The adaptive biasing force method: Everything you always wanted to know but were afraid to ask

Journal of Physical Chemistry B

Volumn 119, Issue 3, 2015, Pages 1129-1151

  Comer, Jeffrey ^a   Gumbart, James C ^b   Hénin, Jérôme ^c   Lelievre, Tony ^d   Pohorille, Andrew ^e,f   Chipot, Christophe ^a,g  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^c INST DE BIOLOGIE PHYSICO CHIMIQUE   (France)

^d UNIVERSITÉ PARIS EST   (France)

^e NASA AMES RESEARCH CENTER   (United States)

^f UNIVERSITY OF CALIFORNIA   (United States)

^g University of Illinois at Urbana Champaign   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EQUATIONS OF MOTION; IMPORTANCE SAMPLING; MATHEMATICAL TRANSFORMATIONS;

ADAPTIVE BIASING FORCES; CONVERGENCE PROPERTIES; EFFICIENCY AND RELIABILITY; FREE ENERGY LANDSCAPE; FREE-ENERGY CALCULATIONS; FREE-ENERGY DIFFERENCE; GEOMETRIC TRANSFORMATIONS; METHODOLOGICAL ASPECTS;

FREE ENERGY;

EID: 84921469425     PISSN: 15206106     EISSN: 15205207     Source Type: Journal    
DOI: 10.1021/jp506633n     Document Type: Article

References (125)

