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Michaelis-Menten Kinetics of a Two-Substrate Reaction

For a two-substrate reaction in the absence of products Eqn (13) simplifies to the following equation:

........ (14)

If the concentration of one substrate, known as the constant substrate, is held constant, while that of the other, known as the variable substrate, is varied, the rate is of the form of the Michaelis-Menten equation in terms of the variable substrate, because Eqn (14) can be rearranged to

........ (15)

(cf. Eqn 10), where

........ (16)

is known as the apparent catalytic constant, and

........ (17)

is known as the apparent specificity constant for A. It follows from Eqns (16) and (17) that = k ₀ and = k _A when [B] is extrapolated to an infinite value. This relationship provides the basis for defining the catalytic constant and the specificity constants in reactions with more than one substrate: in general, the catalytic constant of an enzyme is the value of v /[E]₀ obtained by extrapolating all substrate concentrations to infinity; for any substrate A the specificity constant is the apparent value when all other substrate concentrations are extrapolated to infinity.

Eqn (14) may also be rearranged into a form resembling Eqn (11), as follows:

........ (18)

in which V = k ₀[E]₀ is the limiting rate, which may also, subject to the reservations noted in section 4.1, be called the maximum rate or maximum velocity and symbolized as V max; K mA = k ₀/ k _A is the Michaelis constant for A; K _mB = k ₀/ k _B is the Michaelis constant for B; and K _iA = k _B/ k _AB is the inhihition constant for A. In some mechanisms K _iA is equal to the true dissociation constant for the EA complex: when this is the case the alternative symbol K _iA and the name substrate-dissociation constant for A (cf. section 3.2) may be used. If Eqn(18) is interpreted operationally rather than as the equation for a particular mechanism it is arbitrary whether the constant in the denominator is written with K _iA K _mB (as shown) or as K _mA K _iB, where K _iB = k _A/ k _AB. However, for some mechanisms only one of the two ratios k _B/ k _AB and k _A/ k _AB has a simple mechanistic interpretation and this may dictate which inhibition constant it is appropriate to define.

The term inhibition constant and the symbol K _iA derive from the fact that the quantity concerned is related to (and in the limiting cases equal to) the inhibition constant K _ic or K _iu (as defined below in Section 6.4) measured in experiments where the substrate is treated as an inhibitor of the reverse reaction. However, the relationships are not always simple and quantities such as K _iA in Eqn (18) can be and nearly always are defined and measured without any reference to inhibition experiments. For these reasons some members of the panel feel that the symbolism and terminology suggested are not completely satisfactory. No alternative system has so far gained wide support, however.

An apparent Michaelis constant for A (and similarly for B) may be defined by dividing Eqn (16) by Eqn (17):

........ (19)

This equation provides the basis for defining the Michaelis constant for any substrate in a reaction with more than one substrate: the Michaelis constant for A, K _mA, is the value of the apparent Michaelis constant for A when the concentrations of all substrates except A are extrapolated to infinity. This definition applies to reactions with any numbers of substrates, as also does the definition of the limiting rate V as k ₀ [E]₀, but in other respects it becomes very cumbersome to define constants resembling K _iA for reactions with more than two substrates. The symbolism of Eqn (13) (or the alternative in terms of Dalziel coefficients) is readily extended to reactions with three or more substrates, however.

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