Metadynamics#Developments since 2015

The technique builds on a large number of related methods including (in a chronological order) the

deflation,{{cite journal | journal = Proceedings of the National Academy of Sciences | volume = 64 | issue = 1 | title = Minimization of polypeptide energy. 8. Application of the deflation technique to a dipeptide | first1 = Gordon M. | last1 = Crippen | first2 = Harold A. | last2 = Scheraga | year = 1969 | pages = 42–49 | pmid = 5263023 | pmc = 286123 |bibcode = 1969PNAS...64...42C |doi = 10.1073/pnas.64.1.42 | doi-access = free }}

tunneling,{{cite journal|journal=SIAM J. Sci. Stat. Comput.|last1=Levy|first1=A.V.|last2=Montalvo|first2=A.|volume=6|pages=15–29|year=1985|doi=10.1137/0906002|title=The Tunneling Algorithm for the Global Minimization of Functions}}

tabu search,{{cite journal | journal = ORSA Journal on Computing | volume = 1 | issue = 3 | title = Tabu Search—Part I | first1 = Fred | last1 = Glover | year = 1989 | pages = 190–206 | doi = 10.1287/ijoc.1.3.190 | s2cid = 5617719 }}

local elevation,{{cite journal|journal=J. Comput.-Aided Mol. Des.|last1=Huber|first1=T.|last2=Torda|first2=A.E.|last3=van Gunsteren|first3=W.F.|volume=8|pages=695–708|year=1994|doi=10.1007/BF00124016|pmid=7738605|title=Local elevation: A method for improving the searching properties of molecular dynamics simulation|issue=6|bibcode=1994JCAMD...8..695H|citeseerx=10.1.1.65.9176|s2cid=15839136}}

conformational flooding,{{cite journal|journal=Phys. Rev. E|last1=Grubmüller|first1=H.|volume=52|pages=2893–2906|year=1995|bibcode = 1995PhRvE..52.2893G |doi = 10.1103/PhysRevE.52.2893|pmid=9963736|title=Predicting slow structural transitions in macromolecular systems: Conformational flooding|issue=3 |hdl=11858/00-001M-0000-000E-CA15-8|hdl-access=free}}

Engkvist-Karlström{{cite journal|journal=Chem. Phys.|last1=Engkvist|first1=O.|last2=Karlström|first2=G.|volume=213|issue=1|pages=63–76|year=1996|bibcode = 1996CP....213...63E |doi = 10.1016/S0301-0104(96)00247-9|title=A method to calculate the probability distribution for systems with large energy barriers }} and

Adaptive Biasing Force methods.{{cite journal|journal=J. Chem. Phys.|last1=Darve|first1=E.|last2=Pohorille|first2=A.|volume=115|pages=9169|year=2001|bibcode = 2001JChPh.115.9169D |doi = 10.1063/1.1410978|title=Calculating free energies using average force|issue=20 |hdl=2060/20010090348|s2cid=5310339|hdl-access=free}}

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{{dead link|date=January 2018 |bot=InternetArchiveBot |fix-attempted=yes }} 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{{Cite journal|last1 = Barducci|first1 = A.|last2 = Bussi|first2 = G.|last3 = Parrinello|first3 = M.|title = Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method|journal = Physical Review Letters|volume = 100|issue = 2|pages = 020603|year = 2008|pmid = 18232845|doi = 10.1103/PhysRevLett.100.020603|bibcode = 2008PhRvL.100b0603B|arxiv = 0803.3861|s2cid = 13690352}} is used to change the Gaussian size adaptively. Also, the Gaussian width can be adapted with the adaptive Gaussian metadynamics.{{Cite journal|title = Metadynamics with Adaptive Gaussians|journal = Journal of Chemical Theory and Computation|date = 2012-06-04|pages = 2247–2254|volume = 8|issue = 7|doi = 10.1021/ct3002464|pmid = 26588957|language = EN|first1 = Davide|last1 = Branduardi|first2 = Giovanni|last2 = Bussi|first3 = Michele|last3 = Parrinello|arxiv = 1205.4300|s2cid = 20002793}}

