Mathematical Genetics
LMS Durham Symposium July 2004
The Ewens sampling formula and
related formulas: combinatorial proofs,
extensions to variable population size
and applications to ages of alleles
Robert C. GRIFFITHS
University of Oxford
Sabin LESSARD
Université de Montréal
1
THEoRETTCALPOPULATTONBrOLocy 3, 87-172 (1972)
The Sampling
Theoryof Selectively
NeutralAlleles*
W. J. EwnNsr
Department oJ Zoology, Uniaertity of Texas at Austin, Austin,
Texas, TBZ\2
Received August 17, 7971
DEDICATED TO THE MEMORY OF KEN KOJIMA
In this paper a beginning is made on the sampling theory of neutral alleles.
That is, we consider deductive and subsequently inductive questions relating
to a sample of genes from a selectively neutral locus. The inductions concern
estimation, confidence intervals and hypothesis testing. In particular the test of
the hypothesis that the alleles being sampled are indeed selectively neutral will
be considered. In view of the large âmount of data currently being obtained by
electrophoretic methods on allele frequencies and numbers, and the current
interest in the possibility of extensive "non-Darwinian"
evolution, such a
sampling theory seems necessary. However, a large number of unsolved
problems in this area remain, a partial listing being given towards the end of
this paper.
MenrEuerrcal
TnEony
A number of quantities will be consideredin this paper and it is useful to
gather together the notations that will be used consistently throughout. W'e
define.
N - number of individuals in the parent (diploid) population (normally
unknown)
N, :
effectivesize of parent population (normally unknown)
:
u
mutation rate to entirely new alleles (normally unknown)
z : number of individuals in sample (taken in generation/)
K - number of alleles in the population in generation I (an unknown
random variable)
À : nurnber of different allelesobservedin the sample (a realizedvalue of
a random variable)
* Supportedby USPHSGrantGM-15769.
t Current address: Mathematical Research Center, University of 'Wisconsin, Madison,
WI 53706. Permanent address: Department of Mathematics, La Trobe University,
Bundoora 3083, Victoria, Australia.
87
@ l97Z by AcademicPress,Inc.
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,F.
POPULATION GENETICS THEORY
TIIE PAST AND THE FUTURE
\V.J. Ewens
Department of Mathematics
Monash UniversitY
Clayton, Victoria 3168
Australia
and
DePartrnent of BiologY
University of PennsYlvania
PhiladelPhia, PA 19104
U.S.A.
ABSTRACT. Classical population genetics ttreory was largely directed lolv-ardsprocessesrelating to the
future. present ft*ry, ùy'contrast, focuses on the past, and in particular is motivated by the desire to
make inferences aboui'thé evolutionary processeswhich have led to the presently observed patterns and
nature of genetic variation. There are inany connections between the classical prospective Ft9.y ld.the
n.* r"t ofoctive theory. However, the retrospective theory introduces ideas not appearing in the classical
theory, particularly those concerning the ancestry of the glnes in a sample or in the entire population- It
also iitrbduces tw-oimportanr ne* disttibutions into ttre scientific literature, namely the Poisson-Dirichlet
and the GEM: theseàre important not only in population genetics, but also in a very wide range in
science and mathemarics. Sôme of these are diicussed. Population genetics theory has been grgatly
à*i.tr"tUy the introduction of many new conceptsrelating to the past evolution of biological populations.
1.
Introduction
These notes are based on lectures given to an audience consisting of biologists,
statsticians and mathematicians,and ttrey reflect the breadth of interests of the participants.
I have prefened to seek out connectionsand analogiesbetween these disciplines rather
than to pursue any topic in depth, to sho'whow questionsof interest to geneticistshave led
to mathematical developmentsin areas quite different from biology, and how in tum
various mathematical developments lead to a more complete understanding of the
evolutionary process.
The title does not imply an ambitious attempt to give an overview of population
genetics theory. Rather, it can be interpreted in two different and more specific ways.
First, it is intended to suggestthe view that even the most recent researchhas its origins in,
and often borrows results from, the classical theory. Secondly, it reflects the view (the
closing theme in Ewens (1979)) that the direction of interest in population geneticstheory
is changing from the prospective to the retrospective. The classical theory, aiming to
prove the validity of Darwinian evolution as a Mendelian process, was prospective and
177
S. lzssard (ed.), Matlunutical and Statistical Devetoprnentsof Evolwionary Theory, 177-227
@ 1990by Kluwer Acalemic Publishers.
EWENS’ FORMULA
n sampled genes
n!
n1 !···nk !
assignments of k types
1
× b !···b
if types are unlabelled with bj being
n!
