Physics:Metadynamics

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Short description: Scientific computer simulation method

File:MetaD.webm 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[1] and is usually applied within molecular dynamics simulations. MTD closely resembles a number of newer methods such as adaptively biased molecular dynamics,[2] adaptive reaction coordinate forces[3] and local elevation umbrella sampling.[4] More recently, both the original and well-tempered metadynamics[5] were derived in the context of importance sampling and shown to be a special case of the adaptive biasing potential setting.[6] MTD is related to the Wang–Landau sampling.[7]

Introduction

The technique builds on a large number of related methods including (in a chronological order) the deflation,[8] tunneling,[9] tabu search,[10] local elevation,[11] conformational flooding,[12] Engkvist-Karlström[13] and Adaptive Biasing Force methods.[14]

Metadynamics has been informally described as "filling the free energy wells with computational sand".[15] 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.[1] Usually, the well-tempered metadynamics[5] is used to change the Gaussian size adaptively. Also, the Gaussian width can be adapted with the adaptive Gaussian metadynamics.[16]

Metadynamics has the advantage, upon methods like adaptive umbrella sampling, of not requiring an initial estimate of the energy landscape to explore.[1] 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,[17] Sketch-Map,[18] and non-linear data-driven collective variables.[19]

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,[20] the parallel tempering MTD,[21] the bias-exchange MTD,[22] and the collective-variable tempering MTD.[23] 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[24] and Suwa-Todo[25] algorithms give better replica exchange rates.[26]

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.[27] The required memory to store the grid increases exponentially with the number of dimensions (CVs) too.

A high-dimensional generalization of metadynamics is NN2B.[28] 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.[28][29]

An alternative method, exploiting ANN for the adaptive bias potential, uses mean potential forces for the estimation.[30] This method is also a high-dimensional generalization of the Adaptive Biasing Force (ABF) method.[31] Additionally, the training of ANN is improved using Bayesian regularization,[32] and the error of approximation can be inferred by training an ensemble of ANNs.[30]

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.[33]

In 2020, an evolution of metadynamics was proposed, the on-the-fly probability enhanced sampling method (OPES),[34][35][36] which is now the method of choice of Michele Parrinello's research group.[37] The OPES method has only a few robust parameters, converges faster than metadynamics, and has a straightforward reweighting scheme.[38] OPES has been implemented in the PLUMED library since version 2.7.[39]

Algorithm

Assume we have a classical [math]\displaystyle{ N }[/math]-particle system with positions at [math]\displaystyle{ \{ \vec r_i \} }[/math] [math]\displaystyle{ (i \in 1...N) }[/math] in the Cartesian coordinates [math]\displaystyle{ (\vec r_i \in \mathbb{R}^3) }[/math]. The particle interaction are described with a potential function [math]\displaystyle{ V \equiv V(\{ \vec r_i \}) }[/math]. 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.

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 [math]\displaystyle{ H }[/math] with a bias potential [math]\displaystyle{ V_\text{bias} }[/math]:

[math]\displaystyle{ H = T + V + V_\text{bias} }[/math].

The bias potential is a function of collective variables [math]\displaystyle{ (V_\text{bias} \equiv V_\text{bias}(\vec s\,)) }[/math]. A collective variable is a function of the particle positions [math]\displaystyle{ (\vec s \equiv \vec s(\{ \vec r_i\})) }[/math]. The bias potential is continuously updated by adding bias at rate [math]\displaystyle{ \omega }[/math], where [math]\displaystyle{ \vec s_t }[/math] is an instantaneous collective variable value at time [math]\displaystyle{ t }[/math]:

[math]\displaystyle{ \frac{\partial V_\text{bias}(\vec s\,)}{\partial t} = \omega\, \delta(|\vec s - \vec s_t|) }[/math].

