Chapter 3 Dynamics of Genes in Populations ChauTi

Chapter 3 Dynamics of Genes in Populations Chau-Ti Ting ctting@ntu. edu. tw Unless noted, the course materials are licensed under Creative Commons Attribution-Non. Commercial-Share. Alike 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, A 1 and A 2 A 1 A 1 A 1 A 1 A 2 A 2 A 1 A 1 f(A 1) = p f(A 2) = q p+q=1 A 1 A 2 National Taiwan University Chau-Ti Ting A 1 A 1= 6 A 1 A 2= 3 A 2 A 2= 1 f(A 1) = (6 x 2 + 3) / 2 x 10 = 0. 75 f(A 2) = (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 number of MN = + 2 x total number = f(MM) + (1/2) f(MN) 6

2 x number of NN + number of MN q = f(N) = 2 x total number 2 x number of NN 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 p 2, 2 pq, and q 2. Wikipedia Source: http: //en. wikipedia. org/wiki/Hardy%E 2%80%93 Weinberg_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, A 1 and A 2 f(A 1) = p f(A 2) = q A 1 A 1 National Taiwan University Chau-Ti Ting f( A 1 A 1 ) = p x p = p 2 f( A 1 A 2 ) = p x q + p x q = 2 pq f( A 2 A 2 ) = q x q = q 2 10

A f (A) = p A AA f (A) = p f (AA) = p 2 a Aa f (a) = q f (Aa) = pq a f (a) = q Aa f (Aa) = pq aa f (aa) = q 2 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 Ｘ 2 XY XZ ＸY Y 2 YZ XZ YZ Z 2 Total (X+Y+Z)2 AA Ｘ 2 (1/2)XY (1/4)Y 2 [X+(1/2)Y]2 Aa aa (1/2)XY XZ (1/2)XY (1/2)Y 2 (1/2)YZ XZ (1/2)YZ 2[X+(1/2)Y][Z+(1/2)Y] (1/4)Y 2 (1/2)YZ Z 2 [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) = X 2 + (1/2)XY + (1/4)Y 2 = [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 = p 2 13

Offspring Mating Proportion AA × AA AA × Aa AA × aa Aa × AA Aa × Aa Aa × aa aa × AA aa × Aa aa × aa Ｘ 2 XY XZ ＸY Y 2 YZ XZ YZ Z 2 Total (X+Y+Z)2 AA Ｘ 2 (1/2)XY (1/4)Y 2 [X+(1/2)Y]2 Aa aa (1/2)XY XZ (1/2)XY (1/2)Y 2 (1/2)YZ XZ (1/2)YZ 2[X+(1/2)Y][Z+(1/2)Y] (1/4)Y 2 (1/2)YZ Z 2 [Z+(1/2)Y]2 National Taiwan University Chau-Ti Ting p 2 2 pq q 2 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) = p 2 f(Aa) = 2 pq f(aa) = q 2 f(A) = f(AA) + (1/2) f(Aa) = p 2 + (1/2) 2 pq = p 2 + pq = p (p + q ) =p 15

Testing for fit to Hardy-Weinberg Equilibrium MM MN NN SUM Observed # Expected freq. 114 p 2 0. 5776 76 2 pq 0. 3648 10 q 2 0. 0576 200 1. 0 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 p 2 2 pq q 2 Using X 2 test of Goodness-of-fit to the Hardy-Weinberg propotions 16

Testing for fit to Hardy-Weinberg Equilibrium MM MN NN SUM Observed # Expected freq. 152 p 2 0. 5776 0 2 pq 0. 3648 48 q 2 0. 0576 200 1. 0 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 Ø 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, A 1 and A 2 f(A 1) = p f(A 2) = q A 1 A 1 National Taiwan University Chau-Ti Ting f( A 1 A 1 ) = p 2 f( A 1 A 2 ) = 2 pq f( A 2 A 2 ) = q 2 19

Genptype A A p 2 A A 2 pq A A q 2 Fitness p 2 2 pq q 2 National Taiwan University Chau-Ti Ting qt+1 = pq +q 2 p 2 +2 pq +q 2 20

qt+1 = pq +q 2 p 2 +2 pq +q 2 ∆q = qt+1 – qt= pq [p ( – )+q ( – )] p 2 +2 pq +q 2 21

Consider a recessive allele, A 1, double the fitness of its carrier If f(A 2) = 0. 2 in this generation, what is it in next generation? Genptype A A A (0. 8)2 2(0. 8)(0. 2)2 Fitness 0. 5 p 2 2 pq 0. 5 q 2 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 p 2 A A 2 pq A A q 2 Fitness s t p 2 2 pq s q 2 t National Taiwan University Chau-Ti Ting qt+1 = pq s +q 2 t p 2 + 2 pq s + q 2 t 23

qt+1 = pq s +q 2 t p 2 + 2 pq s + q 2 t ∆q = qt+1 – qt= = pq s +q 2 t p 2 + 2 pq s + q 2 t –q -pq (2 sq – tq – s) p 2 +2 pq �����+q 2 (1+ t) 24

