Metadynamics: A Unified Framework for Accelerating Rare Events and Sampling Thermodynamics and Kinetics

Reference work entry


Metadynamics is an enhanced sampling algorithm in which the normal evolution of the system is biased by a history-dependent potential constructed as a sum of Gaussians centered along the trajectory followed by a suitably chosen set of collective variables. The sum of Gaussians forces the system to escape from local free energy minima and is used to iteratively build an estimator of the free energy. This original idea has been developed and improved over the years in several variants, which nowadays allow addressing in a unified framework some of the most important tasks of molecular simulations: computing the free energy as a function of the collective variables, accelerating rare events, and estimating unbiased kinetic rate constants. This chapter provides a survey of the many formulations of metadynamics with an emphasis on the underlying theoretical concepts and some hints on the appropriate manner of using this approach for solving complicated real-world problems.


  1. Baftizadeh F, Cossio P, Pietrucci F, Laio A (2012) Protein folding and ligand-enzyme binding from bias-exchange meta-dynamics simulations. Curr Phys Chem 2:79–91CrossRefGoogle Scholar
  2. Barducci A, Bussi G, Parrinello M (2008) Well-tempered metadynamics: a smoothly converging and tunable free-energy method. Phys Rev Lett 1(2):020603. CrossRefGoogle Scholar
  3. Barducci A, Bonomi M, Parrinello M (2011) Metadynamics. Wiley Interdiscip Rev Comput Mol Sci 1(5):826–843CrossRefGoogle Scholar
  4. Behler J, Martonak R, Donadio D, Parrinello M (2008) Metadynamics simulations of the high-pressure phases of silicon employing a high-dimensional neural network potential. Phys Rev Lett 100(18):185501. ADSCrossRefGoogle Scholar
  5. Biarnés X, Pietrucci F, Marinelli F, Laio A (2012) METAGUI. A VMD interface for analyzing metadynamics and molecular dynamics simulations. Comput Phys Commun 183(1):203–211. ADSCrossRefGoogle Scholar
  6. Bonomi M, Barducci A, Parrinello M (2009a) Reconstructing the equilibrium Boltzmann distribution from well-tempered metadynamics. J Comput Chem 30(11):1615–1621CrossRefGoogle Scholar
  7. Bonomi M, Branduardi D, Bussi G, Camilloni C, Provasi D, Raiteri P, Donadio D, Marinelli F, Pietrucci F, Broglia RA et al (2009b) Plumed: a portable plugin for free-energy calculations with molecular dynamics. Comput Phys Commun 180(10):1961–1972ADSCrossRefGoogle Scholar
  8. Boyer MJ, Vilčiauskas L, Hwang GS (2016) Structure and li+ ion transport in a mixed carbonate/lipf 6 electrolyte near graphite electrode surfaces: a molecular dynamics study. Phys Chem Chem Phys 18(40):27868–27876CrossRefGoogle Scholar
  9. Branduardi D, Bussi G, Parrinello M (2012) Metadynamics with adaptive gaussians. J Chem Theory Comput 8(7):2247–2254CrossRefGoogle Scholar
  10. Bui T, Phan A, Cole DR, Striolo A (2017) Transport mechanism of guest methane in water-filled nanopores. J Phys Chem C 121(29):15675–15686CrossRefGoogle Scholar
  11. Bussi G, Gervasio FL, Laio A, Parrinello M (2006a) Free-energy landscape for beta hairpin folding from combined parallel tempering and metadynamics. J Am Chem Soc 128(41):13435–13441. CrossRefGoogle Scholar
  12. Bussi G, Laio A, Parrinello M (2006b) Equilibrium free energies from nonequilibrium metadynamics. Phys Rev Lett 96(9):090601. ADSCrossRefGoogle Scholar
