Page 289 - Introduction to chemical reaction engineering and kinetics

270 Chapter 10: Biochemical Reactions: Enzyme Kinetics

activation, which decrease and increase the observable enzyme activity, respectively.
These effects may arise from features inherent in the enzyme-substrate system (“in-
ternal” or substrate effects), or from other substances which may act on the enzyme
as poisons (inhibitors) or as coenzymes or cofactors (activators). In the following two
sections, we consider examples of these substrate and external effects in turn, by in-
troducing simple extensions of the model, and interpreting the resulting rate laws to
account for inhibition and activation.

10.4.1 Substrate Effects

Substrates may affect enzyme kinetics either by activation or by inhibition. Substrate
activation may be observed if the enzyme has two (or more) binding sites, and sub-
strate binding at one site enhances the affinity of the substrate for the other site(s). The
result is a highly active ternary complex, consisting of the enzyme and two substrate
molecules, which subsequently dissociates to generate the product. Substrate inhibition
may occur in a similar way, except that the ternary complex is nonreactive. We consider
first, by means of an example, inhibition by a single substrate, and second, inhibition by
multiple substrates.

10.4.1.1 Single-Substrate Inhibition

The following mechanism relates to an enzyme E with two binding sites for the substrate
S. Two complexes are formed: a reactive binary complex ES, and a nonreactive ternary
complex ESS .

E+S&ES (1)
k-l
ES + +ESS: K2 = k-,lk, (2)
k
’
ES’-E+P (3)
(a) Derive the rate law for this mechanism.
(b) Show that inhibition occurs in the rate of formation of P, relative to that given by the
Michaelis-Menten equation for the two-step mechanism for a single binding site in
which only ES is formed.
(c) What is the maximum rate of reaction (call it the apparent Vmax, or Vmax,app), and
how does it compare with the parameter V,,, = k,c,,? At what value of cs does
it occur?
(d) Sketch rp versus cs for comparison with Figure 10.1.

SOLUTZON
(a) We apply the SSH to the complexes ES and ESS; in the latter case, this is equivalent to
assumption of equilibrium with the dissociation constant for ESS given by K2 = k-,/k,:

r,, = k,c,cs - k-,c,, - k$&s + k-,cEss - krcES = 0 (1)
rESS = k$&s - k-,c,,, = 0 (2)
CEO = cE + cES + CESS (3)
r, = kr% (4)