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Copyright Ó 2010 by the Genetics Society of America
DOI: 10.1534/genetics.110.116756
Gene Duplication, Gene Conversion and the Evolution
of the Y Chromosome
Tim Connallon1 and Andrew G. Clark
Department of Molecular Biology and Genetics, Cornell University, Ithaca, New York 14853-2703
Manuscript received March 17, 2010
Accepted for publication May 31, 2010
ABSTRACT
Nonrecombining chromosomes, such as the Y, are expected to degenerate over time due to reduced
efficacy of natural selection compared to chromosomes that recombine. However, gene duplication,
coupled with gene conversion between duplicate pairs, can potentially counteract forces of evolutionary
decay that accompany asexual reproduction. Using a combination of analytical and computer simulation
methods, we explicitly show that, although gene conversion has little impact on the probability that
duplicates become fixed within a population, conversion can be effective at maintaining the functionality
of Y-linked duplicates that have already become fixed. The coupling of Y-linked gene duplication and
gene conversion between paralogs can also prove costly by increasing the rate of nonhomologous
crossovers between duplicate pairs. Such crossovers can generate an abnormal Y chromosome, as was
recently shown to reduce male fertility in humans. The results represent a step toward explaining some of
the more peculiar attributes of the human Y as well as preliminary Y-linked sequence data from other
mammals and Drosophila. The results may also be applicable to the recently observed pattern of
tetraploidy and gene conversion in asexual, bdelloid rotifers.
N
ONRECOMBINING chromosomes are often associated with genetic degradation and a loss of
functional genes, and nowhere is this pattern more
exaggerated than on the Y chromosome (Charlesworth
and Charlesworth 2000; Bachtrog 2006). However,
in addition to the more widely recognized pattern of
gene loss, genome sequences of mammals and Drosophila are also yielding evidence for Y-linked functional
gene gain followed by amplification of duplicate genes
(Skaletsky et al. 2003; Koerich et al. 2008; Carvalho
et al. 2009; Krsticevic et al. 2009; Hughes et al. 2010).
Duplication and retention of functional Y-linked gene
copies is somewhat surprising because evolutionary theory predicts an opposing pattern. First, to the extent
that gene duplicates are fixed via positive selection,
they are less likely to become fixed on nonrecombining
relative to recombining chromosomes (Otto and
Goldstein 1992; Clark 1994; Yong 1998; Otto and
Yong 2002; Tanaka and Takahasi 2009). Second,
regardless of whether Y-linked duplicates become fixed
via genetic drift or by natural selection, the actions of
Muller’s ratchet, genetic hitchhiking, and background
selection are expected to greatly increase the probability that Y-linked genes degenerate into nonfunctional
Supporting information is available online at http://www.genetics.org/
cgi/content/full/genetics.110.116756/DC1.
1
Corresponding author: Department of Molecular Biology and Genetics,
Cornell University, Biotechnology Bldg. (Room 227), Ithaca, NY 148532703. E-mail: [email protected]
Genetics 186: 277–286 (September 2010)
pseudogenes (Charlesworth and Charlesworth
2000; Bachtrog 2006; Engelstadter 2008).
The issue is more complex when one considers data
from the well-characterized human Y chromosome. A
majority of functional Y-linked genes are members of
duplicate gene pairs residing within large palindromes
and are almost exclusively testis expressed (Skaletsky
et al. 2003). In contrast to many of the single-copy genes
with X-linked homologs, members of Y-linked gene
families are apparently not degenerating, but rather
have become fixed and maintained over many millions
of years (Skaletsky et al. 2003; Yu et al. 2008). Although
Y chromosomes are not well characterized in other taxa,
currently available data suggest that duplication is a
common feature of Y chromosomes in other mammal
species as well as Drosophila (Rozen et al. 2003; Verkaar
et al. 2004; Murphy et al. 2006; Alföldi 2008; Wilkerson
et al. 2008; Krsticevic et al. 2009; Geraldes et al. 2010).
Thus, patterns of gene duplication and retention, for at
least a subset of Y-linked genes, may be a general rule of
Y chromosome evolution.
Another attribute of the mammalian Y appears to be
relevant for duplicate gene evolution. Comparative
analysis between humans and chimpanzees suggests
ongoing recombination between the gene duplicate
pairs that reside on the same Y chromosome. Such
‘‘intrachromosomal’’ recombination includes both nonreciprocal (gene conversion) and reciprocal exchange
(crossing over) between gene duplicate pairs (Rozen
et al. 2003; Lange et al. 2009). Gene conversion between
278
T. Connallon and A. G. Clark
the duplicates potentially maintains gene function by
counteracting stochastic forces of Y chromosome degeneration (Rozen et al. 2003; Charlesworth 2003;
Noordam and Repping 2006). The rationale behind
this hypothesis is subtle. As with other clonally inherited
chromosomes, each evolutionary lineage of the Y is physically coupled to, and its evolutionary fate is influenced
by, the presence of deleterious mutations. Mutationbearing lineages represent evolutionary dead ends unless
they can somehow remove or compensate for deleterious
mutations. Recombination between duplicates can ‘‘rescue’’ functionality via gene conversion between functional and nonfunctional copies.
