1. The Biological ESTEEM Project: Linear Algebra, Population Genetics, and Microsoft Excel p’ = p (pW AA + qW AS ) /W Anton E. Weisstein, Truman State University

2. BIO 2010: Transforming Undergraduate Education for Future Research Biologists National Research Council (2003) Recommendation #2: “Concepts, examples, and techniques from mathematics…should be included in biology courses. …Faculty in biology, mathematics, and physical sciences must work collaboratively to find ways of integrating mathematics…into life science courses…” Recommendation #1: “Those selecting the new approaches should consider the importance of mathematics...”

3. BIO 2010: Transforming Undergraduate Education for Future Research Biologists National Research Council (2003) Specific strategies: A strong interdisciplinary curriculum that includes physical science, information technology, and math. Meaningful laboratory experiences.

4. The Biological ESTEEEM Project Homepage 55 modules: Broad range of both biological and mathematical topics http://bioquest.org/esteem

5. Biological Topics Spread of infectious diseases Tree growth Enzyme kinetics Population genetics

6. Mathematical Topics Random walks Optimi- zation Linear algebra Graph theory

7. Unpacking “ESTEEM” Excel: ubiquitous, easy, flexible, non-intimidating Exploratory: apply to real-world data; extend & improve Experiential: students engage directly with the math

8. Three Boxes Black box: Hide the model ? y = ax b Glass box: Study the model y = ax b No box: Build the model! How do students interact with the mathematical model underlying the biology?

9. Copyleft download use modify share Users may freely the software, w/proper attribution More info available at Free Software Foundation website

10. 3. Survival of the Slightly Better: Exploring an Evolutionary Paradox with Linear Algebra 1. Intro to Population Genetics: Hardy-Weinberg Equilibrium and the Binomial Theorem 2. Evolutionary Analysis: Microevolution, Statistics, and Stability Analysis Synthesizing and Applying Math Concepts Using Biological Cases

11. Definitions Allele: One variant of a specific gene. Genotype: The set of alleles carried by an individual. Phenotype: The detectable manifestations of a specific genotype. Example: ABO blood type IAIA IBIB i I A I A  Type A I B I B  Type B ii  Type O I A i  Type A I A I B  Type AB I B i  Type B

12. Life Cycle Gametes (eggs & sperm) Zygotes (fertilized eggs) Juveniles (reproductively immature) Adults (reproductively mature)

13. Life Cycle Gametes (eggs & sperm) Zygotes (fertilized eggs) Juveniles (reproductively immature) Adults (reproductively mature)

14. Life Cycle Gametes (eggs & sperm) Zygotes (fertilized eggs) Juveniles (reproductively immature) Adults (reproductively mature)

15. Life Cycle Gametes (eggs & sperm) Zygotes (fertilized eggs) Juveniles (reproductively immature) Adults (reproductively mature)

16. Recursion Equations Let x = # AA adults; y = # Aa adults; z = # aa adults. Define p = # A gametes = x + y/2 ; q = # a gametes = y/2 + z. Determine expected # adults of each genotype in next generation. (For now, feel free to make any simplifying assumptions.)

17. Hardy-Weinberg Equilibrium Genotypes reach ratios p 2 : 2pq : q 2 in one generation, then stay there forever! Assumptions? Gametes combine at random All individuals have equal chance of survival Each gen. a perfectly representative sample of the previous

18. 3. Survival of the Slightly Better: Exploring an Evolutionary Paradox with Linear Algebra 1. Intro to Population Genetics: Hardy-Weinberg Equilibrium and the Binomial Theorem 2. Evolutionary Analysis: Microevolution, Statistics, and Stability Analysis Synthesizing and Applying Math Concepts Using Biological Cases

19. The Case of the Sickled Cell The S allele for sickle-cell anemia has a frequency of ~11% in some African populations. Why is it so common? If it provides a selective advantage, why isn’t its frequency 100%?

20. Definitions Reproductive fitness: The average number of offspring produced by an organism in a specific environment. Examples: Antibiotic resistance Camouflage Resistance to infectious diseases Natural selection: An evolutionary mechanism that tends to increase the freq. of traits that increase an organism’s fitness. Source: Jeffrey Jeffords, DiveGallery.com

21. Selection and Sickle-Cell Alleles: A: “normal” hemoglobin S: sickle-cell hemoglobin GenotypeFitness AAW AA = 0.9 ASW AS = 1.0 SSW SS = 0.2 Natural selection: Sickle-cell anemia: ~20% survive to reproductive age Malaria susceptibility: ~90% survive to reproductive age

22. Recursion Equations p = # A gametes; q = # S gametes. Life stageSS (W = 0.2) AS (W = 1.0) AA (W = 0.9) Juvenileq2q2 2pqp2p2 Adultq 2 W SS 2pqW AS p 2 W AA Zygoteq2q2 2pqp2p2 WWW p’ = p (pW AA + qW AS ) /W W= p 2 W AA + 2pqW AS + q 2 W SS Normalization:

23. Selection and Sickle-Cell GenotypeFitness AAW AA = 0.9 ASW AS = 1.0 SSW SS = 0.2 Biological Question: How will this population evolve over time? p’ = p (pW AA + qW AS ) /W Mathematical Question: What are the equilibria for this recursion equation?

