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Foundations of Molecular Modeling and Simulation pp 123–141Cite as

Efficient Sampling of High-Dimensional Free Energy Landscapes: A Review of Parallel Bias Metadynamics

Part of the Molecular Modeling and Simulation book series (MMAS)

Abstract

A number of enhanced sampling methods have been developed to overcome the length and time scale barriers of classical simulations. Metadynamics has made considerable strides in the last decade as a technique for constructing free energy landscapes as a function of a few low-dimensional descriptors of atomic positions, commonly referred to as collective variables (CVs). In particular, parallel bias metadynamics (PBMetaD) and its variants enable the sampling of many CVs without prohibitively increasing simulation time. This parallelizable scheme allows for its implementation in systems that necessitate the use of more than a few CVs, or in the case that the CVs corresponding to the slowest modes of a process are not easily identifiable. Here, we present a review of notable enhanced sampling schemes with a focus on the underlying theory of PBMetaD. We discuss various examples, where this method has been applied and future areas of impact.

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References

  1. Barducci A, Bonomi M, Parrinello M (2011) Metadynamics. Wiley Interdiscip Rev Comput Mol Sci 1:826–843

    CrossRef  CAS  Google Scholar 

  2. 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:e1000452

    CrossRef  PubMed  PubMed Central  CAS  Google Scholar 

  3. van Gunsteren WF et al (2006) Biomolecular modeling: goals, problems perspectives. Angew Chemie Int Ed 45:4064–4092

    CrossRef  CAS  Google Scholar 

  4. Deighan M, Pfaendtner J (2013) Exhaustively sampling peptide adsorption with metadynamics. Langmuir 29:7999–8009

    CrossRef  CAS  PubMed  Google Scholar 

  5. Abrams C, Bussi G (2014) Enhanced sampling in molecular dynamics using metadynamics, replica-exchange, and temperature-acceleration. Entropy 16:163–199

    CrossRef  CAS  Google Scholar 

  6. Bernardi RC, Melo MCR, Schulten K (2015) Enhanced sampling techniques in molecular dynamics simulations of biological systems. Biochim Biophys Acta Gen Subj 1850:872–877

    CrossRef  CAS  Google Scholar 

  7. Incerti M et al (2017) Metadynamics for perspective drug design: computationally driven synthesis of new protein-protein interaction inhibitors targeting the EphA2 receptor. J Med Chem 60:787–796

    CrossRef  CAS  PubMed  Google Scholar 

  8. Clark AJ et al (2016) Prediction of protein-ligand binding poses via a combination of induced fit docking and metadynamics simulations. J Chem Theory Comput 12:2990–2998

    CrossRef  CAS  PubMed  Google Scholar 

  9. Amaro RE et al (2018) Ensemble docking in drug discovery. Biophys J 114:2271–2278

    CrossRef  CAS  PubMed  PubMed Central  Google Scholar 

  10. Invernizzi M, Valsson O, Parrinello M (2017) Coarse graining from variationally enhanced sampling applied to the Ginzburg-Landau model. Proc Natl Acad Sci 114:3370–3374

    CrossRef  CAS  PubMed  Google Scholar 

  11. Fiore CE, da Luz MGE (2010) Comparing parallel- and simulated-tempering-enhanced sampling algorithms at phase-transition regimes. Phys Rev E 82:031104

    CrossRef  CAS  Google Scholar 

  12. Sosso GC et al (2016) Crystal nucleation in liquids: open questions and future challenges in molecular dynamics simulations. Chem Rev 116:7078–7116

    CrossRef  CAS  PubMed  PubMed Central  Google Scholar 

  13. Giberti F, Salvalaglio M, Parrinello M (2015) Metadynamics studies of crystal nucleation. IUCrJ 2:256–266

    CrossRef  CAS  PubMed  PubMed Central  Google Scholar 

  14. Mandal S, Debnath J, Meyer B, Nair NN (2018) Enhanced sampling and free energy calculations with hybrid functionals and plane waves for chemical reactions. J Chem Phys 149:144113

    CrossRef  PubMed  CAS  Google Scholar 

  15. Debnath J, Invernizzi M, Parrinello M (2019) Enhanced sampling of transition states. J Chem Theory Comput. doi:https://doi.org/10.1021/acs.jctc.8b01283