1. Kirkwood, J. G. Statistical Mechanics of Fluid Mixtures J. Chem. Phys. 1935, 3, 300-313
2. Landau, L. D. Statistical Physics; The Clarendon Press: Oxford, U.K., 1938.
3. Zwanzig, R. W. High-Temperature Equation of State by a Perturbation Method. I. Nonpolar Gases J. Chem. Phys. 1954, 22, 1420-1426
4. McDonald, I. R.; Singer, K. Machine Calculation of Thermodynamic Properties of a Simple Fluid at Supercritical Temperatures J. Chem. Phys. 1967, 47, 4766-4772
5. McDonald, I. R.; Singer, K. Calculation of Thermodynamic Properties of Liquid Argon from Lennard-jones Parameters by a Monte Carlo Method Discuss. Faraday Soc. 1967, 43, 40-49
6. Chipot, C. Frontiers in Free-Energy Calculations of Biological Systems Wiley Interdiscip. Rev. Comput. Mol. Sci. 2014, 4, 71-89
7. Chipot, C.; Pohorille, A., Eds. Free Energy Calculations. Theory and Applications in Chemistry and Biology; Springer Verlag: Berlin, 2007.
8. Lelievre, T.; Stoltz, G.; Rousset, M. Free Energy Computations: A Mathematical Perspective; Imperial College Press: London, 2010.
9. Jorgensen, W. L.; Ravimohan, C. Monte Carlo Simulation of Differences in Free Energies of Hydration J. Chem. Phys. 1985, 83, 3050-3054
10. Kong, X.; Brooks, C. L., III. Dynamics: A New Approach to Free Energy Calculations J. Chem. Phys. 1996, 105, 2414-2423
11. Torrie, G. M.; Valleau, J. P. Nonphysical Sampling Distributions in Monte Carlo Free Energy Estimation: Umbrella Sampling J. Comput. Phys. 1977, 23, 187-199
12. Carter, E.; Ciccotti, G.; Hynes, J. T.; Kapral, R. Constrained Reaction Coordinate Dynamics for the Simulation of Rare Events Chem. Phys. Lett. 1989, 156, 472-477
13. Straatsma, T. P.; Zacharias, M.; McCammon, J. A. Holonomic Constraint Contributions to Free Energy Differences from Thermodynamic Integration Molecular Dynamics Simulations Chem. Phys. Lett. 1992, 196, 297-302
14. Pearlman, D. A. Determining the Contributions of Constraints in Free Energy Calculations: Development, Characterization, And Recommendations J. Chem. Phys. 1993, 98, 8946-8957
15. Huber, T.; Torda, A. E.; van Gunsteren, W. F. Local Elevation: A Method for Improving the Searching Properties of Molecular Dynamics Simulation J. Comput. Aided Mol. Des. 1994, 8, 695-708
16. Grubmuller, H. Predicting Slow Structural Transitions in Macromolecular Systems: Conformational Flooding Phys. Rev. E 1995, 52, 2893-2906
17. Engkvist, O.; Karlstrom, G. A Method to Calculate the Probability Distribution for Systems with Large Energy Barriers Chem. Phys. 1996, 213, 63-76
18. Bartels, C.; Schaeffer, M.; Karplus, M. Determination of Equilibrium Properties of Biomolecular Systems Using Multidimensional Adaptive Umbrella Sampling J. Chem. Phys. 1999, 111, 8048-8067
19. Darve, E.; Pohorille, A. Calculating Free Energies Using Average Force J. Chem. Phys. 2001, 115, 9169-9183
20. Laio, A.; Parrinello, M. Escaping Free Energy Minima Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 12562-12565
21. Park, S.; Khalili-Araghi, F.; Tajkhorshid, E.; Schulten, K. Free Energy Calculation from Steered Molecular Dynamics Simulations Using Jarzynskis Equality J. Chem. Phys. 2003, 119, 3559-3566
22. Maragakis, P.; Spichty, M.; Karplus, M. Optimal Estimates of Free Energies from Multistate Nonequilibrium Work Data Phys. Rev. Lett. 2006, 96, 100602
23. Babin, V.; Roland, C.; Sagui, C. Adaptively Biased Molecular Dynamics for Free Energy Calculations J. Chem. Phys. 2008, 128, 134101
24. Barducci, A.; Bussi, G.; Parrinello, M. Well-Tempered Metadynamics: a Smoothly Converging and Tunable Free-Energy Method Phys. Rev. Lett. 2008, 100, 020603
25. Maragliano, L.; Vanden-Eijnden, E.; Roux, B. Free Energy and Kinetics of Conformational Transitions from Voronoi Tessellated Milestoning with Restraining Potentials J. Chem. Theory Comput. 2009, 5, 2589-2594
26. Wang, F.; Landau, D. Determining the Density of States for Classical Statistical Models: A Random Walk Algorithm to Produce a Flat Histogram Phys. Rev. E 2001, 64, 056101
