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Deterministic genetic models

Deterministic genetic models. Terminology. Allele Chromosomes Diploid Dominant Gamete Gene Genotype. Haploid Heterozygous (genotype) Homologous chromosomes Homozygous (genotype) Locus Meiosis Mitosis Panmixia Phenotype. Recessive Recombination Segregation Zygote.

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Deterministic genetic models

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  1. Deterministic genetic models

  2. Terminology Allele Chromosomes Diploid Dominant Gamete Gene Genotype

  3. Haploid Heterozygous (genotype) Homologous chromosomes Homozygous (genotype) Locus Meiosis Mitosis Panmixia Phenotype

  4. Recessive Recombination Segregation Zygote

  5. Mendel’s Laws • Law of segregation • Law of independent assortment

  6. Hardy – Weinberg Principle Two alleles A and B: Relative frequencies: pA, pB Frequencies of genotypes in offspring are: AA BB AB (pA)2 (pB)2 2pApB

  7. Two loci - Recombination Two loci – each with two alleles: A a, B, b Discrete generations, random mating Allele frequencies: pA, pa, pB, pb remain constant over time r – recombination probability pAB(n) – probability of A, B in gener. no n

  8. Two loci - Recombination pAB(n+1)=(1-r) pAB(n)+r pA pB pAB(n+1) - pA pB =(1-r) [pAB(n)- pA pB] pAB(n+1) - pA pB =(1-r)n [pAB(1)- pA pB]

  9. Selection at single locus One locus with two alleles: A, a Discrete generations Random mating Selection, fitness coefficients: fAA, fAa, faa

  10. Allele frequencies in generation no n : pA(n), pa(n) pA(n)+pa(n)=1, Zygote frequencies: pAA(n)=[pA(n)]2, pAa(n)=2 pA(n) pa(n), paa(n)=[pa(n)]2

  11. Zygote freq. with fitness taken into account: p’AA(n)=fAA [pA(n)]2, p’Aa(n)=2 fAa pA(n) pa(n), p’aa(n)=faa [pa(n)]2 Allele frequencies in generation n+1 :

  12. Normalizing factor must be: fAA [pA(n)]2 + 2 fAa pA(n) pa(n) + faa [pa(n)]2 - average fitness in generation no n. No need for two equations. Equation for pA

  13. Equation for evolution pA(n+1)=F[pA(n)] where

  14. Fundamental Theorem of Natural Selection (Fisher, 1930) Average fitness: fAA [pA(n)]2 + 2 fAa pA(n) pa(n) + faa [pa(n)]2 always increases in evolution, or remains constant, if equilibrium is attained.

  15. Equilibria • pAeq=0 • pAeq=1 if belongs to <0,1>

  16. Possible scenarios fAA < fAa < faa - A dies out, a becomes fixed faa < fAa < fAA - a dies out, A becomes fixed Underdominance: fAa < faa , fAA - A1 dies out, A2 becomes fixed if p(0) < peq otherwise A2 dies out, A1 becomes fixed Overdominance fAa > faa , fAA - peq is a stable equilibrium

  17. Example of overdominance Sickle cell anaemia and malaria Two alleles HBA – normal HBS – mutant Homozygotic genotype HBS HBS - lethal Heterozygotic genotype HBA HBS – protects against malaria

  18. Weak selection Transition from difference to differential equation Assume: fAA=1-sAA, fAa=1-sAa, faa=1-saa where  is small. Continuous time dt= , which means that t is measured in units of 1/  generations

  19. Differential equation or

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