Download Darwin`s Four Postulates V a ria tio n is re d u c e d e a c h g e n e ra

The most serious challenge to Darwin
came from “Genetics”
Darwin’s Four Postulates
Hereditary mechanism was unknown, but a key issue was how was
VARIATION maintained?
Variation
Observation:
Inheritance
a
TALL parent mates with a SHORT parent
Competition
offspring tend to be
Survival & reproduction are non-random
INTERMEDIATE in height
thus offspring traits were thought to be a blend of parent traits
Variation is reduced each generation
Under blending inheritance, size of kids is the average on the allele from their
parents. Thus if mom = 15 & dad = 25, then kid = 20.
With the recognition that genetics was
Mendialian, not blending, the challenge of
maintaining variation across generations was
no longer a challenge to Darwinian evolution.
But what happens to genetic variation ?
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Offspring Population
Parent Population
Aa Aa
aa
AA
aa
Aa aa aa
AA
AA
AA
AA
mating and
reproduction
Aa Aa
AA
aa aa
aa
Aa aa
AA
AA
AA AA
Is genetic composition of a population
maintained across generations?
Two ways to describe genetic composition a population:
1) compute frequency of different genotypes
AA, Aa, aa
2) compute frequency of different alleles
A, a
What is the relationship
between the genetic composition
of the Parent Population
and of the Offspring Population?
So let’s compute these frequencies in parents and
in their offspring, to see whether they change
compute frequency of different alleles in parents
compute frequency of different genotypes (parents)
Genotypes
N(100)
AA
Aa
9
42
49
42/100
49/100
Genotype 9/100
frequency
GAA + GAa + Gaa = 1
aa
Genotypes
N(100)
AA
Aa
aa
9
42
49
.42
.49
Genotype freq. .09
Allele frequency:
A
a
GAA + 1/2 GAa = .09 +.21 = .3
Gaa + 1/2 GAa
= .49 + .21= .7
PA + Pa = 1
2
offspring population
parent population
Aa Aa
aa
AA
aa
aa
Aa aa
AA
A
AA A
AA
mating and
reproduction
Aa
Aa
AA
aa aa
aa
Aa aa
AA
A
AA A
AA
{GAA, GAa, Gaa},
{PA, Pa}
{GAA, GAa, Gaa},
{PA, Pa}
Aa
Aa
Aa Aa
aa
AA
Aa aa
aa aa
aaAa
aa
Aa aa aa
AA
Aa AA Aaa
A
AA
AA AA
offspring population
AA
Aa
Aa
Aa
AA
aa aa aa aa
Aa
AA AA Aa
AA
Aa Aa
AA
Aa
Aa
AA
Aa aa
aa aa
Aa
aa
Aa aa
aaaa
AA
AA
Aa
AA
AA
{GAA, G Aa, G aa},
{PA , Pa}
{GAA =.09, Gaa = .42, Gaa = .49},
{PA = .3, Pa = .7}
a
AA
Aa
Aa
aa
{GAA, GAa, Gaa},
{PA, Pa}
a
aaAa
random mating
A
a A Aa A A
A a a
a
a A
A
A
offspring population
parent population
A AA
Aa a
Aa
AA
aa aa aa aa
Aa
AA AA Aa
AA
AA
AA
Aa
Aa aa
AA
Aa aa
aa aa
Aa
aa
Aa aa
aaaa
AA
AA
Aa
AA
AA
{GAA, G Aa, G aa},
{PA , Pa}
{GAA =.09, Gaa = .42, Gaa = .49},
{PA = .3, Pa = .7}
female gametes
male gametes
male gametes
A
a
Aa
AA
aa aa
aa
Aa aa
AA
A
AA A
AA
ga
egmet
gs es
+ s:
pe
rm
female gametes
A
Aa
{GAA, GAa, Gaa},
{PA, Pa}
What is the relationship
between the genotype and allele frequencies
of the Parent Population
and of the Offspring Population?
parent population
offspring population
parent population
A
PA = .3
a
PA = .7
A
PA = .3
a
PA = .7
AA
.09
Aa
.21
Aa
.21
aa
.49
3
offspring population
parent population
A AA
Aa a
Aa
AA
aa aa aa aa
Aa
AA AA Aa
AA
AA
AA
Aa
Aa aa
AA
Aa aa
aa aa
Aa
aa
Aa aa
aa
AA
Aa AA aa
AA
AA
{GAA =.09, Gaa = .42, Gaa = .49},
{PA = .3, Pa = .7}
{GAA =.09, Gaa = .42, Gaa = .49},
{PA = .3, Pa = .7}
offspring population
parent population
A AA
Aa a
Aa
AA
aa aa aa aa
Aa
AA AA Aa
AA
AA
AA
Aa
Aa aa
AA
Aa aa
aa aa
Aa
aa
Aa aa
aa
AA
Aa AA aa
AA
AA
AA
.09
Aa
.21
Aa
.21
aa
.49
A
PA = .3
a
PA = .7
A
PA = .3
a
PA = .7
A
PA = .3
a
PA = .7
AA
.09
Aa
.21
Aa
.21
aa
.49
offspring population
parent population
A AA
Aa a
Aa
AA
aa aa aa aa
Aa
AA AA Aa
AA
AA
AA
Aa
Aa aa
AA
Aa aa
aa aa
Aa
aa
Aa aa
aaaa
AA
AA
Aa
AA
AA
{GAA = .09, Gaa = .42, Gaa = .49},
{PA = .3, Pa = .7}
{GAA = .3, G aa = 0, Gaa = .7},
{PA = .3, Pa = .7}
female gametes
male gametes
female gametes
male gametes
a
PA = .7
A
PA = .3
a
PA = .7
A
PA = .3
a
PA = .7
AA
.09
Aa
.21
Aa
.21
aa
.49
Frequency of genotype
male gametes
female gametes
A
PA = .3
{GAA = ?, G aa = ?, Gaa = ?},
{PA = ?, P a = ?}
{GAA = .3, G aa = 0, Gaa = .7},
{PA = .3, Pa = .7}
aa = q2
AA = p2
Aa = 2pq
Frequency of A
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Two Key Observations
Hardy-Weinberg Equilibrium
1) Allele frequencies don’t change across
generations simply as a result of
reproduction -- so are in “equilibrium”
generation
PA
1
2
3
4
5
.3
.3
.3
.3
.3
2) Genotype frequencies don’t change across
generations after one generation -- so are also in
equilibrium after one generation of random mating
generation
GAA
1
2
3
4
5
.3
.09
.09
.09
.09
OK, so what’s the big deal?
