ENZYME KINETICS: THEORY. A. Introduction

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1 ENZYME INETICS: THEORY A. Introduction Enzyes are protein olecules coposed of aino acids and are anufactured by the living cell. These olecules provide energy for the organis by catalyzing various biocheical reactions. If enzyes were not present in cells, ost of the cheical reactions would not proceed at easurable rates at the teperatures of living systes. Each enzye has at least a single active site which is the location where the enzye binds to the substrate. In this way the substrate is held rigidly in the ost favorable orientation. Within the active site there are various cheical groups that are involved in the reaction. It is iportant to reeber that enzyatic reactions usually result in the addition or reoval of soe olecule or radical such as H2O, -OH, -H, -NH2, etc. Each enzye possesses a ph and a teperature optiu for its activity. This optiu ph and teperature can be easily deterined in the laboratory by carrying out the reaction in buffers over a wide range of ph or conducting tests at different teperatures. Enzyes deonstrate a rather high degree of specificity with respect to their substrates. The degree of specificity varies fro enzye to enzye: soe enzyes carry out a reaction in only one direction (e.g., dehydrogenation) but soe will catalyze a reaction in both forward and reverse directions although usually at greatly different rates (e.g., hydrogenation in addition to dehydrogenation). Soe enzyes will accept only one or two specific substrate olecules; others accept whole classes or subclasses of olecules as substrates. This specificity is the basis for enzye noenclature: according to the kind of reaction perfored (e.g., hydrogenase) or, even ore specifically, according to the substrate acted upon (e.g., succinic dehydrogenase). The siplest possible case of an enzye (E)-catalyzed reaction involves a single substrate (S) olecule giving rise to one product (P): E + S ---> P + E (1) and we find that the aount of P fored increases with tie until a plateau is reached as shown in Figure 4. Figure 4. Product concentration as a function of tie for an enzye catalyzed reaction. (Square brackets denote olar concentration.) 1

2 We are usually concerned with the initial rate (or initial velocity) value ( 0 ) which is the slope easured very near t=0; it is an iportant value for characterizing an enzyatic reaction. It is observed that the velocity depends on the concentration of E as depicted in Figure 5. Figure 5. Product concentration as a function of tie and enzye concentration. A ore critical deonstration of the effect of enzye concentration upon the reaction in question is given by varying [E] and holding tie constant. This produces a curve siilar to that of Figure 4 but with [E] replacing t at the abscissa. Now we ay ask what happens when [E] is held constant and [S] is varied. Here we will obtain a curve as in Figure 6. In this graph, the rate of product foration is called the velocity of the reaction. Notice how the velocity ( O.D. easured by a spectrophotoeter) reaches a iu as substrate concentration reaches a saturation level. Figure 6. elocity of an enzyatic reaction as a function of substrate concentration. Doubling [E], under certain conditions, doubles, but the reaction rate always reaches a plateau at high [S]. Actually, E and S interact to for a coplex, fro which P eerges: 2

3 (2) This forulation is for a siple enzyatic reaction where k1, k2, and k3 denote the kinetic constants for individual reaction coponents indicated by the arrows. The possible back reaction fro E + P ---> ES is assued to be negligible and so no k4 is required for accurate description of the equilibriu. Enzye reactions often require the participation of soe sall olecules such as ATP, NAD, Mg ++ or other cofactors. Concerning equation (2), we can say that: 1. The rate of foring ES is k1 [Ef] [S], where Ef (free enzye) = ([E] - [ES]), that is total E inus bound E. Thus, the rate of foring ES = k1 ([E] - [ES]) [S]. 2. The rate of losing ES is the su of 2 pathways: k2 [ES] (the rate of going back to E + S) k3 [ES] (the rate of going forward to E + P). Thus, the rate of losing ES = (k2+k3) [ES]. 3. Looking at the two above parts, we see that the reaction in equation (2) when the forward and backward rates around ES are in equilibriu, we have: which can be written as: k1 ([E] - [ES]) [S] = (k2+k3) [ES] k 2 + k 3 = E k 1 ( [ ES ])[ S ] [ ES ] " = # $ [ E ] ES % ' S 1 & (3) The Geran biocheist, Leonor Michaelis stated that the ratio of (k2 + k3)/k1, denoted by, is a characteristic quantity for a given enzye on a given substrate under fixed conditions of ph, teperature, etc. Thus: = k 2 + k 3 k 1 (4) is called the Michaelis constant and is the easure of the efficiency of the enzye. When k2 > k1, P foration slows down and is large. In contrast, when k1 > k2, P foration is accelerated and is a saller value. This eans that the ore efficient an enzye is, the lower its value. Michaelis also found that for the echanis indicated in equation (2), the rate of production of P, or velocity () of the reaction can be written as: and = k3 [ES] (5) = k3 [ES] 3

4 But for a given [E], the fastest rate of the reaction would be when all the E is coplexed. This occurs when [S] is uch uch greater than [E]; i.e., every enzye olecule easily finds a substrate olecule, so that: [ES] = [E] and = k3 [E] (6) Then, by dividing (5) by (6) we get: = ES E (7) Therefore, Or We note fro equation (3) that: B. Lineweaver-Burk equation " $ = # %$ S [ E] ES 1 & $ ' ($ S = [ E ] [ ES] 1 [ E] [ ES] = S If we take the reciprocal of equation (7) we obtain: +1 (8) [ E] [ ES] = (9) Note that the left sides of equations (8) and (9) are equal. Thus: = [ S] +1 If we divide the above equation by, we get: 1 = 1 [ S] + 1 (10) which is called the Lineweaver-Burk equation and is represented graphically in Figure 7. In this graph, when x = 0, y = 1/ and when y = 0, x = -1/. This plot is created using the sae data fro which Figure 6 was constructed, by siply taking the reciprocals of all (O.D. values) and [S] values. The solid line is drawn through the experiental points and the dashed line 4

