Food Chemistry

2.5 Kinetics of Enzyme-Catalyzed Reactions 123

substrate reaction is expressed in the following
form:

υ 0 =

V

1 +

Ka

(A 0 )

(2.57)

The constants Kaand Kbin Equation 2.56 are de-
fined analogously to Km, i. e. they yield the con-
centrations 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 con-
stants, like Km(cf. Equation 2.43), is composed
of several rate constants. Kiais the inhibitor con-
stant for A.
When the binding of one substrate is not influ-
enced by the other, each substrate occupies its
own binding locus on the enzyme and the sub-
strates form a ternary enzyme-substrate complex
in a defined order (“ordered mechanism”), the fol-
lowing is valid:

Kia·Kb=Ka·Kb (2.58)

or from Equation 2.56:

υ 0 =

V

1 +

Ka

(A 0 )

+

Kb

(B 0 )

+

Ka·Kb

(A 0 )(B 0 )

(2.59)

However, when only a binary enzyme-substrate
complex is formed, i. e. one substrate or one prod-
uct is bound to the enzyme at a time by a “ping
pong mechanism”, the denominator term Kia·Kb
must be omitted since no ternary complex exists.
Thus, Equation 2.56 is simplified to:

υ 0 =

V

1 +

Ka

(A 0 )

+

Kb

(B 0 )

(2.60)

For the determination of rate constants, the ini-
tial rate of catalysis is measured as a function of
the concentration of substrate B (or A) for sev-
eral concentrations of A (or B). Evaluation can be
done using theLineweaver–Burkplot. Reshaping
Equation 2.56 for a “random mechanism” leads
to:

1

υ 0

=

[

Kb

V

+

Kia·Kb

(A 0 )V

]

1

(B 0 )

+

[

1 +

Ka

(A 0 )

]

1

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 corre-

sponding slopes and ordinate intercepts are taken

from the straight lines obtained at various values

for[A 0 ](Fig. 2.25):

Slope=

Kb

V

+

KiaKb

V

·

1

(A 0 )

Ordinate intercept=

1

V

+

Ka

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 pro-

vide data for calculating constantsKa,Kb,Kia,

and the maximum velocity, V. If the catalysis pro-

ceeds through a “ping pong mechanism”, then

plotting 1/v 0 versus 1/[B 0 ]yields a family of par-

allel 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 bi-

nary enzyme-substrate complex. However, it is

not possible to differentiate an “ordered” from a

“random” reaction mechanism by this means.

2.5.1.3 AllostericEnzymes......................................

We are already acquainted with some enzymes

consisting of several protomers (cf. Table 1.26).

When the protomer activities are independent