Steady state kinetic schemes

1 Simple steady state kinetics and Michaelis-Menten equation 

For the purposes of the following analysis, biocatalysis is assumed to be irreversible and each biocatalyst E possesses only a single catalytic site. Hence, if the total concentration of biocatalyst is [E]o, then this must be the sum of free enzyme, [E], and Michaelis complex, [ES], concentrations as indicated in 

If Equation (8.1) is then divided through by Equation (8.2) we arrive at 

In order to solve Equation (8.3), expressions for [E] and [ES] are required that are obtained by applying Briggs-Haldane steady state principles. According to these principles, biocatalysis rapidly attains a condition of stasis under which all biocatalyst species are at a constant equilibrium concentration. In other words [E] and [ES] are constant with time. Stasis is reflected by 

substituting for [ES] in Equation (8.3) followed by rationalisation of terms, we arrive at 

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Biocatalyst inhibitors I are substrate-like molecules that interact with a given biocatalyst and interfere with the progress of biocatalysis. Inhibitors usually act in one of three ways, either by competitive inhibition, non-competitive inhibition or uncompetitive inhibition. The mode of inhibition is different in each case and as a result a different steady state kinetic scheme is required to account for each mode of inhibition. Consequently, each mode of inhibition is characterised by a different steady state kinetic equation that gives rise to a different graphical output of V versus [S] data, as we will show below. These substantial differences in graphical output can be used to diagnose the type of inhibition if unknown. 

In this section, we shall begin to see how the Briggs-Haldane steady state approach can be enlarged to derive steady state kinetics equations appropriate to more complex kinetic schemes. In doing this, there will be some pleasant surprises in that the form of these new steady state kinetic equations will follow the form of Michaelis-Menten equation (8.8) with a few adaptations not unlike those seen in the Uni Uni steady state kinetic scheme adapted to fit the presence of inhibitors (see Section 8.2.4). 

The reader can show that, with the steady-state approximation for [Tl2+], this scheme agrees with Eq. (6-14), with the constants k = k i and k = k j/k g. Of course, as is usual with steady-state kinetics, only the ratio of the rate constants for the intermediate can be determined. Subsequent to this work, however, Tl2+ has been generated by pulse radiolysis (Chapter 11), and direct determinations of k- and k g have been made.5... 

Fig. 9. The MoFe protein cycle of molybdenum nitrogenase. This cycle depicts a plausible sequence of events in the reduction of N2 to 2NH3 + H2. The scheme is based on well-characterized model chemistry (15, 105) and on the pre-steady-state kinetics of product formation by nitrogenase (102). The enzymic process has not been chsiracter-ized beyond M5 because the chemicals used to quench the reactions hydrolyze metal nitrides. As in Fig. 8, M represents an aji half of the MoFe protein. Subscripts 0-7 indicate the number of electrons trsmsferred to M from the Fe protein via the cycle of Fig. 8.

The reduction of 7,8-dihydrofolate (H2F) to 5,6,7,8-tetrahydrofolate (H4F) has been analyzed extensively14 26-30 and a kinetic scheme for E. Coli DHFR was proposed in which the steady-state kinetic parameters as well as the full time course kinetics under a variety of substrate concentrations and pHs were determined. From these studies, the pKa of Asp27 is 6.5 in the ternary complex between the enzyme, the cofactor NADPH and the substrate dihydrofolate. The second observation is that, contrary to earlier results,27 the rate determining step involves dissociation of the product from the enzyme, rather than hydride ion transfer from the cofactor to the substrate. 

The additional reactive intermediate responsible for the curvature was postulated17,33 to be a CAC.30 The mechanism of Scheme 2 was proposed, in which carbene 10a was in equilibrium with the CAC. Thus, styrenes 11a and 12a can be formed by two pathways from the free carbene (kj) and from the CAC (k-). A steady-state kinetic analysis of Scheme 2 affords Eq. 11, which predicts that a correlation of rearr/addn with l/[alkene] should be linear the behavior actually observed by Tomioka and Liu.17,33 The CAC mechanism also accounts for the observation that the lla/12a product ratio depends upon the identity and concentration of the added alkene both k[ and k2, which define the Y-intercept of Eq. 11, depend on the added alkene. The dependence has been observed,19,33-37 albeit with only small variations in the Y-intercepts. 

