Analysis of derivative binding isotherms. Theoretical considerations

Abstract The basic theoretical groundwork for the use of derivative binding isotherms in the analysis of ligand binding is presented. The derivative binding isotherm is defined as Γ (Y) = df / dy where f = fractional degree of saturation and y = natural logarithm of the free ligand concentration. Since Γ (y) is a positive function, which goes to zero as y → ±∞, the mean value of y, 〈y〉, and the second and third moments, μ 2 and μ 3 about 〈y〉 are well defined. For a macromolecular system consisting of N equivalent and independent binding sites, Γ (y) is a symmetrical bell‐shaped function with one maximum. The maximum occurs when y = −ln K assoc ; μ 2 = π 2 /3, and μ 3 = 0. For multiple sets of independent binding sites, Γ (y) is a superposition of Γ‐type functions. If the sets are sufficiently well separated in binding free energy, multiple extrema may be seen at positions corresponding to the logarithms of the dissociation constants for the individual sets. In any case, 〈y〉 is equal to the mean value of the logarithms of the dissociation constants for the sets; μ 2 > π 2 /3 and equal to π 2 /3 plus the variance of the logarithms of the dissociation constants about their mean value; and μ 3 is, except by coincidence, not equal to zero and equals the third moment of the distribution of logarithms of the dissociation constants about their mean value. Analysis of Γ(y) for the case of cooperative interactions within a set of binding sites was investigated by examining (1) the Hill model (whose mathematical representation is equivalent to that used to describe antibody heterogeneity except that in the latter case the parameter a , the Sips, constant, is constrained (0 < a ≤1);(2) a common model for cooperativity in which the cooperative free energy is a linear function of the fraction bound; and (3) a general representation of cooperative interactions within a set of sites in terms of ϕ( f ), a smooth function that gives the interaction free energy in units of RT . For the Hill model (or Sips model) Γ(y) is a symmetrical function with one maximum at y = (−1)/ a )ln K , μ 2 = π 2 /3 a 2 ; and μ 3 = 0. For the case in which the cooperative free energy is a linear function of f [ϕ( f ) = cf ], 〈y〉 = −ln K 0 + ( c /2); μ 2 = (π 2 /3) + c [( c /12) + 1] where c > −4; and μ 3 = 0. General expressions for the moments in terms of ϕ( f ) are derived. In general, μ 2 < (π 2 /3) for positive cooperativity and μ 2 > (π 2 /3) for negative for negative cooperativity. Γ(y) will be symmetrical if and only if the cooperative free energy is introduced symmetrically about f = 0.5.