Download Document

Chapter 3
Dynamics of Genes in Populations
Chau-Ti Ting
[email protected]
Unless noted, the course materials are licensed under Creative
Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan (CC
BY-NC-SA 3.0)
1
Hardy-Weinberg Equilibrium Principle
A species is defined as a group of organisms
potentially capable of inbreeding.
A population is a group of individuals that can
inbreed and their offspring.
A deme is a partial isolated inbreeding group.
Adults  gametes  zygotes
2
Population is a collection of organisms
3
One locus with two alleles, A1
and A2
A1A1 A1A1
A1A2
A1A2
A2A2
A1A1
A1A1
f(A1) = p
f(A2) = q
A1A1
A1A1
p+q=1
A1A2
National Taiwan University Chau-Ti Ting
A1A1= 6
A1A2= 3
A2A2= 1
f(A1) = (6 x 2 + 3) / 2 x 10 = 0.75
f(A2) = (1 x 2 + 3) / 2 x 10 = 0.25
4
Calculating allelic frequencies
Frequency of a allele =
number of a alleles
total number of alleles
Alternatively, allelic frequencies can be
calculated based on the knowledge of the
genotypic frequencies.
Frequency of AA =
number of AA
total number
f(MM) = 114/200 = 0.57
f(MN) = 76/200 = 0.38
f(NN) = 10/200 = 0.05
5
p = f(M) =
2 x number of MM + number of MN
2 x total number
=
2 x number of MM
2 x total number
+
number of MN
2 x total
number
= f(MM) + (1/2) f(MN)
6
q = f(N) =
2 x number of NN + number of MN
2 x total number
=
2 x number of NN
2 x total number
+
number of MN
2 x total number
= f(NN) + (1/2) f(MN)
7
The Hardy-Weinberg equilibrium principle yields
two fundamental conclusions:
1. The allele frequencies in a population will
remain constant, generation after generation
2. If the allele frequencies in a population are
given by p and q, the genotype frequencies will
be p2, 2pq, and q2.
Wikipedia
Source: http://en.wikipedia.org/wiki/Hardy%E2%80%93Weinberg_principle
8
The crucial assumptions are:
1. No selection
2. No mutation
3. No migration
4. No chance events (infinitely large
population)
5. Random mating
The Hardy-Weinberg conclusions no longer hold
when one or more of these assumptions is
violated.
9
The Hardy-Weinberg Principle
One locus with two alleles, A1 and A2
f(A1) = p
f(A2) = q
A1A1
National Taiwan University
Chau-Ti Ting
f( A1A1 ) = p x p = p2
f( A1A2 ) = p x q + p x q = 2pq
f( A2A2 ) = q x q = q2
10
A
f (A) = p
A
AA
f (A) = p
f (AA) = p2
a
Aa
f (a) = q
f (Aa) = pq
a
f (a) = q
Aa
f (Aa) = pq
aa
f (aa) = q2
National Taiwan University Chau-Ti Ting
11
Alternative way to calculate…
f(AA) = X
f(Aa) = Y
f(aa) = Z
Offspring
Mating
Proportion
AA × AA
AA × Aa
AA × aa
Aa × AA
Aa × Aa
Aa × aa
aa × AA
aa × Aa
aa × aa
Ｘ2
XY
XZ
ＸY
Y2
YZ
XZ
YZ
Z2
Total
(X+Y+Z)2
AA
Ｘ2
(1/2)XY
(1/2)XY
(1/4)Y2
[X+(1/2)Y]2
Aa
aa
(1/2)XY
XZ
(1/2)XY
(1/2)Y2
(1/2)YZ
XZ
(1/2)YZ
2[X+(1/2)Y][Z+(1/2)Y]
(1/4)Y2
(1/2)YZ
(1/2)YZ
Z2
[Z+(1/2)Y]2
National Taiwan University Chau-Ti Ting
12
Alternative way to calculate…
f(AA) = X
f(Aa) = Y
f(aa) = Z
f(AA) = X2 + (1/2)XY + (1/2)XY + (1/4)Y2
= [X + (1/2)Y]2
f(A) = f(AA) + (1/2) f(Aa)
= X + (1/2)Y
If p = f(A), then [X + (1/2)Y]2 = p2
13
Offspring
Mating
Proportion
AA × AA
AA × Aa
AA × aa
Aa × AA
Aa × Aa
Aa × aa
aa × AA
aa × Aa
aa × aa
Ｘ2
XY
XZ
ＸY
Y2
YZ
XZ
YZ
Z2
Total
(X+Y+Z)2
AA
Ｘ2
(1/2)XY
(1/2)XY
(1/4)Y2
[X+(1/2)Y]2
Aa
aa
(1/2)XY
XZ
(1/2)XY
(1/2)Y2
(1/2)YZ
XZ
(1/2)YZ
2[X+(1/2)Y][Z+(1/2)Y]
(1/4)Y2
(1/2)YZ
(1/2)YZ
Z2
[Z+(1/2)Y]2
National Taiwan University Chau-Ti Ting
p2
2pq
q2
Genotype frequencies do not change from generation
after generation
14
How about allelic frequency?
