Nonequilibrium Methods for Equilibrium Free Energy Calculations

  • Gerhard Hummer
Part of the Springer Series in CHEMICAL PHYSICS book series (CHEMICAL, volume 86)


Free Energy Coupling Parameter Liouville Operator Free Energy Difference Free Energy Calculation 
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  1. 1.
    Jarzynski, C., Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 1997,78,2690-2693CrossRefGoogle Scholar
  2. 2.
    Jarzynski, C., Equilibrium free energy differences from nonequilibrium measurements. A master-equation approach, Phys. Rev. E 1997, 56, 5018-5035CrossRefGoogle Scholar
  3. 3.
    Hummer, G.; Szabo, A., Free energy reconstruction from nonequilibrium single-molecule pulling experiments, Proc. Natl Acad. Sci. USA 2001, 98, 3658-3661CrossRefGoogle Scholar
  4. 4.
    Liphardt, J.; Dumont, S.; Smith, S. B.; Tinoco, I.; Bustamante, C., Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality, Science 2002, 296, 1832-1835 CrossRefGoogle Scholar
  5. 5.
    Noy, A., Direct determination of the equilibrium unbinding potential profile for a short DNA duplex from force spectroscopy data, Appl. Phys. Lett. 2004, 85, 4792-4794CrossRefGoogle Scholar
  6. 6.
    Trepagnier, E. H.; Jarzynski, C.; Ritort, F.; Crooks, G. E.; Bustamante, C. J.; Liphardt, J., Experimental test of Hatano and Sasa’s nonequilibrium steady-state equality, Proc. Natl Acad. Sci. USA 2004, 101, 15038-15041CrossRefGoogle Scholar
  7. 7.
    Born, M., Volumen und Hydratationsw ärme der Ionen, Z. Phys. 1920, 1, 45-48CrossRefGoogle Scholar
  8. 8.
    Postma, J. P. M.; Berendsen, H. J. C.; Haak, J. R., Thermodynamics of cavity formation in water. A molecular dynamics study, Faraday Symp. Chem. Soc. 1982, 17, 55CrossRefGoogle Scholar
  9. 9.
    Straatsma, T. P.; Berendsen, H. J. C.; Postma, J. P. M., Free energy of hydrophobic hydration. A molecular dynamics study of noble gases in water, J. Chem. Phys. 1986, 85,6720-6727CrossRefGoogle Scholar
  10. 10.
    Wood, R. H.; M ühlbauer, W. C. F.; Thompson, P. T., Systematic errors in free energy perturbation calculations due to a finite sample of configuration space. Sample-size hys-teresis, J. Phys. Chem. 1991, 95, 6670-6675CrossRefGoogle Scholar
  11. 11.
    Hermans, J., Simple analysis of noise and hysteresis in (slow-growth) free energy simulations, J. Phys. Chem. 1991, 95, 9029-9032CrossRefGoogle Scholar
  12. 12.
    Zwanzig, R. W., High-temperature equation of state by a perturbation method. I. Non-polar gases, J. Chem. Phys. 1954, 22, 1420-1426CrossRefGoogle Scholar
  13. 13.
    Oberhofer, H.; Dellago, C.; Geissler, P. L., Biased sampling of nonequilibrium trajectories. Can fast switching simulations outperform conventional free energy cal-culation methods, J. Phys. Chem. B 2005, 109, 6902-6915CrossRefGoogle Scholar
  14. 14.
    Roepstorff, G., Path Integral Approach to Quantum Physics, Springer: Berlin, Heidelberg, New York, 1994Google Scholar
  15. 15.
    Hummer, G.; Szabo, A., Free energy surfaces from single-molecule force spectroscopy, Acc. Chem. Res. 2005, 38, 504-513CrossRefGoogle Scholar
  16. 16.
    Crooks, G. E., Path-ensemble averages in systems driven far from equilibrium, Phys. Rev. E 2000, 61, 2361-2366CrossRefGoogle Scholar
  17. 17.
    Hatano, T.; Sasa, S., Steady-state thermodynamics of Langevin systems, Phys. Rev. Lett. 2001,86,3463-3466CrossRefGoogle Scholar
  18. 18.
    Crooks, G. E., Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems, J. Stat. Phys. 1998, 90, 1481-1487CrossRefGoogle Scholar
  19. 19.
    Crooks, G. E., Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 1999, 60, 2721-2726CrossRefGoogle Scholar
  20. 20.
    Hummer, G., Fast-growth thermodynamic integration error and efficiency analysis, J. Chem. Phys. 2001, 114, 7330-7337CrossRefGoogle Scholar
  21. 21.
    Bennett, C. H., Efficient estimation of free energy differences from Monte Carlo data, J. Comput. Phys. 1976, 22, 245-268CrossRefGoogle Scholar
  22. 22.
    Shirts, M. R.; Bair, E.; Hooker, G.; Pande, V. S., Equilibrium free energies from non-equilibrium measurements using maximum-likelihood methods, Phys. Rev. Lett. 2003, 91,140601CrossRefGoogle Scholar
  23. 23.