27. Wang, F. G.; Landau, D. P. Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States Phys. Rev. Lett. 2001, 86, 2050-2053
28. Lee, E. H.; Hsin, J.; Mayans, O.; Schulten, K. Secondary and Tertiary Structure Elasticity of Titin Z1Z2 and a Titin Chain Model Biophys. J. 2007, 93, 1719-1735
29. Ivanov, I.; Cheng, X.; Sine, S. M.; McCammon, J. A. Barriers to Ion Translocation in Cationic and Anionic Receptors from the Cys-Loop Family J. Am. Chem. Soc. 2007, 129, 8217-8224
30. Hénin, J.; Tajkhorshid, E.; Schulten, K.; Chipot, C. Diffusion of Glycerol through Escherichia coli Aquaglyceroporin GlpF Biophys. J. 2008, 94, 832-839
31. Wei, C.; Pohorille, A. Permeation of Membranes by Ribose and Its Diastereomers J. Am. Chem. Soc. 2009, 131, 10237-10245
32. Gumbart, J. C.; Roux, B.; Chipot, C. Efficient Determination of Protein-Protein Standard Binding Free Energies from First Principles J. Chem. Theory Comput. 2013, 9, 3789-3798
33. Sprik, M.; Ciccoti, G. Free Energy from Constrained Molecular Dynamics J. Chem. Phys. 1998, 109, 7737-7744
34. Ciccotti, G.; Kapral, R.; Vanden-Eijnden, E. Blue Moon Sampling, Vectorial Reaction Coordinates, and Unbiased Constrained Dynamics ChemPhysChem 2005, 6, 1809-1814
35. Otter, W. K. d.; Briels, W. J. The Calculation of Free-Energy Differences by Constrained Molecular-Dynamics Simulations J. Chem. Phys. 1998, 109, 4139-4146
36. Hénin, J.; Fiorin, G.; Chipot, C.; Klein, M. L. Exploring Multidimensional Free Energy Landscapes Using Time-dependent Biases on Collective Variables J. Chem. Theory Comput. 2010, 6, 35-47
37. Hénin, J.; Chipot, C. Overcoming Free Energy Barriers Using Unconstrained Molecular Dynamics Simulations J. Chem. Phys. 2004, 121, 2904-2914
38. Phillips, J. C.; Braun, R.; Wang, W.; Gumbart, J.; Tajkhorshid, E.; Villa, E.; Chipot, C.; Skeel, L.; Kalé, R. D.; Schulten, K. Scalable Molecular Dynamics With NAMD J. Comput. Chem. 2005, 26, 1781-1802
39. Fiorin, G.; Klein, M. L.; Hénin, J. Using Collective Variables to Drive Molecular Dynamics Simulations Mol. Phys. 2013, 111, 3345-3362
40. den Otter, W. K. Thermodynamic Integration of the Free Energy along a Reaction Coordinate in Cartesian Coordinates J. Chem. Phys. 2000, 112, 7283-7292
41. Ciccotti, G. E.; Kapral, R.; Vanden-Eijnden, E. Blue Moon Sampling, Vectorial Reaction Coordinates, And Unbiased Constrained Dynamics ChemPhysChem 2005, 6, 1809-1814
42. Darve, E.; Rodriguez-Gomez, D.; Pohorille, A. Adaptive Biasing Force Method for Scalar and Vector Free Energy Calculations J. Chem. Phys. 2008, 128, 144120
43. Basner, J. E.; Jarzynski, C. Binless Estimation of the Potential of Mean force J. Phys. Chem. B 2008, 112, 12722-12729
44. Tan, Z.; Gallicchio, E.; Lapelosa, M.; Levy, R. M. Theory of Binless Multi-State Free Energy Estimation with Applications to Protein-Ligand Binding J. Chem. Phys. 2012, 136, 144102
45. Voter, A. F. Hyperdynamics: Accelerated Molecular Dynamics of Infrequent Events Phys. Rev. Lett. 1997, 78, 3908-3911
46. Roux, B. The Calculation of the Potential of Mean Force Using Computer Simulations Comput. Phys. Commun. 1995, 91, 275-282
47. Chipot, C.; Pohorille, A. Structure and Dynamics of Small Peptides at Aqueous Interfaces. A Multi-Nanosecond Molecular Dynamics Study J. Mol. Struct. (THEOCHEM) 1997, 398/399, 529-535
48. Lelievre, T.; Rousset, M.; Stoltz, G. Computation of Free Energy Profiles with Adaptive Parallel Dynamics J. Chem. Phys. 2007, 126, 134111
49. Bussi, G.; Laio, A.; Parrinello, M. Equilibrium Free Energies from Nonequilibrium Metadynamics Phys. Rev. Lett. 2006, 96, 090601
50. Fort, G.; Moulines, E.; Priouret, P. Convergence of Adaptive and Interacting Markov Chain Monte Carlo Algorithms Ann. Stat. 2012, 39, 3262-3289
51. Marsili, S.; Barducci, A.; Chelli, R.; Procacci, P.; Schettino, V. Self-Healing Umbrella Sampling: A Non-equilibrium Approach for Quantitative Free Energy Calculations J. Phys. Chem. B 2006, 110, 14011-14013
