Metadynamics (MTD; also abbreviated as METAD or MetaD) is a computer simulation method in computational physics, chemistry and biology. It is used to estimate the free energy and other state functions of a system, where ergodicity is hindered by the form of the system's energy landscape. It was first suggested by Alessandro Laio and Michele Parrinello in 2002 and is usually applied within molecular dynamics simulations. MTD closely resembles a number of newer methods such as adaptively biased molecular dynamics, adaptive reaction coordinate forces and local elevation umbrella sampling. More recently, both the original 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. MTD is related to the Wang–Landau sampling.
Introduction The technique builds on a large number of related methods including (in a chronological order) the deflation, tunneling, tabu search, local elevation, conformational flooding, Engkvist-Karlström and Adaptive Biasing Force methods. Metadynamics has been informally described as "filling the free energy wells with computational sand". The algorithm assumes that the system can be described by a few collective variables (CV). 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. Usually, the well-tempered metadynamics is used to change the Gaussian size adaptively. Also, the Gaussian width can be adapted with the adaptive Gaussian metadynamics. Metadynamics has the advantage, upon methods like adaptive umbrella sampling, of not requiring an initial estimate of the energy landscape to explore. However, it is not trivial to choose proper collective variables for a complex simulation. Typically, it requires several trials to find a good set of collective variables, but there are several automatic procedures proposed: essential coordinates, Sketch-Map, and non-linear data-driven collective variables.
Multi-replica approach Independent metadynamics simulations (replicas) can be coupled together to improve usability and parallel performance. There are several such methods proposed: the multiple walker MTD, the parallel tempering MTD, the bias-exchange MTD, and the collective-variable tempering MTD. The last three are similar to the parallel tempering method and use replica exchanges to improve sampling. Typically, the Metropolis–Hastings algorithm is used for replica exchanges, but the infinite swapping and Suwa-Todo algorithms give better replica exchange rates.
High-dimensional approach Typical (single-replica) MTD simulations can include up to 3 CVs, even using the multi-replica approach, it is hard to exceed 8 CVs in practice. This limitation comes from the bias potential, constructed by adding Gaussian functions (kernels). It is a special case of the kernel density estimator (KDE). The number of required kernels, for a constant KDE accuracy, increases exponentially with the number of dimensions. So MTD simulation length has to increase exponentially with the number of CVs to maintain the same accuracy of the bias potential. Also, the bias potential, for fast evaluation, is typically approximated with a regular grid. The required memory to store the grid increases exponentially with the number of dimensions (CVs) too. A high-dimensional generalization of metadynamics is NN2B. It is based on two machine learning algorithms: the nearest-neighbor density estimator (NNDE) and the artificial neural network (ANN). NNDE replaces KDE to estimate the updates of bias potential from short biased simulations, while ANN is used to approximate the resulting bias potential. ANN is a memory-efficient representation of high-dimensional functions, where derivatives (biasing forces) are effectively computed with the backpropagation algorithm. An alternative method, exploiting ANN for the adaptive bias potential, uses mean potential forces for the estimation. This method is also a high-dimensional generalization of the Adaptive Biasing Force (ABF) method. Additionally, the training of ANN is improved using Bayesian regularization, and the error of approximation can be inferred by training an ensemble of ANNs.
Developments since 2015 In 2015, White, Dama, and Voth introduced experiment-directed metadynamics, a method that allows for shaping molecular dynamics simulations to match a desired free energy surface. This technique guides the simulation towards conformations that align with experimental data, enhancing our understanding of complex molecular systems and their behavior. In 2020, an evolution of metadynamics was proposed, the on-the-fly probability enhanced sampling method (OPES), which is now the method of choice of Michele Parrinello's research group. The OPES method has only a few robust parameters, converges faster than metadynamics, and has a straightforward reweighting scheme. In 2024, a replica-exchange variant of OPES was developed, named OneOPES, designed to exploit a thermal gradient and multiple CVs to sample large biochemical systems with several degrees of freedom. This variant aims to address the challenge of describing such systems, where the numerous degrees of freedom are often difficult to capture with only a few CVs. OPES has been implemented in the PLUMED library since version 2.7.
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