Metadynamics is a technique in computational physics, chemistry and biology, used to improve sampling of a system (or different systems) where ergodicity is hindered by the form of the system's energy landscape. It was first suggested by Laio and Parrinello in 2002[1] and is usually applied within molecular dynamics simulations. It closely resembles a number of recent methods such as adaptively biased molecular dynamics,[2] adaptive reaction coordinate forces[3] and local elevation umbrella sampling.[4] More recently, both the standard and well-tempered metadynamics were derived in the context of importance sampling and shown to be a special case of the adaptive biasing potential setting.[5] The Metadynamics algorithm is related to the Wang-Landau Sampling.[6]

## Algorithm

The technique builds on a large number of related methods including (in a chronological order) the deflation,[7] tunneling,[8] tabu search,[9] local elevation,[10] conformational flooding,[11] Engkvist-Karlström[12] and adaptive biasing force methods.[13]

Metadynamics has been informally described as "filling the free energy wells with computational sand".[14] The algorithm assumes that the system can be described by a few collective variables. During the simulation, the location of the system in the space determined by the collective variables is calculated and a positive Gaussian potential is added to the real energy landscape of the system. In this way the system is discouraged to come back to the previous point. During the evolution of the simulation, more and more Gaussians sum up, thus discouraging more and more the system to go back to its previous steps, until the system explores the full energy landscape -at this point the modified free energy becomes a constant as a function of the collective variables which is the reason for the collective variables to start fluctuating heavily. At this point the energy landscape can be recovered as the opposite of the sum of all Gaussians.

The time interval between the addition of two Gaussian functions, as well as the Gaussian height and Gaussian width, are tuned to optimize the ratio between accuracy and computational cost. By simply changing the size of the Gaussian, metadynamics can be fitted to yield very quickly a rough map of the energy landscape by using large Gaussians, or can be used for a finer grained description by using smaller Gaussians.[1]

Metadynamics has the advantage, upon methods like adaptive umbrella sampling, of not requiring an initial estimate of the energy landscape to explore.[1]

### Variants

Further refinements of the algorithm have been developed, like well-tempered metadynamics,[15] parallel-tempered metadynamics,[16] and bias-exchange metadynamics.[17] Regarding the choice of collective variables, using essential coordinates has been proposed.[18]

## Applications

Metadynamics has been used to study, among other things, protein folding,[17] chemical reactions,[19] molecular docking[20][21] and phase transitions.[22]

## Implementations

A public implementation of the metadynamics algorithm is PLUMED. It is released as a plugin for several molecular dynamics software packages.[23] The open-source MD programs LAMMPS, NAMD, ORAC and CP2K[24] include metadynamics. Schrödinger is also distributing its implementation of metadynamics within Desmond.