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) (pB) pApB

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)] 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)] 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
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 equationor