On the other hand, double-strand DNA breaks, which
precede gene conversion events (Marais 2003), also
precede crossing over. Crossovers between Y-linked
genes can generate acentric and dicentric Y chromosomes, resulting in infertility and disruption of the
sex determination pathway (e.g., Repping et al. 2002;
Heinritz et al. 2005; Lange et al. 2009). Considering
both gene conversion and crossing over on the Y, recombination can be viewed as a factor that either constrains (via gene conversion) or promotes (via crossing
over) Y chromosome degeneration.
These observations concerning Y chromosome gene
content and recombination raise interesting questions
that have not been formally addressed by evolutionary
theory (but see the recent study by Marais et al. 2010).
First, what conditions favor the evolutionary invasion of
Y-linked gene duplicates, and does recombination influence the probability that duplicates eventually become
fixed within a population? Second, what affect does
recombination have on Y-linked fitness and the maintenance of functional duplicate genes? To address these
questions, we develop and analyze a series of populationgenetic models of Y chromosome evolution. We show
that, when direct selection on gene duplicates is weak,
biased gene conversion can increase, whereas crossing
over will decrease, their probability of fixation. For duplicates with larger fitness effects, the probability of fixation
is largely independent of Y-linked recombination. Finally,
gene conversion has a major impact on the retention of
functional Y-linked genes that are already fixed within the
population and maintains multiple gene copies with or
without selection favoring these duplicates.
develop and analyze a diffusion approximation and
perform stochastic simulations to examine the probability that a rare gene duplicate eventually becomes
fixed within a population of small size.
Invasion of a new gene duplicate: Consider a single
Y-linked locus with a functional allele, A, and a nonfunctional allele, a. Mutation from A to a occurs at rate u
per generation and there is no back mutation. By
introducing a duplication of the locus, the population is
expanded to include five genotypic classes: the original
single-copy classes (A and a), those with two functional
gene copies (AA), those with one functional and one
nonfunctional copy (Aa), and those with two nonfunctional copies (aa). As in the single-locus case, transitions
between states (AA / Aa or aA; Aa or aA / aa) can occur
by mutation, at rate of u per locus; because there are now
two loci, the mutation rate per chromosome is 2u.
For Y chromosomes carrying duplicates, recombination (crossing over and gene conversion) can potentially occur between loci. Throughout our analysis, we
examine cases where recombination occurs at a rate of d
per paralog pair, per generation. The probability that a
single recombination event is a crossover, which generates an abnormal (sterile) Y chromosome (e.g.,
Repping et al. 2002; Heinritz et al. 2005; Lange et al.
2009), is equal to the constant c . The remainder of
recombination events (1 c) represent gene conversion
events between duplicate pairs. Gene conversion involving Aa or aA individuals yields AA or aa sperm at rate
b and 1 b, respectively. Thus, b can be viewed as a
biased gene conversion parameter, where the functional
copy A preferentially replaces the nonfunctional a
whenever b . 0.5 (there is no bias when b ¼ 0.5).
Compared to individuals with two functional gene
copies, individuals with zero functional copies suffer a
fitness reduction of s, while those with one functional
copy suffer a reduction of sh, where h is equivalent to a
dominance coefficient. Complete masking of a nonfunctional allele occurs when h ¼ 0, and there is no
direct fitness benefit of carrying two vs. one functional
gene. Partial masking occurs when 1 . h . 0; in such
cases, there is a fitness benefit of having two functional
copies. Genotypes, genotypic fitness, and zygotic frequencies are described in Table 1.
