Download How to Model Microevolution How to Model Microevolution

How to Model Microevolution
Evolution is a change over time in the frequency of alleles or allele
combinations in the gene pool, so any model of evolution must
include at the minimum the passing of genetic material from one
generation to the next. Hence, our fundamental time unit will be
the transition between two consecutive generations at comparable
stages. All such trans-generational models of microevolution have
to make assumptions about three major mechanisms:
•
•
•
Mechanisms of producing gametes
Mechanisms of uniting gametes
Mechanisms of developing phenotypes.
How to Model Microevolution
In order to specify how gametes are produced, we have to
specify the genetic architecture.
Genetic architecture refers to the number of loci and their
genomic positions, the number of alleles per locus, the
mutation rates, and the mode and rules of inheritance of the
genetic elements.
For example, the first model we will develop assumes a
single autosomal locus with two alleles with no mutation.
Under this genetic architecture, we need only to use
Mendel’s first law of inheritance to specify how genotypes
produce gametes.
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Demes and Gene Pools
• Meiosis Interconnects the Deme to the Gene
Pool
• Therefore, Given Mendel’s Laws and Normal
Meiosis, You Can Always Calculate the Allele
Frequencies in the Gene Pool From the
Genotype Frequencies in the Deme
• Can You Predict the Deme (Genotype
Frequencies) from the Gene Pool (Allele
Frequencies)?
Demes
AA
1/
2
aa
1/
2
1
AA
1/
4
1
A
1/
2
a
1/
2
Aa
1/
2
1/
1
A
1/
2
2
aa
1/
4
1/
1
2
a
1/
2
Gene Pools
2
Hardy (and Weinberg)
Solution
FERTILIZATION
To Go From Gene Pool (Gametes) to Deme
(Initially Zygotes),Need to Specify
The Rules by Which Gametes Unite (Fertilization)
Population Structure
Population Structure is the
rules at the level of the deme
by which gametes are united
in fertilization, thereby
defining the transition from
haploidy to diploidy.
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Models in Population Genetics Minimally
Specify How To Go From One Generation To
The Next
Deme of Adult
Diploid Individuals
Meiosis
Gene Pool of
Haploid Gametes
Need to Specify Genotype Frequencies,
And Therefore Genetic Architecture
(Number of Loci, Alleles per Locus,
Linkage, Rules of Inheritance, etc.).
Need to Specify Population Structure.
Fertilization
Deme of Adult
Diploid Individuals
Need to Specify How Individuals
Develop Phenotypes.
Assumptions of Hardy-Weinberg
Mechanisms of Producing Gametes (Genetic Architecture)
One Autosomal Locus
Two Alleles
No Mutation
Mendel’s First Law (50:50 Segregation in heterozygotes)
Mechanisms of Uniting Gametes (Population Structure)
System of Mating: Random
Size of Population: Infinite
Genetic Exchange: None (One Isolated Population)
Age Structure: None (Discrete Generations)
Mechanisms of Developing Phenotypes
All Genotypes Have Identical Phenotypes With Respect to
Their Ability for Replicating Their DNA
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Random Mating
Random Mating occurs when both of the
gametes united in a zygote are drawn at
random and independently from the
gene pool.
This means that the probability of a
gamete bearing a specific allele = the
frequency of that allele in the gene pool,
and this is true for all gametes involved
in fertilization.
