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Michaelis-Menton Meets the Market. Group Members. Jeff Awe Jacob Dettinger John Moe Kyle Schlosser. Overview. Introduction to continuous modeling Biological Modeling Nutrient absorption The Michaelis-Menton equation Deriving the Michaelis-Menton equation

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Presentation Transcript
group members
Group Members
  • Jeff Awe
  • Jacob Dettinger
  • John Moe
  • Kyle Schlosser
overview
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
what are continuous models
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
setting up differential equations using compartmental analysis
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
example use of compartmental analysis
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

Immigration

ix

Deaths

Births

bx

dx

Emigration

ex

example use of compartmental analysis7
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)]
continuous models in biology

Continuous Models in Biology

The Michaelis-Menton Equation

continuous models can be used and applied almost everywhere you look

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
continuous models for determining drinking laws

Legal Blood Alcohol Content

Continuous Models for Determining Drinking Laws

  • Recommended Rate of Consumption
  • Charts to determine this are developed from a form of the Michaelis-Menton Equation
drinking charts

Inputs to the Charts are the same as inputs to the equation

Drinking Charts

  • Gender
  • Body Weight
  • Consumption Rate
  • Alcohol Concentration
the michaelis menton equation

The Michaelis-Menton Equation

  • This is the specific model used to determine the medication and alcohol absorption rates
slide14

Nutrients must pass through the cell wall using receptors

  • There are a finite number of receptors
  • 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
slide15

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
slide16

The following Reaction Equations represent the process of passing nutrient molecules into a cell:

and

  • = Unoccupied Receptor
  • = Occupied Receptor
  • = Nutrient Molecule
  • = Product of a successful transportation
slide17

Compartmental Analysis

Let following symbols denote concentrations:

,

,

, and

  • 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.
slide20

Differential Equations

As you can see,

Which implies, is a constant.

Thus let .

slide21

Differential Equations

Substitute into our differential equations to eliminate :

Assume that we are at a steady state, thus,

and

slide22

Differential Equations

Solve for and plug into

slide23

Differential Equations

Which gives us our Michaelis-Menton equation:

, and

Where:

,

slide25

•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.

slide26

Facts:

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.

slide27

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.

slide28

Drinking

Your Body

Metabolizing

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.

slide29

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.

slide30

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)

slide32

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.htmlDr. Deckelman (a source of tons of information)