MODEL AND RESULTS
TABLE 1
Gene conversion and the invasion of new gene
duplicates: We first consider conditions favoring the
evolutionary invasion of new Y-linked duplicate genes at
low initial frequency within the population. Deterministic invasion dynamics are described for a two-locus
model, and it is shown separately that the two-locus
model characterizes duplicate gene invasion conditions
on a Y chromosome carrying an arbitrary number of
genes (see supporting information, File S1). We then
Parameterization for the gene duplicate invasion model
Genotype
AA
Aa, aA
A
aa
a
Abnormal Y
Frequency
x11
x10
x1
x00
x0
xs
Fitness
1
1
1
1
1
sh
sh
s
s
0
Gene Conversion and Y Evolution
For a sequence of events of (i) birth, (ii) selection,
(iii) mutation, (iv) recombination, and (v) random
mating (and ignoring factors of u2), the frequency
change of each genotype, per generation, is given by
the following six recursions,
x11
ð1 2uÞð1 dcÞ
x119¼
w
x11
x10 ð1 shÞ
2udð1 cÞb 1
ð1 uÞdð1 cÞb
1
w
w
x109 ¼
x009 ¼
x11
x10 ð1 shÞ
2uð1 dÞ 1
ð1 uÞð1 dÞ
w
w
x11
x10 ð1 shÞ
2udð1 cÞð1 bÞ 1
uð1 dcÞ
w
w
x10 ð1 shÞ
x00 ð1 sÞ
ð1 uÞdð1 cÞð1 bÞ 1
ð1 dcÞ
1
w
w
xs 9 ¼
x11
x10 ð1 shÞ
x00 ð1 sÞ
dc 1
dc 1
dc
w
w
w
x1 9 ¼
x1 ð1 shÞ
ð1 uÞ
w
@l 1 d 1 Oðd 2 Þ 1 1 dð2b 1Þ;
@d d¼0
d¼0
ð1 2uÞð1 dcÞ 1 2udð1 cÞb 1 ð1 shÞð1 uÞð1 dÞ
l¼
2ð1 shÞð1 uÞ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
u fð1 2uÞð1 dcÞ 1 2udð1 cÞb 1 ð1 shÞð1 uÞð1 dÞg2
t
4ð1 2uÞð1 dcÞð1 dÞð1 shÞð1 uÞ
:
1
2ð1 shÞð1 uÞ
ð1aÞ
Selection favors the invasion of a duplicate when the
leading eigenvalue is greater than one (Otto and Day
2007). The magnitude of the leading eigenvalue also
represents the strength of selection acting in favor of a rare
duplicate gene [i.e., the probability of fixation is proportional to l (Otto and Bourguet 1999; Otto and
Yong 2002); see below for additional details]. Without
recombination (d ¼ 0), the leading eigenvalue reduces to
l¼
1 1 2u 1 jshð1 uÞ uj
1
2
2ð1 shÞð1 uÞ
when the direct fitness benefit of additional functional
gene copies outweighs the indirect consequences of
doubling the deleterious mutation rate, as previously
reported for both haploid and diploid systems without
recombination (Clark 1994; Otto and Yong 2002; also
see Otto and Goldstein 1992).
How does recombination alter the evolutionary dynamics of Y chromosomes? When duplicates do not
directly increase fitness (sh ¼ 0), and there is no recombination, selection never favors invasion (Equation
1b above). We can ask whether gene conversion expands
the conditions favorable to invasion of a duplicate in a
way that is similar to previous models of gene duplication
with crossing over (Otto and Yong 2002). By permitting
Y-linked recombination between duplicates, and assuming that the crossover rate is zero (dc ¼ 0; hence, all
recombination is by gene conversion), the leading eigenvalue can be approximated for low rates of gene
conversion (d 0, per generation),
l ¼ l
x1 ð1 shÞ
x0 ð1 sÞ
u1
;
x0 9 ¼
w
w
where mean fitness is w
¼ x11 1 ðx10 1 x1 Þð1 shÞ 1
ðx00 1 x0 Þð1 sÞ.
To describe conditions promoting the invasion of
duplicates, we analyzed the stability of an evolutionary
equilibrium in which duplicated genotypes are absent
from the population. Under such a condition, the frequencies x1 and x0 equilibrate to x̂1 ¼ 1 uð1 hsÞ=
½sð1 hÞ ¼ 1 x̂0 and the leading eigenvalue of the
stability matrix is
ð1bÞ
and evolutionary invasion of a duplicate-bearing Y is
favored when sh . u/(1 u). Duplicates are favored
279
ð1cÞ
which indicates that selection favors duplicates (l . 1)
when gene conversion is biased toward transmission of
functional over nonfunctional gene copies (b . 0.5).
Numerical evaluation of Equation 1a indicates that,
although higher rates of gene conversion can increase
the leading eigenvalue (and hence the probability of
invasion), this positive relationship quickly saturates.
Thus, a little bit of gene conversion has about as much
of an impact on the leading eigenvalue as a high rate of
gene conversion does. Nevertheless, the strength of
such positive selection (with magnitude of l 1) is on
the order of the mutation rate (u) and is therefore
extremely weak. Stochastic simulations (see below) show
that the probability of duplicate fixation is marginally
influenced by biased gene conversion alone.
Further analysis of Equation 1a shows that, as with the
case of no recombination (Otto and Yong 2002),
selection will favor duplicates if they directly increase
fitness (sh . 0). Gene conversion (including unbiased
gene conversion: b ¼ 0.5) can increase the strength of
selection favoring invasion of a duplicate (l 1; Figure
1). However, the relative impact of gene conversion is
minor when sh ? u. In other words, when there are
weak direct benefits of having multiple gene copies, the
strength of natural selection favoring Y-linked gene
duplicates will be enhanced by gene conversion between paralogs. This conclusion holds if the crossover
rate between duplicate pairs (dc) is small (Figure 1). As
the rate of crossing over increases, the production of
abnormal Y haplotypes can generate purifying selection
against Y chromosomes that carry gene duplicates.
