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Limiting Kinetics of Enzyme-Catalysed Reactions

At very low concentrations of substrate many enzyme-catalysed reactions display approximately second-order kinetics, with rate given by the following equation:

v = k _A [E]₀ [A]........ (8)

in which the symbol k _A (or, in general, k _R for a reactant R) is the apparent second-order rate constant or specificity constant and [E]₀, which may also be written as [E]_t or [E]_stoich, is the total or stoichiometric concentration of catalytic centres. (This corresponds to the total enzyme concentration only if there is a single catalytic centre per molecule.) The rationale for the subscript 0 is that the total enzyme concentration is normally the concentration at the instant of mixing, i.e. at time zero. Conversely, at very high substrate concentrations the same reactions commonly display approximately first-order kinetics (zero-order with respect to substrate):

v = k ₀ [E]₀........ (9)

in which k ₀, which may also be written as k _cat is the apparent first-order rate constant. Although these limiting types of behaviour are not universally observed, they are more common than Michaelis-Menten kinetics (Section 4.2) and provide a basis for classifying inhibitory and other effects (Section 5) independently of the need for Michaelis-Menten kinetics.

The apparent second-order rate constants k _A and k _B of competing substrates A and B determine the partitioning between competing reactions, regardless of whether the substrate concentrations are very small or not, and it is for this reason that the name specificity constant is proposed for this parameter of enzymic catalysis. The apparent first-order rate constant k ₀ is a measure of the catalytic potential of the enzyme and is called the catalytic constant.

The quantity k ₀[E]₀ is given the symbol V and the name limiting rate. It is particularly useful when k ₀ cannot be calculated because the total catalytic-centre concentration is unknown, as in studies of enzymes of unknown purity, sub-unit structure and molecular mass. The symbol V _max and the names maximum rate and maximum velocity are also in widespread use although under normal circumstances there is no finite substrate concentration at which v = V and hence no maximum in the mathematical sense. The form V _max is convenient in speech as it avoids the need for a cumbersome distinction between 'capital V ' and 'lower case v '. When a true maximum does occur (as insubstrate inhibition; Section 4.3) the symbol v _max (not V _max) and the name maximum rate may be used for the true maximum value of v but care should be taken to avoid confusion with the limiting rate.

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