At infinitely long simulation time [math]\displaystyle{ t_\text{sim} }[/math], the accumulated bias potential converges to free energy with opposite sign (and irrelevant constant [math]\displaystyle{ C }[/math]):

[math]\displaystyle{ V_\text{bias}(\vec s\,) = \!\!\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 }[/math]

For a computationally efficient implementation, the update process is discretised into [math]\displaystyle{ \tau }[/math] time intervals ([math]\displaystyle{ \lfloor\;\rfloor }[/math] denotes the floor function) and [math]\displaystyle{ \delta }[/math]-function is replaced with a localized positive kernel function [math]\displaystyle{ K }[/math]. The bias potential becomes a sum of the kernel functions centred at the instantaneous collective variable values [math]\displaystyle{ \vec s_j }[/math] at time [math]\displaystyle{ \tau j }[/math]:

[math]\displaystyle{ V_\text{bias}(\vec s\,) \approx \tau \!\!\!\sum_{j=0}^{\left\lfloor \frac{t_\text{sim}}{\tau} \right\rfloor}\!\! \omega\, K(|\vec s - \vec s_j|) }[/math].

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

[math]\displaystyle{ V_\text{bias}(\vec s\,) \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) }[/math].

The parameter [math]\displaystyle{ \tau }[/math], [math]\displaystyle{ \omega }[/math], and [math]\displaystyle{ \vec \sigma }[/math] are determined a priori and kept constant during the simulation.

Implementation

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

set initial [math]\displaystyle{ \{\vec r\} }[/math] and [math]\displaystyle{ \{\vec v\} }[/math] 
set [math]\displaystyle{ V_\text{bias}(\vec s\,) := 0 }[/math]

every MD step:
    compute CV values:
        [math]\displaystyle{ \vec s_t := \vec s(\{\vec r\}) }[/math]
    
    every [math]\displaystyle{ n }[/math] MD steps:
        update bias potential:
            [math]\displaystyle{ V_\text{bias}(\vec s\,) := V_\text{bias}(\vec s\,)
             + \tau \omega \exp\!\! \left(\! -\frac{1}{2} \left| \frac{\vec s - \vec s_t}{\vec \sigma} \right|^2 \right) }[/math]
    
    compute atomic forces:
        [math]\displaystyle{ \vec F_i := -\frac{\partial V(\{\vec r\,\})}{\partial \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}} }[/math]
    
    propagate [math]\displaystyle{ \{\vec r\} }[/math] and [math]\displaystyle{ \{\vec v\} }[/math] by [math]\displaystyle{ \Delta t }[/math]

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 [math]\displaystyle{ t_\text{diff} }[/math], when the motion along the collective variable becomes diffusive:

[math]\displaystyle{ \bar F(\vec s) = - \frac{1}{t_\text{sim} - t_\text{diff}} \int^{t_\text{sim}}_{t_\text{diff}} \!\!\!\!\!V_\text{bias}(\vec s, t)\, dt + C }[/math]

Applications

Metadynamics has been used to study:

Implementations

PLUMED

PLUMED[46] is an open-source library implementing many MTD algorithms and collective variables. It has a flexible object-oriented design[47][48] and can be interfaced with several MD programs (AMBER, GROMACS, LAMMPS, NAMD, Quantum ESPRESSO, DL_POLY_4, CP2K, and OpenMM).[49][50]

Other

Other MTD implementations exist in the Collective Variables Module [51] (for LAMMPS, NAMD, and GROMACS), ORAC, CP2K,[52] EDM,[53] and Desmond.