24. Solving for Equilibria Set p’ = p and solve: or Substitute q = 1 – p and factor: or Nontrivial solution:

25. Stability Analysis: NatSelDiffEqns (Tim Comar, Benedictine College) Is q = 0.11 stable or unstable?

26. The Case of the Protective Protein HIV docks with the CCR5 surface protein present on some cells of immune system CCR5  32 allele partially protects against HIV infection Peterson 1999. JYI 2: ?

27. The Case of the Protective Protein Based on genetic evidence,  32 arose ~700 years ago. Present in ~10% of Caucasians; largely absent in other groups. Why? Hypothesis: May also have protected vs. plague and/or smallpox. Biological Question: How much selective advantage must  32 have given to become so common in only 700 years? Mathematical Question: For what fitness values does 700 years lie within the 95% CI of  32’s age?

28. Definitions Examples: Absence of blood type B in Native Americans Northern elephant seal: virtually no genetic variation 100 years after near-extinction Genetic drift: An evolutionary mechanism by which allele frequencies change due to chance alone, independent of those alleles’ effects on fitness.

29. Modeling Genetic Drift Let N = population size (constant). Assume this pop. produces ∞ gametes: f(A) = p, f(B) =q. But only 2N of those gametes (chosen at random) combine to form the zygotes that develop into the next generation! p’ = B(2N, p) 12N12N ≈ N(p, ) pq 2N

30. Genetic Drift as a Random Walk p’ = B(2N, p) 12N12N ≈ N(p, ) pq 2N N = 2000 N = 200 N = 20 Largest fluctuations in small pops. p = 0 and p = 1 are absorbing states

31. Modeling Microevolution: Deme

32. 3. Survival of the Slightly Better: Exploring an Evolutionary Paradox with Linear Algebra 1. Intro to Population Genetics: Hardy-Weinberg Equilibrium and the Binomial Theorem 2. Evolutionary Analysis: Microevolution, Statistics, and Stability Analysis Synthesizing and Applying Math Concepts Using Biological Cases

33. Sickle Cell Strikes Back! In addition to the A and S alleles, there is also a C allele for hemoglobin! C confers even stronger malaria resistance than AS but with no anemia! But C is found only in a few isolated populations. Why might this happen? Extend previous analysis to 3 alleles: some surprising results!

34. Selection and Sickle-Cell Hemoglobin alleles: A, S, C GenotypeFitness AA0.9 AS1.0 AC0.9 SS0.2 SC0.7 CC1.3 Sickle-cell anemia Malaria susceptibility Mild anemia Strong malaria resistance  C is beneficial only when common!

35. Selection and Sickle-Cell p’ = p (pW AA + qW AS + rW AC ) /W q’ = q (pW AS + qW SS + rW SC ) /W r’ = r (pW AC + qW SC + rW CC ) /W Recursion Equations: Equilibria: p = D A / D,q = D S / D, r = D C / D where D A = (W AS – W SS )(W AC – W CC ) – (W AS – W SC )(W AC – W SC ) D S = (W AS – W AA )(W SC – W CC ) – (W AS – W AC )(W SC – W AC ) D C = (W AC – W AA )(W SC – W SS ) – (W AC – W AS )(W SC – W AS ) D = D A + D S + D C

36. Plotting the Adaptive Landscape 2 alleles: Landscape W(p) is a curve in R 2 3 alleles: Landscape W(p, q, r) is a sheet in R 3 Constraint: p + q + r = 1

37. Stability Analysis 1.Re-express W(p, q, r) as W(x, y) 2.Calculate Hessian matrix: where 3. Take the determinant and apply the 2nd derivative test: TV > U 2, T > 0, T+V > 0 Local max TV > U 2, T < 0, T+V < 0 Local min TV < U 2 Saddle point TV = U 2 Higher-order tests needed

38. Survival of the Slightly Better: DeFinetti Global maximum: only C allele present Local maximum: C allele eliminated Saddle point

39. Cases & Mathematics: Explicit Connections Binomial & Normal Distributions Combinatorics Equilibria & Stability Analysis Normalization Recursion & Difference Eqns. Stochasticity Geometry of Curves & Solids Matrix & Linear Algebra Partial Derivatives

40. Cases & Mathematics: CCR5  32 Example Introduce case thru reading & discussion Elicit & explore meaningful biological questions that require mathematical reasoning & analysis How could you experimentally and ethically determine fitness of diff. genotypes?

41. Corn Snake Genetics Corn snakes come in many different color morphs, depending on which pigments a given produces Wild-type snakes  both red and black pigment Anerythristic  black pigment only Wild-type Anerythristic

42. Recursion Equations p = # A gametes; q = # S gametes. Life stageAA (W = 0.9) AS (W = 1.0) SS (W = 0.2) Zygotep2p2 2pqq2q2 Juvenilep2p2 2pqq2q2 Adultp 2 W AA 2pqW AS q 2 W SS

43. Equilibria of Set p’ = p and solve: or Substitute q = 1 – p and factor: