Michaelis-Menten Kinetics: The Enzyme-Catalyzed
Hydrolysis of Sucrose1
Background
Of the numerous chemical reactions whose kinetics have been studied, some of the most
interesting and complex are those which take place in living organisms. These reactions,
when studied under non-biological conditions (lack of organisms and their various
biomolecules), are often exceptionally slow: often hours or days, and sometimes even
years are required before equilibrium is reached. Yet life requires seconds or even
fractions of seconds to produce the same results, and it does this through the use of
biological catalysts called enzymes. Enzymes are proteins with special clefts or pockets in
their structures in which the reactions occur: the tremendous advantage that enzymes
have is that these pockets, by nature of their shape and chemical environment, force the
reactants to form transition states much lower in energy than the transitions states the
reactants would form under other circumstances, say in solution. Thus, life can subsist at
a reasonable pace and it does not take us a week to walk across a room or digest an apple.
Of the many kinetic models for enzymes currently used, the first (and simplest)
developed is the Michaelis-Menten Model, developed in 1913 by Leonor Michaelis and
Maud Menten, which is still quite useful today. One of its early triumphs was the
successful explanation of the kinetics of sucrase, or β-fructofuranidase, an enzyme found
in yeast which cleaves sucrose (common table sugar) into glucose and fructose, and it is
this process that will be the focus of our lab.
Theory
The overall reaction catalyzed by sucrase is:
(1)
where S is sucrose, E is the enzyme, and G and F are glucose and fructose, respectively.
The overall rate law for the rate of reaction, r, predicted by the Michaelis-Menten Model
is:
(2)
where [S] is the instantaneous sucrose concentration, [E]o is the starting enzyme
concentration, Km is the Michaelis-Menten constant and k is known as the turnover
number. Note that if [S] << Km the reaction becomes first order in starting enzyme
concentration and sucrose concentration, and if [S] >> Km the reaction becomes first
order in starting enzyme concentration. In this latter case the reaction rate and initial
enzyme concentration are related through the turnover number, which can be thought of
1
Based on a laboratory exercise in Experiments in Physical Chemistry 6th Ed. (1996) by Shoemaker,
Garland and Nibler.
as the number of sucrose molecules cleaved per unit time per molecule of enzyme when
essentially all of the enzyme is complexed with sucrose and water (note that the rate is
independent of water concentration, which does not appreciably change during the course
of the reaction).
Another catalysis route we will study is the more “mundane” catalysis by protons, i.e. in
acidic solution. This route is expected to obey the rate law:
(3)
Overall Method
The various reactions will be performed in test tubes, and experimental data obtained
from SPEC-20’s. Aliquots of enzyme solution, water, buffer solution (to maintain the pH
at a level for which the acid-catalyzed route is negligible) and sucrose solution. After the
desired amount of time has passed, a solution of NaOH (to kill off the enzyme’s activity)
and 3,5-dinitrosalicylic acid (which acts as an indicator) is added and the tube placed in
boiling water for five minutes. The solution (after being cooled) is diluted and a sample
placed in the SPEC-20 to determine its %T at 540 nm. The amounts required for all the
runs will be provided below, and special instructions (e.g., for solutions requiring other
reagents) will be given as needed. Note: Every run you will be doing requires placing the
sample (with 3,5-dinitrosalicylic acid) in boiling water for five minutes, including
standardization runs.
Experimental
Equipment Which Should Be Present at Each Station:
-
5 2 or 5 mL graduated pipets (labeled “GFS” for glucose-fructose solution,
“S” for the 0.3 M sucrose solution, “N” for the 3,5-dinitrosalicylic acid
solution, “D” for deionized water and “E” for enzyme solution)
1 Plastic graduated pipet (for dispensing buffer solution “B”)
1 15 mL pipet (for the dilution procedure)
1 large pipet bulb
1 small pipet bulb
1 stoppered iodine flask in ice-water bath (labeled “E” for enzyme solution)
4 stoppered Erlenmeyer flasks (labeled “GFS,” “B,” “S,” and “N”, containing
the solutions mentioned above
1 100 mL beaker for deionized water
1 10 mL graduated cylinder for the diluted sucrose solution
3 large test tubes with screw-on caps
3 600 or 800 mL beakers (for waste, holding reaction tubes, and boiling
water)
1 Spec-20
5 small 1 cm path length test tubes with wooden rack
1 stopwatch
Overall Procedure for Doing a Run:
-
Add (in this order) the enzyme solution, DI water, and buffer solution.
Swirl vigorously.
Add the sucrose solution while at the same time starting the stopwatch.
Swirl vigorously after addition is complete and let sit.
After the desired time has passed, add the 3,5-Dinitrosalicylic acid solution
and swirl vigorously again.
Cover test tube loosely with parafilm and place in boiling water for roughly
five minutes.
Cool by placing under a stream of cold water (hold the test tube at an angle)
Add 15 mL of deionized water and shake (with the lid on!) vigorously to
make sure everything is mixed quite thoroughly.
Add a small amount to a small test tube and swirl to “wash” it, dump out and
fill with more solution.
Adjust for 0 and 100%T using DI water as your blank and measure %T of
your solution.
Important Notes:
-
-
BRING AND WEAR YOUR GOGGLES! (3,5-dinitrosalicylic acid is toxic
and we will be dispensing it often, so there are plenty of chances for liquid to
fly up and possibly hit you in the eye.)
Wear gloves, as the compound mentioned above is toxic.
