1 school of science indiana university-purdue university indianapolis carl c. cowen iupui dept of...

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School of ScienceIndiana University-Purdue University Indianapolis

Carl C. Cowen IUPUI Dept of Mathematical Sciences

Connections Between Mathematics and Biology

With thanks for support from With thanks for support from The National Science Foundation IGMS program, The National Science Foundation IGMS program, (DMS-0308897), Purdue University, and the (DMS-0308897), Purdue University, and the Mathematical Biosciences Institute Mathematical Biosciences Institute

Carl C. CowenCarl C. Cowen IUIUPUPUI Dept of Mathematical SciencesI Dept of Mathematical Sciences

Connections Between Connections Between Mathematics and BiologyMathematics and Biology

ProloguePrologue

IntroductionIntroduction

Some areas of applicationSome areas of application

Cellular Transport Cellular Transport

Example from neuroscience:Example from neuroscience: the Pulfrich Effect the Pulfrich Effect

ProloguePrologue• Background to the presentation:Background to the presentation:

US in a crisis in the education of US in a crisis in the education of young people in science, technology, young people in science, technology, engineering, and mathematics (STEM), engineering, and mathematics (STEM), areas central to our future economy!areas central to our future economy!

• Today, want to get you (or help you stay) Today, want to get you (or help you stay) excited about mathematics and the role it excited about mathematics and the role it will play! will play!

“Rising Above The Gathering Storm: Energizing and Employing America for a Brighter Economic Future” www.nap.edu/catalog/11463.html

ProloguePrologue• Background to the presentation:Background to the presentation:

US in a crisis in the education of US in a crisis in the education of young people in science, technology, young people in science, technology, engineering, and mathematics (STEM), engineering, and mathematics (STEM), areas central to our future economy!areas central to our future economy!

• Today, want to get you (or help you stay) Today, want to get you (or help you stay) excited about mathematics and the role it excited about mathematics and the role it will play!will play!

“Rising Above The Gathering Storm: Energizing and Employing America for a Brighter Economic Future” www.nap.edu/catalog/11463.html

IntroductionIntroduction• Explosion in biological research and Explosion in biological research and

progressprogress

• The mathematical sciences will be a partThe mathematical sciences will be a part

• Opportunity: Opportunity: few mathematical scientists are few mathematical scientists are biologically educated biologically educated few biological scientists are few biological scientists are mathematically educated mathematically educated

Colwell: “We're not near the fulfillment of biotechnology's promise. We're just on the cusp of it…”


progressprogress



Report Bio2010: “How biologists design, perform, and analyze experiments is changing swiftly. Biological concepts and models are becoming more quantitative…”


progressprogress



NSF/NIH Challenges: “Emerging areas transcend traditional academic boundaries and require interdisciplinary approaches that integrate biology, mathematics, and computer science.”

Some areas of application of Some areas of application of math/stat in the biosciencesmath/stat in the biosciences• Genomics and proteomicsGenomics and proteomics

• Description of intra- and inter-cellular Description of intra- and inter-cellular processesprocesses

• Growth and morphologyGrowth and morphology

• Epidemiology and population dynamicsEpidemiology and population dynamics

• NeuroscienceNeuroscience

Poincare: “Mathematics is the art of giving the same name to different things.”

Some areas of application of Some areas of application of math in the biosciencesmath in the biosciences• Genomics and proteomicsGenomics and proteomics

• Description of intra- and inter-cellular Description of intra- and inter-cellular processesprocesses

• Growth and morphologyGrowth and morphology

• Epidemiology and population dynamicsEpidemiology and population dynamics

• NeuroscienceNeuroscience

Poincare: “Mathematics is the art of giving the same name to different things.”

Axonal TransportAxonal Transport

General problem: how do things get General problem: how do things get moved around inside cells? moved around inside cells?

Specific problem: how do large Specific problem: how do large molecules get moved from one end molecules get moved from one end of a long axon to the other? of a long axon to the other?


From “Slow axonal transport: stop and go traffic in the axon”, A. Brown, Nature Reviews, Mol. Cell. Biol. 1: 153 - 156, 2000.


• A. Brown, op. cit.

Macroscopic view: Macroscopic view: Neurofilaments Neurofilaments (marked with (marked with radioactive tracer) radioactive tracer) move slowly move slowly toward distal endtoward distal end


• A. Brown, op. cit.

Microscopic view: neurofilaments Microscopic view: neurofilaments moving quickly along axonmoving quickly along axon

QuickTime™ and aCinepak decompressorare needed to see this picture.


Problem: Problem: How can the macroscopic slow How can the macroscopic slow movement be reconciled with the movement be reconciled with the microscopic fast movement? microscopic fast movement?



Plan: (with Chris Scheper)Plan: (with Chris Scheper) • View the axon as a line segment; • View the axon as a line segment; discretize the segment and time. discretize the segment and time. • Describe motion along axon as • Describe motion along axon as a Markov chain. a Markov chain.