Random Mating
A
p
a
q = 1-p
Gene Pool
Paternal Gamete
Maternal Gamete
A
p
a
q
A
p
AA
p×p=p2
Aa
pq
a
q
aA
qp
aa
q×q=q2
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Hardy-Weinberg
Genotype Frequencies
AA
p2
Aa
2pq
aa
q2
Mendelian Probabilities of Offspring (Zygotes)
Weinberg’s
Derivation
Mating Pair
Frequency of Mating Pair
AA
Aa
aa
AA × AA
GAA×GAA = GAA2
1
0
0
AA × Aa
GAA×GAa = GAAGAa
Aa × AA
GAa×GAA = GAAGAa
AA × aa
GAA×Gaa = GAAGaa
0
1
0
aa × AA
Gaa×GAA = GAAGaa
0
1
0
0
0
1
G’AA
G’Aa
G’aa
0
0
2
Aa × Aa
GAa×GAa = GAa
Aa × aa
GAa×Gaa = GAaGaa
0
aa × Aa
Gaa×GAa = GAaGaa
0
aa × aa
Gaa×Gaa = Gaa
2
Total Offspring
Summing Zygotes Over All Mating Types:
G’AA=GAA2 + [2GAAGAa] + GAa2 = [GAA+ GAa]2 = p2
G’Aa= [ 2GAAGAa]+ 2GAAGaa+ GAa2+ [ GAaGaa]= 2[GAA+ GAa][Gaa+ GAa] = 2 p q
G’aa= GAa2 + [ 2GAaGaa] + Gaa2 = [Gaa+ GAa]2 = q2
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The Life Cycle for a Population
AA
GAA
Deme of Diploid
Individuals
Meiosis
Random
Mating
Fertilization
1/
1
Mendelian
Probabilities
Gene Pool of
Haploid Gametes
Aa
GAa
1/
2
1
2
A
a
p=GAA+ 1/2GAa
q=Gaa+ 1/2GAa
p
p
AA
p2
Deme of Diploid
Individuals
aa
Gaa
p
q
q
Aa
2pq
q
aa
q2
Testing for Hardy-Weinberg Genotype Frequencies. E.g., a
Population of Pueblo Indians Scored for the MN Blood Group Type
Blood Type
M
MN
N
Genotype
MM
MN
NN
Number
83
46
11
140
(0.24)2 =
0.06
1
H.-W. Freq.
(0.76)2 = 2(0.76)(0.24)
0.58
= 0.36
Exp. Number
0.58(140)
= 81.2
0.36(140)
= 50.4
(Obs.-Exp.)2
Exp.
(83-81.2)2
81.2
(46-50.4)2
50.4
Sum
0.06(140) 140
= 8.4
(11-8.4)2
8.4
1.23
Degrees of Freedom = 3 Categories - 1 - 1 estimated parameter = 1
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Random Mating Is Locus
Specific
Although the Pueblo Indians are
randomly mating for the MN Blood
Group Locus, They Are Not
Randomly Mating For All Loci,
e.g., the X and Y Chromosomes
Hardy-Weinberg Frequencies Represent An Equilibrium
No Assumption is
AA
Made About These
Genotype Frequencies;
GAA
They May or May
Mendelian
1
Not Be in
Probabilities
Hardy-Weinberg
Random
Mating
One Generation of
Random Mating Insures
These Are Hardy-Weinberg
Genotype Frequencies
Aa
GAa
1/
aa
Gaa
1/
2
1
2
A
a
p=GAA+ 1/2GAa
q=Gaa+ 1/2GAa
p
p
AA
p2
p
q
Aa
2pq
q
q
aa
q2
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Hardy-Weinberg Frequencies Represent An Equilibrium
The Frequency of
The A Allele in the
Next Generation’s
Gene Pool Is:
p’= p2 + 1/22pq
= p2 + pq
= p(p + q)
=p
Therefore, the
Gene Pool Is
Unchanged
There Is
NO
Evolution
Under The
HardyWeinberg
Model
A
p=GAA+
Random
Mating
p
a
1/
p
2 GAa
p
AA
p2
q
q
Aa
2pq
1/
1
Mendelian
Probabilities
q=Gaa+ 1/2GAa
aa
q2
1/
2
a
22pq
=p(p+q) = p
q’=q2+ 1/
22pq
=q(q+p)=q
A
a
p=GAA+ 1/2GAa
q=Gaa+ 1/2GAa
p
p
p
AA
p2
Mendelian
Probabilities
1
2
A
p’=p2+ 1/
Random
Mating
q
q
q
Aa
2pq
1/
1
2
A
p’=p2+ 1/
q
aa
q2
1/
1
2
a
22pq
=p(p+q) = p
q’=q2+ 1/
22pq
=q(q+p)=q
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Importance of Hardy-Weinberg
• Acceptance of Mendelian Genetics (Punnett’s dilemma)
• Resurrection of Natural Selection (Jenkin’s critique)
• A useful null model of evolutionary stasis.
