Download Effective population size N Factors affecting N

9/22/14 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
# 1
1
1 &
Ne = t %
+
+ ...+
(
N t−1 '
$ N 0 N1
²  e.g.,
€
" 1
1
1
1 %
Ne = 4 $
+
+
+
' = 76.2
#1000 2000 20 1000 &
€
1 9/22/14 Factors affecting Ne
v sex ratio (of breeding individuals)
² “bottleneck” through the less numerous sex
Ne = 4
Nm N f
Nm + N f
² e.g.,
Ne = 4 ×
10 ×100
= 36.4
10 +100
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?
Ne =
4N 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
2 9/22/14 Factors affecting Ne
v sex chromosomes
² the “population” size of sex chromosomes is
smaller than for autosomes
² 
Ne for X-linked genes:
Ne =
9N m N f
4N m + 2N f
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 €
3 9/22/14 N = 1000; at time 0, A1 = A2 = 0.5; mt; nuc
1
0.9
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
number of trials
100
80
mt
nuc
60
40
20
0
2000+
generations
10
0
20
0
30
0
40
0
50
0
60
0
70
0
80
0
90
0
10
00
11
00
12
00
13
00
14
00
15
00
16
00
17
00
18
00
19
00
20
00
21
00
frequency of A1
0.8
number of generations to fixation
N = 1000; at time 0, A1 = A2 = 0.5
4 9/22/14 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)
!
1 $
N e =ND #1+
&
" 4Nm %
² …where N is the population size in each of D
demes and m is the migration rate between
demes
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
5 9/22/14 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)
6 9/22/14 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
7 9/22/14 “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…
“Identical by Descent”
Ft =
1 #
1 &
+ %1−
(Ft−1
2N $ 2N '
or
t
#
1 &
Ft = 1- %1−
(
$ 2N '
€
€
8 9/22/14 “Identical by Descent”
Ft =
1 #
1 &
+ %1−
(Ft−1
2N $ 2N '
= 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
€
or
t
#
1 &
Ft = 1- %1−
(
$ 2N '
assuming F0 = 0
€
compare to:
mean time to
fixation for
new mutant =
~4N
9 9/22/14 Random genetic drift in 107 Drosophila populations;
8 males, 8 females each generation
t
"
1 %
H t = $1−
' H0
# 2N e &
≈ H 0e
−t
2 Ne
10 9/22/14 Deriving the population size effect
v Start here:
1 #
1 &
Ft =
+ %1−
(Ft−1
2N $ 2N '
v Finish here:
€
# 1
1
1 &
Ne = t %
+
+ ...+
(
N
N
N
$ 0
1
t−1 '
v other effects on Ne are similarly based on
the logic of identity by descent
(Hartl & Clark: pp. 116-122)
€
11