Relationships between the magnitude of Hill plot slopes, apparent binding constants and factorability of binding polynomials and their hessians

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An algebraic analysis of the factorability of binding polynomials in several variables is given. The relationship between factorability and site-site linkage is explored and the special case of ligands competing for the same site is analysed. Such cases always show heterotropic linkage. Existing knowledge concerning the relationship between factorability and statistical ratios between binding constants is summarised and a number of problems related to co-operativity coefficients are clarified. The theory of multiplicity and distribution of roots of binding polynomials is discussed and it is shown how the use of Sylvester resultants and subresultants facilitates the calculation of Sturm functions. A new bigradient algorithm for counting positive roots of polynomials is presented. The various possible definitions of apparent binding constants are investigated and the relationship of these to Hill plot slope is developed. It is shown that the Hill plot slope can never be greater than that given by separate factors of a binding polynomial. Using this theorem, it follows that a binding polynomial leading to a Hill slope greater than two must have at least one irreducible quadratic factor with roots having positive real parts. This substantiates a postulate of Wyman and Gill concerning physically meaningless binding constants. It is shown that the MWC model leads to a binding polynomial that can always be factored. However, the roots of the irreducible quadratic factors can have either positive or negative real parts and this contradicts a second posulate of Wyman and Gill that molecular models with unambiguous linkage always give rise to quadratic factors with negative linear coefficients. The nature of the roots of binding polynomials for higher analogues of the MWC model involving more than two conformational forms are given and it is shown that the Hessian has no positive roots. Hence this model, like the simple MWC one, always gives positive homotropic co-operativity.