• A valuable springboard for the investigation of many forces of
evolutionary change by relaxing its assumptions.
The First Assumption That We
Will Relax Is the Assumption of
a Genetic Architecture of OneLocus With Two Alleles.
What Happens When We Have
Two Loci, Each With Two
Alleles?
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Two Locus Hardy Weinberg
Gene
Pool
AB
Ab
aB
ab
gAB
gAb
gaB
gab
Mechanisms of
Uniting Gametes
(Random Mating)
Zygotic/Adult
Population
AB/AB
AB/Ab
AB/aB
AB/ab
gAB2
2g AB gAb
2g ABgaB
2g ABgab
Ab/Ab
gAb 2
Ab/aB
Ab/ab
aB/aB
aB/ab
2g Ab gaB
2g Ab gab
gaB 2
2g aB gab
ab/ab
gab2
Mechanisms of
Producing Gametes
(Mendel's First Law
& Recombination)
Gene Pool
of Next
Generation
AB
Ab
aB
ab
g' AB
g'Ab
g' aB
g' ab
Recombination
Occurs in All
Genotypes, But
Can Change
The State of the
Parental
Gametes
Only in Double
Heterozygotes.
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Two Locus Hardy Weinberg
Gene
Pool
AB
Ab
aB
ab
gAB
gAb
gaB
gab
Mechanisms of
Uniting Gametes
(Random Mating)
Zygotic/Adult
Population
AB/AB
AB/Ab
AB/aB
AB/ab
gAB2
2g AB gAb
2g ABgaB
2g ABgab
Ab/Ab
gAb 2
Ab/aB
Ab/ab
aB/aB
aB/ab
2g Ab gaB
2g Ab gab
gaB 2
2g aB gab
ab/ab
gab2
Mechanisms of
Producing Gametes
(Mendel's First Law
& Recombination)
Gene Pool
of Next
Generation
AB
Ab
aB
ab
g' AB
g'Ab
g' aB
g' ab
Double heterozygotes can produce all four gamete types.
Two Locus Hardy Weinberg
2
g'AB = 1" gAB
+ 12 (2gAB gAb )+ 12 (2gAB gaB )+ 12 (1# r )(2gAB gab )+ 12 r(2gAb gaB )
= gAB [gAB + gAb + gaB + (1# r )gab ] + rgAb gaB
= gAB [gAB + gAb + gaB + gab ] + rgAb gaB # rgAB gab
= gAB + r(gAb gaB # gAB gab ) = gAB # rD
!
Where D = gAB gab - gAb gaB
D is called Linkage Disequilibrium
or Gametic Phase Imbalance
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Two Locus Hardy Weinberg
Similarly, can show:
g’AB = gAB - rD
g’Ab = gAb + rD
g’aB = gaB + rD
g’ab = gab - rD
Where D = gAB gab - gAb gaB
D, “linkage disequilibrium”
• It measures the degree of association at
the population level between the two
sites/loci
• D is created by many evolutionary
forces and historical events, including
the very act of mutation because the
new mutant variant initially exists on
only one chromosomal background.
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Two Locus Hardy Weinberg
That is, Evolution Occurs!
Two Locus Hardy Weinberg
D1
The two locus equilibrium
is approached gradually,
at a rate determined by r.
Historical information is
encoded in D (and other
multi-locus/site measures)
that decays gradually with
time! This information
persists for long periods of
time for tightly linked
sites.
