The 3:3 function in enzyme kinetics possible shapes of [formula omitted] and [formula omitted] plots for third degree steady-state rate equations

Abstract

1. (1) Many complex enzyme mechanisms give third degree rate equations (3:3 functions) and, in such circumstances, the double reciprocal plots are non-linear and v S plots non-hyperbolic. 2. (2) Diagnosis of mechanism from curve shape in such situations requires a classification of all possible curve shapes related to magnitudes of coefficients in the rate equation and such an analysis is attempted in this paper. 3. (3) Sign change analysis for the 3:3 case shows that the maximum possible complexity is 1 maximum and 1 minimum in ( 1 v ) ( 1 S ) and v S , five inflexions in v S , 2 tangents from the origin to the curve in v S and ( 1 v ), ( 1 S ) and two inflexions in ( 1 v ) ( 1 S ) . 4. (4) This gives a set of necessary conditions for 38 possible curve shapes in double reciprocal space which, in turn, depend upon inequalities involving 13 epsilon functions. 5. (5) Novel geometric arguments based upon six of these inequalities lead to a set of necessary and sufficient conditions resulting in specific curve shape features. 6. (6) Using a combination of these two approaches, it is shown that 12 of the curve shapes allowed by Descartes rule of signs cannot actually exist. Hence the 3:3 function can only give 26 curve shapes in double reciprocal space of which 21 show convexity reversals and thus cannot be given by the 2:2 function. 7. (7) The significance of this in enzyme kinetics is that, in general, high degree rate equations for complex enzyme mechanisms of degree >2 will usually be such that inflected double reciprocal plots result. Inflexions in ( 1 v ) ( 1 S ) and hence in ( S v ) S and v ( v S ) are thus to be expected in the experimental study of allosterism and are not reliable indications of co-operativity changes. 8. (8) The analysis is briefly extended to the case of independent kinetically distinct sites (isoenzymes) acting on the same substrate and a preliminary classification of v S for the 3:3 function into 9 basic types is described.