Before random mating
f(A) = p = f(AA) + (1/2) f(Aa) = X + (1/2) Y
After random mating
f(AA) = p2
f(Aa) = 2pq
f(aa) = q2
 f(A) = f(AA) + (1/2) f(Aa) = p2 + (1/2) 2pq
= p2 + pq
= p (p + q )
=p
15
Testing for fit to Hardy-Weinberg Equilibrium
MM
MN
NN
SUM
10
q2
0.0576
200
1.0
1.0
Observed #
Expected freq.
114
p2
0.5776
76
2pq
0.3648
Expected #
Chi-square
(O-E)2/E
115.52
72.96
11.52
200
0.020
0.127
0.201
0.348
National Taiwan University Chau-Ti Ting
f(M)
= (114 x 2 + 76) / 2 x 200 = 0.76 = p
f(N)
= (10 x 2 + 76) / 2 x 200 = 0.24 = q
Expected frequencies
p2
2pq
q2
Using X2 test of Goodness-of-fit to the Hardy-Weinberg
propotions
16
Testing for fit to Hardy-Weinberg Equilibrium
MM
MN
NN
SUM
48
q2
0.0576
200
1.0
1.0
Observed #
Expected freq.
152
p2
0.5776
0
2pq
0.3648
Expected #
Chi-square
(O-E)2/E
115.52
72.96
11.52
200
11.52
72.96
115.52
200
National Taiwan University Chau-Ti Ting
17
Natural Selection — the differential reproduction of
genetically distinct individuals or genotypes within
M. Prakash
a population.
Fitness () — a measure of the individual’s ability
to survive and reproduce. M. Prakash
 The evolutionary success of an individual is
determined not by it absolute fitness but by its
relative fitness. M. Prakash
18
The Hardy-Weinberg Rule
One locus with two alleles, A1 and A2
f(A1) = p
f(A2) = q
A1A1
National Taiwan University
Chau-Ti Ting
f( A1A1 ) = p2
f( A1A2 ) = 2pq
f( A2A2 ) = q2
19
Genptype
AA
p2
AA 
2pq
AA
q2
Fitness



p2 
2pq 
q2 
National Taiwan University Chau-Ti Ting
qt+1 =
pq  +q2 
p2  +2pq  +q2 
20
qt+1 =
pq  +q2 
p2  +2pq  +q2 
∆q = qt+1 – qt=
pq [p (– )+q (– )]
p2  +2pq  +q2 
21
Consider a recessive allele, A1, double the fitness of its carrier
If f(A2) = 0.2 in this generation, what is it in next generation?
