furthermore, current potential functions are, in many cases, not sufficiently accurate to reproduce the dynamics of molecular systems, so the much more 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. In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System

Example of a molecular dynamics simulation in a simple system: deposition of a single Cu atom on a Cu (001) surface. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny Miller indices are a notation system in Crystallography for planes and directions in crystal (Bravais lattices In particular a family of Lattice planes Surface science is the study of physical and chemical phenomena that occur at the interface of two phases, including Solid - Liquid Each circle illustrates the position of a single atom; note that the actual atomic interactions used in the simulations are more complex than those of 2-dimensional hard spheres. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny
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.

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. In the context of Molecular mechanics, a force field (also called a forcefield) refers to the functional form and Parameter sets used Quantum chemistry is a branch of Theoretical chemistry, which applies Quantum mechanics and Quantum field theory to address issues and problems in 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 "potential".

Beginning in theoretical physics, the method of MD gained popularity in materials science and since the 1970s also in biochemistry and biophysics. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Materials Science or Materials Engineering is an interdisciplinary field involving the properties of matter and its applications to various areas of Science and This article is about the Decade 1970-1979 For the Year 1970 see 1970. Biochemistry is the study of the chemical processes in living Organisms It deals with the Structure and function of cellular components such as Biophysics (also biological physics) is an Interdisciplinary Science that employs and develops theories and methods of the Physical sciences for In chemistry, MD serves as an important tool in protein structure determination and refinement using experimental tools such as X-ray crystallography and NMR. Proteins are large Organic compounds made of Amino acids arranged in a linear chain and joined together by Peptide bonds between the Carboxyl X-ray crystallography is a method of determining the arrangement of Atoms within a Crystal, in which a beam of X-rays strikes a crystal and scatters It has also been applied with limited success as a method of refining protein structure predictions. Protein structure prediction is one of the most important goals pursued by Bioinformatics and Theoretical chemistry. 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. Nanotechnology, sometimes shortened to nanotech, refers to a field of Applied science whose theme is the control of matter on an Atomic and Molecular

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. Dynamical systems theory is an area of Applied mathematics used to describe the behavior of complex Dynamical systems usually by employing Differential Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics The concepts of energy conservation and molecular entropy come from thermodynamics. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " Some techniques to calculate conformational entropy such as principal components analysis come from information theory. Conformational entropy is the Entropy associated with the physical arrangement of a Polymer chain that assumes a compact or globular state in solution There are close parallels between the mathematical expressions for the thermodynamic Entropy, usually denoted by S, of a physical system in the Statistical thermodynamics Mathematical techniques such as the transfer operator become applicable when MD is seen as a Markov chain. The transfer operator is different from the transfer homomorphism. In Mathematics, a Markov chain, named after Andrey Markov, is a Stochastic process with the Markov property. Also, there is a large community of mathematicians working on volume preserving, symplectic integrators for more computationally efficient MD simulations. In Mathematics, a symplectic integrator (SI is a numerical integration scheme for a specific group of differential equations related to Classical mechanics

MD can also be seen as a special case of the discrete element method (DEM) in which the particles have spherical shape (e. The term discrete element method (DEM is a family of numerical methods for computing the motion of a large number of particles like molecules or grains of sand g. with the size of their van der Waals radii. Van der Waals Volume The van der Waals volume, V, also called the atomic volume or molecular volume, is the atomic property most directly ) Some authors in the DEM community employ the term MD rather loosely, even when their simulations do not model actual molecules.

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. Seconds Years See also Natural history Geologic 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). A nanosecond ( ns) is one billionth of a second See also times of other orders of magnitude. To help compare Orders of magnitude of different Times this page lists times between 10&minus6 seconds and 10&minus5 seconds (1 micro 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 decomposition in LAMMPS. LAMMPS ("Large-scale Atomic/Molecular Massively Parallel Simulator" is a Molecular dynamics program from Sandia National Laboratories.

