5. Enzyme reactions involving more than one substrate

5.1. Michaelis-Menten Kinetics

Regardless of the number of substrates, a reaction is said to obey Michaelis-Menten kinetics if the rate equation can be expressed in the following form:

. . . . . . . . (13)

which can be regarded as a generalization of Eqn (11). (Z is used here as an example of a product as suggested in Section 2.) Each term in the denominator of the rate expression contains unity or any number of product concentrations in its numerator. and a coefficient k and any number of substrate concentrations raised to no higher than the first power in its denominator. The constant k₀ corresponds to k₀ in Eqn (11); each other coefficient is assigned a subscript for each substrate concentration in the denominator of the term concerned and a superscript for each product concentration in the numerator. The term 1/k₀ must be present, together with one term for each substrate of the form 1/k_A [A], but the terms in products of concentrations. such as those shown in Eqn ( 13) with coefficients k_AB and , may or may not be present. It is sometimes convenient to write the equation in a form in which each k is replaced by its reciprocal, symbolized by with the same subscripts and superscripts, i.e. = l/k₀, = l/k_A, = 1/k_AB, , etc. These reciprocal coefficients are called Dalziel coefficients.

Note The conventional Scottish pronunciation of this name may be expressed in the International Phonetic Alphabet as [di:'jel], with only slightly more stress on the second syllable than the first.

Eqn (13) can be applied to reactions with any number of substrates and products and can also be extended to some kinds of inhibition by substrate, i.e. to the simpler kinds of non-Michaelis-Menten kinetics. It is thus an equation of considerable generality. It is simplest, however, to consider terminology in the context of a two-substrate reaction, and this will be done in Section 5.2.

5.2. 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 = k₀ 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 = k₀[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 Vmax; KmA = k₀/k_A is the Michaelis constant for A; K_mB = k₀/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_iAK_mB (as shown) or as K_mAK_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 k₀ [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.