Progress Curves in Enzyme Kinetics: Design and Analysis of Experiments

  • Mario Markus
  • Theodor Plesser


Three shortcomings in progress curve analysis are discussed, and solutions or at least optimal strategies to overcome these problems are presented. (i) Systematic deviations in the progress curve data due to errors in the initial concentrations are taken into account in a linear approach by a proper weighting matrix in the parameter optimization. A transformation matrix, derived from the weighting matrix, leads to uncorrelated errors when applied to the data. The statistical tools developed for independent measurements can thus be applied to the transformed data. (ii) Model development by progress curve analysis is very cumbersome, since a progress curve does not reflect clearly the properties of the kinetics by visual inspection. Furthermore, the integration of the rate law leads to long computation times. These problems can be alleviated by determining the rates from the derivatives of a functional approximation of the progress curves. After developing a model using the rates, parameter refinement can be performed by fitting the original progress curve data. This procedure has the further advantage that optimization using the rate data is much less sensitive to the initial parameter estimates. (iii) The lack of inference from the visual inspection of progress curves also affects their experimental design. Methods mentioned in the literature that are applicable to the design of initial rate experiments, are extended for progress curves. The best results are obtained with a discrimination function that includes the statistical expectations of the minimum sum of squares.


Weighting Matrix Pyruvate Kinase Progress Curve Data Design Estimation Design 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Michaelis, L. and Menten, M.L. (1913) Biochem. Z. 49, 333–369.Google Scholar
  2. 2.
    Markus, M. and Plesser, Th. (1976) Biochem. Soc. Trans. 4, 361–364.Google Scholar
  3. 3.
    Haiwachs, W. (1978) Biotechnol. Bioeng. 20, 281–285.CrossRefGoogle Scholar
  4. 4.
    Yun, S.L. and Suelter, C.H. (1977) Biochim. Biophys. Acta, 480, 1–13.Google Scholar
  5. 5.
    Orsi, B.A. and Tipton, K.F. (1979) In “Methods in Enzymology” (S.P. Colowick and N.O. Kaplan, eds.) Vol. 63, pp. 159–183, Academic Press, New York.Google Scholar
  6. 6.
    Quiroga, O.D., Gottifredi, J.C., Mercado Fuentes, L. and de Castillo, M.E.C. (1977) Lat. am. j. chem. eng. appl. chem. 7, 89–101.Google Scholar
  7. 7.
    Garfinkel, L., Kohn, M.C. and Garfinkel, D. (1977) CRC Crit. Rev. Bioeng. 2, 329–361.Google Scholar
  8. 8.
    Boiteux, A., Markus, M., Plesser, Th. and Hess, B. (1980) In “Kinetic Data Analysis: Design and Analysis of Enzyme and Pharmacokinetic Experiments” (L. Endrenyi, ed.) pp. 341–352, Plenum, New York.Google Scholar
  9. 9.
    Garfinkel, D., Marbach, C.B. and Shapiro, N.Z. (1977) Ann. Rev. Biophys. Bioeng. 6, 525–542.CrossRefGoogle Scholar
  10. 10.
    Balcolm, J.K. and Fitch, W.M. (1970) J. Biol. Chem. 245, 1637–1647.Google Scholar
  11. 11.
    Cornish-Bowden, A.J. (1972) Biochem. J. 130, 637–639.Google Scholar
  12. 12.
    Elmore, D.T., Kingston, A.E. and Shields, D.B. (1963) J. Chem. Soc. London, 2070–2078.Google Scholar
  13. 13.