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. In the context of Molecular mechanics, a force field (also called a forcefield) refers to the functional form and Parameter sets used 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. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments To analyze an Algorithm is to determine the amount of resources (such as time and storage necessary to execute it Electrostatics is the branch of Science that deals with the Phenomena arising from what seems to be stationary Electric charges Since Classical The Van der Waals equation is an Equation of state that can be derived from a special form of the potential between a pair of molecules (hard-sphere repulsion This computational cost can be reduced by employing electrostatics methods such as Particle Mesh Ewald ( O(nlog(n)) ) or good spherical cutoff techniques ( O(n) ). Ewald summation is a method for computing the interaction energies of periodic systems (e

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. In Mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts 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). To help compare Orders of magnitude of different Times this page lists times between 10&minus15 second and 10&minus12 second (1 Femto This value may be extended by using algorithms such as SHAKE, which fix the vibrations of the fastest atoms (e. In Mechanics, a constraint algorithm is a method for satisfying constraints for bodies that obey Newton's equations of motion g. hydrogens) into place. Multiple time scale methods have also been developed, which allow for extended times between updates of slower long-range forces. [5][6][7]

For simulating molecules in a solvent, a choice should be made between explicit solvent and implicit solvent. Implicit solvation (sometimes known as continuum solvation) is a method of representing Solvent as a continuous medium instead of individual “explicit” solvent Explicit solvent particles (such as the TIP3P and SPC/E water models) must be calculated expensively by the force field, while implicit solvents use a mean-field approach. Computational chemistry, classical water models are used for the simulation of Water clusters liquid water, and aqueous solutions with explicit solvent The impact of explicit solvents on CPU-time can be 10-fold or more. But the granularity and viscosity of explicit solvent is essential to reproduce certain properties of the solute molecules.

In all kinds of molecular dynamics simulations, the simulation box size must be large enough to avoid boundary condition artifacts. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints Boundary conditions are often treated by choosing fixed values at the edges, or by employing periodic boundary conditions in which one side of the simulation loops back to the opposite side, mimicking a bulk phase. In Molecular dynamics, periodic boundary conditions (PBC are a set of Boundary conditions used to simulate an effectively infinitely tiled system usually applied

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. This article covers adiabatic processes in Thermodynamics. For adiabatic processes in Quantum mechanics, see Adiabatic process (quantum mechanics 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

$F(X) = -\nabla U(X)=M\dot{V}(t)$
$V(t) = \dot{X} (t)$

The potential energy function U(X) of the system is a function of the particle coordinates X. Newton's notation for differentiation involved placing a dash/dot over the function name which he termed the fluxion. 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). Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the

For every timestep, each particle's position X and velocity V may be integrated with a symplectic method such as Verlet. In Mathematics, a symplectic integrator (SI is a numerical integration scheme for a specific group of differential equations related to Classical mechanics Verlet integration ( is a method used to integrate Newton's Equations of motion. 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. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature 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.

Canonical ensemble (NVT)

In the canonical ensemble, moles (N), volume (V) and temperature (T) are conserved. A canonical ensemble in Statistical mechanics is a Statistical ensemble representing a Probability distribution of microscopic states of the system 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 are available to add and remove energy from the boundaries of an MD system in a realistic way, approximating the canonical ensemble. A canonical ensemble in Statistical mechanics is a Statistical ensemble representing a Probability distribution of microscopic states of the system Popular techniques to control temperature include the Nosé-Hoover thermostat and Langevin dynamics. Langevin dynamics is an approach to mechanics using simplified models and using Stochastic differential equations to account for omitted degrees of freedom.

Isothermal-Isobaric (NPT) ensemble

In the isothermal-isobaric ensemble, moles (N), pressure (P) and temperature (T) are conserved. The isothermal–isobaric ensemble (constant temperature and constant pressure ensemble is a statistical mechanical ensemble that maintains constant temperature T \ 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. Isotropy is uniformity in all directions Precise definitions depend on the subject area For lipid bilayers, pressure control occurs under constant membrane area (NPAT) or constant surface tension "gamma" (NPγT).

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 [8] tries to overcome the multiple-minima problem by exchanging the temperature of non-interacting replicas of the system running at several temperatures.

Potentials in MD simulations

Main Article: Force field

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 the context of Molecular mechanics, a force field (also called a forcefield) refers to the functional form and Parameter sets used The term potential function may refer to A mathematical function whose values are a physical Potential. In chemistry and biology this is usually referred to as a force field. In the context of Molecular mechanics, a force field (also called a forcefield) refers to the functional form and Parameter sets used 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 term molecular mechanics refers to the use of Newtonian mechanics to model Molecular systems Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects A macromolecule is usually flexible and dynamic It can change its shape in response to changes in its environment or other factors each possible shape is called a conformation and a transition A chemical reaction is a process that always results in the interconversion of Chemical substances The substance or substances initially involved in a chemical reaction are called

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. In Quantum chemistry, the computation of the energy and Wavefunction of an average-size Molecule is a formidable task that is alleviated by the Born-Oppenheimer 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. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons QM/MM ( Quantum mechanics / Molecular mechanics) approach is a molecular Simulation method that combines the strength of both QM (accuracy and MM (speed

