Cooperativity and extrema of the Hill slope for symmetric protein-ligand binding polynomials.

The relationships between cooperativity types, both macro- and microscopic, and extrema of the slope of Hill plots for symmetric binding polynomials of degrees three and four are analyzed in detail. It is shown that the Hill plot for 1 + ax + ax 2 + x3 has three extrema for a greater than 15 which occurs in the presence of totally negative microscopic cooperativity sequences. The extrema are a maximum and a pair of minima and the absolute minimum slope is shown to be 3(square root a + 1-2)/(a - 3). For a symmetric binding polynomial of degree four, necessary and sufficient conditions for three extrema of the Hill slope are derived and again it is shown that this phenomenon can occur in the presence of totally negative microscopic cooperativity sequences. An upper bound, which is an improvement on the current best bound of two, is obtained for the absolute maximum slope when a minimum and a pair of maxima occur. A lower bound for the absolute minimum slope is found when a maximum and a pair of minima occur.