1. Bioreactors Engineering
Enzymes
1. Enzyme Kinetics
1.1 Introduction
A mathematical model of the kinetics of single- substrate-enzyme catalyzed reactions
was first developed by V. C.R Henri in 1902 and by L. Michaelis and Menten in 1913.
Kinetics of simple enzyme – catalyzed reactions are often referred to as
Michaelis-Menten kinetics or saturated kinetics.
The model is based on data from batch reactors with constant liquid volume in which
the initial substrate [So] and enzyme [Eo] concentrations are known.
More complicated enzyme-substrate interactions such as multisubstrate-multienzymes
can take place in biological systems.
Saturation kinetics can be obtained from a simple reaction scheme (Eq. 3.1) that
Involves a reversible step for ES complex formation and a dissociation step of the ES
complex

2. E + S ↔ ES ↔ ES* ↔ EP ↔ E + P Bioreactors Engineering Enzymes
1. Enzyme Kinetics
1.1 Introduction
General Enzyme-Catalyzed Reaction Equation is;
E + S ↔ ES ↔ ES* ↔ EP ↔ E + P
E + S ↔ ES ↔ E + P
Assumptions; the ES complex is established rather rapidly and the rate of the reverse reaction of the second step is negligible
For the enzyme-catalyzed reaction, two approaches are used in developing a rate expression;
1- Rapid- equilibrium approach
2- Quasi- steady state approach.

3. Bioreactors Engineering
Enzymes
1. Enzyme Kinetics
1. 2 Mechanistic Models for Simple Enzyme Kinetics
Both the quasi-steady state approximation and the assumption of rapid equilibrium
share the same few initial steps in deriving a rate expression for the mechanisms in
Eq. 3.1.
The rate of product formation is;
V = rate of product formation or substrate consumption in moles/l-s.
The rate constant K2 = Kcat in the biological literatures.
The rate of variation of the ES complex is;
Since the enzyme is not consumed, the conservation equation on the enzyme yields;
At this point, an assumption is required to achieve an analytical solution.

4. Bioreactors Engineering
Enzymes
1. Enzyme Kinetics
1. 2 Mechanistic Models for Simple Enzyme Kinetics
1.2.1 The rapid equilibrium assumption
Henri and Michaelis and Menten used essentially this approach. Assuming a rapid equilibrium between the E and S to form [ES] complex, we can use the equilibrium coefficient to express [ES] in terms of [S], the equilibrium constant is;

5. Bioreactors Engineering
Enzymes

6. Bioreactors Engineering
Enzymes
1. Enzyme Kinetics
1. 2 Mechanistic Models for Simple Enzyme Kinetics
1.2.2 The quasi-steady-state assumption
In many cases the assumption of rapid equilibrium is not valid. Although the enzyme-substrate reaction still shows saturation –type kinetics.
G. E. Briggs and J. B. S. Haldane first proposed the quasi-steady-state assumption.
A closed system (batch reactor) is used in which the So>>Eo. They suggest that since [Eo] was small, d[ES]/dt=0.
Computer simulations of the actual time course represented by Eqs. 3.2, 3.3 and 3.4 have shown that in a closed system the quasi-steady-state hypothesis holds after a brief transient if [So] >> [Eo] ( for example, 100 times). Fig. 2 displays one such time course.
By applying the quasi-steady-state assumption to eq. 3.3, we find;

7. Bioreactors Engineering
Enzymes

8. Bioreactors Engineering
Enzymes

9. Bioreactors Engineering
Enzymes
HW1 9

10. Bioreactors Engineering
Enzymes
1. Enzyme Kinetics
1.3 Experimentally Determining Rate Parameters for Michaelis-Menten type Kinetics

11. Bioreactors Engineering
Enzymes
1. Enzyme Kinetics
1.3 Experimentally Determining Rate Parameters for Michaelis-Menten type Kinetics
1.3.1 Double-reciprocal plot (Lineweaver-Burk plot)
Eq. 3.12b can be linearized in double-reciprocal form:

12. Bioreactors Engineering
Enzymes
1.3.1 Double-reciprocal plot (Lineweaver-Burk plot)

13. Bioreactors Engineering
Enzymes
1. Enzyme Kinetics
1.3 Experimentally Determining Rate Parameters for Michaelis-Menten type Kinetics
1.3.2 Eadie-Hofstee plot
Rearrangement of Eq. 3.12b yields;

14. Bioreactors Engineering
Enzymes
1. Enzyme Kinetics
1.3 Experimentally Determining Rate Parameters for Michaelis-Menten type Kinetics
1.3.3 Hanes-Woolf plot
Rearrangement of Eq. 3.12b yields;

15. Bioreactors Engineering
Enzymes
1. Enzyme Kinetics
1.3 Experimentally Determining Rate Parameters for Michaelis-Menten type Kinetics
1.3.4 Batch kinetics
The time course of variation of [S] in a batch enzymatic reaction can be determined from;
By integration yield;

16. Bioreactors Engineering
Enzymes
End of Enzyme kinetics