log v^0
Vmaxv 0¼hlog½SþlogK ð 15 : 17 Þwherehis theHill constantorcoefficient, andKis anoverall binding constant
related to the individual binding constants fornsites. The Hill constant, which is
equal to the slope of the plot, is a measure of the cooperativity between the sites such
that: ifh¼1, binding is non-cooperative and normal Michaelis–Menten kinetics
exist; ifh>1, binding is positively cooperative; and ifh<1, binding is negatively
cooperative. At very low substrate concentrations that are insufficient to fill more
than one site and at high concentrations at which most of the binding sites are
occupied, the slopes of Hill plots tend to a value of 1. The Hill coefficient is therefore
taken from the linear central portion of the plot. One of the problems with Hill plots is
the difficulty of estimatingVmaxaccurately.
The Michaelis constantKmis not used with allosteric enzymes. Instead, the term
S0.5, which is the substrate concentration required to produce 50% saturation of the
enzyme, is used. It is important to appreciate that sigmoidal kinetics do not confirm
the operation of allosteric effects because sigmoidicity may be the consequence of the
enzyme preparation containing more than one enzyme capable of acting on the
substrate. It is easy to establish the presence of more than one enzyme, as there will
be a discrepancy between the amount of substrate consumed and the expected amount
of product produced.
Two classical models have been proposed to interpret allosteric regulation. They are
both based on the assumption that the allosteric enzyme consists of a number of
subunits (protomers) each of which can bind substrate and exist in two conformations
referred to as the R (relaxed) and T (tense)states. It is assumed that the substrate binds
more tightly to the R form. The first such model was due to Jacques Monod, Jeffries
Wyman and Jean-Pierre Changeux, and is referred to as thesymmetry model.It
assumes that conformational change between the R and T states is highly coupled so
that all subunits must exist in the same conformation. Thus binding of substrate to a
T state protomer, causing it to change conformation to the R state, will automatically
switch the other protomers to the R form, thereby enhancing reactivity (Fig. 15.10).
The second model of Daniel Koshland, known as theinduced-fitorsequential model,
does not assume the tightly coupled concept and hence allows protomers to exist in
different conformations but in such a way that binding to one protomer modifies the
reactivity of others.
Recent research has shown that allosterism is not confined to oligomeric proteins
since some monomeric proteins may also display the behaviour (Fig. 15.10). The
emerging opinion is that allostery is a consequence of the flexibility of proteins such
that the continuous folding and unfolding in localised regions of the protein gives rise
to a population of conformations that interconvert on various timescales, which differ
in their affinity for certain ligands and, in the case of enzymes, in their catalytic
activity. The interconverting conformations have similar energies and their mixture
constitutes the ‘native state’ of the protein. The binding of an allosteric effector at its
distinct site results in the redistribution of the conformational ensembles as a result of
the alteration of their rates of interconversion, and as a consequence, the confor-
mation and hence activity of the active site is modified. Integral to this redistribution600 Enzymes