Skip to main content
NCBI home page
Biophysical Journal logoLink to Biophysical Journal
. 2002 Apr;82(4):2244–2255. doi: 10.1016/S0006-3495(02)75570-7

Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding.

Peter J Steinbach 1, Roxana Ionescu 1, C Robert Matthews 1
PMCID: PMC1302017  PMID: 11916879

Abstract

A hybrid analysis that combines the maximum entropy method (MEM) with nonlinear least squares (NLS) fitting has been developed to interpret a general time-dependent signal. Data that include processes of opposite sign and a slow baseline drift can be inverted to obtain both a continuous distribution of lifetimes and a sum of discrete exponentials. Fits by discrete exponentials are performed with initial parameters determined from the distribution of lifetimes obtained with the MEM. The regularization of the parameter space achieved by the MEM stabilizes the introduction of each successive exponential in the NLS fits. This hybrid approach is particularly useful when fitting by a large number of exponentials. Revision of the MEM "prior" based on features in the data can improve the lifetime distribution obtained. Standard errors in the mean are estimated automatically for raw data. The results presented for simulated data and for fluorescence measurements of protein folding illustrate the utility and accuracy of the hybrid algorithm. Analysis of the folding of dihydrofolate reductase reveals six kinetic processes, one more than previously reported.

Full Text

The Full Text of this article is available as a PDF (173.4 KB).

Selected References

These references are in PubMed. This may not be the complete list of references from this article.

