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

2. With thanks for support from The National Science Foundation IGMS program, (DMS-0308897), Purdue University, and the Mathematical Biosciences Institute Carl C. Cowen IUPUI Dept of Mathematical Sciences IUPUI Dept of Mathematical Sciences Connections Between Mathematics and Biology

3. Prologue Introduction Some areas of application Cellular 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!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!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!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!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 progressExplosion in biological research and progress The mathematical sciences will be a partThe mathematical sciences will be a part Opportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educatedOpportunity: 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 progressExplosion in biological research and progress The mathematical sciences will be a partThe mathematical sciences will be a part Opportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educatedOpportunity: 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 progressExplosion in biological research and progress The mathematical sciences will be a partThe mathematical sciences will be a part Opportunity: few mathematical scientists are biologically educated few biological scientists are mathematically educatedOpportunity: 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 proteomicsGenomics and proteomics Description of intra- and inter-cellular processesDescription of intra- and inter-cellular processes 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.”

10. Some areas of application of math in the biosciences Genomics and proteomicsGenomics and proteomics Description of intra- and inter-cellular processesDescription of intra- and inter-cellular processes 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.”

11. Some areas of application of math in the biosciences Genomics and proteomicsGenomics and proteomics Description of intra- and inter-cellular processesDescription of intra- and inter-cellular processes 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.”

12. Some areas of application of math in the biosciences Genomics and proteomicsGenomics and proteomics Description of intra- and inter-cellular processesDescription of intra- and inter-cellular processes 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.”

13. Some areas of application of math in the biosciences Genomics and proteomicsGenomics and proteomics Description of intra- and inter-cellular processesDescription of intra- and inter-cellular processes 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.”

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

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

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 1922 F. Fertsch experimented, showed Pulfrich why it happened, and was given the credit for it by Pulfrich

23. The Pulfrich Effect 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 by neuro-physiologists:

24. The Pulfrich Effect filter

25. The Pulfrich Effect filter

26. The Pulfrich Effect filter

27.  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

28.  s d x s  x + s = tan   d

29.  s d x s  x - s = tan   d

30.  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   )

31.  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 )

32.  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- 

33.  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)

34.  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)

35.  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)

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 phenomenaMathematical models can be useful descriptions of biological phenomena Models can be used as evidence to support or refute biological hypothesesModels 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 inaccessibleModels can suggest new experiments, simulate experiments or treatments that have not yet been carried out, or estimate 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“Seeing in Depth, Volume 2: Depth Perception” by Ian P. Howard and Brian J. Rogers, I Porteus, 2002. Chapter 28: The Pulfrich effect