2009_Book_FoodChemistry

2.5 Kinetics of Enzyme-Catalyzed Reactions 123
substrate reaction is expressed in the following
form:
υ 0 =
V
1 + K a
(A 0 )
(2.57)
The constants K a and K b in Equation 2.56 are defined
analogously to K m , i. e. they yield the concentrations
of A or B for v 0 = V/2 assuming that,
at any given moment, the enzyme is saturated by
the other substrate (B or A). Each of the constants,
like K m (cf. Equation 2.43), is composed
of several rate constants. K ia is the inhibitor constant
for A.
When the binding of one substrate is not influenced
by the other, each substrate occupies its
own binding locus on the enzyme and the substrates
form a ternary enzyme-substrate complex
in a defined order (“ordered mechanism”), the following
is valid:
K ia · K b = K a · K b (2.58)
or from Equation 2.56:
υ 0 =
V
1 + K a
(A 0 ) + K b
(B 0 ) + K a · K b
(A 0 )(B 0 )
(2.59)
However, when only a binary enzyme-substrate
complex is formed, i. e. one substrate or one product
is bound to the enzyme at a time by a “ping
pong mechanism”, the denominator term K ia · K b
must be omitted since no ternary complex exists.
Thus, Equation 2.56 is simplified to:
υ 0 =
V
1 + K a
(A 0 ) + K b
(B 0 )
(2.60)
For the determination of rate constants, the initial
rate of catalysis is measured as a function of
the concentration of substrate B (or A) for several
concentrations of A (or B). Evaluation can be
done using the Lineweaver–Burk plot. Reshaping
Equation 2.56 for a “random mechanism” leads
to:
1
υ 0
=
[
Kb
V + K ] [
ia · K b 1
(A 0 )V (B 0 ) + 1 + K ]
a 1
(A 0 ) V
(2.61)
Fig. 2.25. Evaluation of a two-substrate reaction,
proceeding through a ternary enzyme-substrate
complex (according to Lineweaver and Burk).
[A 0 ] 4 > [A 0 ] 3 > [A 0 ] 2 > [A 0 ] 1
First, 1/v 0 is plotted against 1/[B 0 ]. The corresponding
slopes and ordinate intercepts are taken
from the straight lines obtained at various values
for [A 0 ] (Fig. 2.25):
Slope = K b
V + K iaK b
V · 1
(A 0 )
Ordinate intercept = 1 V + K a
V ·
1
(A 0 )
(2.62)
and are then plotted against 1/[A 0 ].Inthisway
two straight lines are obtained (Fig. 2.26a and
b), with slopes and ordinate intercepts which provide
data for calculating constants K a , K b , K ia ,
and the maximum velocity, V. If the catalysis proceeds
through a “ping pong mechanism”, then
plotting 1/v 0 versus 1/[B 0 ] yields a family of parallel
lines (Fig. 2.27) which are then subjected to
the calculations described above.
A comparison of Figs. 2.25 and 2.27 leads to
the conclusion that the dependence of the initial
catalysis rate on substrate concentration allows
the differentiation between a ternary and a binary
enzyme-substrate complex. However, it is
not possible to differentiate an “ordered” from a
“random” reaction mechanism by this means.
2.5.1.3 Allosteric Enzymes
We are already acquainted with some enzymes
consisting of several protomers (cf. Table 1.26).
When the protomer activities are independent