The Kinetics of the Enzyme- Catalyzed Inversion of Sucrose

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1 The Kinetics of the Enzyme- Catalyzed Inversion of Sucrose Recommended Preparatory Reading The subheadings Enzyme Catalysis and The Michaelis- Menten Mechanism in section 13.1 of Mortimer. Theory Enzymes are an important class of proteins that catalyze virtually all biochemical reactions. One such reaction, and the one that we will study in this experiment, is the inversion of sucrose, catalyzed by the enzyme invertase that is derived from yeast: HO HO O HO O O O HO + H 2 O Invertase HO HO O + HO Sucrose Glucose Fructose The basic mechanism by which an enzyme E converts a reactant S (usually called a substrate) into products P was first proposed by Michaelis and Menten. Their mechanism was confirmed using the inversion of sucrose and involves the following steps with rate constants kk: Step 1: E + S ES Step 2: ES E + P k 2 k -2 k 1 k -1 The first step in the mechanism, in which the enzyme- substrate complex ES is formed, is fast. The second step, involving the decomposition of the complex to form products, is relatively slow and, hence, is rate- determining step. We can ignore the reverse reaction in step two because (1) the rate constant for this reverse reaction is usually small and (2) [P] is small because we will be studying only the initial stage of this reaction. In many enzyme- catalyzed reactions there are numerous steps and complexes formed, however, the simple mechanism presented here is sufficient to describe the kinetics of the inversion of sucrose. If step two is the rate- determining step and we can ignore the reverse reaction, you should know from first- year that the reaction is first order with respect to ES and the rate law is rrrrrrrr = kk ES Equation 1 Page 1 of 11

2 Before proceeding any further, we need to express ES in terms of the enzyme and substrate concentrations. Michaleis and Mentin used the steady- state approximation for the complex ES. That is, they assumed that dd ES dddd, the net rate of change of ES, is very small compared to the rates of formation or destruction of ES. Thus, we can write the following expression: dd ES dddd = kk E S kk ES kk ES = 0 Equation 2 Solving this expression for ES yields ES = kk E S kk + kk = E S KK Equation 3 where KK is the Michaelis- Menten constant defined by KK = kk + kk kk. Note that if kk kk, KK is simply the equilibrium constant for the dissociation of the ES complex. We cannot determine the enzyme concentration at any particular time during the reaction, however, this will not present a problem because the enzyme is conserved; its total concentration (in the free form and in the complexed form) is always equal to the initial enzyme concentration E : E = E + ES Equation 4 Combining equations (3), (5), and (6) we obtain an equation for the rate of reaction as a function of substrate (sucrose) concentration: rrrrrrrr = kk E S KK + S Equation 5 Studies have shown that the kinetics of the enzyme- catalyzed inversion of sucrose are first order with respect to S when the concentration of sucrose is low: rrrrrrrr = kk E KK S. When the sucrose concentration is high the kinetics become zero order and the rate approaches a limiting value: rrrrrrrr = kk E. Thus, a determination of the dependence of rate on substrate concentration will yield both kk E and KK values. Further, if we know the value of E, kk can be calculated. kk is called the turnover number. It is simply the number of sucrose molecules hydrolyzed per second per molecule of enzyme (when all of the enzyme is in the complexed ES form). In this experiment, we will measure the initial rate of reaction as a function of E and S. Usually, we try to analyze the data by plotting the results in such a way so as to yield a straight- line relationship. Equation (7) can be rearranged into two useful forms: Page 2 of 11

3 Lineweaver Burke: 1 = 1 + KK 1 rrrrrrrr kk EE kk EE Equation 6 S Eadie Hofstee: rrrrrrrr S Equation 7 = kk EE KK rrrrrrrr KK Referencing equation (8), a plot of 1 rrrrrrrr versus 1 S is called a Lineweaver- Burk plot. kk E can be determined from the intercept then KK from the slope. An Eadie- Hoftsee plot can be constructed from equation 9. It is a plot of rrrrrrrr S versus rrrrrrrr, from which the slope can be used to determine KK and the intercept used to determine kk E. Acid- Catalyzed Reaction Many enzyme- catalyzed reactions can be carried out under non- physiological conditions by using H + ions as a less efficient catalyst. In the case of the inversion of sucrose, the acid- catalyzed reaction rate law has the form rrrrrrrr = + dd P dddd = dd S dddd = kk H Equation 8 S This rate law will be studied in the absence of invertase and at H concentrations in the range of 0.1 to 0.5 M, which have been found to give suitable values for the rate. On the other hand, the enzyme- catalyzed rate will be studied in a buffer solution that maintains H at ~10-5 M. This effectively illuminates the acid- catalyzed path during those runs because H is very low. Activation Energy Rate constants are frequently represented by the Arrhenius equation, kk = AA ee Equation 9 where the activation energy EE represents an energy barrier that must be overcome before the reaction, in this case, between sucrose and invertase can take place. In first- year chemistry you were told that invertase, or any other catalyst for that matter, causes the rate of the reaction to increase because it creates a different mechanism for the reaction; one that has lower activation energy. One of the objectives of this experiment is to determine EE for the enzyme and acid- catalyzed inversion of sucrose from the temperature dependence of the reaction rate. From these results you will be able to decide whether the higher efficiency of invertase can be attributed simply to a lowering of the activation energy barrier. Page 3 of 11

