Steady state enzyme turnover kinetics

In fact, certain enzymatic reactions in cells may involve only a handful of enzyme molecules, with as few as several hundred or several thousand enzyme molecules present compared to many more substrate and product molecules. This condition implies that the substrate molecules compete for binding to the relatively few enzyme molecules, while the enzymes function more or less independently of one another. The assumption that the molecules of a particular enzyme act independently significantly simplifies the modeling of biochemical kinetics. In addition, if reactants are present in quantities much greater than enzymes then one can reasonably treat enzyme turnover kinetics by assuming that the substrate concentration remains effectively constant over the timescale of enzyme turnover. 

The quasi-steady approximation is generally valid when the amount that enzyme complex concentrations change is less than the amount that reactant concentrations change over the timescale of interest. This is true, for example, in Section 3.1.3.2 as long as dc/dt C Jmm- Thus the stricter condition that reactant concentrations are large compared to enzyme concentrations (a condition that is by no means universally true in vivo) is not necessarily required to apply the approximation. 

Consider again the simple irreversible Michaelis-Menten scheme  

+i[S] serves as the apparent mass-action rate constant for the conversion E — ES. Each time an enzyme cycles from state E to ES and back to E again, one molecule of S is converted to P. If the rate of turnover of the catalytic cycle is significantly greater than the rate of change of reactant (S and P) concentrations, then the apparent mass-action constant +i[S] in Equation (4.2) remains effectively constant over the timescale of the catalytic cycle. This is true, for example, when the enzyme concentration is small compared to reactant concentrations, many catalytic cycles are required to produce a significant change in reactant concentrations. 

Since the catalytic cycle operates with relatively rapid kinetics, E and ES will obtain a steady state governed by Equations (4.2) and (4.3) and the quasi-steady state concentrations of enzyme and complex will change rapidly in response to relatively slow changes in [S]. Thus the quasi-steady approximation is justified based on a difference in timescales between the catalytic cycle kinetics and the overall rate of change of biochemical reactions. 

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Often the key entity one is interested in obtaining in modeling enzyme kinetics is the analytical expression for the turnover flux in quasi-steady state. Equations (4.12) and (4.38) are examples. These expressions are sometimes called Michaelis-Menten rate laws. Such expressions can be used in simulation of cellular biochemical systems, as is the subject of Chapters 5, 6, and 7 of this book. However, one must keep in mind that, as we have seen, these rates represent approximations that result from simplifications of the kinetic mechanisms. We typically use the approximate Michaelis-Menten-type flux expressions rather than the full system of equations in simulations for several reasons. First, often the quasi-steady rate constants (such as Ks and K in Equation (4.38)) are available from experimental data while the mass-action rate constants (k+i, k-i, etc.) are not. In fact, it is possible for different enzymes with different detailed mechanisms to yield the same Michaelis-Menten rate expression, as we shall see below. Second, in metabolic reaction networks (for example), reactions operate near steady state in vivo. Kinetic transitions from one in vivo steady state to another may not involve the sort of extreme shifts in enzyme binding that have been illustrated in Figure 4.7. Therefore the quasi-steady approximation (or equivalently the approximation of rapid enzyme turnover) tends to be reasonable for the simulation of in vivo systems. 

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... 

Steady state kinetics and protein-protein binding measurements have also been reported for the interaction of these mutant cytochromes with bovine heart cytochrome c oxidase [120]. The binding of cytochrome c variants to the oxidase occurred with increasing values of Kj in the order He (3 x 10 Mol L ) < Leu = Gly < wild-type < Tyr < Ser (3 x 10 molL ). Steady-state kinetic analysis indicated that the rate of electron transfer with cytochrome c oxidase increased in the order Ser < He < Gly < Leu < Tyr < wild-type, an order notably different from that observed for a related analysis of the oxidation of these mutants by cytochrome c peroxidase [85]. This difference in order of mutant turnover by the oxidase and peroxidase may arise from differences in the mode of interaction of the cytochrome with these two enzymes. 

A method for deriving enzyme-rate expressions combining both rapid equilibrium and steady-state procedures first illustrated by Chak With this method, demonstrated by Fromm and Huang, a different rate expression will be obtained depending on which steps are chosen to be in rapid equilibrium and which steps are not. See Enzyme Kinetic Derivations Turnover Number S. Cha (1988) J. Biol. Chem. 243, 820. 

Johnson and Fierke Hammes have presented detailed accounts of how rapid reaction techniques allow one to analyze enzymic catalysis in terms of pre-steady-state events, single-turnover kinetics, substrate channeling, internal equilibria, and kinetic partitioning. See Chemical Kinetics Stopped-Flow Techniques... 

A number of cases are known in which the properties of an enzyme are markedly altered by interaction with a membrane. Of course, in some cases the normal function of an enzyme is destroyed when it is removed from the membrane. For example, the mitochondrial coupling factor cannot synthesize ATP when removed from the membrane, since coupling to a proton gradient is required. The portion of the coupling factor that is easily solubilized (F,) is an ATPase. The steady-state kinetic properties of this solubilized ATPase are appreciably changed when it is reconstituted with mitochondrial membranes The turnover numbers and pH dependencies are different the solubilized enzyme is strongly inhibited by ADP, whereas the reconstituted enzyme is not and the reconstituted enzyme is inhibited by oligomycin, whereas the solubilized enzyme is not. 

