Self-Assembly Principles of Helicates - Metallofoldamers: Supramolecular Architectures from Helicates to Biomimetics - page 108

system indicates negative cooperativity. Further, one would expect a straight line and a
concave downward curve, respectively, in the absence of interactions and for positive
cooperativity.
Although these tests are truly applicable for some supramolecular systems [41], their
general use in supramolecular chemistry is not appropriate. In spite of this, the above
graphical tests were originally applied to double- and triple-stranded helicates and
revealed a strong positive cooperativity [42-44]. These confusing analyses were identified
in 2003 by Ercolani [26], who showed the application of the Scatchard plot was
inappropriate in self-assembled compounds, where both (virtually non-equivalent) inter-
and intramolecular processes operate.
In order to reliably detect cooperativity in multicomponent assemblies, Ercolani
developed the concept of statistical repetitive binding [26], where the microscopic
cumulative constant of a multicomponent assembly

M ; L

b is obtained as the product of

mn

mn , the microscopic equilibrium constants k M;L

the degeneracy factor v

M;L

inter for inter-

molecular reactions and the microscopic constants k M;L

intra associated with intramolecular

reactions (Equation 3.9).

Y Y

k M;L k M;L

intra

M;L M;L

mn

m M þ n L н M m L n b

mn ¼ v inter

ð 3 : 9 Þ

inter intra

The present description is adequate for self-assembly processes: (i) in the absence of
interligand and intermetallic interactions and (ii) if the microscopic constants do not vary
within related complexes. The application of this modelling to a simple dinuclear assem-
bly with two ligands is given in Figure 3.13. The first three steps refer to strictly inter-
molecular binding events between metal ions and ligand coordination sites characterized
with k M;L

inter . The fourth step corresponds to the intramolecular macrocyclization associated
with k M;L

intra . If the microscopic stability constants k M;L inter and k M;L

intra can be independently

STEP 1 STEP 2

M,L M,L M,L M,L M,L

( k inter ) 2

M,L

β 11 : ω 11 ⋅ k inter β 21 = ω 21 ⋅

STEP 3

STEP 4

M,L M,L

( k inter ) 3

M,L M,L

β 2 M,L

M,L

( k inter ) 3

M,L

β 22 = ω 22 ⋅ ⋅ ( k intra ) (open) = ω 22 (open) ⋅

Figure 3.13 Statistical repetitive binding for a dinuclear complex [M 2 L 2 ]. [45] Adapted by

permission of The Royal Society of Chemistry.