OSCILLATORY ENZYME
REACTIONS AND
MICHAELIS-MENTEN
KINETICS
DANIELLE COPE
K U A N Y S H D O S M U R AT O V
M I L E N A ST E FA N OV I C
NGUYEN “NICOLE” LAM
All work not explicitly cited has been derived from the article itself.
All MATLAB code is available in the “notes” section at the bottom of the
slide to allow copy & paste capabilities.
1
DR. ALBERT GOLDBETER
• The author of our paper, and a
professor at Université Libre de
Bruxelles in Belgium, from which he
has also received several degrees.
• He is a distinguished researcher,
having published dozens of papers,
books, and audiocasts.
• His research interests include
• Modeling molecular mechanisms and
rhythms on a molecular level
• Sleep-wake cycle of certain microorganisms
• Phenomena in enzyme regulation
• Bistability (the existence of multiple steady
states at the same time)
Biographical information from:
http://www.ulb.ac.be/sciences/utc/GOLDBETER/agoldbet.html
Figure 1: Albert Goldbeter
http://www.smb.org/publications/newsletter/bios/vol24no1_goldbeter.pdf
2
PURPOSE OF THE ARTICLE
• Study the relationships between MichaelisMenten kinetics, and the endogenous oscillations
that occur in enzymatic reactions.
• Determine if enzymes which behave like the
Michaelis-Menten model are able to produce
biochemical oscillations.
• Determine the effect that these enzymes have on
oscillations in systems where the periodic
behavior relies on allosteric (two or more binding
sites) enzyme regulation.
• Determine if the Michaelis-Menten model can be
used to accurately represent systems of coupled
enzyme reactions.
Models of periodic behavior in systems of
phosphofructokinase in glycolysis and kinases
which are dependent on cyclin in their cell cycle
were used to test these questions because they
are the systems best known for biochemical
oscillations which occur because of enzymatic
regulation.
Figure 2: http://imgc.allpostersimages.com/images/P-473-48890/64/6476/X8F6100Z/posters/rieder-khodjakov-fluorescent-micrograph-of-anamphibian-cell-during-metaphase.jpg
3
MICHAELIS-MENTEN MODEL
• Developed by Leonor Michaelis and Maud •
Leonora Menten.1
• Furthered the work of Victor Henri.
• Explains the plateauing behavior of enzyme
reaction rates when the substrate is at a
saturated concentration.
• Models dynamics of different enzyme
systems. In this article it was used to model
oscillatory behavior of an enzyme system.
• Assumptions: there is excess substrate, all
active sites used, no product at t=0, and
total enzyme is conserved
The equation developed as part of this model
allows the calculation of the initial rate of
reaction, and the substrate and product
concentrations as functions of time throughout
the course of the reaction.
1http://chemwiki.ucdavis.edu/Biological_Chemistry/Catalysts/Enzymatic_Kinetics/Michaelis-Menten_Kinetics
Figure 3 (Left Image):
http://www.chemheritage.org/Images/Main-Images-250x290/Discover/Themes/Biomolecules/menten1.jpg
Figure 4 (Right Image):
4
http://www.chemheritage.org/Discover/Online-Resources/Chemistry-in-History/Themes/Biomolecules/Proteins-and-Sugars/asset_upload_file390_61288_thumbnail.jpg
PRACTICAL EXAMPLE:
CLASSICAL MICHAELIS-MENTEN
KINETICS
The data below could be obtained from a
batch reactor operating at constant
enzyme concentration.
1
2
3
5
7
10
15
20
0.2
0.22
0.3
0.45
0.41
0.5
0.4
0.33
V vs.S
Figure 6
0.6
0.5
Mmol/L.min
Substrate S
(mmol/L)
Initial Reaction
Rate V
(mmol/L.min)
0.4
0.3
0.2
0.1
0
0
Figure 5 (Image): http://www.caliberbio.com/facilities.html
Data from Dr. Arul Jayaraman
5
10
15
20
25
Mmol/L
5
PRACTICAL EXAMPLE:
CLASSICAL MICHAELIS-MENTEN
KINETICS
The Michaelis-Menten parameters can be evaluated using a linear transformation
and curve fitting:
1/v vs. 1/s
6
5
1/v
4
3
y = 3.0387x + 2.2249
R² = 0.7826
2
1
0
0
Figure 7
0.2
0.4
0.6
0.8
1
1.2
1/s
6
METHODOLOGY
• We read the article to understand
the background, and what the author
was trying to convey.
• We concluded that the best way to
enhance this article using numerical
methods was to write code to graph
the models that the author was
publishing.
• Then we utilized our knowledge of
numerical methods to write
programs which would reproduce the
author’s models
Figure 8 (Top Image):
http://us.123rf.com/400wm/400/400/hugofelix/hugofelix1202/hugofelix120200030/
12529129-young-scientist-at-lab-isolated-over-white.jpg
Figure 9 (Bottom image):
http://us.123rf.com/400wm/400/400/dmitrimaruta/dmitrimaruta1201/dmitrimaruta120100233
/12106981-programmer-working-with-a-touch-screen-interface-isolated-on-white.jpg
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GLYCOLIC OSCILLATIONS
• Glycolic oscillations are considered a
good example of period behavior.
• They were first studied in yeast, but
have since been expanded to
describe oscillatory behavior that
many different microorganisms
exhibit.
• A non-equilibrium system is
considered to be the best for
displaying such behavior graphically.
• The behavior of both the product and
substrate concentrations may be
graphically displayed.