Why should gene conversion broaden duplicate invasion conditions under weak selection? An intuitive
explanation can be reached by considering the recursion
280
T. Connallon and A. G. Clark
the duplicate is favored when ½1 2uð1 dbÞ=w
. 1.
Invasion is clearly facilitated by gene conversion (db .
0). Nevertheless, because the term 2u(1 db) is extremely
small, gene conversion will marginally influence the
probability of fixation whenever sh ? u.
Probability of duplicate fixation: The deterministic
model presented above can be modified to describe
the evolutionary dynamics in finite populations. Following Otto and Bourguet (1999) and Otto and Yong
(2002), the selection coefficient for a rare gene duplicate can be approximated as l 1, where l is the
leading eigenvalue of the stability matrix (Equation 1a,
above). Given this selection coefficient, the probability
that a rare duplicate is eventually fixed can be estimated
by diffusion approximation (Kimura 1957, 1962), with
drift and diffusion coefficients M ¼ (l 1)x(1 x) and
V ¼ x(1 x)/N, respectively, where x is the frequency of
a duplicate-bearing Y haplotype and N is the Y chromosome effective population size. For an initial frequency
of 1/N, the probability that a duplicate is fixed will be
PrðfixationÞ ¼
Figure 1.—Gene conversion can enhance the strength of
positive selection for rare duplicate genes, whereas crossovers
select against duplicates. Selection coefficient approximations (l 1) are based on the leading eigenvalue (Equation
1a), as described and justified in the text, and are presented as
a ratio of selection with (d . 0) vs. without recombination
(d ¼ 0). Representative results are presented for u ¼ 105
and assume that there is no gene conversion bias (i.e., b ¼ 0.5).
dynamics for a population fixed for the single-gene haplotype. Because this explanation is heuristic, we ignore
crossovers and assume that they do not occur (c ¼ 0). The
rate of increase for a rare haplotype with two functional
gene copies depends on its relative competitiveness
against the resident, single-copy haplotype. For initial
condition x11 ¼ 1/N and x10 ¼ x00 ¼ 0, the expected
proportion of functional duplicate haplotypes (x11) within
and
the gamete pool is E½x11 9 ¼ x11 ½1 2uð1 dbÞ=w,
1 e 2ð1lÞ
2ðl 1Þ
:
2N ð1lÞ
1 e 2N ð1lÞ
1e
ð2Þ
To assess the validity of Equation 2, we conducted
computer simulations that incorporate mutation, selection, and genetic drift. Each simulation was initiated at
x11 ¼ 1/N, x0 ¼ u(1 hs)/(s sh), and x1 ¼ 1 x11 x0.
To generate genotypic frequencies for the next generation, N genotypes were randomly drawn from a
multinomial distribution, after selection, from the six
genotypes described above. Mutation–selection–drift
recursions were iterated until the duplicate genotype
was either fixed or lost from the population. Equation 2
provides a good approximation for the probability of
duplicate fixation over a broad range of parameter
space (Figure 2 and Figure S1). As direct selection on a
duplicate approaches zero (sh / 0), the probability of
fixation approaches 1/N. As direct selection increases in
strength (1 ? 1 l ? 1/N), the probability of fixation
approaches 2(l 1).
Gene conversion had little impact on the probability
of duplicate fixation (see Figure S1). As shown above,
the leading eigenvalue of the stability matrix is not
substantially influenced by gene conversion unless sh is
of similar order to u. Even though the selection coefficient approximation (l 1) can increase with gene
conversion, its absolute magnitude under weak direct
selection (sh 0) will generally be too small for natural
selection to be effective, unless of course Nu . 1, which
is particularly unlikely for Y-linked loci. Thus, gene
conversion is unlikely to significantly enhance the rate
of duplicate gene fixation, but can potentially reduce
the fixation rate of duplicates if the rate of deleterious
crossovers between paralogs is high.
Gene conversion and the maintenance of gene
duplicates: A major hypothesis inspired by the human Y
chromosome is that gene conversion between duplicates
Gene Conversion and Y Evolution
Figure 2.—The probability of fixation for Y-linked duplicate genes. The solid line depicts the analytical approximation from Equation 2. Circles represent the proportion of
duplicate genotypes (out of 100,000 replicate simulations
for each data point) that eventually become fixed within
the population. Results are shown for d ¼ 0, N ¼ 1000, and
u ¼ 105, per locus, per generation. Values of d . 0 yield approximately the same results (see Figure S1).
may prevent the accumulation of mutations and ultimately prevent or slow down Y chromosome degeneration due to Muller’s ratchet (Charlesworth 2003;
Rozen et al. 2003; Noordam and Repping 2006). To formally evaluate this possibility, we considered two models
for the maintenance of functional Y-linked genes. We first
conducted simulations of our two-locus model with initial
condition x11 ¼ 1 (a pair of functional duplicates is initially fixed within the population) and analyzed whether
gene conversion prevented the loss of one or both of the
functional gene copies. Gene conversion between Y-linked
paralogs decreased the rate of gene loss under a wide
range of fitness conditions, including the extreme case
where there was no direct benefit of having two, as
opposed to one, functional gene copies (Figure S2).