  16. Zheng S, Pfaendtner J (2015) Enhanced sampling of chemical and biochemical reactions with metadynamics. Mol Simul 41

    Google Scholar 

  17. Awasthi S, Nair NN (2017) Exploring high dimensional free energy landscapes: temperature accelerated sliced sampling. J Chem Phys 146

    Google Scholar 

  18. Miroliaei M, Nemat-Gorgani M (2002) Effect of organic solvents on stability and activity of two related alcohol dehydrogenases: a comparative study. Int J Biochem Cell Biol 34:169–175

    CrossRef  CAS  PubMed  Google Scholar 

  19. Wu D, Fajer MI, Cao L, Cheng X, Yang W (2016) Generalized ensemble sampling of enzyme reaction free energy pathways. Methods Enzymol 577:57–74

    CrossRef  CAS  PubMed  PubMed Central  Google Scholar 

  20. Bolhuis PG, Chandler D, Dellago C, Geissler PL (2002) Transition path sampling: throwing ropes over rough mountain passes, in the dark. Annu Rev Phys Chem 53:291–318

    CrossRef  CAS  PubMed  Google Scholar 

  21. Nakamura M, Obata M, Morishita T, Oda T (2014) An ab initio approach to free-energy reconstruction using logarithmic mean force dynamics. J Chem Phys 140

    Google Scholar 

  22. Moritsugu K, Terada T, Kidera A (2010) Scalable free energy calculation of proteins via multiscale essential sampling. J Chem Phys 133:224105

    CrossRef  PubMed  CAS  Google Scholar 

  23. Darve E, Pohorille A (2001) Calculating free energies using average force. J Chem Phys 115:9169–9183

    CrossRef  CAS  Google Scholar 

  24. Henkelman G, Jónsson H (2000) Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. J Chem Phys 113:9978–9985

    CrossRef  CAS  Google Scholar 

  25. Fujisaki H, Shiga M, Kidera A (2010) Onsager-Machlup action-based path sampling and its combination with replica exchange for diffusive and multiple pathways. J Chem Phys 132:134101

    CrossRef  PubMed  CAS  Google Scholar 

  26. Glowacki DR, Paci E, Shalashilin DV (2009) Boxed molecular dynamics: a simple and general technique for accelerating rare event kinetics and mapping free energy in large molecular systems. J Phys Chem B 113:16603–16611

    CrossRef  CAS  PubMed  Google Scholar 

  27. Hansmann UHE, Wille LT (2002) Global optimization by energy landscape paving. Phys Rev Lett 88:068105

    CrossRef  PubMed  CAS  Google Scholar 

  28. Weinan W, Ren W, Vanden-Eijnden E (2002) String method for the study of rare events. Phys Rev B 66:052301

    Google Scholar 

  29. Dellago C, Bolhuis PG, Advanced computer simulation approaches for soft matter sciences, vol 3, Springer, Berlin, pp 167–233. doi:https://doi.org/10.1007/978-3-540-87706-6_3

  30. Fujisaki H, Moritsugu K, Matsunaga Y, Morishita T, Maragliano L (2015) Extended phase-space methods for enhanced sampling in molecular simulations: a review. Front Bioeng Biotechnol 3:1–10

    CrossRef  Google Scholar 

  31. Laio A, Parrinello M, Computer simulations in condensed matter systems: from materials to chemical biology, vol 1. Springer, Berlin, pp 315–347. doi:https://doi.org/10.1007/3-540-35273-2_9

  32. Rodriguez-Gomez D, Darve E, Pohorille A (2004) Assessing the efficiency of free energy calculation methods. J Chem Phys 120:3563–3578

    CrossRef  CAS  PubMed  Google Scholar 

  33. Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680

    Google Scholar 

  34. Hansmann UHE (1997) Parallel tempering algorithm for conformational studies of biological molecules. Chem Phys Lett 281:140–150

    CrossRef  CAS  Google Scholar 

  35. Sugita Y, Okamoto Y (1999) Replica-exchange molecular dynamics method for protein folding. Chem Phys Lett. https://doi.org/10.1016/S0009-2614(99)01123-9

