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Mathematica Slovaca

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Volume 59, Issue 5 (Oct 2009)


Estimation, model discrimination, and experimental design for implicitly given nonlinear models of enzyme catalyzed chemical reactions

Anna Siudak / Eric Lieres / Christine Müller
Published Online: 2009-10-27 | DOI: https://doi.org/10.2478/s12175-009-0150-3


Many nonlinear models as e.g. models of chemical reactions are described by systems of differential equations which have no explicit solution. In such cases the statistical analysis is much more complicated than for nonlinear models with explicitly given response functions. Numerical approaches need to be applied in place of explicit solutions. This paper describes how the analysis can be done when the response function is only implicitly given by differential equations. It is shown how the unknown parameters can be estimated and how these estimations can be applied for model discrimination and for deriving optimal designs for future research. The methods are demonstrated with a chemical reaction catalyzed by the enzyme Benzaldehyde lyase.

MSC: Primary 62J02, 62H12, 62K05, 62P10, 62P30

Keywords: implicitly given nonlinear model; differential equations; estimation; least trimmed squares; model discrimination; experimental design

  • [1] DEMIR, A. S.— EREN, E.— HOSRIK, B.— Şeşeoglu, Ö.— POHL, M.— JANZEN, E.— KOLTER, D.— FELDMANN, R.— DÜNKELMANN, P.— MÜLLER, M.: Enantioselective Synthesis of α-Hydroxy Ketones via Benzaldehyde Lyase-Catalyzed C-C Bond Formation Reaction, Advanced Synthesis & Catalysis 344(1), (2002), 96–103. http://dx.doi.org/10.1002/1615-4169(200201)344:1<96::AID-ADSC96>3.0.CO;2-ZCrossrefGoogle Scholar

  • [2] DORMAND, J. R.— PRINCE, P. J.: A family of embedded Runge-Kutta formulae, J. Comput. Math. 10, (1958), 517–534. Google Scholar

  • [3] HURVICH, C. M.— TSAI, C. L.: Regression and time series model selection in small samples, Biometrika 76, (1989), 297–307. http://dx.doi.org/10.1093/biomet/76.2.297CrossrefGoogle Scholar

  • [4] STROMBERG, A. J.— RUPPERT, D.: Breakdown in nonlinear regression, J. Amer. Statist. Assoc. 87 (1992), 991–997. http://dx.doi.org/10.2307/2290636CrossrefGoogle Scholar

  • [5] ATKINSON, A. C.— DONEV, A. N.: Optimum Experimental Designs. Oxford Statistical Science Series, Oxford University Press, Oxford, 1992. Google Scholar

  • [6] BUNKE, H.— BUNKE, O.: Nonlinear Regression, Functional Relations and Robust Methods, John Wiley & Sons, Inc., Berlin, 1989. Google Scholar

  • [7] EFRON, B.— TIBSHIRANI, R. J.: An Introduction to the Bootstrap. Monogr. Statist. Appl. Probab. 57, Chapman & Hall/CRC, Boca Raton, 1998. Google Scholar

  • [8] FAHRMEIR, L.— HAMERLE, A.: Multivariate statistische Verfahren, Walter deGruyter & Co., Berlin, 1995. Google Scholar

  • [9] KÜHL, S.: Enzymkatalysierte C-C Knüpfung: Reaktionstechnische Untersuchungen zur Synthese pharmazeutischer Intermediate, Universitt Bonn, Bonn, 2007. Google Scholar

  • [10] PÁZMAN, A.: Foundations of Optimum Experimental Design, Reidel, Dordrecht, 1986. Google Scholar

  • [11] PÁZMAN, A.: Nonlinear Statistical Models, Kluwer, Dordrecht, 1993. Google Scholar

  • [12] ROUSSEEUW, P. J.— LEROY, A. M.: Robust Regression and Outlier Detection, Wiley, New York, 1987. http://dx.doi.org/10.1002/0471725382CrossrefGoogle Scholar

  • [13] SEBER, G. A. F.— WILD, C. J.: Nonlinear Regression, John Wiley & Sons, Inc., Hoboken, NJ, 2003. Google Scholar

  • [14] STREHMEL, K.— WEINER, R.: Numerik gewhnlicher Differentialgleichungen, B. G. Teubner, Stuttgart, 1995. Google Scholar

About the article

Published Online: 2009-10-27

Published in Print: 2009-10-01

Citation Information: Mathematica Slovaca, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.2478/s12175-009-0150-3.

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© 2009 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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