Simplifying the Analysis of Enzyme Kinetics of Cytochrome *c* Oxidase by the Lambert-W Function

Affiliation(s)

Department of Molecular Spectroscopy, Max Planck Institute for Polymer Research, Mainz, Germany.

Department of Physics, Experimental Molecular Biophysics, Freie Universitaet Berlin, Berlin, Germany.

Department of Chemistry, Physical and Biophysical Chemistry, Bielefeld University, Bielefeld, Germany.

Department of Molecular Spectroscopy, Max Planck Institute for Polymer Research, Mainz, Germany.

Department of Physics, Experimental Molecular Biophysics, Freie Universitaet Berlin, Berlin, Germany.

Department of Chemistry, Physical and Biophysical Chemistry, Bielefeld University, Bielefeld, Germany.

ABSTRACT

Conventional analysis of enzyme-catalyzed reactions uses a set of initial rates of product formation or substrate decay at a variety of substrate concentrations. Alternatively to the conventional methods, attempts have been made to use an integrated Michaelis-Menten equation to assess the values of the Michaelis-Menten K_{M} and turnover *k*_{cat} constants directly from a single time course of an enzymatic reaction. However, because of weak convergence, previous fits of the integrated Michaelis-Menten equation to a single trace of the reaction have no proven records of success. Here we propose a reliable method with fast convergence based on an explicit solution of the Michaelis-Menten equation in terms of the Lambert-W function with transformed variables. Tests of the method with stopped-flow measurements of the catalytic reaction of cytochrome *c* oxidase, as well as with simulated data, demonstrate applicability of the approach to de termine K_{M} and *k*_{cat } constants free of any systematic errors. This study indicates that the approach could be an alternative solution for the characterization of enzymatic reactions, saving time, sample and efforts. The single trace method can greatly assist the real time monitoring of enzymatic activity, in particular when a fast control is mandatory. It may be the only alternative when conventional analysis does not apply, e.g. because of limited amount of sample.

Conventional analysis of enzyme-catalyzed reactions uses a set of initial rates of product formation or substrate decay at a variety of substrate concentrations. Alternatively to the conventional methods, attempts have been made to use an integrated Michaelis-Menten equation to assess the values of the Michaelis-Menten K

Cite this paper

M. Schleeger, J. Heberle and S. Kakorin, "Simplifying the Analysis of Enzyme Kinetics of Cytochrome*c* Oxidase by the Lambert-W Function," *Open Journal of Biophysics*, Vol. 2 No. 4, 2012, pp. 117-129. doi: 10.4236/ojbiphy.2012.24015.

M. Schleeger, J. Heberle and S. Kakorin, "Simplifying the Analysis of Enzyme Kinetics of Cytochrome

References

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[2] S. Schnell and P. K. Maini, “A Century of Enzyme Kinetics: Reliability of the KM and Vmax Estimates,” Comments on Theoretical Biology, Vol. 8, No. 2, 2003, pp. 169-187. doi:10.1080/08948550302453

[3] G. L. Atkins and I. A. Nimmo, “A Comparison of 7 Methods for Fitting the Michaelis-Menten Equation,” Biochemical Journal, Vol. 149, No. 3, 1975, pp. 775-777.

[4] R. Walsh, E. Martin and S. Darvesh, “A Versatile Equation to Describe Reversible Enzyme Inhibition and Activation Kinetics: Modeling β-Galactosidase and Butyrylcholinesterase,” Biochimica et Biophysica Acta, Vol. 1770, No. 5, 2007, pp. 733-746. doi:10.1016/j.bbagen.2007.01.001

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[7] H. Lineweaver and D. Burk, “The Determination of Enzyme Dissociation Constants,” Journal of the American Chemical Society, Vol. 56, No. 3, 1934, pp. 658-666. doi:10.1021/ja01318a036

[8] G. S. Eadie, “The Inhibition of Cholinesterase by Physostigmine and Prostigmine,” Journal of Biological Chemistry, Vol. 146, No. 1, 1942, pp. 85-93.

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[11] R. G. Duggleby and J. F. Morrison, “The Analysis of Progress Curves for Enzyme-Catalyzed Reactions by Non-Linear Regression,” Biochimica et Biophysica Acta, Vol. 481, No. 2, 1977, pp. 297-312. doi:10.1016/0005-2744(77)90264-9

[12] F. Liao, X.-Y. Zhu, Y.-M. Wang and Y.-P. Zuo, “The Comparison of the Estimation of Enzyme Kinetic Parameters by Fitting Reaction Curve to the Integrated Michaelis- Menten Rate Equations of Different Predictor Variables,” Journal of Biochemical and Biophysical Methods, Vol. 62, No. 1, 2005, pp. 13-24. doi:10.1016/j.jbbm.2004.06.010

