Metadynamics (MTD; also abbreviated as METAD or MetaD) is a
computer simulation
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
method in
computational physics
Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, ...
,
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
and
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
. It is used to
estimate
Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...
the
free energy and other
state functions of a
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
, where
ergodicity
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...
is hindered by the form of the system's
energy landscape
An energy landscape is a mapping of possible states of a system. The concept is frequently used in physics, chemistry, and biochemistry, e.g. to describe all possible conformations of a molecular entity, or the spatial positions of interacting m ...
. It was first suggested by
Alessandro Laio and
Michele Parrinello in 2002
and is usually applied within
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
simulations. MTD closely resembles a number of recent 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 Tabu search is a metaheuristic search method employing local search methods used for mathematical optimization. It was created by Fred W. Glover in 1986
and formalized in 1989.
Local (neighborhood) searches take a potential solution to a pro ...
,
local elevation
Local elevation is a technique used in computational chemistry or physics, mainly in the field of molecular simulation (including molecular dynamics (MD) and Monte Carlo (MC) simulations). It was developed in 1994 by Huber, Torda and van Gunstere ...
,
conformational flooding,
Engkvist-Karlström
and
Adaptive Biasing Force methods.
Metadynamics has been informally described as "filling the free energy wells with computational sand".
[http://www.grs-sim.de/cms/upload/Carloni/Presentations/Marinelli.ppt] 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
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
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
Umbrella sampling is a technique in computational physics and chemistry, 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 Torrie and ...
, 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 procedure 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
Parallel tempering in physics and statistics, is a computer simulation method typically used to find the lowest free energy state of a system of many interacting particles at low temperature. That is, the one expected to be observed in reality ...
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
A regular grid is a tessellation of ''n''-dimensional Euclidean space by congruent parallelotopes (e.g. bricks).
Its opposite is irregular grid.
Grids of this type appear on graph paper and may be used in finite element analysis, finite vol ...
. The required
memory
Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered, ...
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
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...
algorithms: the
nearest-neighbor density estimator (NNDE) and the
artificial neural network
Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains.
An ANN is based on a collection of connected unit ...
(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
In machine learning, backpropagation (backprop, BP) is a widely used algorithm for training feedforward artificial neural networks. Generalizations of backpropagation exist for other artificial neural networks (ANNs), and for functions gener ...
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 the Bayesian regularization, and the error of approximation can be inferred by training an ensemble of ANNs.
Recent developments
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. OPES has been implemented in the
PLUMED library since version 2.7.
Algorithm
Assume, we have a
classical -particle system with positions at
in the
Cartesian coordinates . The particle interaction are described with a
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
function
. The potential function form (e.g. two local minima separated by a high-energy barrier) prevents an
ergodic sampling with
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
or
Monte Carlo
Monte Carlo (; ; french: Monte-Carlo , or colloquially ''Monte-Carl'' ; lij, Munte Carlu ; ) is officially an administrative area of the Principality of Monaco, specifically the ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is ...
methods.
Original metadynamics
A general idea of MTD is to enhance the system sampling by discouraging revisiting of sampled states. It is achieved by augmenting the system
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
with a bias potential
:
:
.
The bias potential is a function of
collective variables . A collective variable is a function of the particle positions
. The bias potential is continuously updated by adding bias at rate
, where
is an instantaneous collective variable value at time
:
:
.
At infinitely long simulation time
, the accumulated bias potential converges to
free energy with opposite sign (and irrelevant constant
):
:
For a computationally efficient implementation, the update process is
discretised into
time intervals (
denotes the
floor function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
) and
-function is replaced with a localized positive
kernel function In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving ...
. The bias potential becomes a sum of the kernel functions centred at the instantaneous collective variable values
at time
:
:
.
Typically, the kernel is a
multi-dimensional Gaussian function, whose covariance matrix has diagonal non-zero elements only:
:
.
The parameter
,
, and
are determined ''a priori'' and kept constant during the simulation.
