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Oscillatory enzyme reactions and Michaelis-Menten Kinetics. All work not explicitly cited has been derived from the article itself. Danielle Cope Kuanysh Dosmuratov Milena Stefanovic Nguyen “Nicole” Lam. - PowerPoint PPT Presentation
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DA N I E L L E C O P EKU A N Y S H D O S M U RAT O V
M I L E N A S T E FA N O V I CN G U Y E N “ N I C O L E ” L A M
OSCILLATORY ENZYME REACTIONS AND
MICHAELIS-MENTEN KINETICS
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)
2
Figure 1: Albert Goldbeterhttp://www.smb.org/publications/newsletter/bios/vol24no1_goldbeter.pdfBiographical information from:
http://www.ulb.ac.be/sciences/utc/GOLDBETER/agoldbet.html
3
PURPOSE OF THE ARTICLE• Study the relationships between
Michaelis-Menten 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.
Figure 2: http://imgc.allpostersimages.com/images/P-473-488-90/64/6476/X8F6100Z/posters/rieder-khodjakov-fluorescent-micrograph-of-an-amphibian-cell-during-metaphase.jpg
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.
4
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_KineticsFigure 3 (Left Image):http://www.chemheritage.org/Images/Main-Images-250x290/Discover/Themes/Biomolecules/menten1.jpgFigure 4 (Right Image):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 KINETICSThe data below could be obtained from a batch reactor operating at constant enzyme concentration.
0 5 10 15 20 2500.10.20.30.40.50.6
V vs.S
Mmol/L
Mm
ol/L
.min
Figure 6
Substrate S (mmol/L)
Initial Reaction Rate V
(mmol/L.min)1 0.22 0.223 0.35 0.457 0.41
10 0.515 0.420 0.33
Figure 5 (Image): http://www.caliberbio.com/facilities.htmlData from Dr. Arul Jayaraman 5
6
PRACTICAL EXAMPLE:CLASSICAL MICHAELIS-MENTEN KINETICSThe Michaelis-Menten parameters can be evaluated using a linear transformation and curve fitting:
0 0.2 0.4 0.6 0.8 1 1.20
1
2
3
4
5
6
f(x) = 3.03867281573322 x + 2.224903446203R² = 0.782638553434082
1/v vs. 1/s
1/s
1/v
Figure 7
7
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 modelsFigure 8 (Top Image):
http://us.123rf.com/400wm/400/400/hugofelix/hugofelix1202/hugofelix120200030/12529129-young-scientist-at-lab-isolated-over-white.jpgFigure 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
8
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.jpgFigure 11 (Bottom Image): From article
More active enzyme
Less active enzyme
9
PRODUCT CONCENTRATIONSProduct 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 ) 5L=106
σ=5.075 s-1
q=3ks=3.81 s-1
v=0.5 s-1
n=2S(0)=P(0)=0
Figure 12 (Graph A): from article
10
SUBSTRATE CONCENTRATIONSSubstrate 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 ) 5L=106
σ=5.075 s-1
q=3ks=3.81 s-1
v=0.5 s-1
n=2S(0)=P(0)=0
Figure 13 (Graph A): from article
11
NUMERICAL METHODSFigure 14 (dydtsys3): Chapra p. 576)
Figure 15 (command window)
12
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)
13
OSCILLATORY ACTIVITY OF CYCLIN-DEPENDENT 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 phosphorylation-dephosphorylation cascade.”2
• Positive feedback also occurs but has a much lesser, even negligent effect on this particular behavior.2 “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.jpgFigure 21(Bottom Image): From article
14
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
15
NUMERICAL ANALYSISFirst we programmed the differential equations into a function file
• 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
Figure 23 (dydtsys2): Chapra p. 576
16
NUMERICAL ANALYSIS
We programmed a function file for the classical forth-order Runge-Kutta method
Figure 24 (rk4sys): Chapra p. 576
NUMERICAL ANALYSIS
17Figure 25 (rk4sys): Chapra p. 576
18
NUMERICAL ANALYSIS
We also wrote a function file for Euler’s method based on the RK4 methodFigure 26 (eulersys): Chapra p. 576
19
NUMERICAL ANALYSIS
Figure 27 (command window)
Figure 28 (phase plot)
20
NUMERICAL ANALYSIS
Script file to plot graphs
Figure 29 (command window)
21
NUMERICAL ANALYSISMatlab Graphs
Figure 34 (Graph from Article)
Figure 30-33 (MATLAB Graphs)
22
NUMERICAL ANALYSIS
• 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
Figure 35 (Command Window)
Figure 36 (MATLAB Graph)
23
CONCLUSIONS AND FUTURE WORK
Figure 37: http://www.chercheinfo.com/uploads/1951-5258c2c0f0.gif
• This paper and our numerical analysis confirms the validity of the Michaelis-Menten 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
24
REFERENCESArticle: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/
S0014579313005607)Figure 38: http://www.onlinecolleges.net/wp-content/uploads/2012/04/computer-science-online/computer-programmer.jpg
25
REFERENCESTextbook: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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