52. Barducci, A.; Bussi, G.; Parrinello, M. Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method Phys. Rev. Lett. 2008, 100, 020603
53. Grubmuller, H.; Heymann, B.; Tavan, P. Ligand binding: Molecular Mechanics Calculation of the Streptavidin-Biotin Rupture force Science 1996, 271, 997-999
54. Izrailev, S.; Stepaniants, S.; Balsera, M.; Oono, Y.; Schulten, K. Molecular Dynamics Study of Unbinding of the Avidin-biotin Complex Biophys. J. 1997, 72, 1568-1581
55. Izrailev, S.; Stepaniants, S.; Isralewitz, B.; Kosztin, D.; Lu, H.; Molnar, F.; Wriggers, W.; Schulten, K. In Computational Molecular Dynamics: Challenges, Methods, Ideas; Deuflhard, P.; Hermans, J.; Leimkuhler, B.; Mark, A. E.; Skeel, R.; Reich, S., Eds.; Lecture Notes in Computational Science and Engineering; Springer Verlag: Berlin, 1998; Vol. 4, pp 39-65.
56. Jarzynski, C. Nonequilibrium Equality for Free Energy Differences Phys. Rev. Lett. 1997, 78, 2690-2693
57. Crooks, G. Nonequilibrium Measurements of Free Energy Differences for Microscopically Reversible Markovian Systems J. Stat. Phys. 1998, 90, 1481-1487
58. Hénin, J.; Pohorille, A.; Chipot, C. Insights into the Recognition and Association of Transmembrane -Helices. The Free Energy of -Helix Dimerization in Glycophorin A J. Am. Chem. Soc. 2005, 127, 8478-8484
59. Yu, Y. M.; Cai, W. S.; Chipot, C.; Shao, X. G. Molecular Dynamics Study of the Inclusion of Cholesterol into Cyclodextrins J. Phys. Chem. B 2006, 110, 6372-6378
60. Valleau, J. P.; Card, D. N. Monte Carlo Estimation of the Free Energy by Multistage Sampling J. Chem. Phys. 1972, 57, 5457-5462
61. Bolhuis, P. G.; Chandler, D.; Dellago, C.; Geissler, P. Transition Path Sampling: Throwing Ropes over Mountain Passes, In the Dark Annu. Rev. Phys. Chem. 2002, 59, 291-318
62. Ma, A.; Dinner, A. R. Automatic Method for Identifying Reaction Coordinates in Complex Systems J. Phys. Chem. B 2005, 109, 6769-6779
63. Best, R. B.; Hummer, G. Reaction Coordinates and Rates from Transition Paths Proc. Natl. Acad. Sci. U. S. A. 2005, 102, 6732-6737
64. E, W.; Ren, W.; Vanden-Eijnden, E. Transition Pathways in Complex Systems: Reaction Coordinates, Isocommittor Surfaces, And Transition Tubes Chem. Phys. Lett. 2005, 413, 242-247
65. Chipot, C.; Hénin, J. Exploring the Free Energy Landscape of a Short Peptide Using an Average Force J. Chem. Phys. 2005, 123, 244906
66. Hashemian, B.; Millán, D.; Arroyo, M. Modeling and Enhanced Sampling of Molecular Systems with Smooth and Nonlinear Data-Driven Collective Variables J. Chem. Phys. 2013, 139, 214101
67. Lelievre, T.; Rousset, M.; Stoltz, G. Long-time Convergence of an Adaptive Biasing Force Method Nonlinearity 2008, 21, 1155-1181
68. Lelievre, T.; Minoukadeh, K. Long-time Convergence of an Adaptive Biasing Force Method: the Bi-channel Case Arch. Ration. Mech. Anal. 2011, 202, 1-34
69. Jourdain, B.; Lelievre, T.; Roux, R. Existence, Uniqueness and Convergence of a Particle Approximation for the Adaptive Biasing Force Process ESAIM: Math. Modell. Numer. Anal. 2010, 44, 831-865
70. Fort, G.; Jourdain, B.; Kuhn, E.; Lelievre, T.; Stoltz, G. Convergence of the Wang-Landau Algorithm Math. Comput. 2014, in press
71. Fort, G.; Jourdain, B.; Kuhn, E.; Lelievre, T.; Stoltz, G. Efficiency of the Wang-Landau Algorithm: a Simple Test Case Appl. Math. Res. Express 2014, 2, 275-311
72. Wood, R. H. Estimation of Errors in Free Energy Calculations Due to the Lag between the Hamiltonian and the System Configuration J. Phys. Chem. 1991, 95, 4838-4842
73. Lu, N.; Kofke, D. A. Accuracy of Free-Energy Perturbation Calculations in Molecular Simulation. I. Modeling J. Chem. Phys. 2001, 114, 7303-7312
74. Lu, N.; Kofke, D. A. Accuracy of Free-Energy Perturbation Calculations in Molecular Simulation. II. Heuristics J. Chem. Phys. 2001, 115, 6866-6875