Genptype
AA 
AA
AA
(0.8)2 2(0.8)(0.2) (0.2)2
Fitness

p2 
0.5
0.5
2pq 
q2 
National Taiwan University Chau-Ti Ting
q’ =
0.16 x 0.5 + 0.04 x 0.5
0.64 + 0.32 x 0.5 + 0.04 x 0.5
=0.12
22
For mathematical convenience,
we shall assign a relative fitness value of 1
= 1, = 1 + s, and = 1 + t
Genptype
A A
p2
AA
2pq
AA
q2
Fitness

s
t
p2 
2pq s
q2 t
National Taiwan University Chau-Ti Ting
qt+1 =
pq s +q2 t
p2 + 2pq s + q2 t
23
qt+1 =
pq s +q2 t
p2 + 2pq s + q2 t
∆q = qt+1 – qt=
=
pq s +q2 t
p2 + 2pq s + q2 t
–q
-pq (2 sq – tq – s)
p2 +2pq
+q2 (1+ t)
24
At equilibrium, ∆q = 0
-pq (2 sq – tq – s)
p2 +2pq
=0
+q2 (1+ t)
2 sq – tq – s = 0
s
q=
2s–t
25
Negative selection
Most new mutations arising in a population reduce the fitness of
their carries. Such mutations are called deleterious and they will
be selection against and eventually removed from the population.
This type of selection is called negative or purifying selection.
Dan Graur and Wen-Hsiung Li
A
A
A A
A
A
A
A
C
A
A
Negative Selection
A
A
A A
AA
A
A
A
A
A
A
A
National Taiwan University Chau-Ti Ting
26
Positive selection
In exceedingly rare cases, a mutation may arise that increases the
fitness of its carriers. Such a mutation is called advantageous, and
it will be subjected to positive or advantageous selection.
Dan Graur and Wen-Hsiung Li
A
A
A A
A
A
A
A
B
A
A
Positive Selection
A
B
A A
B
B
B
B
B
B
B
B
National Taiwan University Chau-Ti Ting
27
Random Genetic Drift
Allele frequency changes can also occur by chance, in which case
the changes are not directional but random. The process of changes
in allele frequency due solely to chance effects is called random
genetic drift.
f (white)
0.5
0.5
National Taiwan University Chau-Ti Ting
28
f (white)
0.5
G0
0.5
G1
0.6
National Taiwan University Chau-Ti Ting
29
f (white)
0.6
G1
0.6
G2
0.8
National Taiwan University Chau-Ti Ting
30
f (white)
0.8
G1
0.8
G2
0.4
National Taiwan University Chau-Ti Ting
31
A diploid population with N individuals
 at any given locus, there are 2N genes
one locus with two alleles, A1 and A2
f(A1) = p
f(A2) = 1 – p = q
When 2N gametes are sampled from the infinite gamete pool,
the probability, Pi, that the sample contains exactly i alleles of
genotype A1 is given by the binomial probability function
M. Prakash
Pi =
(2N)!
i! (2N – i)!
pi q2N-i
32
Census population size, N, defined as the total number of
individuals in a population.
From the point of view of population genetics and evolution,
however, the relevant number of individuals to be considered
consists of only those individuals that actively participate in
reproduction. This part is called the effective population size and is
denoted by Ne.
Dan Graur and Wen-Hsiung Li
33
In general, Ne is smaller, sometimes much smaller, than N.
Reduction in the effective population size can occurred if:
1) in a population with overlap generations, at any given time
part of the population will be consist of individuals in either
their prereproductive or postreproductive stage;
2) the number of males involved in reproduction is different
from the number of females;
3) the population size is fluctuated from generation to
generation.
M. Prakash
34
In a population consist of Nm males and Nf females (N = Nm +
Nf ), Ne is given by
M. Prakash
Ne =
4 Nm Nf
Nm + Nf
The long-term effective population size in a species for a
M. Prakash
period of n generations is given by
n
Ne =
1/N1 + 1/N2 + ····· + 1/Nn
Where Ni is the population size of the ith generation. In other
words, Ne equals the harmonic mean of the Ni values, and
consequently it is closer to the smallest value of Ni than to the
largest one.
