The determination of positive and negative co-operativity with allosteric enzymes and the interpretation of sigmoid curves and non-linear double reciprocal plots for the MWC and KNF models

Abstract

(i) It is proved that only four independent constants can ever be obtained by extrapolation procedures applied to non-hyperbolic steady-state or binding data, (ii) Analysis of the algebraic graphs y x , (1/y) (1/x) , y ( y x ) and ( x y )/x is shown to require a knowledge of the sign of six curve shape determinants. In each case, the sign is a necessary and sufficient condition for a specific curve shape feature, (iii) The precise graphical effect of positive and negative co-operativity then requires the definition of two reference curves, the osculating hyperbola at zero substrate concentration, OH(0), and the osculating hyperbola at infinite substrate concentration OH(∞). These are better first order approximations than the Hill equation, (iv) Rules for determining unambiguously the sign of initial, final and overall co-operativity coefficients by inspection of non-hyperbolic binding curves are then possible, (v) These rules require that saturation data for: y= ∑ i=1 n a ix i ∑ i=0 n β ix i be fitted by computer for low concentrations to the hyperbola: OH(o)= ( -a 1 2 ψ 11 20 )x [( -a 1β 0 ψ 11 20 )+x] while regression of high substrate concentration data is to: OH(∞)= ( a n β n )x [( φ n,n-1 a nβ n )+x] . Comparisons of the best fit pseudo-kinetic constants then gives the type of co-operativity present in an unambiguous way with no assumptions as to molecular mechanism, (vi) These rules are then applied to the MWC and KNF allosteric models of ligand binding and the constraints necessary for specific curve shape effects are given, (vii) The graphical expression of positive or negative final co-operativity depends only on events at high substrate concentration but overall and initial co-operativities produce specific geometric effects depending upon the difference between behaviour of saturation data at both extremes of concentration, (viii) This apparent anomaly is explained by a discussion of the relationships between the osculating hyperbolae, the theoretical parent hyperbola and the Hill plot asymptotes.