Problem with plan: Problem with plan: Matrix describing Markov chain is Matrix describing Markov chain is very large, and eigenvector matrix very large, and eigenvector matrix is ill-conditioned! is ill-conditioned! Traditional approach to Markov Traditional approach to Markov Chains will not work! Chains will not work! Need to find alternative approach to Need to find alternative approach to

analyze model -- work in progress! analyze model -- work in progress!



If it cannot, it would throw doubt on If it cannot, it would throw doubt on Brown’s hypothesis about how axonal Brown’s hypothesis about how axonal transport works -- and there is a transport works -- and there is a competing hypothesis suggested by competing hypothesis suggested by another researcher!another researcher!

The Pulfrich EffectThe Pulfrich Effect

An experiment!An experiment!

Carl Pulfrich (1858-1927)Carl Pulfrich (1858-1927) reported effect and gave explanation reported effect and gave explanation in 1922 in 1922

F. Fertsch experimented, showedF. Fertsch experimented, showed Pulfrich why it happened, and was Pulfrich why it happened, and was given the credit for it by Pulfrich given the credit for it by Pulfrich


• The brain processes signals together that arrive from the two eyes at the same time

• The signal from a darker image is sent later than the signal from a brighter image, that is, signals from darker images are delayed

Hypothesis suggested Hypothesis suggested by neuro-physiologists: by neuro-physiologists:


filter

s

d

x

s

• x, d, , and are all functions of time, but we’ll skip that for now

• s is fixed: you can’t move your eyeballs further apart •The brain “knows” the values of , , and s

• The brain “wants to calculate” the values of x and d

s

d

x

s

• x + s = tan d

s

d

x

s

• x + s = tan d

• x - s = tan d

s

d

x

s

• x + s = tan d

• x - s = tan d

• 2s = tan d - tan d

• d = 2s/(tan - tan )

• 2x = tan d + tan d

• x = d(tan + tan )/2

• x = s(tan + tan ) / (tan - tan )

s

d

x

s

• x + s = tan d

• x - s = tan d

• tan d = x + s

• tan = (x + s)/d

• = arctan( (x + s)/d )

• = arctan( (x - s)/d )

s

d

x(t)

s

• = arctan( (x(t-) + s)/d )

• = arctan( (x(t) - s)/d )

x(t-)

• x(t),d = actual position at time t

• x(t-),d = actual position

at earlier time t-

s

d

y(t)

s

• = arctan( (x(t-) + s)/d )

• = arctan( (x(t) - s)/d )

• e(t) = 2s / (tan - tan )

• y(t) = s(tan + tan ) / (tan - tan )

• x(t),d = actual position at time t

• x(t-),d = actual position at earlier time t-• y(t),e(t) = apparent position at time t

e(t)

s

d

y(t)

s

• e(t) = 2s / (tan - tan ) = 2sd / (x(t-) - x(t) + 2s)

• y(t) = s(tan + tan ) / (tan - tan ) = s(x(t-) + x(t)) / (x(t-) - x(t) + 2s)

• y(t),e(t) = apparent position at time t

• = arctan( (x(t-) + s)/d )

• = arctan( (x(t) - s)/d )

e(t)

s

d

y(t)

s

• The predicted curve traversed by the apparent position is approximately an ellipse

• The more the delay (darker filter), the greater the apparent difference in depth

• If the moving object is the bob on a swinging pendulum x(t) = sin(t)

• y(t),e(t) = apparent position at time t

e(t)

The Pendulum without filterThe Pendulum without filter

QuickTime™ and a decompressorare needed to see this picture.

The Pendulum with filterThe Pendulum with filter


The Pulfrich Effect (second try)The Pulfrich Effect (second try)


ConclusionsConclusions• Mathematical models can be useful Mathematical models can be useful

descriptions of biological phenomenadescriptions of biological phenomena

• Models can be used as evidence to Models can be used as evidence to support or refute biological hypothesessupport or refute biological hypotheses

• Models can suggest new experiments, Models can suggest new experiments, simulate experiments or treatments that simulate experiments or treatments that have not yet been carried out, orhave not yet been carried out, orestimate parameters that are estimate parameters that are experimentally inaccessibleexperimentally inaccessible

ConclusionsConclusions

Working together, biologists, Working together, biologists, statisticians, and statisticians, and

mathematicians can contribute mathematicians can contribute more to science than any group more to science than any group

can contribute separately.can contribute separately.

ReferenceReference

• ““Seeing in Depth, Volume 2: Depth Seeing in Depth, Volume 2: Depth Perception” by Ian P. Howard and Perception” by Ian P. Howard and Brian J. Rogers, I Porteus, 2002.Brian J. Rogers, I Porteus, 2002.Chapter 28: The Pulfrich effectChapter 28: The Pulfrich effect

1 school of science indiana university-purdue university indianapolis carl c. cowen iupui dept of...

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