  13. Bussi G, Branduardi D et al (2015) Free-energy calculations with metadynamics: theory and practice. Rev Comput Chem 28:1–49Google Scholar
  14. Camilloni C, Provasi D, Tiana G, Broglia RA (2008) Exploring the protein G helix free-energy surface by solute tempering metadynamics. Proteins Struct Funct Bioinf 71(4):1647–1654. CrossRefGoogle Scholar
  15. Car R, Parrinello M (1985) Unified approach for molecular-dynamics and density-functional theory. Phys Rev Lett 45:2471ADSCrossRefGoogle Scholar
  16. Carter EA, Ciccotti G, Hynes JT, Kapral R (1989) Constrained reaction coordinate dynamics for the simulation of rare events. Chem Phys Lett 156:472–477ADSCrossRefGoogle Scholar
  17. Ceriani C, Laio A, Fois E, Gamba A, Martonak R, Parrinello M (2004) Molecular dynamics simulation of reconstructive phase transitions on an anhydrous zeolite. Phys Rev B 70(11):113403. ADSCrossRefGoogle Scholar
  18. Cheng T, Goddard WA, An Q, Xiao H, Merinov B, Morozov S (2017) Mechanism and kinetics of the electrocatalytic reaction responsible for the high cost of hydrogen fuel cells. Phys Chem Chem Phys 19(4):2666–2673CrossRefGoogle Scholar
  19. Crespo Y, Marinelli F, Pietrucci F, Laio A (2010) Metadynamics convergence law in a multidimensional system. Phys Rev E 81:055701. ADSCrossRefGoogle Scholar
  20. Cunha RA, Bussi G (2017) Unraveling Mg2+–Rna binding with atomistic molecular dynamics. RNA 23(5):628–638CrossRefGoogle Scholar
  21. Cvijovic D, Klinowski J (1995) Taboo search – an approach to the multiple minima problem. Science 267:664–666ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. Dama JF, Parrinello M, Voth GA (2014) Well-tempered metadynamics converges asymptotically. Phys Rev Lett 112(24):240602ADSCrossRefGoogle Scholar
  23. Dellago C, Bolhuis P, Csajka FS, Chandler D (1998) Transition path sampling and the calculation of rate constants. J Chem Phys 108:1964–1977ADSCrossRefGoogle Scholar
  24. Dellago C, Bolhuis P, Geissler P (2002) Transition path sampling. Adv Chem Phys 123:1–78Google Scholar
  25. Di Pietro E, Pagliai M, Cardini G, Schettino V (2006) Solid-state phase transition induced by pressure in LiOH center dot H2O. J Phys Chem B 110(27):13539–13546. CrossRefGoogle Scholar
  26. Donadio D, Bernasconi M (2005) Ab initio simulation of photoinduced transformation of small rings in amorphous silica. Phys Rev B 71(7):073307. ADSCrossRefGoogle Scholar
  27. Donadio D, Raiteri P, Parrinello M (2005) Topological defects and bulk melting of hexagonal ice. J Phys Chem B 109:5421–5424CrossRefGoogle Scholar
  28. Ensing B, De Vivo M, Liu Z, Moore P, Klein ML (2006) Metadynamics as a tool for exploring free energy landscapes of chemical reactions. Acc Chem Res 39(2):73–81CrossRefGoogle Scholar
  29. Fiorin G, Klein ML, Hénin J (2013) Using collective variables to drive molecular dynamics simulations. Mol Phys 111(22–23):3345–3362ADSCrossRefGoogle Scholar
  30. Fitzner M, Sosso GC, Pietrucci F, Pipolo S, Michaelides A (2017) Pre-critical fluctuations and what they disclose about heterogeneous crystal nucleation. Nat Commun 8(1):2257ADSCrossRefGoogle Scholar
  31. Fleming KL, Tiwary P, Pfaendtner J (2016) New approach for investigating reaction dynamics and rates with ab initio calculations. J Phys Chem A 120(2):299–305CrossRefGoogle Scholar
  32. Fu CD, Pfaendtner J (2018) Lifting the curse of dimensionality on enhanced sampling of reaction networks with parallel bias metadynamics. J Chem Theory Comput 14:2516–2525CrossRefGoogle Scholar