M. Prakash
35
Gene Substitution
Gene substitution is defined as the process whereby a
mutant allele completely replaces the predominant or wild type
allele in a population. In this process, a mutant allele arises in a
population as a single copy and become fixed after a certain
number of generations.
Dan Graur and Wen-Hsiung Li
A
A
A A
A
A
A
A
B
A
A
B
A
B
B
B
B
B
B
B
B
National Taiwan University Chau-Ti Ting
36
Fixation probability
The probability that a particular allele will become fixed in a
population depends on 1) its initial frequency, 2) its selective
advantage or disadvantage, s, and 3) the effective population size, Ne.
M. Prakash
In the following, we shall consider that the relative fitness of the
three genotypes A1A1, A1A2, and A2A2 are 1, 1+ s, and 1+ 2s,
respectively. Kimura (1962) showed that the probability of fixation
of A2 is
M. Prakash
P=
1– e- 4Nesq
1– e- 4Nes
37
P=
1– e- 4Nesq
1– e- 4Nes
Since e-x ≈ 1- x for small values of x, P equals to q as s
approaches 0. Thus for a neutral allele, the fixation
probability equals its initial frequency in the population.
We note that a new mutant arising as a single copy in a
diploid population of size N has an initial frequency of 1/(2N).
For a neutral mutation, i.e., s = 0, the fixation probability
1
P=
2N
38
When s ≠ 0,
P=
1– e- (2Nes/N)
1– e- 4Nes
If the population size is equal to the effective population size,
P=
1– e- 2s
1– e- 4Nes
If the absolute value of s is small, we obtain
P=
2s
1– e- 4Nes
For positive value of s and large value of N,
P ≈ 2s
39
Fixation time
The time required for the fixation or loss of an allele
depends on 1) the frequency of the allele, 2) its selective
advantage, and 3) the size of the population.
In the following, we deal with the mean fixation time of those
mutations that will eventually become fixed in the population.
This variable is call conditional fixation time. In the case of a
new mutation whose initial frequency in a diploid population is
be definition q = 1/(2N), the mean conditional fixation time, t,
was calculated by Kimura and Ohta (1969). For a neutral
mutation, it is approximated by
t = 4N generations
and for a mutation with a selective advantage of s, it is
approximated by
Dan Graur and Wen-Hsiung Li
t = (2/s) ln(2N) generations
40
Neutral mutations
allele frequency
t
Time
allele frequency
Advantageous mutations
t
Time
National Taiwan University Chau-Ti Ting
41
Rate of gene substitution
Rate of gene substitution: the number of mutations reaching fixation
per unit time.
Neutral mutations: If neutral mutations occur at a locus in a
diploid population of size N is 2Nu per generation. Since the
probability of fixation for each of these mutations is 1/(2N), we
obtain the rate of substitution of neutral alleles by multiplying the
total number of mutations by the probability of their fixation:
Dan Graur and Wen-Hsiung Li
1
K = 2Nu
=u
2N
42
Advantageous mutation: the rate of substitution can also
be obtained by multiply the rate of mutation by the
probability of fixation for advantageous alleles. When s > 0
Dan Graur and Wen-Hsiung Li
K = 2Nu x 2s = 4Nsu
43
Genetic Polymorphism
A population is monomorphic at a locus if there
exists only one allele at the locus. A locus is said to be
polymorphic if two or more alleles coexist in the population.
Dan Graur and Wen-Hsiung Li
44
An appropriate measure of genetic variability is the
mean expected heterozygosity, or gene diversity.
This measure
1) does not depend on an arbitrary delineation of
polymorphism
2) can be computed directly from knowledge of the
allele frequencies
3) is less affected by sampling effects. Gene
diversity at a locus is defined as
Dan Graur and Wen-Hsiung Li
45
The average of h values over all the loci studied, H,
can be used as an estimate of the extent of genetic
variability within the population. That is,
where hi is the gene diversity at locus i, and n is
number of loci.
