1. Chapter 3 Dynamics of Genes in Populations
Chau-Ti Ting
Unless noted, the course materials are licensed under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Taiwan (CC BY-NC-SA 3.0)

2. Adults  gametes  zygotes
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

3. Population is a collection of organisms

4. One locus with two alleles, A1 and A2
National Taiwan University Chau-Ti Ting
f(A1) = p
f(A2) = q
p + q = 1
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

5. total number of alleles
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

6. 2 x number of MM + number of MN
p = f(M) =
2 x number of MM + number of MN
2 x total number =
2 x number of MM +
number of MN =
f(MM) + (1/2) f(MN)

7. 2 x number of NN + number of MN
q = f(N) =
2 x number of NN + number of MN
2 x total number =
2 x number of NN +
number of MN =
f(NN) + (1/2) f(MN)

8. 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:

9. 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.

10. The Hardy-Weinberg Principle
One locus with two alleles, A1 and A2
f(A1) = p
f(A2) = q
A1A1
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f( A1A1 ) = p x p = p2
f( A1A2 ) = p x q + p x q = 2pq
f( A2A2 ) = q x q = q2

11. A a f (A) = p f (a) = q AA f (AA) = p2 Aa f (Aa) = pq aa f (aa) = q2
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12. Alternative way to calculate… f(AA) = X f(Aa) = Y f(aa) = Z
Offspring AA Aa aa
Mating
Proportion
AA × AA
AA × Aa
AA × aa
Aa × AA
Aa × Aa
aa × AA
aa × Aa
aa × aa Ｘ2 XY XZ Y2 ＸY
Aa × aa YZ Z2
Total
(1/2)XY
(1/4)Y2
(1/2)Y2
(1/2)YZ
(X+Y+Z)2
[X+(1/2)Y]2
2[X+(1/2)Y][Z+(1/2)Y]
[Z+(1/2)Y]2
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13. 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

14. Offspring AA Aa aa
Mating
Proportion
AA × AA
AA × Aa
AA × aa
Aa × AA
Aa × Aa
aa × AA
aa × Aa
aa × aa Ｘ2 XY XZ Y2 ＸY
Aa × aa YZ Z2
Total
(1/2)XY
(1/4)Y2
(1/2)Y2
(1/2)YZ
(X+Y+Z)2
[X+(1/2)Y]2
2[X+(1/2)Y][Z+(1/2)Y]
[Z+(1/2)Y]2
National Taiwan University Chau-Ti Ting p2
2pq q2
Genotype frequencies do not change from generation after generation

15. 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

16. Testing for fit to Hardy-Weinberg Equilibrium
f(M) = (114 x ) / 2 x 200 = 0.76 = p
f(N) = (10 x ) / 2 x 200 = 0.24 = q
Expected frequencies
p2 2pq q2
Using X2 test of Goodness-of-fit to the Hardy-Weinberg propotions MM MN NN
Observed #
Expected freq.
Expected #
Chi-square
(O-E)2/E
114 p2
0.5776
0.020
115.52 76
2pq q2
0.0576
72.96
11.52
0.127
0.201
0.348
SUM 10
200
1.0
0.3648
National Taiwan University Chau-Ti Ting

17. Testing for fit to Hardy-Weinberg EquilibriumMM MN NN
Observed #
Expected freq.
Expected #
Chi-square
(O-E)2/E
152 p2
0.5776
11.52
115.52
2pq q2
0.0576
72.96
SUM 48
200
1.0
0.3648
National Taiwan University Chau-Ti Ting
72.96
200

18. Natural Selection — the differential reproduction of genetically distinct individuals or genotypes within a population.
Fitness () — a measure of the individual’s ability to survive and reproduce.
The evolutionary success of an individual is determined not by it absolute fitness but by its relative fitness.
M. Prakash
M. Prakash
M. Prakash

19. The Hardy-Weinberg Rule
One locus with two alleles, A1 and A2
f(A1) = p
f(A2) = q
A1A1
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f( A1A1 ) = p2
f( A1A2 ) = 2pq
f( A2A2 ) = q2

20. p2  +2pq  +q2  pq  +q2 
Fitness
p2  2pq  q2 
A1A1
A1A
AA
Genptype p2
2pq q2
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p2  +2pq  +q2 
pq  +q2 
qt+1 =

21. pq [p (– )+q (– )]
p2  +2pq  +q2 
pq  +q2 
qt+1 =
∆q = qt+1 – qt=
pq [p (– )+q (– )]
p2  +2pq  +q2 

22. 1 0.5 Fitness p2  2pq  q2  A1A1 A1A AA Genptype (0.8)2
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? 1
0.5
Fitness
p2  2pq  q2 
A1A1
A1A
AA
Genptype
(0.8)2
2(0.8)(0.2)
(0.2)2
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x x 0.5
0.16 x x 0.5
q’ =
=0.12

23. p2 + 2pq s+ q2 t pq s+q2 t
For mathematical convenience,
we shall assign a relative fitness value of 1 = 1, = 1 + s, and = 1 + t 1
1+s
1+t
Fitness
p pq (1+s) q2 (1+t)
A1A1
A1A
AA
Genptype p2
2pq q2
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p2 + 2pq s+ q2 t
pq s+q2 t
qt+1 =

24. p2 + 2pq s+ q2 t
pq s+q2 t
qt+1 =
∆q = qt+1 – qt=
p2 + 2pq s+ q2 t
pq s+q2 t
– q
-pq (2 sq – tq – s)
p2 +2pq   +q2 (1+ t) =

25. At equilibrium, ∆q = 0
-pq (2 sq – tq – s)
= 0
p2 +2pq   +q2 (1+ t)
2 sq – tq – s = 0
q =
2 s – t s

26. 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 C
Negative Selection
National Taiwan University Chau-Ti Ting
M. Prakash
2007. Fundamentals of Gene Evolution, p Discovery Publishing House.