  33. Gil-Ley A, Bussi G (2015) Enhanced conformational sampling using replica exchange with collective-variable tempering. J Chem Theory Comput 11(3):1077–1085CrossRefGoogle Scholar
  34. Gil-Ley A, Bottaro S, Bussi G (2016) Empirical corrections to the amber RNA force field with target metadynamics. J Chem Theory Comput 12(6):2790–2798CrossRefGoogle Scholar
  35. Giorgino T, Laio A, Rodriguez A (2017) METAGUI 3: a graphical user interface for choosing the collective variables in molecular dynamics simulations. Comput Phys Commun 217:204–209ADSCrossRefGoogle Scholar
  36. Grubmüller H (1995) Predicting slow structural transitions in macromolecular systems: conformational flooding. Phys Rev E 52(3):2893ADSCrossRefGoogle Scholar
  37. Hasell T, Miklitz M, Stephenson A, Little MA, Chong SY, Clowes R, Chen L, Holden D, Tribello GA, Jelfs KE et al (2016) Porous organic cages for sulfur hexafluoride separation. J Am Chem Soc 138(5):1653–1659CrossRefGoogle Scholar
  38. Henin J, Fiorin G, Chipot C, Klein ML (2009) Exploring multidimensional free energy landscapes using time-dependent biases on collective variables. J Chem Theory Comput 6(1):35–47CrossRefGoogle Scholar
  39. Hosek P, Toulcova D, Bortolato A, Spiwok V (2016) Altruistic metadynamics: multisystem biased simulation. J Phys Chem B 120(9):2209–2215CrossRefGoogle Scholar
  40. Hu XL, Piccinin S, Laio A, Fabris S (2012) Atomistic structure of cobalt-phosphate nanoparticles for catalytic water oxidation. ACS Nano 6(12):10497CrossRefGoogle Scholar
  41. Huber T, Torda A, van Gunsteren W (1994) Local elevation: a method for improving the searching properties of molecular dynamics simulation. J Comput Aided Mol Des 8:695–708ADSCrossRefGoogle Scholar
  42. Iannuzzi M, Parrinello M (2004) Proton transfer in heterocycle crystals. Phys Rev Lett 93: 025901ADSCrossRefGoogle Scholar
  43. Iannuzzi M, Laio A, Parrinello M (2003) Efficient exploration of reactive potential energy surfaces using Car-Parrinello molecular dynamics. Phys Rev Lett 90:238302ADSCrossRefGoogle Scholar
  44. Karamertzanis PG, Raiteri P, Parrinello M, Leslie M, Price SL (2008) The thermal stability of lattice-energy minima of 5-fluorouracil: metadynamics as an aid to polymorph prediction. J Phys Chem B 112(14):4298–4308. CrossRefGoogle Scholar
  45. Kevrekidis IG, Gear CW, Hummer G (2004) Equation-free: the computer-aided analysis of comptex multiscale systems. AIChE J 50(7):1346–1355CrossRefGoogle Scholar
  46. Kumar S, Rosenberg JM, Bouzida D, Swendsen RH, Kollman PA (1995) Multidimensional free-energy calculations using the weighted histogram analysis method. J Comput Chem 16:1339–1350CrossRefGoogle Scholar
  47. Laino T, Donadio D, Kuo IFW (2007) Migration of positively charged defects in alpha-quartz. Phys Rev B 76(19):195210. ADSCrossRefGoogle Scholar
  48. Laio A, Gervasio FL (2008) Metadynamics: a method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science. Rep Prog Phys 71(12):126601ADSCrossRefGoogle Scholar
  49. Laio A, Parrinello M (2002) Escaping free energy minima. Proc Natl Acad Sci USA 99:12562–12566ADSCrossRefGoogle Scholar
  50. Laio A, Rodriguez-Fortea A, Gervasio FL, Ceccarelli M, Parrinello M (2005) Assessing the accuracy of metadynamics. J Phys Chem B 109:6714–6721CrossRefGoogle Scholar
