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Systematic Finite-Sampling Inaccuracy in Free Energy Differences and Other Nonlinear Quantities

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Abstract

Systematic inaccuracy is inherent in any computational estimate of a non-linear average, such as the free energy difference ΔF between two states or systems, because of the availability of only a finite number of data values, N. In previous work, we outlined the fundamental statistical description of this “finite-sampling error.” We now give a more complete presentation of (i) rigorous general bounds on the free energy and other nonlinear averages, which generalize Jensen's inequality; (ii) asymptotic N→∞ expansions of the average behavior of the finite-sampling error in ΔF estimates; (iii) illustrative examples of large-N behavior, both in free-energy and other calculations; and (iv) the universal, large-N relation between the average finite-sampling error and the fluctuation in the error. An explicit role is played by Lévy and Gaussian limiting distributions.

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REFERENCES

  1. S. B. Singh, Ajay, D. E. Wemmer, and P. A. Kollman, Proc. Nat. Acad. Sci. (USA)91:7673(1994).

  2. C. Jarzynski, Phys. Rev. Lett. 78:2690(1997).

    Google Scholar 

  3. C. Jarzynski, Phys. Rev. E 56:5018(1997).

    Google Scholar 

  4. A. D. Bruce, N. B. Wilding, and G. J. Ackland, Phys. Rev. Lett. 79:3002(1997).

    Google Scholar 

  5. N. Lu and D. A. Kofke, J. Chem. Phys. 111:4414(1999).

    Google Scholar 

  6. N. Lu and D. A. Kofke, J. Chem. Phys. 114:7303(2001).

    Google Scholar 

  7. N. Lu and D. A. Kofke, J. Chem. Phys. 115:6866(2001).

    Google Scholar 

  8. M. de Koning, A. Antonelli, and S. Yip, Phys. Rev. Lett. 83:3973(1999).

    Google Scholar 

  9. G. Hummer and A. Szabo, Proc. Nat. Acad. Sci. (USA) 98:3658(2001).

    Google Scholar 

  10. C. Jarzynski, Proc. Nat. Acad. Sci. (USA) 98:3636(2001).

    Google Scholar 

  11. B. Isralewitz, M. Gao, and K. Schulten, Curr. Opin. Struc. Bio. 11:224(2001).

    Google Scholar 

  12. J. Liphardt et al., Science 296:1832(2002).

    Google Scholar 

  13. D. M. Zuckerman and T. B. Woolf, Chem. Phys. Lett. 351:445(2002).

    Google Scholar 

  14. D. M. Zuckerman and T. B. Woolf, Phys. Rev. Lett. 89:180602(2002).

    Google Scholar 

  15. W. P. Reinhardt and J. E. Hunter, J. Chem. Phys. 97:1599(1992).

    Google Scholar 

  16. B. L. Tembe and J. A. McCammon, Computers Chem. 8:281(1984).

    Google Scholar 

  17. D. Beveridge and F. DiCapua, Ann. Rev. Biophys. Biophys. Chem. 18:431(1989).

    Google Scholar 

  18. J. A. McCammon, Curr Opin. Struc. Bio. 2:96(1991).

    Google Scholar 

  19. P. A. Kollman, Chemical Reviews 93:2395(1993).

    Google Scholar 

  20. B. Widom, J. Chem. Phys. 39:2808(1963).

    Google Scholar 

  21. J. P. Valleau and D. N. Card, J. Chem. Phys. 57:5457(1972).

    Google Scholar 

  22. C. H. Bennett, J. Comp. Phys. 22:245(1976).

    Google Scholar 

  23. W. L. Jorgensen and C. Ravimohan, J. Chem. Phys. 83:3050(1985).

    Google Scholar 

  24. A. J. Cross, Ann. N.Y. Acad. Sci. 482:89(1986).

    Google Scholar 

  25. M. Watanabe and W. P. Reinhardt, Phys. Rev. Lett. 65:3301(1990).

    Google Scholar 

  26. R. H. Wood, J. Phys. Chem. 95:4838(1991).

    Google Scholar 

  27. J. Hermans, J. Phys. Chem. 95:9029(1991).

    Google Scholar 

  28. A. E. Mark, W. F. van Gunsteren, and H. J. C. Berendsen, J. Chem. Phys. 94:3808(1991).

    Google Scholar 

  29. M. Mezei, Molec. Sim. 10:225(1993).

    Google Scholar 

  30. A. Hodel, T. Simonson, R. O. Rox, and A. T. Brü;nger, J. Phys. Chem. 97:3409(1993).

    Google Scholar 

