On series approximations of Michaelis-Menten kinetics

Simon Brown, David Simcock


The Michaelis-Menten reaction mechanism yields a system of nonlinear differential equations that have recently been approximated using various series approaches and, in some cases, [L/L] Padé approximants.  The coefficients of the series obtained from these different techniques are identical, so any differences in perfomance of the approximations are determined by the number of terms or the form of the approximant employed.  Furthermore, the approximations successfully represent only the initial phase of the reaction and do not describe the entire timecourse.  Some improvement can be obtained using Padé approximants of a different form and, from these, we provide an estimate of the time at which the steady-state occurs.


Enzyme Kinetics; Mathematics


Michaelis L, Menten ML. 1913. Die Kinetik der Invertinwirkung. Biochem Z 49:333-369

Hill CM, Waight RD, Bardsley WG. 1977. Does any enzyme follow the Michaelis-Menten equation? Mol Cell Biochem 15(3):173-178, http://dx.doi.org/10.1007/BF01734107

Murray JD. 2002. Mathematical biology. Berlin: Springer-Verlag.

Schnell S, Maini PK. 2003. A century of enzyme kinetics: reliability of the KM and vmax estimates. Comments Theor Biol 8(2-3):169-187, http://dx.doi.org/10.1080/08948550302453

Brown S. 2010. Developing the enzyme-machine analogy: a non-mathematical approach to teaching Michaelis-Menten kinetics. Orbital 2(1):92-100, http://www.orbital.ufms.br/index.php/Chemistry/article/view/80

Briggs GE, Haldane JBS. 1925. A note on the kinetics of enzyme action. Biochem J 19(2):338-339, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1259181/pdf/biochemj01157-0177.pdf

Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE. 1996. On the Lambert W function. Adv Comput Math 5(1):329-359, http://dx.doi.org/10.1007/BF02124750

Atkins GL, Nimmo IA. 1973. The reliability of Michaelis constants and maximum velocities estimated by using the integrated Michaelis-Menten equation. Biochem J 135(4):779-784, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1165894/pdf/biochemj00596-0216.pdf

Darvey IG, Shrager R, Kohn LD. 1975. Integrated steady state rate equations and the determination of individual rate constants. J Biol Chem 250(12):4696-4701, http://www.jbc.org/content/250/12/4696.full.pdf

Duggleby RG, Wood C. 1989. Analysis of progress curves for enzyme-catalysed reactions. Automatic construction of computer programs for fitting integrated rate equations. Biochem J 258(2):397-402, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1138375/pdf/biochemj00212-0088.pdf

Franco R, Aran JM, Canela EI. 1991. Fitting integrated enzyme rate equations to progress curves with the use of a weighting matrix. Biochem J 274:509-511,

Kellershohn N, Laurent M. 1985. Analysis of progress curves for a highly concentrated Michaelian enzyme in the presence of absence of product inhibition. Biochem J 231(1):65-74, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1152704/pdf/biochemj00294-0064.pdf

London JW, Shaw LM, Garfinkel D. 1977. Progress curve algorithm for calculating enzyme activities from kinetic assay spectrophotometric measurements. Anal Chem 49(12):1716-1719, http://dx.doi.org/10.1021/ac50020a021

Nimmo IA, Atkins GL. 1974. A comparison of two methods for fitting the integrated Michaelis-Menten equation. Biochem J 141(3):913-914, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1168197/pdf/biochemj00577-0312.pdf

Russell RW, Drane JW. 1992. Improved rearrangement of the integrated Michaelis-Menten equation for calculating in vivo kinetics of transport and metabolism. J Dairy Sci 75(12):3455-3464, http://dx.doi.org/10.3168/jds.S0022-0302(92)78121-1

Schwert G. 1969. The estimation of kinetic constants for the lactate dehydrogenase system by the use of integrated rate equations. J Biol Chem 244(5):1285-1290, http://www.jbc.org/content/244/5/1285.full.pdf

Schwert G. 1969. Use of integrated rate equations in estimating the kinetic constants of enzyme-catalyzed reactions. J Biol Chem 244(5):1278-1284, http://www.jbc.org/content/244/5/1278.full.pdf

St Maurice M, Bearne SL. 2002. Kinetics and thermodynamics of mandelate racemase catalysis. Biochem 41(12):4048-4058, http://dx.doi.org/10.1021/bi016044h

Chance B. 1943. The kinetics of the enzyme-substrate compound of peroxidase. J Biol Chem 151(2):553-577, http://www.jbc.org/content/151/2/553.full.pdf