Implementation
Below there is a
pseudocode of MTD base on
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
(MD), where
and
are the
-particle system positions and velocities, respectively. The bias
is updated every
MD steps, and its contribution to the system forces
is
.
set initial
and
set
every MD step:
compute CV values:
every
MD steps:
update bias potential:
compute atomic forces:
propagate
and
by
Free energy estimator
The finite size of the kernel makes the bias potential to fluctuate around a mean value. A converged free energy can be obtained by averaging the bias potential. The averaging is started from
, when the motion along the collective variable becomes diffusive:
:
Applications
Metadynamics has been used to study:
*
protein folding
Protein folding is the physical process by which a protein chain is translated to its native three-dimensional structure, typically a "folded" conformation by which the protein becomes biologically functional. Via an expeditious and reproduci ...
*
chemical reactions
A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking ...
*
molecular docking
*
phase transitions
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
.
* encapsulation of DNA onto hydrophobic and hydrophilic single-walled carbon nanotubes.
Implementations
PLUMED
PLUMED is an
open-source library
A library is a collection of materials, books or media that are accessible for use and not just for display purposes. A library provides physical (hard copies) or digital access (soft copies) materials, and may be a physical location or a vir ...
implementing many MTD algorithms and
collective variables. It has a flexible
object-oriented
Object-oriented programming (OOP) is a programming paradigm based on the concept of " objects", which can contain data and code. The data is in the form of fields (often known as attributes or ''properties''), and the code is in the form of p ...
design and can be interfaced with several MD programs (
AMBER
Amber is fossilized tree resin that has been appreciated for its color and natural beauty since Neolithic times. Much valued from antiquity to the present as a gemstone, amber is made into a variety of decorative objects."Amber" (2004). In ...
,
GROMACS,
LAMMPS
Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is a molecular dynamics program from Sandia National Laboratories. LAMMPS makes use of Message Passing Interface (MPI) for parallel communication and is free and open-source softw ...
,
NAMD
Nanoscale Molecular Dynamics (NAMD, formerly Not Another Molecular Dynamics Program) is computer software for molecular dynamics simulation, written using the Charm++ parallel programming model. It is noted for its parallel efficiency and is often ...
,
Quantum ESPRESSO
Quantum ESPRESSO is a suite for first-principles electronic-structure calculations and materials modeling, distributed for free and as free software under the GNU General Public License. It is based on density-functional theory, plane wave basis ...
, DL_POLY_4,
CP2K
CP2K is a freely available ( GPL) quantum chemistry and solid state physics program package, written in Fortran 2008, to perform atomistic simulations of solid state, liquid, molecular, periodic, material, crystal, and biological systems. It prov ...
, and OpenMM).
Other
Other MTD implementations exist in th
Collective Variables Module (for
LAMMPS
Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is a molecular dynamics program from Sandia National Laboratories. LAMMPS makes use of Message Passing Interface (MPI) for parallel communication and is free and open-source softw ...
,
NAMD
Nanoscale Molecular Dynamics (NAMD, formerly Not Another Molecular Dynamics Program) is computer software for molecular dynamics simulation, written using the Charm++ parallel programming model. It is noted for its parallel efficiency and is often ...
, and
GROMACS),
ORAC ORAC or Orac may refer to:
* Oxygen radical absorbance capacity, a scalar value derived in the laboratory for comparing the antioxidant content of different foods or nutritional supplements
* Office of the Registrar of Aboriginal Corporations, for ...
,
CP2K
CP2K is a freely available ( GPL) quantum chemistry and solid state physics program package, written in Fortran 2008, to perform atomistic simulations of solid state, liquid, molecular, periodic, material, crystal, and biological systems. It prov ...
,
and
Desmond.
External links
Introduction to metadynamicsPLUMEDColvars module website (NAMD, LAMMPS, Gromacs)Visual movie of metadynamics
See also
*
Local elevation
Local elevation is a technique used in computational chemistry or physics, mainly in the field of molecular simulation (including molecular dynamics (MD) and Monte Carlo (MC) simulations). It was developed in 1994 by Huber, Torda and van Gunstere ...
*
Parallel tempering
Parallel tempering in physics and statistics, is a computer simulation method typically used to find the lowest free energy state of a system of many interacting particles at low temperature. That is, the one expected to be observed in reality ...
*
Umbrella sampling
Umbrella sampling is a technique in computational physics and chemistry, 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 Torrie and ...
References
{{reflist, 2
Molecular dynamics
Computational chemistry
Theoretical chemistry