Enzyme activation rate equation

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. 

If the three-parameter Michaelis-Menten equation is divided by C i, it becomes the same as the three-parameter Langmuir-I linshelwood equation where 1/Cm = Ka. Both these rate equations can become quite complex when more than one species is competing with the reactant(s) for the enzyme or active sites on the solid catalyst. 

Most biological reactions fall into the categories of first-order or second-order reactions, and we will discuss these in more detail below. In certain situations the rate of reaction is independent of reaction concentration hence the rate equation is simply v = k. Such reactions are said to be zero order. Systems for which the reaction rate can reach a maximum value under saturating reactant conditions become zero ordered at high reactant concentrations. Examples of such systems include enzyme-catalyzed reactions, receptor-ligand induced signal transduction, and cellular activated transport systems. Recall from Chapter 2, for example, that when [S] Ku for an enzyme-catalyzed reaction, the velocity is essentially constant and close to the value of Vmax. Under these substrate concentration conditions the enzyme reaction will appear to be zero order in the substrate. 

In the case of competitive inhibition (Fig. 9A), the inhibitor I competes with the substrate S for the active site of the enzyme. Setting Kj —> oo, the corresponding rate equation is... 

When binding of a substrate molecule at an enzyme active site promotes substrate binding at other sites, this is called positive homotropic behavior (one of the allosteric interactions). When this co-operative phenomenon is caused by a compound other than the substrate, the behavior is designated as a positive heterotropic response. Equation (6) explains some of the profile of rate constant vs. detergent concentration. Thus, Piszkiewicz claims that micelle-catalyzed reactions can be conceived as models of allosteric enzymes. A major factor which causes the different kinetic behavior [i.e. (4) vs. (5)] will be the hydrophobic nature of substrate. If a substrate molecule does not perturb the micellar structure extensively, the classical formulation of (4) is derived. On the other hand, the allosteric kinetics of (5) will be found if a hydrophobic substrate molecule can induce micellization. 

A more realistic but still relatively simple model of enzyme catalysis includes binding of both substrate and product as described by Equation 11.9. This reaction is characterized by five individual rate constants k and k2, and k4 and k5, correspond to the forward and reverse binding steps of the substrate S and product P to the enzyme E, respectively, while k3 expresses the irreversible chemical conversion at the enzyme active site ... 

In order for an equilibrium to exist between E -E S and ES, the rate constant kp would have to be much smaller than k i However, for the majority of enzyme activities, this assumption is unlikely to hold true. Nevertheless, the rapid equilibrium approach remains a most useful tool since equations thereby derived often have the same form as those derived by more correct steady-state approaches (see later), and although steady-state analyses of very complex systems (such as those displaying cooperative behavior) are almost impossibly complicated, rapid equilibrium assumptions facilitate relatively straightforward derivations of equations in such cases. 

The activation function of the biochemical neuron is defined by the reaction mechanism and the pertinent rate equations. This function is actually a set of differential equations derived from mass balances for the components taking part in the enzymic reactions in each biochemical neuron (see Section 4.1.3). 

This equation will describe the loss of enzyme activity, if the affinity label is binding at the active site. Moreover, reversibly bound ligands capable of occupying the same site as the affinity label will competitively inhibit the rate of enzyme inactivation. 

Unconsumed substrates are treated as substrates or essential activators in deriving rate equations and studying detailed mechanisms. Nonetheless, one must indicate whether an unconsumed substrate (U) remains bound to the enzyme or not (in this case, U also becomes an unaltered product) in the reaction scheme. In practice, unconsumed substrates are likely to be involved in all the typical multisubstrate kinetic mechanisms Only one case is illustrated here, namely that the unconsumed substrate Su activates catalysis when bound in a rapid-equilibrium ordered mechanism ... 

Symbol for maximal velocity of an enzyme-catalyzed reaction, usually expressed as the molarity change in product per unit time (usually, second). Fmax must not be confused with or specific activity the former has dimensions of time, and the latter is usually expressed as micromol product per unit time per milligram of protein. See Michaelis-Menten Equation Enzyme Rate Equations (1. The Basics)... 

Regulation of enzymic activity occurs via two modes (cf. Ref. 50) alteration of the substrate binding process and/or alteration of the catalytic efficiency (turnover number) of the enzyme. The initial rate of a simple enzymatic reaction v is governed by the Michaelis-Menten equation... 

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... 

In this mechanism, the catalytically active rhodium(I) derivative is a bimetallic compound thereby providing active sites on two adjacent rhodium atoms. The methods of enzyme kinetics (12) yield the following rate equation for this bimetallic mechanism ... 

The maximum rate of a reaction (V ) is attained when all the enzyme active sites are saturated with substrate molecules. For the rate to approach VmiX> the substrate concentration must be high in fact, it must be much greater than KM. When [S] > Ku, the Michaelis-Menten equation becomes... 

