PowerPoint Presentation - Mathematics in Biology??!!
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Mathematics in Biology??!!
Mark Pernarowski
Montana State University, Bozeman
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 Problems
Predator-Prey
X=hare population
Y=lynx population
Spread of Disease
Hong Kong Flu (1968-69 NYC).
Pre-vacinne flu related pneumonia
SIR Model
Susceptibles S
Infected I
Recovered R
How do leeches swim?
Physics of Fluids
F(x,t)=leech force
u=fluid velocity
p=pressure
QuickTime™ and a
Animation decompressor
are needed to see this picture.
Patterns in Nature
Chemicals that react
and diffuse in animal
coats make visible
patterns
c(x,t) concentration at
time t location x.
Embryo Development -genes
Young embryos form a body axis early on
Why? How? Chemicals cause cells to move.
Models Reproduce
Experiments
n=cell density
u=chemical
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)
Lot’s of different kinds of math models:
Visual System
The electrical
firing rate of
your visual
system neurons
changes with
stimulus angle!
Each neuron can be modeled.
Ions move across
membrane
Voltage V
changes in time
Models of Visual System Orientation Patterns
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.
Diabetes
An increase in glucose (sugar) causes pancreas cells to
make insulin. Calcium plays a key role.
Bursting Electrical Activity
Calcium makes cells voltage “burst”
Same kind of math models as neurons.
World’s smallest engine!
Molecules which make “ATP” (your energy)
QuickTime™ and a
Video decompressor
are needed to see this picture.
A simpler math model of animal populations
is population at year n=1,2,3,….
Less animals next year if population is too high. Why?
Maybe population levels out
http://www.math.montana.edu/frankw/ccp/modeling/discrete/logistic/appwindow.htm
Maybe population alternates (period 2)
Maybe population alternates (period 4)
Maybe population chaotic
Conclusion
There will be more math in biology and
medicine in your lifetime.
Unlike other science the math models are
being discovered right now!