75. Zuckerman, D. M.; Woolf, T. B. Theory of a Systematic Computational Error in Free Energy Differences Phys. Rev. Lett. 2002, 89, 180602
76. Zuckerman, D. M.; Woolf, T. B. Overcoming Finite-Sampling Errors in Fast-Switching Free-Energy Estimates: Extrapolative Analysis of a Molecular System Chem. Phys. Lett. 2002, 351, 445-453
77. Shirts, M. R.; Bair, E.; Hooker, G.; Pande, V. S. Equilibrium Free Energies from Nonequilibrium Measurements Using Maximum-Likelihood Methods Phys. Rev. Lett. 2003, 91, 140601
78. Lu, N.; Kofke, D. A.; Woolf, T. B. Improving the Efficiency and Reliability of Free Energy Perturbation Calculations Using Overlap Sampling Methods J. Comput. Chem. 2004, 25, 28-39
79. Shirts, M. R.; Pande, V. S. Solvation Free Energies of Amino Acid Side Chain Analogs for Common Molecular Mechanics Water Models J. Chem. Phys. 2005, 122, 134508
80. Minh, D. D. L.; Chodera, J. D. Optimal estimators and asymptotic variances for nonequilibrium path-ensemble averages J. Chem. Phys. 2009, 131, 134110
81. Pohorille, A.; Jarzynski, C.; Chipot, C. Good Practices in Free-Energy Calculations J. Phys. Chem. B 2010, 114, 10235-10253
82. Kofke, D.; Cummings, P. Precision and Accuracy of Staged Free-Energy Perturbation Methods for Computing the Chemical Potential by Molecular Simulation Fluid Phase Equilib. 1998, 150, 41-49
83. Rodriguez-Gomez, D.; Darve, E.; Pohorille, A. Assessing the Efficiency of Free Energy Calculation Methods J. Chem. Phys. 2004, 120, 3563-3578
84. Taylor, J. R. An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2 nd ed.; University Science Books: Sausalito, CA, 1997.
85. Comer, J.; Schulten, K.; Chipot, C. Calculation of Lipid-Bilayer Permeabilities Using an Average force J. Chem. Theory Comput. 2014, 10, 554-564
86. Comer, J. R.; Schulten, K.; Chipot, C. Diffusive Models of Membrane Permeation with Explicit Orientational Freedom J. Chem. Theory Comput. 2014, 10, 2710-2718
87. Wilson, M. A.; Wei, C.; Bjelkmar, P.; Wallace, B. A.; Pohorille, A. Molecular Dynamics Simulation of the Antiamoebin Ion Channel: Linking Structure and Conductance Biophys. J. 2011, 100, 2394-2402
88. Zheng, L.; Chen, M.; Yang, W. Random Walk in Orthogonal Space to Achieve Efficient Free-Energy Simulation of Complex Systems Proc. Natl. Acad. Sci. U. S. A. 2008, 105, 20227-20232
89. Gumbart, J. C.; Roux, B.; Chipot, C. Standard Binding Free Energies from Computer Simulations: What Is the Best Strategy? J. Chem. Theory Comput. 2013, 9, 794-802
90. Swendsen, R. H.; Wang, J. S. Replica Monte Carlo Simulation of Spin Glasses Phys. Rev. Lett. 1986, 57, 2607-2609
91. Marinari, E.; Parisi, G. Simulated Tempering-A new Monte Carlo Scheme Europhys. Lett. 1992, 19, 451-458
92. Lyubartsev, A.; Martsinovski, A.; Shevkunov, S.; Vorontsov-Velyaminov, P. New Approach to Monte Carlo Calculation of the Free Energy: Method of Expanded Ensembles J. Chem. Phys. 1992, 96, 1776-1783
93. Sugita, Y.; Okamoto, Y. Replica-Exchange Molecular Dynamics Method for Protein Folding Chem. Phys. Lett. 1999, 314, 141-151
94. Sugita, Y.; Kitao, A.; Okamoto, Y. Multidimensional Replica-Exchange Method for Free-Energy Calculations J. Chem. Phys. 2000, 113, 6042-6051
95. Faraldo-Gomez, J. D.; Roux, B. Characterization of Conformational Equilibria through Hamiltonian and Temperature Replica-Exchange Simulations: Assessing Entropic and Environmental Effects J. Comput. Chem. 2007, 28, 1634-1647
96. Fajer, M.; Swift, R. V.; McCammon, J. A. Using Multistate Free Energy Techniques to Improve the Efficiency of Replica Exchange Accelerated Molecular Dynamics J. Comput. Chem. 2009, 30, 1719-1725
97. Minoukadeh, K.; Chipot, C.; Lelievre, T. Potential of Mean Force Calculations: A Multiple-Walker Adaptive Biasing Force Approach J. Chem. Theory Comput. 2010, 6, 1008-1017
98. Dehez, F.; Tarek, M.; Chipot, C. Energetics of Ion Transport in a Peptide Nanotube J. Phys. Chem. B 2007, 111, 10633-10635
