Connections Between Mathematics and Biology

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School of Science Indiana University-Purdue
University Indianapolis
Connections Between Mathematics and Biology
Carl C. Cowen IUPUI Dept of Mathematical
Sciences
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Connections Between Mathematics and Biology
Carl C. Cowen IUPUI Dept of Mathematical
Sciences
With thanks for support from The National
Science Foundation IGMS program,
(DMS-0308897), Purdue University, and the
Mathematical Biosciences Institute
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PrologueIntroductionSome areas of
applicationCellular Transport Example from
neuroscience the Pulfrich Effect
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Prologue

• Background to the presentation US in a
  crisis in the education of young people in
  science, technology, engineering, and mathematics
  (STEM), areas central to our future economy!
• Today, want to get you (or help you stay) excited
  about mathematics and the role it will play!

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

• Background to the presentation US in a
  crisis in the education of young people in
  science, technology, engineering, and mathematics
  (STEM), areas central to our future economy!
• Today, want to get you (or help you stay) excited
  about mathematics and the role it will play!

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

• Explosion in biological research and progress
• The mathematical sciences will be a part
• Opportunity few mathematical scientists are
  biologically educated few biological
  scientists are mathematically educated

Colwell We're not near the fulfillment of
biotechnology's promise. We're just
on the cusp of it
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Introduction

• Explosion in biological research and progress
• The mathematical sciences will be a part
• Opportunity few mathematical scientists are
  biologically educated few biological
  scientists are mathematically educated

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

• Explosion in biological research and progress
• The mathematical sciences will be a part
• Opportunity few mathematical scientists are
  biologically educated few biological
  scientists are mathematically educated

NSF/NIH Challenges Emerging areas transcend
traditional academic boundaries and require
interdisciplinary approaches that integrate
biology, mathematics, and computer science.
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Some areas of application of math/stat in the
biosciences

• Genomics and proteomics
• Description of intra- and inter-cellular
  processes
• Growth and morphology
• Epidemiology and population dynamics
• Neuroscience

Poincare Mathematics is the art of giving the
same name to different things.
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Some areas of application of math in the
biosciences

• Genomics and proteomics
• Description of intra- and inter-cellular
  processes
• Growth and morphology
• Epidemiology and population dynamics
• Neuroscience

Poincare Mathematics is the art of giving the
same name to different things.
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Some areas of application of math in the
biosciences

• Genomics and proteomics
• Description of intra- and inter-cellular
  processes
• Growth and morphology
• Epidemiology and population dynamics
• Neuroscience

Poincare Mathematics is the art of giving the
same name to different things.
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Some areas of application of math in the
biosciences

• Genomics and proteomics
• Description of intra- and inter-cellular
  processes
• Growth and morphology
• Epidemiology and population dynamics
• Neuroscience

Poincare Mathematics is the art of giving the
same name to different things.
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Some areas of application of math in the
biosciences

• Genomics and proteomics
• Description of intra- and inter-cellular
  processes
• Growth and morphology
• Epidemiology and population dynamics
• Neuroscience

Poincare Mathematics is the art of giving the
same name to different things.
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Axonal Transport
General problem how do things get moved
around inside cells?Specific problem how do
large molecules get moved from one end
of a long axon to the other?
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Axonal Transport
From Slow axonal transport stop and go traffic
in the axon, A. Brown, Nature Reviews, Mol.
Cell. Biol. 1 153 - 156, 2000.
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Axonal Transport
Macroscopic view Neurofilaments (marked with
radioactive tracer) move slowly toward distal end

• A. Brown, op. cit.

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Axonal Transport
Microscopic view neurofilaments moving quickly
along axon

• A. Brown, op. cit.

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Axonal Transport
Problem How can the macroscopic slow
movement be reconciled with the microscopic
fast movement?
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Axonal Transport
Problem How can the macroscopic slow
movement be reconciled with the microscopic
fast movement?Plan (with Chris Scheper)
View the axon as a line segment
discretize the segment and time. Describe
motion along axon as a Markov chain.
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Axonal Transport
Problem with plan Matrix describing Markov
chain is very large, and eigenvector matrix
is ill-conditioned! Traditional
approach to Markov Chains will not work!
Need to find alternative approach to analyze
model -- work in progress!
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Axonal Transport
Problem How can the macroscopic slow
movement be reconciled with the microscopic
fast movement?If it cannot, it would throw
doubt on Browns hypothesis about how axonal
transport works -- and there is a competing
hypothesis suggested by another researcher!
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The Pulfrich Effect
An experiment!Carl Pulfrich (1858-1927)
reported effect and gave explanation in
1922F. Fertsch experimented, showed Pulfrich
why it happened, and was given the credit for
it by Pulfrich
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The Pulfrich Effect
Hypothesis suggested by neuro-physiologists

• 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

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The Pulfrich Effect
filter
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The Pulfrich Effect
filter
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The Pulfrich Effect
filter
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x

• x, d, q1 , and q2 are all functions of time,
  but well skip that for now
• s is fixed you cant move your eyeballs
  further apart
• The brain knows the values of q1 , q2 , and
  s
• The brain wants to calculate the values
  of x and d

d
q2
q1
s
s
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x

• x s tan q1 d

d
q2
q1
s
s
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x

• x s tan q1 d
• x - s tan q2 d

d
q2
q1
s
s
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x

• x s tan q1 d
• x - s tan q2 d
• 2s tan q1 d - tan q2 d
• d 2s/(tan q1 - tan q2 )

d
q2
q1
s
s

• 2x tan q1 d tan q2 d
• x d(tan q1 tan q2 )/2
• x s(tan q1 tan q2 ) / (tan q1 - tan q2 )

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x

• x s tan q1 d
• x - s tan q2 d
• tan q1 d x s
• tan q1 (x s)/d
• q1 arctan( (x s)/d )
• q2 arctan( (x - s)/d )

d
q2
q1
s
s
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x(t)
x(t-D)

• x(t),d actual position at time t
• x(t-D),d actual position at
  earlier time t-D

d
q2
q1
s
s

• q1 arctan( (x(t-D) s)/d )
• q2 arctan( (x(t) - s)/d )

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y(t)

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

d
e(t)
q2
q1
s
s

• q1 arctan( (x(t-D) s)/d )
• q2 arctan( (x(t) - s)/d )
• e(t) 2s / (tan q1 - tan q2 )
• y(t) s(tan q1 tan q2 ) / (tan q1 - tan q2 )

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y(t)

• y(t),e(t) apparent position at time t
• q1 arctan( (x(t-D) s)/d )
• q2 arctan( (x(t) - s)/d )

d
e(t)
q2
q1
s
s

• e(t) 2s / (tan q1 - tan q2 ) 2sd
  / (x(t-D) - x(t) 2s)
• y(t) s(tan q1 tan q2 ) / (tan q1 - tan q2
  ) s(x(t-D) x(t)) / (x(t-D) - x(t)
  2s)

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y(t)

• If the moving object is the bob on a swinging
  pendulum x(t) a sin(bt)
• y(t),e(t) apparent position at time t

d
e(t)
q2
q1
s
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

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The Pendulum without filter
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The Pendulum with filter
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The Pulfrich Effect
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The Pulfrich Effect (second try)
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Conclusions

• Mathematical models can be useful descriptions of
  biological phenomena
• Models can be used as evidence to support or
  refute biological hypotheses
• Models can suggest new experiments, simulate
  experiments or treatments that have not yet been
  carried out, orestimate parameters that are
  experimentally inaccessible

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Conclusions
Working together, biologists, statisticians, and
mathematicians can contribute more to science
than any group can contribute separately.
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Reference

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