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

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.

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.

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;

Bioreactors Engineering Enzymes

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;

Bioreactors Engineering Enzymes

Bioreactors Engineering Enzymes

Bioreactors Engineering Enzymes HW1 9

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

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:

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

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;

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;

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;

Bioreactors Engineering Enzymes End of Enzyme kinetics