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Modeling Evolution: The Wright-Fisher Model

Modeling Evolution: The Wright-Fisher Model. Rick Stuart and Samantha Biehl. Introduction to Evolution. Change over time Natural Selection Variation Inheritance Differential Survival Alleles Dominant Recessive. Introduction to Evolution. Genetic Drift Extinction Fixation

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Modeling Evolution: The Wright-Fisher Model

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  1. Modeling Evolution: The Wright-Fisher Model Rick Stuart and Samantha Biehl

  2. Introduction to Evolution • Change over time • Natural Selection • Variation • Inheritance • Differential Survival • Alleles • Dominant • Recessive

  3. Introduction to Evolution • Genetic Drift • Extinction • Fixation • Increasing variability • Migration • Mutation • Fitness

  4. The Basic Wright-Fisher Model • Stochastic model • Genetic drift • Finite population • Allele frequencies • Two or more alleles

  5. Definition of Variables • Number of individuals • Number of alleles (type A and B) • Number of type A alleles at step • Number of type B alleles at step • Number of type A alleles at step • Probability of getting type A allele • Probability of getting type B allele

  6. The Basic Wright-Fisher Model • Probability of having A alleles • Markov Chain • The transition matrix T

  7. The Basic Wright-Fisher Model • Absorbing States • All type B • All type A • Once in absorbing state, will stay in that state • Given enough time, will enter one of the absorbing states • Extinction • Fixation

  8. The Basic Wright-Fisher Model • Binomial distribution • By induction it can be shown , for all

  9. The Basic Wright-Fisher Model • This makes the process a bounded martingale • Bounded martingale property, converges as

  10. The Basic Wright-Fisher Model • simulations • trials • , (A allele) • Expect number of A alleles to fixate • Simulations results • Mean of fixations per trials

  11. The Basic Wright-Fisher Model • Assumptions • Generations do not overlap • Population size remains constant • No mutation or recombination occurring • Gender of individuals ignored • Familial ties ignored • No selection pressure acting

  12. The Basic Wright-Fisher Model Individuals or Alleles Randomly choose A B B A B A B A B B B A A A A A B B B A A

  13. The Basic Wright-Fisher Model Individuals or Alleles Alleles from step are randomly chosen to produce alleles in step A B B A B A B A B B B A A A A A B B B A A A A B B A A A B A B A B A A A A B B B

  14. The Basic Wright-Fisher Model1 Simulation, 2N=20

  15. The Basic Wright-Fisher Model10 Simulations, 2N=20

  16. The Basic Wright-Fisher Model10 Simulations, 2N=1000

  17. The Basic Wright-Fisher Model5 Simulations, 2N=100,000

  18. The Wright-Fisher Model with Fitness Individuals or Alleles A B B A B A B A B B B A A A A A B B B A 0 1 CRV

  19. The Wright-Fisher Model with Fitness10 Simulations, 2N=20 wa=1.0 wb=0.9

  20. The Wright-Fisher Model with FitnessSimulations, wa=1.00 wb=0.95

  21. The Wright-Fisher Model with FitnessSimulations, wa=1.00 wb=0.95

  22. Wright-Fisher with Fitness and Mutation • Wright-Fisher model with fitness • Mutation rate • Mutation fitness • Unique, random • Simple model of natural selection

  23. Wright-Fisher with Fitness and Mutation Individuals or Alleles A B B A B A H A B B B A A A D D B B B A 0 1 CRV

  24. Wright-Fisher with Fitness and Mutation10 Simulations, 2N=100

  25. Conclusion • Objective • Study a stochastic model and report findings • Basic Wright-Fisher Model • Genetic drift • Modifications • Fitness • Competition of alleles • Mutation • Natural selection • Variation

  26. References • “Genetic Drift.” Wikipedia: The Free Encyclopedia. 3 May 2013. Web. http://en.wikipedia.org/wiki/Genetic_drift • "Martingale Convergence Theorem.” Planetmath.org. N.p., n.d. Web. 06 May 2013. http://planetmath.org/martingaleconvergencetheorem • Mitrofanova, Antonina. "Lecture 2: Absorbing States in Markov Chains. Mean Time to Absorption. Wright-Fisher Model. Moran Model." Lecture. Lecture Notes. NYU, Department of Computer Science, New York. 18 Dec. 2007. Web. http://cs.nyu.edu/mishra/COURSES/09.HPGP/scribe2

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