Fromm and Rudolph have discussed the practical limitations on interpreting product inhibition experiments. The table below illustrates the distinctive kinetic patterns observed with bisubstrate enzymes in the absence or presence of abortive complex formation. It should also be noted that the random mechanisms in this table (and in similar tables in other texts) are usually for rapid equilibrium random mechanism schemes. Steady-state random mechanisms will contain squared terms in the product concentrations in the overall rate expression. The presence of these terms would predict nonhnearity in product inhibition studies. This nonlin-earity might not be obvious under standard initial rate protocols, but products that would be competitive in rapid equilibrium systems might appear to be noncompetitive in steady-state random schemes , depending on the relative magnitude of those squared terms. See Abortive Complex... 

This reaction cycle has more steps than the simple Michaelis-Menten scheme. Nonetheless, the steady-state rate equations describing these reaction cycles have indistinguishable functions, and one cannot determine the number of intermediary steps by steady-state kinetics alone. 

The steady-state kinetics of a simple single-substrate, single-binding site, single-intermediate-enzyme catalysed reaction in the presence of competitive inhibitor are shown in Scheme A5.5.1. 

For linear mechanisms we have obtained structurized forms of steady-state kinetic equations (Chap. 4). These forms make possible a rapid derivation of steady-state kinetic equations on the basis of a reaction scheme without laborious intermediate calculations. The advantage of these forms is, however, not so much in the simplicity of derivation as in the fact that, on their basis, various physico-chemical conclusions can be drawn, in particular those concerning the relation between the characteristics of detailed mechanisms and the observable kinetic parameters. An interesting and important property of the structurized forms is that they vividly show in what way a complex chemical reaction is assembled from simple ones. Thus, for a single-route linear mechanism, the numerator of a steady-state kinetic equation always corresponds to the kinetic law of the overall reaction as if it were simple and obeyed the law of mass action. This type of numerator is absolutely independent of the number of steps (a thousand, a million) involved in a single-route mechanism. The denominator, however, characterizes the "non-elementary character accounting for the retardation of the complex catalytic reaction by the initial substances and products. 

Unlike the standard protocols for steady-state kinetic analysis, transient kinetic analysis is dependent on the availability of signals to measure individual steps of the reaction. Moreover, the observable kinetics change and can be complex or deceivingly simple depending on the relative magnitudes of sequential steps in a pathway. The rule of thumb is that one exponential phase exists in the time dependence of a reaction for each step in the pathway. For example, the kinetics of signals observable according to Scheme 2 will follow a triple exponential function Y =Ai e A- + A2 - A3 -f c. Moreover,... 

In the scheme above, linear photooxidation requires that d[Y00H]/dt = 0. With this assumption, and assuming steady state kinetics in both Y and Y00 it can be shown (-U2) that the overall rate of photooxidation, W, is given by,... 

The two kinetic constants, and itcat. are most often misinterpreted as the substrate dissociation constant and the rate of the chemical reaction, respectively. However, this is not always the case, and /Cm can be greater than, less than, or equal to the true substrate dissociation constant, K. The steady-state kinetic parameters only provide information sufficient to describe a minimal kinetic scheme. In terms of measurable steady-state parameters, a reaction sequence must be reduced to a minimal mechanism (Scheme I),... 

Thus kcai and are a function of all the rate constants in the pathway and any simplifying assumptions concerning individual rate constants are likely to be inaccurate. Moreover, the three reaction pathways shown in Schemes I and 11, and 111 are indistinguishable by steady-state methods. Although product inhibition patterns provide evidence for the E-P state, individual kinetic constants cannot be resolved. Schemes 11 and 111 reduce to Scheme 1 under the conditions where ki, k2- Steady-state kinetics cannot resolve the three reaction mechanisms because the form of the equation for steady-state kinetics is identical for each mechanism (v = rate) ... 

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