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

Mendel’s Laws Law of segregation Law of independent assortment

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

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

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]

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

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

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 :

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

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

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.

Equilibria pAeq=0 pAeq=1 if belongs to <0,1>

Possible scenarios 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

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

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

Differential equation or