    Shirts, M. R.; Pande, V. S., Comparison of efficiency and bias of free energies computed by exponential averaging. The Bennett acceptance ratio and thermodynamic integration, J. Chem. Phys. 2005, 122, 144107CrossRefGoogle Scholar
  24. 24.
    Allen, M. P.; Tildesley, D. J., Computer Simulation of Liquids, Clarendon: Oxford, UK, 1987Google Scholar
  25. 25.
    Lechner, W.; Oberhofer, H.; Dellago, C.; Geissler, P. L., Equilibrium free energies from fast-switching trajectories with large time steps, J. Chem. Phys. 2006, 124, 044113CrossRefGoogle Scholar
  26. 26.
    Reinhardt, W. P.; Hunter III, J. E., Variational path optimization and upper and lower bounds for the free energy via finite time minimization of the external work, J. Chem. Phys. 1992, 97, 1599-1601 CrossRefGoogle Scholar
  27. 27.
    Miller, M. A.; Reinhardt, W. P., Efficient free energy calculations by variationally op-timized metric scaling concepts and applications to the volume dependence of cluster free energies and to solid-solid phase transitions, J. Chem. Phys. 2000, 113, 7035-7046CrossRefGoogle Scholar
  28. 28.
    Best, R. B.; Hummer, G., Reaction coordinates and rates from transition paths, Proc. Natl Acad. Sci. USA 2005, 102, 6732-6737CrossRefGoogle Scholar
  29. 29.
    Chandler, D., Statistical mechanics of isomerization dynamics in liquids and the transition state approximation, J. Chem. Phys. 1978, 68, 2959-2970CrossRefGoogle Scholar
  30. 30.
    Hummer, G., From transition paths to transition states and rate coefficients, J. Chem. Phys. 2004, 120, 516-523CrossRefGoogle Scholar
  31. 31.
    Jarzynski, C., Rare events and the convergence of exponentially averaged work values, Phys. Rev. E 2006, 73, 046105CrossRefGoogle Scholar
  32. 32.
    Frenkel, D., Free-energy computation and first-order phase transitions. In Molecular Dynamics Simulations of Statistical Mechanical Systems. Proceedings of the Enrico Fermi Summer School, Varenna, 1985 (Amsterdam, 1986), Ciccotti, G.; Hoover, W. G., Eds., North-Holland, pp. 151-188Google Scholar
  33. 33.
    Gore, J.; Ritort, F.; Bustamante, C., Bias and error in estimates of equilibrium free-energy differences from nonequilibrium measurements, Proc. Natl Acad. Sci. USA 2003,100,12564-12569CrossRefGoogle Scholar
  34. 34.
    Zuckerman, D. M.; Woolf, T. B., Theory of a systematic computational error in free energy differences, Phys. Rev. Lett. 2002, 89, 180602CrossRefGoogle Scholar
  35. 35.
    Wu, D.; Kofke, D. A., Asymmetric bias in free-energy perturbation measurements using two Hamiltonian-based models, Phys. Rev. E 2004, 70, 066702CrossRefGoogle Scholar
  36. 36.
    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. 2002,351,445-453CrossRefGoogle Scholar
  37. 37.
    Ytreberg, F. M.; Zuckerman, D. M., Efficient use of nonequilibrium measurement to estimate free energy differences for molecular systems, J. Comp. Chem. 2004, 25, 1749-1759CrossRefGoogle Scholar
  38. 38.
    Rodriguez-Gomez, D.; Darve, E.; Pohorille, A., Assessing the efficiency of free energy calculation methods, J. Chem. Phys. 2004, 120, 3563-3578CrossRefGoogle Scholar
  39. 39.
    Hummer, G.; Szabo, A., Calculation of free energy differences from computer simulations of initial and final states, J. Chem. Phys. 1996, 105, 2004-2010CrossRefGoogle Scholar
  40. 40.
    Grubm üller, H.; Heymann, B.; Tavan, P., Ligand binding molecular mechanics calcula-tion of the streptavidin biotin rupture force, Science 1996, 271, 997-999CrossRefGoogle Scholar
  41. 41.
    Izrailev, S.; Stepaniants, S.; Balsera, M.; Oono, Y.; Schulten, K., Molecular dynamics study of unbinding of the avidin-biotin complex, Biophys. J. 1997, 72, 1568-1581CrossRefGoogle Scholar
  42. 42.
    Paci, E.; Karplus, M., Forced unfolding of fibronectin type 3 modules. An analysis by biased molecular dynamics simulations, J. Mol. Biol. 1999, 288, 441-459CrossRefGoogle Scholar
  43. 43.
    Park, S.; Schulten, K., Calculating potentials of mean force from steered molecular dynamics simulations, J. Chem. Phys. 2004, 120, 5946-5961CrossRefGoogle Scholar
  44. 44.
    Ferrenberg, A. M.; Swendsen, R. H., Optimized Monte Carlo data analysis, Phys. Rev. Lett. 1989, 63, 1195-1198CrossRefGoogle Scholar
  45. 45.
