Some mathematical results concerning hessians of binding polynomials and co-operativity coefficients

A number of miscellaneous results concerning binding polynomials, saturation functions, Hill plot slopes, co-operativity coefficients and steady-state rate equations are presented. A theorem of Newton and Sylvester is of value in understanding the dependence of the nature of the zeros of binding polynomials on the signs of successive co-operativity coefficients. Real rational functions of degree n : n + r with positive coefficients can have at most n turning points for r > 0 or n − 1 for r = 0. For n binding sites the Hill plot can have at most n − 2 changes in sign of homotropic co-operativity and for this it is necessary but not sufficient for co-operativity coefficients to alternate in sign. The geometric significance of zeros of Hessians is given. When Hill plots have inflexions of slope unity, then the Adair constants satisfy an algebraic equation of order 6( n−3).