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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]


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]


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]


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


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.

External links[edit]

See also[edit]


  1. ^ a b c 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. arXiv:cond-mat/0208352. Bibcode:2002PNAS...9912562L. doi:10.1073/pnas.202427399. PMC 130499. PMID 12271136.  edit
  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. Bibcode:2008JChPh.128b4101A. doi:10.1063/1.2821102. 
  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. Bibcode:2009MolPh.107.1243B. doi:10.1080/00268970902852608. 
  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. ^ Dickson, B.M. (2011). "Approaching a parameter-free metadynamics". Phys. Rev. E 84: 037701–037703. arXiv:1106.4994. Bibcode:2011PhRvE..84c7701D. doi:10.1103/PhysRevE.84.037701. 
  6. ^ Junghans, Christoph, 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.
  7. ^ Crippen, G.M.; Scheraga, H.A. (1969). Chemistry 64: 42–49.  Missing or empty |title= (help)
  8. ^ 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. 
  9. ^ Glover, F. (1989). ORSA J. Comput. 1: 190–206.  Missing or empty |title= (help)
  10. ^ 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. Bibcode:1994JCAMD...8..695H. doi:10.1007/BF00124016. PMID 7738605. 
  11. ^ Grubmüller, H. (1995). "Predicting slow structural transitions in macromolecular systems: Conformational flooding". Phys. Rev. E 52 (3): 2893–2906. Bibcode:1995PhRvE..52.2893G. doi:10.1103/PhysRevE.52.2893. 
  12. ^ Engkvist, O.; Karlström, G. (1996). "A method to calculate the probability distribution for systems with large energy barriers". Chem. Phys. 213: 63–76. Bibcode:1996CP....213...63E. doi:10.1016/S0301-0104(96)00247-9. 
  13. ^ Darve, E.; Pohorille, A. (2001). "Calculating free energies using average force". J. Chem. Phys. 115 (20): 9169. Bibcode:2001JChPh.115.9169D. doi:10.1063/1.1410978. 
  14. ^
  15. ^ Barducci, A.; Bussi, G.; Parrinello, M. (2008). "Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method". Physical Review Letters 100 (2): 020603. arXiv:0803.3861. Bibcode:2008PhRvL.100b0603B. doi:10.1103/PhysRevLett.100.020603. PMID 18232845.  edit
  16. ^ 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. 
  17. ^ a b 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.  edit
  18. ^ 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.  edit
  19. ^ 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.  edit
  20. ^ 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.  edit
  21. ^ 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. PMC 2566863. PMID 18801848.  edit
  22. ^ 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. arXiv:cond-mat/0411559. Bibcode:2005ZK....220..489M. doi:10.1524/zkri.220.5.489.65078.  edit
  23. ^ Bonomi, M.; Branduardi, D.; Bussi, G.; Camilloni, C.; Provasi, D.; Raiteri, P.; Donadio, D.; Marinelli, F.; Pietrucci, F.; Broglia, R. A.; Parrinello, M. (2009). "PLUMED: A portable plugin for free-energy calculations with molecular dynamics". Computer Physics Communications 180 (10): 1961. arXiv:0902.0874. Bibcode:2009CoPhC.180.1961B. doi:10.1016/j.cpc.2009.05.011.  edit.
  24. ^