Evaluation of Hill Slopes and Hill Coefficients when the Saturation Binding or Velocity is not Known

Abstract

1. The Hill coefficient (nH), an often-used measure of deviations from hyperbolic behaviour (nonhyperbolicity) in kinetic and binding systems, is usually estimated from the maximum or minimum slope of the Hill plot. The method depends strongly on the assumed magnitude of the asymptotic velocity (V) or binding (P) whose evaluation may be difficult in nonlinear/co-operative systems. Therefore, alternative procedures were devised for the estimation nH which do not require the prior knowledge of V or P. 2. When pairs of velocity/binding readings (v and w) are obtained at concentrations of c and alpha c, respectively (where alpha is a fixed constant), then the relation between w and v is described by a hyperbola, provided that Hill's equation is valid. In this case, linearizing plots, v/w versus v, w versus, w/v, and 1/w versus 1/v, can be used for estimation of the degree of the equation. However, if the Hill expression is applicable, these methods are not efficient and traditional procedures, particularly nonlinear regression, should be used. 3. The 'linearizing' plots of the Hill equation can be applied advantageously for the evaluation of the Hill slope and of nH also in the general case, when the Hill expression is actually not valid, provided that deviations from hyperbolic behaviour are positive. Appropriately extrapolated intercepts of the first two plots estimate alphanH. Furthermore, the slope of the third plot yields, similarly to the method of Kurganov et al., a continuous measure of the Hill slope (including its maximum) at all concentrations. The agreement is, at positive nonhyperbolicities, excellent theoretical values of Hill slopes and coefficients and those estimated by the proposed methods. 4. A coefficient of nonhyperbolicity (theta) is defined for 2nd-degree rate equations which provides a quantitative measure of positive or negative deviation from first-degree, hyperbolic characteristics. It is closely related to the Hill coefficient.