1. Mathematics in Biology??!! Mark Pernarowski Montana State University, Bozeman Mark Pernarowski Montana State University, Bozeman

2. 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

3. Population Problems  Predator-Prey  X=hare population  Y=lynx population  Predator-Prey  X=hare population  Y=lynx population

4. 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

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

6. 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

7. 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.

8. 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.

9. Models Reproduce Experiments  n=cell density  u=chemical  n=cell density  u=chemical

10. 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)

11. Lot’s of different kinds of math models:

12. Visual System  The electrical firing rate of your visual system neurons changes with stimulus angle!

13. Each neuron can be modeled.  Ions move across membrane  Voltage V changes in time  Ions move across membrane  Voltage V changes in time

14. Models of Visual System Orientation Patterns  Activity a(r,t) at time t, position r in your brain.

15.  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.

16. Diabetes  An increase in glucose (sugar) causes pancreas cells to make insulin. Calcium plays a key role.

17. 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.

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

19. 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?

20. Maybe population levels out http://www.math.montana.edu/frankw/ccp/modeling/discrete/logistic/appwindow.htm

21. Maybe population alternates (period 2)

22. Maybe population alternates (period 4)

23. Maybe population chaotic

24. 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!