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Bioprocess Engineering

, Volume 10, Issue 4, pp 155–159 | Cite as

Multisubstrate Michaëlis-Menten kinetics: Explicit dependence of substrate concentration on time for batch reactors

  • R. M. Costa
  • F. X. Malcata
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Abstract

The problem of expressing the concentration of each of a set of substrates in a batch reactor as an explicit function of time in the case the multisubstrate system is described by Michaelis-Menten kinetics is tackled in dimensionless form via expansion as a Taylor series. The general analytical form for the coefficients of this expansion is obtained and estimates of the average quadratic error associated with increasing number of terms of the expansion is calculated. Considering the situation where the concentration of all substrates varies in a similar fashion, one concludes that use of four terms yields errors well within the range acceptable for predesign steps. In addition to other applications, the reasoning developed here finds applications in the statistical design and analysis of experiments aiming at the nonlinear fitting of kinetic parameters to actual data.

Keywords

Waste Water Water Management Water Pollution Kinetic Parameter Substrate Concentration 
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.

List of Symbols

an

coefficient of the Taylor expansion

CE

molar concentration of enzyme

CSi

molar concentration of substrate of the i-th type

E

enzyme molecule

ESi

complex of enzyme and substrate of the i-th type

Êq, M

integral quadratic error of the Taylor expansion

kcat, i

kinetic constant associated with production of product of the i-th type

Km, i

Michaëlis-Menten constant associated with substrate of the i-th type

M

number of terms prior to truncature of the Taylor expansion

n

dummy integer variable

N

total number of substrates

Pi

product molecule of the i-th type

ri

rate of reaction involving transformation of the i-th type of substrate

Si

substrate molecule of the i-th type

t

time of reaction

vmax, i

reaction rate under saturating conditions of the enzyme with substrate of the i-th type

Subscripts

o

at the initial conditions

f

at the final conditions

Superscript

dimensionless variable or parameter

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References

  1. 1.
    Bailey, J. E.; Ollis, D. F.: Biochemical engineering fundamentals, p. 100. McGraw-Hill New York (1986)Google Scholar
  2. 2.
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  3. 3.
    Bates, D. M.; Watts, D. G.: Nonlinear regression analysis and its applications, p. 124. Wiley, New York (1988)Google Scholar
  4. 4.
    Malcata, F. X.: Starting D-optimal designs for batch kinetic studies of enzyme-catalyzed reactions in the presence of enzyme deactivation. Biometrics 48 (1992) 929PubMedGoogle Scholar
  5. 5.
    Gradshteyn, I. S.; Ryzhik, I. M.: Table of integrals, series, and products, p. 19. Academic Press, San Diego (1980)Google Scholar
  6. 6.
    Freitas, A. C.; Malcata, F. X.: Michaëlis-Menten kinetics: explicit dependence of substrate concentration on reaction time. Int. J. Math. Ed. Sci. Technol. (in press)Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • R. M. Costa
    • 1
  • F. X. Malcata
    • 1
  1. 1.Escola Superior de BiotecnologiaUniversidade Católica PortuguesaPortoPortugal

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