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. 

In disagreement with the above indications was the finding of Aldridge et al. (146) that for enzyme which was phosphorylated at pH 5.5 with inorganic phosphate and rapidly mixed with buffer at pH 8.4, the rate of dephosphorylation was twice as fast as the turnover of the enzyme at pH 8.0. Also, transient state kinetic studies by Femley and Walker (99, 110) showed a rapid release (burst) of phenol followed by a steady state release of phenol, only at pH < 7. Thus, these data would seem to indicate that at pH >7 the rate determining step is phosphorylation. 

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. 

It is tempting to try to explain the halophilic features of ADHFR even in the absence of a detailed kinetic scheme for this enzyme, assuming that the main features of the kinetic scheme of the non-halophilic enzymes hold true also for the halophilic enzyme. The salt concentration might have an effect on the rates of binding or dissociation of the various substrates or on the rate of the hydride transfer reaction. Because, as we saw, the hydride transfer reaction is largely dependent on the protonation of Asp-27, it becomes the rate-limiting step at pH values higher than the pKa of this residue. The effect of salt concentration on the steady-state turnover can be ex-... 

Vmax is the velocity of an enzyme-catalyzed reaction when the enzyme is saturated with all of its substrates and is equal to the product of the rate constant for the rate-limiting step of the reaction at substrate saturation (kCiU) times the total enzyme concentration, Ex, expressed as molar concentration of enzyme active sites. For the very simple enzyme reaction involving only one substrate described by Equation II-4, kCM = . Elowever, more realistic enzyme reactions involving two or more substrates, such as described by Equations II-11 and 11-12, require several elementary rate constants to describe their mechanisms. It is not usually possible to determine by steady-state kinetic analysis which elementary rate constant corresponds to kcat. Nonetheless, it is common to calculate kcat values for enzymes by dividing the experimentally determined Fmax, expressed in units of moles per liter of product formed per minute (or second), by the molar concentration of the enzyme active sites at which the maximal velocity was determined. The units of cat are reciprocal time (min -1 or sec - x) and the reciprocal of cat is the time required for one enzyme-catalyzed reaction to occur. kcat is also sometimes called the turnover number of the enzyme. 

As we have seen, the catalytic cycle flux provides a useful metric for analyzing enzyme kinetics. In this section, we analyze the turnover time for catalytic cycles and show that the quasi-steady rate law arises from the mean cycle time [151]. In addition, we show that for arbitrary mechanisms for a single-substrate reaction, the steady state rate law can always be expressed using the Michaelis-Menten form... 

The physiological pathway of electron transfer in flavocytochrome is from bound lactate to FMN, then FMN to 52-heme, and finally 52-heme to cytochrome c (Fig. 9) (2,11, 80,102). The first step, oxidation of L-lactate to pyruvate with concomitant electron transfer to FMN, is the slowest step in the enzyme turnover (103). With the enzyme from S. cerevisiae, a steady-state kinetic isotope effect (with ferricyanide as electron acceptor) of around 5 was obtained for the oxidation of dl-lactate deuterated at the C position, consistent with the major ratedetermining step being cleavage of the C -H bond (103). Flavocytochrome 52 reduction by [2- H]lactate measured by stopped-flow spectrophotometry resulted in isotope effects of 8 and 6 for flavin and heme reduction, respectively, indicating that C -H bond cleavage is not totally rate limiting (104). 

Kinetic analysis of TD-62 under steady-state conditions with ferricyanide as electron acceptor showed that kcat and Km values for L-lactate (165 sec and 0.96 mM, at 25°C, 10 mM Tris-HCl, pH 7.5, / = 0.10 M NaCl) were not dramatically different from the values determined for wild-type enzyme (200 sec and 0.49 mM, conditions as above) 144). However, the behavior of TD-62 under assay conditions was unusual in that the rate of lactate oxidation decreased long before either L-lactate or ferricyanide became depleted. The decrease in reaction rate occurred as a biphasic process, leading eventually to complete loss of activity. This deactivation of TD-62 was only observed under turnover conditions. Deactivation was independent of ferricyanide and was also observed when cytochrome c was used as electron acceptor (150). 

The simplest vay to measure an isotope effect is the noncompetitive technique, in vhich the rate (kn) with fully protiated substrate ( H labeled), is compared to the rate (kn) at which deuterium labeled substrate ( H labeled) reacts [28]. The label may be in the primary or a secondary position, yielding the primary or secondary KIE, respectively. Steady-state noncompetitive measurements yield the isotope effect on the rate constants or k st/Ku, but suffer from the requirement of both high substrate purity and isotopic enrichment, and from a large uncertainty in the KIE (ca. 5-10%) due to propagated errors. Single-turnover experiments can yield noncompetitive KIEs on the chemical step, but also generally have large uncertainties. Nevertheless, noncompetitive measurements are the only way to obtain KIEs on kcat, which for certain enzymes may be the sole kinetic parameter that reflects the chemical step(s). 

Finally, the effects of mechanistic complexity must be addressed in any study of tunneling, particularly for enzyme-catalyzed reactions. There is no simple way to avoid the complications from multiple rate-limiting steps - they may appear in rapid-mix experiments, relaxation kinetics, and steady-state turnovers. There is good reason to believe, however, that with sufficient numbers of isotopic probes, many interesting mechanistic details can be resolved. 

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