    Park, S.; Khalili-Araghi, F.; Tajkhorshid, E.; Schulten, K., Free energy calculation from steered molecular dynamics simulations using Jarzynski’s equality, J. Chem. Phys. 2003,119,3559-3566CrossRefGoogle Scholar
  46. 46.
    Sun, S. X., Equilibrium free energies from path sampling of nonequilibrium trajectories, J. Chem. Phys. 2003, 118, 5769-5775CrossRefGoogle Scholar
  47. 47.
    Ytreberg, F. M.; Zuckerman, D. M., Single-ensemble nonequilibrium path-sampling estimates of free energy differences, J. Chem. Phys. 2004, 120, 10876-10879 CrossRefGoogle Scholar
  48. 48.
    Hendrix, D. A.; Jarzynski, C., A fast growth method of computing free energy differences, J. Chem. Phys. 2001, 114, 5974-5981CrossRefGoogle Scholar
  49. 49.
    Hummer, G., Fast-growth thermodynamic integration results for sodium ion hydration, Mol. Simul. 2002, 28, 81-90CrossRefGoogle Scholar
  50. 50.
    Hu, H.; Yun, R. H.; Hermans, J., Reversibility of free energy simulations slow growth may have a unique advantage with a note on use of Ewald summation, Mol. Simul. 2002, 28,67-80CrossRefGoogle Scholar
  51. 51.
    Darve, E.; Pohorille, A., Calculating free energies using average force, J. Chem. Phys. 2001,115,9169-9183CrossRefGoogle Scholar
  52. 52.
    Marszalek, P. E.; Lu, H.; Li, H. B.; Carrion-Vazquez, M.; Oberhauser, A. F.; Schulten, K.; Fernandez, J. M., Mechanical unfolding intermediates in titin modules, Nature 1999, 402, 100-103CrossRefGoogle Scholar
  53. 53.
    Jensen, M. O.; Park, S.; Tajkhorshid, E.; Schulten, K., Energetics of glycerol conduction through aquaglyceroporin glpf, Proc. Natl Acad. Sci. USA 2002, 99, 6731-6736CrossRefGoogle Scholar
  54. 54.
    Amaro, R.; Luthey-Schulten, Z., Molecular dynamics simulations of substrate channeling through an alpha-beta barrel protein, Chem. Phys. 2004, 307, 147-155CrossRefGoogle Scholar
  55. 55.
    Mukamel, S., Quantum extension of the Jarzynski relation analogy with stochastic de-phasing, Phys. Rev. Lett. 2003, 90, 170604CrossRefGoogle Scholar
  56. 56.
    Jarzynski, C.; Wojcik, D. K., Classical and quantum fluctuation theorems for heat exchange, Phys. Rev. Lett. 2004, 92, 230602CrossRefGoogle Scholar
  57. 57.
    De Roeck, W.; Maes, C., Quantum version of free-energy-irreversible-work relations, Phys. Rev. E 2004, 69, 026115CrossRefGoogle Scholar
  58. 58.
    Atilgan, E.; Sun, S. X., Equilibrium free energy estimates based on nonequilibrium work relations and extended dynamics, J. Chem. Phys. 2004, 121, 10392-10400CrossRefGoogle Scholar
  59. 59.
    Ytreberg, F. M.; Zuckerman, D. M., Peptide conformational equilibria computed via a single-stage shifting protocol, J. Phys. Chem. B 2005, 109, 9096-9103CrossRefGoogle Scholar
  60. 60.
    Chernyak, V.; Chertkov, M.; Jarzynski, C., Dynamical generalization of nonequilibrium work relation, Phys. Rev. E 2005, 71, 025102CrossRefGoogle Scholar
  61. 61.
    Rodinger, T.; Pom ès, R., Enhancing the accuracy the efficiency and the scope of free energy simulations, Curr. Opin. Struct. Biol. 2005, 15, 164-170CrossRefGoogle Scholar
  62. 62.
    De Koning, M., Optimizing the driving function for nonequilibrium free-energy calculations in the linear regime. A variational approach, J. Chem. Phys. 2005, 122, 104106CrossRefGoogle Scholar
  63. 63.
    Lua, R. C.; Grosberg, A. Y., Practical applicability of the Jarzynski relation in statistical mechanics. A pedagogical example, J. Phys. Chem. B 2005, 109, 6805-6811CrossRefGoogle Scholar
  64. 64.
    Adib, A. B., Entropy and density of states from isoenergetic nonequilibrium processes, Phys. Rev. E 2005, 71, 056128CrossRefGoogle Scholar
  65. 65.
    Collin, D.; Ritort, F.; Jarzynski, C.; Smith, S.B.; Tinoco, I.; Bustamante, C. Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature 2005,437,231-234CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Gerhard Hummer
    • 1
  1. 1.Laboratory of Chemical PhysicsNational Institute of Diabetes and Digestive and Kidney Diseases National Institutes of HealthBethesda

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