An evaluation of the cooperativity effect in metal-ligand and protein-ligand complexes can be made by using dimensionless parameters Kγ (homotropic), and Kγ' (heterotropic). Starting from the relation ‾n = .vdelta. ln ΣM/.vdelta. ln[A] between formation function ‾n and partition function ΣM = 1 + β1[A] + β2[A]2 + ...βi[A]i... + βt[A]t, it is shown that on the Bjerrum plane ‾n = f(ln[A]), the std. free-energy ΔG° and therefore the std. chem. potentials Δμγ° = -TT ln Kγ and Δμγ'° = -RT ln Kγ' can be obtained exactly from a convoluted or satn. function FMc, which has a property of being 1 when Δμ° = 0, as do the equil. consts. The std. convoluted function FMc° for multistep equil. coincides with the max. term, βt, of ΣM and can be calcd. on the Bjerrum plane as balance between 2 integrals, one integral giving the contribution to free-energy of the "assocn." partition function, ΣM, the other giving the contribution to free-energy of the "dissocn." partition function, ΣMD. The FMc can be calcd. as the product of stepwise equil. consts. K1,K2,...,Kt. Differences between areas on the Bjerrum plane measure Δμγ°/RT and Δμγ'°/RT. The statistical factors for homotropic and heterotropic complexes are equal. The chem. potential changes due to the cooperativity effect can be better evaluated from K‾γ = βi1/i/(K1kst(‾γ)) (av. cooperativity effect) rather than by Kγ = (βi/βi-2)1/2/(kst(γ)βi-1/βi-2) (stepwise cooperativity effect). The former is well correlated with the successive addns. 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 < |Δμγ°| < 6 kJ/mol), is significantly greater than the exptl. error. The cooperativity effect in homotropic and heterotropic metal complexes is of the same order of magnitude as the cooperativity effect in macromol.-ligand equil.
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