Although gene conversion can substantially reduce the
rate of gene loss, the results indicate that loss of completely
redundant genes (where sh ¼ 0) will persist under gene
conversion, albeit at a substantially reduced rate.
Prior models of Muller’s ratchet generally find that
the rate at which deleterious mutations become fixed
depends upon both the strength of purifying selection
and the number of loci evolving on an asexual chromosome (Charlesworth and Charlesworth 2000;
Bachtrog 2008). To account for selection and gene
conversion across many loci, we extended our model to
describe the degeneration of Y chromosomes carrying
an arbitrary number of genes. To permit gene conversion, we assumed that each Y initially carries n distinct
gene types, each with a duplicate copy (for a total of 2n
loci). Because the increased number of genes greatly
expands the number of possible genotypic and fitness
states (and consequently the matrix of transition prob-
281
abilities between states), we made a simplifying assumption that each of the n gene types represents an essential
male fertility factor. Males lacking a functional copy of
one or more gene types are sterile and comprise a
heterogeneous genotypic class with reproductive success of zero. Although the essentiality assumption is
useful for modeling purposes, it will often be biologically reasonable because Y-linked genes, at least in
mammals and Drosophila, are often essential for male
fertility. For example, human Y chromosome microdeletions within Y-palindromic regions are often associated with spermatogenic failure (Noordam and Repping
2006; Lange et al. 2009). In Drosophila melanogaster,
mutations in at least three of seven currently Y-annotated
genes (kl-2, kl-3, and kl-5, as well as an additional set of
unannotated genes: kl-1, ks-1, and ks-2; data obtained
from http://flybase.org/) are known to cause male-sterile
phenotypes. Nevertheless, the overall agreement between our multilocus and two-locus results (the latter
does not assume essentiality; see Figure S2) suggests that
a violation of the essentiality assumption is unlikely to
strongly affect our conclusions.
For each paralog pair, there are three possible
genotypes: both loci functional, one functional and
one nonfunctional, and both nonfunctional. Transitions between genotypic states can occur by mutation,
by gene conversion, or by crossing over, with crossover
yielding an abnormal Y chromosome. For individuals
carrying a structurally normal Y, fitness follows the
function w ¼ (1 sh)k(0)j, where j refers to the number
of gene pairs with both copies nonfunctional, and k
refers to the number of pairs where one of the two gene
copies is functional (0 # k # n). Individuals with j . 0
and individuals carrying abnormal Y chromosomes are
sterile. After selection, the reproductive contribution of
an individual with k Y-linked mutations is
x kS ¼
xk wk
;
w
where xk is the zygotic frequency of k-bearing males, wk ¼
and
(1 sh)k is the fitness of a male with k mutations,
P
mean male fitness with respect to the Y is w
¼ nk¼0 xk wk .
(The reproductive contribution of sterile individuals is
zero.)
To facilitate analytical tractability, we assume that the
rates of recombination and mutation are both small
enough to ignore multiple mutation and multiple recombination events per generation. In other words, there
is a zero probability of an individual with k mutations
producing a fertile son with k 2 or k 1 2 mutations. This
assumption is justified as long as 2nu > 1 and nd > 1,
which requires that the mutation and recombination rate
per locus is small, and the number of loci mutable to a
nonfunctional allele is much smaller than the reciprocal
of the mutation or gene conversion rate: n > min[1/u, 1/
d]. Because n represents a small fraction of Y-linked
nucleotides (i.e., it represents a very specific functional
282
T. Connallon and A. G. Clark
class), this assumption is biologically reasonable. Nevertheless, a violation of these assumptions is expected to
make our results conservative by downwardly biasing the
speed of Muller’s ratchet (which is enhanced by a higher
mutation rate) and minimizing the positive effect of gene
conversion (higher gene conversion rates increasingly
counteract Muller’s ratchet). Extending across the 2n
loci, the probability that a Y chromosome experiences
one mutation is Pr(M ¼ 1) ¼ 2nu ¼ U. The probability
that zero mutations occur is Pr(M ¼ 0) ¼ 1 U. The
probability of a recombination event between one of the n
paralog pairs is Pr(R ¼ 1) ¼ nd ¼ D. The probability of no
recombination is Pr(R ¼ 0) ¼ 1 D.
