9/27/14 Coalescent Theory
v the Wright-Fisher model considers
changes in the ideal population as time
moves forward
v coalescent theory (~1980+) looks
backwards in time
v how long does it take for k alleles to
coalesce to k - 1 alleles, then k - 2, k - 3,
…, and finally a single ancestral allele?
Stochastic elements of the coalescent
v alleles randomly sample their parents in the
previous generation
²  results in variation in offspring number
v sample of loci from the genome
²  different loci have different genealogical histories
v sample of alleles from the population
²  different samples of the same locus may have
different coalescent trees
v distribution of mutations on the genealogy
²  mutations allow estimates of coalescence times
1 9/27/14 Rosenberg & Nordborg 2002 Nat Rev Gen
2 9/27/14 Coalescent probabilities 1
v the present is time 0 (zero)
v probability that two alleles had a common
ancestor in generation 1
1
=
2N
v probability that two alleles did not have a
common ancestor in generation 1
"
1 %
= $1−
'
# 2N &
Coalescent probabilities 2
v probability that two alleles have still not
coalesced by generation t
t
( " 1 %+
−t ( 2 N )
= *1− $
'- ≈ e
) # 2N &,
3 9/27/14 Coalescent probabilities 3
v probability that two alleles had a common
ancestor in generation t+1
t
1 ) # 1 &,
1 −t ( 2N )
=
1−
≈
e
%
(
+
.
$
'
2N *
2N - 2N
€
probability of
coalescence in
generation t + 1
probability of NO
coalescence in
generations 1
through t
N = 20, k = 2
0.01
exact coal
exp coal
0.0095
Probability of
Coalescence
0.009
0.0085
0.008
0
5
10
15
20
Generation
4 9/27/14 Coalescent probabilities 4
v can we randomly choose a coalescence time
from the exponential distribution?
²  need to solve for t as a function of a random
variable from 0 to 1
PNC ≈ e
−t ( 2 N )
ln ( PNC ) ≈ −t ( 2N )
ln ( PNC ) × 2N ≈ −t
t ≈ − ln ( PNC ) × 2N
Coalescent probabilities 5
v what if we consider k alleles and not just 2?
v what is the probability that k alleles had k
distinct parental alleles the previous
generation?
k−1
"
i % " ( 2k ) %
Pr ( k ) = ∏ $1−
' ≈ $1−
'
#
&
2N
2N
#
&
i=0
5 9/27/14 N = 20 to 10,000, k = 6
1
0.95
0.9
0.85
Probability of
Non-coalescence
0.8
0.75
Exact Probability
0.7
Approximate
0.65
0.6
10
100
1000
10000
Population Size
Coalescent probabilities 6
v probability that k alleles do not coalesce for t
generations
) k ,
t
( ) t..
+ 2
+−
+* 2 N
" ( 2k ) %
PNC = $1−
' ≈e
# 2N &
.-
2N
t ≈ − ln ( PNC ) × $ k ' = − ln ( PNC ) ×
&&
))
% 2 (
4N
k ( k −1)
6 9/27/14 Coalescent probabilities 7
v probability that k alleles do not coalesce for t
generations, and then one pair coalesces to
" k $
give k - 1 alleles at t + 1 generations
( )e
= Pr ( k ) "#1− Pr ( k )$% ≈
t
k
2
( ) t((
' 2
'−
' 2N
#
(
%
2N
v distribution has mean and variance:
Mean =
€
4N
16N 2
2
generations Var =
2 generations
k(k −1)
[k (k -1)]
€
7 9/27/14 expected coalescence times for
k = 10 and N = 10,000
40000
time to k-1 alleles
number of generations
35000
cumulative generations
" 1%
t = 4N $1− '
# k&
30000
25000
20000
15000
10000
5000
0
10
9
8
7
6
5
4
3
2
k remaining alleles
8