External links

See also

References

  1. ↑ 1.0 1.1 1.2 Laio, A.; Parrinello, M. (2002). "Escaping free-energy minima". Proceedings of the National Academy of Sciences of the United States of America 99 (20): 12562–12566. doi:10.1073/pnas.202427399. PMID 12271136. Bibcode: 2002PNAS...9912562L. 
  2. ↑ Babin, V.; Roland, C.; Sagui, C. (2008). "Stabilization of resonance states by an asymptotic Coulomb potential". J. Chem. Phys. 128 (2): 134101/1–134101/7. doi:10.1063/1.2821102. PMID 18205437. Bibcode: 2008JChPh.128b4101A. 
  3. ↑ Barnett, C.B.; Naidoo, K.J. (2009). "Free Energies from Adaptive Reaction Coordinate Forces (FEARCF): An application to ring puckering". Mol. Phys. 107 (8): 1243–1250. doi:10.1080/00268970902852608. Bibcode: 2009MolPh.107.1243B. https://zenodo.org/record/966244. 
  4. ↑ Hansen, H.S.; Hünenberger, P.H. (2010). "Using the local elevation method to construct optimized umbrella sampling potentials: Calculation of the relative free energies and interconversion barriers of glucopyranose ring conformers in water". J. Comput. Chem. 31 (1): 1–23. doi:10.1002/jcc.21253. PMID 19412904. 
  5. ↑ 5.0 5.1 Barducci, A.; Bussi, G.; Parrinello, M. (2008). "Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method". Physical Review Letters 100 (2): 020603. doi:10.1103/PhysRevLett.100.020603. PMID 18232845. Bibcode: 2008PhRvL.100b0603B. 
  6. ↑ Dickson, B.M. (2011). "Approaching a parameter-free metadynamics". Phys. Rev. E 84 (3): 037701–037703. doi:10.1103/PhysRevE.84.037701. PMID 22060542. Bibcode: 2011PhRvE..84c7701D. 
  7. ↑ Christoph Junghans, Danny Perez, and Thomas Vogel. "Molecular Dynamics in the Multicanonical Ensemble: Equivalence of Wang–Landau Sampling, Statistical Temperature Molecular Dynamics, and Metadynamics." Journal of Chemical Theory and Computation 10.5 (2014): 1843-1847. doi:10.1021/ct500077d
  8. ↑ Crippen, Gordon M.; Scheraga, Harold A. (1969). "Minimization of polypeptide energy. 8. Application of the deflation technique to a dipeptide". Proceedings of the National Academy of Sciences 64 (1): 42–49. doi:10.1073/pnas.64.1.42. PMID 5263023. Bibcode: 1969PNAS...64...42C. 
  9. ↑ Levy, A.V.; Montalvo, A. (1985). "The Tunneling Algorithm for the Global Minimization of Functions". SIAM J. Sci. Stat. Comput. 6: 15–29. doi:10.1137/0906002. 
  10. ↑ Glover, Fred (1989). "Tabu Search—Part I". ORSA Journal on Computing 1 (3): 190–206. doi:10.1287/ijoc.1.3.190. 
  11. ↑ Huber, T.; Torda, A.E.; van Gunsteren, W.F. (1994). "Local elevation: A method for improving the searching properties of molecular dynamics simulation". J. Comput.-Aided Mol. Des. 8 (6): 695–708. doi:10.1007/BF00124016. PMID 7738605. Bibcode: 1994JCAMD...8..695H. 
  12. ↑ Grubmüller, H. (1995). "Predicting slow structural transitions in macromolecular systems: Conformational flooding". Phys. Rev. E 52 (3): 2893–2906. doi:10.1103/PhysRevE.52.2893. PMID 9963736. Bibcode: 1995PhRvE..52.2893G. 
  13. ↑ Engkvist, O.; Karlström, G. (1996). "A method to calculate the probability distribution for systems with large energy barriers". Chem. Phys. 213 (1): 63–76. doi:10.1016/S0301-0104(96)00247-9. Bibcode: 1996CP....213...63E. 
