enzyme concentration. A description of the transformation of substrate to product generally shows V as a hyperbolic function of S concentration with V increasing asymptotically toward Vmax as the active site becomes saturated with S.
Even in this simple case, the extraction of the magnitude of the four specific rate constants requires numerical analysis, with additional complexity being introduced by the appearance of intermediates or the requirement for a second or third substrate. These complications lead to equations in which the additional rate constants cannot be calculated from steady-state data. However, the analogous terms for KM and kcat can be calculated and hold similar meanings. Perhaps the most useful application of steady-state kinetics at this level is the recognition of diagnostic patterns in the reciprocal replots of the initial velocity data as a function of substrate concentration. Two-substrate reactions fall into two general classes represented by
The difference is that in the former process a fragment X of substrate A is transferred covalently to the enzyme and then to the second substrate, whereas in the latter no free enzyme bearing a covalently linked fragment X is formed. Within the second pathway, the addition of A and B, and similarly the release of products C and D, can be ordered (as written) or random. The two pathways give rise to the representative graphs shown in Fig. 1.
As one might imagine, the kinetic rate laws associated with these mechanisms are generally too complex for dissection of single steps or their evaluation. Moreover, the method provides evidence for only the minimal number of intermediates in a pathway since the form of the equations is unchanged by including multiple species.
Steady-state kinetic parameters such as kcat and KM can vary when they are studied as a function of pH. After one corrects for ionizations of the substrate and controls for possible effects on the native structure of the enzyme, variations in kcat and KM can often be assigned to ionizations of acid/base groups at the active site of the enzyme. The term kcat/KM reflects the proton dissociation constants of the free enzyme, provided that the proton transfers remain fast relative to other steps in the pathway. In the simple one-intermediate kinetic sequence expanded to implicate two ionizations, the term kcat/KM would display pKa and pKb; the term kcat would reflect pK'b a and pK'a b. The pH dependence of the kcat parameter affords information about the substrate-bound state.
Materials that bind to the enzyme either at the active site or at a distal site and slow the turnover of the enzyme but are not themselves transformed act as inhibitors. These compounds may or may not be structurally similar to the substrate; nevertheless, their binding, particularly at the active site, often provides important complexes for structure determination. The most commonly studied type of inhibition is termed competitive, which means that the substrate and the inhibitor compete directly for the active site of the enzyme. The effect of this type of inhibitor on the steady-state kinetic parameters is to alter the graphical evaluation of the Michaelis constant but not the value of Vmax, which can still be attained in the presence of the inhibitor provided that the substrate concentration is high enough. Binding of the inhibitor to regions divorced from that binding the substrate always affects the evaluation of Vmax because no concentration of substrate is sufficient to displace the inhibitor.
The most useful approaches for obtaining information regarding the existence of intermediates and their lifetimes are fast reaction methods that mix enzyme and substrate within milliseconds, which permits the observation of single turnover events by various spectroscopic methods. Alternatively the reaction is rapidly quenched at known time intervals and its progress is analyzed chromatographically. In many cases in which an intermediate accumulates to the level of the enzyme concentration, such methods reveal the presence of “burst kinetic” that feature the rapid buildup of the intermediate in the transient phase followed by its slower rate of formation/decay in the steady state. The simplest kinetic scheme consistent with this phenomenon is given by
where the rate constants are in the order k1[S]>k2 >k3. The amplitude of the burst can provide the concentration of active sites in an enzyme preparation. By varying the concentration of S, one can find values for k−1/k1, k2, and k3. There are many variations on transient kinetics, as will be illustrated in our case studies of individual enzymes.
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