99. Comer, J.; Dehez, F.; Cai, W.; Chipot, C. Water Conduction Through a Peptide Nanotube J. Phys. Chem. C 2013, 117, 26797-26803
100. Alrachid, H.; Lelievre, T. Long-time Convergence of an Adaptive Biasing Force Method: Variance Reduction by Helmholtz Projection. Manuscript in preparation.
101. Chipot, C.; Lelievre, T. Enhanced Sampling of Multidimensional Free-energy Landscapes Using Adaptive Biasing Forces SIAM J. Appl. Math. 2011, 71, 1673-1695
102. Le Bris, C.; Lelievre, T.; Maday, Y. Results and Questions on a Nonlinear Approximation Approach for Solving High-dimensional Partial Differential Equations Constructive Approximation 2009, 30, 621-651
103. Andersen, H. C. Rattle: a Velocity Version of the Shake Algorithm for Molecular Dynamics Calculations J. Comput. Phys. 1983, 52, 24-34
104. Dickson, B.; Legoll, F.; Lelievre, T.; Stoltz, G.; Fleurat-Lessard, P. Free Energy Calculations: An Efficient Adaptive Biasing Potential Method J. Phys. Chem. B 2010, 114, 5823-5830
105. Hummer, G. Position-Dependent Diffusion Coefficients and Free Energies from Bayesian Analysis of Equilibrium and Replica Molecular Dynamics Simulations New J. Phys. 2005, 7, 34
106. Ermak, D.; McCammon, J. Brownian Dynamics with Hydrodynamic Interactions J. Chem. Phys. 1978, 69, 1352
107. Grassia, P.; Hinch, E.; Nitsche, L. Computer Simulations of Brownian Motion of Complex Systems J. Fluid Mech. 1995, 282, 373-403
108. Turkcan, S.; Alexandrou, A.; Masson, J. B. A Bayesian Inference Scheme to Extract Diffusivity and Potential Fields from Confined Single-Molecule Trajectories Biophys. J. 2012, 102, 2288-2298
109. Ljubetic, A.; Urbancic, I.; Strancar, J. Recovering Position-Dependent Diffusion from Biased Molecular Dynamics Simulations J. Chem. Phys. 2014, 140, 084109
110. Woolf, T. B.; Roux, R. Proc. Natl. Acad. Sci. U. S. A. 1994, 91, 11631
111. Marrink, S. J.; Berendsen, H. J. C. Simulation of Water Transport through a Lipid Membrane J. Phys. Chem. 1994, 98, 4155-4168
112. + inside Gramicidin A: a Comparative Study of Four Computational Methods Biophys. Chem. 2006, 124, 268-278
113. Forney, M.; Janosi, L.; Kosztin, I. Calculating Free-energy Profiles in Biomolecular Systems from Fast Nonequilibrium Processes Phys. Rev. E 2008, 78, 051913
114. Micheletti, C.; Bussi, G.; Laio, A. Optimal Langevin Modeling of Out-of-Equilibrium Molecular Dynamics Simulations J. Chem. Phys. 2008, 129, 074105
115. Hegger, R.; Stock, G. Multidimensional Langevin Modeling of Biomolecular Dynamics J. Chem. Phys. 2009, 130, 034106
116. Holland, B. W.; Gray, C. G.; Tomberli, B. Calculating Diffusion and Permeability Coefficients with the Oscillating Forward-Reverse Method Phys. Rev. E 2012, 86, 036707
117. Comer, J.; Dehez, F.; Cai, W.; Chipot, C. Water Conduction through a Peptide Nanotube J. Phys. Chem. C 2013, 117, 26797-26803
118. Zwanzig, R. Diffusion in a Rough Potential Proc. Natl. Acad. Sci. U. S. A. 1988, 85, 2029
119. von Toussaint, U. Bayesian Inference in Surface Physics Rev. Mod. Phys. 2011, 83, 943-999
120. Dose, V. Bayesian Inference in Physics: Case Studies Rep. Prog. Phys. 2003, 66, 1421
121. Best, R.; Hummer, G. Diffusion Models of Protein Folding Phys. Chem. Chem. Phys. 2011, 13, 16902-16911
122. Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. Equation of State Calculations by Fast Computing Machines J. Chem. Phys. 1953, 21, 1087-1092
123. Darve, E.; Wilson, M.; Pohorille, A. Calculating Free Energies Using Scaled-Force Molecular Dynamics Mol. Sim. 2002, 28, 113-144
124. Chopin, N.; Lelievre, T.; Stoltz, G. Free Energy Methods for Bayesian Inference: Efficient Exploration of Univariate Gaussian Mixture Posteriors Stat. Comput. 2012, 22, 897-916
125. Cao, L.; Stoltz, G.; Lelievre, T.; Marinica, M.; Athenes, M. Free Energy Calculations From Adaptive Molecular Dynamics Simulations With Adiabatic Reweighting J. Chem. Phys. 2014, 140, 104108