  51. Maragliano L, Vanden-Eijnden E (2006) A temperature accelerated method for sampling free energy and determining reaction pathways in rare events simulations. Chem Phys Lett 426(1):168–175ADSCrossRefGoogle Scholar
  52. Marinari E, Parisi G (1992) Simulated tempering: a new Monte Carlo scheme. EPL (Europhys Lett) 19(6):451ADSCrossRefGoogle Scholar
  53. Marinelli F, Faraldo-Gómez JD (2015) Ensemble-biased metadynamics: a molecular simulation method to sample experimental distributions. Biophys J 108(12):2779–2782CrossRefGoogle Scholar
  54. Marinelli F, Pietrucci F, Laio A, Piana S (2009) A kinetic model of trp-cage folding from multiple biased molecular dynamics simulations. PLOS Comput Biol 5:1–18. MathSciNetCrossRefGoogle Scholar
  55. Martoňák R, Laio A, Parrinello M (2003) Predicting crystal structures: the Parrinello-Rahman method revisited. Phys Rev Lett 90:75503ADSCrossRefGoogle Scholar
  56. Martoňák R, Laio A, Bernasconi M, Ceriani C, Raiteri P, Parrinello M (2005) Simulation of structural phase transitions by metadynamics. Z Krist 220:489–498Google Scholar
  57. Martoňák R, Donadio D, Oganov AR, Parrinello M (2006) Crystal structure transformations in SiO2 from classical and ab initio metadynamics. Nat Mater 5(8):623–626. ADSCrossRefGoogle Scholar
  58. Martoňák R, Donadio D, Oganov AR, Parrinello M (2007) From four- to six-coordinated silica: transformation pathways from metadynamics. Phys Rev B 76(1):014120. ADSCrossRefGoogle Scholar
  59. McCarty J, Parrinello M (2017) A variational conformational dynamics approach to the selection of collective variables in metadynamics. J Chem Phys 147(20):204109. ADSCrossRefGoogle Scholar
  60. Mendels D, McCarty J, Piaggi PM, Parrinello M (2018) Searching for entropically stabilized phases: the case of silver iodide. J Phys Chem C 122(3):1786–1790CrossRefGoogle Scholar
  61. Merlitz H, Wenzel W (2002) Comparison of stochastic optimization methods for receptor-ligand docking. Chem Phys Lett 362:271–277ADSCrossRefGoogle Scholar
  62. Michael M, de Pablo J (2013) A boundary correction algorithm for metadynamics in multiple dimensions. J Chem Phys 139:084102ADSCrossRefGoogle Scholar
  63. Micheletti C, Laio A, Parrinello M (2004) Reconstructing the density of states by history-dependent metadynamics. Phys Rev Lett 92:170601ADSCrossRefGoogle Scholar
  64. Munro CJ, Hughes ZE, Walsh TR, Knecht MR (2016) Peptide sequence effects control the single pot reduction, nucleation, and growth of au nanoparticles. J Phys Chem C 120(33): 18917–18924CrossRefGoogle Scholar
  65. Oganov A, Martonak R, Laio A, Raiteri P, Parrinello M (2005) Anisotropy of Earth’s D‘’ layer and stacking faults in the MgSiO3 post-perovskite phase. Nature 438(7071):1142–1144. ADSCrossRefGoogle Scholar
  66. Oliveira LF, Fu CD, Pfaendtner J (2018) Density functional tight-binding and infrequent metadynamics can capture entropic effects in intramolecular hydrogen transfer reactions. J Chem Phys 148(15):154101ADSCrossRefGoogle Scholar
  67. Palafox-Hernandez JP, Tang Z, Hughes ZE, Li Y, Swihart MT, Prasad PN, Walsh TR, Knecht MR (2014) Comparative study of materials-binding peptide interactions with gold and silver surfaces and nanostructures: a thermodynamic basis for biological selectivity of inorganic materials. Chem Mater 26(17):4960–4969CrossRefGoogle Scholar
  68. Park S, Pande VS (2007) Choosing weights for simulated tempering. Phys Rev E 76(1):016703ADSCrossRefGoogle Scholar
