Connections Between Mathematics and Biology

1. Connections Between Mathematics and Biology Carl CowenPurdue University and theMathematical Biosciences Institute

2. IntroductionSome areas of applicationExample from neuroscience: the Pulfrich Effect

3. 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…”

4. 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…”

5. 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.”

6. 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.”

7. 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.”

8. 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.”

9. 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.”

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. 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

12. 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

13. The Pulfrich Effect filter

14. The Pulfrich Effect filter

15. 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

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

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

18. 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 )

19. 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

20. 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 )

21. 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 )

22. 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)

23. 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

24. 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

25. Conclusions Working together, biologists and mathematicians can contribute more to science than either group can contribute separately.

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