Mathematics in Biology??!!Mathematics in Biology??!!
Mark Pernarowski
Montana State University, Bozeman
Mark Pernarowski
Montana State University, Bozeman
What Kind of Biology?What Kind of Biology?
Population of organisms - i.e., animals Spread of Disease - i.e., Flu, vaccines, How muscles work - even the heart Diabetes - insulin and glucose Patterns in natures - zebra stripes, leopard spots Brain electrical behavior - sight, sound Organism growth - embryos Genetics …..and much more
Population of organisms - i.e., animals Spread of Disease - i.e., Flu, vaccines, How muscles work - even the heart Diabetes - insulin and glucose Patterns in natures - zebra stripes, leopard spots Brain electrical behavior - sight, sound Organism growth - embryos Genetics …..and much more
Population ProblemsPopulation Problems
Predator-Prey X=hare population Y=lynx population
Predator-Prey X=hare population Y=lynx population
Spread of Disease Spread of Disease Hong Kong Flu (1968-69 NYC). Pre-vacinne flu related pneumonia
Hong Kong Flu (1968-69 NYC). Pre-vacinne flu related pneumonia
SIR ModelSIR Model
Susceptibles S Infected I Recovered R
Susceptibles S Infected I Recovered R
How do leeches swim?How do leeches swim?
Physics of Fluids F(x,t)=leech force u=fluid velocity p=pressure
Physics of Fluids F(x,t)=leech force u=fluid velocity p=pressure QuickTime™ and a
Animation decompressorare needed to see this picture.
Patterns in NaturePatterns in Nature
Chemicals that react and diffuse in animal coats make visible patterns
c(x,t) concentration at time t location x.
Chemicals that react and diffuse in animal coats make visible patterns
c(x,t) concentration at time t location x.
Embryo Development -genesEmbryo Development -genes Young embryos form a body axis early on Why? How? Chemicals cause cells to move.
Young embryos form a body axis early on Why? How? Chemicals cause cells to move.
Models Reproduce Experiments
Models Reproduce Experiments
n=cell density u=chemical
n=cell density u=chemical
Some Biology is “Stimulus/Response”Some Biology is “Stimulus/Response”
A stimulus I causes an output u
Sometimes get an output even if there is no stimulus (I=0) (i.e., people who talk to you even when you don’t talk to them)
A stimulus I causes an output u
Sometimes get an output even if there is no stimulus (I=0) (i.e., people who talk to you even when you don’t talk to them)
Lot’s of different kinds of math models:Lot’s of different kinds of math models:
Visual SystemVisual System
The electrical firing rate of your visual system neurons changes with stimulus angle!
The electrical firing rate of your visual system neurons changes with stimulus angle!
Each neuron can be modeled.Each neuron can be modeled.
Ions move across membrane
Voltage V changes in time
Ions move across membrane
Voltage V changes in time
Models of Visual System Orientation PatternsModels of Visual System Orientation Patterns
Activity a(r,t) at time t, position r in your brain. Activity a(r,t) at time t, position r in your brain.
On - neurons fire immediately
Off- neurons fire after light turned off
Math models explain both and more.
On - neurons fire immediately
Off- neurons fire after light turned off
Math models explain both and more.
DiabetesDiabetes An increase in glucose (sugar) causes pancreas cells to
make insulin. Calcium plays a key role.
An increase in glucose (sugar) causes pancreas cells to make insulin. Calcium plays a key role.
Bursting Electrical ActivityBursting Electrical Activity Calcium makes cells voltage “burst” Same kind of math models as neurons.
Calcium makes cells voltage “burst” Same kind of math models as neurons.
World’s smallest engine!World’s smallest engine! Molecules which make “ATP” (your energy) Molecules which make “ATP” (your energy)
QuickTime™ and aVideo decompressor
are needed to see this picture.
A simpler math model of animal populationsA simpler math model of animal populations
is population at year n=1,2,3,…. Less animals next year if population is too high. Why?
is population at year n=1,2,3,…. Less animals next year if population is too high. Why?
Maybe population levels outMaybe population levels out
http://www.math.montana.edu/frankw/ccp/modeling/discrete/logistic/appwindow.htm
Maybe population alternates (period 2)Maybe population alternates (period 2)
Maybe population alternates (period 4)Maybe population alternates (period 4)
Maybe population chaoticMaybe population chaotic
ConclusionConclusion
There will be more math in biology and medicine in your lifetime.
Unlike other science the math models are being discovered right now!
There will be more math in biology and medicine in your lifetime.
Unlike other science the math models are being discovered right now!