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,{{Cite journal|last1 = Spiwok|first1 = V.|last2 = Lipovová|first2 = P.|last3 = Králová|first3 = B.|title = Metadynamics in essential coordinates: free energy simulation of conformational changes|journal = The Journal of Physical Chemistry B|volume = 111|issue = 12|pages = 3073–3076|year = 2007|pmid = 17388445|doi = 10.1021/jp068587c}} Sketch-Map,{{Cite journal|title = Demonstrating the Transferability and the Descriptive Power of Sketch-Map|journal = Journal of Chemical Theory and Computation|date = 2013-02-22|pages = 1521–1532|volume = 9|issue = 3|doi = 10.1021/ct3010563|pmid = 26587614|language = EN|first1 = Michele|last1 = Ceriotti|first2 = Gareth A.|last2 = Tribello|first3 = Michele|last3 = Parrinello| s2cid=20432114 |url = https://pure.qub.ac.uk/portal/en/publications/demonstrating-the-transferability-and-the-descriptive-power-of-sketchmap(ea20f98a-bb39-48a0-ac61-fe8bfe7299e2).html}} and non-linear data-driven collective variables.{{Cite journal|title = Modeling and enhanced sampling of molecular systems with smooth and nonlinear data-driven collective variables|journal = The Journal of Chemical Physics|date = 2013-12-07|issn = 0021-9606|pages = 214101|volume = 139|issue = 21|doi = 10.1063/1.4830403|pmid = 24320358|first1 = Behrooz|last1 = Hashemian|first2 = Daniel|last2 = Millán|first3 = Marino|last3 = Arroyo|bibcode = 2013JChPh.139u4101H |hdl = 2117/20940|hdl-access = free}}

Independent metadynamics simulations (replicas) can be coupled together to improve usability and parallel performance. There are several such methods proposed: the multiple walker MTD,{{Cite journal|title = Efficient Reconstruction of Complex Free Energy Landscapes by Multiple Walkers Metadynamics †|journal = The Journal of Physical Chemistry B|date = 2005-10-28|pages = 3533–3539|volume = 110|issue = 8|doi = 10.1021/jp054359r|pmid = 16494409|language = en|first1 = Paolo|last1 = Raiteri|first2 = Alessandro|last2 = Laio|first3 = Francesco Luigi|last3 = Gervasio|first4 = Cristian|last4 = Micheletti|first5 = Michele|last5 = Parrinello|s2cid = 15595613}} the parallel tempering MTD,{{cite journal|last=Bussi|first=Giovanni|author2=Gervasio, Francesco Luigi |author3=Laio, Alessandro |author4= Parrinello, Michele |title=Free-Energy Landscape for β Hairpin Folding from Combined Parallel Tempering and Metadynamics|journal=Journal of the American Chemical Society|date=October 2006|volume=128|issue=41|pages=13435–13441|doi=10.1021/ja062463w|pmid=17031956}} the bias-exchange MTD,{{Cite journal

| last1 = Piana | first1 = S.

| last2 = Laio | first2 = A.

