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School of Science Indiana University-Purdue University Indianapolis

School of Science Indiana University-Purdue University Indianapolis. Connections Between Mathematics and Biology. Carl C. Cowen IU PU I Dept of Mathematical Sciences. 1. Connections Between Mathematics and Biology. Carl C. Cowen IU PU I Dept of Mathematical Sciences.

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School of Science Indiana University-Purdue University Indianapolis

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  1. School of Science Indiana University-Purdue University Indianapolis Connections Between Mathematics and Biology Carl C. Cowen IUPUI Dept of Mathematical Sciences 1

  2. 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

  3. PrologueIntroductionSome areas of applicationCellular Transport Example from neuroscience: the Pulfrich Effect

  4. 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

  5. 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

  6. 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…”

  7. 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…”

  8. 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.”

  9. 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.”

  10. 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.”

  11. 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.”

  12. 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.”

  13. 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.”

  14. 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?

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

  16. Axonal Transport Macroscopic view: Neurofilaments (marked with radioactive tracer) move slowly toward distal end A. Brown, op. cit.

  17. Axonal Transport Microscopic view: neurofilaments moving quickly along axon A. Brown, op. cit.

  18. Axonal Transport Problem: How can the macroscopic slow movement be reconciled with the microscopic fast movement?

  19. 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.

  20. 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!

  21. 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!

  22. 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

  23. 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

  24. The Pulfrich Effect filter

  25. The Pulfrich Effect filter

  26. The Pulfrich Effect filter

  27. x 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 d q2 q1 s s

  28. x x + s = tan q1 d d q2 q1 s s

  29. x x + s = tan q1 d x - s = tan q2 d d q2 q1 s s

  30. 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 )

  31. 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

  32. 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 )

  33. 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 )

  34. 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)

  35. 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

  36. The Pendulum without filter

  37. The Pendulum with filter

  38. The Pulfrich Effect

  39. The Pulfrich Effect (second try)

  40. 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

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

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

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