Mathematics in Biology??!! Mark Pernarowski Montana State University, Bozeman 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 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  Predator-Prey  X=hare population  Y=lynx population

Spread of Disease  Hong Kong Flu ( NYC).  Pre-vacinne flu related pneumonia  Hong Kong Flu ( NYC).  Pre-vacinne flu related pneumonia

SIR Model  Susceptibles S  Infected I  Recovered R  Susceptibles S  Infected I  Recovered R

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

Patterns 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 -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  n=cell density  u=chemical  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)  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  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.  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.  Calcium makes cells voltage “burst”  Same kind of math models as neurons.

World’s smallest engine!  Molecules which make “ATP” (your energy)

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?  is population at year n=1,2,3,….  Less animals next year if population is too high. Why?

Maybe population levels out

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!  There will be more math in biology and medicine in your lifetime.  Unlike other science the math models are being discovered right now!