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Guidelines for the analysis of free energy calculations

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Abstract

Free energy calculations based on molecular dynamics simulations show considerable promise for applications ranging from drug discovery to prediction of physical properties and structure-function studies. But these calculations are still difficult and tedious to analyze, and best practices for analysis are not well defined or propagated. Essentially, each group analyzing these calculations needs to decide how to conduct the analysis and, usually, develop its own analysis tools. Here, we review and recommend best practices for analysis yielding reliable free energies from molecular simulations. Additionally, we provide a Python tool, alchemical-analysis.py, freely available on GitHub as part of the pymbar package (located at http://github.com/choderalab/pymbar), that implements the analysis practices reviewed here for several reference simulation packages, which can be adapted to handle data from other packages. Both this review and the tool covers analysis of alchemical calculations generally, including free energy estimates via both thermodynamic integration and free energy perturbation-based estimators. Our Python tool also handles output from multiple types of free energy calculations, including expanded ensemble and Hamiltonian replica exchange, as well as standard fixed ensemble calculations. We also survey a range of statistical and graphical ways of assessing the quality of the data and free energy estimates, and provide prototypes of these in our tool. We hope this tool  and discussion will serve as a foundation for more standardization of and agreement on best practices for analysis of free energy calculations.

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Notes

  1. Annihilation and decoupling can be thought to differ primarily in how they handle charge and Lennard-Jones parameters. Specifically, annihilation involves actually setting solute partial charges to zero, while decoupling involves turning off charge interactions with environment. Likewise, annihilation involves actually setting the Lennard-Jones parameters to zero, while decoupling involves turning off interactions with environment. Our current explanation is specific to the more general case, annihilation, but in case of decoupling no gas transformation \(1\rightarrow 3\) is needed and the overall transformation reduces to the single leg \(2\rightarrow 4\), i.e. the hydration free energy change is found as the negative of \(\varDelta G_{decoupling}^{water}\), with the possible exception of an anaytical standard state correction depending on the experimental reference state employed.

  2. Whenever there is an additional field corresponding to the pV energy term it will be added to the potential energy of corresponding state.

  3. Our Python tool does not currently separate out a restraining component of the free energy, because restraining transformations are not always separable from other transformations. Unlike Coulombic transformations, most of the other transformation types can be (and are [25, 34]) performed simultaneously (to decrease the number of the simulation runs), i.e. they are coupled, which makes component separation impossible.

  4. Remember, the time-reversed \(\varDelta G\) estimates are plotted in a backward manner, so that if the point in question is encountered at the time \(t'=0.6 t_{total}\), the portion of the data to be discarded as non-equilibrated is from \(t=0\) up to \(t = t_{total} - t' = 0.4 t_{total}\).

References

  1. Bennett CH (1976) Efficient estimation of free energy differences from Monte Carlo data. J Comput Phys 22(2):245–268

    Article  Google Scholar 

  2. Beutler TC, Mark AE, van Schaik RC, Gerber PR, van Gunsteren WF (1994) Avoiding singularities and numerical instabilities in free energy calculations based on molecular simulations. Chem Phys Lett 222(6):529–539

    Article  CAS  Google Scholar 

  3. Beveridge DL, DiCapua FM (1989) Free energy via molecular simulation: applications to chemical and biomolecular systems. Annu Rev Biophys Biophys Chem 18:431–492

    Article  CAS  Google Scholar 

  4. Boyce SE, Mobley DL, Rocklin GJ, Graves AP, Dill KA, Shoichet BK (2009) Predicting ligand binding affinity with alchemical free energy methods in a polar model binding site. J Mol Biol 394(4):747–763

    Article  CAS  Google Scholar 

  5. Buelens FP, Grubmüller H (2012) Linear-scaling soft-core scheme for alchemical free energy calculations. J Comput Chem 33(1):25–33

    Article  CAS  Google Scholar 

  6. Case DA, Babin V, Berryman JT, Betz RM, Cai Q, Cerutti DS, Cheatham III TE, Darden TA, Duke RE, Gohlke H, Goetz AW, Gusarov S, Homeyer N, Janowski P, Kaus J, Kolossváry I, Kovalenko A, Lee TS, LeGrand S, Luchko T, Luo R, Madej B, Merz KM, Paesani F, Roe DR, Roitberg A, Sagui C, Salomon-Ferrer R, Seabra G, Simmerling CL, Smith W, Swails J, Walker RC, Wang J, Wolf RM, Wu X, Kollman PA (2014) Amber 14. University of California, San Francisco

  7. Chipot C (2014) Frontiers in free-energy calculations of biological systems. Wiley Interdiscip Rev Comput Mol Sci 4(1):71–89

    Article  CAS  Google Scholar 

  8. Chodera JD, Swope WC, Pitera JW, Seok C, Dill KA (2007) Use of the weighted histogram analysis method for the analysis of simulated and parallel tempering simulations. J Chem Theory Comput 3(1):26–41