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. In the context of Molecular mechanics, a force field (also called a forcefield) refers to the functional form and Parameter sets used A chemical bond is the physical process responsible for the attractive interactions between Atoms and Molecules and which confers stability to diatomic and polyatomic In Aerospace engineering, the Dihedral is the Angle between the two wings see Dihedral. The Van der Waals equation is an Equation of state that can be derived from a special form of the potential between a pair of molecules (hard-sphere repulsion Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. 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. Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Molecular geometry, bond length or bond distance is the average distance between nuclei of two bonded Atoms in a Molecule. Spectroscopy was originally the study of the interaction between Radiation and Matter as a function of Wavelength (λ

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). In the context of Molecular mechanics, a force field (also called a forcefield) refers to the functional form and Parameter sets used Ewald summation is a method for computing the interaction energies of periodic systems (e For the aircraft see Martin P3M Particle-Particle Particle Mesh ( P3M) is a specialized hybrid algorithm for calculating potentials in

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. Ab initio quantum chemistry methods are Computational chemistry methods based on Quantum chemistry. 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. Bond order potentials are a class of empirical (analytical potentials used e Examples of such potentials include the Brenner potential[9] for hydrocarbons and its further developments for the C-Si-H and C-O-H systems. Bond order potentials are a class of empirical (analytical potentials used e The ReaxFF potential[10] can be considered a fully reactive hybrid between bond order potentials and chemistry force fields. ReaxFF (for “reactive force field” is a force field developed by Adri van Duin William A

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. A pair of neutral atoms or molecules is subject to two distinct forces in the limit of large separation and small separation an attractive force at long ranges ( van der Waals force, or

$U(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]$

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. ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form Usually, a simulation only includes the dipolar term, although sometimes the quadrupolar term is included as well.

$U_{ij}(r_{ij}) = \sum \frac {z_i z_j}{4 \pi \epsilon_0} \frac {1}{r_{ij}} + \sum A_l \exp \frac {-r_{ij}}{p_l} + \sum C_l r_{ij}^{-n_j} + \cdots$

In many-body potentials, the potential energy includes the effects of three or more particles interacting with each other. The n -body problem is the problem of finding given the initial positions masses and velocities of n bodies their subsequent motions as determined by 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[11], 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. Bond order potentials are a class of empirical (analytical potentials used e Carbon (kɑɹbən is a Chemical element with the symbol C and its Atomic number is 6 Silicon (ˈsɪlɪkən or /ˈsɪlɪkɒn/ silicium is the Chemical element that has the symbol Si and Atomic number 14 Germanium (dʒɚˈmeɪniəm is a Chemical element with the symbol Ge and Atomic number 32 Other examples are the embedded-atom method (EAM)[12] and the Tight-Binding Second Moment Approximation (TBSMA) potentials[13], 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.

Semi-empirical potentials

Semi-empirical potentials make use of the matrix representation from quantum mechanics. Semi-empirical Quantum chemistry methods are based on the Hartree-Fock formalism but make many approximations and obtain some parameters from empirical data 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. In the tight binding model for a solid-state lattice of atoms it is assumedthat the full Hamiltonian H of the system may be approximated by theHamiltonian of an isolated

Polarizable potentials

Main article: Force field

Most classical force fields implicitly include the effect of polarizability, e. In the context of Molecular mechanics, a force field (also called a forcefield) refers to the functional form and Parameter sets used Polarizability is the relative tendency of a charge distribution like the Electron cloud of an Atom or Molecule, to be distorted from its normal shape 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. Drude particles are model Oscillators used to simulate the effects of electronic Polarizability in the context of a classical Molecular mechanics force 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[14]. Some promising results have also been achieved for proteins[15]. However, it is still uncertain how to best approximate polarizability in a simulation.

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 Quantum chemistry, the computation of the energy and Wavefunction of an average-size Molecule is a formidable task that is alleviated by the Born-Oppenheimer If excited states, chemical reactions or a more accurate representation is needed, electronic behavior can be obtained from first principles by using a quantum chemical method, such as Density Functional Theory. Density functional theory (DFT is a quantum mechanical theory used in Physics and Chemistry to investigate the Electronic structure (principally 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. Ab Initio Software Corporation was founded in the mid 1990's by the former CEO of Thinking Machines Corporation Sheryl Handler, and several other former employees Quantum chemistry is a branch of Theoretical chemistry, which applies Quantum mechanics and Quantum field theory to address issues and problems in A potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in Quantum mechanics and Statistical mechanics This calculation is usually made in the close neighborhood of the reaction coordinate. In Chemistry, a reaction coordinate is an abstract one-dimensional Coordinate which represents progress along a Reaction pathway. 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. 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. The Car-Parrinello method is a type of Ab initio (first principles Molecular dynamics, usually employing periodic Boundary conditions, planewave basis Density functional theory (DFT is a quantum mechanical theory used in Physics and Chemistry to investigate the Electronic structure (principally

