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Copyright Ó 2007 by the Genetics Society of America
DOI: 10.1534/genetics.106.068874
An Exact Nonparametric Method for Inferring Mosaic Structure
in Sequence Triplets
Maciej F. Boni,*
,†,1
David Posada
‡
and Marcus W. Feldman
†
*Stanford Genome Technology Center, Palo Alto, California 94304,
†
Department of Biological Sciences, Stanford University, Stanford,
California 94305 and
‡
Department of Biochemistry, Genetics and Immunology, University of Vigo, Vigo 36310, Spain
Manuscript received November 27, 2006
Accepted for publication March 18, 2007
ABSTRACT
Statistical tests for detecting mosaic structure or recombination among nucleotide sequences usually
rely on identifying a pattern or a signal that would be unlikely to appear under clonal reproduction.
Dozens of such tests have been described, but many are hampered by long running times, confounding of
selection and recombination, and/or inability to isolate the mosaic-producing event. We introduce a test
that is exact, nonparametric, rapidly computable, free of the infinite-sites assumption, able to distinguish
between recombination and variation in mutation/fixation rates, and able to identify the breakpoints and
sequences involved in the mosaic-producing event. Our test considers three sequences at a time: two par-
ent sequences that may have recombined, with one or two breakpoints, to form the third sequence (the
child sequence). Excess similarity of the child sequence to a candidate recombinant of the parents is a
sign of recombination; we take the maximum value of this excess similarity as our test statistic D
m,n,b
.We
present a method for rapidly calculating the distribution of D
m,n,b
and demonstrate that it has comparable
power to and a much improved running time over previous methods, especially in detecting recombina-
tion in large data sets.
M
OSAIC structure exists in a nucleotide sequence
if different segments of the sequence descend
from different ancestors. A nucleotide sequence can be
a mosaic of other sequences as a result of recombina-
tion or gene conversion; mosaic structure in bacterial
DNA can also result from transduction, transformation,
or conjugation, which are collectively referred to as hor-
izontal gene transfer. The detection of mosaic structure
has received much attention over the past two decades
as a result of both a proliferation of sequence data and
leaps in computing power, which together have allowed
for the inference of multiple ancestral contributions to
a nucleotide sequence. The biological questions at the
source of this recent attention range from interest in
the evolution of pathogens (Awadalla 2003; Moya
et al. 2004; Wilson et al. 2005) and the characterization
of linkage disequilibrium in large genomes (Pritchard
and Przeworski 2001; Ardlie et al. 2002; Gabriel et al.
2002) to theoretical questions about clonality and
the definitions of clonal and nearly clonal organisms
(Maynard Smith et al. 1993; Halkett et al. 2005). For
reviews on the methods and results in this field, see
Posada et al. (2002) and Stumpf and McVean (2003).
Maynard Smith (1999) recognized that the contin-
uum between completely clonal and freely recombining
organisms naturally gives rise to two distinct problems:
determining whether recombination occurs and mea-
suring its frequency. In this investigation, we focus on
the former. Detecting recombination usually involves
searching groups of sequences for candidate recombi-
nants or recombination signals and testing whether
these represent statistically significant departures from
expectation under a null hypothesis of no recombina-
tion. Dozens of statistical tests have been developed
(Stephens 1985; Sawyer 1989; Balding et al. 1992;
Karlin and Brendel 1992; Maynard Smith 1992;
Takahata 1994; Sneath 1995; Goss and Lewontin
1996; Jakobsen and Easteal 1996; Grassly and Holmes
1997; Maynard Smith and Smith 1998; Sneath 1998;
Awadalla et al. 1999; Crandall and Templeton 1999;
Holmes et al. 1999; Maynard Smith 1999; Wall 1999;
Gibbs et al. 2000; Martin and Rybicki 2000; Worobey
2001; Bruen et al. 2006) and evaluated (Wall 2000;
Brown et al. 2001; Posada and Crandall 2001; Wiuf
et al. 2001; Posada 2002) in this endeavor, none of
which has yet emerged as the single standard test to
be used for identifying recombination. In addition to
testing for the existence of recombination, certain
methods are also able to locate recombination break-
points and, sometimes, the parent sequences involved in
the recombination event, although the latter can be
quite difficult. Methods that do not focus on parent
sequences and breakpoints usually rely on detecting
a recombination signal—for example, a phylogenetic
incongruence or an excess of homoplasies—but may
have trouble isolating the actual recombination event,
1
Corresponding author: Stanford Genome Technology Center, 855
S. California Ave., Palo Alto, CA 94304.
E-mail: maciek@charles.stanford.edu
Genetics 176: 1035–1047 ( June 2007)
which entails identifying particular parent sequences
that recombined at particular breakpoints to form a
recombinant offspring sequence.
Some methods (Takahata 1994; Robertson et al.
1995; Crandall and Templeton 1999; Holmes et al.
1999; Gibbs et al. 2000; Martin and Rybicki 2000;
Martin et al. 2005) perform tests on three sequences at
a time, which allows them to posit candidate parent
sequences and candidate breakpoints. The proposed
arrangement is then tested with a likelihood analysis, by
visual detection of similarity in different sequence
regions, or against a null distribution that would be ex-
pected under clonal evolution. The most common among
these triplet tests—the Chimaera method (Posada
and Crandall 2001; Posada 2002), which is based on
a x
2
-statistic (Maynard Smith 1992), and the Martin–
Rybicki (MR) binomial distribution test (Martin and
Rybicki 2000)—identify unusually high levels of se-
quence similarity inside a predefined window or on
either side of a candidate breakpoint. We also take this
approach by introducing a simple and intuitive statistic
describing how identity varies along a sequence within a
sequence triple. Our test statistic D
m,n,b
is discrete and
nonparametric. Describing its distribution, in principle,
would require a computing time that grows exponentially
with the number of informative sites (a subset of the poly-
morphisms) in the given sequence triple; to avoid this
costly brute-force computation, we introduce a method
for computing probabilities and P-values in polynomial
time. Our method is memory intensive but very fast:
computation of exact P-values takes seconds on a per-
sonal computer when there are ,250 informative sites
in the proposed sequence triple.
Our triplet test represents an advance over Chimaera
and the MR method in that we eliminate the need for a
sliding window, use a nonparametric statistic, and in-
troduce a computation scheme that is exact and orders
of magnitude faster. In evaluating our method’s power
to detect recombination in sequence triplets, we find
that we always have higher power than the MR method
and comparable power to Chimaera. In repeated appli-
cations of our triplet test to data sets with more than
three sequences, we show that our method is among the
most powerful of 16 previously tested methods.
