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##### 10 Equations in Biology: Michaelis-Menten Kinetics

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**Outline**1. Brief intro to enzyme kinetics 2. From intuitive conceptions to a formal model 3. Building interactive models in Excel 4. From a mathematical model to biological insights**Intro to Enzyme Kinetics**kf kcat E + S ES E + P kr Variables [E]: free enzyme molecules [S]: free substrate molecules [ES]: enzyme-substrate complexes [P]: free product molecules Parameters kf , kr , kcat : reaction rates**$ in checking account on Sept. 1st**= Writing “Word Equations” • A word equation is a qualitative description of the system’s major processes • Goal: help students express their proto-mathematical understanding of a system before introducing formal math. Example: Balancing a checkbook $ in checking account on Aug. 1st deposits made in August withdrawals made in August ? + –**Word Equations for Enzyme Kinetics Model**# of free S molecules at time t # of free S molecules binding to free E # of ES complexes splitting into E + S # of free S molecules at time t +1 = – + Write similar word equations for the other 3 variables.**Translate Word Equations into Formal Equations**# of free S molecules at time t = [S]t conc. of enzyme-substrate complexes # of ES complexes splitting into E + S = kr[ES]t conc. of free enzyme conc. of free substrate # of free S molecules binding to free E = kf[E]t [S]t by Law of Mass Action**Modeling in Excel**Why Excel rather than more powerful tools? • easily available on most campuses • widely used = transferable proficiency • nonthreatening to students • makes students work directly with the equations: no dodges via the user interface!**Implementing a Model in Excel**✓ 1. Write a series of recursion equations. 2. Set up Excel sheet with time in Column A, other variables in next columns. 3. Enter and name model parameters; enter variables’ starting values. 4. Enter recursion equations for each variable. 5. Copy and paste for subsequent time steps. 6. Graph results appropriately.**Structured Exploration: Understandthe Model**1. How would you measure the reaction’s velocity? How does velocity change over the course of the reaction, and why? 2. How does each rate constant influence the reaction’s velocity? Why?**Further Exploration**Enzyme Kinetics – Multiple Runs.xlsx • What happens to velocity as [S]0 increases? Why? • Estimatevmax for this enzyme.**Further Exploration**• Biological interpretation of KM? • Characterize the Michaelis-Menten curves of enzymes w/low, medium, and highKM.**Estimating Reaction Constants**• What is the shape of the M-M curve? • Nonlinear regression: Fit the appropriate curve to the M-M plot by minimizing the sum of squared residuals. • Use built-in scrollbars to adjust estimates of vmax and Km.**Estimating Reaction Constants:Linear Methods**Goal: Convert the Michaelis-Menten equation into a form that linearizes the best-fit curve. Example Lineweaver-Burk plot: 1/v vs. 1/[S]0 Formula = …? Should be able to write similar formulas for other plots (H-W, E-H)**Estimating Reaction Constants:Evaluating Diff. Methods**On what basis might one method be considered “better” than others? • Each method’s sensitivity to estimation error? • Each method’s ability to detect such error?**Beyond Simple Enzyme Kinetics**How could you modify the model to describe… • Influx or efflux of reaction components? • Reversible, competing, or multi-step reactions? • Totally different biological systems (pop. dynamics, epidemiology, pharmacokinetics, …)?**Preview: Agent-Based Models**• Simulate & visualize behavior of each indiv. molecule • Focus of next week’s seminar (Nernst equation)**Take-home message**• Build equations to clarify each term’s meaning and identify key assumptions • Explore models to test understanding of a system, and to ground abstract discussions • Lesson structure: what aspects of this seminar did you find most/least crucial?