    Article  CAS  Google Scholar 

  9. Deng Y, Roux B (2009) Computations of standard binding free energies with molecular dynamics simulations. J Phys Chem B 113(8):2234–2246

    Article  CAS  Google Scholar 

  10. Flyvbjerg H, Petersen HG (1989) Error estimates on averages of correlated data. J Chem Phys 91(1):461

    Article  CAS  Google Scholar 

  11. Frenkel D, Smit B (2001) Understanding molecular simulation. From algorithms to applications. Academic Press, London

    Google Scholar 

  12. Gapsys V, Seeliger D, de Groot BL (2012) New soft-core potential function for molecular dynamics based alchemical free energy calculations. J Chem Theory Comput 8(7):2373–2382

    Article  CAS  Google Scholar 

  13. Hummer G, Pratt LR, García AE (1997) Multistate Gaussian model for electrostatic solvation free energies. J Am Chem Soc 119(36):8523–8527

    Article  CAS  Google Scholar 

  14. Hummer G, Pratt LR, García AE, Berne BJ, Rick SW (1997) Electrostatic potentials and free energies of solvation of polar and charged molecules. J Phys Chem B 101(16):3017–3020

    Article  CAS  Google Scholar 

  15. Hunter JD (2007) Matplotlib: a 2D graphics environment. Comput Sci Eng 9(3):90–95

    Article  Google Scholar 

  16. Kenneth M, Merz J, Ringe D, Reynolds CH (2010) Drug design. Structure- and ligand-based approaches. Cambridge University Press, Cambridge

    Google Scholar 

  17. Kirkwood JG (1935) Statistical mechanics of fluid mixtures. J Chem Phys 3(5):300–313

    Article  CAS  Google Scholar 

  18. Klimovich PV, Mobley DL (2010) Predicting hydration free energies using all-atom molecular dynamics simulations and multiple starting conformations. J Comput Aided Mol Des 24(4):307–316

    Article  CAS  Google Scholar 

  19. Kumar S, Bouzida D, Swendsen RH, Kollman PA, Rosenberg JM (1992) The weighted histogram analysis method for free-energy calculations on biomolecules. 1. The method. J Comput Chem 13(8):1011–1021

    Article  CAS  Google Scholar 

  20. Lu H, Schulten K (1999) Steered molecular dynamics simulations of force-induced protein domain unfolding. Proteins 35(4):453–463

    Article  Google Scholar 

  21. Lyubartsev AP, Laaksonen A, Vorontsov-Velyaminov PN (1994) Free energy calculations for Lennard-Jones systems and water using the expanded ensemble method A Monte Carlo and molecular dynamics simulation study. Mol Phys 82(3):455–471

    Article  CAS  Google Scholar 

  22. Lyubartsev AP, Martsinovski AA, Shevkunov SV, Vorontsov-Velyaminov PN (1992) New approach to Monte Carlo calculation of the free energy: method of expanded ensembles. J Chem Phys 96(3):1776–1783

    Article  CAS  Google Scholar 

  23. Marszalek PE, Lu H, Li H, Carrion-Vazquez M, Oberhauser AF, Schulten K, Fernandez JM (1999) Mechanical unfolding intermediates in titin modules. Nature 402(6757):100–103

    Article  CAS  Google Scholar 

  24. Martínez-Veracoechea FJ, Escobedo FA (2008) Variance minimization of free energy estimates from optimized expanded ensembles. J Phys Chem B 112(27):8120–8128

    Article  Google Scholar 

  25. Mobley DL, Bayly CI, Cooper MD, Shirts MR, Dill KA (2009) Small molecule hydration free energies in explicit solvent: an extensive test of fixed-charge atomistic simulations. J Chem Theory Comput 5(2):350–358

    Article  CAS  Google Scholar 

  26. Monroe JI, Shirts MR (2014) Converging free energies of binding in cucurbit[7]uril and octa-acid host-guest systems from SAMPL4 using expanded ensemble simulations. J Comput Aided Mol Des 28(4):401–415

  27. Naden LN, Shirts MR, A linear basis function approach to efficient alchemical free energy calculations. 2. Inserting and deleting charged molecules (submitted)

  28. Paliwal H, Shirts MR (2011) A benchmark test set for alchemical free energy transformations and its use to quantify error in common free energy methods. J Chem Theory Comput 7(12):4115–4134

    Article  CAS  Google Scholar 

  29. Paluch AS, Mobley DL, Maginn EJ (2011) Small molecule solvation free energy: enhanced conformational sampling using expanded ensemble molecular dynamics simulation. J Chem Theory Comput 7(9):2910–2918

    Article  CAS  Google Scholar 

  30. Pham TT, Shirts MR (2011) Identifying low variance pathways for free energy calculations of molecular transformations in solution phase. J Chem Phys 135(3):034,114

    Article  Google Scholar 

  31. Pitera JW, van Gunsteren WF (2002) A comparison of non-bonded scaling approaches for free energy calculations. Mol Simul 28(1–2):45–65

