8. PH effects

The rates of enzyme-catalysed reactions vary with pH and often pass through a maximum as the pH is varied. If the enzyme obeys Michaelis-Menten kinetics the kinetic parameters k₀ and k_A often behave similarly. The pH at which the rate or a suitable parameter is a maximum is called the pH optimum and the plot of rate or parameter against pH is called a pH profile. Neither the pH optimum nor the pH profile of an enzyme has any absolute significance and both may vary according to which parameter is plotted and according to the conditions of the measurements.

If the pH is changed and then brought back to its original value, the behaviour is said to be reversible if the original properties of the enzyme are restored; otherwise it is irreversible. Reversible pH behaviour may occur over a narrow range of pH, but effects of large changes in pH are in most cases irreversible.

The diminution in rate as the pH is taken to the acid side of the optimum can be regarded as inhibition by hydrogen ions. The diminution in rate on the alkaline side can be regarded as inhibition by hydroxide ions. The equations describing pH effects are therefore analogous to inhibition equations. For single-substrate reactions the pH behaviour of the parameters k₀ and k_A can sometimes be represented by an equation of the form

. . . . . . . . (24)

in which k represents k₀ or k_A, and is the value of the same parameter that would be observed if the enzyme existed entirely in the optimal state of protonation; it may be called the pH-independent value of the parameter. The constants K₁ and K₂ can sometimes be identified as acid dissociation constants for the enzyme. substrates or other species in the reaction mixture. The identification is, however, never straight forward and has to be justified by independent evidence. The behaviour is frequently much more complicated than represented by Eqn (24).

It is not accidental that this section has referred exclusively to pH dependences of k₀ and k_A. The pH dependence of the initial rate or, worse, the extent of reaction after a given time is rarely meaningful; the pH dependence of the Michaelis constant is often too complex to be readily interpretable.

9. PRE-STEADY-STATE KINETICS

The pre-steady-state or transient phase in enzyme-catalysed reactions occupies very short periods (usually fractions of a second) and very low product concentrations. Special techniques therefore have to be used. For reactions that are not too fast the stopped-flow technique, in which the enzyme and reactants are rapidly mixed and the flow stopped. is commonly used.

For reactions that have to be studied over periods of less than 1 ms relaxation techniques are used. In these techniques the system is disturbed, usually but not necessarily from a state of equilibrium, after which it relaxes to equilibrium or a new steady state. In the temperature-jump (T jump) technique the temperature is increased rapidly and the system relaxes to a new state of equilibrium or a new steady state at the final temperature. In the pressure-jump technique the pressure is rapidly changed.

The relaxation time of a reaction is the time it takes for the extent of reaction to change by a proportion (1 - e^-1) of the total change during the relaxation process (e = 2.71828 . . . is the base of natural logarithms). For composite mechanisms, such as those that occur with enzyme-catalysed reactions, relaxation experiments usually reveal more than one relaxation time. These relaxation times can be related to the rate constants of the elementary steps in the mechanism, but the relationships are usually complicated.

10. 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.