| title = A bias-exchange approach to protein folding

| journal = The Journal of Physical Chemistry B

| volume = 111

| issue = 17

| pages = 4553–4559

| year = 2007

| pmid = 17419610

| doi = 10.1021/jp067873l

| hdl = 20.500.11937/15651

| hdl-access = free

}} and the collective-variable tempering MTD.{{Cite journal|title = Enhanced Conformational Sampling Using Replica Exchange with Collective-Variable Tempering|journal = Journal of Chemical Theory and Computation|date = 2015-02-19|pmc = 4364913|pmid = 25838811|pages = 1077–1085|volume = 11|issue = 3|doi = 10.1021/ct5009087|language = EN|first1 = Alejandro|last1 = Gil-Ley|first2 = Giovanni|last2 = Bussi|arxiv = 1502.02115}} 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{{Cite journal|title = An infinite swapping approach to the rare-event sampling problem|journal = The Journal of Chemical Physics|date = 2011-10-07|issn = 0021-9606|pages = 134111|volume = 135|issue = 13|doi = 10.1063/1.3643325|pmid = 21992286|first1 = Nuria|last1 = Plattner|first2 = J. D.|last2 = Doll|first3 = Paul|last3 = Dupuis|first4 = Hui|last4 = Wang|first5 = Yufei|last5 = Liu|first6 = J. E.|last6 = Gubernatis|arxiv = 1106.6305 |bibcode = 2011JChPh.135m4111P |s2cid = 40621592}} and Suwa-Todo{{Cite journal|title = Markov Chain Monte Carlo Method without Detailed Balance|journal = Physical Review Letters|date = 2010-01-01|volume = 105|issue = 12|pages = 120603|doi = 10.1103/PhysRevLett.105.120603|pmid = 20867621|first = Hidemaro|last = Suwa|arxiv = 1007.2262 |bibcode = 2010PhRvL.105l0603S |s2cid = 378333}} algorithms give better replica exchange rates.{{Cite journal|title = Replica state exchange metadynamics for improving the convergence of free energy estimates|journal = Journal of Computational Chemistry|date = 2015-07-15|issn = 1096-987X|pages = 1446–1455|volume = 36|issue = 19|doi = 10.1002/jcc.23945|pmid = 25990969|language = en|first1 = Raimondas|last1 = Galvelis|first2 = Yuji|last2 = Sugita|s2cid = 19101602}}

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.{{Cite web|url=https://plumed.github.io/doc-v2.4/user-doc/html/_metadyn.html|title=PLUMED: Metadynamics|website=plumed.github.io|access-date=2018-01-13}} The required memory to store the grid increases exponentially with the number of dimensions (CVs) too.

A high-dimensional generalization of metadynamics is NN2B.{{Cite journal|last1=Galvelis|first1=Raimondas|last2=Sugita|first2=Yuji|date=2017-06-13|title=Neural Network and Nearest Neighbor Algorithms for Enhancing Sampling of Molecular Dynamics|journal=Journal of Chemical Theory and Computation|volume=13|issue=6|pages=2489–2500|doi=10.1021/acs.jctc.7b00188|pmid=28437616|issn=1549-9618}} 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.{{Cite journal|last1=Schneider|first1=Elia|last2=Dai|first2=Luke|last3=Topper|first3=Robert Q.|last4=Drechsel-Grau|first4=Christof|last5=Tuckerman|first5=Mark E.|date=2017-10-11|title=Stochastic Neural Network Approach for Learning High-Dimensional Free Energy Surfaces|journal=Physical Review Letters|volume=119|issue=15|pages=150601|doi=10.1103/PhysRevLett.119.150601|pmid=29077427|bibcode=2017PhRvL.119o0601S|doi-access=free}}

An alternative method, exploiting ANN for the adaptive bias potential, uses mean potential forces for the estimation.{{cite journal|last1=Zhang|first1=Linfeng|last2=Wang|first2=Han|last3=E|first3=Weinan|date=2017-12-09|title=Reinforced dynamics for enhanced sampling in large atomic and molecular systems. I. Basic Methodology|journal=The Journal of Chemical Physics|volume=148|issue=12|pages=124113|arxiv=1712.03461|doi=10.1063/1.5019675|pmid=29604808|s2cid=4552400}} This method is also a high-dimensional generalization of the Adaptive Biasing Force (ABF) method.{{Cite journal|last1=Comer|first1=Jeffrey|last2=Gumbart|first2=James C.|last3=Hénin|first3=Jérôme|last4=Lelièvre|first4=Tony|last5=Pohorille|first5=Andrew|last6=Chipot|first6=Christophe|date=2015-01-22|title=The Adaptive Biasing Force Method: Everything You Always Wanted To Know but Were Afraid To Ask|journal=The Journal of Physical Chemistry B|volume=119|issue=3|pages=1129–1151|doi=10.1021/jp506633n|pmid=25247823|pmc=4306294|issn=1520-6106}} Additionally, the training of ANN is improved using Bayesian regularization,{{cite journal|last1=Sidky|first1=Hythem|last2=Whitmer|first2=Jonathan K.|date=2017-12-07|title=Learning Free Energy Landscapes Using Artificial Neural Networks|journal=The Journal of Chemical Physics|volume=148|issue=10|pages=104111|arxiv=1712.02840|doi=10.1063/1.5018708|pmid=29544298|s2cid=3932640}} and the error of approximation can be inferred by training an ensemble of ANNs.

