Comparison of Sampling Methods via Robust Free Energy Inference: Application to Calmodulin

Comparison of Sampling Methods via Robust Free Energy Inference: Application to Calmodulin


A free energy landscape estimation-method based on Bayesian inference is presented and used for comparing the efficiency of thermally enhanced sampling methods with respect to regular molecular dynamics, where the simulations are carried out on two binding states of calmodulin. The proposed free energy estimation method (the GM method) is compared to other estimators using a toy model showing that the GM method provides a robust estimate not subject to overfitting. The continuous nature of the GM method, as well as predictive inference on the number of basis functions, provide better estimates on sparse data. We find that the free energy diffusion properties determine sampling method effectiveness, such that the diffusion dominated apo-calmodulin is most efficiently sampled by regular molecular dynamics, while the holo with its rugged free energy landscape is better sampled by enhanced methods.

Free energy estimation, REST, temperature replica exchange

supplementary_material KTH]Science for Life Laboratory, Department of Physics, KTH Royal Institute of Technology, Box 1031, SE-171 21 Solna KTH]Science for Life Laboratory, Department of Physics, KTH Royal Institute of Technology, Box 1031, SE-171 21 Solna SU]Science for Life Laboratory, Department of Biochemistry and Biophysics, Stockholm University, Box 1031, SE-171 21 Solna KTH]Science for Life Laboratory, Department of Physics, KTH Royal Institute of Technology, Box 1031, SE-171 21 Solna \abbreviationsFE,GM,KDE,kNN, REST


introduction.tex \subfilemethod.tex \subfileresults.tex


The simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC Centre for High Performance Computing (PDC-HPC). CB acknowledges the Knut and Alice Wallenberg foundation (1484505) and the Carl Trygger foundation (CTS-15:298) for funding. Furthermore, the authors thank Berk Hess for insightful comments during the writing process.


The code for estimating FE landscapes with the GM method is available free of charge at The supplementary information contains tables with the number of estimated basis functions in the GM method (Tables LABEL:table:inferred_bf-LABEL:tab:nGaussApo) and figures comparing the end results from using the GM method and Bayesian inference on a step function (Figures LABEL:fig:supp:1D-LABEL:fig:supp:DRID50).