1
the number of types represented j times
×n! orders of loss backward in time by mutation or coalescence, with the convention that
one of the genes at random is lost when two
coalesce
2
θ
× i(θ+i−1)
if the ith lost is the last of its type
j−1
or i(θ+i−1)
if it is the jth last of its type, for
i = 1, ..., n, since we have the rates
•iθ/2 for mutation among i genes
•i(i − 1)/2 for coalescence among i genes
•θ/2 for loss by mutation of a given gene
•(j − 1)/2 for loss by coalescence of a given
gene with one of j − 1 genes of the same type
The multiplication of all terms gives
n!
b
1 1 ···nbn
1
θk
· b !···b
·
n ! θ(θ+1)···(θ+n−1)
1
for the probability of having k types with bj
types represented j times.
3
WATTERSON’S FORMULA
(KINGMAN’S FORMULA IF θ = 0)
n labelled genes traced back to m ancestral
genes of types 1, ..., m
n1 · · · nm possible ancestral genes
×(n − m)! orders of loss of the younger genes
j−1
θ
× i(θ+i−1)
or i(θ+i−1)
for i = m + 1, ..., n,
and this gives
Qk
n
!
l=1 l
l=m+1 (nl −1)!
i=m+1 i(θ+i−1)
(n−m)!θk−m
Qn
Qm
for the probability of having nl given genes of
type l for l = 1, ..., m, m + 1, ..., k.
4
VARIABLE POPULATION SIZE
WITH AGE-ORDERED TYPES
TYPE 1 BEING THE OLDEST
AND TYPE k THE YOUNGEST
n!
n1 !···nk !
assignments of k types
Q
P
P
× kl=1 nl · ( lν=1 nν − il ) · · · ( lν=1 nν − il+1 + 2)
(ai) orders of loss with il genes remaining just
before type l is lost by mutation for l = 1, ..., k
(j−1)/λ(T )
θ
i
× i[θ+(i−1)/λ(T
or
i[θ+(i−1)/λ(Ti)]
i )]
given a time back of loss Ti when i genes remain and then a population size λ(Ti) relative
to the present size, for i = 1, ..., n, which gives
k−1
P
³Qθ
´ i aiE
k
l=1 nl
(
Qk
Qn
l=2 λ(Til )
)
i=2 [θλ(Ti )+i−1]
5
LADDER INDICES AND HEIGHTS
IN AN URN MODEL
The probability of having age-ordered frequencies n1, ..., nk in a sample given mutation events
when i1, ..., ik genes remain is
Qk
Pm
k
Y
( ν=1 nν − im)!
l=2 (il − 1) ·
Pm−1
(n − 1)!
(
m=1
ν=1 nν − im + 1)!
This is also the probability that the successive
maxima obtained by drawing balls labelled from
1 to n, at random and without replacement,
have increments n1, ..., nk given that the successive maxima occur at the drawing numbers i1, ..., ik .
6
POPULATION DISTRIBUTION OF
AGE-ORDERED FREQUENCIES
As the sample size goes to infinity, the relative
frequency of the lth oldest type, nl /n, given
i1...., ik , is distributed as
Q∞
ξl−1 ν=l (1 − ξν )
where all ξl are independent and have densities
(il+1 − 1)(1 − z)il+1−2, 0 < z < 1.
Moments and Laplace transforms of these frequencies can be deduced and then, in particular, the probability that the oldest type in the
sample is the oldest in the population
¸
³ ´ ·Q
Pn
θλ(T
)
j
k
∞ (1 −
k−1 n E
(−1)
·
j=2
k=1
k
k+j−1 θλ(Tj )+j−1 )
All this is consistent with the GEM distribution for a population of constant size.
7
GENEALOGY OF A DERIVED TYPE
A derived type represented r times in a sample
of size n is lost by mutation at time back Tm+1
when there remain m+1 genes with probability
P
pm+1/( n−r
j=1 pj+1 ) where pm+1 is
Ã
(n−1)!θ
r
·
n−m−1
r−1
Ã
n−1
r
!
!
½
·E
Qn λ(Tm+1)
i=2 [θλ(Ti )+i−1]
¾
Then the distribution of the age of the derived
type can be studied, by conditioning on m + 1,
under assumptions about the population size
and be approximated under conditions such as
low θ, large n or small r, and similarly for the
coalescence times of the genes of the derived
type.
8
CONCLUSION
Combinatorial arguments coupled with the coalescent approach may simplify the proofs of
known results and lead to some new ones.
9