Dependence of Enzyme Reaction Rate on Substrate Concentration

2 DEPENDENCE OF ENZYME REACTION RATE ON SUBSTRATE CONCENTRATION 

When [S]o the enzyme concentration, is usually directly proportional to the enzyme concentration in the reaction mixture, and for most enzymes v0 is a rectangular hyperbolic function 

The equation describing the rectangular hyperbola that usually represents enzyme-reaction data (e.g., Fig. 9-1) is called the Michaelis-Menten equation  

The equation has the property that when [S]0 is very large, v0 = Kmax (the so-called maximal velocity), also when u0 = Fmax/2, the value of [S]0 is Km, the so-called Michaelis constant. 

---

Dependence of Enzyme Reaction Rate on Substrate Concentration... 

Allosteric enzymes are oligomeric proteins exhibiting a sigmoidal dependence of the reaction rate on substrate concentration instead of a normal , hyperbolic (classical Michaelis-Menten) one. At first the rate increases only slightly with increasing substrate concentration there is then a rapid increase until near the maximum rate. This behaviour results from the presence of a regulatory allosteric center located on the same or another subunit as the catalytic center. This center may interact... 

Fig. 6.38 Dependence of the reaction rate on substrate concentration for competitive inhibition. (From http //cbc.arizona.edu/dasses/bloc462/462a/NOTES/ENZYMES/RawnFig7 32Compet.gif).

Parameters for the simple model were determined graphically by Eadie-Hofstee plotting of initial reaction rates and substrate concentrations. Details are given elsewhere (30). As has been observed in hydrolysis of other solid substrates, a residue of non-lysed substrate was found at extended reaction times, when dY/dt tended toward zero. The extent of reaction was strongly dependent on initial substrate and enzyme concentrations (33,34). An empirical funciton for Y was fitted to the ultimate turbidity data for lysis runs at a variety of initial yeast and enzyme concentrations using a least squares method. The calculated values for Yco were used in the simulations (30). Figure 3 shows results of the simple model. 

Fii. ee.e. Dependence of enzyme-catalyzed reaction rate on substrate concentration. At high concentrations, the enzyme becomes saturated with substrate and the reaction rate becomes maximum and constant since [ES] becomes constant (Equation 22.13). 

The dependence of an enzyme-catalyzed reaction rate on substrate concentration is illustrated in Figure 22.2. An enzyme is characterized by the number of molecules of substrate it can complex per unit time and convert to product, that is, the turnover number. As long as the substrate concentration is small enough with respect to the enzyme concentration that the turnover number is not exceeded, the reaction rate is directly proportional to substrate concentration, that is, it is first order with respect to substrate (Equation 22.13). If the enzyme concentration is held constant, then the overall reaction is first order and directly proportional to substrate concentration (k[E] = constant in Equation 22.13). This serves as the basis for substrate determination. However, if the amount of substrate exceeds the turnover number for the amount of enzyme present, the enzyme becomes saturated with respect to the number of molecules it can complex (saturated with respect to substrate), and the reaction rate reaches a maximum value. At this point, the reaction becomes independent of further substrate concentration increases, that is, becomes pseudp zeroj5r if the enzyme concentration is constant (Figure 22.2) in Equation 22.13, [ES] becomes constant and R = constant. 

What this really means is that the ternary complex has such a transitory existence that it never makes up a significant fraction of the total amount of enzyme. Steady-state kinetics concerns itself only with those complexes which, by their existence, detectably alter the pattern of dependence of reaction rate on substrate concentration. The Theorell-Chance mechanism may be seen perhaps as a manifestation of highly effective catalysis. Certainly, in the case of the enzyme for which it was first described, horse liver alcohol dehydrogenase, the mechanism is obeyed for good substrates i.e. short-chain primary alcohols with secondary alcohols, which are poor substrates, the ternary complex becomes kinetically significant - because it works less well [44]. 

Figure 1 Dependence of enzyme-catalyzed reaction rate on substrate concentration.

