Cooperativity effect in metal-ligand and macromolecule-ligand equilibria

Abstract

An evaluation of the cooperativity effect in metal-ligand and protein-ligand complexes can be made by means of dimensionless parameters K γ (homotropic), and K γ′ (heterotropic). Starting from the relation n = ∂ ln ϵ M/∂ ln [A] between formation function n and partition function ϵ M = 1 + β 1[A] + β 2 [A] 2 + …β i [A] i… + β t [A] t it is shown that on the Bjerrum plane n = f(ln [A]), the standard free-energy ▵ G° and therefore the standard chemical potentials ▵μ° γ = RT ln K γ and ▵μ° γ′=  RT ln K γ′ can be obtained exactly from a convoluted or saturation function F c M, which has the property of being 1 when ▵μ° = O, as do the equilibrium constants. The standard convoluted function F c M° for multi- step equilibria coincides with the maximum term, /gB t , of ϵ M and can be calculated on the Bjerrum plane as balance between two integrals, one integral giving the contribution to free-energy of the ‘association’ partition function, ϵ M, the other giving the contribution to free-energy of the ‘dissociation’ partition function, ϵ D M. F c M can be calculated as the product of stepwise equilibrium constants K1 , K 2,…, K t . Differences between areas on the Bjerrum plane measure ▵μ° γ/ RT and ▵μ° γ′/RT. The statistical factors for homotropic and heterotropic complexes are equal. The chemical potential changes due to the cooperativity effect can be better evaluated from K γ = β i 1/ i /( K 1 k st( γ ) ) (average cooperativity effect) rather than by Kγ = (β i /β i -2 ) 1/2/( k st(γ)β i-1 /β i-2 ) (stepwise cooperativity effect). The former is well correlated with the successive additions of ligands. The cooperativity effect is strongly influenced by changes in ionic strength, thus confirming the interpretation of the phenomenon as due to ligand-ligand external interactions. The cooperativity effect, although small (0 < ▵μ° γ▵ < 6 kJ/mol), is significantly greater than the experimental error. The cooperativity effect in homotropic and heterotropic metal complexes is of the same order of magnitude as the cooperativity effect in macromolecule-ligand equilibria.