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
• Applying the Michaelis-Menton equation to
a blood alcohol model
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
• Method used to set up equations where
variables are
– Independent
– Increasing or decreasing
• Examples
– Modeling populations
– Modeling nutrition absorption
Example use of compartmental
analysis
• Populations are affected by:Immigration
–
–
–
–
ix
Immigration (i)
Emigration (e)
Births (b)
Deaths (d)
Deaths
Births
bx
dx
• Resulting equation is
dx/dt=bx+ix-dx-ex
Emigration
ex
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
The Michaelis-Menton
Equation
•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
•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
•Gender
•Body Weight
•Consumption Rate
•Alcohol Concentration
The Michaelis-Menton Equation
Kn
K ( n) 
A n
•This is the specific model used to
determine the medication and alcohol
absorption rates
Bacterial Growth Models
•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
The Michaelis-Menton Equation
•Let
n be the concentration of the nutrient
•Then the growth rate, as a function of the
concentration can be expressed by this
equation
Kn
K ( n) 
A n
•Where K and A are positive constants
The following Reaction Equations represent the
process of passing nutrient molecules into a cell:
N  X0

 X 1

k1
k 1
and
X 1  P  X 0
k2
•
X 0 = Unoccupied Receptor
•
X1 = Occupied Receptor
N
• P
•
= Nutrient Molecule
= Product of a successful transportation
Compartmental Analysis
Let following symbols denote concentrations:
x0 , x1 , p , and n
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.
Compartmental Analysis


k 1 x1
n
 k1nx0

dn
  k1nx0  k 1 x1
dt


k 2 x1


k 1 x1
x0
 k1nx0

dx0
 k1nx0  k 1 x1  k 2 x1
dt
Compartmental Analysis


k 2 x1
p
dp
 k 2 x1
dt


k1nx0
 k 1 x1

x1 
k x

2 1
dx1
 k1nx0  k 1 x1  k 2 x1
dt
Differential Equations
dx0
dn
 k1nx0  k 1 x1  k 2 x1
  k1nx0  k 1 x1
dt
dt
dp
 k 2 x1
dt
dx1
 k1nx0  k 1 x1  k 2 x1
dt
dx
dx
0
As you can see,
 1 0
dt
dt
 x1 is a constant.
Thus let r  x0  x1.
Which implies, x0
Differential Equations
Substitute x0  r  x1 into our differential
equations to eliminate x0 :
dn
 k1nr  (k 1  k1n) x1
dt
dx1
 k1rn  (k 1  k 2 x1  k1n) x1
dt
Assume that we are at a steady state, thus,
dx1
 0 and k1rn  (k1  k2 x1  k1n) x1  0
dt
Differential Equations
dn
 k1nr  (k 1  k1n) x1
Solve for x1 and plug into
dt
k1rn
x1 
(k 1  k 2  k1n)
dn
(k 1  k1n)( k1rn )
 k1nr 
dt
(k 1  k 2  k1n)
 k1nrk 1  k1nrk 2  k1nrk1n  k 1k1rn  k1nk1rn

(k 1  k 2  k1n)
Differential Equations
k 2 rn
k1k 2 nr


k 1  k 2
k 1  k 2  k1n
n
k1
Which gives us our Michaelis-Menton equation:
Kn
K ( n) 
A n
k 1  k 2
dn
Where: K ( n ) 
, A
dt
k1
K  k2 r
, and
Your BAC
(and the squirrel)
John Moe
•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.
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.
Kn
K ( n) 
A n
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.
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.
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.
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)
On to Excel…
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)