4 Method For this example of the Michaelis- Menten mechanism, we hypothesize that a complex (ES) is formed between one molecule of sucrose (S) and one molecule of invertase (E). The complex then reacts with a water molecule to yield glucose (product P ), and fructose (product P ), and regenerates one molecule of invertase (see the chemical equation given at the outset). The rate of product formation is given by rrrrrrrr = dd P dddd = dd P = kk dddd H O ES Equation 10 where kk is the rate constant of the rate- determining step. Remember that for reactions occurring in aqueous solution, the concentration of water remains constant, therefore, kk H O can be replaced by a new constant kk. This makes equation (13) identical to equation (3). We can monitor the progress of the reaction taking advantage of the fact that glucose and fructose are monosaccharides, and all monosaccharides are active reducing agents, while the sucrose from which they are formed is not a reducing sugar. Thus, any measure of the reducing capacity of the reaction mixture becomes, in effect, a measure of the extent to which the sucrose has been hydrolyzed in that mixture. We will use a 3,5- dinitrosalicyclic acid regent to assay the reducing capacity of the reaction mixtures. The oxidized and reduced form of this reagent absorb light at different wavelengths. Measurement of the absorbance of 540 nm light can be used to measure the extent to which sucrose was hydrolyzed when this reagent has been added to the reaction mixture. When you perform the experiment you will prepare reaction systems containing various amounts of sucrose, invertase, and buffer solution. The sucrose is allowed to hydrolyze for a specified period of time. The hydrolysis is then quenched (stopped) by addition of sodium hydroxide (contained in the 3,5- dinitrosalicyclic acid reagent solution). The mixture is heated to develop the color of the reduced form of 3,5- dinitrosalicyclic acid and the intensity of the color is determined using a spectrophotometer. After calibration of the spectrophometer, the reading directly establishes the number of moles of glucose and fructose present in the solution. Knowing the time that the mixture was allowed to react, we can calculate the rate at which sucrose was converted to glucose and fructose. Because we only study the reaction during the first five minutes, the rate calculated approximates the initial reaction rate, rrrrrrrr. Experimental Safety Information This is a relatively safe experiment. There are no major safety concerns associated with its performance. MSDS sheets for all chemicals used can be found in the experiment s folder placed near experimental setup in C Please wear appropriate eye protection and a lab coat for the duration of this experiment. Disposable gloves are not required but are available if you wish to use them. Page 4 of 11

