Chemistry Reference
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ligand to a receptor induces allosteric effects, which changes the binding affinity of the
identical receptor site toward the second ligand. The allosteric cooperativity is quantita-
tively described with an interaction parameter
that may be extracted from the ratio of
statistically corrected experimental stability constants for successive binding events
a ¼
a
K
n
, whereby the constant
K
n
þ1
is the examined constant and
K
n
is taken as the
reference. For non-interacting binding sites (non-cooperative), the successive constants
K
n
and
K
n
þ1
can be expressed with the microscopic association constant
K
multiplied by
statistical factors. Positive cooperativity is attributed to the affinity enhancement (
K
n
þ1
=
a >
1),
while the opposite situation holds for negative cooperativity (
1).
The cooperativity assessment can be clearly illustrated for the binding of a monovalent
ligand
L
to a divalent receptor
R
possessing two equivalent binding sites in Equation 3.4,
whereby each equilibrium is characterized with the experimental microscopic binding
constant (
K
1
and
K
2
, respectively) multiplied by the statistical factor.
a >
1
=
2
K
2
L-
R
-L
2
K
1
L-
R
-
-
R
-
þ
2L
)
)
ð
3
:
4
Þ
The reference microscopic affinity
K
for the binding of L to
R
can be: (i) evaluated
independently for a model system or alternatively (ii) assigned to
K
1
. Therefore, in the
absence of interactions between binding sites,
K
¼
K
1
¼
K
2
, and the cooperativity factor
2
a ¼
1. In the presence of cooperativity, the cooperativity factor will thus
differ from unity. Deviations from the statistical behaviour may find their origin in elec-
trostatic, conformational or steric interactions, for instance. The associated free energy
contribution expressed as a Boltzman factor can be obtained using the relation
D
E
a
¼
K
2
K
1
=ð
K
Þ
¼
. Allosteric cooperativity is currently encountered in classical coordina-
tion chemistry, and the tutorial example of [Ni(NH
3
)
6
]
2þ
complexes is treated in [13,15a].
In this system, the anticooperative fixation of NH
3
is attributed to repulsive interligand
interactions. Analogously, the presence of cooperativity can be tested for simple mono-
nuclear complexes. Although the allosteric cooperativity can also be detected in poly-
nuclear assemblies (pair homocomponent interactions, i.e., between coordinated ligand
moieties), its reliable assessment is more complicated and requires alternative
approaches, such as thermodynamic modelling (see below).
Alternatively, allosteric cooperativity may arise when metal ions bind to a receptor.
This situation is often encountered in coordination chemistry and can be described with
Equation 3.4, if L accounts for a metal ion and
R
for a receptor of cations. The coopera-
tivity factor is accessed analogously with the procedure applied for the ligand binding.
Accordingly, the assessment of cooperativity is described in the literature for coordination
complexes with simple receptors, for instance those in Figure 3.7.
RT
ln
a
3.3.2 Chelate Cooperativity
While allosteric cooperativity is restricted to intermolecular interactions, chelate coopera-
tivity operates with both inter- and intramolecular interactions. It refers not only to the
chelate effect (Section 3.2.1) but also to the macrocyclization reactions described in Sec-
tion 3.2.2, where the presence of cooperativity may be of high importance for the forma-
tion of supramolecular assemblies. To describe this kind of cooperativity in the correct
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