Figure 10 (Top Image):
http://www.kurzweilai.net/images/Yeast.jpg
Figure 11 (Bottom Image): From article
More
active
enzyme
Less
active
enzyme
8
PRODUCT CONCENTRATIONS
Product concentrations may be
plotted using the equations to the top
right. For the data used in this paper
to yield the graph on the bottom right
the following values should be used:
Km=(a) 80 (b) 40 (c ) 20 (d) 10 (e ) 5
L=106
σ=5.075 s-1
q=3
ks=3.81 s-1
v=0.5 s-1
n=2
S(0)=P(0)=0
Figure 12 (Graph A): from article
9
SUBSTRATE CONCENTRATIONS
Substrate concentrations may be
plotted using the equations to the top
right. For the data used in this paper
to yield the graph on the bottom right
the following values should be used:
Km=(a) 80 (b) 40 (c ) 20 (d) 10 (e ) 5
L=106
σ=5.075 s-1
q=3
ks=3.81 s-1
v=0.5 s-1
n=2
S(0)=P(0)=0
Figure 13 (Graph A): from article
10
NUMERICAL METHODS
Figure 14 (dydtsys3): Chapra p. 576)
Figure 15 (command window)
11
NUMERICAL METHODS
As you can see the graphs that
resulted from our code does not
match the graphs in the published
paper. We used the same method that
we used in the graphs on slide 20;
therefore, we believe that the author
left out vital information about the
way he plotted these graphs.
Figures 16-19 (MATLAB generated graphs)
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OSCILLATORY ACTIVITY OF CYCLINDEPENDENT KINASES IN THE CELL
CYCLE
• An interesting example is the
oscillatory enzyme reactions that are
caused by the periodic behavior of
cyclin-dependent kinases, which is
part of their cell cycle
• These oscillating cell cycles occur
because of “negative auto-regulation
in a phosphorylationdephosphorylation cascade.”2
• Positive feedback also occurs but has
a much lesser, even negligent effect
on this particular behavior.
“Oscillatory enzyme reactions and Michaelis–Menten kinetics” by Albert Goldbeter
Figure 20 (Top Image) credit:
http://us.123rf.com/400wm/400/400/eraxion/eraxion0806/
eraxion080600296/3241269-human-active-nerve-cell.jpg
Figure 21(Bottom Image): From article
2
13
PERIODIC PHENOMENA AND
MICHAELIS-MENTEN KINETICS
• This model focuses on bicyclic
phosphorylation-dephosphorylation
cascade.
• Assume: Cdc2 kinase is activated by E1,
inactivated by E2 and that E3 is directly
activated by reversible phosphorylation.
Figure 22 (Image): http://static.guim.co.uk/sys-images/Guardian/Pix/pictures/2008/04/10/cancer460x276.jpg
14
NUMERICAL ANALYSIS
First we programmed the differential equations into a function file
Figure 23 (dydtsys2): Chapra p. 576
• Governing equations for the system are a set of three ordinary differential equations
• Values for constants and initial conditions could change based on different reaction
types and experimental data
• Codes have been written for easy updating depending on specific circumstances 15
NUMERICAL ANALYSIS
Figure 24 (rk4sys): Chapra p. 576
We programmed a function file for the classical forth-order Runge-Kutta method
16
NUMERICAL ANALYSIS
Figure 25 (rk4sys): Chapra p. 576
17
NUMERICAL ANALYSIS
Figure 26 (eulersys): Chapra p. 576
We also wrote a function file for Euler’s method based on the RK4 method
18
NUMERICAL ANALYSIS
Figure 27 (command window)
Figure 28 (phase plot)
19
NUMERICAL ANALYSIS
Figure 29 (command window)
Script file to plot graphs
20
NUMERICAL ANALYSIS
Matlab Graphs
Figure 34 (Graph from Article)
Figure 30-33 (MATLAB Graphs)
21
NUMERICAL ANALYSIS
Figure 35 (Command Window)
Figure 36 (MATLAB Graph)
• Three numerical methods
were tested to solve these
equations: Runge-Kutta,
Euler, and the built-in
Matlab function ode113
• All three give very
similar values, as the
graph on the right
shows
22
CONCLUSIONS AND FUTURE WORK
• This paper and our numerical analysis
confirms the validity of the MichaelisMenten equations and their
applicability to dynamic, oscillatory
systems
• It also confirms the accuracy of the
RK4 method, Euler’s method, and the
ode113 build-in Matlab command
• Further improvements could be made
by contacting the author and figuring
out his missing information
Figure 37: http://www.chercheinfo.com/uploads/1951-5258c2c0f0.gif
23
REFERENCES
Article:
Albert Goldbeter, Oscillatory enzyme
reactions and Michaelis–Menten
kinetics, FEBS Letters, Volume
587, Issue 17, 2 September
2013, Pages 2778-2784,
ISSN 0014-5793,
http://dx.doi.org/10.1016/
j.febslet.2013.07.031.
(http://www.sciencedirect.com/
science/article/pii/
Figure 38:
S0014579313005607)
http://www.onlinecolleges.net/wp-content/uploads/2012/04/computer-science-online/
computer-programmer.jpg
24
REFERENCES
Textbook:
Chapra, Steven C. Applied Numerical
Methods with MATLAB for
Engineers and Scientists. Boston:
McGraw-Hill Higher Education,
2008. Print.
Figure 40: http://www.mhhe.com/engcs/general/chapra/Chapra3e.jpg
Figure 39: http://static.guim.co.uk/sys-images/Guardian/Pix/pictures/2008/04/27/cancer10a.jpg
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