Steady state enzyme reactions

W.J. Ray, Jr., Rate-Limiting Step - A Quantitative Definition - Application to Steady-State Enzymic Reactions, Biochem. 22 (1983) 4625-4637. 

Finally, yet another issue enters into the interpretation of nonlinear Arrhenius plots of enzyme-catalyzed reactions. As is seen in the examples above, one typically plots In y ax (or. In kcat) versus the reciprocal absolute temperature. This protocol is certainly valid for rapid equilibrium enzymes whose rate-determining step does not change throughout the temperature range studied (and, in addition, remains rapid equilibrium throughout this range). However, for steady-state enzymes, other factors can influence the interpretation of the nonlinear data. For example, for an ordered two-substrate, two-product reaction, kcat is equal to kskjl ks + k ) in which ks and k are the off-rate constants for the two products. If these two rate constants have a different temperature dependency (e.g., ks > ky at one temperature but not at another temperature), then a nonlinear Arrhenius plot may result. See Arrhenius Equation Owl Transition-State Theory van t Hoff Relationship... 

The derivation of the steady-state enzyme rate equation for the single substrate enzyme-catalyzed reaction is provided in the entry entitled Enzyme Rate Equations (L The Basics). 

Fig. 3.3. Tentative mechanism of reduction of dioxygen. The scheme shows some of the more significant reaction steps at the haem iron-Cug centre of cytochrome oxidase. The reaction may be initiated by delivery of dioxygen to the reduced enzyme (in anaerobiosis top of figure). An initially formed oxy intermediate is normally extremely short-lived, but can be stabilised and identified in artificial conditions (see Refs. 92, 99,129, 134). Concerted transfer of two electrons from Fe and Cu to bound dioxygen yields a peroxy intermediate. This, or its electronic analogue, is stabilised in the absence of electron donors (ferrocytochrome a and/or reduced Cu ), and has been termed Compound C [129,130,132). It may also be observed at room temperature, and is then probably generated from the oxidised state by partial oxidation of water in the active site, in an energy-linked reversed electron transfer reaction [29] (see also Refs. 92, 99). Also the ferryl intermediate [92,99,100] has been tentatively observed in such conditions [29]. In aerobic steady states the reaction is thought to involve the cycle of intermediates in the centre of the figure (dark frames). The irreversible step is probably the conversion of g = 6 (see Refs. 98, 133) to peroxy .

Bloomfield, V., Peller, L., Alberty, R. A. (l%2b). Multiple intermediates in steady-state enzyme kinetics III. Analysis of the kinetics of some reactions catalyzed by dehydrogenases. J. Amer. Chem. Soc. 84,4375-4384. 

In any particular metabolic pathway in a steady state, the reaction carried out by the rate-limiting enzyme is slower than the nonrate-limiting enzyme reactions in that pathway. 

Based on these assnmptions, the following steady-state diffusion-reaction equations for mass balances of substrate concentrations within an immobilized enzyme can be written as... 

Perusal of the physicochemical chapters of textbooks on physiology and biochemistry, published up to the 1950s, reveals an overwhelming concern with the analysis of equilibria. This interest in the presentation of detailed and useful accounts of ionic processes and the energy balance of metabolic pathways left little space for attention to rate processes. Briggs Haldane (1925) introduced the steady state treatment of simple enzyme reactions, as opposed to the earlier, unrealistic, equilibrium approach (see section 3.3). Since then, and especially from the 1950s onwards, there has been more appreciation of the fact that cellular processes are in a constant state of flux or are in a steady state. Individual reactions may be at or near equilibrium, but for the cell as a whole equilibrium is death. 

The catalytic function of an enzyme is described by enzyme kinetics usually determined under steady-state conditions. A steady state refers to a complete balance of a particular quantity between its rate of formation and its rate of disappearance. In steady-state enzyme kinetics, the concentrations of enzyme-bound intermediates are meant to be in a steady state. On mixing an enzyme with a large excess of substrates, there is an initial period, known as a presteady state, during which the concentrations of the intermediates build up to a maximal level under the reaction conditions. Then the reaction rate changes relatively slowly with time and the intermediates are considered to be at steady-state concentrations. Note that the steady state is an approximation because the substrate is gradually depleted during the course of reaction. Therefore, steady-state kinetic measurements should be performed in a relatively short time interval over which the... 