Gutfreund H. 1955. Steps in the formation and decomposition of some enzyme-substrate complexes. Disc Faraday Soc 20:167-173, http://dx.doi.org/10.1039/df9552000167

Albe KR, Butler MH, Wright BE. 1990. Cellular concentrations of enzymes and their substrates. J theor Biol 143(2):163-195, http://dx.doi.org/10.1016/S0022-5193(05)80266-8

Minton AP, Wilf J. 1981. Effect of macromolecular crowing upon the structure and function of an enzyme: glyceraldehyde-3-phosphate dehydrogenase. Biochem 20(17):4821-4826, http://dx.doi.org/10.1021/bi00520a003

Segel LA, Slemrod M. 1989. The quasi-steady-state assumption: a case study in perturbation. SIAM Rev 31(3):446-477, http://dx.doi.org/10.1137/1031091

Khader MM. 2013. On the numerical solutions to nonlinear biochemical reaction model using Picard-Padé technique. World J Model Simulation 9(1):38-46, http://worldacademicunion.com/journal/1746-7233WJMS/wjmsvol09no01paper03.pdf

Arafa AAM, Rida SZ, Mohamed H. 2001. An application of the homotopy analysis method to the transient behavior of a biochemical reaction model. Inf Sci Lett 3(1):29-33, http://dx.doi.org/10.12785/isl/030104

Manimozhi P, Rajendran L. 2013. Analytical expression of substrate and enzyme concentration in the Henri-Michaelis-Menten model using homotopy analysis method. Int J Math Arch 4(10):204-214, http://ijma.info/index.php/ijma/article/view/2421

Sen AK. 1988. An application of the Adomian decomposition method to the transient behavior of a model biochemical reaction. J Math Anal Appl 131(1):232-245, http://dx.doi.org/10.1016/0022-247X(88)90202-8

Batiha A-M, Batiha B. 2011. Differential transformation method for a reliable treatment of the nonlinear biochemical reaction model. Adv Stud Biol 3(8):355-360, http://www.m-hikari.com/asb/asb2011/asb5-8-2011/batihaASB5-8-2011.pdf

Picard É. 1893. Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires. J Math Pures Appl 9:217-272, http://sites.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf

Adomian G. 1983. Stochastic systems. New York: Academic Press, Inc.

Liao S. 2004. Beyond perturbation: an introduction to homotopy analysis method. Boca Raton: Chapman Hall/CRC Press.

Pukhov GE. 1980. Differentsial'nye preobrazovaniya funktsiii i urabnenii [Differential transformations of functions and equations]. Kiev: Naukova Dumka.

Thongmoon M, Pusjuso S. 2010. The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations. Nonlin Anal Hybrid Syst 4:425-431, http://dx.doi.org/10.1016/j.nahs.2009.10.006

Eslami M, Zareamoghaddam H. 2011. Differential transform method for Abel differential equation. World Appl Sci J 13(5):1012-1015, http://www.idosi.org/wasj/wasj13(5)/7.pdf

Idrees M, Mahood F, Ali A, Zaman G. 2013. Exact solution for a class of stiff systems by differential transform method. Appl Math 4:440-444, http://dx.doi.org/10.4236/am.2013.43065

Press WH, Teukolsky SA, Vetterling WT, Flannery BP. 1992. Numerical Recipes in C. The Art of Scientific Computing. Cambridge: Cambridge University Press. 994

Vazquez-Leal H, Benhammouda B, Filobello-Nino U, Sarmiento-Reyes A, Jimenez-Fernandez VM, Gacia-Gervacio JL, Huerta-Chua J, Morales-Mendoza LJ, Gonzalez-Lee M. 2014. Direct application of Padé approximant for solving nonlinear differential equations. SpringerPlus 3:563, http://dx.doi.org/10.1186/2193-1801-3-563

Baker GA, Jr, Graves-Morris P. 1981. Padé approximants. Reading: Addison-Wesley Publishing Company.

Shannon AG, Clarke JH. 1983. Polynomial truncation. Math Gaz 67(442):278-280, http://dx.doi.org/10.2307/3617265

Full Text: PDF

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.

SCIENTIA International Identifiers: ISSN: 2282-2119 . DOI prefix: 10.12969 . EAN: 977-2282-211-00-9 . Handle (hdl) prefix: 11167

Scientia ISSN code - Fax: +039 050 7620351