School of Science
Indiana University-Purdue University Indianapolis
Connections Between
Mathematics and Biology
Carl C. Cowen
IUPUI Dept of Mathematical Sciences
1
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
Prologue
Introduction
Some areas of application
Cellular Transport
Example from neuroscience:
the Pulfrich Effect
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
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
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…”
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…”
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.”
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.”
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.”
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.”
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.”
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.”
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?
Axonal Transport
From “Slow axonal transport: stop and go
traffic in the axon”, A. Brown, Nature
Reviews, Mol. Cell. Biol. 1: 153 - 156, 2000.
Axonal Transport
Macroscopic view:
Neurofilaments
(marked with
radioactive tracer)
move slowly
toward distal end
• A. Brown, op. cit.
Axonal Transport
QuickTime™ and a Cinepak decompressor are needed to see this picture.
Microscopic view: neurofilaments
moving quickly along axon
• A. Brown, op. cit.
Axonal Transport
Problem:
How can the macroscopic slow
movement be reconciled with the
microscopic fast movement?
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.
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!
Axonal Transport
Problem:
How can the macroscopic slow
movement be reconciled with the
microscopic fast movement?
If it cannot, it would throw doubt on
Brown’s hypothesis about how axonal
transport works -- and there is a
competing hypothesis suggested by
another researcher!
The Pulfrich Effect
An experiment!
Carl Pulfrich (1858-1927)
reported effect and gave explanation
in 1922
F. Fertsch experimented, showed
Pulfrich why it happened, and was
given the credit for it by Pulfrich
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
The Pulfrich Effect
filter
The Pulfrich Effect
filter
The Pulfrich Effect
filter
x
d
q2
q1
s
s
• x, d, q1 , and q2 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 q1 , q2 , and s
• The brain “wants to
calculate” the values
of x and d
x
• x + s = tan q1 d
d
q2
q1
s
s
x
• x + s = tan q1 d
• x - s = tan q2 d
d
q2
q1
s
s
x
d
q2
q1
s
s
• x + s = tan q1 d
• x - s = tan q2 d
• 2s = tan q1 d - tan q2 d
• d = 2s/(tan q1 - tan q2 )
• 2x = tan q1 d + tan q2 d
• x = d(tan q1 + tan q2 )/2
• x = s(tan q1 + tan q2 ) / (tan q1 - tan q2 )
x
d
q2
q1
s
• x + s = tan q1 d
• x - s = tan q2 d
• tan q1 d = x + s
• tan q1 = (x + s)/d
s
• q1 = arctan( (x + s)/d )
• q2 = arctan( (x - s)/d )
x(t-D) x(t)
d
q2
q1
s
• x(t),d = actual position at
time t
• x(t-D),d = actual position
at earlier time t-D
s
• q1 = arctan( (x(t-D) + s)/d )
• q2 = arctan( (x(t) - s)/d )
y(t)
d
e(t)
q2
q1
s
s
• 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
• 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 )
y(t)
• 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 )
= 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)
y(t)
d
e(t)
q2
q1
s
s
• 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
• 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
The Pendulum without filter
QuickTime™ and a decompressor are needed to see this picture.
The Pendulum with filter
QuickTime™ and a decompressor are needed to see this picture.
The Pulfrich Effect
QuickTime™ and a decompressor are needed to see this picture.
The Pulfrich Effect (second try)
QuickTime™ and a decompressor are needed to see this picture.
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, or
estimate parameters that are
experimentally inaccessible
Conclusions
Working together, biologists,
statisticians, and
mathematicians can contribute
more to science than any group
can contribute separately.
Reference
• “Seeing in Depth, Volume 2: Depth
Perception” by Ian P. Howard and
Brian J. Rogers, I Porteus, 2002.
Chapter 28: The Pulfrich effect