Fio. 2. Proposed kinetic model for yeast inorganic pyrophosphatase. Here M represents Mg + but may also apply to any divalent cation with which the enzyme is active. In the rate equation A represents all mono-magnesium PPi complexes, B represents the di-magnesium complex, and I represents free PPi. Hydrogen ion equilibria are not considered. Kinetic runs were done at pH 7.4, 30° (9). Best values for kinetic constants were obtained from a computer program for nonlinear regression (9S). 

Applicable rate equations, with accompanying definitions of kinetic expressions, for the various activities of the partially purified rat liver microsomal enzyme at pH 6.0, derived on the basis of the mechanism in Fig. 4, are as follows (40) (to simplify notation, fc5 X [H20] is set =... 

Concentration of A Arrhenius constants Arrhenius constant Constant in equation 5.82 Surface area per unit volume Parameter in equation 5.218 Cross-sectional area Concentration of B Stoichiometric constants Parameter in equation 5.218 Concentration of gas in liquid phase Saturation concentration of gas in liquid Concentration of G-mass Concentration of D-mass Dilution rate DamkOhler number Critical dilution rate for wash-out Effective diffusion coefficient Dilution rate for maximum biomass production Dilution rate for CSTF 1 Dilution rate for CSTF 2 Activation energy Enzyme concentration Concentration of active enzyme Active enzyme concentration at time t Initial active enzyme concentration Concentration of inactive enzyme Total enzyme concentration Concentration of enzyme-substrate complex with substance A... 

Enzyme assays The amount of enzyme protein present can be determined (assayed) in terms of the catalytic effect it produces, that is the conversion of substrate to product. In order to assay (monitor the activity of) an enzyme, the overall equation of the reaction being catalyzed must be known, and an analytical procedure must be available for determining either the disappearance of substrate or the appearance of product. In addition, one must take into account whether the enzyme requires any cofactors, and the pH and temperature at which the enzyme is optimally active (see Topic C3). For mammalian enzymes, this is usually in the range 25-37°C. Finally, it is essential that the rate of the reaction being assayed is a measure of the enzyme activity present and is not limited by an insufficient supply of substrate. Therefore, very high substrate concentrations are generally required so that the initial reaction rate, which is determined experimentally, is proportional to the enzyme concentration (see Topic C3). 

Enzyme activity may be inhibited by substances that inactivate the enzyme or occupy the active site of the enzyme before the substrate has a chance. As a result, the rate of transformation of the substrate to product is slowed. In competitive inhibition, similar substrates (or analogs) can bind to the same active site on the enzyme. Therefore, they compete with each other for the same active sites. This inhibition process is reversible and can be prevented or slowed by increasing the substrate concentration or by diluting the inhibitor in the solution. In this case, the enzyme already bound to the substrate is not inhibited. The effect of the competitive inhibitor (I) on the rate of enzyme reaction in Equation (5.129), Equation (5.130), Equation (5.131), and Equation (5.132) yields ... 

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. 

A further complication arises if a significant fraction of the total catalyst material may be present in the form of reaction intermediates rather than as the free catalyst. If the catalyst is highly active, a minute amount suffices to produce a high reaction rate, and even a trace-level intermediate may then contain a large fraction or possibly most of the catalyst material. Such behavior is typical for enzyme catalysis, but not confined to it. In such cases, the concentration of free catalyst may vary with conversion, may not be known, and may be very difficult to measure. Rather, what is known is the total amount of catalyst material added, and rate equations in terms of the latter are therefore needed. Such systems will be discussed in the later sections of this chapter. 

The presence of enzyme activators such as metal ions and coenzymes can be determined by using enzymes. For the assay enzyme the metal ion or cofactor is essential for activity. The amount of activator in a sample is determined from the rate enhancement of the reaction. Guilbault (37) lists 12 metal ions and 10 cofactors which can be determined enzymatically. Three of these substances can be determined with luciferase (Equation 23). 

When the equilibrium for reversible E-I complex formation ( Ti) is rapid, and the rate of dissociation of the E-I complex (fcoff) is fast relative to kjuact (the most common situation), then kinact is the rate-determining step, and time-dependent loss of enzyme activity occurs. Under these conditions, when [I] [Eq], then Kitz and Wilson (65) described kapp by Equation 19. Two... 

According to Equation 17.47, an affinity label should exhibit time- and concentration-de-pendent inactivation. The rate of inactivation is proportional to low concentrations of inhibitor, whereas at high inhibitor concentrations saturation occurs and no further increase in the rate of inactivation is observed. A typical pseudo first-order plot of log enzyme activity vs. time is illustrated in Fig. 17.26. In some cases nonlinear plots may be obtained, particularly for mechanism-based inhibitors (166, 167). 

This approach was used in the study of the properties of D-amino acid oxidase isolated or fixed in cells of Trigonopsis variabilis and entrapped in calcium pectate or polyacrylamide gel [28]. The approach of a differential reactor (low enzyme activity in the packed bed) was applied. The experimental thermometric data, ATr, were transformed to reaction rates, vobs, according to Eq. (21), whereas parameter a was determined by the calibration shown in Fig. 5. The data were described by the equation... 

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