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Effective population size N e 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, smaller than the census size
N e
is
²   often substantially smaller!
Factors affecting N e v   changes in population size across generations
²   N e
= harmonic mean population size
N e
= t
#
%
$ N
1
0
+
1
N
1
+ ...
+
1
N t − 1
&
(
'
²   e.g.,
€
N e
= 4
1
1000
+
1
2000
+
1
20
+
1
1000
= 76.2
1

Factors affecting N e v   sex ratio (of breeding individuals)
²   “ bottleneck ” through the less numerous sex
N e
N
= 4
N m m
+
N
N f f
²   e.g.,
N e
= 4
10
100
10 + 100
= 36.4
Factors affecting N e v   variation in “family size” (i.e., fitness)
²
What is N e
in a haploid population in which every individual leaves one offspring?
²   Diploid?
N e
=
4 N t − 1 var( k ) + k 2
k
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
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Factors affecting N e v   sex chromosomes
²   the “ population ” size of sex chromosomes is smaller than for autosomes
²   N e
for X-linked genes:
N e
=
4
9 N m
N
N m
+ 2 f
N f
²   if N m
= N f
, then N e
= 3 N /4
²   mtDNA and Y-chromosome and females
N e
=
N /4
²   assuming equal variance in RS between males
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N = 1000; at time 0, A1 = A2 = 0.5; mt ; nuc
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199 number of generations x 10
606 trials
140
120
100
80
60
40
20 mt nuc
0
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100
2000+ generations number of generations to fixation
N = 1000; at time 0, A1 = A2 = 0.5
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Factors affecting N e 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)
N e
= ND
!
1+
"
1
4 Nm
$
&
%
²   …where N is the population size in each of D demes and demes m is the migration rate between
Factors affecting N e 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
?
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“ Identical by Descent ” v   concept used to derive theoretical expectations for N e v   Hartl & Clark develop the concept using a structured population model (Fig. 3.9)
6

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, F t
= 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/(2 N ) , so…
“ Identical by Descent ”
F t
=
1
2 N
+
1
1
2 N
F
t − 1 or
F t
= 1- 1
1
2 N
t
8

“ Identical by Descent ”
F t
=
1
2 N
+
1
1
2 N
F
t − 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
or
F t
= 1- 1
1
2 N
t assuming F
0
= 0
compare to: mean time to fixation for new mutant =
~4 N
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Random genetic drift in 107 Drosophila populations;
8 males, 8 females each generation
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H t
=
1
1
2 N e
t
H
0
H
0 e
− t
2 N e
10

v
Start here:
F t
1
=
2 N
+
1
1
2 N
F
t − 1 v
Finish here: v
N e
= t
$
#
%
N e
N
1
0
+
1
N
1
+ ...
+
N
1 t − 1
(
'
&
are similarly based on the logic of identity by descent
(Hartl & Clark: pp. 116-122)
€
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