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!