Factorability of the Hessian of the binding polynomial. The central issues concerning statistical ratios between binding constants, hill plot slope and positive and negative co-operativity

Abstract

1. (1) It is demonstrated that arguments as to co-operativity based upon the sign of co-operativity coefficients, factorability of the binding polynomial, sigmoid binding isotherms, concavity of double reciprocal plots or other graphical features are ambiguous. 2. (2) The sole unambiguous test for positive or negative co-operativity in the base of ligand binding to a macromolecule is the magnitude of the Hill plot slope with respect to unity as a function of ligand concentration. 3. (3) This is entirely dependent on the factorability of the Hessian of the binding polynomial where there is no aggregation of the macromolecule and in any graphical space changes in co-operativity occur at positive roots of the Hessian. Each graphical space will then have a unique geometric feature associated with these roots and this will constitute the test for positive or negative co-operativity in that space. 4. (4) The algebraic properties of the Hessian are described and the relationship between the roots of a polynomial and its concomitants are given and related to roots of the Hessian for the binary cubic and quartic. 5. (5) This study leads to the conclusion that co-operativity has not yet been satisfactorily defined for steady-state systems but for homotropic effects resulting from ligand binding to a single protein species the sign of co-operativity is the same as the sign of the Hessian of the allosteric binding polynomial. Although this can be determined experimentally, it is proved that, in the fourth degree case, it is not always then possible to infer the sign of the co-operativity coefficients or nature of the factorability of the binding polynomial since, unlike the second and third degree cases, the fourth degree Hill plot shapes are ambiguous. 6. (6) When the macromolecule aggregates in addition to binding ligand there is no binding polynomial and no Hessian but in these circumstances there is always a corresponding function, the tact invariant, the sign of which is the same as the sign of the homotropic co-operativity. The reason that the Hessian is of importance when there is a binding polynomial is because in this case the Hessian functions as the tact-invariant of the actual binding isotherm and the reference curve corresponding to ligand binding to a single site at the corresponding degree of saturation.