    CrossRef  Google Scholar 

  36. Affentranger R, Tavernelli I, Di Iorio EE (2006) A novel hamiltonian replica exchange md protocol to enhance protein conformational space sampling. J Chem Theory Comput 2:217–228

    CrossRef  CAS  PubMed  Google Scholar 

  37. Liu P, Kim B, Friesner RA, Berne BJ (2005) Replica exchange with solute tempering: A method for sampling biological systems in explicit water. Proc Natl Acad Sci 102:13749–13754

    CrossRef  CAS  PubMed  Google Scholar 

  38. 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 Chem Phys 116:4389–4402

    CrossRef  CAS  Google Scholar 

  39. 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 426:168–175

    CrossRef  CAS  Google Scholar 

  40. Earl DJ, Deem MW (2005) Parallel tempering: theory, applications, and new perspectives. Phys Chem Chem Phys 7:3910

    CrossRef  CAS  PubMed  Google Scholar 

  41. Meloni S, Ciccotti G (2015) Free energies for rare events: temperature accelerated MD and MC. Eur Phys J Spec Top 224:2389–2407

    CrossRef  CAS  Google Scholar 

  42. Luitz M, Bomblies R, Ostermeir K, Zacharias M (2015) Exploring biomolecular dynamics and interactions using advanced sampling methods. J Phys Condens Matter 27:323101

    CrossRef  PubMed  CAS  Google Scholar 

  43. Sega M, Autieri E, Pederiva F (2011) Pickett angles and Cremer-Pople coordinates as collective variables for the enhanced sampling of six-membered ring conformations. Mol Phys 109:141–148

    CrossRef  CAS  Google Scholar 

  44. Peters B (2016) Reaction coordinates and mechanistic hypothesis tests. Annu Rev Phys Chem 67:669–690

    CrossRef  CAS  PubMed  Google Scholar 

  45. Torrie GM, Valleau JP (1977) Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling. J Comput Phys 23:187–199

    CrossRef  Google Scholar 

  46. Kumar S, Rosenberg JM, Bouzida D, Swendsen RH, Kollman PA (1992) THE weighted histogram analysis method for free-energy calculations on biomolecules. I. The method. J Comput Chem 13:1011–1021

    CrossRef  CAS  Google Scholar 

  47. Grossfield A. WHAM: the weighted histogram analysis method, version 2.0.10. http://membrane.urmc.rochester.edu/wordpress/?page_id=126

  48. Shirts MR, Chodera JD (2008) Statistically optimal analysis of samples from multiple equilibrium states. J Chem Phys 129:124105

    CrossRef  PubMed  PubMed Central  CAS  Google Scholar 

  49. Schlitter J, Engels M, Krüger P, Jacoby E, Wollmer A (1993) Targeted molecular dynamics simulation of conformational change-application to the T ↔ R transition in insulin. Mol Simul 10:291–308

    CrossRef  CAS  Google Scholar 

  50. Wang F, Landau DP (2001) Efficient, multiple-range random walk algorithm to calculate the density of states. Phys Rev Lett 86:2050–2053

    CrossRef  CAS  PubMed  Google Scholar 

  51. Valsson O, Parrinello M (2014) Variational approach to enhanced sampling and free energy calculations. Phys Rev Lett 113:1–5

    CrossRef  CAS  Google Scholar 

  52. Laio A, Parrinello M (2002) Escaping free-energy minima. PNAS 99:12562

    CrossRef  CAS  PubMed  Google Scholar 

  53. Micheletti C, Laio A, Parrinello M (2004) Reconstructing the density of states by history-dependent metadynamics. Phys Rev Lett 92:170601

    CrossRef  PubMed  CAS  Google Scholar 

  54. Piana S, Laio A (2007) A bias-exchange approach to protein folding. J Phys Chem B 111

    Google Scholar 

  55. Vargiu AV, Ruggerone P, Magistrato A, Carloni P (2008) Dissociation of minor groove binders from DNA: insights from metadynamics simulations. Nucleic Acids Res 36:5910–5921