The enzyme is easily chewed up by bacteria and other bugs in the surrounding
area, so the enzyme solution must be kept as sterile as possible and chilled at
all times. Try to have the flask stoppered whenever it is not in use and never
remove solution with a pipet other then the one specified for this purpose.
Please place all liquid samples in your waste beaker and add to the class waste
bottle as it gets full.
Experimental Runs
A. Standardization Runs
As we will be using absorbances to determine concentrations, we will need to
create a calibration plot, and the runs in this section are designed to do just
that. No enzyme is involved, so just add all of the amounts of the reagents
indicated in the table below as well as 2 mL of the 3,5-dinitrosalicylic acid
reagent, mix thoroughly, place in boiling water for five minutes and dilute
before analysis in the Spec-20:
B.-C. Progress of the Reaction With Time
For these runs, follow the procedure outlined in the “Overall Procedure” listed
above. The first run of each set (C0 and D0) is a “zero-time blank” in which the
3,5-dinitrosalicylic acid solution is added before the sucrose and the mixture is
swirled and analyzed without having to wait for a given time interval. Asterisks
have been added to mark these runs as different from the others. For the 0.03 M
solutions in set D, take 1 mL of the 0.3 M sucrose solution and dilute to 10 mL
with DI water in the graduated cylinder.
Helpful Hint: If you do a little planning beforehand, you can do several runs in a
staggered fashion, one after the other: prepare the three tubes, without the sucrose,
and add sucrose to the first tube and start the stopwatch. One minute (or however
long you prefer) later, add sucrose to the next tube, and repeat for the third
another minute after that. Instead of focusing on one tube at a time, you can do
three runs in a relatively short space of time and with little difficulty.
Of the next two sets of runs, only one is required (you can pick which one you want
to do). However, you may do both for extra credit if there’s enough time.
D. Dependence of Initial Rate on Sucrose Concentration
You should make a fresh 0.03 M solution for this set. Note that the first run is a
zero-time blank run!
E. Acid-Catalyzed Hydrolysis of Sucrose
For these runs, you will be using hydrochloric acid rather than sucrase as the
catalyst.
Extra Equipment Needed:
-
1 Erlenmeyer flask with 1 M hydrochloric acid
1 Erlenmeyer flask with 1 M sodium hydroxide
1 5 mL transfer pipet
1 9 mL transfer pipet
Procedure:
-
Add the sucrose solution and DI water to a tube.
Add the acid, starting the watch as you do so.
Wait for 5 minutes, then add 5 mL of sodium hydroxide to kill the catalysis.
Add 3,5-dinitrosalicylic acid, put in boiling water for 5 minutes as before
Dilute with 9 mL DI water, not 15
Determine %T, A at 540 nm
Calculations to be Performed Before Dry Lab
Calibration Plot
The concentrations of the absorbing species in this lab will be such that Beer’s
Law, A=εbc will hold. However, there are two products formed in the hydrolysis
reaction, both of which react with 3,5-dinitrosalicylic acid to form absorbing
species. The value of c will be proportional to the number of moles of the two
products, n1 + n2. Since both products are formed in equal amounts, we can set
them equal to a variable x, which would be the number of moles of sucrose
hydrolyzed. Therefore c is also proportional to x and Beer’s Law can be rewritten
as
(4)
The concentration of glucose will be equal to that of fructose, or a general
“product” concentration [P]= [F]= [G], which can be related to absorbance
through
(5)
where V is the volume of the reaction mixture in the tube, not that of the solution
placed in the spectrometer.
For determining a calibration plot, set x = (n1+n2)/2, where n1 and n2 are the
number of moles of glucose and fructose added to the tube. Since the amounts of
the two compounds in the stock solution may not be the same (since the weights
of the two samples were probably slightly different), it is a good idea to use the
average of the two. The molecular weight of both compounds is 180.16 g/mol.
Variation of Reaction Rate with Time
Do you expect to see the reaction be first-order for the sucrose concentrations
from either set B or set C? If so, which one? What about zero-order dependence?
Using x as defined earlier, try plotting x/0.003 vs. t for sets B and C to check for
zero-order dependence (0.003 being the volume of the reaction mixture in L), as
well as ln[a/(a-x)] vs. t to check for first-order dependence (a being the number of
moles of starting sucrose). What are your findings?
Dependence of Initial Rate on Sucrose Concentration
Data from set E can be used to determine Km and k used in (2). Two different
linear relationships may be used to do this: both employ the starting sucrose
concentration [S]0 and initial reaction rate r0 (essentially the same as r). These are
the Lineweaver-Burk plot (6) and the Eadie-Hofstee plot (7):
(6)
(7)
The Lineweaver-Burk plot is a plot of 1/r0 vs. 1/[S]0 while the Eadie-Hofstee plot
has r0/[S]0 and r0 as the dependent and independent variables, respectively. In
both cases k[E]0 is determined from the intercept and Km from the slope. Using
the concentration of enzyme solution (in g / L) that the TA will give you, as well
as the assumption that the molecular weight of sucrase is roughly 100,000 g / mol,
determine [E]0 and k. Which of the two plots seems to give better results?
Acid Catalyzed Sucrose Hydrolysis
Do your data appear to be in agreement with the proposed rate law? Determine kH
as well as a turnover number (number of molecules of sucrose hydrolyzed per
proton per second). According to the rate law this depends on [S]. Is this the case?
Compare the enzyme turnover number with a “normal” hydrogen ion turnover
number. What does this say about the effectiveness of enzymes?