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. 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

The interpretations of Michaelis and Menten were refined and extended in 1925 by Briggs and Haldane, by assuming the concentration of the enzyme-substrate complex ES quickly reaches a constant value in such a dynamic system. That is, ES is formed as rapidly from E + S as it disappears by its two possible fates dissociation to regenerate E + S, and reaction to form E + P. This assumption is termed the steady-state assumption and is expressed as... 

FIGURE 16.21 Burst kinetics observed iu the chymotrypsiii reaction. A burst of nitrophe-nolate production is followed by a slower, steady-state release. After an initial lag period, acetate release is also observed. This kinetic pattern is consistent with rapid formation of an acyl-enzyme intermediate (and the burst of nitrophenolate). The slower, steady-state release of products corresponds to rate-limiting breakdown of the acyl-enzyme intermediate. 

FIGURE 18.12 The use of inhibitors to reveal the sequence of reactions in a metabolic pathway, (a) Control Under normal conditions, the steady-state concentrations of a series of intermediates will be determined by the relative activities of the enzymes in the pathway, (b) Plus inhibitor In the presence of an inhibitor (in this case, an inhibitor of enzyme 4), intermediates upstream of the metabolic block (B, C, and D) accumulate, revealing themselves as intermediates in the pathway. The concentration of intermediates lying downstream (E and F) will fall. 

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-... 

S] + K )] for the hexokinase-catalyzed phosphorylation reactions of 2DG and D-glucose, respectively [S (substrate) + E (enzyme) — ES— -I- P (product)]. This constant (LC) accounts for the ratio of the arteriovenous extraction fraction (by transport and phosphorylation) of 2DG to that of D-glucose (LC= 1) under steady-state conditions. This concept can be directly applied to the case of 2DFG by employing the LC (-0.5) for 2DFG. 

The activity of carbamoyl phosphate synthase I is determined by A -acetylglutamate, whose steady-state level is dictated by its rate of synthesis from acetyl-CoA and glutamate and its rate of hydrolysis to acetate and glutamate. These reactions are catalyzed by A -acetylglu-tamate synthase and A -acetylglutamate hydrolase, respectively. Major changes in diet can increase the concentrations of individual urea cycle enzymes 10-fold to 20-fold. Starvation, for example, elevates enzyme levels, presumably to cope with the increased production... 

Unlike the case of a surface reaction, there is no need to specify the units of measurement because all reacting species are in the three-dimensional space of the reaction medium, and all relevant rates are in concentration per unit of time. The unknown concentration of the unoccupied enzymes follows by assuming that the reaction is at steady state ... 

Reaction of purified Ca " -ATPase with 0.3 mM NBD-Cl in the presence of 1 mM AMP-PNP and 1 mM CaCl2 caused inhibition of ATPase activity with the incorporation of 2= 15 nmol NBD-Cl per mg protein [335]. The inhibition was attributed to the binding of 7-8 nmol NBD-Cl/mg enzyme protein, corresponding to = 1 mol NBD-Cl per mol ATPase. The NBD-labeled enzyme was digested with pepsin and several NBD-labeled peptides were isolated [335]. All peptides contained the Gly-X (Cys) sequence that occurs only in one place in the Ca -ATPase, i.e., at Gly343-Cys344. Therefore NBD-Cl reacts with the same cysteine 344 residue that is also modified by maleimide derivatives [319]. The NBD modified enzyme had only 5-10% of the ATPase activity of the control ATPase, but the steady state concentration of the phosphoenzyme intermediate was only slightly reduced [335]. The Ca ... 

Enzymes that catalyze redox reactions are usually large molecules (molecular mass typically in the range 30-300 kDa), and the effects of the protein environment distant from the active site are not always well understood. However, the structures and reactions occurring at their active sites can be characterized by a combination of spectroscopic methods. X-ray crystallography, transient and steady-state solution kinetics, and electrochemistry. Catalytic states of enzyme active sites are usually better defined than active sites on metal surfaces. 

DIFFUSION AND REACTION IN A POROUS ENZYME CARRIER STEADY-STATE SPLIT BOUNDARY SOLUTION... 

According to the preceding results we cannot determine the steady state of the system using the sequential approach suggested by Woodley [27]. This method involves sequential study of two phenomena reactant transfer in biphasic medium and enzyme kinetics in the aqueous medium. In the steady state, substrate transfer rate is equal to the reaction rate. 

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