STATISTICAL TESTS
We begin with three homologous sequences of the
same length. The relationship among these three se-
quences is similar in practice to the relationship formu-
lated by Crandall and Templeton (1999, pp. 166–167)
among networks of sequences. From our three sequen-
ces, we designate one as the child sequence and inves-
tigate whether it could be a recombinant of the other
two sequences, which we call parent sequences. We first
present the simple case of a single-breakpoint recombi-
nant but later focus on the more interesting and realistic
case of a double-breakpoint recombinant. Considering
our sequence triple, we ask whether one can reject the
null hypothesis that the evolutionary history among the
three sequences was completely clonal.
We call our parent sequences p and q and our child
sequence c. For sequence length L, we can represent
our three sequences as vectors of nucleotides: p ¼ (p
1
,
p
2
, ..., p
L
), q ¼ (q
1
, q
2
, ..., q
L
), and c ¼ (c
1
, c
2
, ..., c
L
).
A single-breakpoint recombinant between the parent
sequences at position l can be denoted
ðpqÞ
l
¼ðp
1
; ...; p
l
; q
l11
; ...; q
L
Þ;
with 0 # l # L.
Writing jp qj as the number of nucleotide differ-
ences between sequences p and q, we say that the most
likely recombination breakpoint l minimizes j(pq)
l
cj ,
the number of differences between the observed child
sequence and a possible recombinant of the parent se-
quences. If this candidate recombinant is much closer
(than either parent) to the child sequence, then we may
have reason to believe that the evolutionary history of
sequence c is better explained by a recombination or a
gene conversion than by strictly clonal reproduction. If
the candidate recombinant (pq)
l
is only slightly closer
than the parents to the child sequence, then the can-
didate recombinant’s additional sequence similarity may
simply be an accident of how mutations accumulated on
either side of the breakpoint l. Assessing whether the
locations of the mutations (relative to the breakpoint)
are significantly nonrandom is the foundation for the
maximum x
2
-test (Maynard Smith 1992), the Chimaera
method (Posada and Crandall 2001; Posada 2002),
the exact test based on the binomial distribution sug-
gested by Martin and Rybicki (2000), and the heuristic
test suggested by Crandall and Templeton (1999); it
is also the focus of our analysis.
We introduce a nonparametric statistic slightly differ-
ent from the ones above, but one that is more direct at
detecting potential mosaics. Let
d
NoRec
¼ minfjp cj; jq cjg ð1Þ
be the minimum distance from the child to either of the
parents, and let
d
Rec;1
¼ min
0#l#L
fjðpqÞ
l
cjg ð2Þ
be the minimum distance from the child to a candidate
recombinant of the parents (including the boundary
case recombinants, which are just the parents them-
selves); the subscript ‘‘1’’ indicates that there is just one
breakpoint in the recombinant. Then, we define
D
1
¼ d
NoRec
d
Rec;1
: ð3Þ
The quantity D
1
describes the difference, between
clonal evolution and nonclonal evolution, in the num-
ber of mutations needed to describe the evolutionary
1036 M. F. Boni, D. Posada and M. W. Feldman
history between the child and the closer parent; by
nonclonal evolution we mean, here, an evolutionary his-
tory that allows for a single recombination event with a
single breakpoint. Clearly D
1
$ 0, and even if there had
truly been no recombination or gene conversion among
the sequences, a particular sequence triple could give the
appearance of recombination with a high value of D
1
if,
by chance, the pattern of mutations was such that the
left side of the child sequence appeared to be more
closely related to parent p and the right side appeared to
be closer to parent q. The distribution of this recombi-
nation signal D
1
under the null hypothesis of clonal
reproduction can be easily computed (see next section).
The difference in (3) is affected only by informative
sites of the sequence triple (p, q, c). For our purposes,
we define informative sites as those where the child’s
nucleotide matches exactly one of the parents’ nucleo-
tides. Uninformative sites are sites where (i) all three
sequences agree, (ii) all three sequences differ, or (iii)
the parents have matching (i.e., identical) nucleotides
that differ from the child’s. Our definition of informa-
tive sites is identical to that used in the Chimaera method
and to the sister groups defined by Takahata (1994).
Suppose that there are m informative sites where p
and c match and n informative sites where q and c
match. The quantity D
1
in (3) is then more precisely
defined as D
m,n,1
. Under the null hypothesis of clonal
evolution among sequences p, q, and c, D
m,n,1
is a
random variable that describes the maximum number
of mutation events one could ‘‘explain away’’ by recom-
bining p with q at a single breakpoint.
A two-breakpoint recombinant of sequences p and q
can be described by
ðpqpÞ
ij
¼ðp
1
; ...; p
i
; q
i11
; ...; q
j
; p
j11
; ...; p
L
Þ;
where i # j. Letting
d
Rec;2
¼ min
0#i#j#L
fjðpqpÞ
ij
cjg; ð4Þ
we define
D
m;n;2
¼ d
NoRec
d
Rec;2
; ð5Þ
where m and n are again the numbers of the two types of
informative sites.
D
m,n,1
and D
m,n,2
are random variables that describe
single-breakpoint and double-breakpoint recombina-
tion signals, respectively, under the null hypothesis of
no recombination. They are discrete random variables
with range 0 #D
m,n,b
# min {m, n}, where b is the number
of breakpoints. Observed D-quantities can be quickly
calculated [in OðLÞ-time, for any b] from sequence data,
and the null hypothesis of clonal evolution can be
rejected if they are too high. In the next two sections, we
review what is already known about the distribution of
D
m,n,1
and present a method for calculating the distri-
bution of D
m,n,2
.
Single-breakpoint recombinant: Consider a sequence
triple (p, q, c) with m informative sites where p and c
match and n informative sites where q and c match.
Moving left to right across the informative sites on the
child sequence, we can assign each informative site a
letter based on probable ancestry (determined by the
parent to which it is identical) and obtain a sequence
such as PPPQPPPQQQQ, where a P denotes an in-
formative site at which the child sequence and parent
p share a nucleotide, and Q denotes an informative site
at which the child sequence and parent q share a nu-
cleotide. Under the null hypothesis of clonal reproduc-
tion, the placement of P’s and Q’s in the sequence
should be completely random; i.e., each of the (m 1
n)!/(m!n!) possibilities has equal probability. In the
example sequence above, it appears that the P’s cluster
toward the left side of the sequence and the Q’s to the
right side; therefore, this sequence may be a true (sta-
tistically significant) recombinant.