  14. ↑ Darve, E.; Pohorille, A. (2001). "Calculating free energies using average force". J. Chem. Phys. 115 (20): 9169. doi:10.1063/1.1410978. Bibcode: 2001JChPh.115.9169D. 
  15. ↑ http://www.grs-sim.de/cms/upload/Carloni/Presentations/Marinelli.ppt[yes|permanent dead link|dead link}}]
  16. ↑ Branduardi, Davide; Bussi, Giovanni; Parrinello, Michele (2012-06-04). "Metadynamics with Adaptive Gaussians" (in EN). Journal of Chemical Theory and Computation 8 (7): 2247–2254. doi:10.1021/ct3002464. PMID 26588957. 
  17. ↑ Spiwok, V.; Lipovová, P.; Králová, B. (2007). "Metadynamics in essential coordinates: free energy simulation of conformational changes". The Journal of Physical Chemistry B 111 (12): 3073–3076. doi:10.1021/jp068587c. PMID 17388445. 
  18. ↑ Ceriotti, Michele; Tribello, Gareth A.; Parrinello, Michele (2013-02-22). "Demonstrating the Transferability and the Descriptive Power of Sketch-Map" (in EN). Journal of Chemical Theory and Computation 9 (3): 1521–1532. doi:10.1021/ct3010563. PMID 26587614. https://pure.qub.ac.uk/portal/en/publications/demonstrating-the-transferability-and-the-descriptive-power-of-sketchmap(ea20f98a-bb39-48a0-ac61-fe8bfe7299e2).html. 
  19. ↑ Hashemian, Behrooz; Millán, Daniel; Arroyo, Marino (2013-12-07). "Modeling and enhanced sampling of molecular systems with smooth and nonlinear data-driven collective variables". The Journal of Chemical Physics 139 (21): 214101. doi:10.1063/1.4830403. ISSN 0021-9606. PMID 24320358. Bibcode: 2013JChPh.139u4101H. 
  20. ↑ Raiteri, Paolo; Laio, Alessandro; Gervasio, Francesco Luigi; Micheletti, Cristian; Parrinello, Michele (2005-10-28). "Efficient Reconstruction of Complex Free Energy Landscapes by Multiple Walkers Metadynamics †" (in en). The Journal of Physical Chemistry B 110 (8): 3533–3539. doi:10.1021/jp054359r. PMID 16494409. 
  21. ↑ Bussi, Giovanni; Gervasio, Francesco Luigi; Laio, Alessandro; Parrinello, Michele (October 2006). "Free-Energy Landscape for β Hairpin Folding from Combined Parallel Tempering and Metadynamics". Journal of the American Chemical Society 128 (41): 13435–13441. doi:10.1021/ja062463w. PMID 17031956. 
  22. ↑ 22.0 22.1 Piana, S.; Laio, A. (2007). "A bias-exchange approach to protein folding". The Journal of Physical Chemistry B 111 (17): 4553–4559. doi:10.1021/jp067873l. PMID 17419610. 
  23. ↑ Gil-Ley, Alejandro; Bussi, Giovanni (2015-02-19). "Enhanced Conformational Sampling Using Replica Exchange with Collective-Variable Tempering" (in EN). Journal of Chemical Theory and Computation 11 (3): 1077–1085. doi:10.1021/ct5009087. PMID 25838811. 
  24. ↑ Plattner, Nuria; Doll, J. D.; Dupuis, Paul; Wang, Hui; Liu, Yufei; Gubernatis, J. E. (2011-10-07). "An infinite swapping approach to the rare-event sampling problem". The Journal of Chemical Physics 135 (13): 134111. doi:10.1063/1.3643325. ISSN 0021-9606. PMID 21992286. Bibcode: 2011JChPh.135m4111P. 
  25. ↑ Suwa, Hidemaro (2010-01-01). "Markov Chain Monte Carlo Method without Detailed Balance". Physical Review Letters 105 (12): 120603. doi:10.1103/PhysRevLett.105.120603. PMID 20867621. Bibcode: 2010PhRvL.105l0603S. 
