Enzyme reactions steady state kinetics

A recent detailed study of the kinetic mechanism of the E. coli enzyme using steady-state kinetics and ITC shows that the enzyme has a random bi-bi mechanism in which a ternary complex of enzyme, ATP, and phosphopantetheine are formed. The following apparent kinetic parameters for the forward reaction were determined 220 10 pmoll and cat 1.59 0.01 s (for ATP), and 4.7 0.5 pmolP and ifcat... 

Kinetic studies involving enzymes can principally be classified into steady and transient state kinetics. In tlie former, tlie enzyme concentration is much lower tlian that of tlie substrate in tlie latter much higher enzyme concentration is used to allow detection of reaction intennediates. In steady state kinetics, the high efficiency of enzymes as a catalyst implies that very low concentrations are adequate to enable reactions to proceed at measurable rates (i.e., reaction times of a few seconds or more). Typical enzyme concentrations are in the range of 10 M to 10 ], while substrate concentrations usually exceed lO M. Consequently, tlie concentrations of enzyme-substrate intermediates are low witli respect to tlie total substrate (reactant) concentrations, even when tlie enzyme is fully saturated. The reaction is considered to be in a steady state after a very short induction period, which greatly simplifies the rate laws. 

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 mechanism of the first half-reaction has been studied by a combination of reductive titrations with CO and sodium dithionite and pre-steady-state kinetic studies by rapid freeze quench EPR spectroscopy (FQ-EPR) and stopped-flow kinetics 159). These combined studies have led to the following mechanism. The resting enzyme is assumed to have a metal-bound hydroxide nucleophile. Evidence for this species is based on the similarities between the pH dependence of the EPR spectrum of Cluster C and the for the for CO, deter-... 

Figure 2.8 Relationship between steady state kinetic constants and specific portions of the enzyme reaction pathway.

In this chapter we described the thermodynamics of enzyme-inhibitor interactions and defined three potential modes of reversible binding of inhibitors to enzyme molecules. Competitive inhibitors bind to the free enzyme form in direct competition with substrate molecules. Noncompetitive inhibitors bind to both the free enzyme and to the ES complex or subsequent enzyme forms that are populated during catalysis. Uncompetitive inhibitors bind exclusively to the ES complex or to subsequent enzyme forms. We saw that one can distinguish among these inhibition modes by their effects on the apparent values of the steady state kinetic parameters Umax, Km, and VmdX/KM. We further saw that for bisubstrate reactions, the inhibition modality depends on the reaction mechanism used by the enzyme. Finally, we described how one may use the dissociation constant for inhibition (Kh o.K or both) to best evaluate the relative affinity of different inhibitors for ones target enzyme, and thus drive compound optimization through medicinal chemistry efforts. 

A steady-state kinetics study for Hod was pursued to establish the substrate binding pattern and product release, using lH-3-hydroxy-4-oxoquinoline as aromatic substrate. The reaction proceeds via a ternary complex, by an ordered-bi-bi-mechanism, in which the first to bind is the aromatic substrate then the 02 molecule, and the first to leave the enzyme-product complex is CO [359], Another related finding concerns that substrate anaerobically bound to the enzyme Qdo can easily be washed off by ultra-filtration [360] and so, the formation of a covalent acyl-enzyme intermediate seems unlikely in the... 

Kinetics of O-Methylaiion. The steady state kinetic analysis of these enzymes (41,42) was consistent with a sequential ordered reaction mechanism, in which 5-adenosyl-L-methionine and 5-adenosyl-L-homocysteine were leading reaction partners and included an abortive EQB complex. Furthermore, all the methyltransferases studied exhibited competitive patterns between 5-adenosyl-L-methionine and its product, whereas the other patterns were either noncompetitive or uncompetitive. Whereas the 6-methylating enzyme was severely inhibited by its respective flavonoid substrate at concentrations close to Km, the other enzymes were less affected. The low inhibition constants of 5-adenosyl-L-homocysteine (Table I) suggests that earlier enzymes of the pathway may regulate the rate of synthesis of the final products. 

When the enzyme is first mixed with a large excess of substrate, there is an initial period, the pre-steady state, during which the concentration of ES builds up. This period is usually too short to be easily observed, lasting just microseconds. The reaction quickly achieves a steady state in which [ES] (and the concentrations of any other intermediates) remains approximately constant over time. The concept of a steady state was introduced by G. E. Briggs and Haldane in 1925. The measured V0 generally reflects the steady state, even though V0 is limited to the early part of the reaction, and analysis of these initial rates is referred to as steady-state kinetics. 

FIGURE 6-19 Pre-steady state kinetic evidence for an acyl-enzyme intermediate. The hydrolysis of p-nitrophenylacetate by chymotrypsin is measured by release of p-nitrophenoi (a colored product). Initially, the reaction releases a rapid burst of p-nitrophenol nearly stoichiometric with the amount of enzyme present. This reflects the fast acylation phase of the reaction. The subsequent rate is slower, because enzyme turnover is limited by the rate of the slower deacylation phase. 

One less kinetic parameter can be obtained from an analysis of the data for a ping-pong mechanism than can be obtained for ordered reactions. Nevertheless, in Eq. 9-47, twelve rate constants are indicated. At least this many steps must be considered to describe the behavior of the enzyme. Not all of these constants can be determined from a study of steady-state kinetics, but they may be obtained in other ways. 

