1. Michaelis-Menton Meets the Market

2. Group Members Jeff Awe Jacob Dettinger John Moe Kyle Schlosser

3. Overview Introduction to continuous modeling Biological Modeling –Nutrient absorption –The Michaelis-Menton equation Deriving the Michaelis-Menton equation Applying the Michaelis-Menton equation to a blood alcohol model

4. What are Continuous Models? Used when we want to treat the independent variable as continuous Represent rate of change as a derivative –Basically this means that continuous modeling is modeling with differential equations We will use a technique called compartmental analysis to derive these differential equations

5. Setting up Differential Equations using Compartmental Analysis Method used to set up equations where variables are –Independent –Increasing or decreasing Examples –Modeling populations –Modeling nutrition absorption

6. Example use of compartmental analysis Populations are affected by: –Immigration (i) –Emigration (e) –Births (b) –Deaths (d) Resulting equation is dx/dt=bx+ix-dx-ex Births bx Immigration ix Emigration ex Deaths dx

7. Example use of compartmental analysis Derived differential equation is dx/dt=bx+ix-dx-ex We could solve this equation using a technique called separation Resulting equation represents the population as a function of time. [x(t)]

8. Continuous Models in Biology The Michaelis-Menton Equation

9. Continuous Models can be used and applied almost everywhere you look Medications and their Dosages Amounts Dosage Intervals Finding your body’s absorption rate of the Medication

10. Continuous Models for Determining Drinking Laws Legal Blood Alcohol Content Recommended Rate of Consumption Charts to determine this are developed from a form of the Michaelis-Menton Equation

11. Drinking Charts Inputs to the Charts are the same as inputs to the equation Gender Body Weight Consumption Rate Alcohol Concentration

12. The Michaelis-Menton Equation This is the specific model used to determine the medication and alcohol absorption rates

13. Bacterial Growth Models

14. When the nutrient concentration is low, the bacterial growth rate is proportional to the concentration When the nutrient level is high, the growth rate is constant Nutrients must pass through the cell wall using receptors There are a finite number of receptors

15. The Michaelis-Menton Equation Let be the concentration of the nutrient Then the growth rate, as a function of the concentration can be expressed by this equation Where K and A are positive constants

16. = Unoccupied Receptor = Occupied Receptor = Nutrient Molecule = Product of a successful transportation The following Reaction Equations represent the process of passing nutrient molecules into a cell: and

17. Let following symbols denote concentrations: We observe two laws governing compartmental diagrams: For a single reactant, the rate of the reaction is proportional to the concentration of the reactant. For two reactants, the rate of the reaction is proportional to the product of the concentrations.,,, and Compartmental Analysis

20. Differential Equations As you can see, Which implies, is a constant. Thus let.

21. Differential Equations Substitute into our differential equations to eliminate : Assume that we are at a steady state, thus, and

22. Differential Equations Solve for and plug into

23. Differential Equations Which gives us our Michaelis-Menton equation: Where:,, and

24. Your BAC (and the squirrel) John Moe

25. Blood alcohol concentrations are complicated and vary from person to person. The state trooper is probably unlikely to accept as an excuse that John said it would be OK in his math models presentation. It is also a bad idea to feed a squirrel beer.

26. BAC is measured as grams of alcohol per 100 mL of blood. Alcohol is distributed evenly in all of the water in a person’s body. Blood is 81.57% water. Facts:

27. With blood alcohol, the concentration of alcohol is much higher than the number of receptors, so the rate of alcohol elimination is basically a constant. The average person eliminates alcohol at the rate of about 7.5 grams / hour, although it can range from 4 - 12.

28. If you drink at a constant rate the amount of alcohol in your body would then be just the amount your are drinking per hour minus the amount your are metabolizing per hour multiplied by the number of hours you’ve been at it. All we need now to calculate BAC is the amount of water in your body. DrinkingMetabolizingYour Body

29. The amount of water in a person is roughly proportional to their weight. This constant is then adjusted because blood is not 100% water. For males, you divide by 3.1 times weight in pounds. For females it is 2.5.

30. So if an average male person of weight w pounds averages c grams of alcohol per hour, then their BAC at time t would be: (c-7.5)t/(3.1w) For a female(because they have less water on average) it would be: (c-7.5)t/(2.5w) (c≥7.5 and t≥0)

31. On to Excel…

32. A course in Mathematical Modeling by Douglas Mooney & Randell Swift, MAA Publications 1999 “The Calculation of Blood Alcohol Concentration” http://www.vicroads.vic.gov.au/road_safe/safe_first/ breath_test/BAC/BACReport.html Dr. Deckelman (a source of tons of information)