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


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.


Waste Water Water Management Water Pollution Kinetic Parameter Substrate Concentration 
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List of Symbols


coefficient of the Taylor expansion


molar concentration of enzyme


molar concentration of substrate of the i-th type


enzyme molecule


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


number of terms prior to truncature of the Taylor expansion


dummy integer variable


total number of substrates


product molecule of the i-th type


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


substrate molecule of the i-th type


time of reaction

vmax, i

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



at the initial conditions


at the final conditions


dimensionless variable or parameter


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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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