DANIELLE COPE KUANYSH DOSMURATOV MILENA STEFANOVIC NGUYEN “NICOLE” LAM 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 Goldbeter Biographical information from:

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: 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. 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. 1 Figure 3 (Left Image): Figure 4 (Right Image): 4

PRACTICAL EXAMPLE: CLASSICAL MICHAELIS-MENTEN KINETICS The data below could be obtained from a batch reactor operating at constant enzyme concentration. Substrate S (mmol/L) Initial Reaction Rate V (mmol/L.min) Figure 5 (Image): Data from Dr. Arul Jayaraman 5

PRACTICAL EXAMPLE: CLASSICAL MICHAELIS-MENTEN KINETICS The Michaelis-Menten parameters can be evaluated using a linear transformation and curve fitting: 6 Figure 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 models Figure 8 (Top Image): young-scientist-at-lab-isolated-over-white.jpg Figure 9 (Bottom image): / programmer-working-with-a-touch-screen-interface-isolated-on-white.jpg 7

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): 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: K m =(a) 80 (b) 40 (c ) 20 (d) 10 (e ) 5 L=10 6 σ=5.075 s -1 q=3 k s =3.81 s -1 v=0.5 s -1 n=2 S(0)=P(0)=0 9 Figure 12 (Graph A): from article

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: K m =(a) 80 (b) 40 (c ) 20 (d) 10 (e ) 5 L=10 6 σ=5.075 s -1 q=3 k s =3.81 s -1 v=0.5 s -1 n=2 S(0)=P(0)=0 10 Figure 13 (Graph A): from article

NUMERICAL METHODS 11 Figure 14 (dydtsys3): Chapra p. 576) Figure 15 (command window)

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. 12 Figures (MATLAB generated graphs)

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: eraxion / human-active-nerve-cell.jpg Figure 21(Bottom Image): From article 13

PERIODIC PHENOMENA AND MICHAELIS-MENTEN KINETICS This model focuses on bicyclic phosphorylation-dephosphorylation cascade. Assume: Cdc2 kinase is activated by E 1, inactivated by E 2 and that E 3 is directly activated by reversible phosphorylation. Figure 22 (Image): 14

NUMERICAL ANALYSIS First 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 15 Figure 23 (dydtsys2): Chapra p. 576

NUMERICAL ANALYSIS We programmed a function file for the classical forth-order Runge-Kutta method 16 Figure 24 (rk4sys): Chapra p. 576

NUMERICAL ANALYSIS 17 Figure 25 (rk4sys): Chapra p. 576

NUMERICAL ANALYSIS We also wrote a function file for Euler’s method based on the RK4 method 18 Figure 26 (eulersys): Chapra p. 576

NUMERICAL ANALYSIS 19 Figure 27 (command window) Figure 28 (phase plot)

NUMERICAL ANALYSIS Script file to plot graphs 20 Figure 29 (command window)

NUMERICAL ANALYSIS Matlab Graphs Figure 34 (Graph from Article) 21 Figure (MATLAB Graphs)

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 22 Figure 35 (Command Window) Figure 36 (MATLAB Graph)

CONCLUSIONS AND FUTURE WORK Figure 37: 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 23

REFERENCES Article: Albert Goldbeter, Oscillatory enzyme reactions and Michaelis–Menten kinetics, FEBS Letters, Volume 587, Issue 17, 2 September 2013, Pages , ISSN , j.febslet ( science/article/pii/ S ) 24 Figure 38: computer-programmer.jpg

REFERENCES 25 Textbook: Chapra, Steven C. Applied Numerical Methods with MATLAB for Engineers and Scientists. Boston: McGraw-Hill Higher Education, Print. Figure 40: Figure 39: