[TOC] | Appendix A Monod
Kinetics and Competitive Inhibition |
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Before describing the particular implementations of the various types of kinetics, it is necessary to review the theory behind Monod kinetics. Monod kinetics are different from, but still based upon Michaelis-Menten kinetics for enzymes. One can think of Monod Kinetics as describing a chain of enzymatically mediated reactions with a limiting step described by Michaelis-Menten kinetics. This is why the equations for both kinetic models are identical. The following paragraphs describe the development and theory behind Michaelis-Menten kinetics.

The basic assumption behind Michaelis-Menten Enzyme kinetics is that enzymes catalyze reactions by first forming an enzyme-substrate complex (Grady and Lim, 1975). This substrate complex will either decay back to enzyme and substrate (the reverse of the previously mentioned reaction) or irreversibly decay to enzyme and product. These chemical reactions for complex formation and product formation respectively are:

(A.1) | |

(A.2) |

Where: S= substrateE= enzymeES= enzyme-substrate complexP= productk= rate constant for complex formation_{1}k= rate constant for reverse complex formation_{2}k= rate constant for product formation_{3}

The rates for the above reactions would be as follows:

(A.3) | |

(A.4) | |

(A.5) |

Where: k= rate constant for complex formation_{1}k= rate constant for complex reverse formation_{2}k= rate constant for product formation_{3}{ S} = concentration of substrate{ E} = concentration of free enzyme{ ES} = concentration of substrate-enzyme complex{ P} = concentration of product concentration

Furthermore, it is assumed the above set of equations are in equilibrium such that:

(A.6) |

Therefore:

(A.7) |

A mass balance on the total enzyme is given as:

(A.8) |

Combining equations (A.7) and (A.8) and substituting into equation (A.5) gives:

(A.9) |

and

(A.10) |

so let

(A.11) |

therefore

(A.12) |

Where: { P} = concentration of product{ ES} = concentration of enzyme-substrate complex{ S} = concentration of substrateE= total complexed and un-complexed enzyme_{T}k= rate constant for complex formation_{1}k= rate constant for reverse complex formation_{2}k= rate constant for product formation_{3}k= "half-saturation" concentration_{m}

Which is analogous to Monod kinetics, *k _{3}* is analogous to the maximum
specific substrate utilization rate,

In competitive inhibition an inhibitory complex can combine with the controlling enzyme in addition to the reaction equation (A.1). This additional complex prohibits the enzyme from forming the complex with the substrate of interest.

(A.13) |

Where: I= inhibitorE= enzymeEI= enzyme-substrate complexP= productk= rate constant for complex formation_{4}k= rate constant for reverse complex formation_{5}

Equation (A.8) now looks like:

(A.14) |

Where: E= total complexed and un-complexed enzyme_{T}{ E} = concentration of free enzyme{ ES} = concentration of substrate-enzyme complex{ EI} = concentration of inhibitor-enzyme complex

After substitution of equation (A.13) (with the assumption of equilibrium) equation (A.14) becomes:

(A.15) |

The same derivation for Michaelis-Menten Kinetics as presented above applies:

(A.16) | |

(A.17) |

so let

(A.18) |

therefore

(A.19) |

Where: { P} = concentration of product{ ES} = concentration of enzyme-substrate complex{ S} = concentration of substrate{ I} = concentration of inhibitorE= total complexed and un-complexed enzyme_{T}k= rate constant for complex formation_{1}k= rate constant for reverse complex formation_{2}k= rate constant for product formation_{3}k= "half-saturation" concentration_{m}k= "saturation" constant for inhibitor_{I}

There are other types of inhibition, such as un-competitive, and substrate inhibition which are not presented here.

Semprini (1991) studied the competitive inhibition of TCE degradation by methane. A double Monod form of inhibition kinetics was used:

(A.20) |

Where: C= concentration of contaminant_{c}C= concentration of the inhibitor_{i}C= concentration of the electron acceptor_{A}K= saturation constant for contaminant_{Sc}K= saturation constant for the electron acceptor_{A}k= maximum transformation rate_{c}K= inhibition constant_{i}

It should be noted equation (A.20) is in the form of double Monod kinetics, however, the first term in the equation is the same form as equation (A.19). The second order electron acceptor term was included since the presence of an electron acceptor was required for the contaminant transformation.

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