Given a sequence of events of (i) birth, (ii) selection,
(iii) mutation, (iv) recombination, and (v) random mating, the frequency of fertile males in the next generation
follows the recursion
xk 9 ¼
x
shÞk1 U ðn k 1 1Þ
w
n
xk ð1 shÞk Uk 1 2nð1 U Þ 1
2n
w
Dð1 cÞðn kÞ
11 D
3
n
x ð1 shÞk U ðn kÞ x ð1 shÞk11
k
k11
1
1
w
w
n
U ðk 1 1Þ 1 2nð1 U Þ 3
2n
Dbð1 cÞðk 1 1Þ
:
3
n
k1 ð1
The ‘‘least-loaded’’ (k ¼ 0) and ‘‘most-loaded’’ (k ¼ n)
classes of fertile males follow the recursion
x0 ð1 U Þð1 DcÞn 1 UDð1 cÞb
x0 9 ¼
w
n
x1 ð1 shÞ U
Dð1 cÞb
1
11 U
w
2n
n
and
xn 9 ¼
xn1 ð1 shÞn1 U ð1 DÞ
w
n
xn ð1 shÞn ð2 U Þð1 DÞ
;
1
w
2
respectively. The frequency of sterile males in the next
generation (via crossover, mutation, or gene conversion) will be
xs 9 ¼ 1 n
X
xk 9:
k¼0
Deterministic equilibria and mean fitness of the Y: When
there is no recombination between duplicates (D ¼ 0),
Figure 3.—Gene conversion increases the frequency of Y
chromosome haplotypes that carry zero deleterious mutations (i.e., the ‘‘least-loaded’’ genotypic class). The cost of a
mutation eliminating function of a copy of each duplicate
pair is represented by sh (this cost increases from left to right
on the x-axis). The relative proportion of mutation-free Y
chromosomes in recombining vs. nonrecombining populations is presented as a ratio of the two scenarios (gene conversion increases the proportion of mutation-free Y’s when this
ratio is greater than one). The number of distinct, Y-linked
genes is represented by n. Results are presented for c ¼ 0,
b ¼ 0.5, and u ¼ 5 3 104, per locus, per generation, and
D ¼ U ¼ 2nu. Additional results are presented in Figure S3.
mean Y chromosome fitness as well as the distribution of
mutations among individuals can be analytically determined. If mutations that eliminate duplicate gene
function are deleterious (sh . 0), and the number of
unique Y-linked genes is large (n ? U/sh), the population approaches the equilibrium: x̂k PoisðU=shÞ,
1 U . This is analogous to the case of
x̂s 0, and w
mutation–selection balance with incomplete dominance (sh . 0), with a Y-linked genetic load of L ¼ U 1 eU (e.g., Haldane 1937; Kimura and Maruyama
1966; Kondrashov and Crow 1988). When knocking
out a duplicate yields no fitness effect (sh ¼ 0), or the
number of Y-linked genes is small (n > U/sh), the
population approaches the equilibrium: x̂n 1 U=2,
1 U =2. Under this scenario, the
x̂s U =2, and w
genetic load is reduced by a factor of 2, to L ¼ U/2 1 eU/2 (Haldane 1937).
Gene conversion between duplicates increases the
frequency of the least-mutated class (Figure 3 and Figure
S3), whether or not there is a gene conversion bias
favoring functional over nonfunctional loci. The frequency of the least-loaded class represents a quantity of
particular importance for adaptation on clonally transmitted chromosomes such as the Y (Charlesworth
and Charlesworth 2000). Without recombination,
the unit of selection is the chromosome rather than the
locus. Beneficial mutations that are associated with
mutation-free genetic backgrounds are relatively likely
Gene Conversion and Y Evolution
to become fixed (Peck 1994; Orr and Kim 1998) and do
not permit hitchhiking of deleterious mutations during
a selective sweep (Rice 1987). However, as the frequency
of the least-loaded class becomes small, virtually all
beneficial mutations will arise in inferior genetic backgrounds. This will limit the adaptive potential of the
Y chromosome. Because it increases the fraction of
mutant-free Y chromosomes, gene conversion is expected to enhance the fixation probability for beneficial
mutations and can reduce the deleterious consequences of hitchhiking.
By shifting the mutational distribution toward relatively mutation-free genotypes, gene conversion also
increases mean Y chromosome fitness. This effect does
not depend on a gene conversion bias, but can become
exacerbated when conversion events favor functional
over nonfunctional variants (for models yielding similar
conclusions about the genetic load, albeit by different
approaches, see Bengtsson 1986, 1990, and especially
Ohta 1989).
These long-term effects of gene conversion can be
accounted for by a straightforward explanation. When
the fitness cost of silencing both copies of a duplicate
pair is much greater than the cost of silencing one of the
copies (when duplicates partially or completely mask
deleterious mutations: h , 0.5), selection across Y chromosomes mimics truncation selection, which is particularly efficient at removing deleterious alleles (e.g.,
Kondrashov 1988; Ohta 1989). Truncation selection
arises because mutations on a relatively mutation-free Y
will generally affect one copy of a pair, with the second,
functional copy compensating for loss of the first. As the
number of mutations on a Y increases, so does the
probability of silencing the second copy of a pair. Consequently, the deleterious effect of each mutation increases faster than linearly with the number of mutations
carried on a Y.