    CrossRef  CAS  PubMed  PubMed Central  Google Scholar 

  56. 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 39:73

    CrossRef  CAS  PubMed  Google Scholar 

  57. Gervasio FL, Laio A, Parrinello M (2005) Flexible docking in solution using metadynamics. J Am Chem Soc 127:2600

    CrossRef  CAS  PubMed  Google Scholar 

  58. Limongelli V, Bonomi M, Parrinello M (2013) Funnel metadynamics as accurate binding free-energy method. Proc Natl Acad Sci. https://doi.org/10.1073/pnas.1303186110

    CrossRef  PubMed  Google Scholar 

  59. Martoňák R et al (2005) Simulation of structural phase transitions by metadynamics. Zeitschrift Für Krist Mater 220:489

    Google Scholar 

  60. Laio A, Rodriguez-Fortea A, Gervasio FL, Ceccarelli M, Parrinello M (2005) Assessing the accuracy of metadynamics. J Phys Chem B 109:6714

    CrossRef  CAS  PubMed  Google Scholar 

  61. Barducci A, Bussi G, Parrinello M (2008) Well-tempered metadynamics: a smoothly converging and tunable free-energy method. Phys Rev Lett 100:020603

    CrossRef  PubMed  CAS  Google Scholar 

  62. Dama JF, Parrinello M, Voth GA (2014) Well-tempered metadynamics converges asymptotically. Phys Rev Lett 112

    Google Scholar 

  63. Singh S, Chiu C-C, De Pablo JJ (2011) Flux tempered metadynamics. J Stat Phys 145:932–945

    CrossRef  Google Scholar 

  64. Bonomi M, Camilloni C, Vendruscolo M (2016) Metadynamic metainference: Enhanced sampling of the metainference ensemble using metadynamics. Sci Rep 6:31232

    CrossRef  CAS  PubMed  PubMed Central  Google Scholar 

  65. Branduardi D, Bussi G, Parrinello M (2012) Metadynamics with adaptive gaussians. J Chem Theory Comput 8:2247–2254

    CrossRef  CAS  PubMed  Google Scholar 

  66. Dama JF, Rotskoff G, Parrinello M, Voth GA (2014) Transition-tempered metadynamics: robust, convergent metadynamics via on-the-fly transition barrier estimation. J Chem Theory Comput 10:3626–3633

    CrossRef  CAS  PubMed  Google Scholar 

  67. Oliver WC, Pharr GM (1992) An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J Mater Res 7:1564–1583

    CrossRef  CAS  Google Scholar 

  68. Min D, Liu Y, Carbone I, Yang W (2007) On the convergence improvement in the metadynamics simulations: a Wang-Landau recursion approach. J Chem Phys 126:194104

    CrossRef  PubMed  CAS  Google Scholar 

  69. 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

    CrossRef  CAS  PubMed  Google Scholar 

  70. Singh S, Chopra M, de Pablo JJ (2012) Density of states-based molecular simulations. Annu Rev Chem Biomol Eng 3:369–394

    CrossRef  PubMed  CAS  Google Scholar 

  71. Valsson O, Tiwary P, Parrinello M (2016) Enhancing important fluctuations: rare events and metadynamics from a conceptual viewpoint. Annu Rev Phys Chem 67:159–184

    CrossRef  CAS  PubMed  Google Scholar 

  72. Zhang Y, Voth GA (2011) Combined metadynamics and umbrella sampling method for the calculation of ion permeation free energy profiles. J Chem Theory Comput 7:2277–2283

    CrossRef  CAS  PubMed  PubMed Central  Google Scholar 

  73. White AD, Dama JF, Voth GA (2015) Designing free energy surfaces that match experimental data with metadynamics. J Chem Theory Comput 11:2451–2460

    CrossRef  CAS  PubMed  Google Scholar 

  74. Pfaendtner J (2018) Biomolecular simulations: methods in molecular biology. In: Bonomi M, Camilloni C (eds) In Press, Springer

    Google Scholar 

  75. Bussi G, Gervasio FL, Laio A, Parrinello M (2006) Free-energy landscape for hairpin folding from combined parallel tempering and metadynamics. J Am Chem Soc 128:13435

    CrossRef  CAS  PubMed  Google Scholar 

  76. Bonomi M, Parrinello M (2010) Enhanced sampling in the well-tempered ensemble. Phys Rev Lett 104:1–4

    CrossRef  CAS  Google Scholar 

  77. Gil-Ley A, Bussi G (2015) Enhanced conformational sampling using replica exchange with collective-variable tempering. J Chem Theory Comput 11:1077