This sequence of P’s and Q’s is most easily visualized
as a random walk on a set of axes where P is a step up and
Q is a step down. This is not a traditional random walk
since the number of up steps is known to be m, the
number of down steps is known to be n, and the only
randomness is the order in which they appear. After s
steps, the height X
s
of the random walk is distributed
quasi-hypergeometrically [the quantity (X
s
1 s)/2 is
distributed hypergeometrically]. The probability of
being at height h after s steps, when jhj # s and 0 # s #
m 1 n,is
PðX
s
¼ hÞ¼
m
s 1 h
2
!
n
s h
2
!
m 1 n
s
1
if h 1 s is even; P(X
s
¼ h) ¼ 0ifh 1 s is odd. This type of
finite stochastic process can be called a hypergeometric
random walk (HGRW). HGRWs have been previously
analyzed in the probability literature in the form of
ballot problems (Feller 1957), wherein one candidate
in an election receives m votes, the second candidate
receives n votes, and the order in which the votes are
counted is of interest. We denote a hypergeometric
random walk with m up steps and n down steps by the
random variable H
m,n
. Given data, we refer to an ob-
served walk diagrammed from the informative sites of
a sequence triple; examples of observed walks dia-
grammed from real data are in Figure 1.
Given our sequence triple with m 1 n informative sites
and allowing only one breakpoint in a putative re-
combinant, the observed value D
m,n,1
is related to the
maximum height of the walk diagrammed from the
informative sites of sequences p, q, and c, by the relation
D
m;n;1
¼ max H
m;n
1 min f0; n mg:
Using results from ballot theory (Barton and Mallows
1965) and gambling problems (Whitworth 1901,
prop. 39, pp. 116–117), it can be shown that
Exact Tests for Mosaic Structure 1037
PðD
m;n;1
$ kÞ¼
m 1 n
n 1 k
m 1 n
n
when m # n
m 1 n
m 1 k
m 1 n
n
when m . n
;
8
>
>
>
<
>
>
>
:
ð6Þ
or equivalently that
Pðmax H
m;n
$ kÞ¼
m 1 n
n 1 k
m 1 n
n
: ð7Þ
From the observed maximum height of the dia-
grammed walk of the informative sites of a sequence
triple, the null hypothesis of clonal reproduction can be
rejected at the level P as calculated in (6) or (7). This
is implicitly a one-tailed test with rejection of the null
hypothesis of clonal evolution when the observed D
m,n,1
(or the maximum height of the observed walk) is large
relative to m and n. An HGRW with a statistically im-
probable maximum height will have its up steps clus-
tered toward the beginning (left side) of the walk and its
down steps clustered toward the end (right side) of the
walk. This is precisely a mosaic pattern in a nucleotide
sequence: a child sequence having ancestry in p in the
left-hand side of its sequence and ancestry in q in the
right-hand side of its sequence.
Double-breakpoint recombinant: Identifying mosaics
with two breakpoints is the more relevant and interest-
ing problem since in long sequence regions, converted
tracts of DNA or horizontally transferred segments will
usually have both breakpoints present. Identification of
two breakpoints also allows for the removal of the hor-
izontally acquired segment; the remaining segment(s)
can then be tested again for clonal evolution, and multi-
breakpoint mosaics could be inferred by repeating such
a process. Note that the two-breakpoint case subsumes
the one-breakpoint case since a one-breakpoint recom-
binant can be viewed as having two breakpoints where
one breakpoint is on the end of the sequence.
Again, considering only the informative sites of the se-
quence triple (p, q, c) and viewing their ordering in the
context of a hypergeometric random walk, the quantity
D
m,n,2
can be calculated by identifying the maximum
descent (md) of the walk constructed from the arrange-
ment of informative sites. Letting X
s
be the height of
H
m,n
at step s, the maximum descent is defined as
md H
m;n
¼ max
0#s#t#m 1n
ðX
s
X
t
Þ;
and it can be shown that
PðD
m;n;2
¼ kÞ¼
Pðmd H
m;n
¼ kÞ whenm $ n
Pðmd H
m;n
¼ k 1 n mÞ whenm , n
:
Statistical theory underlying a general class of statistics
based on partial sum processes (Siegmund 1988; Karlin
et al. 1990), change-point problems (Siegmund 1986),
and maximal segmental sums (Karlin and Dembo
1992) provides asymptotic approximations that could
be applied to calculate the probability that md H
m,n
is
large relative to m and n. Notably, Lemmas 3 and 4 in
Siegmund (1988) and Theorems 2 and 3 in Hogan and
Siegmund (1986) contain the appropriate constructions
to approximate probabilities of maximum descents in
HGRWs. In the theory on ballot problems, the maximum
descent of an HGRW represents the maximum lead
change (in one direction only) when counting ballots
in a two-candidate election; as far as we are aware, this
distribution has not been calculated with the combina-
torial methods and reflection techniques usually applied
Figure 1.—Observed walks diagrammed from the informa-
tive sites of sequence triples. (A) The walk is diagrammed
from Neisseria data (from the fourth row of Table 1). (B) The
walk is diagrammed from influenza data (from the first row
of Table 2). The circles indicate the beginning and end of
the maximum descent in each walk, and in both cases the be-
ginning of the maximum descent is also the maximum height
of the walk. The dotted line in each diagram denotes the ex-
pected location of the hypergeometric random walk. The
shaded areas in each diagram show the range of 100 simu-
lated HGRWs.
1038 M. F. Boni, D. Posada and M. W. Feldman
in ballot problems. Below, we provide a method for cal-
culating this distribution exactly.
We use the shorthand x
m,n,k
¼ P(md H
m,n
¼ k ), and for
j, k $ 0, we define
y
m;n;k;j
¼ P ðmd H
m;n
¼ k \ min H
m;n
¼jÞ:
Then,
x
m;n;k
¼
X
k
j¼0
y
m;n;k;j
; ð8Þ
and the y-probabilities can be obtained by solving the
recursions
j ¼ 0 : y
m;n;k;0
¼
m
m 1 n
½ y
m1;n;k;1
1 y
m1;n;k;0
ð9Þ
j . k $ 0 : y
m;n;k;j
¼ 0 ð10Þ
j ¼ k . 0 : y
m;n;j;j
¼
n
m 1 n
½y
m;n1; j 1; j1
1 y
m;n1; j ; j1
ð11Þ
k . j . 0 : y
m;n;k;j
¼
m
m 1 n
y
m1;n;k;j11
1
n
m 1 n
y
m;n1;k;j1
;
ð12Þ
with boundary conditions
y
m;0;k;j
¼
1 for k ¼ j ¼ 0
0 otherwise
ð13Þ
y
0;n;k;j
¼
1 for k ¼ j ¼ n
0 otherwise
ð14Þ
y
m;n;0;0
¼
1 for n ¼ 0
0 otherwise
ð15Þ
y
m;n;k;j
¼ 0 when k . n or k , n m ð16Þ
y
m;n;k;j
¼ 0 when j . n or j , n m: ð17Þ
All of the above recursions can be proved with a simple
but careful first-step analysis of the random walk H
m,n
.