  26. ↑ Galvelis, Raimondas; Sugita, Yuji (2015-07-15). "Replica state exchange metadynamics for improving the convergence of free energy estimates" (in en). Journal of Computational Chemistry 36 (19): 1446–1455. doi:10.1002/jcc.23945. ISSN 1096-987X. PMID 25990969. 
  27. ↑ "PLUMED: Metadynamics". https://plumed.github.io/doc-v2.4/user-doc/html/_metadyn.html. 
  28. ↑ 28.0 28.1 Galvelis, Raimondas; Sugita, Yuji (2017-06-13). "Neural Network and Nearest Neighbor Algorithms for Enhancing Sampling of Molecular Dynamics". Journal of Chemical Theory and Computation 13 (6): 2489–2500. doi:10.1021/acs.jctc.7b00188. ISSN 1549-9618. PMID 28437616. 
  29. ↑ Schneider, Elia; Dai, Luke; Topper, Robert Q.; Drechsel-Grau, Christof; Tuckerman, Mark E. (2017-10-11). "Stochastic Neural Network Approach for Learning High-Dimensional Free Energy Surfaces". Physical Review Letters 119 (15): 150601. doi:10.1103/PhysRevLett.119.150601. PMID 29077427. Bibcode: 2017PhRvL.119o0601S. 
  30. ↑ 30.0 30.1 Zhang, Linfeng; Wang, Han; E, Weinan (2017-12-09). "Reinforced dynamics for enhanced sampling in large atomic and molecular systems. I. Basic Methodology". The Journal of Chemical Physics 148 (12): 124113. doi:10.1063/1.5019675. PMID 29604808. 
  31. ↑ Comer, Jeffrey; Gumbart, James C.; Hénin, Jérôme; Lelièvre, Tony; Pohorille, Andrew; Chipot, Christophe (2015-01-22). "The Adaptive Biasing Force Method: Everything You Always Wanted To Know but Were Afraid To Ask". The Journal of Physical Chemistry B 119 (3): 1129–1151. doi:10.1021/jp506633n. ISSN 1520-6106. PMID 25247823. 
  32. ↑ Sidky, Hythem; Whitmer, Jonathan K. (2017-12-07). "Learning Free Energy Landscapes Using Artificial Neural Networks". The Journal of Chemical Physics 148 (10): 104111. doi:10.1063/1.5018708. PMID 29544298. 
  33. ↑ White, Andrew D.; Dama, James F.; Voth, Gregory A. (2015). "Designing Free Energy Surfaces That Match Experimental Data with Metadynamics". Journal of Chemical Theory and Computation 11 (6): 2451–2460. doi:10.1021/acs.jctc.5b00178. PMID 26575545. 
  34. ↑ Invernizzi, Michele; Parrinello, Michele (2020-04-02). "Rethinking Metadynamics: From Bias Potentials to Probability Distributions". The Journal of Physical Chemistry Letters 11 (7): 2731–2736. doi:10.1021/acs.jpclett.0c00497. ISSN 1948-7185. PMID 32191470. 
  35. ↑ Invernizzi, Michele; Piaggi, Pablo M.; Parrinello, Michele (2020-07-06). "Unified Approach to Enhanced Sampling". Physical Review X 10 (4): 41034. doi:10.1103/PhysRevX.10.041034. ISSN 2160-3308. Bibcode: 2020PhRvX..10d1034I. 
  36. ↑ Invernizzi, Michele; Parrinello, Michele (2022-06-14). "Exploration vs Convergence Speed in Adaptive-Bias Enhanced Sampling". Journal of Chemical Theory and Computation 18 (6): 3988–3996. doi:10.1021/acs.jctc.2c00152. ISSN 1549-9618. PMID 35617155. 
  37. ↑ Parrinello, Michele (2022-01-12). "Breviarium de Motu Simulato Ad Atomos Pertinenti". Israel Journal of Chemistry 62 (1–2): e202100105. doi:10.1002/ijch.202100105. ISSN 0021-2148. https://onlinelibrary.wiley.com/doi/10.1002/ijch.202100105. Retrieved 2022-12-06. 
