A method has been proposed to assess cooperativity in self-assembly processes. The method is based on a clear distinction between intermolecular and intramolecular processes which are compared with the corresponding reference reactions. It has been applied to two classical cases, namely the self-assembly of helicates and of porphyrin ladders, by using data previously published by the groups of Lehn and Anderson, respectively. Contrarily to the conclusions of the authors, pointing out self-assembly processes driven by positive cooperativity, the method here presented indicates in both cases the absence of cooperative effects. The methods previously used to assess cooperativity, in particular Scatchard plot and/or Hill plot, are criticized as being inappropriate for self-assembly, because they are pertinent to a specific case only, namely the intermolecular binding of a monovalent ligand L to a multivalent receptor M, a case very different from self-assembly which involves both inter- and intramolecular interactions. The present method underscores the fact that positive cooperativity in artificial self-assembling systems is probably much more rare than it was previously thought.

A general treatment of macrocyclization reactions occurring under thermodynamic control is presented. The fundamental quantities on which the treatment is based are the effective molarities of the cyclic oligomers and the equilibrium constant for the intermolecular model reaction between monofunctional reactants (Kinter). Four typical cases have been considered, namely, addition and condensation of a monomer of the type A-B, addition of A-A, and addition of A-A + B-B. A critical comparison with the classical theory of Jacobson and Stockmayer is presented.It is shown that the phenomenon characterized by the critical monomer concentration (cut-off point) is a limiting phenomenon which would occur only for infinitely large values of Kin,,,. The treatment has been successfully applied to the DOS/DTC-induced cyclooligomerization of /3-propiolactone in CDC13 solution that yields well-behaved ringchain equilibrates closely adhering to the theoretical model. Best fit of the experimental product distributions to the general equations gave the equilibrium constant (Kinter) of the intermolecular model reaction, as well as the effective molarities (EMi) for the cyclic oligomers from trimer to octamer. The EMi values decrease in proportion of the -2.5 power of the oligomerization degree, thus providing a strong indication that the oligomeric polylactones are essentially strainless. The extremely low value of Kin,,, (2.5) is responsible for the absence of a cut-off point, which is usually present in ring-chain polymeric equilibrates.

The distinction between different types of cooperativity is essential for understanding the fundamentals involved. The three title cooperative effects arise from the interplay of intermolecular binding interactions, the presence of one or more intramolecular binding interactions, and, in the latter case, their possible interplay. A master equation for the stability of an assembly is outlined that takes into account all of the three possible types of cooperativit

Evaluation of statistical factors in self-assembly processes is not a firmly settled question. As a contribution to solve this problem, a critical re-examination of the symmetry number method and generalization of the direct count method are presented. The two approaches, producing the same results, mutually reinforce their role with respect to other discordant methods whose results cannot be independently checked. The direct count method moreover serves as a rationale for the apparently odd results the symmetry number method sometimes provides. The two methods thus turn out to be complementary to each other. Discussion of some exemplary cases points to the importance and subtlety of the role played by the geometrical features of assemblies involving intramolecular bonds.

The translocation of biopolymers through pores and channels plays a fundamental role in numerous biological processes. We describe here the mechanism of the threading of a series of polymer chains through a synthetic macrocycle, which mimics these natural processes. The threading of polymers involves a kinetically favorable "entron" effect, which is associated with the initial filling of the cavity by the end of the polymer. A preassociation between the outside of the macrocycle and the polymer induces a process in which the polymer end loops back into the cavity of the macrocycle. This looping mechanism results in accelerated threading rates and unidirectional motion and is reminiscent of the protein translocation through membrane pores.

A theoretical treatment of self-assembly macrocyclizations occurring under thermodynamic control is presented. The fundamental quantities on which the treatment is based are the effective molarity of the self-assembling cyclic n-mer (EM n ) and the equilibrium constant for the intermolecular model reaction between monofunctional reactants (K inter ). Knowledge of these quantities allows the evaluation of the distribution curve of the selfassembling macrocycle. In order for effective self-assembly to take place two conditions are required: (i) the self-assembling macrocycle must have an EM much larger than that of the other cyclic oligomers; (ii) the product EM n K inter must be not lower than 185r, where r is the number of bonds that hold together the monomer units in the cyclic oligomer, the higher the better. It is shown that in the limit of an infinite value of K inter there is a critical monomer concentration (cmc ) nEM n ) below which the system is virtually composed of the self-assembling macrocycle only and above which the concentration of the latter remains constant and the excess monomer produces acyclic species only. In general the optimum monomer concentration for self-assembly is slightly more than one-tenth of the cmc. Deviation from this value is less and less important the higher the value of EM n K inter ; however, the concentration of the initial monomer should not be outside the range defined by the lower self-assembly concentration and the cmc. Previous conclusions about self-assembly macrocyclizations drawn by Hunter et al. (J. Chem. Soc., Chem. Commun. 1995, 2563 are criticized in the light of the present approach.

The supramolecular polymerization of two AB-type monomers capable of hydrogen-bond-mediated A x B heterocoupling and A x A homocoupling is discussed. The AB-type supramolecular polymerization is based on the strong interaction between self-dimerizing 2-ureido-pyrimidinone (UPy) and 2,7-diamido-1,8-naphthyridine (NaPy). In an effort to reduce the "self-stoppered" effect that is inherently present in these supramolecular polymerizations we used a novel ureido-pyrimidinone substituted with a dibutylamino group at the pyrimidinone ring. As a result of the substitution, the dimerization constant of the novel UPy unit is lowered compared to the previous UPy unit while the heterodimerization strength is retained. Unexpectedly, the increased selectivity toward heteroassociation not only influences the concentration-dependent degree of polymerization due to reduction of the "self-stoppered" effect but also has a pronounced effect on the ring-chain equilibrium by increasing the tendency to cyclize. In order to quantitatively explain our results, a model was developed that accurately predicts the degree of polymerization by taking into account homo- and heterodimerization as well as cyclization. Finally, molecular weight distributions for noncyclizing AB supramolecular polymerizations with and without a reversible A x A interaction are calculated. It is found that the molecular weight distribution becomes narrower when A x A interactions are present.

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