Below, the random variables H
m1,n
and H
m,n1
refer to
the subwalk of H
m,n
that starts after the first step of H
m,n
.
As an example, recursion (11) can be proved by not-
ing that the event {md H
m,n
¼ j \ min H
m,n
¼j} implies
that the first step of H
m,n
must be down (X
1
¼1) and
that md H
m,n1
must be either j or j 1. Thus,
Pðmd H
m;n
¼ j \ min H
m;n
¼jÞ
¼ P ðmd H
m;n
¼ j \ min H
m;n
¼j
\ X
1
¼1 \ md H
m;n1
¼ jÞ
1 P ðmd H
m;n
¼ j \ min H
m;n
¼j
\ X
1
¼1 \ md H
m;n1
¼ j 1Þ: ð18Þ
In both summands of the right-hand side of (18), the
last three events imply the first. We can rewrite the right-
hand side of (18) as
Pðmin H
m;n
¼j \ X
1
¼1 \ md H
m;n1
¼ jÞ
1 P ðmin H
m;n
¼j \ X
1
¼1 \ md H
m;n1
¼ j 1Þ:
ð19Þ
The events
fmin H
m;n
¼j \ X
1
¼1g
[fmin H
m;n1
¼ðj 1Þ\X
1
¼1gð20Þ
are identical; one occurs if and only if the other occurs.
Using this identity, we substitute into (19) and obtain
PðminH
m;n1
¼ðj 1Þ\X
1
¼1 \ mdH
m;n1
¼ jÞ
1 P ðmin H
m;n1
¼ðj 1Þ\X
1
¼1 \ mdH
m;n1
¼ j 1Þ:
ð21Þ
By independence of the first step X
1
¼1 from the
subwalk H
m,n1
, this becomes
PðX
1
¼1ÞPðminH
m;n1
¼ðj 1Þ\mdH
m;n1
¼ jÞ
1 P ðX
1
¼1ÞP ðminH
m;n1
¼ðj 1Þ\mdH
m;n1
¼ j 1Þ;
ð22Þ
which is
n
m 1 n
½ y
m;n1;j; j 1
1 y
m;n1; j1; j1
:
The other recursions can be proven similarly, and the
boundary cases (13)–(17) are easily verifiable.
The computation time for any y
m,n,k,j
is bounded
above by mn
3
, which is the maximum table size required
in memory to solve recursions (9)–(12); k 1 1 y-values
must be computed to calculate x
m,n,k
via Equation 8. On
a single 3-GHz processor with access to 2 GB RAM, the
worst-case x-calculations for 250 informative sites take
,3 sec; most x-probabilities can be calculated in ,1 min
for up to 400 informative sites. All calculations pre-
sented in this article (except where noted) were done
on a 3.2-GHz Linux laptop with 1 GB of RAM and 750
MB of virtual memory. C11 source code for calculating
the x- and y-variables is available from the authors.
For a given sequence triple in which we observe a
D
m,n,2
¼ k, with a P-value of
P
n
j¼k
x
m;n;j
we can reject the
null hypothesis of completely clonal reproduction in
favor of an evolutionary history that includes a two-
breakpoint recombination event.
APPLICATIONS
The following are two simple examples that use the
distributions D
m,n,1
and D
m,n,2
to test for mosaic struc-
ture among three sequences.
Exact Tests for Mosaic Structure 1039
Neisseria: We considered a classic example from the
genus Neisseria and applied our tests to its argF gene,
which is widely believed to have mosaic structure as a
result of horizontal gene transfer among different spe-
cies (Zhou and Spratt 1992; Grassly and Holmes
1997; Husmeier and McGuire 2003). Zhou and Spratt
(1992)foundregionsofclusteredpolymorphisminacom-
parison between the argF genes of a Neisseria meningitidis
isolate and a N. gonorrhoeae isolate and deduced that this
region of clustered polymorphisms had likely ancestry in
the species N. cinerea (since N. meningitidis and N. cinerea
were nearly identical in this region). The authors noted
that there were two regions in N. meningitidis that could
have arisen by horizontal gene transfer, one of which
might have been the result of variation in mutation rates
or fixation rates (usually called ‘‘rate variation’’). Further
studies (Grassly and Holmes 1997; Husmeier and
McGuire 2003) suggested that additional regions in
the argF gene may have arisen by recombination.
We used three of the Neisseria sequences, one of each
species, from the studies mentioned above (GenBank
accession nos. X64860, X64866, and X64869; 787 nt in
length) and tested whether there is any parent–parent–
child relationship among them that lends support to
one sequence being a mosaic of the other two. Table 1
shows that of the six possible arrangements, one has a
highly significant (P ¼ 10
12
) single-breakpoint recom-
bination signal, while the other five have none. This oc-
curs because the first 202 nucleotides of N. meningitidis
cluster significantly with N. cinerea (3.5% divergent,
while N. meningitidis and N. gonorrhoeae are 13% di-
vergent in this region) and the final 585 nucleotides of
N. meningitidis cluster significantly with N. gon orrhoeae
(2.9% divergent, while N. meningitidis and N. cinerea are
15% divergent in this region). This indicates that the
first 202 nucleotides of N. meningitidis have probable
ancestry in N. cinerea while the final 585 nucleotides of
N. meningitidis have probable ancestry in N. gonorrhoeae,a
view that is supported by the last two columns of Table 1,
which allow for two breakpoints in the child sequence’s
composition but support a mosaic structure almost iden-
tical to the one-breakpoint case.
Influenza A: Gibbs et al. (2001) found evidence for
recombination in the hemagglutinin gene of the 1918
‘‘Spanish’’ influenza strain, but their results were later
refuted by Worobey et al. (2002) and Strimmer et al.
(2003). We reanalyzed the five sequences presented by
Gibbs that were the candidate recombiners and recom-
binants: two swine sequences (A/swine/Iowa/15/30 and
A/swine/Wisconsin/1/61) and three human sequen-
ces (A/South Carolina/1/18, A/Kiev/59/79, and A/
Alma Ata/1417/84), where the last two numbers in the
sequence names indicate the year the sequence was iso-
lated. In Table 2 we show the results obtained using our
D-method on the significant relationships presented in
Figure 1 of Gibbs et al. (2001).
