again invoking the quasisteady-state assumption, we obtain

One important thing to note about this case is that the rate depends on the product concentration as well as substrate concentration, and that the rate decreases with increasing product concentration.

Two other single-substrate concentration effects are worth noting. Substrate activation describes the situation observed in Figure 5.2, in which the effect of substrate concentration on reaction rate shows a sigmoidal shape. It can be modeled by adding a reaction to (5.28) and (5.29) in which the free enzyme is in equilibrium with an inactive form. Activation is different from enzyme induction (discussed in Section 6.2.2). In induction the presence of substrate actually stimulates the organism to produce the enzyme; that is, the enzyme is absent when not needed.

Substrate inhibition is the case when adding substrate beyond an optimum amount causes a reduction in the reaction rate (Figure 5.2). It can be modeled by assuming that a second substrate molecule complexes reversibly with the enzyme but that this new complex does not produce product directly. The resulting rate equation is r - k[E]t (5.38)

where Km and K are constants. This expression is known as the Haldane equation, and is equivalent in form to the Andrews equation (Section 11.7.7), often used to model micro-bial biodegradation of toxic substances. Equation (5.38) has a maximum at the substrate concentration:

Many industrial organic chemicals are both biodegradable and toxic to microorganisms, and thus their biodegradation may be modeled by the Haldane equation. Examples of such compounds include benzene and phenol.

Inhibition can be important in the normal control of metabolism. For example, the amino acid isoleucine inhibits one of the enzymes involved in its formation, preventing an oversupply from being produced. This is an example of feedback inhibition.

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