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Free R value: a novel statistical quantity for assessing the accuracy of crystal structures


THE determination of macromolecular structure by crystallography involves fitting atomic models to the observed diffraction data1. The traditional measure of the quality of this fit, and presumably the accuracy of the model, is theR value. Despite stereochemical restraints2, it is possible to overfit or 'misfit' the diffraction data: an incorrect model can be refined to fairly good R values as several recent examples have shown3. Here I propose a reliable and unbiased indicator of the accuracy of such models. By analogy with the cross-validation method4,5 of testing statistical models I define a statistical quantity (RfreeT) that measures the agreement between observed and computed structure factor amplitudes for a 'test' set of reflections that is omitted in the modelling and refinement process. As examples show, there is a high correlation between RfreeT and the accuracy of the atomic model phases. This is useful because experimental phase information is usually inaccurate, incomplete or unavailable. I expect that RfreeT will provide a measure of the information content of recently proposed models of thermal motion and disorder6–8, time-averaging9 and bulk solvent10.

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  1. Jensen, L. H. Meth. Enzym. 115, 227–234 (1985).

    CAS  Article  Google Scholar 

  2. Hendrickson, W. A. Meth. Enzym. 115, 252–270 (1985).

    CAS  Article  Google Scholar 

  3. Bränden, C. I. & Jones, A. Nature 343, 687–689 (1990).

    ADS  Article  Google Scholar 

  4. Mosteller, F. & Tukey, J. W. Data Analysis and Regression: a Second Course in Statistics (Addison-Wesley, Reading, Massachusetts, 1977).

    Google Scholar 

  5. Efron, B. & Tibshirani, R. Science 253, 390–395 (1991).

    ADS  CAS  Article  Google Scholar 

  6. Diamond, R. Acta crystallogr. A46, 425–435 (1990).

    CAS  Article  Google Scholar 

  7. Kuriyan, J. & Weis, W. I. Proc. natn. Acad. Sci. U.S.A. 88, 2773–2777 (1991).

    ADS  CAS  Article  Google Scholar 

  8. Kuriyan, J. et al. Proteins 10, 340–358 (1991).

    CAS  Article  Google Scholar 

  9. Gros, P., Gunsteren, W. F. & Hol, W. G. J. Science 249, 1149–1152 (1990).

    ADS  CAS  Article  Google Scholar 

  10. Badger, J. & Caspar, D. L. D. Proc. natn. Acad. Sci. U.S.A. 88, 622–626 (1991).

    ADS  CAS  Article  Google Scholar 

  11. Stout, G. H. & Jensen, L. H. in X-ray Structure Determination, a Practical Guide 2nd edn 343–378 (Wiley, New York, 1989).

    Google Scholar 

  12. Press, W. H., Flannery, B. P., Teukolosky, S. A. & Vetterling, W. T. in Numerical Recipes 498–504 (Cambridge University Press, 1986).

    Google Scholar 

  13. Hamilton, W. C. Acta crystallogr. 18, 502–510 (1965).

    CAS  Article  Google Scholar 

  14. Hsu, I.-N., Delbare, L. T. J., James, M. N. G. & Hofmann, T. Nature 266, 140–145 (1977).

    ADS  CAS  Article  Google Scholar 

  15. James, M. N. G. & Sielecki, A. R. J. molec. Biol. 163, 299–361 (1983).

    CAS  Article  Google Scholar 

  16. Brünger, A. T., Kuriyan, J. & Karplus, M. Science 235, 458–460 (1987).

    ADS  Article  Google Scholar 

  17. Curmi, P. A. M., Schreuder, H., Cascio, D., Sweet, R. M. & Eisenberg, D. J. biol. Chem. (in the press).

  18. Chapman, M. S. et al. Science 241, 71–74 (1988).

    ADS  CAS  Article  Google Scholar 

  19. Brünger, A. T. A. Rev. Phys. Chem. 42, 197–233 (1991).

    ADS  Article  Google Scholar 

  20. Jack, A. & Levitt, M. Acta crystallogr. A34, 931–935 (1978).

    ADS  CAS  Article  Google Scholar 

  21. Brooks, B. R. et al. J. comp. Chem. 4, 187–217 (1983).

    CAS  Article  Google Scholar 

  22. Konnert, J. H. & Hendrickson, W. A. Acta crystallogr. A36, 344–349 (1980).

    ADS  CAS  Article  Google Scholar 

  23. Ernst, R. R., Bodenhausen, G. & Wokaun, A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Clarendon, Oxford, 1987).

    Google Scholar 

  24. Wüthrich, K., NMR of Proteins and Nucleic Acids (Wiley, New York, 1986).

    Book  Google Scholar 

  25. Brünger, A. T. J. molec. Biol. 203, 803–816 (1988).

    Article  Google Scholar 

  26. Brünger, A. T., Krukowski, A. & Erickson, J. Acta crystallogr. A46, 585–593 (1900).

    Google Scholar 

  27. Brünger, A. T. X-PLOR, Version 2.1, Yale University (1990).

  28. Read, R. Acta crystallogr. A42, 140–149 (1986).

    CAS  Article  Google Scholar 

  29. Clore, G. M. & Gronenborn, A. M. Science 252, 1390–1399 (1991).

    ADS  CAS  Article  Google Scholar 

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Brünger, A. Free R value: a novel statistical quantity for assessing the accuracy of crystal structures. Nature 355, 472–475 (1992).

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