5 represents extrapolation to deterine the x and y intercepts. and are readily calculated fro the values of these intercepts. Figure 7. Experiental data graphed on a Lineweaver-Burk plot. C. Enzye inhibitors One of the iportant facets of enzye kinetics is the phenoenon of enzye inhibition. Substances can be found which when added to a reaction ixture result in lower velocities; i.e., the reaction is inhibited. Two coon echaniss are copetitive inhibition and noncopetitive inhibition. If the inhibitor (I) is a substance whose shape perits it to bind at the substrate binding site of E, then copetition occurs between S and I leading to utually exclusive states. The inhibitor cobines with free enzye. It cannot cobine with the enzye when the substrate is already attached to E, hence the ter copetitive inhibition. Since the interaction between E and S is being interfered with, one ight expect the for the inhibited reaction to be different (larger). You reeber that the ore efficient the enzye, the lower would be the. In the presence of an inhibitor, the enzye works slower, so should be higher (larger). But since EI ay dissociate, excess S can overcoe the inhibition by effectively excluding I and so one ay also expect that the of the uninhibited reaction ay ultiately be attained. However, if the inhibitor attaches to E at soe site other than the active site, and by this binding causes a shape change in E that renders E catalytically inactive or less active, then we have noncopetitive inhibition: 5

6 The interaction of E and I does not interfere with the S binding to E, so that ESI ay be fored in addition to EI. But ES is sensitive to the presence of I, rendering ESI catalytically inactive or less active. Since the interaction between E and S is not being interfered with, any E not bound by I carries out its noral reaction at its noral rate so that one does not expect to be altered. But, since there is no way to exclude the binding of I, the inhibitor actually lowers the effective enzye concentration and this will, of course, alter (lower). D. Enzye assays Iplicit in all of the preceding discussions has been the idea that we can soehow isolate enzyes at will for study. In practice this is not always so easy. Biocheists obtain enzyes and easure their activities by various ethods. The enzye to be investigated is first extracted fro soe living aterial. It ay then be assayed in the crude extract or after various degrees of purification, using standard ethods for fractionating proteins. The processes of extraction and purification ay vary fro fairly siple to quite difficult, depending upon the particular enzye and its purity. An enzye assay consists of ixing the enzye with a substrate in a solution of controlled ph with any additional substance whose effect is to be tested, incubating the reaction ixture at a suitable teperature for a suitable tie, stopping the reaction precisely, and then soehow easuring the aount of reaction that has occurred. In general, the aount of reaction that has taken place ay be quantified in one of two ways: in ters of the disappearance of substrate or the appearance of product, depending upon which is cheically the ore advantageous. The actual ethod of easureent depends on soe biocheical or biophysical property of the olecule being assayed. For exaple, titration of acid or base or changes in absorption or rotation of light in the reaction ixture as soe olecule decreases or increases in concentration, easureent of evolved gas (a product), or transfer of a radioactive isotope fro substrate to product can all be used. Frequently, soe reagent is used which cobines with a product of the enzye reaction to produce a color. The intensity of the color can then be easured spectrophotoetrically and copared with color produced by known aounts of product to provide a basis for expressing units of enzye activity. ********** The preceding discussion is presented as a theoretical introduction underlying the study of any enzye. In the exercises that follow, we will take one enzye in particular (naely acid phosphatase) and apply the principles that we have discussed so far on it. There are two general classes of phosphatases found variously distributed in plant and anial tissues. These enzyes hydrolyze a wide variety of phosphoric onoesters. One class, tered acid phosphatase (the one we will be using in our exercise), has a ph optiu in the neighborhood of ph 5 and is inhibited by fluoride but not by etal chelating agents. The other, tered alkaline phosphatase, has a ph optiu around ph 9 and is generally not inhibited by fluoride but is inhibited by etal chelating agents. 6

7 Wheat ger (the wheat ebryo that is reoved fro the wheat grain before illing so that the flour will keep better) contains a phosphatase that can be extracted readily in water. The enzye is not as specific as soe other enzyes and will act on a wide range of possible substrates. We will use p-nitrophenyl phosphate (NPP) as substrate because it allows us a siple assay procedure: wheat ger phosphatase acting on NPP produces p-nitrophenol (PNP) as the product and PNP is yellow at alkaline ph. By adding Na2CO3 to our reaction ixtures, the ph can be raised enough to stop the reaction (an iportant requireent in enzye studies) and to ake the product colored for spectrophotoetric quantification. The reaction can be written as: E. Construction and use of a standard graph You reeber that if we easure the optical density (O.D.) of a solution at all the possible wavelengths and plot the data of wavelengths (λ) vs. O.D., we will obtain a bell-shaped curve with a single peak. The wavelength at which the peak occurs, called labda- (λ) is the wavelength at which the olecules have the highest absorption of light. Every solution has its own characteristic λ. For exaple, the λ of PNP is 420 n. According to Beer's Law, if we prepare a solution of a substance at a known concentration, dilute it to several additional known concentrations, easure the absorbance of each of these saples at the λ of the substance and plot optical density as a function of concentration, we should obtain a straight line that passes through the origin. This line is called the standard graph of the substance. Consequently, if soeone handed us a solution of the substance of unknown concentration, we could quickly deterine its concentration by easuring its absorbance. This could be done for as any unknowns as we wish. Reading such data fro a standard graph ay be done by siple inspection aided by a ruler and pencil. But a ore precise ethod would be to establish an equation of two ratios with one unknown using an unabiguous point fro the standard graph. Use of any section of this Lab Manual without the written consent of, Dept. of Biology, University of Pennsylvania is strictly prohibited. 7

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