5 The following aqueous solutions have been prepared for your use: 1. Invertase solution ( g/l). This solution must be kept chilled in an ice- water bath for the duration of the experiment! 2. Glucose- fructose standard solution (0.900 g/l glucose and g/l fructose) 3. 3,5- Dinitrosalicyclic acid reagent solution (10 g/l 3,5- dinitrosalicyclic acid, 300 g/l potassium sodium tartarate, and 0.4 mol/l sodium hydroxide) 4. Acetate buffer solution, ph (4.10 g/l sodium acetate and 2.65 ml/l glacial acetic acid) You will need to record the specific activity of the invertase used in this experiment as measured by the manufacturer (Sigma- Aldrich). This value can be found on the specification sheet located in the experiment s folder. Be sure to note the units and the temperature at which it was measured. Sucrose Solution Preparation In addition to the solutions listed above, you will also need a 0.3 M standard sucrose solution. Sucrose solutions are susceptible to attack by microorganisms, therefore, this solution cannot be made in advance. On an analytical balance, tare a 100 ml volumetric flask with a glass funnel in the top. Add sucrose to the funnel until the balance reads 10.2 to 10.3 g. When stable, record the actual mass added (to at least 4 decimal places) in your notebook. Wash any sucrose in the funnel into the flask with a gentle stream of distilled water from a wash bottle. Continue filling the flask until it is ½ - ¾ full then swirl until the sucrose is completely dissolved. Carefully fill with water until the bottom of the meniscus is sitting perfectly on the calibration mark on the neck of the flask (a dropping pipette should be used to precisely add the last few drops of water). There should be no water droplets above the calibration mark. Cap the flask tightly with a plastic stopper and invert it 25 times. Standard Assay Procedure 1. Using the adjustable Eppendorf pipettes, into a clean test tube dispense in the order given the appropriate volumes of enzyme solution, distilled water, buffer solution and sucrose solution. Your instructor will demonstrate how to use the pipettes properly. IT IS ESSENTIAL THAT YOU USE A NEW CLEAN PIPETTE TIP FOR EACH DIFFERENT SOLUTION! Swirl the test tube vigorously before and immediately after the addition of sucrose. 2. Begin timing when the sucrose solution is added. Let the reaction proceed for exactly 5 minutes (or whatever time period is specified for the particular run) 3. At the end of the time period, quench the reaction by adding 2.0 ml of 3,5- dinitrosalicycic acid reagent. Once again, swirl the test tube vigorously after adding this reagent. 4. Cover the test tube with a plastic cap and place the test tube in a boiling water bath for 5 minutes. 5. Cool the reaction mixture in the tube by holding it under cold running tap water. 6. Dilute with mixture with 15.0 ml of distilled water from a pipette. Cover tightly and shake to ensure through mixing. 7. Measure and record the absorbance using the HP 8452A diode array spectrophotometer (your instructor will explain its use). Page 5 of 11

6 A. Blank Runs We will use this set of runs to prove that the 3,5- dinitrosalicyclic acid reagent gives no appreciable test in the absence of the reducing sugar products (glucose and fructose). To establish this, we must examine the action of 3,5- dinitrosalicyclic acid on each of the other components of the reaction mixture (invertase, water and buffer solution). Label 4 test tubes A0, A1, A2 and A3. Begin the standard assay procedure above using the reagent volumes in the table below. You should initially place only the enzyme, water and buffer solutions in each test tube. Note in the table that an asterisk (*) is placed before the first volume of sucrose. This indicates that the 3,5- dinotrosalicyclic acid reagent is to be added to this test tube before adding the sucrose solution. This is no need to time run A0 after adding the sucrose solution it acts as a zero time blank. Begin the run by adding 1.0 ml of sucrose solution to tube A2 and starting the timer. After exactly 5 minutes quench (stop) the reaction by adding 2.0 ml of 3,5- dinotrosalicyclic acid reagent to tube A2 then to tubes A1 and A3. Pick up the standard assay procedure at step 3. Run Enzyme Water Buffer 0.3 M Sucrose (ml) (ml) (ml) (ml) A * 1.0 A A A Table 1 B. Standardization Runs Now we must measure the absorbance at 540 nm produced by the action of 3,5- dinotrosalicyclic acid on known amounts of glucose and fructose. Label a set of test tubes B1 B5. Add the indicated volumes of enzyme solution, water, buffer, sucrose solution and glucose- fructose solution to each tube. There is no need to time these runs since, in the absence of enzyme, no reaction will occur. Add 2 ml of 3-5- dinitrosalicyclic acid reagent to each tube and pick up the standard assay procedure at step 4. Run Enzyme (ml) Water (ml) Buffer (ml) 0.3 M Sucrose (ml) Glucose- Fructose Standard Solution (ml) B B B B B Table 2 Page 6 of 11