Without recombination, the accumulation of mutations is unidirectional, and the population will tend to
evolve toward the edge of the truncation point (n
mutations at distinct genes), particularly if masking by
duplicates is strong (i.e., having two functional copies
provides the same fitness as one copy). At the extreme of
sh ¼ 0 (complete masking), the population evolves to
contain n functional genes, each distinct. Gene conversion restores variability by permitting bidirectional
transitions (e.g., k to k 1 and k 1 1 mutations). Y
chromosomes that are closer to the truncation point
have a higher probability of transitioning (by mutation
or recombination) beyond the truncation point where
they are removed by selection. Consequently, the
population distribution shifts toward fewer mutations
per Y. However, if selection in favor of functional
duplicates is strong relative to the number Y-linked
genes (sh . 0; n large), most individuals will carry few
mutations, the truncation point becomes irrelevant to Y
chromosome evolution, selection shifts toward multi-
283
plicative epistasis, and gene conversion does not
strongly influence mean fitness or the distribution of
mutations among Y chromosomes. This explanation
accounts for the decreased impact of gene conversion
on mutation-free Y chromosomes, as the strength of
selection (sh) increases (Figure 3 and Figure S3).
Muller’s ratchet and the accumulation of nonfunctional
genes: The deterministic results (presented above) represent an upper limit for Y chromosome fitness. In finite
populations, where Muller’s ratchet operates, mean
fitness can further decrease with each successive loss of
‘‘mutation-free’’ individuals. Once lost from the population, mutation-free genotypes are unlikely to be recovered by back mutation or positive selection because they
must initially arise within the current least-loaded class
and subsequently avoid stochastic loss (Peck 1994; Orr
and Kim 1998; Gordo and Charlesworth 2000).
To explore the influence of gene conversion on the
rate and severity of Y chromosome degeneration via
Muller’s ratchet, we conducted a series of stochastic
simulations, varying the selection and recombinational
parameters (u, h, n, d, c, b). We first use the recursions
presented above to bring the frequencies of each
genotypic class to deterministic equilibrium. Convergence to equilibrium is followed by 100,000 generations
of simulation under a mutation–selection–drift model
and constant male population size. For each generation,
genotype frequencies were sampled from a pseudorandom multinomial distribution (pseudorandom numbers generated with R; R Development Core Team
2005), with genotypes randomly sampled after selection, mutation, and recombination.
When there is no gene conversion between duplicates, Muller’s ratchet can operate rapidly, causing
Y-linked fitness decay and loss of functional genes.
Representative simulation results are shown in Figures
4 and 5. In agreement with previous theory (Haigh 1978;
Gordo and Charlesworth 2000; Bachtrog 2008), the
impact of the ratchet is strongest when the ancestral Y
carries many functional gene duplicates and when
mutations have small individual fitness effects. Relatively
low rates of gene conversion can rescue Y-linked genes
from stochastic loss via Muller’s ratchet and thereby
increase mean fitness of the Y (Figures 4 and 5).
Increasing the total mutation and gene conversion rates
on the Y (U and D, respectively) amplifies the differences
between recombining and nonrecombining chromosomes, whereas a decrease in these compound parameters (U, D / 0) eliminates these long-term evolutionary
differences. This effect occurs both with and without
biased gene conversion between duplicates.
Gene conversion appears to constrain accumulation
of deleterious mutations in a way that is identical to
crossing over in traditional models of Muller’s ratchet.
Under both models, the rate at which the ratchet
‘‘clicks’’—the least mutated class of individuals is lost—is
highest when individual mutations are weakly deleterious
284
T. Connallon and A. G. Clark
Figure 4.—Intrapalindrome gene conversion
prevents the erosion of Y chromosome gene content and enhances adaptation on the Y. N represents the Y-linked effective size, sh is the fitness
cost associated with mutations to one copy of
each duplicate pair, t refers to the generation
within the simulation, and n is the number of distinct genes on the chromosome (including duplicates, each Y carries 2n genes). Results are
presented for c ¼ 0, b ¼ 0.5, and u ¼ 5 3 104,
per locus, per generation. Each data point represents the average of 10 simulation replicates.
Since estimates of gene conversion from human–
chimp comparisons suggest that D may be considerably higher than the mutation rate (Rozen
et al. 2003), the results, if anything, will underestimate the impact of gene conversion on functional gene retention.
and/or the chromosome-wide mutation rate (an increasing function of the mutation rate per locus and
the number of loci) is high (Charlesworth and
Charlesworth 2000; Bachtrog 2008). The similar
consequences of gene conversion and crossing over are
not surprising: both processes permit chromosomal
transitions from more to fewer mutations and this,
along with purifying selection, can counteract the
steady accumulation of new deleterious mutations
within a population.