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.{{cite journal

| last1 = White

| first1 = Andrew D.

| last2 = Dama

| first2 = James F.

| last3 = Voth

| first3 = Gregory A.

| title = Designing Free Energy Surfaces That Match Experimental Data with Metadynamics

| journal = Journal of Chemical Theory and Computation

| volume = 11

| issue = 6

| pages = 2451–2460

| year = 2015

| doi = 10.1021/acs.jctc.5b00178

| pmid = 26575545

| osti = 1329576

}}

In 2020, an evolution of metadynamics was proposed, the on-the-fly probability enhanced sampling method (OPES),{{Cite journal| doi = 10.1021/acs.jpclett.0c00497| issn = 1948-7185| volume = 11| issue = 7| pages = 2731–2736| last1 = Invernizzi| first1 = Michele| last2 = Parrinello| first2 = Michele| title = Rethinking Metadynamics: From Bias Potentials to Probability Distributions| journal = The Journal of Physical Chemistry Letters| date = 2020-04-02| pmid = 32191470| arxiv = 1909.07250| s2cid = 202577890}}{{Cite journal| doi = 10.1103/PhysRevX.10.041034| issn = 2160-3308| volume = 10| issue = 4| pages = 41034| last1 = Invernizzi| first1 = Michele| last2 = Piaggi| first2 = Pablo M.| last3 = Parrinello| first3 = Michele| title = Unified Approach to Enhanced Sampling| journal = Physical Review X| date = 2020-07-06| arxiv = 2007.03055| bibcode = 2020PhRvX..10d1034I| s2cid = 220381217}}{{Cite journal| doi = 10.1021/acs.jctc.2c00152| issn = 1549-9618| volume = 18| issue = 6| pages = 3988–3996| last1 = Invernizzi| first1 = Michele| last2 = Parrinello| first2 = Michele| title = Exploration vs Convergence Speed in Adaptive-Bias Enhanced Sampling| journal = Journal of Chemical Theory and Computation| date = 2022-06-14| pmid = 35617155| pmc = 9202311}} which is now the method of choice of Michele Parrinello's research group.{{Cite journal| doi = 10.1002/ijch.202100105| issn = 0021-2148| volume = 62| issue = 1–2| pages = e202100105| last = Parrinello| first = Michele| title = Breviarium de Motu Simulato Ad Atomos Pertinenti| journal = Israel Journal of Chemistry| accessdate = 2022-12-06| date = 2022-01-12| s2cid = 245916578| url = https://onlinelibrary.wiley.com/doi/10.1002/ijch.202100105| url-access = subscription}} The OPES method has only a few robust parameters, converges faster than metadynamics, and has a straightforward reweighting scheme.{{Cite web|title = On-the-fly Probability Enhanced Sampling (OPES)|url = https://parrinello.ethz.ch/research/opes.html |website = www.parrinello.ethz.ch|access-date = 2022-06-12}} In 2024, a replica-exchange variant of OPES was developed, named OneOPES,{{Cite journal |last1=Rizzi |first1=Valerio |last2=Aureli |first2=Simone |last3=Ansari |first3=Narjes |last4=Gervasio |first4=Francesco Luigi |date=2023-09-12 |title=OneOPES, a Combined Enhanced Sampling Method to Rule Them All |journal=Journal of Chemical Theory and Computation |language=en |volume=19 |issue=17 |pages=5731–5742 |doi=10.1021/acs.jctc.3c00254 |issn=1549-9618 |pmc=10500989 |pmid=37603295}} 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.{{Cite web|title = PLUMED - OPES|url = https://www.plumed.org/doc-master/user-doc/html/_o_p_e_s.html |website = www.plumed.org|access-date = 2022-06-12}}

Assume we have a classical $N$-particle system with positions at $\backslash \{\; \backslash vec\; r\_i\; \backslash \}$ $(i\; \backslash in\; 1...N)$ in the Cartesian coordinates $(\backslash vec\; r\_i\; \backslash in\; \backslash mathbb\{R\}^3)$. The particle interaction are described with a potential function $V\; \backslash equiv\; V(\backslash \{\; \backslash vec\; r\_i\; \backslash \})$. The potential function form (e.g. two local minima separated by a high-energy barrier) prevents an ergodic sampling with molecular dynamics or Monte Carlo methods.