    CrossRef  CAS  PubMed  PubMed Central  Google Scholar 

  78. Pfaendtner J, Bonomi M (2015) Efficient sampling of high-dimensional free-energy landscapes with parallel bias metadynamics. J Chem Theory Comput 11:5062–5067

    CrossRef  CAS  PubMed  Google Scholar 

  79. Prakash A, Fu CD, Bonomi M, Pfaendtner J (2018) Biasing smarter, not harder, by partitioning collective variables into families. J Chem Theory Comput. doi:https://doi.org/10.1021/acs.jctc.8b00448

  80. Prakash A, Baer MD, Mundy CJ, Pfaendtner J (2018) Peptoid backbone flexibilility dictates its interaction with water and surfaces: a molecular dynamics investigation. Biomacromol 19:1006–1015

    CrossRef  CAS  Google Scholar 

  81. Tiwary P, Parrinello M (2015) A time-independent free energy estimator for metadynamics. J Phys Chem B 119:14

    CrossRef  CAS  Google Scholar 

  82. Fu CD, Pfaendtner J (2018) Lifting the curse of dimensionality on enhanced sampling of reaction networks with parallel bias metadynamics. J Chem Theory Comput 14:2516–2525

    CrossRef  CAS  PubMed  Google Scholar 

  83. Pietrucci F, Andreoni W (2011) Graph theory meets Ab initio molecular dynamics: atomic structures and transformations at the nanoscale. Phys Rev Lett 107:085504

    CrossRef  PubMed  CAS  Google Scholar 

  84. Arsiccio A, McCarty JJ, Pisano R, Shea J-E (2018) The effect of surfactants on surface-induced denaturation of proteins: evidence of an orientation-dependent mechanism. J Phys Chem B. doi:https://doi.org/10.1021/acs.jpcb.8b07368

  85. Prakash A, Sprenger KG, Pfaendtner J (2018) Essential slow degrees of freedom in protein-surface simulations: a metadynamics investigation. Biochem Biophys Res Commun 498:274–281

    CrossRef  CAS  PubMed  Google Scholar 

  86. Löhr T, Jussupow A, Camilloni C (2017) Metadynamic metainference: Convergence towards force field independent structural ensembles of a disordered peptide. J Chem Phys 146:165102

    CrossRef  PubMed  CAS  Google Scholar 

  87. Nava M, Palazzesi F, Perego C, Parrinello M (2017) DImer metadynamics. J Chem Theory Comput 13:425–430

    CrossRef  CAS  PubMed  Google Scholar 

  88. Tribello GA, Ceriotti M, Parrinello M (2010) A self-learning algorithm for biased molecular dynamics. Proc Natl Acad Sci USA 107:17509–17514

    Google Scholar 

  89. Tiwary P, Berne BJ (2016) Spectral gap optimization of order parameters for sampling complex molecular systems. Proc Natl Acad Sci 113:2839 LP-2844

    Google Scholar 

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Acknowledgements

This work was supported in part by NSF award MCB-1715123 and NIH award 1R21DE026959-01

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  1. Department of Chemical Engineering, University of Washington, Seattle, WA, 98195, USA

    Sarah Alamdari, Janani Sampath, Arushi Prakash, Luke D. Gibson & Jim Pfaendtner

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  1. Sarah Alamdari
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  2. Janani Sampath
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  3. Arushi Prakash
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Correspondence to Jim Pfaendtner .

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  1. Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, IN, USA

    Prof. Edward J. Maginn

  2. Chemical and Biological Engineering, University at Buffalo, Buffalo, NY, USA

    Prof. Jeffrey Errington

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Alamdari, S., Sampath, J., Prakash, A., Gibson, L.D., Pfaendtner, J. (2021). Efficient Sampling of High-Dimensional Free Energy Landscapes: A Review of Parallel Bias Metadynamics. In: Maginn, E.J., Errington, J. (eds) Foundations of Molecular Modeling and Simulation. Molecular Modeling and Simulation. Springer, Singapore. https://doi.org/10.1007/978-981-33-6639-8_6

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