Dan Graur and Wen-Hsiung Li
46
Nucleotide diversity
For DNA sequence data, a more appropriate measure
of polymorphism in a population is the average number of
nucleotide differences per site between any two randomly
chosen sequenes. This measure is called nucleotide
diversity and its denoted by 
 = xixjij
ij
where xixj are the frequencies of the ith and jth type of DNA
sequences, respectively, and ij is the proportion of the
different alleles between the ith and jth types.
Dan Graur and Wen-Hsiung Li
47
Copyright Declaration
Work
Licensing
Author/Source
Open Clip Art Library bitterjug
http://openclipart.org/detail/28419/magnifying-glass-by-bitterjug
2012/04/30 visited
Page
P3
P3, P18, P33, P44
Open Clip Art Library danko
http://openclipart.org/detail/4152/friendly-rabbit-by-danko
2012/04/30 visited
P4
National Taiwan University Chau-Ti Ting
The Hardy-Weinberg ….If
the allele frequencies in a
population are given by p and
q, the genotype frequencies
will be p2, 2pq, and q2.
Wikipedia
P8
http://en.wikipedia.org/wiki/Hardy%E2%80%93Weinberg_principle
2012/04/30 visited
P10, P19
National Taiwan University Chau-Ti Ting
P11
National Taiwan University Chau-Ti Ting
P12, P14
National Taiwan University Chau-Ti Ting
48
Work
Licensing
Author/Source
Page
P16
National Taiwan University Chau-Ti Ting
P17
National Taiwan University Chau-Ti Ting
Natural Selection — the
differential reproduction of
genetically distinct
individuals or genotypes
within a population.
M. Prakash
2008. Molecular Biology of Evolution, p. 204. Discovery Publishing House.
http://books.google.com.tw/books?id=6x2UmpU6grsC&printsec=frontcover&
hl=zh-TW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P18
Fitness () — a measure of
the individual’s ability to
survive and reproduce.
M. Prakash
2007. Fundamentals of Gene Evolution, p. 133. Discovery Publishing House.
http://books.google.com.tw/books?id=T9aNqdaXrX8C&printsec=frontcover&
hl=zh-TW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P18
The evolutionary success of
an individual is determined
not by it absolute fitness but
by its relative fitness
M. Prakash
2007. Fundamentals of Gene Evolution, p. 133. Discovery Publishing House.
http://books.google.com.tw/books?id=T9aNqdaXrX8C&printsec=frontcover&
hl=zh-TW#v=onepage&q&f=false
2012/04/30 visited
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P18
P18, P44
Modified from Open Clip Art Library danko
http://openclipart.org/detail/4152/friendly-rabbit-by-danko
2012/04/30 visited
P20
National Taiwan University Chau-Ti Ting
49
Work
Licensing
Author/Source
Page
P22
National Taiwan University Chau-Ti Ting
P23
National Taiwan University Chau-Ti Ting
Most new mutations
arising …of selection is
called negative or purifying
selection.
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 41. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P26
P26
National Taiwan University Chau-Ti Ting
In exceedingly rare cases, a
mutation may…and it will be
subjected to positive or
advantageous selection.
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 41. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P27
P27
National Taiwan University Chau-Ti Ting
Allele frequency changes can
also occur by chance, …in
allele frequency due solely to
chance effects is called
random genetic drift.
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 47. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P28
50
Work
Licensing
Author/Source
Page
P28
National Taiwan University Chau-Ti Ting
P29
National Taiwan University Chau-Ti Ting
P30
National Taiwan University Chau-Ti Ting
P31
National Taiwan University Chau-Ti Ting
When 2N gametes are
sampled …i alleles of
genotype A1 is given by the
binomial probability function
M. Prakash
2008. Molecular Biology of Evolution, p. 209. Discovery Publishing House.
http://books.google.com.tw/books?id=6x2UmpU6grsC&printsec=frontcover&h
l=zh-TW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P32
Census population size, N,
defined as the total number of
individuals in a population….
is denoted by Ne.
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 52. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P33
In general, Ne is smaller,
sometimes much
smaller, …fluctuated from
generation to generation.