Steady state kinetics may be used to distinguish between the various mechanisms mentioned above. Under the appropriate conditions, their application can determine the order of addition of substrates and the order of release of products from the enzyme during the reaction. For this reason, the term mechanism when used in steady state kinetics often refers just to the sequence of substrate addition and product release. 

Steady state kinetic measurements on an enzyme usually give only two pieces of kinetic data, the KM value, which may or may not be the dissociation constant of the enzyme-substrate complex, and the kcM value, which may be a microscopic rate constant but may also be a combination of the rate constants for several steps. The kineticist does have a few tricks that may be used on occasion to detect intermediates and even measure individual rate constants, but these are not general and depend on mechanistic interpretations. (Some examples of these methods will be discussed in Chapter 7.) In order to measure the rate constants of the individual steps on the reaction pathway and detect transient intermediates, it is necessary to measure the rate of approach to the steady state. It is during the time period in which the steady state is set up that the individual rate constants may be observed. 

As we discussed in Chapter 3, the KM for an enzymatic reaction is not always equal to the dissociation constant of the enzyme-substrate complex, but may be lower or higher depending on whether or not intermediates accumulate or Briggs-Haldane kinetics hold. Enzyme-substrate dissociation constants cannot be derived from steady state kinetics unless mechanistic assumptions are made or there is corroborative evidence. Pre-steady state kinetics are more powerful, since the chemical steps may often be separated from those for binding. 

The enzyme-product complexes of the yeast enzyme dissociate rapidly so that the chemical steps are rate-determining.31 This permits the measurement of kinetic isotope effects on the chemical steps of this reaction from the steady state kinetics. It is found that the oxidation of deuterated alcohols RCD2OH and the reduction of benzaldehydes by deuterated NADH (i.e., NADD) are significantly slower than the reactions with the normal isotope (kn/kD = 3 to 5).21,31 This shows that hydride (or deuteride) transfer occurs in the rate-determining step of the reaction. The rate constants of the hydride transfer steps for the horse liver enzyme have been measured from pre-steady state kinetics and found to give the same isotope effects.32,33 Kinetic and kinetic isotope effect data are reviewed in reference 34 and the effects of quantum mechanical tunneling in reference 35. 

The calculation of rate constants from steady state kinetics and the determination of binding stoichiometries requires a knowledge of the concentration of active sites in the enzyme. It is not sufficient to calculate this specific concentration value from the relative molecular mass of the protein and its concentration, since isolated enzymes are not always 100% pure. This problem has been overcome by the introduction of the technique of active-site titration, a combination of steady state and pre-steady state kinetics whereby the concentration of active enzyme is related to an initial burst of product formation. This type of situation occurs when an enzyme-bound intermediate accumulates during the reaction. The first mole of substrate rapidly reacts with the enzyme to form stoichiometric amounts of the enzyme-bound intermediate and product, but then the subsequent reaction is slow since it depends on the slow breakdown of the intermediate to release free enzyme. 

The intermediate reacts sufficiently rapidly to be on the reaction pathway. These criteria require that pre-steady state kinetics be used at some stage in order to measure the relevant formation and decomposition rate constants of the intermediate. But the rapid reaction measurements are not sufficient by themselves, since the rate constants must be shown to be consistent with the activity of the enzyme under steady state conditions. Hence the power, and the necessity, of combining the two approaches. 

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. 

The Henri-Michaelis-Menten Treatment Assumes That the Enzyme-Substrate Complex Is in Equilibrium with Free Enzyme and Substrate Steady-State Kinetic Analysis Assumes That the Concentration of the Enzyme-Substrate Complex Remains Nearly Constant Kinetics of Enzymatic Reactions Involving Two Substrates... 

It should be noted that this solution procedure requires the knowledge of elementary rate constants, klt k2, and k3. The elementary rate constants can be measured by the experimental techniques such as pre-steady-state kinetics and relaxation methods (Bailey and Ollis, pp. 111 -113, 1986), which are much more complicated compared to the methods to determine KM and rmax. Furthermore, the initial molar concentration of an enzyme should be known, which is also difficult to measure as explained earlier. However, a numerical solution with the elementary rate constants can provide a more precise picture of what is occurring during the enzyme reaction, as illustrated in the following example problem. 

The most important observation in the pre-steady-state kinetics of the CN system is that after a short lag (100 msec) there is a phase (lasting about 3 sec) where the evolution of H2 is linear and only after these 3 sec does CN reduction occur. This long lag prior to CN reduction would correspond to 18 to 20 electron transfer steps from the Fe protein. More realistically this delay probably involves a CN -induced modification of the enzyme, such as a ligand substitution reaction (this modified state of the enzyme is designated as. E in Figure 21). However, this modification step is too slow to be part of the steady-state cycle for CN reduction. Also, it cannot be a slow activation of the enzyme prior to turnover, since the onset of H2 evolution is the same in both the presence and the absence of CN . Steady-state observations indicate that cyanide binds to a more oxidized form of the MoFe protein than binds N2, but that state cannot be defined because of the long lag phase. 

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