At equilibrium, ∆q = 0 -pq (2 sq – tq – s) =0 p 2 +2 pq �����+q 2 (1+ t) 2 sq – tq – s = 0 s q= 2 s–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 C A A Negative Selection A A AA 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 B A A Positive Selection A B A A 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 National Taiwan University Chau-Ti Ting 28

f (white) 0. 5 G 0 0. 5 G 1 0. 6 National Taiwan University Chau-Ti Ting 29

f (white) 0. 6 G 1 0. 6 G 2 0. 8 National Taiwan University Chau-Ti Ting 30

f (white) 0. 8 G 1 0. 8 G 2 0. 4 National Taiwan University Chau-Ti Ting 31

A diploid population with N individuals at any given locus, there are 2 N genes one locus with two alleles, A 1 and A 2 f(A 1) = p f(A 2) = 1 – p = q When 2 N gametes are sampled from the infinite gamete pool, the probability, Pi, that the sample contains exactly i alleles of genotype A 1 is given by the binomial probability function M. Prakash Pi = (2 N)! i! (2 N – i)! pi q 2 N-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/N 1 + 1/N 2 + ····· + 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 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, N e. M. Prakash In the following, we shall consider that the relative fitness of the three genotypes A 1 A 1, A 1 A 2, and A 2 A 2 are 1, 1+ s, and 1+ 2 s, respectively. Kimura (1962) showed that the probability of fixation of A 2 is M. Prakash P= 1– e- 4 Nesq 1– e- 4 Nes 37

P= 1– e- 4 Nesq 1– e- 4 Nes 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/(2 N). For a neutral mutation, i. e. , s = 0, the fixation probability 1 P= 2 N 38

When s ≠ 0, P= 1– e- (2 Nes/N) 1– e- 4 Nes If the population size is equal to the effective population size, P= 1– e- 2 s 1– e- 4 Nes If the absolute value of s is small, we obtain P= 2 s 1– e- 4 Nes For positive value of s and large value of N, P ≈ 2 s 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/(2 N), the mean conditional fixation time, t, was calculated by Kimura and Ohta (1969). For a neutral mutation, it is approximated by t = 4 N 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(2 N) 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: 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 2 Nu per generation. Since the probability of fixation for each of these mutations is 1/(2 N), 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 = 2 Nu =u 2 N 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 = 2 Nu x 2 s = 4 Nsu 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 Open Clip Art Library danko http: //openclipart. org/detail/4152/friendly-rabbit-by-danko 2012/04/30 visited Page P 3, P 18, P 33, P 44 P 4 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 p 2, 2 pq, and q 2. Wikipedia P 8 http: //en. wikipedia. org/wiki/Hardy%E 2%80%93 Weinberg_principle 2012/04/30 visited P 10, P 19 National Taiwan University Chau-Ti Ting P 11 National Taiwan University Chau-Ti Ting P 12, P 14 National Taiwan University Chau-Ti Ting 48

Work Licensing Author/Source Page P 16 National Taiwan University Chau-Ti Ting P 17 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=6 x 2 Ump. U 6 grs. C&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 P 18 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=T 9 a. Nqda. Xr. X 8 C&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 P 18 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=T 9 a. Nqda. Xr. X 8 C&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 P 18, P 44 Modified from Open Clip Art Library danko http: //openclipart. org/detail/4152/friendly-rabbit-by-danko 2012/04/30 visited P 20 National Taiwan University Chau-Ti Ting 49

Work Licensing Author/Source Page P 22 National Taiwan University Chau-Ti Ting P 23 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 P 26 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 P 27 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 P 28 50

Work Licensing Author/Source Page P 28 National Taiwan University Chau-Ti Ting P 29 National Taiwan University Chau-Ti Ting P 30 National Taiwan University Chau-Ti Ting P 31 National Taiwan University Chau-Ti Ting When 2 N gametes are sampled …i alleles of genotype A 1 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=6 x 2 Ump. U 6 grs. C&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 P 32 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 P 33 In general, N e 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=6 x 2 Ump. U 6 grs. C&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 P 34 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=T 9 a. Nqda. Xr. X 8 C&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 P 35 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=T 9 a. Nqda. Xr. X 8 C&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 P 35 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=T 9 a. Nqda. Xr. X 8 C&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 P 35 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 P 36 National Taiwan University Chau-Ti Ting The probability that a particular allele will become fixed in a population depends on…the effective population size, N e. M. Prakash 2008. Molecular Biology of Evolution, p. 215. Discovery Publishing House. http: //books. google. com. tw/books? id=6 x 2 Ump. U 6 grs. C&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 P 37 In the following, we shall consider that the relative…respectively. Kimura (1962) showed that the probability of fixation of A 2 is M. Prakash 2007. Fundamentals of Gene Evolution, p. 144. Discovery Publishing House. http: //books. google. com. tw/books? id=T 9 a. Nqda. Xr. X 8 C&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 P 37 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=T 9 a. Nqda. Xr. X 8 C&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 P 38 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=T 9 a. Nqda. Xr. X 8 C&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 P 38 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 P 40 P 41 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 P 42 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 P 43 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 P 44 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 P 45 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 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 P 46 P 47 54