7 C- D. Progress of the Reaction with Time In a small 50 ml beaker, prepare some 0.03 M sucrose solution by pipetting 2.0 ml of 0.3 M sucrose solution into the tube and adding 18.0 ml of water (using the graduated pipette). Runs C are performed using the 0.3 M solution and runs D are performed using the dilute 0.03 M solution. Label two sets of test tubes as indicated in the tables below. Add ONLY the enzyme, water and buffer to each tube. Remember, tubes C0 and D0 get 2.0 ml of dinitrosalicyclic acid reagent before adding the sucrose and do not need to be timed. Begin the series of runs by adding the sucrose solution to tube C20 and simultaneously starting the timer. Add the sucrose solution to the other tubes at the times noted under the Start column. End the reaction by adding 2.0 ml of 3,5- dintrosalicyclic acid reagent to the tubes at the times indicate under the End column. Pick up the standard assay procedure at step 4. Run Enzyme Water Buffer 0.3 M Sucrose Time Start End (ml) ml) (ml) (ml) (min) (min:sec) (min:sec) C * C :00 5:00 C :00 6:00 C :00 7:00 C :00 11:00 C :00 20:00 Table 3 Run Enzyme Water Buffer 0.03 M Sucrose Time Start End (ml) (ml) (ml) (ml) (min) (min:sec) (min:sec) D * D :30 5:30 D :30 6:30 D :30 7:30 D :30 11:30 D :30 20:30 Table 4 E. Dependence of the Initial Rate on Substrate (sucrose) Concentration Label a set of test tubes. Notice that you must use the 0.3 M sucrose solution for tubes E0 E2 and the dilute solution for tubes E3 E6. Remember, E0 gets the 3,5- dinotrosalicyclic acid reagent before adding the sucrose and does not need to be timed. Start the runs at the time indicated in the table by adding the sucrose solution to tube E1. After 5 minutes, end the runs by adding 2.0 ml of 3,5- dinitrosalicyclic acid reagent. Pick up the standard assay procedure at step 4. Run Enzyme (ml) Water (ml) Buffer (ml) Sucrose (ml) Start End E * 1.0 (0.3 M) E (0.3 M) 0:00 5:00 E (0.3 M) 0:30 5:30 E (0.03 M) 1:00 6:00 E (0.03 M) 1:30 6:30 E (0.03 M) 2:00 7:00 E (0.03 M) 2:30 7:30 Table 5 Page 7 of 11

8 F. Nonenzymatic Hydrolysis of Sucrose In this part of the procedure we will perform hydrolysis runs using H + ions as the catalyst. Label the test tubes. Pipette the indicated volumes of 0.3 M sucrose solution and water into the test tubes. Add the required volume of 1 M HCl to tube F1 and start the timer. Add the acid to the other tubes at 1 min intervals. Allow the mixture in each test tube to react for exactly 5 minutes then quench the reaction by adding 5.0 ml of 0.1 M Na. Add 2.0 ml of dinitrosalicyclic acid reagent and heat the stoppered tubes in boiling water for 5 mins, cool, and dilute by adding 9 ml of water (not 15 ml as previously) so that the final volume is 20 ml. Measure the absorbance at 540 mn. Run 0.3 M Sucrose (ml) Water (ml) 1 M HCl (ml) F F F F F Table 6 G. Dependence of the Rate on Temperature The reaction will be carried out at several different temperatures. Water baths thermostated at 0, 12, 25, 35 and 45 o C have been set up around the lab. Your instructor will point out their locations. Choose, from runs E, a set of reactant volumes that gives an absorbance of ~0.5. You will use these volumes for all of the G runs. Label a set of test tubes G0, G12, G25, G35 and G45. Place the required volumes of enzyme, buffer and water in each tube. Label 5 more test tubes each with the letter S. Place 2 dropping pipette fulls of 0.3 M sucrose solution into each of the S tubes. Clamp tube G0 and one S tube into the 0 o C bath, tube G12 and one S tube into the 12 o C bath, and so on. Leave the tubes in the baths for at least 5 min before proceeding. Pipette 1.0 ml of the sucrose solution into the tube containing enzyme, water and buffer and start the timer, leaving the reaction mixture in the constant temperature bath. After exactly 5 min, quench the reaction by adding 2.0 ml of 3,5- dinitrosalicyclic acid reagent. Complete the assay by picking up the standard assay procedure at step 4. Page 8 of 11