  31. G. Hummer and A. Szabo, J. Chem. Phys. 105:2004(1996).

    Google Scholar 

  32. J. C. Schö;n, J. Chem. Phys. 105:10072(1996).

    Google Scholar 

  33. X. Kong and C. M. Brooks, J. Chem. Phys. 105:2414(1996).

    Google Scholar 

  34. C. Jarque and B. Tidor, J. Phys. Chem. B 101:9402(1997).

    Google Scholar 

  35. M. A. Miller and W. P. Reinhardt, J. Chem. Phys. 113:7035(2000).

    Google Scholar 

  36. S. Shobana, B. Roux, and O. S. Andersen, J. Phys. Chem. B 104:5179(2000).

    Google Scholar 

  37. G. Hummer, J. Chem. Phys. 114:7330(2001).

    Google Scholar 

  38. J. W. Pitera and W. F. van Gunsteren, J. Phys. Chem. B 105:11264(2001).

    Google Scholar 

  39. H. Hu, R. H. Yun, and J. Hermans, Molec. Sim. 28:67(2002).

    Google Scholar 

  40. R. H. Wood, W. C. F. Mü;hlbauer, and P. T. Thompson, J. Phys. Chem. 95:6670(1991).

    Google Scholar 

  41. A. D. Stone and J. D. Joannopoulos, Phys. Rev. E 25:2400(1982).

    Google Scholar 

  42. A. M. Ferrenberg, D. P. Landau, and K. Binder, J. Stat. Phys. 63:867(1991).

    Google Scholar 

  43. D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University, Cambridge, 2000).

    Google Scholar 

  44. H. Nanda and T. B. Woolf, in preparation. (unpublished).

  45. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2 (Wiley, New York, 1971).

    Google Scholar 

  46. V. M. Zolotarev, One-dimensional Stable Distributions (American Mathematical Soc., Providence, 1986).

    Google Scholar 

  47. V. V. Uchaikin and V. M. Zolotarev, Chance and Stability: Stable Distributions and Their Applications (VSP, Utrecht, 1999).

    Google Scholar 

  48. M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch, Lévy Flights and Related Topics in Physics (Springer, Berlin, 1995).

    Google Scholar 

  49. J. G. Kirkwood, J. Chem. Phys. 3:300(1935).

    Google Scholar 

  50. K. S. Shing and K. E. Gubbins, Molec. Phys. 46:1109(1982).

    Google Scholar 

  51. G. H. Hardy, J. E. Littlewood, and G. Pó;lya, Inequalities (Cambridge University Press, Cambridge, 1967).

    Google Scholar 

  52. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, New York, 1991).

    Google Scholar 

  53. R. W. Ash, Basic Probability Theory (Wiley, New York, 1970).

    Google Scholar 

  54. G. Christoph and W. Wolf, Convergence Theorems with a Stable Limit Law (Akadmie-Verlag, Berlin, 1992).

    Google Scholar 

  55. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

    Google Scholar 

  56. V. V. Petrov, Limit Theorems of Probability Theory (Oxford University Press, Oxford, 1995).

    Google Scholar 

  57. H. Meirovitch, J. Chem. Phys. 111:7215(1999).

    Google Scholar 

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Authors and Affiliations

  1. Center for Computational Biology & Bioinformatics, University of Pittsburgh, 200 Lothrop Street, Pittsburgh, Pennsylvania, 15261

    Daniel M. Zuckerman

  2. Department of Environmental & Occupational Health, Graduate School of Public Health, University of Pittsburgh, USA

    Daniel M. Zuckerman

  3. Department of Physiology and Department of Biophysics, Johns Hopkins University School of Medicine, Baltimore, Maryland, 21205

    Thomas B. Woolf

Authors
  1. Daniel M. Zuckerman
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  2. Thomas B. Woolf
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Zuckerman, D.M., Woolf, T.B. Systematic Finite-Sampling Inaccuracy in Free Energy Differences and Other Nonlinear Quantities. Journal of Statistical Physics 114, 1303–1323 (2004). https://doi.org/10.1023/B:JOSS.0000013961.84860.5b

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  • DOI: https://doi.org/10.1023/B:JOSS.0000013961.84860.5b

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