Not all enzymes obey Michaelis-Menten kinetics. The experimental dependence of the reaction rate on the substrate concentration is different from Michaehs-Menten type of concentration-substrate curves and follows a sigmoidal, or S-shaped, curve. 

Substrate concentration is yet another variable that must be clearly defined. The hyperbolic relationship between substrate concentration ([S ) and reaction velocity, for simple enzyme-based systems, is well known (Figure C1.1.1). At very low substrate concentrations ([S] ATm), there is a linear first-order dependence of reaction velocity on substrate concentration. At very high substrate concentrations ([S] A m), the reaction velocity is essentially independent of substrate concentration. Reaction velocities at intermediate substrate concentrations ([S] A"m) are mixed-order with respect to the concentration of substrate. If an assay is based on initial velocity measurements, then the defined substrate concentration may fall within any of these ranges and still provide a quantitative estimate of total enzyme activity (see Equation Cl. 1.5). The essential point is that a single substrate concentration must be used for all calibration and test-sample assays. In most cases, assays are designed such that [S] A m, where small deviations in substrate concentration will have a minimal effect on reaction rate, and where accurate initial velocity measurements are typically easier to obtain. 

The electron transfer from cytochrome c to O2 catalyzed by cytochrome c oxidase was studied with initial steady state kinetics, following the absorbance decrease at 550 nm due to the oxidation of ferrocyto-chrome c in the presence of catalytic amounts of cytochrome c oxidase (Minnart, 1961 Errede ci a/., 1976 Ferguson-Miller ei a/., 1976). Oxidation of cytochrome c oxidase is a first-order reaction with respect to ferrocytochrome c concentration. Thus initial velocity can be determined quite accurately from the first-order rate constant multiplied by the initial concentration of ferrocytochrome c. The initial velocity depends on the substrate (ferrocytochrome c) concentration following the Michaelis-Menten equation (Minnart, 1961). Furthermore, a second catalytic site was found by careful examination of the enzyme reaction at low substrate concentration (Ferguson-Miller et al., 1976). The Km value was about two orders of magnitude smaller than that of the enzyme reaction previously found. The multiphasic enzyme kinetic behavior could be interpreted by a single catalytic site model (Speck et al., 1984). However, this model also requires two cytochrome c sites, catalytic and noncatalytic. 

The equations of enzyme kinetics provide a quantitative way of desaibing the dependence of enzyme rate on substrate concentration. The simplest of these equations, the Michaelis-Menten equation, relates the initial velocity (Vj) to the concentration of substrate [S] and the two parametCTS and (Equation 9.1) The of the enzyme is the maximal velocity that can be achieved at an infinite concentration of substrate, and the of the enzyme for a substrate is the concentration of substrate required to reach Vz V iax- The Michaelis-Menten model of enzyme kinetics applies to a simple reaction in which the enzyme and substrate form an enzyme-substrate complex (ES) that can dissociate back to the free enzyme and substrate. The initial velocity of product formation, Vj, is proportionate to the concentration of enzyme-substrate complexes [ES]. As substrate concentration is increased, the concentration of enzyme-substrate complexes increases, and the reaction rate inaeases proportionately. 

Figure 5 Michaeiis-Menten dependence of the reaction rate (v) on the substrate concentration [S], The hyperbolic relationship is shown in (A) for three different concentrations of enzymes (E, Ea, and E3). A typicai piot for the determination of the enzyme concentration or activity is shown in (B).

In the measurement of enzyme activity, a high substrate concentration that is greatly in excess of the Km value is always used, and the enzyme sample to be investigated is correspondingly diluted vmder the conditions, the rate of the enzyme-catalyzed reaction depends only on the enzyme concentration, i.e., it is a zero order reaction. Even under conditions of substrate saturation, the measured catalytic activities are influenced by slight differences in reaction conditions, such as the temperature, composition and concentration of the buffer, pH value, nature of the substrate and its concentration, coenzymes, and protein content in the sample. Therefore, the results of measurement of the catalytic activity of an enzyme are in principle method dependent direct comparison of the results between laboratories is made difficult by the use of different methods in different laboratories. 

