Self-catalyzed nucleated assembly

Figure 2 Chemical reaction models for (a) isodesmic and (b) nucleated supramolecular assembly. 1 and K 2>1 are equilibrium constants for the elongation reactions, and Ka 1 and K a those for the conversion between assembly active and inactive forms of the monomer units. If /f aKa, then the nucleated assembly is self-catalyzed ( autosteric ) and if K a = Ka this is not so.

Let us start with the nucleated assembly that is not self-catalyzed. It turns out useful to distinguish between the mean aggregation number of all the material in the solution, N, from that in which only the activated species is considered and that we denote by Na. If we define the equilibrium constant K = exp [—.%] with g as the binding free energy, and introduce the nucleation constant Ka = exp[—/3ya], then under conditions of thermodynamic equilibrium, mass action gives (Aggeli, 2001 Ciferri, 2005 Nyrkova et al., 2000 Tobolsky and Eisenberg, 1960)... 

If the nucleated assembly is self-catalyzed, then this modifies only the role of the equilibrium constant K in the non-self-catalyzed model and has to be replaced by the ratio K/Ka. This means that we again obtain for the fraction polymerized material f=KaNa/(1 + KaNa) but that the degree of polymerization averaged over the active material only now obeys (Ciferri, 2005)... 

IfX 1, disordered (nonhelical) assemblies do not form in any appreciable quantities. For h< 0, there is a polymerization transition from monomers to helical assemblies that is of the self-catalyzed nucleation type provided flj 3> 1. In the language of the coarse-grained self-catalyzed nucleated assembly model, the transition takes place near Xp exp [f3h] and we are able to assign an activation constant Ka exp [ 3j + 311], The theory of Section 2 approximately applies. 

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