# 9.2: Michaelis-Menten Kinetics

In Sec. 8.4 we presented a brief analysis of the cell growth phenomena that occurs in a chemostat. In addition we presented the well-known Monod equation that has been used extensively to model macroscopic cell growth. In this section, we briefly explore an enzyme-catalyzed reaction that occurs in all cellular systems. Within a cell, such as the one illustrated in Figure $$\PageIndex{1}$$, hundreds of reactions occur.

To appreciate the complexity of cells, we note that a typical eukaryotic cell contains the following subcellular components: nucleolus, nucleus, ribosome, vesicle, rough endoplasmic reticulum, Golgi apparatus, Cytoskeleton, smooth endoplasmic reticulum, mitochondria, vacuole, cytoplasm, lysosome, and centrioles within centrosome13. Obviously the cell is a busy place, and much of that business is associated with the enzyme-catalyzed reactions that produce intracellular material represented by species $$D$$ and extracellular material represented by species $$C$$. Species $$D$$ provides the material that leads to cell growth as described in Sec. 8.4, while species $$C$$ provides desirable products to be harvested by chemical engineers and others.

## Catalysts

A catalyst is an agent that causes an increase in the reaction rate without undergoing any permanent change, and the enzymes represented by species $$E$$ in Figure $$\PageIndex{1}$$ perform precisely that function in the production of intracellular and extracellular material. Enzymes are global proteins that bind substrates (reactants) in particular configurations that enhance the reaction rate. The simplest description of this process is due to Michaelis and Menten14 who proposed a two-step process in which a substrate first binds reversibly with an enzyme and then reacts irreversibly to form a product. In this development we first consider the substrate $$A$$, the enzyme $$E$$, and the product $$D$$. To begin with, the enzyme $$E$$ forms a complex with substrate $$A$$ in a reversible manner as indicated by Eqs. \ref{64a} and \ref{65a}.

Elementary chemical kinetic schema I:

$E+A \stackrel{k_{I}}{\longrightarrow} E A \label{64a}$

Elementary stoichiometry I:

$R_{E}^{I}=-R_{E A}^{I}, \quad R_{E}^{I}=R_{A}^{I} \label{64b}$

Elementary chemical reaction rate equation I:

$R_{E}^{I}=-k_{I} c_{E} c_{A} \label{64c}$

Elementary chemical kinetic schema II:

$E A \quad \stackrel{k_{II}}{\longrightarrow} E+A \label{65a}$

Elementary stoichiometry II:

$R_{E A}^{II}=-R_{E}^{II}, \quad R_{E A}^{II}=-R_{A}^{II} \label{65b}$

Elementary chemical kinetic rate equation II:

$R_{E A}^{II}=-k_{II} c_{E A} \label{65c}$

In the final step, the complex $$EA$$ reacts irreversibly to form the product $$D$$ and the enzyme $$E$$ according to

Elementary chemical kinetic schema III:

$E A \stackrel{k_{III}}{\longrightarrow} E+D \label{66a}$

Elementary stoichiometry III:

$R_{E A}^{\mathrm{III}}=-R_{E}^{III}, \quad R_{E A}^{III}=-R_{D}^{III} \label{66b}$

Elementary chemical reaction rate equation III:

$R_{E A}^{III}=-k_{III} c_{E A} \label{66c}$

In all these elementary steps we assume that the stoichiometric schemata are identical in form to the chemical kinetic schemata. In the shorthand nomenclature of biochemical engineering, Michaelis-Menten kinetics are represented by

$E + A \quad \overset{\stackrel{k_{I}}{\longrightarrow}}{\stackrel{\longleftarrow}{k_{II}}} \quad EA \stackrel{k_{III}}{\longrightarrow} E + D \label{67}$

Our objective at this point is to develop an expression for the net rate of production of species $$D$$ in terms of the concentration of species $$A$$. The net rate of production for species $$D$$ takes the form

$R_{D}= R_{D}^{I} + R_{D}^{II} + R_{D}^{III} = k_{III} c_{EA} \label{68}$

and the net rates of production of the other species are given by

$R_{A}= R_{A}^{I} + R_{A}^{II} + R_{A}^{III} = -k_{I} c_{E}c_{A} + k_{II} c_{EA} \label{69a}$

$R_{E}= R_{E}^{I} + R_{E}^{II} + R_{E}^{III} = -k_{I} c_{E}c_{A} + k_{II} c_{EA} + k_{III} c_{EA} \label{69b}$

$R_{EA}= R_{EA}^{I} + R_{EA}^{II} + R_{EA}^{III} = k_{I} c_{E}c_{A} - k_{II} c_{EA} - k_{III} c_{EA} \label{69c}$

Since a catalyst only facilitates a reaction and is neither consumed nor produced by the reaction, we can assume that the total concentration of the enzyme catalyst is constant. We express this idea as

$c_E + c_{EA} = c^{\circ}_{E} \label{70}$

in which $$c^{\circ}_{E}$$ is the initial concentration of the enzyme in the reactor. In addition, the net rate of production of the enzyme catalyst should be zero and we express this idea as

$R_E = 0 \label{71}$

Use of Equation \ref{71} with Equation \ref{69c} leads to a constraint on the rates of reaction given by

$0 = -k_{I} c_{E}c_{A} + k_{II} c_{EA} + k_{III} c_{EA} \label{72}$

This can be arranged in the form

$\frac{k_I}{k_{II}+k_{III}} c_E c_A = c_{EA} \label{73}$

and use of the constraint on the total concentration of enzyme given by Equation \ref{70} provides

$\frac{k_I}{k_{II}+k_{III}} \left( c^{\circ}_E - c_{EA} \right) c_A = c_{EA} \label{74}$

Solving for the concentration of the enzyme complex gives

$c_{EA} = \frac{c^{\circ}_E c_A}{[(k_{II} + k_{III})/k_I] + c_A} \label{75}$

This result can be used in Equation \ref{68} to represent the net rate of production of the desired product as

$R_D = \frac{(k_{III}c^{\circ}_E) c_A}{K_A + c_A} \label{76}$

in which $$K_A$$ is defined by

$K_A = (k_{II} + k_{III})/k_I \label{77}$

The maximum net rate of production of species $$D$$ occurs when $$c_A >> K_A$$ and this suggests that Equation \ref{76} can be expressed as

Michaelis-Menten kinetics:

$R_D = \frac{\mu_{max} c_A}{K_A + c_A} \label{78}$

This microscopic result is identical in form to the macroscopic Monod equation for cell mass production (see Eq. $$(8.4.27)$$); however, the production of cells (see Figure $$8.4.1$$) is not the same as the production of species $$D$$ illustrated in Figure $$\PageIndex{1}$$. Certainly there is a connection between the production of cells and the production of intercellular material, and this connection has been explored by Ramkrishna and Song15.

It is of some interest to note that when the rate of production of species $$D$$ is completely controlled by the reaction illustrated by Equation \ref{66a}, we have a situation in which

$k_{III} << k_{II} \label{79}$

and the parameter $$K_A$$ in Equation \ref{77} simplifies to

$K_A \to k_{II}/k_I = K^{-1}_{eq}, \quad k_{III} << k_{II} \label{80}$

In this case $$K_A$$ becomes the inverse of a true equilibrium coefficient. Since the imposition of Equation \ref{79} has no effect on the form of Equation \ref{78} there is often confusion concerning the precise nature of $$K_A$$.