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 $H$ with a bias potential $V\_\backslash text\{bias\}$:

:$H\; =\; T\; +\; V\; +\; V\_\backslash text\{bias\}$.

The bias potential is a function of collective variables $(V\_\backslash text\{bias\}\; \backslash equiv\; V\_\backslash text\{bias\}(\backslash vec\; s\backslash ,))$. A collective variable is a function of the particle positions $(\backslash vec\; s\; \backslash equiv\; \backslash vec\; s(\backslash \{\; \backslash vec\; r\_i\backslash \}))$. The bias potential is continuously updated by adding bias at rate $\backslash omega$, where $\backslash vec\; s\_t$

is an instantaneous collective variable value at time $t$

:

:$\backslash frac\{\backslash partial\; V\_\backslash text\{bias\}(\backslash vec\; s\backslash ,)\}\{\backslash partial\; t\}\; =$

\omega\, \delta(|\vec s - \vec s_t|)

.

At infinitely long simulation time $t\_\backslash text\{sim\}$, the accumulated bias potential converges to free energy with opposite sign (and irrelevant constant $C$):

:$V\_\backslash text\{bias\}(\backslash vec\; s\backslash ,)\; =$

\!\!\int_0^{t_\text{sim}} \!\!\!\omega\, \delta(|\vec s - \vec s_t|)\; dt

\quad\Rightarrow\quad

F(\vec s\,) =

-\!\!\!\!\lim_{t_\text{sim} \to \infty}\!\! V_\text{bias}(\vec s\,) + C

For a computationally efficient implementation, the update process is discretised into $\backslash tau$ time intervals ($\backslash lfloor\backslash ;\backslash rfloor$ denotes the floor function) and $\backslash delta$-function is replaced with a localized positive kernel function $K$. The bias potential becomes a sum of the kernel functions centred at the instantaneous collective variable values $\backslash vec\; s\_j$ at time $\backslash tau\; j$

:

:$V\_\backslash text\{bias\}(\backslash vec\; s\backslash ,)\; \backslash approx$

\tau \!\!\!\sum_{j=0}^{\left\lfloor \frac{t_\text{sim}}{\tau} \right\rfloor}\!\!

\omega\, K(|\vec s - \vec s_j|)

.

Typically, the kernel is a multi-dimensional Gaussian function, whose covariance matrix has diagonal non-zero elements only:

:$V\_\backslash text\{bias\}(\backslash vec\; s\backslash ,)\; \backslash approx$

\tau \!\!\!\sum_{j=0}^{\left\lfloor \frac{t_\text{sim}}{\tau} \right\rfloor}\!\!

\omega \exp\!\! \left(\! -\frac{1}{2} \left| \frac{\vec s - \vec s_j}{\vec \sigma} \right|^2 \right)

.

The parameter $\backslash tau$, $\backslash omega$, and $\backslash vec\; \backslash sigma$ are determined a priori and kept constant during the simulation.

Below there is a pseudocode of MTD base on molecular dynamics (MD), where $\backslash \{\backslash vec\; r\backslash \}$ and $\backslash \{\backslash vec\; v\backslash \}$ are the $N$-particle system positions and velocities, respectively. The bias $V\_\backslash text\{bias\}$ is updated every $n\; =\; \backslash tau/\backslash Delta\; t$ MD steps, and its contribution to the system forces $\backslash \{\backslash vec\; F\backslash ,\backslash \}$ is $\backslash \{\backslash vec\; F\_\backslash text\{bias\}\backslash \}$.

set initial $\backslash \{\backslash vec\; r\backslash \}$ and $\backslash \{\backslash vec\; v\backslash \}$

set $V\_\backslash text\{bias\}(\backslash vec\; s\backslash ,)\; :=\; 0$

every MD step:

compute CV values:

$\backslash vec\; s\_t\; :=\; \backslash vec\; s(\backslash \{\backslash vec\; r\backslash \})$

every $n$ MD steps:

update bias potential:

