The reduction in degree of allosteric and other complex rate equations using sylvester's dialytic method of elimination

Abstract

1. (1) For all enzyme mechanisms there exists the possibility of reduction in degree of the steady-state rate equation when numerator and denominator are not relatively prime. 2. (2) For some mechanisms, this would occur under specific experimental conditions or at certain special rate constant values resulting in a reduced set of possible curve shapes. 3. (3) In realistic allosteric mechanisms, the introduction of catalytic rate constants may, in some circumstances, increase the number of nodes involving substrate addition leading to an artificially inflated degree. 4. (4) The algebraic theory necessary to study this problem is presented in a form appropriate to the routine testing of rate equations and all functions up to degree 4 : 4 are analysed using Sylvester's Dialytic method of elimination. 5. (5) Formulae are presented that give necessary and sufficient conditions for the reduction of y (n:n+r)= ∑ 1 n λ 1x 1 ∑ 0 n+r β 1x i to the proper, irreducible form y( n : n + r n − k : n + r − k )= ∑ 1 n−k a ix 1 ∑ 0 n+r−k b ix i for n > k thus allowing the routine testing of calculated high degree rate equations for a reduced spectrum of curve profiles. 6. (6) The conditions resulting in reduction in degree are shown to be related to the inequalities dictating curve shape in many instances. 7. (7) It is proved that all n : n rate equations give the same 1 : 1 function on full reduction and this is referred to as the theoretical parent hyperbola defined by y n : n 1 : 1 = λ 1β nx (λ nβ 0+λ 1β nx) 8. (8) All curves given by n : n rate equations can be characterized by reference to a unique 1 : 1 function called the experimental parent hyperbola. This is defined by y = x/[( solβ n α n ) + ( solβ n α n )x] where ( β 0/ α 1) and ( solβ n α n ) are experimentally determined parameters. The possible use of this concept in the study of allosterism and complex kinetics in general is explored. The discovery that an increasing number of enzymes deviate from Michaelis-Menten kinetics makes a thorough understanding of high degree rate equations a matter of some concern. Although at the present time there is considerable controversy over the precise interpretation of non-hyperbolic steady-state kinetics, there is, fortunately, resulting from the studies of King & Altman (1956), Wong & Hanes (1962), Vol'kenstein & Gol'shtein (1965) and others, a unifying reliable theory upon which theoretical studies can be based except for polymerizing systems. This should prove a durable concept and is now formulated explicitly.