    Article  CAS  Google Scholar 

  32. Pohorille A, Jarzynski C, Chipot C (2010) Good practices in free-energy calculations. J. Phys. Chem. B 114(32):10235–10253

    Article  CAS  Google Scholar 

  33. Pronk S, Pall S, Schulz R, Larsson P, Bjelkmar P, Apostolov R, Shirts MR, Smith JC, Kasson PM, van der Spoel D, Hess B, Lindahl E (2013) GROMACS 4.5: a high-throughput and highly parallel open source molecular simulation toolkit. Bioinformatics 29(7):845–854

    Article  CAS  Google Scholar 

  34. de Ruiter A, Boresch S, Oostenbrink C (2013) Comparison of thermodynamic integration and Bennett acceptance ratio for calculating relative protein-ligand binding free energies. J Comput Chem 34(12):1024–1034

    Article  Google Scholar 

  35. Shirts MR, Chodera JD (2008) Statistically optimal analysis of samples from multiple equilibrium states. J Chem Phys 129(12):124,105

    Article  Google Scholar 

  36. Shirts MR, Pitera JW, Swope WC, Pande VS (2003) Extremely precise free energy calculations of amino acid side chain analogs: comparison of common molecular mechanics force fields for proteins. J Chem Phys 119(11):5740–5761

    Article  CAS  Google Scholar 

  37. Shivakumar D, Williams J, Wu Y, Damm W (2010) Prediction of absolute solvation free energies using molecular dynamics free energy perturbation and the OPLS force field. J Chem Theory Comput 6(5):1509–1519

    Article  CAS  Google Scholar 

  38. Sotomayor M, Corey DP, Schulten K (2005) In search of the hair-cell gating spring elastic properties of ankyrin and cadherin repeats. Structure 13(4):669–682

    Article  CAS  Google Scholar 

  39. Steinbrecher T, Mobley DL, Case DA (2007) Nonlinear scaling schemes for Lennard-Jones interactions in free energy calculations. J Chem Phys 127(21):214,108

    Article  Google Scholar 

  40. Straatsma TP, Berendsen HJC, Postma JPM (1986) Free energy of hydrophobic hydration: a molecular dynamics study of noble gases in water. J Chem Phys 85(11):6720–6727

    Article  CAS  Google Scholar 

  41. Yang W, Bitetti-Putzer R, Karplus M (2004) Free energy simulations: use of reverse cumulative averaging to determine the equilibrated region and the time required for convergence. J Chem Phys 120(6):2618–2628

    Article  CAS  Google Scholar 

  42. Zacharias M, Straatsma TP, McCammon JA (1994) Separation-shifted scaling, a new scaling method for Lennard-Jones interactions in thermodynamic integration. J Chem Phys 100(12):9025

    Article  CAS  Google Scholar 

  43. Zhao G, Perilla JR, Yufenyuy EL, Meng X, Chen B, Ning J, Ahn J, Gronenborn AM, Schulten K, Aiken C, Zhang P (2013) Mature HIV-1 capsid structure by cryo-electron microscopy and all-atom molecular dynamics. Nature 497(7451):643–646

    Article  CAS  Google Scholar 

  44. Zwanzig RW (1954) High-temperature equation of state by a perturbation method. I. Nonpolar gases. J Chem Phys 22(8):1420–1426

    Article  CAS  Google Scholar 

Download references

Acknowledgments

We acknowledge the financial support of the National Institutes of Health (1R15GM096257-01A1, 1R01GM108889-01) and the National Science Foundation (CHE 1352608) and computing support from the UCI GreenPlanet cluster, supported in part by NSF Grant CHE-0840513. We thank Shuai Liu (UCI), Hannes Loeffler (STFC), Stefano Bosisio (University of Edinburgh), and Shun Zhu (Fudan University) for providing data to test the script, Nathan Lim (UCI) and Adam van Wart (UCI) for valuable comments on the draft, and Kyle Beauchamp (Memorial Sloan Kettering Cancer Center) for maintaining the PyMBAR project.

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Authors and Affiliations

  1. Department of Pharmaceutical Sciences and Department of Chemistry, University of California, Irvine, 147 Bison Modular, Irvine, CA, 92697, USA

    Pavel V. Klimovich & David L. Mobley

  2. Department of Chemical Engineering, University of Virginia, Charlottesville, VA, 22094, USA

    Michael R. Shirts

  3. Department of Chemistry, University of New Orleans, 2000 Lakeshore Drive, New Orleans, LA, 70148, USA

    David L. Mobley

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  1. Pavel V. Klimovich
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  2. Michael R. Shirts
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Correspondence to David L. Mobley.

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Klimovich, P.V., Shirts, M.R. & Mobley, D.L. Guidelines for the analysis of free energy calculations. J Comput Aided Mol Des 29, 397–411 (2015). https://doi.org/10.1007/s10822-015-9840-9

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