For the reaction S in equilibrium P catalysed by a dimeric enzyme, the reaction schemes are considered on the basis of the KNF model. For each of the ten possible schemes, the rate equation is derived on the basis of the combined steady-state and rapid-equilibrium assumptions. The curves of the plots of initial velocity v versus the substrate concentration [S] and the Hill coefficients h calculated from the rate equations depend strongly on the reaction scheme and the parameter X1. This parameter is defined by log (KS2/KS1) and is a measure of the relative affinities of the first and second protomers for the substrate. When X1 less than 0, v-[S] curves for some schemes exhibit negative co-operativity (h less than 1.0) and v-[S] curves for other schemes are similar to that of the Michaelis-Menten scheme, indicating that, even if there is interaction between the distinct protomers, sigmoidal rate behaviour is not necessarily observed. When X1 greater than 0, all the reaction schemes except one, which shows substrate-inhibition kinetic behaviour, exhibit sigmoidal kinetic behaviour (h greater than 1.0), and at the limit of X1 much greater than 0 the Hill coefficients attain the maximum possible value of 2.0. Furthermore, we have found that, even if X1 = 0, the v-[S] curve for almost all the schemes considered in the present work does not necessarily agree with that for the Michaelis-Menten scheme. This means that the deviation of the v-[S] curve from a hyperbola can be observed even if there is no interaction between the distinct protomers.

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