$V\_\backslash text\{bias\}(\backslash vec\; s\backslash ,)\; :=\; V\_\backslash text\{bias\}(\backslash vec\; s\backslash ,)$

+ \tau \omega \exp\!\! \left(\! -\frac{1}{2} \left| \frac{\vec s - \vec s_t}{\vec \sigma} \right|^2 \right)

compute atomic forces:

$\backslash vec\; F\_i\; :=\; -\backslash frac\{\backslash partial\; V(\backslash \{\backslash vec\; r\backslash ,\backslash \})\}\{\backslash partial\; \backslash vec\; r\_i\}$

\overbrace{\left. -\frac{\partial V_\text{bias}(\vec s\,)}{\partial\vec s} \right|_{\vec s_t}\!\!\!

\frac{\partial \vec s(\{\vec r\,\})}{\partial \vec r_i}}^{\vec F_{\text{bias},i}}

propagate $\backslash \{\backslash vec\; r\backslash \}$ and $\backslash \{\backslash vec\; v\backslash \}$ by $\backslash Delta\; t$

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 $t\_\backslash text\{diff\}$, when the motion along the collective variable becomes diffusive:

:$\backslash bar\; F(\backslash vec\; s)\; =\; -\; \backslash frac\{1\}\{t\_\backslash text\{sim\}\; -\; t\_\backslash text\{diff\}\}$

\int^{t_\text{sim}}_{t_\text{diff}} \!\!\!\!\!V_\text{bias}(\vec s, t)\, dt + C

Metadynamics has been used to study:

• protein folding
• chemical reactions{{Cite journal| last1 = Ensing | first1 = B.| last2 = De Vivo | first2 = M.| last3 = Liu | first3 = Z.| last4 = Moore | first4 = P.| last5 = Klein | first5 = M.| title = Metadynamics as a tool for exploring free energy landscapes of chemical reactions| journal = Accounts of Chemical Research| volume = 39| issue = 2| pages = 73–81| year = 2006| pmid = 16489726| doi = 10.1021/ar040198i}}
• molecular docking{{Cite journal| last1 = Gervasio | first1 = F.| last2 = Laio | first2 = A.| last3 = Parrinello | first3 = M.| title = Flexible docking in solution using metadynamics| journal = Journal of the American Chemical Society| volume = 127| issue = 8| pages = 2600–2607| year = 2005| pmid = 15725015| doi = 10.1021/ja0445950| s2cid = 6304388}}{{Cite journal | last1 = Vargiu | first1 = A. V. | last2 = Ruggerone | first2 = P. | last3 = Magistrato | first3 = A. | last4 = Carloni | first4 = P. | title = Dissociation of minor groove binders from DNA: insights from metadynamics simulations | journal = Nucleic Acids Research | volume = 36 | issue = 18 | pages = 5910–5921 | year = 2008 | pmid = 18801848 | pmc = 2566863 | doi = 10.1093/nar/gkn561}}
• phase transitions.{{Cite journal| last1 = Martoňák | first1 = R.| last2 = Laio | first2 = A.| last3 = Bernasconi | first3 = M.| last4 = Ceriani | first4 = C.| last5 = Raiteri | first5 = P.| last6 = Zipoli | first6 = F.| last7 = Parrinello | first7 = M.| title = Simulation of structural phase transitions by metadynamics| journal = Zeitschrift für Kristallographie| volume = 220| issue = 5–6| pages = 489| year = 2005|arxiv=cond-mat/0411559| doi = 10.1524/zkri.220.5.489.65078|bibcode = 2005ZK....220..489M | s2cid = 96851280}}
• encapsulation of DNA onto hydrophobic{{citation |author1=Cruz, F.J.A.L. |author2=de Pablo, J.J. |author3=Mota, J.P.B. |title=Endohedral confinement of a DNA dodecamer onto pristine carbon nanotubes and the stability of the canonical B form |journal=J. Chem. Phys. |volume=140 |issue=22 |year=2014 |pages=225103 |doi=10.1063/1.4881422|pmid=24929415 |arxiv=1605.01317 |bibcode=2014JChPh.140v5103C |s2cid=15149133 }} and hydrophilic{{citation |author1=Cruz, F.J.A.L. |author2=Mota, J.P.B. |title=Conformational Thermodynamics of DNA Strands in Hydrophilic Nanopores |journal=J. Phys. Chem. C |volume=120 |issue=36 |year=2016 |pages=20357–20367 |doi=10.1021/acs.jpcc.6b06234}} single-walled carbon nanotubes.

