Enzyme kinetics substrate concentration variation

Two other general ways of treating micellar kinetic data should be noted. Piszkiewicz (1977) used equations similar to the Hill equation of enzyme kinetics to fit variations of rate constants and surfactant concentration. This treatment differs from that of Menger and Portnoy (1967) in that it emphasizes cooperative effects due to substrate-micelle interactions. These interactions are probably very important at surfactant concentrations close to the cmc because solutes may promote micellization or bind to submicellar aggregates. Thus, eqn (1) and others like it do not fit the data for dilute surfactant, especially when reactants are hydrophobic and can promote micellization. 

Allosteric enzymes do not follow the Michaelis-Menten kinetic relationships between substrate concentration Fmax and Km because their kinetic behaviour is greatly altered by variations in the concentration of the allosteric modulator. Generally, homotrophic enzymes show sigmoidal behaviour with reference to the substrate concentration, rather than the rectangular hyperbolae shown in classical Michaelis-Menten kinetics. Thus, to increase the rate of reaction from 10 per cent to 90 per cent of maximum requires an 81-fold increase in substrate concentration, as shown in Fig. 5.34a. Positive cooperativity is the term used to describe the substrate concentration-activity curve which is sigmoidal an increase in the rate from 10 to 90 per cent requires only a nine-fold increase in substrate concentration (Fig. 5.346). Negative cooperativity is used to describe the flattening of the plot (Fig. 5.34c) and requires requires over 6000-fold increase to increase the rate from 10 to 90 per cent of maximum rate. 

Multiple substrate mechanisms follow Michaelis-Menten kinetics. Experiments are performed with constant concentrations of the enzyme and one substrate with variation of the second substrate concentration ([S2]). (Note that the second substrate concentration [S2] is not the same as a deceptively similar term, the square of the substrate concentration [S]2.) Plotting V against [S2] gives a hyperbolic curve and allows determination of Km for the second substrate. The Km values for all substrates may be found in a similar fashion. 

For packed-bed enzyme reactors, these results show that low flow rates ensure quantitative conversion of substrate into detectable products. The variation of K m with flow rate indicates that lower flow rates produce higher K m values, so that the linear region of the saturation kinetic curve extends to higher substrate concentrations at lower flow rates. This effect becomes significant when complete conversion does not occur during the residence time of the analyte. 

In studying the kinetics of an enzyme-catalyzed reaction the most reliable procedure is first to make rate measuremertts in the early stages of the reaction. If we can follow, by spectrophotometry or in any other way, the change in concentration of a substrate or a product, the initial slope of a concentration-time curve leads at once to the initial rate of the reaction (see Figure 9.4). The variation of this initial rate with substrate concentration can then be investigated. This is a much more reliable procedure than to allow the reaction to proceed to a greater extent and to analyze the concentration-time curves by the method of integration. The difficulty with this... 

Enzyme systems do not always follow Michaelis-Menten kinetics. Sometimes they show a sigmoidal variation of rate with substrate concentration, as shown schematically in Figure 10.12. There has recently been considerable interest in this kind of behavior, and it turns out that a number of different types of mechanisms can give rise to it. 

The variations in the initial reaction velocity with substrate concentrations in the case of most enzyme systems are explained by Michaelis-Menten kinetics. If the effect of pH, non-substrate components and buffers are neglected, the following simplified form can represent the... 

This occurs during a substantial part of the reaction time-course over a wide range of kinetic rate constants and substrate concentrations and at low to moderate enzyme concentrations. An illustration of this assumption is depicted in Figure 4 where the variation in [ES], d[ES]/d, [S], and [P] with the time-course of the reaction is simulated for a set of values. As seen in the figure, the simplified equation is valid throughout most of the reaction. 

The kinetic behaviour of a bound enzyme inhibited by the substrate shows multiple steady states within a particular range of concentrations of the substrate in the macroenvironment, provided diffusion to the enzyme is slow. When the substrate has to cross a membrane to reach the enzyme, regulatory schemes showed that the enzyme s activity can be changed dramatically by small variations in substrate concentration, membrane permeability, and the kinetic constants of the enzyme. It was concluded that the regulatory properties of an enzyme in a cell are more effective than suggested by experiments with the enzyme in solution. 

A coating bearing one enzyme (papain) is produced on the surface of a glass pH electrode by the method previously introduced (co-crosslinking). The papain reaction decreases the pH, and the pH-activity variation gives an autocatalytic effect for pH values greater than the optimum under zero-order kinetics for the substrate (benzoyl arginine ethyl ester) the pH inside the membrane is studied as a function of the pH in the bulk solution in which the electrode is immersed. A hysteresis effect is observed and the enzyme reaction rate depends not only on the metabolite concentrations, but also on the history of the system. 

The parameters that are plotted versus pH are (1) og(VIK) for each substrate, (2) log(V), (3) pA i (logarithm to the base 10 of the reciprocal of the dissociation constant) for a competitive inhibitor or a substrate not adding last to the enzyme, and (4) pAj or pA , for metal ion activators. It is particularly important to consider the pH variation of VIK and V, the two independent kinetic constants, and not simply to determine the rate at some arbitrary concentration of each substrate. The Michaelis constant is merely the ratio of V and V/K, so its pH profile is a combination of effects on V and V/K. Although we shall discuss the shapes of pH profiles, the reader should remember that graphical plotting is for a preliminary look at the data, and that the data must be fitted to the appropriate rate equation by the least-squares method to obtain reliable estimates of kinetic parameters, pA values, and their standard errors (5). Because pH profiles commonly show decreases of a factor of 10 per pH unit over portions of the pH range, the fits are always made in the log form [i.e., log(V), log(V/A), or pAj versus pH],... 

If an enzyme obeys Eqn. 16, i.e. if it shows linear kinetics with respect to both substrates, then by varying the concentration of one substrate with several fixed concentrations of the other, and measuring the enzymatic rate with each combination of concentrations, one may evaluate all 4 constants. Thus if [B] is fixed at several values, judiciously chosen after a preliminary experiment to establish the approximate range of values of [B] required to give rate variation, one would obtain a family of lines as shown in Fig. 11. 

In contrast with Michaelian enzymes, which have hyperbolic kinetics, allosteric enzymes, thanks to their sigmoidal kinetics, possess an enhanced sensitivity towards variations in the concentration of an effector or of the substrate. This is the reason why many enzymes that play an important role in the control of metabolism are of the allosteric type. 

In any enzyme system, activating or inhibitory effects are measured in terms of variations of the two kinetic constants, Km and as a function of the concentrations of substrate. A, or effectors, F. Two classes of effects may then be expected in allosteric enzymes. 

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