CHEM 331L
Physical Chemistry Laboratory
Revision 2.0
Determination of the Michaelis-Menten Parameters for
the Enzyme Alkaline Phosphatase
In this laboratory exercise we will determine the Michaelis-Menten kinetic parameters for the
catalytic enzyme Alkaline Phosphatase acting on the reactant p-Nitrophenyl Phosphate. The first
of these parameters determines how tightly the enzyme binds to the reactant. The second
determines the maximal rate at which the enzyme can catalyze the conversion of reactant into
product.
Enzymes are large proteins or protein complexes that catalyze biologically important reactions.
Like other catalysts, they speed-up chemical reactions, but are neither a reactant nor a product of
the overall reaction. The speed-up, or Reaction Rate increase, can be quite dramatic. In the case
of Catalase, an enzyme that catalyzes the decomposition of Hydrogen Peroxide, the enzyme
catalyzed reaction exhibits a 107-fold increase in Reaction Rate over the uncatalyzed reaction.
Historically, Louis Pasteur suspected some “vital force” was responsible for the ability of Yeast
cells to ferment Sugar into Alcohol. In 1878 the physiologist Wilhelm Kuhne coined the term
enzyme (fom the Greek ενζυμον for “in leaven”) to describe this “vital force”. Edward Buchner
later discovered a non-living extract of Yeast cells could catalyze this fermentation process; in
other words, enzymes are not “vital” in nature. Within a quarter century James Sumner showed
the enzyme Urease, an enzyme that catalyzes the conversion of Urea into Ammonia, is a protein
and isolated and crystallized it.
Crystals of the Protein Urease
http://sandwalk.blogspot.com/2007/10/nobel-laureate-james-batcheller-sumner.html
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Living cells contain thousands of different enzymes, each of which catalyzes a specific
biological reaction or reaction type; each enzyme being very specific for its reactant(s),
otherwise known as the Substrates. The substrate binds to a region of the enzyme known as the
Active Site.
The first working model of enzyme kinetics was put forth in 1913 by Leonor Michaelis and
Maud Menten.
Leonor Michaelis
http://en.wikipedia.org/wiki/File:Leonor_Michaelis.jpg
Maud Menten
http://en.wikipedia.org/wiki/File:Maud_Menten.jpg
In the Michaelis-Menten model, a large excess of Substrate (S) reversibly binds to the Enzyme
(E) to form an Enzyme-Substrate Complex (ES):
E + S
ES
Occasionally the substrate in this complex undergoes conversion to Product (P):
(Eq. 1)
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ES
E + P
(Eq. 2)
Since the second step in this Mechanism (Eq. 2) is slow, the Rate Law for the reaction is given
by:
Rate = k2 [ES]
(Eq. 3)
This rate law suffers from the fact that it contains the concentration of a mechanistic
Intermediate; ES. Since the concentration of an intermediate is typically difficult to measure, it
is desirable to eliminate it from the rate law. To do this, Michaelis-Menten applied the SteadyState Approximation to the ES concentration; in other words, it is assumed the ES concentration
is reasonably constant during much of the reaction period. Hence,
~ 0
(Eq. 4)
(As an example of a steady state process, think of a grocery check-out line in which 3 people are
being processed by the checkers. Imagine a situation in which customers arrive at the check-out
stand at exactly the same rate as happy customers leave. In this case we have a Steady-State of 3
customers at the check-out counter at all times. Our situation is slightly different because
customers arriving at the check-out counter can decide to return to the grocery aisles without
being checked-out.) The Steady-State Approximation is not always applicable, but it is
frequently useful.