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[14] C. T. Goudar, J. R. Sonnad and R. G. Duggleby, “Parameter Estimation Using a Direct Solution of the Integrated Michaelis-Menten Equation,” Biochimica et Biophysica Acta, Vol. 1429, No. 2, 1999, pp. 377-383. doi:10.1016/S0167-4838(98)00247-7

[15] C. T. Goudar, S. K. Harris, M. J. McInerney and J. M. Suflita, “Progress Curve Analysis for Enzyme and Microbial Kinetic Reactions Using Explicit Solutions Based on the Lambert W Function,” Journal of Microbiological Methods, Vol. 59, No. 3, 2004, pp. 317-326. doi:10.1016/j.mimet.2004.06.013

[16] C. T. Goudar and T. G. Ellis, “Explicit Oxygen Concentration Expression for Estimating Biodegradation Kinetics from Respirometric Experiments,” Biotechnology and Bioengineering, Vol. 75, No. 1, 2001, pp. 74-81. doi:10.1002/bit.1166

[17] R. Walsh, E. Martin and S. Darvesh, “A Method to Describe Enzyme-Catalyzed Reactions by Combining Steady State and Time Course Enzyme Kinetic Parameters,” Biochimica et Biophysica Acta, Vol. 1800, No. 1, 2010, pp. 1-5. doi:10.1016/j.bbagen.2009.10.007

[18] C. E. Cooper, “The Steady-State Kinetics of Cytochrome c Oxidation by Cytochrome Oxidase,” Biochimica et Biophysica Acta, Vol. 1017, No. 3, 1990, pp. 187-203. doi:10.1016/0005-2728(90)90184-6

[19] H.-M. Lee, T. K. Das, D. L. Rousseau, D. Mills, S. Ferguson-Miller and R. B. Gennis, “Mutations in the Putative H-Channel in the Cytochrome c Oxidase from Rhodobacter Sphaeroides Show That This Channel is not Important for Proton Conduction but Reveal Modulation of the Properties of Heme a,” Biochemistry, Vol. 39, No. 11, 2000, pp. 2989-2996. doi:10.1021/bi9924821

[20] B. Chance, “Techniques for the Assay of the Respiratory Enzymes,” Methods in Enzymology, Vol. 4, 1957, pp. 273- 329. doi:10.1016/0076-6879(57)04060-4

[21] M. V. Putz, A. M. Lacrama and V. Ostafe, “Full Analytic Progress Curves of Enzymic Reactions in Vitro,” International Journal of Molecular Sciences, Vol. 7, No. 11, 2006, pp. 469-484. doi:10.3390/i7110469

[22] A. R. Tzafriri and E. R. Edelman, “The Total Quasi-Steady-State Approximation is Valid for Reversible Enzyme Kinetics,” Journal of Theoretical Biology, Vol. 226, No. 3, 2004, pp. 303-313. doi:10.1016/j.jtbi.2003.09.006

[23] S. Schnell and C. Mendoza, “Enzyme Kinetics of Multiple Alternative Substrates,” Journal of Mathematical Chemistry, Vol. 27, No. 1-2, 2000, pp. 155-170. doi:10.1023/A:1019139423811

[1] L. Michaelis and M. L. Menten, “Die Kinetik der Invertinwirkung,” Biochemische Zeitschrift, Vol. 49, 1913, pp. 333-369.

[2] S. Schnell and P. K. Maini, “A Century of Enzyme Kinetics: Reliability of the KM and Vmax Estimates,” Comments on Theoretical Biology, Vol. 8, No. 2, 2003, pp. 169-187. doi:10.1080/08948550302453

[3] G. L. Atkins and I. A. Nimmo, “A Comparison of 7 Methods for Fitting the Michaelis-Menten Equation,” Biochemical Journal, Vol. 149, No. 3, 1975, pp. 775-777.

[4] R. Walsh, E. Martin and S. Darvesh, “A Versatile Equation to Describe Reversible Enzyme Inhibition and Activation Kinetics: Modeling β-Galactosidase and Butyrylcholinesterase,” Biochimica et Biophysica Acta, Vol. 1770, No. 5, 2007, pp. 733-746. doi:10.1016/j.bbagen.2007.01.001

[5] B. H. J. Hofstee, “Non-Inverted Versus Inverted Plots in Enzyme Kinetics,” Nature, Vol. 184, 1959, pp. 1296-1298. doi:10.1038/1841296b0

[6] G. E. Briggs and J. B. Haldane, “A Note on the Kinetics of Enzyme Action,” Biochemical Journal, Vol. 19, No. 2, 1925, pp. 338-339.

[7] H. Lineweaver and D. Burk, “The Determination of Enzyme Dissociation Constants,” Journal of the American Chemical Society, Vol. 56, No. 3, 1934, pp. 658-666. doi:10.1021/ja01318a036

[8] G. S. Eadie, “The Inhibition of Cholinesterase by Physostigmine and Prostigmine,” Journal of Biological Chemistry, Vol. 146, No. 1, 1942, pp. 85-93.