  69. Patey GN, Valleau JP (1975) Monte-Carlo method for obtaining interionic potential of mean force in ionic solution. J Chem Phys 63:2334–2339ADSCrossRefGoogle Scholar
  70. Pfaendtner J, Bonomi M (2015) Efficient sampling of high-dimensional free-energy landscapes with parallel bias metadynamics. J Chem Theory Comput 11(11):5062–5067CrossRefGoogle Scholar
  71. Piaggi PM, Valsson O, Parrinello M (2017) Enhancing entropy and enthalpy fluctuations to drive crystallization in atomistic simulations. Phys Rev Lett 119(1):015701ADSCrossRefGoogle Scholar
  72. Piana S, Laio A (2007) A bias-exchange approach to protein folding. J Phys Chem B 111(17):4553–4559. CrossRefGoogle Scholar
  73. Pietrucci F, Gerra G, Andreoni W (2010) CdTe surfaces: characterizing dynamical processes with first-principles metadynamics. Appl Phys Lett 97(14):141914ADSCrossRefGoogle Scholar
  74. Pitera JW, Chodera JD (2012) On the use of experimental observations to bias simulated ensembles. J Chem Theory Comput 8(10):3445–3451CrossRefGoogle Scholar
  75. Quigley D, Rodger PM (2008) Metadynamics simulations of ice nucleation and growth. J Comput Phys 128(15):154518. Google Scholar
  76. Raiteri P, Laio A, Gervasio FL, Micheletti C, Parrinello M (2006) Efficient reconstruction of complex free energy landscapes by multiple walkers metadynamics. J Phys Chem B 110:3533–3539CrossRefGoogle Scholar
  77. Risken H (1989) The Fokker-Planck equation. SpringerzbMATHCrossRefGoogle Scholar
  78. Rohrdanz MA, Zheng W, Clementi C (2013) Discovering mountain passes via torchlight: methods for the definition of reaction coordinates and pathways in complex macromolecular reactions. Ann Rev Phys Chem 64:295–316ADSCrossRefGoogle Scholar
  79. Rosso L, Mináry P, Zhu Z, Tuckerman ME (2002) On the use of the adiabatic molecular dynamics technique in the calculation of free energy profiles. J Comput Phys 116(11):4389–4402Google Scholar
  80. Salvalaglio M, Tiwary P, Parrinello M (2014) Assessing the reliability of the dynamics reconstructed from metadynamics. J Chem Theor Comput 10(4):1420–1425. CrossRefGoogle Scholar
  81. Salvalaglio M, Perego C, Giberti F, Mazzotti M, Parrinello M (2015) Molecular-dynamics simulations of urea nucleation from aqueous solution. Proc Natl Acad Sci 112(1):E6–E14ADSCrossRefGoogle Scholar
  82. Salvalaglio M, Tiwary P, Maggioni GM, Mazzotti M, Parrinello M (2016) Overcoming time scale and finite size limitations to compute nucleation rates from small scale well tempered metadynamics simulations. J Chem Phys 145(21):211925ADSCrossRefGoogle Scholar
  83. Sidky H et al, (2018) SSAGES: Software Suite for Advanced General Ensemble Simulations. J Chem Phys 148:044104 ADSCrossRefGoogle Scholar
  84. Sultan MM, Pande VS (2017) Tica-metadynamics: accelerating metadynamics by using kinetically selected collective variables. J Chem Theory Comput 13(6):2440–2447., pMID:28383914CrossRefGoogle Scholar
  85. Theodoropoulos C, Qian Y, Kevrekidis IG (2000) Coarse stability and bifurcation analysis using time-steppers: a reaction-diffusion example. Proc Natl Acad Sci USA 97:9840–9843ADSzbMATHCrossRefGoogle Scholar
  86. Tiwary P (2017) Molecular determinants and bottlenecks in the dissociation dynamics of biotin-streptavidin. J Phys Chem B 121(48):10841–10849. CrossRefGoogle Scholar
  87. Tiwary P, Berne B (2016a) How wet should be the reaction coordinate for ligand unbinding? J Chem Phys 145(5):054113ADSCrossRefGoogle Scholar