“saves” Darwinian Natural Selection
variation is preserved across generation
(e.g., gene frequencies don’t change)
relates the allele frequencies in the parents
to the genotype frequencies of the offspring
and is stable across generations
{(PA)2, (2)(PA)(Pa), (Pa)2 }
{GAA , GAa, Gaa}
Map parent gene frequencies to offspring genotype frequencies
(assumptions: random mating, no natural or sexual selection,
no mutation, no migration,
and no genetic drift
But if we observe a difference in allele frequency
between Parents to Offspring, then
something (= “evolution”) must be happening
we measure evolution as the change (∆) in allele
frequency between generations, from parent to offspring:
evolution (i.e., a change in gene frequency) will not
occur unless our assumptions are violated
evolution = PA – PA
Thus the “null hypothesis” is that no evolution will occur
∆ PY ≠ 0
(assumptions: random mating, no natural or sexual
selection,no mutation, no migration, and no genetic drift
(infinite population size)
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What can cause?
OR if we observe a difference in genotype frequency
from that expected by H-W, then
something (= “evolution”) is happening
∆ PY ≠ 0
PA = 0.3, Pa = 0.7
AA
Aa
H-W expectation 0.09
observed
0.03
0.42
0.49
0.55
0.42
Simulating gene frequencies across generations
with finite population sizes
gene frequency Gen 1
PA = 0.3
gene frequency Gen 2
aa
PA = ?
(assumptions: random mating, no natural or sexual
selection, no mutation, no migration, and no genetic drift
(infinite population size)
Draw from infinite pool of genes, PA = .5 = Pa
a
a
Aa a A a A
a A a
aaAa
A
aAa
A
A
a
a A Aa AA A A a AA AA A
A a a
aA
A a aa
a
a AA a a
a
a A
a A a
A
A
A
What will be PA′ in your sample?
gene frequency Gen 3 PA = ?
Does variability in replicate samples of PA′ depend
on sample size?
Repeatedly draw samples of size = N,
compute average & standard deviation of PA′
does average and SD vary with N?
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Standard deviation in p’
Population size
Standard deviation in p’
Magnitude of “drift” inversely proportional to N
Simulating drift in gene frequencies iterated across
Generations with finite population sizes
gene frequency Gen 1
σ=
1
2 √N
PA = 0.3
gene frequency Gen 2
PA = ?
gene frequency Gen 3 PA = ?
Population size
7
What will happen if Hardy-Weinberg holds?
N = 10,000/generation, 100 populations, initial p = 0.5, no selection
ftp://evolution.gs.washington.edu/pub/popgen/popg.html
PopG genetics simulation
N = 100/generation, 100 populations, initial p = 0.5, no selection
N = 25/generation, 100 populations, initial p = 0.5, no selection
ftp://evolution.gs.washington.edu/pub/popgen/popg.html
PopG genetics simulation
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N = 100/generation, 100 populations, initial p = 0.9, no selection
What is the probability that a given allele
will drift to fixation (P = 1)
P
˜
frequency (A)
Probability of
fixation
Frequency of A
How LONG will it take a given allele
to drift to fixation on average? f (p, N)
How LONG will it take a given allele
to drift to fixation? f (p, N)
N = 10,000
Mean time
to fixation
Mean time
to fixation
N = 5,000
N = 1,000
Kimura & Ohta (1971)
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Population Bottlenecks
& Founder Effect
population crash
Population Bottlenecks
Pop size
Mainland population
colonizes
Island
population
time
A genetic bottleneck in
Drosophila subobscura
D. subobscura chromosomes
from Krimbas 1993
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A genetic bottleneck in
Drosophila subobscura
N of inversions in Old World = 86
N of inversions in New World = 19 (only 22%)
But which inversions were lost? Rarest?
Probability of occurrence in New World
Drosophila subobscura inversions
1.0
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0.4
0.6
0.8
0.6
0.4
0.2
0.0
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0.0
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0.2
rare
Overall frequency in Old World
common
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