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.

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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

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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.

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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

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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

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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

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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.jpgFigure 11 (Bottom Image): From article

More active enzyme

Less active enzyme

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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

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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

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NUMERICAL METHODSFigure 14 (dydtsys3): Chapra p. 576)

Figure 15 (command window)

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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 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

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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

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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

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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)

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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)

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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

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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

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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