  1. Frauenfelder, H. Conformational Substates in Proteins. Annu. Rev. Biophys. Biophys. Chem. 1988, 17, 451–479.
  2. Marinari, E.; Parisi, G. Simulated Tempering: A New Monte Carlo Scheme. EPL (Europhysics Letters) 1992, 19, 451.
  3. Sugita, Y.; Okamoto, Y. Replica-exchange molecular dynamics method for protein folding. Chem. Phys. Lett. 1999, 314, 141–151.
  4. Sanbonmatsu, K.; Garcia, A. Structure of Met-enkephalin in explicit aqueous solution using replica exchange molecular dynamics. Proteins: Struct., Funct., Bioinf. 2002, 46, 225–234.
  5. Zhang, W.; Wu, C.; Duan, Y. Convergence of replica exchange molecular dynamics. J. Chem. Phys. 2005, 123, 154105.
  6. Periole, X.; Mark, A. E. Convergence and sampling efficiency in replica exchange simulations of peptide folding in explicit solvent. J. Chem. Phys. 2007, 126, 014903.
  7. Rhee, Y. M.; Pande, V. S. Multiplexed-Replica Exchange Molecular Dynamics Method for Protein Folding Simulation. Biophys. J. 2003, 84, 775–786.
  8. Liu, P.; Kim, B.; Friesner, R. A.; Berne, B. J. Replica exchange with solute tempering: A method for sampling biological systems in explicit water. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 13749–13754.
  9. Wang, L.; Friesner, R. A.; Berne, B. J. Replica Exchange with Solute Scaling: A More Efficient Version of Replica Exchange with Solute Tempering (REST2). J. Phys. Chem. B 2011, 115, 9431–9438.
  10. Huang, X.; Hagen, M.; Kim, B.; Friesner, R. A.; Zhou, R.; Berne, B. J. Replica Exchange with Solute Tempering: Efficiency in Large Scale Systems. J. Phys. Chem. B 2007, 111, 5405–5410.
  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. Zwanzig, R. High-Temperature Equation of State by a Perturbation Method. I. Nonpolar Gases. J. Chem. Phys. 1954, 22, 1420–1426.
  13. Gapsys, V.; Michielssens, S.; Seeliger, D.; de Groot, B. L. Accurate and Rigorous Prediction of the Changes in Protein Free Energies in a Large-Scale Mutation Scan. Angewandte Chemie (International Ed. in English) 2016, 55, 7364–7368.
  14. Darve, E.; Pohorille, A. Calculating free energies using average force. J. Chem. Phys. 2001, 115, 9169–9183.
  15. Berg, B. A.; Neuhaus, T. Multicanonical algorithms for first order phase transitions. Phys. Lett. B 1991, 267, 249–253.
  16. Wang, F.; Landau, D. P. Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States. Phys. Rev. Lett. 2001, 86, 2050–2053.
  17. Lidmar, J. Improving the efficiency of extended ensemble simulations: The accelerated weight histogram method. Phys. Rev. E 2012, 85, 056708.
  18. Lindahl, V.; Lidmar, J.; Hess, B. Accelerated weight histogram method for exploring free energy landscapes. J. Chem. Phys. 2014, 141, 044110.
  19. Grubmuller, H. Predicting slow structural transitions in macromolecular systems: Conformational flooding. Phys. Rev. E 1995, 52, 2893–2906.
  20. Laio, A.; Parrinello, M. Escaping free-energy minima. Proc Natl Acad Sci U S A 2002, 99, 12562–12566.
  21. Barducci, A.; Bussi, G.; Parrinello, M. Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method. Phys. Rev. Lett. 2008, 100.
  22. Bonomi, M.; Parrinello, M. Enhanced Sampling in the Well-Tempered Ensemble. Phys. Rev. Lett. 2010, 104, 190601.
  23. Gil-Ley, A.; Bussi, G. Enhanced Conformational Sampling Using Replica Exchange with Collective-Variable Tempering. J. Chem. Theory Comput. 2015, 11, 1077–1085.
  24. Habeck, M. Bayesian estimation of free energies from equilibrium simulations. Phys. Rev. Lett. 2012, 109, 100601.
  25. 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.
  26. Kumar, S.; Rosenberg, J. M.; Bouzida, D.; Swendsen, R. H.; Kollman, P. A. THE weighted histogram analysis method for free-energy calculations on biomolecules. I. The method. J. Comput. Chem. 1992, 13, 1011–1021.
  27. Parzen, E. On Estimation of a Probability Density Function and Mode. The Annals of Mathematical Statistics 1962, 33, 1065–1076.
  28. Wasserman, L. All of Statistics; Springer Texts in Statistics; Springer New York, 2004; pp 303–326.