Under this Approximation, the Rate of Formation of ES (Eq. 1 – forward rxn) is exactly balanced
by the Rate of Decomposition of ES (Eqs. 1 – reverse rxn and 2). Thus, we can re-write (Eq. 4) as:
0 ~ k1 [E] [S] - k-1 [ES] - k2 [ES]
(Eq. 5)
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which can be solved for the [ES] concentration:
[ES] =
(Eq. 6)
This result can be inserted into the Rate Law, (Eq. 3), to give:
Rate =
(Eq. 7)
At this point we introduce the first Michaelis-Menten parameter, KM, which measures the affinity
of the enzyme for the substrate. It is defined as:
KM =
(Eq. 8)
Large values of KM represent a weak binding of the substrate to the enzyme before reaction
occurs, and vice-versa for small values.
This allows us to re-write the Rate Law of (Eq. 7) as:
Rate =
(Eq. 9)
Now another complication presents itself. In almost all cases, it is impossible to measure the free
enzyme concentration [E]. Usually we know only how much total enzyme is available; [E]o.
This means:
[E]o = [E] + [ES]
(Eq. 10)
Solving (Eq. 10) for [ES], inserting the result into (Eq. 6) and solving for [E], gives us:
[E] =
(Eq. 11)
Placing this in the rate law of (Eq. 9), gives us a result which is entirely in terms of
experimentally determinable parameters:
Rate =
(Eq. 12)
If the Substrate concentration is large, [S] >> KM, this result reduces to:
Rate = vm = k2 [E]o
(Eq. 13)
(Eq. 13) represents the maximal rate at which the Enzyme can operate and vm is the second
Michaelis-Menten parameter. (Again think of our check-out counter at the grocery store.
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Consider a situation in which the store is flooded with customers; i.e., the substrate concentration
is high. The number of checkers remains constant. More customers entering the store will not
cause the checkers to process customers any faster. The checkers are working as fast as they
can. They can only check-out customers at a given maximal rate.)
Hence, we write the rate law in its final form:
v =
(Eq. 14)
where v denotes the reaction rate.
This can be inverted to give:
=
+
x
(Eq. 15)
Because the Rate, v, and Substrate concentration, [S], will typically decrease as the reaction
proceeds, these quantities are usually measured over a short period after the reaction initially
starts; v ≈ vo and [S] ≈ [S]o. Under this constraint, we have the Lineweaver-Burke form of the
Michaelis-Menten Rate Law, and (whew!) we are finally finished:
=
x
+
(Eq. 16)
If we measure vo at a series of different initial substrate concentrations [S]o we will obtain data
that form a straight line when plotted as 1/vo vs. 1/[S]o.
Plotting data in this form allows us to determine the Michaelis-Menten kinetic parameters,
because:
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slope = KM / vm
(Eq. 17)
intercept = 1 / vm
(Eq. 18)
and:
We will determine these parameters for the enzyme Alkaline Phosphatase, which removes
Phosphate (PO43-) from various substrates in an Alkaline, or basic (OH-), environment. This
enzyme is found throughout the body; liver, bone and intestine. It is a dimeric enzyme
consisting of two subunits, each with a MW = 69,000 g/mole and containing two Zinc (Zn) ions.
Its optimal pH for activity is in the range of pH = 8.0 – 10.5.
Ball and Stick Model of
Alkaline Phosphatase
The basic reaction catalyzed by this enzyme can be represented as:
Substrate-O-PO32-
Substrate + PO43-
Our particular substrate will be p-Nitrophenyl Phosphate:
(Eq. 19)
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The catalyzed dephosphorylation reaction for this substrate is written as:
p-Nitrophenyl-OPO32-(aq) + H2O
p-Nitrophenol(aq) + PO43-(aq)
(Eq. 20)
This substrate is particularly nice because the product is colored, so the Rate of Reaction v can be
measured by following how fast the color appears. And, this can be quantified by relating the
solution's absorbance (A) to its concentration (c) via the Beer-Lambert Law:
A = bc
(Eq. 21)
where  is the molar absorptivity of the absorbing species and b is the path length of the
spectrophotometer cell used in measuring the absorbance.
Thus, we will measure the Michaelis-Menten parameters KM and vm for Alkaline Phosphatase
using the substrate p-Nitrophenyl Phosphate. The needed Reaction Rate vo for the LineweaverBurke plot, (Eq. 16), will be determined by measuring how fast the solution becomes colored as a
result of the production of the product p-Nitrophenol.