With any type of analysis, detecting recombination in
ancient influenza sequences is a challenge because of the
high mutation rates in RNA viruses. A recombination
that occurred 90 years ago would have its recombination
signal obscured by mutations that accumulated after the
recombination event. The relationship specified by the
first two rows in Table 2, for example, requires a mini-
mum of 104 years of evolution after the posited recom-
bination event (61 years between the South Carolina
and the Kiev strains and 43 years between the South
Carolina and Wisconsin strains). Our five influenza se-
quences are on average 10% divergent (range: 2.4–
18.3%), which means that detecting recombination events
should be easy if the events were recent but difficult if
they were ancient. On the timescale of influenza evolu-
tion, the hypothesized recombination events in Table 2
would be quite ancient.
Nevertheless, our method does detect weak recombi-
nation signals in the 1918 and 1984 human influenza
strains. It is important to note that we are performing
TABLE 1
Mosaic structure in Neisseria argF gene
pqcNull
Observed
maximum P-value
Observed maximum
descent P-value
N. men. N. cin. N. gon. H
84,6
78 1 2 0.30
N. cin. N. men. N. gon. H
6,84
0 1 78 1
N. gon. N. cin. N. men. H
84,32
52 1 19 8.93 3 10
11
N. cin. N. gon. N. men. H
32,84
19 1.42 3 10
12
71 2.50 3 10
11
N. gon. N. men. N. cin. H
6,32
0 1 26 1
N. men. N. gon. N. cin. H
32,6
26 1 2 0.60
The first three columns show a candidate parent–parent–child configuration that is tested for recombination; the fourth col-
umn shows the null distribution for the ordering of informative sites in the given sequence triple. The 1-breakpoint recombinant
in the fourth row can be achieved with three different breakpoints, at positions 201, 202, and 203 (a breakpoint at position 201
indicates a breakpoint after the 201st nucleotide). The 2-breakpoint recombinant in the fourth row can be achieved with 66 dif-
ferent pairs of breakpoints: the first is always one of 202–204 while the second is one of 742–759/784–787. The 2-breakpoint
recombinant in the third row can be achieved with 21 different pairs of breakpoints: the first is always one of 0–6 while the second
is one of 202–204. N. men., N. meningitidis; N. cin., N. cinerea; N. gon., N. gonorrhoeae.
1040 M. F. Boni, D. Posada and M. W. Feldman
post hoc tests on previously analyzed sequences for which
Gibbs et al. (2001) obtained statistically significant re-
combination signals. Given these same five sequences
without any a priori knowledge about their relationships,
we might compute P-values for all 60 possible parent–
parent–child relationships among these sequences. The
last two columns of Table 2 show which of these com-
parisons would still be significant after a Dunn–S
ˇ
ida´k
correction for 60 comparisons. The Dunn–S
ˇ
ida´k cor-
rection is, of course, extremely conservative, especially
since the D-values from our comparisons are positively
correlated. A more accurate correction for multiple com-
parisons would take into account that we have multiple
significant results. Using an exact binomial test, the
probability under H
0
that $3 of 60 comparisons would
be significant at the 10
3
level is P ¼ 3.3 3 10
5
.Tobe
slightly more conservative, we could say that the two
P-values in rows 1 and 2 of Table 2 that are ,10
3
are in
fact manifestations of the same arrangement of strains
(Kiev, Wisconsin, and South Carolina); then, the prob-
ability that $2 of 60 comparisons would be significant
at the 10
3
level is P ¼ 1.7 3 10
3
.
Although it has been long believed that intragenic
(homologous) recombination does not occur in influ-
enza (Kilbourne 1978), the occurrence of nonhomol-
ogous recombination (Khatchikian et al. 1989; Orlich
et al. 1994; Suarez et al. 2004) together with the data pre-
sented by Gibbs suggests that homologous recombina-
tion in influenza may be possible. However, as pointed
out by Worobey et al. (2002), the observed substitution
pattern in the influenza hemagglutinin can also be
explained by within-sequence rate variation that varies
across the different branches of the phylogeny (lineage-
specific rate variation). Using pairwise comparisons
among human sequences of the influenza A hem-
agglutinin, Worobey et al. described the HA1 region
(nucleotide sites 151–920) as evolving more quickly than
the HA2 region (sites 1–150 and 921–1695) in humans.
If the opposite can be shown to be true for swine hem-
agglutinin sequences—that the HA2 evolves more quickly
than the HA1—then the detected mosaicism in the 1918
human influenza hemagglutinin would be best explained
by lineage-specific rate variation. This type of rate var-
iation has also been called heterotachy (Lopez et al.
2002), and it was first introduced in the context of a
changing set of concomitantly variable codons by Fitch
and Markowitz (1970). It has been suggested that, for
influenza A viruses, heterotachous or lineage-specific
rate variation is a more likely evolutionary history than
an intragenic recombination event (E. C. Holmes, per-
sonal communication).
SIMULATIONS
In addition to our D-method’s theoretical appeal of
being exact and nonparametric we show that it has the
practical advantages of speed, power, and a low false-
positive rate.
Power and false positives: We compared the power
and false-positive rates of our D-method to the 14 methods
TABLE 2
Mosaic structure in influenza A hemagglutinin gene
Observed
maximum
Observed
maximum
descent
Dunn–S
ˇ
ida´k
pqcNull P-value P-value Maximum md
1979h 1961s 1918h H
148,159
7 0.45 46
a
4.00 3 10
4
0.03
1961s 1979h 1918h H
159,148
39
b
7.85 3 10
4
30
c
5.22 3 10
3
0.05 NS
1979h 1961s 1930s H
94,218
0 1 124 1
1961s 1979h 1930s H
218,94
124 1 10
d
9.06 3 10
3
NS
1979h 1961s 1984h H
92,220
0 1 128 1
1961s 1979h 1984h H
220,92
128 1 12
e
8.88 3 10
4
0.06
The first three columns refer to the five influenza sequences mentioned in the Influenza A section. Here, the sequences are
referred to by year and whether the sequence is human (h) or swine (s). The last two columns show the Dunn–S
ˇ
ida´k corrected
P-values given that without any knowledge about which sequences are recombinant, 60 comparisons would have to be made to test
all parent–parent–child combinations. The breakpoint descriptions listed in footnotes a–e refer to a gapped alignment of length
1778 nt; 80 positions are gapped.
a
There are 90 pairs of breakpoints that result in a maximum descent of 46 units in the diagrammed walk from these three
sequences. The first breakpoint is in position 242–247, while the second one is in 953–955/971–982.
b
The maximum height of 39 for this triple can be attained by 15 different breakpoints, at positions 952–954/970–981.
c
There are 30 pairs of breakpoints that result in a maximum descent of 30 for this sequence triple. The first breakpoint is in
position 953–955/971–982; the second breakpoint is either in position 1653 or in position 1654.
d
There are 45 pairs of breakpoints that result in a maximum descent of 10 for this sequence triple. The first breakpoint is in
position 953–955/971–982; the second breakpoint is in position 1049–1051.
e
There are 9 pairs of breakpoints that result in a maximum descent of 12 for this sequence triple. The first breakpoint is in
position 953–955; the second breakpoint is in position 1049–1051.