[9] G. L. Atkins and I. A. Nimmo, “The Reliability of Michaelis Constants and Maximum Velocities Estimated by Using the Integrated Michaelis-Menten Equation,” Biochemical Journal, Vol. 135, No. 4, 1973, pp. 779-784.

[10] H. N. Fernley, “Statistical Estimations in Enzyme Kinetics. The Integrated Michaelis Equation,” European Journal of Biochemistry, Vol. 43, No. 2, 1974, pp. 377-378. doi:10.1111/j.1432-1033.1974.tb03423.x

[11] R. G. Duggleby and J. F. Morrison, “The Analysis of Progress Curves for Enzyme-Catalyzed Reactions by Non-Linear Regression,” Biochimica et Biophysica Acta, Vol. 481, No. 2, 1977, pp. 297-312. doi:10.1016/0005-2744(77)90264-9

[12] F. Liao, X.-Y. Zhu, Y.-M. Wang and Y.-P. Zuo, “The Comparison of the Estimation of Enzyme Kinetic Parameters by Fitting Reaction Curve to the Integrated Michaelis- Menten Rate Equations of Different Predictor Variables,” Journal of Biochemical and Biophysical Methods, Vol. 62, No. 1, 2005, pp. 13-24. doi:10.1016/j.jbbm.2004.06.010

[13] S. Schnell and C. Mendoza, “Closed Form Solution for Time-Dependent Enzyme Kinetics,” Journal of Theoretical Biology, Vol. 187, No. 2, 1997, pp. 207-212. doi:10.1006/jtbi.1997.0425

[14] C. T. Goudar, J. R. Sonnad and R. G. Duggleby, “Parameter Estimation Using a Direct Solution of the Integrated Michaelis-Menten Equation,” Biochimica et Biophysica Acta, Vol. 1429, No. 2, 1999, pp. 377-383. doi:10.1016/S0167-4838(98)00247-7

[15] C. T. Goudar, S. K. Harris, M. J. McInerney and J. M. Suflita, “Progress Curve Analysis for Enzyme and Microbial Kinetic Reactions Using Explicit Solutions Based on the Lambert W Function,” Journal of Microbiological Methods, Vol. 59, No. 3, 2004, pp. 317-326. doi:10.1016/j.mimet.2004.06.013

[16] C. T. Goudar and T. G. Ellis, “Explicit Oxygen Concentration Expression for Estimating Biodegradation Kinetics from Respirometric Experiments,” Biotechnology and Bioengineering, Vol. 75, No. 1, 2001, pp. 74-81. doi:10.1002/bit.1166

[17] R. Walsh, E. Martin and S. Darvesh, “A Method to Describe Enzyme-Catalyzed Reactions by Combining Steady State and Time Course Enzyme Kinetic Parameters,” Biochimica et Biophysica Acta, Vol. 1800, No. 1, 2010, pp. 1-5. doi:10.1016/j.bbagen.2009.10.007

[18] C. E. Cooper, “The Steady-State Kinetics of Cytochrome c Oxidation by Cytochrome Oxidase,” Biochimica et Biophysica Acta, Vol. 1017, No. 3, 1990, pp. 187-203. doi:10.1016/0005-2728(90)90184-6

[19] H.-M. Lee, T. K. Das, D. L. Rousseau, D. Mills, S. Ferguson-Miller and R. B. Gennis, “Mutations in the Putative H-Channel in the Cytochrome c Oxidase from Rhodobacter Sphaeroides Show That This Channel is not Important for Proton Conduction but Reveal Modulation of the Properties of Heme a,” Biochemistry, Vol. 39, No. 11, 2000, pp. 2989-2996. doi:10.1021/bi9924821

[20] B. Chance, “Techniques for the Assay of the Respiratory Enzymes,” Methods in Enzymology, Vol. 4, 1957, pp. 273- 329. doi:10.1016/0076-6879(57)04060-4

[21] M. V. Putz, A. M. Lacrama and V. Ostafe, “Full Analytic Progress Curves of Enzymic Reactions in Vitro,” International Journal of Molecular Sciences, Vol. 7, No. 11, 2006, pp. 469-484. doi:10.3390/i7110469

[22] A. R. Tzafriri and E. R. Edelman, “The Total Quasi-Steady-State Approximation is Valid for Reversible Enzyme Kinetics,” Journal of Theoretical Biology, Vol. 226, No. 3, 2004, pp. 303-313. doi:10.1016/j.jtbi.2003.09.006

[23] S. Schnell and C. Mendoza, “Enzyme Kinetics of Multiple Alternative Substrates,” Journal of Mathematical Chemistry, Vol. 27, No. 1-2, 2000, pp. 155-170. doi:10.1023/A:1019139423811