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The Biological ESTEEM Project: Linear Algebra, Population Genetics, and Microsoft Excel

p’ = p ( pW AA + qW AS ) /. W. The Biological ESTEEM Project: Linear Algebra, Population Genetics, and Microsoft Excel. Anton E. Weisstein, Truman State University. BIO 2010: Transforming Undergraduate Education for Future Research Biologists National Research Council (2003).

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The Biological ESTEEM Project: Linear Algebra, Population Genetics, and Microsoft Excel

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

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

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

  4. BiologicalTopics Spread of infectious diseases Tree growth Enzyme kinetics Population genetics

  5. MathematicalTopics Graph theory Random walks Linear algebra Optimi-zation

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

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

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

  9. Synthesizing and Applying Math Concepts Using Biological Cases 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

  10. IAIA Type A IA IAIB Type AB IAi Type A i IB IBi Type B ii Type O IBIB Type B 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

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

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

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

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

  15. 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.)

  16. Hardy-Weinberg Equilibrium Genotypes reach ratios p2 : 2pq : q2 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

  17. Synthesizing and Applying Math Concepts Using Biological Cases 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

  18. 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%?

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

  20. Selection and Sickle-Cell Alleles: A: “normal” hemoglobin S: sickle-cell hemoglobin Natural selection: Malaria susceptibility: ~90% survive to reproductive age Sickle-cell anemia: ~20% survive to reproductive age

  21. Zygote p2 2pq q2 Juvenile p2 2pq q2 Adult p2WAA 2pqWAS q2WSS Normalization: W = p2WAA + 2pqWAS + q2WSS W W W p’ = p (pWAA + qWAS) / W Recursion Equations p = # A gametes; q = # S gametes. Life stage AA (W = 0.9) AS (W = 1.0) SS (W = 0.2)

  22. p’ = p (pWAA + qWAS) / W Selection and Sickle-Cell Biological Question: • How will this population evolve over time? Mathematical Question: What are the equilibria for this recursion equation?

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

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

  25. 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: ?

  26. 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?

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

  28. pq 2N 1 2N ≈ N(p, ) p’ = B(2N, p) 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!

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

  30. Modeling Microevolution:Deme

  31. Synthesizing and Applying Math Concepts Using Biological Cases 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

  32. 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!

  33. Selection and Sickle-Cell Hemoglobin alleles: A, S, C Malaria susceptibility Malaria susceptibility Sickle-cell anemia Mild anemia Strong malaria resistance  C is beneficial only when common!

  34. p’ = p (pWAA + qWAS + rWAC) / q’ = q (pWAS + qWSS + rWSC) / r’ = r (pWAC + qWSC + rWCC) / W W W p = DA / D, q = DS / D, r = DC / D Selection and Sickle-Cell Recursion Equations: Equilibria: where DA = (WAS – WSS)(WAC – WCC) – (WAS – WSC)(WAC – WSC) DS = (WAS – WAA)(WSC – WCC) – (WAS – WAC)(WSC – WAC) DC = (WAC – WAA)(WSC – WSS) – (WAC – WAS)(WSC – WAS) D = DA + DS + DC

  35. 2 alleles: Landscape W(p) is a curve in R2 3 alleles: Landscape W(p, q, r) is a sheet in R3 Plotting the Adaptive Landscape Constraint: p + q + r = 1

  36. where Stability Analysis Re-express W(p, q, r) as W(x, y) Calculate Hessian matrix: 3. Take the determinant and apply the 2nd derivative test:

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

  38. 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

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