A comparison of Figs. 2.25 and 2.27 leads to the conclusion that the dependence of the initial catalysis rate on substrate concentration allows the differentiation between a ternary and a binary enzyme-substrate complex. However, it is not possible to differentiate an ordered from a random reaction mechanism by this means. 

The kinetics of enzyme reactions were first studied by the German chemists Leonor Michaelis and Maud Menten in the early part of the twentieth century. They found that, when the concentration of substrate is low, the rate of an enzyme-catalyzed reaction increases with the concentration of the substrate, as shown in the plot in Fig. 13.41. However, when the concentration of substrate is high, the reaction rate depends only on the concentration of the enzyme. In the Michaelis-Menten mechanism of enzyme reaction, the enzyme, E, and substrate, S, reach a rapid preequilibrium with the bound enzyme-substrate complex, ES ... 

Another special case deserves comment—zero-order reactions. For a reaction that is zero-order with respect to a given substrate, the velocity does not depend on the concentration of substrate. We see zero-order behavior at Vmax the reaction is zero-order in the concentration of substrate. Note that even at Vmax, the reaction is full first order in the concentration of enzyme (the rate increases as the enzyme concentration increases). 

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. 

To purify a protein, it is essential to have a way of detecting and quantifying that protein in the presence of many other proteins at each stage of the procedure. Often, purification must proceed in the absence of any information about the size and physical properties of the protein or about the fraction of the total protein mass it represents in the extract. For proteins that are enzymes, the amount in a given solution or tissue extract can be measured, or assayed, in terms of the catalytic effect the enzyme produces—that is, the increase in the rate at which its substrate is converted to reaction products when the enzyme is present. For this purpose one must know (1) the overall equation of the reaction catalyzed, (2) an analytical procedure for determining the disappearance of the substrate or the appearance of a reaction product, (3) whether the enzyme requires cofactors such as metal ions or coenzymes, (4) the dependence of the enzyme activity on substrate concentration, (5) the optimum pH, and (6) a temperature zone in which the enzyme is stable and has high activity. Enzymes are usually assayed at their optimum pH and at some convenient temperature within the range... 

Equations (5) and (10) imply that the velocity of an uncatalyzed reaction increases indefinitely with an increase in the concentration of the reactants. With enzyme-catalyzed reactions, something very different is observed. The rate usually increases linearly with substrate concentration at low concentrations, but then levels off and becomes independent of the concentration at high concentrations (fig. 7.6). The explanation for this hyperbolic dependence on substrate concentration is straightforward. For an enzyme to affect AG, the substrate must bind to a special site on the protein, the active site (fig. 7.7). At very low concentrations of substrate, the active sites of most of the enzyme molecules in the solution are unoccupied. Increasing the substrate concentration brings more enzyme molecules into play, and the reaction speeds up. At high concentrations, on the other hand, most of the enzyme molecules have their active sites occupied, and the observed rate depends only on the rate at which the bound reactants are converted into products. Further increases in the substrate concentration then have little effect. 

While the majority of these concepts are introduced and illustrated based on single-substrate single-product Michaelis-Menten-like reaction mechanisms, the final section details examples of mechanisms for multi-substrate multi-product reactions. Such mechanisms are the backbone for the simulation and analysis of biochemical systems, from small-scale systems of Chapter 5 to the large-scale simulations considered in Chapter 6. Hence we are about to embark on an entire chapter devoted to the theory of enzyme kinetics. Yet before delving into the subject, it is worthwhile to point out that the entire theory of enzymes is based on the simplification that proteins acting as enzymes may be effectively represented as existing in a finite number of discrete states (substrate-bound states and/or distinct conformational states). These states are assumed to inter-convert based on the law of mass action. The set of states for an enzyme and associated biochemical reaction is known as an enzyme mechanism. In this chapter we will explore how the kinetics of a given enzyme mechanism depend on the concentrations of reactants and enzyme states and the values of the mass action rate constants associated with the mechanism. 

---