  88. Tiwary P, Berne BJ (2016b) Kramers turnover: from energy diffusion to spatial diffusion using metadynamics. J Chem Phys 144(13):134103–134106ADSCrossRefGoogle Scholar
  89. Tiwary P, Berne BJ (2016c) Spectral gap optimization of order parameters for sampling complex molecular systems. Proc Natl Acad Sci 113(11):2839–2844. ADSCrossRefGoogle Scholar
  90. Tiwary P, Berne BJ (2017) Predicting reaction coordinates in energy landscapes with diffusion anisotropy. J Chem Phys 147(15):152701ADSCrossRefGoogle Scholar
  91. Tiwary P, Parrinello M (2013) From metadynamics to dynamics. Phys Rev Lett 111:230602–230606. ADSCrossRefGoogle Scholar
  92. Tiwary P, Parrinello M (2014) A time-independent free energy estimator for metadynamics. J Phys Chem B 119(3):736–742CrossRefGoogle Scholar
  93. Tiwary P, van de Walle A (2016) A review of enhanced sampling approaches for accelerated molecular dynamics. In: Multiscale materials modeling for nanomechanics. Springer, New York pp 195–221. Google Scholar
  94. Tiwary P, Dama JF, Parrinello M (2015a) A perturbative solution to metadynamics ordinary differential equation. J Chem Phys 143(23):234112ADSCrossRefGoogle Scholar
  95. Tiwary P, Mondal J, Morrone JA, Berne B (2015b) Role of water and steric constraints in the kinetics of cavity–ligand unbinding. Proc Natl Acad Sci 112(39):12015–12019ADSCrossRefGoogle Scholar
  96. Tribello GA, Ceriotti M, Parrinello M (2012) Using sketch-map coordinates to analyze and bias molecular dynamics simulations. Proc Natl Acad Sci 109(14):5196–5201ADSCrossRefGoogle Scholar
  97. Tribello GA, Bonomi M, Branduardi D, Camilloni C, Bussi G (2014) Plumed 2: new feathers for an old bird. Comput Phys Commun 185(2):604–613ADSCrossRefGoogle Scholar
  98. Trudu F, Donadio D, Parrinello M (2006) Freezing of a Lennard-Jones fluid: from nucleation to spinodal regime. Phys Rev Lett 97(10):105701. ADSCrossRefGoogle Scholar
  99. Valsson O, Tiwary P, Parrinello M (2016) Enhancing important fluctuations: rare events and metadynamics from a conceptual viewpoint. Ann Rev Phys Chem 67(1):159–184ADSCrossRefGoogle Scholar
  100. VandeVondele J, Rothlisberger U (2002) Canonical adiabatic free energy sampling (cafes): a novel method for the exploration of free energy surfaces. J Phys Chem B 106(1):203–208CrossRefGoogle Scholar
  101. Vartak S, Roudgar A, Golovnev A, Eikerling M (2013) Collective proton dynamics at highly charged interfaces studied by ab initio metadynamics. J Phys Chem B 117(2):583–588CrossRefGoogle Scholar
  102. Voter AF (1997) Hyperdynamics: accelerated molecular dynamics of infrequent events. Phys Rev Lett 78:3908–3911. ADSCrossRefGoogle Scholar
  103. Wang F, Landau DP (2001) Efficient, multiple-range random walk algorithm to calculate the density of states. Phys Rev Lett 86:2050ADSCrossRefGoogle Scholar
  104. White AD, Dama JF, Voth GA (2015) Designing free energy surfaces that match experimental data with metadynamics. J Chem Theory Comput 11(6):2451–2460CrossRefGoogle Scholar
  105. Zipoli F, Bernasconi M, Martoňák R (2004) Constant pressure reactive molecular dynamics simulations of phase transitions under pressure: the graphite to diamond conversion revisited. Eur Phys J B 39:41–47ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.International Centre for Theoretical Physics (ICTP)TriesteItaly
  3. 3.Department of Chemistry and Biochemistry and Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA

Personalised recommendations