  1. Alcala J. R., Gratton E., Prendergast F. G. Resolvability of fluorescence lifetime distributions using phase fluorometry. Biophys J. 1987 Apr;51(4):587–596. doi: 10.1016/S0006-3495(87)83383-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
  2. Austin R. H., Beeson K. W., Eisenstein L., Frauenfelder H., Gunsalus I. C. Dynamics of ligand binding to myoglobin. Biochemistry. 1975 Dec 2;14(24):5355–5373. doi: 10.1021/bi00695a021. [DOI] [PubMed] [Google Scholar]
  3. Bajzer Z., Prendergast F. G. Maximum likelihood analysis of fluorescence data. Methods Enzymol. 1992;210:200–237. doi: 10.1016/0076-6879(92)10012-3. [DOI] [PubMed] [Google Scholar]
  4. Bicout D. J., Szabo A. Entropic barriers, transition states, funnels, and exponential protein folding kinetics: a simple model. Protein Sci. 2000 Mar;9(3):452–465. doi: 10.1110/ps.9.3.452. [DOI] [PMC free article] [PubMed] [Google Scholar]
  5. Bilsel O., Matthews C. R. Barriers in protein folding reactions. Adv Protein Chem. 2000;53:153–207. doi: 10.1016/s0065-3233(00)53004-6. [DOI] [PubMed] [Google Scholar]
  6. Bilsel O., Zitzewitz J. A., Bowers K. E., Matthews C. R. Folding mechanism of the alpha-subunit of tryptophan synthase, an alpha/beta barrel protein: global analysis highlights the interconversion of multiple native, intermediate, and unfolded forms through parallel channels. Biochemistry. 1999 Jan 19;38(3):1018–1029. doi: 10.1021/bi982365q. [DOI] [PubMed] [Google Scholar]
  7. Brooks C. L., 3rd, Onuchic J. N., Wales D. J. Statistical thermodynamics. Taking a walk on a landscape. Science. 2001 Jul 27;293(5530):612–613. doi: 10.1126/science.1062559. [DOI] [PubMed] [Google Scholar]
  8. Frieden C. Refolding of Escherichia coli dihydrofolate reductase: sequential formation of substrate binding sites. Proc Natl Acad Sci U S A. 1990 Jun;87(12):4413–4416. doi: 10.1073/pnas.87.12.4413. [DOI] [PMC free article] [PubMed] [Google Scholar]
  9. Gillespie B., Plaxco K. W. Nonglassy kinetics in the folding of a simple single-domain protein. Proc Natl Acad Sci U S A. 2000 Oct 24;97(22):12014–12019. doi: 10.1073/pnas.97.22.12014. [DOI] [PMC free article] [PubMed] [Google Scholar]
  10. Hagen S. J., Hofrichter J., Eaton W. A. Protein reaction kinetics in a room-temperature glass. Science. 1995 Aug 18;269(5226):959–962. doi: 10.1126/science.7638618. [DOI] [PubMed] [Google Scholar]
  11. Hogiu S., Enescu M., Pascu M. L. Dynamic and thermodynamic effects of glycerol on bovine serum albumin in aqueous solution: a tryptophan phosphorescence study. J Photochem Photobiol B. 1997 Aug;40(1):55–60. doi: 10.1016/s1011-1344(97)00022-5. [DOI] [PubMed] [Google Scholar]
  12. Jennings P. A., Finn B. E., Jones B. E., Matthews C. R. A reexamination of the folding mechanism of dihydrofolate reductase from Escherichia coli: verification and refinement of a four-channel model. Biochemistry. 1993 Apr 13;32(14):3783–3789. doi: 10.1021/bi00065a034. [DOI] [PubMed] [Google Scholar]
  13. Johnson J. B., Lamb D. C., Frauenfelder H., Müller J. D., McMahon B., Nienhaus G. U., Young R. D. Ligand binding to heme proteins. VI. Interconversion of taxonomic substates in carbonmonoxymyoglobin. Biophys J. 1996 Sep;71(3):1563–1573. doi: 10.1016/S0006-3495(96)79359-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
  14. Kuwata K., Shastry R., Cheng H., Hoshino M., Batt C. A., Goto Y., Roder H. Structural and kinetic characterization of early folding events in beta-lactoglobulin. Nat Struct Biol. 2001 Feb;8(2):151–155. doi: 10.1038/84145. [DOI] [PubMed] [Google Scholar]
  15. Lambright D. G., Balasubramanian S., Boxer S. G. Dynamics of protein relaxation in site-specific mutants of human myoglobin. Biochemistry. 1993 Sep 28;32(38):10116–10124. doi: 10.1021/bi00089a030. [DOI] [PubMed] [Google Scholar]
  16. Lavalette D., Tetreau C., Brochon J. C., Livesey A. Conformational fluctuations and protein reactivity. Determination of the rate-constant spectrum and consequences in elementary biochemical processes. Eur J Biochem. 1991 Mar 28;196(3):591–598. doi: 10.1111/j.1432-1033.1991.tb15854.x. [DOI] [PubMed] [Google Scholar]
  17. Leeson D. T., Gai F., Rodriguez H. M., Gregoret L. M., Dyer R. B. Protein folding and unfolding on a complex energy landscape. Proc Natl Acad Sci U S A. 2000 Mar 14;97(6):2527–2532. doi: 10.1073/pnas.040580397. [DOI] [PMC free article] [PubMed] [Google Scholar]
  18. Livesey A. K., Brochon J. C. Analyzing the distribution of decay constants in pulse-fluorimetry using the maximum entropy method. Biophys J. 1987 Nov;52(5):693–706. doi: 10.1016/S0006-3495(87)83264-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
  19. Morozova-Roche L. A., Jones J. A., Noppe W., Dobson C. M. Independent nucleation and heterogeneous assembly of structure during folding of equine lysozyme. J Mol Biol. 1999 Jun 18;289(4):1055–1073. doi: 10.1006/jmbi.1999.2741. [DOI] [PubMed] [Google Scholar]
  20. Plaza del Pino I. M., Parody-Morreale A., Sanchez-Ruiz J. M. Maximum entropy, analysis of kinetic processes involving chemical and folding-unfolding changes in proteins. Anal Biochem. 1997 Jan 15;244(2):239–255. doi: 10.1006/abio.1996.9873. [DOI] [PubMed] [Google Scholar]
  21. Sabelko J., Ervin J., Gruebele M. Observation of strange kinetics in protein folding. Proc Natl Acad Sci U S A. 1999 May 25;96(11):6031–6036. doi: 10.1073/pnas.96.11.6031. [DOI] [PMC free article] [PubMed] [Google Scholar]
  22. Steinbach P. J., Ansari A., Berendzen J., Braunstein D., Chu K., Cowen B. R., Ehrenstein D., Frauenfelder H., Johnson J. B., Lamb D. C. Ligand binding to heme proteins: connection between dynamics and function. Biochemistry. 1991 Apr 23;30(16):3988–4001. doi: 10.1021/bi00230a026. [DOI] [PubMed] [Google Scholar]
  23. Steinbach P. J., Chu K., Frauenfelder H., Johnson J. B., Lamb D. C., Nienhaus G. U., Sauke T. B., Young R. D. Determination of rate distributions from kinetic experiments. Biophys J. 1992 Jan;61(1):235–245. doi: 10.1016/S0006-3495(92)81830-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
  24. Steinbach P. J. Two-dimensional distributions of activation enthalpy and entropy from kinetics by the maximum entropy method. Biophys J. 1996 Mar;70(3):1521–1528. doi: 10.1016/S0006-3495(96)79714-X. [DOI] [PMC free article] [PubMed] [Google Scholar]
  25. Tian WD, Sage JT, Srajer V, V, Champion PM. Relaxation dynamics of myoglobin in solution. Phys Rev Lett. 1992 Jan 20;68(3):408–411. doi: 10.1103/PhysRevLett.68.408. [DOI] [PubMed] [Google Scholar]
  26. Touchette N. A., Perry K. M., Matthews C. R. Folding of dihydrofolate reductase from Escherichia coli. Biochemistry. 1986 Sep 23;25(19):5445–5452. doi: 10.1021/bi00367a015. [DOI] [PubMed] [Google Scholar]
  27. Walkenhorst W. F., Green S. M., Roder H. Kinetic evidence for folding and unfolding intermediates in staphylococcal nuclease. Biochemistry. 1997 May 13;36(19):5795–5805. doi: 10.1021/bi9700476. [DOI] [PubMed] [Google Scholar]
  28. Wallace Louise A., Robert Matthews C. Highly divergent dihydrofolate reductases conserve complex folding mechanisms. J Mol Biol. 2002 Jan 11;315(2):193–211. doi: 10.1006/jmbi.2001.5230. [DOI] [PubMed] [Google Scholar]

Articles from Biophysical Journal are provided here courtesy of The Biophysical Society

RESOURCES