9 Results NOTE: The concentrations of all solutions used in this experiment are given at the beginning of the experimental section of this outline First, calculate the concentration of the sucrose solution that you prepared at the beginning of the experiment. Now we must convert the measured absorbances at 540 nm into the desired product (P) concentration in the reaction mixture. A calibration is carried out using the results from runs A2 and B1 B5. According to Beer s law, AA = εεεεεε Equation 11 where dd is the path length of the solution and cc is the concentration of the adsorbing species (in this case the reduced form of 3,5- dinotrosalicyclic acid in the final 20 ml solution). The absorption coefficient ε is constant at any specified wavelength. dd will also be constant if matched cuvettes with the same path length are used. The value of cc is proportional to nn + nn where nn and nn are the number of moles of glucose and fructose in the reaction mixture up to the time at which it was quenched. From the reaction stoichiometry, it is obvious that nn = nn = xx, where xx is the number of moles of sucrose hydrolized. Thus, cc is proportional to xx and we can write Equation (14) as AA = αααα Equation 12 where αα is a calibration constant. The product concentration P = P = P is given by P = xx VV = AA αααα Equation 13 where VV is the volume of the reaction mixture (3 ml for enzyme runs and 4 ml for acid runs). For each run, in a spreadsheet tabulate VV, AA at 540 nm, and the run time, tt. In a new column tabulate the volume of invertase solution used, VV. Also, somewhere on the spreadsheet report values of the glucose concentration and the fructose concentration in the glucose- fructose solution and the enzyme stock solution concentration. For the standardization runs calculate nn + nn 2, where nn and nn are the number of moles of glucose and fructose added to the assay tubes in the standardization runs. Plot a graph of AA versus nn + nn 2. Add the best- fit line to the graph, report the slope and give its units. The slope is equal to the calibration constant αα. For each run in parts C, D, E, F and G, use the absorbance value, AA, and your value of αα to calculate xx (the number of moles of sucrose hydrolyzed during the run). In a new spreadsheet column, use the concentration of the enzyme stock solution to calculate the mass of enzyme added to the assay tube for each run, mm, in milligrams. Page 9 of 11

10 Now, for each run in part C calculate the specific activity of the enzyme, then calculate an average value. The specific activity is the number of micromoles of sucrose hydrolyzed, per minute, per milligram of enzyme present: ssssssssssssssss aaaaaaaaaaiiiiii = Equation 14 xx 10 tt mm Of course, the specific activity of an enzyme is a function of the purity of the enzyme. The manufacturer of the enzyme uses purification methods to remove any inactive protein until the specific activity reaches a maximum value. Indicating that the enzyme is ~100% pure. The exact chemical composition of invertase is still unknown, however, its molar mass has been estimated to be ~100,000 g/mol. Using this molar mass and your calculated specific activity, estimate the turn over number, kk, for the enzyme. kk = ssssssssssssssss aaaaaaaaaaaaaaaa 1 mol sucrose 10 µμmol molecules sucrose 1 mol 1 min 60 s Equation mg 1 g invertase 100,000 g invertase 1 mol 1 mol invertase molecules In the introduction, we hypothesized that the reaction was zero- order in sucrose when the concentration is relatively high but becomes first- order when the sucrose concentration is low. Using your results from all of the runs in part C, plot a graph of xx versus tt. On the same graph but using a separate data series, do the same for all of the runs in part D and add best- fit lines to both of the sets of data. Indicate whether the data are consistent with our hypothesis (Hint: the slopes give an indication of the reaction rate). Using the data from runs E, prepare a Lineweaver- Burk plot of 1 rrrrrrrr versus 1 [S] and an Eadie- Hofstee plot of rrrrrrrr [S] versus rrrrrrrr. Note that the initial reaction rate, rrrrrrrr =, and has the units mol (. ) L- 1 min - 1. [S] is the initial sucrose concentration in the reaction mixture and will have the units mol L - 1. Referring to equations 6 and 7, determine the values at room temperature of kk E and KK from both of these plots. Using the estimated molar mass of 100,000 g/mol for invertase, calculate E in mol/l units and obtain the value of kk in clearly stated units. From the data of part F, estimate the rate constant kk for the acid- catalyzed reaction using equation 10. Also calculate the turnover number for the acid- catalyzed reaction, i.e., the number of molecules of sucrose hydrolyzed, per second per hydrogen ion present. You can do this by first finding the specific activity of H (remember, you have already done this calculation for the enzyme) then use conversion factors to arrive at the turnover number. Finally, from the data of part G, determine the rrrrrrrr for each temperature TT and plot log rrrrrrrr From the slope of the line, determine the activation energy from versus 1 TT. dd log rrrrrrrr = EE dd 1 TT RR Equation 16 Page 10 of 11

11 Discussion Compare your experimental specific activity with that measured by Sigma- Aldrich (it can be found in the experiment s folder). Give reasons why your experimental value might not agree with Sigma- Aldrich s. Compare the activation energy for the enzyme- catalyzed reaction with that for the acid- catalyzed reaction. Does the difference in EE values account completely for the ratio of turnover numbers for the enzyme and H? Which of the two types of analysis, Lineweaver- Burk or Eadie- Hofstee, seems to give better results and why? How does the value of kk that you calculated (using results from part E) agree with the estimated value that was based on the specific activity? Page 11 of 11

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