Exact Tests for Mosaic Structure 1041
evaluated in Posada and Crandall (2001). Figure 2
duplicates the conditions of Figure 1 in Posada and
Crandall (2001); in addition, two of the methods de-
scribed by Carvajal-Rodrı
´
guez et al. (2006) are in-
cluded in the top two rows of comparisons in Figure 2.
Power and false-positive rates are tested for different
values of the population-genetic parameter u ¼ 4N
e
mL,
where N
e
is the effective population size, m is the per site
per generation mutation rate, and L is the sequence
length. Power is tested across different values of the re-
combination parameter r ¼ 4N
e
rL, where r is the per site
per generation recombination rate. False-positive rates
are tested for different levels a of rate variation (a is the
shape parameter of a fixed-mean G-distribution of evo-
lutionary rates as in Yang 1996) since, as noted in the
Neisseria and influenza examples, statistical tests for re-
combination can confound recombination and varia-
tion in mutation/fixation rates.
The left column of Figure 2 shows the power of 14 (or
16) other methods as well as the power of our D-method,
which was determined as follows. Each data point cor-
responds to 100 simulated sequence sets with 10 se-
quences in each set (details in Posada and Crandall
2001). In a set of 10 sequences, there are 720 unique
parent–parent–child arrangements; the quantity D
m,n,2
was calculated for each of these 720 triplets and the
P-value associated with that quantity was computed with
recursions (9)–(12). The minimum of these 720 P-values
was corrected with a Dunn–S
ˇ
ida´k correction and then
reported as the P-value for rejecting clonal evolution in
that 10-sequence set. This procedure was implemented
in C11 as a command-line Linux program called
3SEQ; source code is available from the authors. The
number of sets in which clonal evolution could be re-
jected at the 0.05 level was reported as the power of our
D-method. The false-positive rates in the right-hand
column of Figure 2 were computed in the same way.
Figure 2 shows that for a high enough mutation rate,
our method is among the most powerful available for
detecting recombination. For the sequence sets where
u ¼ 10, the mean pairwise distance within each set of
10 sequences ranges from 1 to 30 nt. Using D
m,n,2
to test
for recombination requires a minimum of nine in-
formative sites to reject clonality at the 0.05 level; when
correcting with a Dunn–S
ˇ
ida´k correction for 720 com-
parisons, a minimum of 20 informative sites is needed.
For this reason, our method has low power for data sets
with little polymorphism. For the tested parameter com-
binations, our false-positive rate is at most 2% and
among the lowest of all methods tested. It is important
to note that some of the more powerful methods in the
left-hand column had high false-positive rates in the
right column. The plots in supplemental Figure S1
(http://www.genetics.org/supplemental/) show the ra-
tios of power to false-positive rate for the 16 methods
from Figure 2.
Supplemental Figure S2 at http://www.genetics.
org/supplemental/ shows an additional false-positive
Figure 2.—Power and
false-positive comparisons
to the 14 methods tested in
Posada and Crandall
(2001). The top four graphs
include two additional
LPT methods described in
Carvajal-Rodrı
´
guez et al.
(2006). The graphs in the left
column plot power under dif-
ferent recombination rates,
while the right-hand column
shows false-positive rates
when there is variation in mu-
tation rates but recombina-
tion is not present; a ¼ ‘
means that there is no rate
variation, while lower values
of a indicate higher rate vari-
ation. The red line shows the
power and false-positive rate
of D
m,n,2
in detecting recom-
bination. The gray lines show
the power and false-positive
rates of 14 (or 16) other
methods. a ¼ ‘ in the left
column; r ¼ 0intheright
column.
1042 M. F. Boni, D. Posada and M. W. Feldman
analysis in data sets generated with autocorrelated muta-
tion rates (from Figure 5c of Bruen et al. 2006); our
false-positive rate was never .3.2% for these data sets.
Supplemental Figure S3 at http://www.genetics.org/
supplemental/ shows a power analysis under conditions
with population growth, using the simulated data from
Figure 4 of Bruen et al. (2006). D
m,n,2
is quite powerful
under a scenario of population growth (as long as se-
quence diversity is high enough), and it retains very
high power even when the recombination parameter
r is small.
Since our statistical test is designed for sequence trip-
lets we perform an additional power analysis that focuses
exclusively on detecting recombination in sets of three
sequences. We compare D
m,n,2
to three other common
statistical tests designed to identify recombination in
sequence triplets (a total of eight methods were tested
of which the three most powerful are shown in Figure 3;
details of and results for all eight methods are in
the supplemental materials at http://www.genetics.org/
supplemental/). For each data point in Figure 3, the
program TREEVOLVE (Grassly et al. 1999) was used to
generate 100 replicates of three sequences with the given
population-genetic parameters, using the F84 model of
nucleotide substitution (Felsenstein and Churchill
1996) with p
A
¼ 0.4, p
C
¼ 0.2, p
G
¼ 0.1, p
T
¼ 0.3, and a
transition/transversion ratio of two. The black line in
Figure 3 denotes the power and false-positive rate of a
single-breakpoint version of Chimaera with exact P-value
computations (Posada and Crandall 2001; Spencer
2003), the gray line corresponds to the most recent ver-
sion of Chimaera (Chim-2006), and the blue line corre-
sponds to the Martin–Rybicki method with window size
30 nt and step size 1 nt.
For statistical identification of mosaic structure in
sequence triplets, our D-method is as powerful as the
most powerful methods available. All four methods in
Figure 3 have similar power and false-positive rates, with
Figure 3.—Power and false-positive comparisons with MR and Chimaera on sequence triplets. The red line shows power and
false-positive rates for D
m,n,2
. The black line shows the power and false-positive rates for Chim-Sp, a single-breakpoint no-window
Chimaera implementation (described on p. 14 of the supplemental materials of Posada and Crandall 2001) whose P-values were
calculated using the method of Spencer (2003). The gray line shows the power and false-positive rates of Chim-2006, a new Chi-
maera implementation with a sliding-window and sliding-breakpoint scheme; P-values were computed by permuting alignment
columns 1000 times. The blue line shows the power and false-positive rates for MR-30,1 (Martin–Rybicki method with window size
30 nt and step size 1 nt). The third column shows ratio of power to false-positive rate at a ¼ ‘. False-positive rates at a ¼ ‘ were
calculated with 1000 simulated triplets; all other data points were calculated with 100 simulated triplets. a ¼ ‘ in the left column; r
¼ 0 in the middle column.