M. Prakash
2008. Molecular Biology of Evolution, p. 213. Discovery Publishing House.
http://books.google.com.tw/books?id=6x2UmpU6grsC&printsec=frontcover&h
l=zh-TW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P34
51
Work
Licensing
Author/Source
Page
In a population consist of Nm
males and Nf females (N = Nm +
Nf ), Ne is given by
M. Prakash
2007. Fundamentals of Gene Evolution, p. 143. Discovery Publishing House.
http://books.google.com.tw/books?id=T9aNqdaXrX8C&printsec=frontcover&hl=zhTW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P35
The long-term effective
population size in a species for a
period of n generations is given
by
M. Prakash
2007. Fundamentals of Gene Evolution, p. 143. Discovery Publishing House.
http://books.google.com.tw/books?id=T9aNqdaXrX8C&printsec=frontcover&hl=zhTW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P35
Where Ni is the population size of
the ith generation….consequently
it is closer to the smallest value of
Ni than to the largest one.
M. Prakash
2007. Fundamentals of Gene Evolution, p. 143. Discovery Publishing House.
http://books.google.com.tw/books?id=T9aNqdaXrX8C&printsec=frontcover&hl=zhTW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P35
Gene Substitution Gene
substitution is defined as the
process whereby a mutant…fixed
after a certain number of
generations.
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 53. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P36
P36
National Taiwan University Chau-Ti Ting
The probability that a particular
allele will become fixed in a
population depends on…the
effective population size, Ne.
M. Prakash
2008. Molecular Biology of Evolution, p. 215. Discovery Publishing House.
http://books.google.com.tw/books?id=6x2UmpU6grsC&printsec=frontcover&hl=zhTW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P37
In the following, we shall
consider that the
relative…respectively. Kimura
(1962) showed that the probability
of fixation of A2 is
M. Prakash
2007. Fundamentals of Gene Evolution, p. 144. Discovery Publishing House.
http://books.google.com.tw/books?id=T9aNqdaXrX8C&printsec=frontcover&hl=zhTW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P37
52
Work
Licensing
Author/Source
Page
Since e-x ≈ 1- x for small
values of x, P equals to q as s
approaches 0. Thus for a
neutral allele, the fixation
probability equals its initial
frequency in the population.
M. Prakash
2007. Fundamentals of Gene Evolution, p. 144. Discovery Publishing House.
http://books.google.com.tw/books?id=T9aNqdaXrX8C&printsec=frontcover&h
l=zh-TW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P38
We note that a new mutant
arising as a single copy
in…For a neutral mutation,
i.e., s = 0, the fixation
probability
M. Prakash
2007. Fundamentals of Gene Evolution, p. 144. Discovery Publishing House.
http://books.google.com.tw/books?id=T9aNqdaXrX8C&printsec=frontcover&h
l=zh-TW#v=onepage&q&f=false
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P38
The time required for the
fixation or loss of an allele
depends on …a selective
advantage of s, it is
approximated by
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 55. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P40
P41
National Taiwan University Chau-Ti Ting
If neutral mutations occur at a
locus in a diploid population
of size … number of
mutations by the probability
of their fixation:
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 57. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P42
the rate of substitution can
also be obtained
by …advantageous alleles.
When s > 0
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 57. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P43
A population is monomorphic
at a locus if there exists …if
two or more alleles coexist in
the population.
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 57. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P44
53
Work
An appropriate measure of
genetic …sampling effects.
Gene diversity at a locus is
defined as
The average of h values over
all the loci studied, H, can be
used as an…
where hi is the gene diversity
at locus i, and n is number of
loci.
Nucleotide diversity…alleles
between the ith and jth types.
Licensing
Author/Source
Page
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 58. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P45
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 58. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P46
Dan Graur and Wen-Hsiung Li
2000. Fundamentals of Molecular Evolution., p. 59. Sinauer Associates, Inc.
Sunderland, MA, USA.
It is used subject to the fair use doctrine of:
•Taiwan Copyright Act Articles 52 & 65
P47
54