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). Arieh Warshel is a professor of Chemistry and Biochemistry at the University of Southern California. The University of Southern California (commonly referred to as USC, SC, Southern California, and incorrectly Duke University is a private Research University located in Durham, North Carolina, United States. The Pennsylvania State University (commonly known as Penn State) is a state-related, land-grant, space grant public research University The University of Minnesota Twin Cities ( U of M or The U) is the oldest and largest part of the University of Minnesota system. The University of Florida ( Florida or UF) is a public land-grant, sea-grant, space-grant major Research

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 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(n3) or worse (Restricted Hartree-Fock calculations have been suggested to scale ~O(n2. In Computational physics and Computational chemistry, the Hartree-Fock ( HF) method is an approximate method for the determination of the ground-state 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 phenomenon 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. Alcohol dehydrogenase (ADH is an enzyme discovered in the mid-1960s in Drosophila melanogaster. In this case, tunneling is important for the hydrogen, as it determines the reaction rate. [16]

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)[17] and Go-models[18]. 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 2-4 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. Protein folding is the physical process by which a Polypeptide folds into its characteristic and functional three-dimensional structure. In a "relaxed" double-helical segment of DNA, the two strands twist around the helical axis once every 10 Deoxyribonucleic acid ( DNA) is a Nucleic acid that contains the genetic instructions used in the development and functioning of all known This article is about a biological infectious particle for other uses see Phage (disambiguation. Ribosomes ( from ribo nucleic acid and "Greek soma ( meaning body") are complexes of RNA and Protein that

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. CHARMM ( Chemistry at HARvard Macromolecular Mechanics) is the name of a widely used set of force fields for Molecular dynamics as well as the

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.

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. CHARMM ( Chemistry at HARvard Macromolecular Mechanics) is the name of a widely used set of force fields for Molecular dynamics as well as the 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. [19]

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 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). The Satellite Tobacco Mosaic Virus or Tobacco mosaic satellivirus was first reported in Nicotiana glauca from southern California U NAMD ( NA noscale M olecular D ynamics1is a free-of-charge Molecular dynamics simulation package written using the Charm++ Molecular dynamics simulations were used to probe the mechanisms of viral assembly. A virus (from the Latin virus meaning Toxin or Poison) is a sub-microscopic infectious agent that is unable 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. For the leaf bug see Miridae. A capsid is the protein shell of a virus. In classical genetics the genome of a Diploid Organism including Eukarya refers to a full set of chromosomes or genes in a Gamete, thereby 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. [20]
• 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. Villin is a 925 kDa tissue-specific Actin-binding protein associated with the actin core bundle of the Brush border. These computers had the folding@home program installed, a large-scale distributed computing effort coordinated by Vijay Pande at Stanford University. Folding@home (sometimes abbreviated as FAH or F@h) is a Distributed computing (DC project designed to perform computationally intensive simulations Vijay S Pande is currently an Associate Professor in the Chemistry Department 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. Phi value analysis is an experimental Protein engineering method used to study the structure of the folding transition state in small protein domains that fold in Protein folding is the physical process by which a Polypeptide folds into its characteristic and functional three-dimensional structure. Each trajectory in a Pfold calculation can be relatively short, but many independent trajectories are needed. [21]

Major software for MD simulations

• ABINIT (DFT)
• AMBER (classical)
• Ascalaph (classical)
• CASTEP (DFT)
• CPMD (DFT)
• CHARMM (classical, the pioneer in MD simulation, extensive analysis tools)
• DL_POLY (classical)
• ESPResSo (classical, coarse-grained, parallel, extensible)
• Fireball (tight-binding DFT)
• GROMACS (classical)
• GROMOS (classical)
• GULP (classical)
• LAMMPS (classical, large-scale with spatial-decomposition of simulation domain for parallelism)
• MDynaMix (classical, parallel)
• MOLDY (classical, parallel)
• MOSCITO (classical)
• NAMD (classical, parallelization with up to thousands of CPU's)
• NEWTON-X (ab initio, surface-hopping dynamics)
• ProtoMol (classical, extensible, includes multigrid electrostatics)
• PWscf (DFT)
• SIESTA (DFT)
• VASP (DFT)
• TINKER (classical)
• YASARA (classical)
• ORAC (classical)
• XMD (classical)