  29. Pohorille, A.; Darve, E. A Bayesian Approach to Calculating Free Energies in Chemical and Biological Systems. 2006; pp 23–30.
  30. Babu, Y. S.; Bugg, C. E.; Cook, W. J. Structure of calmodulin refined at 2.2 A resolution. J. Mol. Biol. 1988, 204, 191–204.
  31. Ishida, H.; Nakashima, K.-i.; Kumaki, Y.; Nakata, M.; Hikichi, K.; Yazawa, M. The solution structure of apocalmodulin from Saccharomyces cerevisiae implies a mechanism for its unique Ca2+ binding property. Biochemistry (Mosc.) 2002, 41, 15536–15542.
  32. Jo, S.; Kim, T.; Iyer, V. G.; Im, W. CHARMM-GUI: A web-based graphical user interface for CHARMM. J. Comput. Chem. 2008, 29, 1859–1865.
  33. Lee, J.; Cheng, X.; Swails, J. M.; Yeom, M. S.; Eastman, P. K.; Lemkul, J. A.; Wei, S.; Buckner, J.; Jeong, J. C.; Qi, Y.; Jo, S.; Vijay, S. P.; Case, D. A.; Brookslll, C. L.; MacKerellJr., A. D.; Klauda, J. B.; Im, W. CHARMM-GUI input generator for NAMD, GROMACS, AMBER, OpenMM, and CHARMM/OpenMM simulations using the CHARMM36 additive force field. J. Chem. Theory Comput. 2015, 12, 405–413.
  34. MacKerell, A. D.; Bashford, D.; Bellott, M.; Dunbrack, R. L.; Evanseck, J. D.; Field, M. J.; Fischer, S.; Gao, J.; Guo, H.; Ha, S.; Joseph-McCarthy, D.; Kuchnir, L.; Kuczera, K.; Lau, F. T.; Mattos, C.; Michnick, S.; Ngo, T.; Nguyen, D. T.; Prodhom, B.; Reiher, W. E.; Roux, B.; Schlenkrich, M.; Smith, J. C.; Stote, R.; Straub, J.; Watanabe, M.; Wiorkiewicz-Kuczera, J.; Yin, D.; Karplus, M. All-atom empirical potential for molecular modeling and dynamics studies of proteins. J. Phys. Chem. B 1998, 102, 3586–3616.
  35. Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. Comparison of simple potential functions for simulating liquid water. J. Chem. Phys. 1983, 79, 926–935.
  36. Liao, J.; Marinelli, F.; Lee, C.; Huang, Y.; Faraldo-Gómez, J. D.; Jiang, Y. Mechanism of extracellular ion exchange and binding-site occlusion in a sodium/calcium exchanger. Nat. Struct. Mol. Biol. 2016, 23, 590–599.
  37. Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. Molecular dynamics with coupling to an external bath. J. Chem. Phys. 1984, 81, 3684–3690.
  38. Darden, T.; York, D.; Pedersen, L. Particle mesh Ewald: An Nlog(N) method for Ewald sums in large systems. J. Chem. Phys. 1993, 98, 10089–10092.
  39. Nose, S. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 1984, 81, 511–519.
  40. Parrinello, M.; Rahman, A. Polymorphic transitions in single crystals: A new molecular dynamics method. J. Appl. Phys. 1981, 52, 7182–7190.
  41. Hess, B.; Bekker, H.; Berendsen, H. J. C.; Fraaije, J. G. E. M. LINCS: A linear constraint solver for molecular simulations. J. Comput. Chem. 1997, 18, 1463–1472.
  42. Patriksson, A.; Spoel, D. v. d. A temperature predictor for parallel tempering simulations. Phys. Chem. Chem. Phys. 2008, 10, 2073–2077.
  43. Tribello, G. A.; Bonomi, M.; Branduardi, D.; Camilloni, C.; Bussi, G. PLUMED 2: New feathers for an old bird. Comput. Phys. Commun. 2014, 185, 604–613.
  44. Abraham, M. J.; Murtola, T.; Schulz, R.; Pall, S.; Smith, J. C.; Hess, B.; Lindahl, E. GROMACS: High performance molecular simulations through multi-level parallelism from laptops to supercomputers. SoftwareX 2015, 1-2, 19–25.
  45. Bussi, G. Hamiltonian replica exchange in GROMACS: a flexible implementation. Mol. Phys. 2014, 112, 379–384.
  46. Babu, Y. S.; Sack, J. S.; Greenhough, T. J.; Bugg, C. E.; Means, A. R.; Cook, W. J. Three-dimensional structure of calmodulin. Nature 1985, 315, 37–40.
  47. Zhou, T.; Caflisch, A. Distribution of Reciprocal of Interatomic Distances: A Fast Structural Metric. J. Chem. Theory Comput. 2012, 8, 2930–2937.
  48. McGibbon, R.; Beauchamp, K.; Harrigan, M.; Klein, C.; Swails, J.; Hernandez, C.; Schwantes, C.; Wang, L.-P.; Lane, T.; Pande, V. MDTraj: A Modern Open Library for the Analysis of Molecular Dynamics Trajectories. Biophys. J. 2015, 109, 1528–1532.
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