Download 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
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:
Discrete generations
Random mating
Selection, fitness coefficients:
fAA, fAa, faa
A, a
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 :
p' AA (n)  0.5 p' Aa (n)
p A (n  1) 
normalizin g factor
p'aa (n)  0.5 p' Aa (n)
pa (n  1) 
normalizin g factor
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
f AA[ p A (n)]2  f Aa p A (n) pa (n)
p A (n  1) 
f AA[ p A (n)]2  2 f Aa p A (n) pa (n)  f aa [ pa (n)]2
pa (n)  1  p A (n)
Equation for evolution
pA(n+1)=F[pA(n)]
where
f AA p 2  f Aa p(1  p)
F ( p) 
2
2
f AA p  2 f Aa p(1  p)  f aa (1  p)
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
•
p Aeq
f Aa  f aa

2 f Aa  f AA  f aa
if belongs to <0,1>
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
Example of overdominance
Sickle cell anaemia and malaria
HBA – normal
HBS – mutant
Homozygotic genotype HBS HBS - lethal
Heterozygotic genotype HBA HBS – protects
against malaria
Two alleles
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
dp A
  p A ( p A  1)[ p A (2s Aa  s AA  saa )  ( s Aa  saa )]
dt
or
dp A
 kpA ( p A  1)( p A  p Aeq )
dt
p Aeq
s Aa  saa

2 s Aa  s AA  saa
k  2s Aa  s AA  saa