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NON-MICHAELIS-MENTEN KINETICS

Some enzymes display non-Michaelis-Menten kinetics that do not approximate to Michaelis-Menten kinetics to any useful extent. In such case there is little value in retaining the terminology and symbolism of Michaelis-Menten kinetics. Instead it is often possible to express the rate as a rational function of the substrate concentration:

........ (25)

(In principle this kind of equation can be generalized to accommodate more than a single substrate but it then becomes highly complicated and only the single-substrate case will be considered here.) A rational function is a ratio of two polynomials. The degree of a polynomial is the largest exponent: thus, the degree of the numerator of the expression in Eqn (25) is n and that of the denominator is m. The rational function as a whole may be described as a ' n:m function'. In general the degree of the numerator of a rate equation does not exceed the degree of the denominator for enzyme-catalysed reactions, but there is no other necessary relationship between n and m and neither bears any necessary relationship to the number of catalytic centres per molecule of enzyme. In the terminology of Eqn (25) any rate equation obeying Michaelis-Menten kinetics can be defined as a 1:1 function.

Any coefficient in the numerator of the right-hand side of Eqn (25) has units (mol dm^-3)^{1- i} s^-1, and any coefficient in the denominator has units (mol^-1 dm³) ⁱ. Similar equations are sometimes written in which the constant 1 in the denominator is replaced with a constant . This practice is discouraged, because the equation then contains a redundant parameter and all of the coefficients , have undefined dimensions.

Under some conditions, which cannot be expressed simply and are not normally obvious from inspection of the coefficients of Eqn (25), the equation may generate a plot of v against [A] that shows a monotonic increase in v towards a limiting value at all positive finite values of [A]. A necessary, but not sufficient, condition is that the degrees of the numerator and denominator be equal, i.e. n = m. It is then meaningful to define a limiting rate V = with the meaning defined in Section 4.1. Moreover, it may also be useful to describe the kinetics quantitatively in terms of the slope of a plot of log[ v /(V - v)] against log[A], which is known as a Hill plot. This slope is called the Hill coefficient and is given the symbol h. In a kinetic context it bears no necessary relationship of any kind to the number of catalytic centres per molecule of enzyme and it should not be given a symbol, such as n that suggests that it does. The symbol n _H has sometimes been used as an alternative to h, but it should be borne in mind that this may cause difficulties in printing when used as an exponent. At any concentration of substrate at which h is greater than unity, the kinetics are said to display co-operativity. In some contexts the more explicit term positive co-operativity may be preferable to avoid ambiguity. At any concentration at which h = 1 the kinetics are said to be non-co-operative and if h is less than unity they are said to be negatively co-operative. In the case of Michaelis-Menten kinetics h = I over the whole concentration range, but in other cases h is not constant and the sign of co-operativity may change one or more times over the range of concentrations. Thus co-operativity is not absolute and in general can only be defined in relation to a particular concentration.

The term co-operativity is sometimes restricted to a purely mechanistic meaning, i.e. it is considered to refer to interactions between distinct sites on the enzyme. In common practice, however, the terms discussed above are frequently applied to enzymes in the absence of clear evidence for such interactions, and the aim therefore has been to legitimize such usage by providing purely operational definitions. In contexts where it is considered necessary to emphasize the operational character of the kind of co-operativity defined in terms of the Hill coefficient it may be qualified as Hill co-operativity or kinetic co-operativity.

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