27. 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 B
Positive Selection
M. Prakash
2007. Fundamentals of Gene Evolution, p Discovery Publishing House.
National Taiwan University Chau-Ti Ting

28. 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
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M. Prakash
2008. Molecular Biology of Evolution, p Discovery Publishing House.

29. f (white)
0.5 G0 G1
0.6
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30. f (white)
0.6 G1 G2
0.8
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31. f (white)
0.8 G1 G2
0.4
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32. 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
(2N)!
i! (2N – i)!
pi q2N-i
Pi =

33. 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

34. the population size is fluctuated from generation to generation.
In general, Ne is smaller, sometimes much smaller, than N. Reduction in the effective population size can occurred if:
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;
the number of males involved in reproduction is different from the number of females;
the population size is fluctuated from generation to generation.
M. Prakash

35. 4 Nm Nf Nm + Nf n 1/N1 + 1/N2 + ····· + 1/Nn
In a population consist of Nm males and Nf females (N = Nm + Nf ), Ne is given by
M. Prakash
4 Nm Nf
Nm + Nf
Ne =
The long-term effective population size in a species for a period of n generations is given by
M. Prakash n
1/N1 + 1/N2 + ····· + 1/Nn
Ne =
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

36. 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 B
National Taiwan University Chau-Ti Ting

37. 1– e- 4Nesq 1– e- 4Nes 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
1– e- 4Nesq
1– e- 4Nes
P =

38. 1– e- 4Nesq
1– e- 4Nes
P =
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 2N
P =

39. 1– e- (2Nes/N) 1– e- 4Nes 1– e- 4Nes 1– e- 2s 1– e- 4Nes 2s
When s ≠ 0,
1– e- (2Nes/N)
1– e- 4Nes
P =
If the population size is equal to the effective population size,
1– e- 4Nes
1– e- 2s
P =
If the absolute value of s is small, we obtain
1– e- 4Nes 2s
P =
For positive value of s and large value of N,
P ≈ 2s

40. t = (2/s) ln(2N) generations
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
t = (2/s) ln(2N) generations
Dan Graur and Wen-Hsiung Li

41. Advantageous mutations
allele frequency
Time t
Neutral mutations
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42. 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 2N
K = 2Nu
= u

43. 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
K = 2Nu x 2s = 4Nsu
Dan Graur and Wen-Hsiung Li

44. 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

45. does not depend on an arbitrary delineation of polymorphism
An appropriate measure of genetic variability is the mean expected heterozygosity, or gene diversity. This measure
does not depend on an arbitrary delineation of polymorphism
can be computed directly from knowledge of the allele frequencies
is less affected by sampling effects. Gene diversity at a locus is defined as
Dan Graur and Wen-Hsiung Li

46. where hi is the gene diversity at locus i, and n is number of loci.
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

47. 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

48. Copyright Declaration
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Page
Open Clip Art Library bitterjug
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National Taiwan University Chau-Ti Ting P4
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
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P12, P14

49. Work Licensing Author/Source Page
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P16
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Natural Selection — the differential reproduction of genetically distinct individuals or genotypes within a population.
M. Prakash
2008. Molecular Biology of Evolution, p Discovery Publishing House.
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.
2007. Fundamentals of Gene Evolution, p Discovery Publishing House.
The evolutionary success of an individual is determined not by it absolute fitness but by its relative fitness
2012/04/30 visited
Modified from Open Clip Art Library danko
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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
In exceedingly rare cases, a mutation may…and it will be subjected to positive or advantageous selection.
P27
Allele frequency changes can also occur by chance, …in allele frequency due solely to chance effects is called random genetic drift.
2000. Fundamentals of Molecular Evolution., p. 47. Sinauer Associates, Inc. Sunderland, MA, USA.
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51. Work Licensing Author/Source Page
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P28
P29
P30
P31
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 Discovery Publishing House.
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.
P33
In general, Ne is smaller, sometimes much smaller, …fluctuated from generation to generation.
2008. Molecular Biology of Evolution, p Discovery Publishing House.
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52. 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 Discovery Publishing House.
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
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.
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.
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.
2008. Molecular Biology of Evolution, p Discovery Publishing House.
P37
In the following, we shall consider that the relative…respectively. Kimura (1962) showed that the probability of fixation of A2 is
2007. Fundamentals of Gene Evolution, p Discovery Publishing House.

53. 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 Discovery Publishing House.
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
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.
P40
National Taiwan University Chau-Ti Ting
P41
If neutral mutations occur at a locus in a diploid population of size … number of mutations by the probability of their fixation:
2000. Fundamentals of Molecular Evolution., p. 57. Sinauer Associates, Inc. Sunderland, MA, USA.
P42
the rate of substitution can also be obtained by …advantageous alleles. When s > 0
P43
A population is monomorphic at a locus if there exists …if two or more alleles coexist in the population.
P44

54. Work Licensing Author/Source Page
An appropriate measure of genetic …sampling effects. Gene diversity at a locus is defined as
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
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.
P46
Nucleotide diversity…alleles between the ith and jth types.
2000. Fundamentals of Molecular Evolution., p. 59. Sinauer Associates, Inc. Sunderland, MA, USA.
P47