PLUMED{{Cite web|title = PLUMED|url = http://www.plumed.org/|website = www.plumed.org|access-date = 2016-01-26}} is an open-source library implementing many MTD algorithms and collective variables. It has a flexible object-oriented design{{Cite journal|title = PLUMED: A portable plugin for free-energy calculations with molecular dynamics|journal = Computer Physics Communications|date = 2009-10-01|pages = 1961–1972|volume = 180|issue = 10|doi = 10.1016/j.cpc.2009.05.011|first1 = Massimiliano|last1 = Bonomi|first2 = Davide|last2 = Branduardi|first3 = Giovanni|last3 = Bussi|first4 = Carlo|last4 = Camilloni|first5 = Davide|last5 = Provasi|first6 = Paolo|last6 = Raiteri|first7 = Davide|last7 = Donadio|first8 = Fabrizio|last8 = Marinelli|first9 = Fabio|last9 = Pietrucci|arxiv = 0902.0874 |bibcode = 2009CoPhC.180.1961B |s2cid = 4852774}}{{Cite journal|title = PLUMED 2: New feathers for an old bird|journal = Computer Physics Communications|date = 2014-02-01|pages = 604–613|volume = 185|issue = 2|doi = 10.1016/j.cpc.2013.09.018|first1 = Gareth A.|last1 = Tribello|first2 = Massimiliano|last2 = Bonomi|first3 = Davide|last3 = Branduardi|first4 = Carlo|last4 = Camilloni|first5 = Giovanni|last5 = Bussi|arxiv = 1310.0980 |bibcode = 2014CoPhC.185..604T |s2cid = 17904052}} and can be interfaced with several MD programs (AMBER, GROMACS, LAMMPS, NAMD, Quantum ESPRESSO, DL_POLY_4, CP2K, and OpenMM).{{Cite web|title = MD engines - PLUMED|url = http://www.plumed.org/md-engines|website = www.plumed.org|access-date = 2016-01-26|url-status = dead|archiveurl = https://web.archive.org/web/20160207130659/http://www.plumed.org/md-engines|archivedate = 2016-02-07}}{{Cite web|title = howto:install_with_plumed [CP2K Open Source Molecular Dynamics ]|url = https://www.cp2k.org/howto:install_with_plumed|website = www.cp2k.org|access-date = 2016-01-26}}

Other MTD implementations exist in the [https://colvars.github.io/ Collective Variables Module] {{cite journal |last1=Fiorin |first1=Giacomo |last2=Klein |first2=Michael L. |last3=Hénin |first3=Jérôme |title=Using collective variables to drive molecular dynamics simulations |journal=Molecular Physics |date=December 2013 |volume=111 |issue=22–23 |pages=3345–3362 |doi=10.1080/00268976.2013.813594 |bibcode=2013MolPh.111.3345F |language=en |issn=0026-8976|doi-access=free }} (for LAMMPS, NAMD, and GROMACS), ORAC, CP2K,{{Cite web | url=http://manual.cp2k.org/trunk/CP2K_INPUT/MOTION/FREE_ENERGY/METADYN.html | title=Cp2K_Input / Motion / Free_Energy / Metadyn}} EDM,https://github.com/whitead/electronic-dance-music Plugin for LAMMPS and Desmond.

• [http://people.sissa.it/~laio/Research/Res_metadynamics.php Introduction to metadynamics]
• [http://www.plumed.org PLUMED]
• [https://colvars.github.io Colvars module website (NAMD, LAMMPS, Gromacs)]
• [https://www.youtube.com/watch?v=IzEBpQ0c8TA Visual movie of metadynamics]
• [https://parrinello.ethz.ch/research/opes.html On-the-fly Probability Enhanced Sampling (OPES)]

• Local elevation
• Parallel tempering
• Umbrella sampling

{{reflist|2}}

Category:Molecular dynamics

Category:Computational chemistry

Category:Theoretical chemistry