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evaluated from the experimental macroscopic stability constants for simple model
reactions, the macroscopic stability constant of any species [M
m
L
n
] can be predicted
with Equation 3.9. The decision about the cooperativity of interactions is based on com-
paring the predicted constants with the experimental ones. Ercolani initially applied his
model to Lehn's double-stranded helicates [26]. The predicted constant of [Cu
3
L
2
]
3þ
is
indeed close to the experimental values, which indicates that statistical repetitive bind-
ing occurs, in contrast to positive cooperativity suggested in the original paper [43]. It is
worth noting that the correct parameterization of
k
M;L
intra
requires the formation of at least
macrobicyclic (i.e., trinuclear) complexes. In this context, the trinuclear lanthanide hel-
icates [45] fulfil this criterion, but escape Ercolani's analysis due to the lack of thermo-
dynamic data.
The present approach has clearly shown that the classical protein/ligand cooperativity
tests do not allow a reliable detection of cooperativity in supramolecular assemblies. In
addition, the results of Ercolani's tests indicate that positive deviations from statistical
repetitive binding are rare in metallo-organic assemblies. Although the model takes into
account intra- and intermolecular reactions, it suffers from a number of limitations (e.g.,
identical binding affinity along the strand, no quantification of interaction parameters) and
is not widely applied for sophisticated assemblies.
3.5.2 Thermodynamic Modelling
3.5.2.1 Site-Binding Model
The difficulties in testing cooperativity stimulated coordination chemists to search for
new tools that can be used to describe and analyse supramolecular compounds [45]. Not
surprisingly, this innovative effort is closely connected with designing new polynuclear
highly charged helicates with controlled properties [44]. Therefore, the initial develop-
ment of thermodynamic models was associated with the description of intermetallic inter-
actions in polynuclear compounds. Inspired by proton-binding models in polyelectrolytes
[46], the site-binding model for metal binding to a multisite receptor was formulated in
2004 [16]. To illustrate the use of this thermodynamic modelling, let us consider a pre-
assembled linear receptor containing
n
adjacent coordination sites that can bind
n
metal
ions according to Equation 3.10, as schematically shown in Figure 3.14.
Y
Y
n
n
b
n
M;R
n
;1
u
M;M
ij
n
M
mþ
þ
R
)
k
i
M
n
R
b
n
¼ v
ð
3
:
10
Þ
i
¼1
i
<
j
The stability constants for these equilibria are modelled by the product of: (i) statistical
factors
M;R
n;1
, (ii) microscopic binding affinities
k
i
and (iii) interaction parameters
u
M;M
v
ij
accounting for intermetallic interactions within the receptor and expressed as the Boltz-
man factor
u
M;M
ij
¼ expðD
E
ij
=
RT
Þ. The associated total free energy of complexation is
given in Equation 3.11.
RT
X
X
D
E
ij
n
RT
ln
v
;1
ln
k
i
þ
D
G
M;R
n
M;R
n
;1
¼
ð
3
:
11
Þ
1
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