  38. ↑ "On-the-fly Probability Enhanced Sampling (OPES)". https://parrinello.ethz.ch/research/opes.html. 
  39. ↑ "PLUMED - OPES". https://www.plumed.org/doc-master/user-doc/html/_o_p_e_s.html. 
  40. ↑ Ensing, B.; De Vivo, M.; Liu, Z.; Moore, P.; Klein, M. (2006). "Metadynamics as a tool for exploring free energy landscapes of chemical reactions". Accounts of Chemical Research 39 (2): 73–81. doi:10.1021/ar040198i. PMID 16489726. 
  41. ↑ Gervasio, F.; Laio, A.; Parrinello, M. (2005). "Flexible docking in solution using metadynamics". Journal of the American Chemical Society 127 (8): 2600–2607. doi:10.1021/ja0445950. PMID 15725015. 
  42. ↑ Vargiu, A. V.; Ruggerone, P.; Magistrato, A.; Carloni, P. (2008). "Dissociation of minor groove binders from DNA: insights from metadynamics simulations". Nucleic Acids Research 36 (18): 5910–5921. doi:10.1093/nar/gkn561. PMID 18801848. 
  43. ↑ Martoňák, R.; Laio, A.; Bernasconi, M.; Ceriani, C.; Raiteri, P.; Zipoli, F.; Parrinello, M. (2005). "Simulation of structural phase transitions by metadynamics". Zeitschrift für Kristallographie 220 (5–6): 489. doi:10.1524/zkri.220.5.489.65078. Bibcode: 2005ZK....220..489M. 
  44. ↑ Cruz, F.J.A.L.; de Pablo, J.J.; Mota, J.P.B. (2014), "Endohedral confinement of a DNA dodecamer onto pristine carbon nanotubes and the stability of the canonical B form", J. Chem. Phys. 140 (22): 225103, doi:10.1063/1.4881422, PMID 24929415, Bibcode: 2014JChPh.140v5103C 
  45. ↑ Cruz, F.J.A.L.; Mota, J.P.B. (2016), "Conformational Thermodynamics of DNA Strands in Hydrophilic Nanopores", J. Phys. Chem. C 120 (36): 20357–20367, doi:10.1021/acs.jpcc.6b06234 
  46. ↑ "PLUMED". http://www.plumed.org/. 
  47. ↑ Bonomi, Massimiliano; Branduardi, Davide; Bussi, Giovanni; Camilloni, Carlo; Provasi, Davide; Raiteri, Paolo; Donadio, Davide; Marinelli, Fabrizio et al. (2009-10-01). "PLUMED: A portable plugin for free-energy calculations with molecular dynamics". Computer Physics Communications 180 (10): 1961–1972. doi:10.1016/j.cpc.2009.05.011. Bibcode: 2009CoPhC.180.1961B. 
  48. ↑ Tribello, Gareth A.; Bonomi, Massimiliano; Branduardi, Davide; Camilloni, Carlo; Bussi, Giovanni (2014-02-01). "PLUMED 2: New feathers for an old bird". Computer Physics Communications 185 (2): 604–613. doi:10.1016/j.cpc.2013.09.018. Bibcode: 2014CoPhC.185..604T. 
  49. ↑ "MD engines - PLUMED". Archived from the original on 2016-02-07. https://web.archive.org/web/20160207130659/http://www.plumed.org/md-engines. 
  50. ↑ "howto:install_with_plumed [CP2K Open Source Molecular Dynamics "]. https://www.cp2k.org/howto:install_with_plumed. 
  51. ↑ Fiorin, Giacomo; Klein, Michael L.; Hénin, Jérôme (December 2013). "Using collective variables to drive molecular dynamics simulations" (in en). Molecular Physics 111 (22–23): 3345–3362. doi:10.1080/00268976.2013.813594. ISSN 0026-8976. Bibcode: 2013MolPh.111.3345F. 
  52. ↑ "Cp2K_Input / Motion / Free_Energy / Metadyn". http://manual.cp2k.org/trunk/CP2K_INPUT/MOTION/FREE_ENERGY/METADYN.html. 
  53. ↑ https://github.com/whitead/electronic-dance-music Plugin for LAMMPS



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