Deterministic genetic models

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Deterministic genetic models
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Terminology

• Allele
• Chromosomes
• Diploid
• Dominant
• Gamete
• Gene
• Genotype

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

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• Recessive
• Recombination
• Segregation
• Zygote

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Mendels Laws

• Law of segregation
• Law of independent assortment

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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

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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

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Two loci - Recombination

• pAB(n1)(1-r) pAB(n)r pA pB
• pAB(n1) - pA pB (1-r) pAB(n)-
  pA pB
• pAB(n1) - pA pB (1-r)n pAB(1)-
  pA pB

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Selection at single locus

• One locus with two alleles A, a
• Discrete generations
• Random mating
• Selection, fitness coefficients
• fAA, fAa, faa

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• 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

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• Zygote freq. with fitness taken into account
• pAA(n)fAA pA(n) 2,
• pAa(n)2 fAa pA(n) pa(n),
• paa(n)faa pa(n)2
• Allele frequencies in generation n1

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• 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

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Equation for evolution

• pA(n1)FpA(n)
• where

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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.

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Equilibria

• pAeq0
• pAeq1

if belongs to lt0,1gt
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Possible scenarios

• fAA lt fAa lt faa - A dies out, a
  becomes fixed
• faa lt fAa lt fAA - a dies out, A
  becomes fixed
• Underdominance
• fAa lt faa , fAA - A1 dies out, A2
  becomes fixed
• if p(0) lt peq
  otherwise A2 dies
• out, A1
  becomes fixed
• Overdominance
• fAa gt faa , fAA - peq is a stable
  equilibrium

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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

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Weak selection

• Transition from difference to differential
  equation
• Assume
• fAA1-?sAA, fAa1-?sAa, faa1-?saa
• where ? is small.
• Continuous time dt ?, which means that t is
  measured in units of 1/ ? generations

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Differential equation
or