2009_Book_FoodChemistry

2.5 Kinetics of Enzyme-Catalyzed Reactions 125
indicative of positive cooperation. Each substrate
molecule, often called an effector, accelerates
the binding of succeeding substrate molecules,
thereby increasing the catalytic activity of the
enzyme (case b in Fig. 2.28). When R s > 81, the
system shows negative cooperation. The effector
(or allosteric inhibitor) decreases the binding of
the next substrate molecule (case c in Fig. 2.28).
Various models have been developed in order
to explain the allosteric effect. Only the
symmetry model proposed by Monod, Wyman
and Changeux (1965) will be described in its
simplified form: specifically, when the substrate
acts as a positive allosteric regulator or effector.
Based on this model, the protomers of an
allosteric enzyme exist in two conformations,
one with a high affinity (R-form) and the other
with a low affinity (T-form) for the substrate.
These two forms are interconvertible. There is an
interaction between protomers. Thus, binding of
the allosteric regulator by one protomer induces
a conformational change of all the subunits and
greatly increases the activity of the enzyme.
Let us assume that the R- and T-forms of an enzyme
consisting of four protomers are in an equilibrium
which lies completely on the side of the
T-form:
(2.64)
Addition of substrate, which here is synonymous
to the allosteric effector, shifts the equilibrium
from the low affinity T-form to the substantially
more catalytically active R-form. Since one substrate
molecule activates four catalytically active
sites, the steep rise in enzyme activity after only
a slight increase in substrate concentration is not
unexpected. In this model it is important that
the RT conformation is not permitted. All subunits
must be in the same conformational state
at one time to conserve the symmetry of the protomers.
The equation given by Hill in 1913, derived
from the sigmoidal absorption of oxygen by
hemoglobin, is also suitable for a quantitative description
of allosteric enzymes with sigmoidal behavior:
υ 0 =
V(A 0) n
K ′ +(A 0 ) n (2.65)
Fig. 2.29. Linear presentation of Hill’s equation
The equation says that the catalytic rate increases
by the nth power of the substrate concentration
when [A 0 ] is small in comparison to K. The Hill
coefficient, n, is a measure of the sigmoidal character
of the curve and, therefore, of the extent
of the enzyme’s cooperativity. For n = 1 (Equation
2.65) the reaction rate is transformed into the
Michaelis–Menten equation, i. e. in which no cooperativity
factor exists.
In order to assess the experimental data, Equation
2.65 is rearranged into an equation of
a straight line:
log = υ 0
= nlog(A 0 ) − logK ′ (2.66)
V − υ 0
The slope of the straight line obtained by
plotting the substrate concentration as log[A 0 ]
versus log[v 0 /(V − v 0 )] is the Hill coefficient, n
(Fig. 2.29). The constant K incorporates all the
individual K m values involved in all the steps of
substrate binding and transformation. The value
of K m is obtained by using the substrate concentration,
denoted as [A 0 ] 0.5 v ,atwhichv 0 = 0.5V.
Under these conditions, the following is derived
from Equation 2.66):
log 0.5V
0.5V = 0 = n · log(A 0) 0.5 V − logK ′ (a)
K ′ =(A 0 ) n 0.5 V
2.5.2 Effect of Inhibitors
(b)
(2.67)
The catalytic activity of an enzyme, in addition
to substrate concentration, is affected by the type