Page 578 - Carrahers_Polymer_Chemistry,_Eighth_Edition

Reactions on Polymers 541

The initial rate of product formation, R , for the Michaelis–Menten model depends only on the
i
rate of complex breakdown, that is,
R = k [EM] (16.37)
i 2
Substitution from Equation 16.36 into Equation 16.37 gives

k 2 [E ][M]
0
R i =
k + [M] (16.38)
This expression is dependent on the concentration of M and describes the initial part of the plot
given in Figure 16.6.
Generally, the concentration of M far exceeds that of the enzyme sites such that essentially all
of the enzyme sites are complexed, that is, [EM] = [E ]. (This is similar to a situation that occurs
o
regularly in south Florida where four- and six-lane roads are funneled into a two-lane section of
road because of a wreck or road construction.) Thus, the rate of product formation is maximized
under these conditions. This maximum rate, R , allows us to substitute [E ] for [EM] in Equation
o
m
16.37 giving
R = k [E ] (16.39)
2
o
m
Since the enzyme concentration is constant, the rate of product formation under these conditions
is independent of [M] and is said to be zero order (Figure 11.6).
The maximum rate is directly related to the rate at which the enzyme “processes” or permits con-
version of the reactant molecule(s). The number of moles of reactants processed per mole of enzyme
per second is called the turnover number. Turnover numbers vary widely. Some are high, such as
for the scavenging of harmful free radicals by catalase, with a turnover number of about 40 million.
Others are small such as the hydrolysis of bacterial cell walls by the enzyme lysozyme, with a turn-
over number of about one-half.
The Michaelis–Menten approach does not describe the behavior of allosteric enzymes, such
as hemoglobin, where rate curves are sigmoidal rather than hyperbolic. A more complex mode is
called for to account for the biofeedback that occurs with allosteroid enzymes. Such affects may be
positive such as those associated with hemoglobin, where binding by one site changes the geom-
etry and electronic environment of the other remaining sites, allowing these addition sites to bind
oxygen under more favorable conditions. The effects may also be negative, such as that of cytidine
triphosphate, which inhibits ATCase and catalyzes the condensation of aspartate and carbamoly
phosphate-forming carbamoyl aspartate.
Two major models are typically used to describe these situations: the concerted model and the
sequential model. In the concerted model, the enzyme has two major conformations—a relaxed
form that can bind the appropriate reactant molecule(s) and a tight form that is unable to tightly bind
the reactant molecule(s). In this model, all subunits containing reactive sites change at the same time
(Figure 16.7). An equilibrium exists between the active and inactive structures. Binding at one of
the sites shifts the equilibrium to favor the active relaxed form.
The major feature in the sequential model is the induction of a conformational change from the
inactive tight form to the active relaxed form as the reacting molecule(s) is bound at one of the sites.
This change from an unfavorable to a favorable structure is signaled to other potentially reactive sites
bringing about a change to the more favored structural arrangement in these other sites (Figure 16.8).
Structural changes can be brought about through simple electrostatic and steric events caused by
the presence of the reacting molecule(s). Structural changes also result as cross-linking and other
primary bonding changes occur.

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