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Procedure
Standard Curve
Our first task is to generate a Standard Curve relating the Concentration (c) of the product to its
Absorbance (A). Hence, we will measure the solution absorbance at a series of increasing
p-Nitrophenol concentrations.
1.
Obtain 7 test tubes and make the following dilutions:
Tube #
1
2
3
4
5
6
7
1 mM p-Nitrophenol (mL)
None
0.1
0.2
0.4
0.6
0.8
1.0
0.2M NaOH (mL)
2.5
2.4
2.3
2.1
1.9
1.7
1.5
Water (mL)
7.5
7.5
7.5
7.5
7.5
7.5
7.5
Mix well.
2.
Transfer 200 L from each test tube to a 96 Well Plate. Measure the Absorbance of each
solution at 420 nm using a Well Plate Reader.
Alkaline Phosphatase Activity
Now we will measure the Rate of Reaction of the Enzyme at various initial Substrate
concentrations.
1.
Obtain 7 test tubes and make the following solutions:
Tube #
1
2
3
4
5
6
7
Mix well.
6 mM p-Nitrophenyl Phos. (mL)
None
0.10
0.20
0.40
0.60
1.00
1.50
Buffer Sol’n (mL)
2.00
1.90
1.80
1.60
1.40
1.00
0.50
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2.
Heat the test tubes in a Warm Water Bath at 37oC.
3.
Dilute the Enzyme stock solution according the directions of the instructor.
4.
Add 0.5mL diluted Enzyme stock to test tube 2 and start timing the reaction. Mix the
solution and return it to the Warm Water Bath.
5.
Subsequently, every 1 minute, add 0.5mL dilute Enzyme stock to each of the reamaining
solutions, mix and return them to the Warm Water Bath.
6.
Stop each reaction after 30 minutes by adding 7.5 mL of 0.2 M NaOH.
7.
Take the tubes out of the Warm Water Bath and allow them to cool.
8.
Transfer 200 L from each test tube into a 96 Well Plate and measure the Absorbance at
420 nm.
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Data Analysis
Standard Curve
1.
Subtract the Absorbance from the “blank”, test tube 1, from all the remaining Absorbances;
test tubes 2-7.
2.
Calculate the concentration of p-Nitrophenol in each of the test tubes 2-7 in M units.
3.
Using a software package such as Excel, plot these Absorbances vs. p-Nitrophenol
concentration in M units. Add a Trendline to the plot. Determine the slope, intercept and
appropriate error estimates.
Alkaline Phosphatase Activity
1.
Subtract the Absorbance from the “blank”, test tube 1, from all the remaining Absorbances;
test tubes 2-7.
2.
Use the Standard Curve Trendline equation to determine the concentration of pNitrophenol [S] at the 30 minute reaction stop time. Include an error estimate.
3.
Determine the p-Nitrophenyl Phosphate concentration, [S]o, for each of test tubes 2-7 in
units of M.
4.
Calculate the Reaction Rate v for each test tube according to:
vo =
(Eq. 22)
5.
Prepare a Lineweaver-Burke plot (Eq. 16) by graphing 1/vo vs. 1/[S]o for the data of test
tubes 2-7 using a software package such as Excel. Add a trendline to the plot; determine
the slope and intercept and their error estimates.
6.
Calculate the parameters KM and vm, along with error estimates.
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References
Cantor, Charles R. and Schimmel, Paul R. (1980) "Biophysical Chemistry, Part I: The
Conformation of Biological Macromolecules," Freeman, San Francisco.
Stryer, Lubert (1995) "Biochemistry," 4th Ed. W.H. Freeman and Company, New York.
van Holde, Kensal E.; Johnson, W. Curtis; and Ho, P. Shing. (2006) “Principles of Physical
Biochemistry,” 2nd Ed. Pearson, New Jersey.