Exact Tests for Mosaic Structure 1043
the distinguishing feature that Chimaera is the least con-
servative method, MR is the most conservative, and D
m,n,2
is somewhere in between. For u$50, D
m,n,2
has the best
combination of power and false-positive rate.
Speed: Table 3 shows the computation times of our
method compared to MR and Chimaera. Our method
has a clear advantage, especially in large data sets, since
P-values are simply read from memory once a table of
y
m,n,k,j
-values is built. For example, analysis of the influ-
enza data (Boni 2007) requires reading 29 million
P-values from memory, which is not a time-consuming
task for a 3.2-GHz processor. Likewise, computing exact
P-values using the method described by Spencer (2003)
is quite fast; this is slightly slower than our D-method
since a new table needs to be built for each P-value com-
putation. On the other hand, performing 14.5 million
sliding-window x
2
-computations on each of 1000 ran-
domized data sets (Chim-2006) or computing 9.6
million P-values from a binomial distribution for each
of 287 possible windows (MR-30,1) can be quite com-
putationally expensive.
Note that nontriplet methods can be much faster
than triplet methods. For example, analyzing the data
in Table 3 with F
w
(Bruen et al. 2006) takes seconds, but
the recombinant sequences cannot be isolated.
DISCUSSION
Comparison: Many statistical methods have already
been developed for detecting recombination from se-
quence data. The usual recombination signals that these
methods attempt to identify are (i) varying patterns of
sequence identity, (ii) phylogenetic incongruencies,
(iii) excess homoplasies, (iv) clustered polymorphism,
and (v) low linkage disequilibrium; our method is of
the first type. Here, we summarize the main similarities/
differences between and advantages/disadvantages of
our method and previous ones.
Most importantly, our method considers three se-
quences at a time using the appropriate mechanistic
framework in which to view mosaic structure: the exis-
tence of one sequence that is a mosaic of a second and
a third. Maynard Smith (1992) also acknowledged this
as the appropriate framework, although the test he de-
veloped is designed for two sequences. Maynard Smith’s
maximum x
2
-method was later reformulated as a proper
three-sequence problem and is now called maximum-
match x
2
or Chimaera (Posada and Crandall 2001;
Posada 2002). Takahata (1994) recognized that one
needed to look at a minimum of three sequences by
focusing on sites that support a particular sister-group
status where exactly two of three nucleotides agree. The
BOOTSCAN search method (Salminem et al. 1995) ex-
amines candidate recombinants to see how different
regions cluster with either of two parental sequences;
bootstrap support, rather than a significance test, pro-
vides a measure of reliability of the proposed clustering.
Recently, Martin et al. (2005) modified the BOOT-
SCAN method to search only sequence triples and to
find recombinants statistically using the binomial test
in Martin and Rybicki (2000). Finally, Holmes et al.
(1999) describe a phylogenetic method called LARD
that considers three sequences at a time and tests the
hypothesis of completely clonal evolution vs. the hypoth-
esis of clonal evolution for segments on either side of a
breakpoint; their problem is formulated similarly to
ours, the main difference being that their method fo-
cuses on phylogeny. It should be noted that some meth-
ods (Robertson et al. 1995; Gibbs et al. 2000) require
four sequences: three involved in a recombination event
and a fourth used as an outgroup.
The mechanistic three-sequence approach contrasts
with approaches that attempt to identify indirect signals
from sequence data, such as an excess of homoplasies
(Hudson and Kaplan 1985; Jakobsen and Easteal
1996; Maynard Smith and Smith 1998; Maynard
Smith 1999; Bruen et al. 2006) or a clustering of poly-
morphisms (Stephens 1985; Maynard Smith 1992;
Martin and Rybicki 2000) that would be indicative of
a recent recombination or gene conversion. While these
methods can be quite effective, one must keep in mind
that polymorphism clustering can be caused by se-
lection or mutational hotspots and that an excess of
homoplasies can be quite difficult to detect in rapidly
mutating organisms such as RNA viruses.
Our method has several technical advantages.
First, we do not use Monte Carlo methods to generate
TABLE 3
Computation times (last four columns) for computing recombination statistics and P-values in large data sets
Data set
Segregating
sites
No.
sequences P-value
D
m,n,2
(min) MR-30,1 Chim-Sp Chim-2006
Dengue E 618 69 3.3 3 10
5
2 86 min 24 min 100 hr
Human mtDNA 1079 262 4.6 3 10
3
6 180 hr 48 hr 550 days
Influenza HA 316 308 1 4 43 hr 9 hr 105 days
All times and estimates are for a single 3.2-GHz processor. Dengue data are serotype 2 from Holmes et al. (1999); human mi-
tochondrial DNA sequences are a subset of distinct strains from Kivisild et al. (2006); influenza seqeunces are New Zealand H3N2
isolates from 2000–2005 analyzed in Boni (2007). The P-value reported in this table is the minimum P-value (testing with D
m,n,2
)
from all comparisons in a data set, corrected with a Dunn–S
ˇ
ida´k correction.
1044 M. F. Boni, D. Posada and M. W. Feldman
P-values, which makes our P-value computations very fast.
Moreover, once a table is built in memory to calculate
a particular x
m,n,k
, successive P-values can simply be ex-
tracted from the table; this means that repeated ap-
plication of our D-tests is limited only by how quickly
the computer’s memory can be accessed. Monte Carlo
methods have the additional disadvantage that the pre-
cision of computed P-values is limited by the number
of permutations that can be done; this could be prob-
lematic in large data sets where precise P-values may be
needed to survive multiple-comparisons corrections. Sec-
ond, we avoid the widely used sliding-window approaches
(Salminem et al. 1995; Siepel et al. 1995; Grassly and
Holmes 1997; Lole et al. 1999; Martin and Rybicki
2000; Strimmer et al. 2003; Martin et al. 2005) that
require the user to define a window size at the scale at
which recombination is believed to have occurred. By
considering all possible breakpoints in expression (4),
we find the optimal ‘‘window size’’ that should be used
for inferring recombination in a particular sequence
triplet. This allows for the detection of recombinant seg-
ments at any scale.
By removing uninformative sites, our D-method
should not confound variation in mutation/fixation
rates with recombination; indeed, the middle column of
Figure 3 and supplemental Figure S2 at http://www.
genetics.org/supplemental/ show that even under high
rate variation our false-positive rate is at most 5% (and
usually ,3%). However, lineage-specific or heterota-
chous rate variation can, in the absence of recombina-
tion, produce the pattern that is meant to be rejected
by our D-distributions. Consider the tree in Figure 4.