Related software

• 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. Verlet integration ( is a method used to integrate Newton's Equations of motion. Beeman's algorithm is a method for numerically integrating Ordinary differential equations generally position and velocity which is closely related to Verlet In Mathematics, a symplectic integrator (SI is a numerical integration scheme for a specific group of differential equations related to Classical mechanics Cell lists (also sometimes referred to as Cell linked-lists) are a tool for finding all atom pairs within a given cut-off distance of each other in Molecular dynamics A Verlet list is a data structure in Molecular dynamics simulations to efficiently maintain a list of all particles within a given cut-off distance of each other Ewald summation is a method for computing the interaction energies of periodic systems (e Implicit solvation (sometimes known as continuum solvation) is a method of representing Solvent as a continuous medium instead of individual “explicit” solvent Ewald summation is a method for computing the interaction energies of periodic systems (e For the aircraft see Martin P3M Particle-Particle Particle Mesh ( P3M) is a specialized hybrid algorithm for calculating potentials in In Mathematics, Numerical analysis, and numerical Partial differential equations domain decomposition methods solve a Boundary value problem Parallel computing is a form of computation in which many instructions are carried out simultaneously operating on the principle that large problems can often ABINIT is a free software package for Physicists distributed under the GNU General Public License, whose main program allows the total Amber is Fossil tree Resin, which is appreciated for its color and beauty CASTEP is a commercial (and academic Software package which uses Density functional theory with a Plane wave basis set to calculate electronic The Carr-Parrinello Molecular Dynamics, better known as CPMD, is a package for performing Ab-initio Quantum mechanical Molecular dynamics (MD CHARMM ( Chemistry at HARvard Macromolecular Mechanics) is the name of a widely used set of force fields for Molecular dynamics as well as the GROMACS ( GROningen MAchine for Chemical Simulations) is a Molecular dynamics simulation package originally developed in the University of Groningen, now GROMOS is a force field for Molecular dynamics Simulation developed at the University of Groningen and the ETH Zurich. LAMMPS ("Large-scale Atomic/Molecular Massively Parallel Simulator" is a Molecular dynamics program from Sandia National Laboratories. Moscito is a simulation software package for Molecular dynamics (MD simulation originally developed at the University of Dortmund. NAMD ( NA noscale M olecular D ynamics1is a free-of-charge Molecular dynamics simulation package written using the Charm++ PWscf ( Plane-Wave Self-Consistent Field) is a set of programs for Electronic structure calculations within Density functional theory and Density functional SIESTA ( Spanish Initiative for Electronic Simulations with Thousands of Atoms) is an original method and a software implementation for performing Electronic structure The Vienna Ab-initio Simulation Package, better known as VASP (or alternatively VAMP) is a package for performing ab initio Quantum mechanical TINKER is a Computer software application for Molecular dynamics simulation with a complete and general package for Molecular mechanics and Molecular XMD is a classical Molecular dynamics software designed tosimulate problems related to Materials science. Visual molecular dynamics ( VMD) is a Molecular modelling and Visualization Computer program. PyMOL is an open-source user-sponsored molecular visualization system created by Warren Lyford DeLano and commercialized by DeLano Scientific LLC which is a private Sirius is a molecular modeling and analysis system developed at San Diego Supercomputer Center.
• esra - Lightweight molecular modeling and analysis library (Java/Jython/Mathematica).
• Molecular Workbench - Interactive molecular dynamics simulations on your desktop

References

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16. ^ Billeter, SR; SP Webb, PK Agarwal, T Iordanov, S Hammes-Schiffer (2001). "Hydride Transfer in Liver Alcohol Dehydrogenase: Quantum Dynamics, Kinetic Isotope Effects, and Role of Enzyme Motion". J Am Chem Soc 123: 11262-11272. doi:doi:10.1021/ja011384b. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
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20. ^ 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
21. ^ The Folding@Home Project and recent papers published using trajectories from it. Vijay Pande Group. Stanford University

General references

• 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.
• Frenkel, Daan; Smit, Berend [2001]. Daan Frenkel (born 1948 Amsterdam) is a Dutch computational physicist. Understanding Molecular Simulation : from algorithms to applications. San Diego, California: Academic Press. Academic Press ( London, New York and San Diego) was an Academic Book Publisher that is now part of Elsevier. 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. See also 2002 (disambiguation Year 2002 ( MMII) was a Common year starting on Tuesday of the Gregorian calendar. 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.
• William Graham Hoover (1991) Computational Statistical Mechanics, Elsevier, ISBN 0-444-88192-1.