    Sørenson, T.S. and Schack, P. (1972) In “Analysis and Simulation of Biochemical Systems” (H.C. Hemker and B. Hess, eds.) pp. 169–195, North-Holland, Amsterdam.Google Scholar
  14. 14.
    Newman, P.F., Atkins, G.L. and Nimmo, I.A. (1974) Biochem. J. 143, 779–781.Google Scholar
  15. 15.
    Rao, C.R. (1973) “Linear Statistical Inference and its Applications”, Wiley, New York.CrossRefGoogle Scholar
  16. 16.
    Patengill, M.D. and Sands, D.E. (1979) J. Chem. Educ. 56, 244–247.CrossRefGoogle Scholar
  17. 17.
    Bartfai, T. and Mannervik, B. (1972) FEBS Lett. 26, 252–256.CrossRefGoogle Scholar
  18. 18.
    Reich, J.G., Winkler, J. and Zinke, I. (1974) Studia biophys. 43, 77–90.Google Scholar
  19. 19.
    Atkins, G.L. (1976) Biochem. Soc. Trans. 4, 357–361.Google Scholar
  20. 20.
    Boiteux, A., Hess, B., Malcovati, M., Markus, M. and Plesser, Th. (1976) X. Congress of Biochemistry, Hamburg, Abstract 07-4-101.Google Scholar
  21. 21.
    Markus, M., Plesser, Th., Boiteux, A. and Hess, B. (1978) 12th FEBS Meeting, Dresden, Abstract 2753.Google Scholar
  22. 22.
    Segel, I.H. (1975) “Enzyme Kinetics”, p. 275, Wiley, New York.Google Scholar
  23. 23.
    Hopper, M.J. (1979) “Harwell Subroutine Library. A Catalogue of Subroutines”. Computer Science and Systems Division, AERE Harwell, Oxfordshire, England.Google Scholar
  24. 24.
    Gear, G.W. (1971) “Numerical Initial Value Problems in Ordinary Differential Equations”, Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
  25. 25.
    Dixon, L.C.W., Gomulka, J. and Szegoe, G.P. (1975) In “Towards Global Optimization” (L.C.W. Dixon and G.P. Szegoe, eds. ) pp. 29–54.Google Scholar
  26. 26.
    Markus, M., Plesser, Th., Boiteux, A., Hess, B. and Malcovati, M. (1960) Biochem. J. 189, 421–433.Google Scholar
  27. 27.
    Reich, J G. (1970) FEBS Lett. 9, 245–251.CrossRefGoogle Scholar
  28. 28.
    Froment, G.F. (1975) AIChE Journal, 21, 1041–1056.CrossRefGoogle Scholar
  29. 29.
    Fedorov, V.V. and Pazman, A. (1968) Fortschr. Physik, 24, 325–355.CrossRefGoogle Scholar
  30. 30.
    Endrenyi, L. (1980) In “Kinetic Data Analysis: Design and Analysis of Enzyme and Pharmacokinetic Experiments”, (L. Endrenyi, ed.), pp. 137–167, Plenum, New York.Google Scholar
  31. 31.
    Kanyár, B. (1978) Acta Biochim. Biophys. Acad. Sci. Hung. 13, 153–160.Google Scholar
  32. 32.
    Pritchard, D.J. and Bacon, D.W. (1978) Chem. Eng. Sci. 33, 1539–1543.CrossRefGoogle Scholar
  33. 33.
    Hurst, R., Pincock, A. and Broekhoven, L.H. (1973) Biochim. Biophys. Acta, 321, 1–26.Google Scholar
  34. 34.
    Mannervik, B. (1975) Biosystems, 7, 101–119.CrossRefGoogle Scholar
  35. 35.
    Fedorov, V.V. (1972) “Theory of Optimal Experiments”, Academic Press, New York.Google Scholar
  36. 36.
    Retzlaff, G., Rust, G. and Waibel, J. (1975) “Statistische Versuchsplanung”, pp. 116–118, Verlag Chemie, Weinheim.Google Scholar
  37. 37.
    Draper, N.R. and Smith, H. (1967) “Applied Regression Analysis” Wiley, New York.Google Scholar

Copyright information

© Plenum Press, New York 1981

Authors and Affiliations

  • Mario Markus
    • 1
  • Theodor Plesser
    • 1
  1. 1.Max-Planck-Institut fuer ErnaehrungsphysiologieDortmund 1Federal Republic of Germany

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