Branch 1 connects the root to sequence p while branch
2 connects the q–c common ancestor to sequence q.
Differential environmental pressures on branches 1 and
2 can create the impression of mosaic structure. Sup-
pose that the organism, during its evolution along branch
1, experiences an environment where the right-hand
side of the sequence evolves rapidly and accumulates
many substitutions while the left-hand side is either con-
served or mutates neutrally. Suppose further that the
organism, during evolution along branch 2, experien-
ces an environment where the left-hand side of the se-
quence evolves rapidly and accumulates substitutions
while the right-hand side is conserved or mutates neu-
trally. Under this scenario of clonal evolution, where en-
vironmental pressure increases substitution rates in the
right part of the sequence on branch 1 and in the left
part of the sequence on branch 2, the resulting se-
quence triple (p, q, c) will give the appearance that a
recombination event occurred. In this case, the right
part of sequence c will be very similar to sequence q
while the left part will be very similar to sequence p. This
type of sequence identity in different sequence regions
is exactly what our D-statistics are designed to reveal.
While this combination of events may seem unlikely,
the influenza sequences described here may have under-
gone just such evolutionary pressures. A key component
in this scenario where mosaic structure is generated
without recombination is that the organism experiences
different selective environments on different branches
of its phylogeny.
General conclusions: We have introduced exact, non-
parametric statistical tests for identifying nucleotide se-
quence mosaic structure with one or two breakpoints.
Our test statistic is a function of a given sequence triple
where one sequence is hypothesized to be a recombi-
nant of the other two. Given a sequence triple, we calcu-
late the difference in proximity (to the child sequence)
between the closer parent sequence and the closest
candidate recombinant sequence. This difference is de-
noted D
m,n,b
—where m and n describe the numbers of
informative sites at which the child sequence clusters
with one or the other parent, and b denotes the number
of breakpoints allowed in a candidate recombinant—
and it is studied as a random variable under the null
hypothesis of clonal evolution. The distribution of D
m,n,1
has been described in the probability literature on bal-
lot problems, while the distribution of D
m,n,2
has been
approximated but not described exactly. With brute-
force methods, exact probabilities of the distribution of
D
m,n,2
would require exponentially growing computation
times that would become unmanageable once m 1 n .
35. To remedy this problem, we derive a set of recursive
equations to calculate the probability mass function
of D
m,n,2
in Oðmn
3
Þ-time. These calculations can be
performed in seconds on a single-processor personal
Figure 4.—Phylogenetic tree that shows a possible clonal
evolutionary history for the sequences p, q, and c. Mutations
occurring in branch 1 will result in an informative site of type
Q, while mutations occurring in branch 2 will result in an in-
formative site of type P. The distributions describing the prob-
ability that the mutations in branch 1 or 2 cluster on either
side of a breakpoint or between some pair of breakpoints
are those of D
m,n,1
and D
m,n,2
.
Exact Tests for Mosaic Structure 1045
computer (3 GHz, 2 GB RAM) as long as m 1 n , 250.
When 250 , m 1 n , 400, most computations are
equally quick although some may require additional
memory or the use of virtual memory.
Our method relies on deducing parent–child se-
quence identity for different parents in different se-
quence regions. If a recombination occurred between
sequences p and q to create the sequence c, then one
segment of sequence c should be more similar to parent
p while the remaining segment(s) of sequence c should
be more similar to parent q. If this pattern is statistically
significant—i.e., if it appears in the far right-hand tail of
the distribution of D
m,n,b
—we deduce that a recombina-
tion occurred.
Our D-method is among the most powerful available
for detecting recombination in sequence data, even in
highly recombinant data sets (generating data sets as in
Figure 2 with r ¼ 128, our method had 100% power for
u$50) or in data sets generated under conditions of
population growth (see supplemental Figure S3 at http://
www.genetics.org/supplemental/). For many of the sim-
ulated data sets in this article, D
m,n,2
appears to have the
best combination of power and low false-positive rate.
With comparable power to the best available methods,
the most immediate practical advantage of using D
m,n,2
over other methods is its speed in large data sets. As can
be seen in Table 3, computing P-values from D
m,n,2
can
be many orders of magnitude faster than other triplet
methods, depending on the number of sequences and
the amount of polymorphism in the data set. For N se-
quences, triplet methods will make on the order of N
3
comparisons, which for N . 1000 can be quite a large
number for a personal computer. For example, 1000 in-
fluenza sequences with a similar level of polymorphism
as in Table 3 would take 137 min to analyze with D
m,n,2
,
while 2000 sequences would take 18 hr. Fortunately, our
method (along with most triplet methods) is completely
parallelizable, which means that as sequence databases
grow we can take advantage of parallel computing to
search for recombinants in very large data sets. Note that
if we have a particular query sequence that we would like
to test for recombination, the number of comparisons is
of order N
2
.
Our choice of applications here represents only a
small sample of the clonal or nearly clonal sequences we
could analyze with our D-statistics. They would also be
quite useful in finding recombinants in human immu-
nodeficiency virus databases and in larger dengue virus
data sets and in analyzing the recently suggested re-
combinants in measles (Schierup et al. 2005). Human
mitochondrial DNA is generally believed to evolve
clonally, although the data set in Table 3 has quite
strong mosaic signals; a reanalysis of other mtDNA data
sets (Piganeau and Eyre-Wal ker 2004; Piganeau et al.
2004) would help determine whether recombination
occurred during the evolution of the mitochondrion.
For the influenza virus, our test could be used on whole
(concatenated) influenza genomes, as in Holmes et al.
(2005), to detect possible reassortment; hundreds of se-
quenced whole influenza genomes have already been
analyzed (Nelson et al. 2006) and thousands more have
been deposited in GenBank. As sequence databases
expand in the genomic era, the D-method presented
here could become one of the most efficient methods
for detecting recombination and finding recombinants
in large data sets.
We thank E. C. Holmes for many discussions especially on the rate
variation scenario for influenza; we thank T. C. Bruen for providing
data sets for power analysis and false-positive analysis; and we thank
N. A. Rosenberg, J. M. Macpherson, and J. Van Cleve for helpful
comments and suggestions. An anonymous editor pointed us to the
known result in Equation 7. This work was funded in part by National
Institutes of Health grants GM28016 (M.F.B., M.W.F.) and HG000205
(M.F.B.). D.P. is funded by grant BFU2004-02700 of the Spanish
Ministry of Education and Science and by the Ramo´n y Cajal program
of the Spanish government.
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Communicating editor: M. K. Uyenoyama
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