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Effective population size Ne

v the number of individuals in a theoretically ideal population having the same magnitude of random genetic drift as the actual population

v for most natural populations, Ne is smaller than the census size ² often substantially smaller!

Factors affecting Ne

v changes in population size across generations

²  Ne = harmonic mean population size

²  e.g.,

€

Ne = t 1N0

+1N1

+ ...+ 1Nt−1

#

$ %

&

' (

€

Ne = 4 11000

+1

2000+120

+1

1000"

# $

%

& ' = 76.2

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Factors affecting Ne

v sex ratio (of breeding individuals) ² “bottleneck” through the less numerous sex

² e.g.,

Ne = 4NmN f

Nm + N f

Ne = 4×10×10010+100

= 36.4

Factors affecting Ne

v variation in “family size” (i.e., fitness) ² What is Ne in a haploid population in which

every individual leaves one offspring? ² Diploid?

² where k is family size for a pair of diploid individuals

² what is k for a stable population? ² poisson distribution: mean = variance

Ne =4Nt−1

var(k)+ k 2 − k

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Factors affecting Ne

v sex chromosomes ² the “population” size of sex chromosomes is

smaller than for autosomes

²  Ne for X-linked genes:

²  if Nm= Nf, then Ne = 3N/4 ² mtDNA and Y-chromosome Ne = N/4

²  assuming equal variance in RS between males and females €

Ne =9NmN f

4Nm + 2N f

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199

number of generations x 10

N = 1000; at time 0, A1 = A2 = 0.5; mt; nuc fre

quen

cy o

f A1

0

20

40

60

80

100

120

140

100200300400500600700800900100011001200130014001500160017001800190020002100

mtnuc

number of generations to fixation

num

ber o

f tria

ls

N = 1000; at time 0, A1 = A2 = 0.5

606 trials

2000+ generations

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Factors affecting Ne

v population structure with gene flow ² population subdivision maintains relatively

greater genetic diversity (slows the process of drift to fixation in the overall population)

² …where N is the population size in each of D demes and m is the migration rate between demes

Ne=ND 1+ 14Nm

!

"#

$

%&

Factors affecting Ne

v breeding sex ratio v fluctuation in population size v variance in family size (or # offspring) v number of gene copies

² sex chromosomes, mtDNA

v population structure

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Fig. 3.4 Random genetic drift in 107 Drosophila populations; 8 males, 8 females each generation

Why different from ideal expectations?

“Identical by Descent”

v concept used to derive theoretical

expectations for Ne v Hartl & Clark develop the concept using a

structured population model (Fig. 3.9)

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v Fig. 3.9

Overall average allele frequency stays the same but heterozygosity declines

Fig. 3.5 Predicted distributions of allele frequencies in replicate populations of N = 16

Fig. 3.9 (last slide) is showing the exact same process as this figure

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“Identical by Descent”

v what is the probability that two randomly sampled alleles are identical by descent (i.e., “replicas of a gene present in a previous generation”)? ²  Wright’s “fixation index” F

v at the start of the process (time 0), “declare” all alleles in the population to be unique or unrelated, Ft = 0 at t = 0

v  in the next generation, the probability of two randomly sampled alleles being copies of the same allele from a single parent = 1/(2N), so…

€

or

Ft =1- 1−12N

#

$ %

&

' ( t

“Identical by Descent”

€

Ft =12N

+ 1− 12N

#

$ %

&

' ( Ft−1

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€

or

Ft =1- 1−12N

#

$ %

&

' ( t

“Identical by Descent”

€

Ft =12N

+ 1− 12N

#

$ %

&

' ( Ft−1

= probability that alleles are copies of the same gene from the immediately preceding generation plus the probability that the alleles are copies of the same gene from an earlier generation

assuming F0 = 0

compare to: mean time to fixation for new mutant = ~4N

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Random genetic drift in 107 Drosophila populations; 8 males, 8 females each generation

Ht = 1− 12Ne

"

#$

%

&'

t

H0

≈ H0e−t2Ne

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Deriving the population size effect

v Start here:

v Finish here:

v other effects on Ne are similarly based on the logic of identity by descent

€

Ft =12N

+ 1− 12N

#

$ %

&

' ( Ft−1

€

Ne = t 1N0

+1N1

+ ...+ 1Nt−1

#

$ %

&

' (

(Hartl & Clark: pp. 116-122)