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Theoretical Decay of LD in a Random-Mating Population
In a genomic region with no recombination,
the LD created by mutation never dissipates.
Two Locus Hardy Weinberg Equilibrium
pA = gAB + gAb
pB = gAB + gaB
so
pA pB = ( gAB + gAb )( gAB + gaB )
2
= gAB
+ gAB gaB + gAB gAb + gAb gaB
= gAB ( gAB + gaB + gAb) + gAb gaB
= gAB (1" gab ) + gAb gaB
!
= gAB " gAB gab + gAb gaB
= gAB " D
As t goes to infinity, D goes to 0 (the equilibrium), so at the
two-locus equilibrium, gAB=pApB, and similarly for the other
! gamete frequencies.
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Two Locus Hardy Weinberg
At equilibrium the two loci associate at random (proportional to
their allele frequencies) in the Population’s Gene Pool:
B
k
b
m
A
p
AB
pk=gAB
Ab
pm=gAb
a
q
aB
qk=gaB
ab
qm=gab
D = gAB gab - gAb gaB=pkqm-pmqk=0
D≠ 0 measures the degree of non-random association at the
population gene pool level between the two sites/loci
Many Factors Create
Disequilibrium, Including the
Very Act Of Mutation
Once created, disequilibrium decays at a
rate determined in part by recombination,
and in part by population structure (as we
will see later).
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Linkage disequilibrium is created when
mutation creates new variation
D = gAB gab - gAb gaB = gAB0 - 0gaB =0
Initial Gene Pool:
A
B
a
B
a
B
Mutation At A Second Site Produces Three Gamete Types:
Gene Pool
After Mutation:
A
B
a
B
a
b
D = gAB gab - gAb gaB = gAB gab - 0gaB = gAB gab ≠ 0
Can see the effects of mutation on Linkage
disequilibrium more clearly through D’
"
D
,D<0
$
min
p
p
,p
p
( A B a b)
$
D '= #
$
D
,D>0
$
% min( pA pb ,pa pB )
D’ varies between -1 and +1, and when mutation first creates
the third gamete type, D’=-1 or +1, so mutation creates
maximal linkage disequilibrium.
!
17
D (or D’) decays with recombination:
A
B
A
B
a
B
a
B
D’=D=0
Mutation At A Second Site Produces Three Gamete Types:
A
B
A
D = gAB gab
D’=1
(pA=gAB, pb=gab)
A
B
B
a
Recombination
Produces Four
Gamete Types
A
b
a
B
a
b
B
a
b
D = gAB gab - gAb gaB< gAB gab, D’<1 (pA=gAB+gAb)
In regions of little to no recombination, the
pattern of disequilibrium is determined primarily
by the historical conditions that existed at the
time of mutation, resulting in little to no
correlation of D with physical distance
Indel
Taq I
Pst I
Sst I
Pvu II
Xmn I
Apo AI
Apo CIII
Apo AIV
Significant linkage disequilibrium
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On larger physical scales, |D| is negatively
correlated with physical distance
0.9
0.8
0.7
|D’| 0.6
0.5
0.4
0.3
0.2
0.1
0
5
10
20
40
80
kb
AllYor
Distance
Utah
Swed
160
S
U
(kb)
YorBot
YorTop
Reich et al. (2001 Nature 411:199-204)
Disequilibrium and Historical
Effects Create Both
Opportunities and Difficulties for
the Analysis of Population
Genetic Data, As We Shall See
19
Some lessons from 1 vs. 2-locus HW:
A seemingly slight change in the model can
create qualitative differences (e.g., no evolution
in 1 locus HW vs. evolution in 2-locus HW;
instantaneous equilibrium in 1 locus HW vs.
gradual or no equibrium in 2 locus HW)
Scale matters (e.g., the relationship between D’
and physical distance on different scales of
physical distance).
The inferences made from a model are often very sensitive
to the assumptions of that model. Generalize with care!
20