DISCUSSION
Previous theory indicates that selection does not
generally favor the invasion of a rare duplicate gene
unless there is a direct benefit of carrying an additional
gene copy (Clark 1994) or there is recombination
between the paralogs (Yong 1998; Otto and Yong
2002; Tanaka and Takahasi 2009). We have shown that
gene conversion between duplicates can broaden the
parameter conditions favoring the invasion of duplicate
genes from low initial frequency. Biased gene conversion,
with conversion favoring undamaged over damaged gene
copies, can generate positive selection for rare duplicates
that do not provide a direct fitness benefit (that is,
individuals with two functional copies have fitness equal
to those with one). However, the strength of positive
selection acting on such duplicates is weak (on the order
of the mutation rate). This result is in agreement with a
recent simulation study, which also found that gene
conversion does not strongly promote the invasion of
new Y-linked duplicates (Marais et al. 2010).
The invasion dynamics of rare duplicate genes bear
some similarities to models of adaptation within gene
families (Walsh 1985; Mano and Innan 2008), which
show that gene conversion can enhance the probability
that a weakly beneficial allele becomes fixed. In our
model, gene conversion alone is unlikely to overpower
genetic drift unless Nu ? 1, yet this condition is rarely (if
ever) expected to arise within animal populations, particularly with respect to Y-linked loci that have reduced
effective size relative to other nuclear genes. Furthermore, there is no biological reason to suspect that gene
conversion will necessarily be biased against mutant
copies of a particular gene. We therefore expect that
Y-linked duplicates will most likely become fixed by
genetic drift, unless they directly increase the fitness of
those who carry them (for additional discussion of
duplicate gene fixation, see Innan and Kondrashov
2010). Likewise, deleterious Y-linked crossover events can
generate selection against gene duplicates. This factor
will have little impact on the probability of fixation or loss
unless the crossover rate is relatively high and direct
selection on the duplicate is weak or absent.
Y chromosome recombination can exert a profound
influence on the retention of functional copies of genes
Gene Conversion and Y Evolution
285
Figure 5.—The proportion of loss-of-function
duplicates following 100,000 generations of mutation, selection, and genetic drift. Parameters
are described in the Figure 4 legend and throughout the text. Results are presented for c ¼ 0, b ¼
0.5, u ¼ 5 3 104, per locus, per generation, and
D on the order of the mutation rate, D ¼ U ¼ 2nu.
Each point represents the average of 10 replicate
simulations.
that have already become fixed within the population.
Our simulations show that low rates of gene conversion
are sufficient to maintain Y-linked genes and counteract
degradation via Muller’s ratchet. These results are
conservative, as higher rates enhance the preservation
of functional gene copies. Thus, once gene conversion
has evolved, it can potentially provide a degree of
stability on an otherwise evolutionarily unstable Y
chromosome. Interestingly, Marais et al. (2010) observed that the rate of invasion for gene conversion
modifier alleles does not greatly exceed neutral expectations unless they greatly increase the gene conversion
rate. This suggests that, while low rates of conversion
may slow the rate of Muller’s ratchet, the evolution of
the gene conversion rate itself may be much more
restrictive.
The large number of genes within the ‘‘ampliconic’’
region of the human Y (Skaletsky et al. 2003) should
provide a large target for mutations, creating an
opportunity for Muller’s ratchet to act. This role of
gene conversion on the Y is therefore likely to explain
patterns of gene retention on the human Y chromosome. It is less clear whether similar patterns characterize other animal species. Current (albeit incomplete)
data suggest that gene family amplification and retention might be common Y chromosome attributes
(Rozen et al. 2003; Verkaar et al. 2004; Murphy et al.
2006; Alföldi 2008; Wilkerson et al. 2008; Krsticevic
et al. 2009), although the prevalence of Y-linked gene
conversion outside the human and chimp lineages is
less clear (but see Geraldes et al. 2010). Future
sequencing efforts, including evidence for gene conversion among Y-linked genes in nonhuman species, will
help to determine the general relevance of the duplication and gene conversion model presented here.
Within-chromosome crossovers can generate an abnormal, sterility-inducing Y (Lange et al. 2009) and
potentially represent a deleterious fitness consequence
of Y-linked recombination. This cost also implies that
the number of Y-linked duplicate genes (or in humans
the size of Y-linked palindromes) will have an upper
limit. As the number of Y-linked loci that interact via
recombination increases, so too should the rate of
deleterious crossovers. This suggests an upper limit to
Y chromosome gene content, where crossing over
becomes unbearably costly. From this perspective,
duplication and recombination represent a costly
mechanism of Y chromosome preservation.
In addition to the Y chromosome, our findings have
implications for asexually reproducing species. Recent
reports suggest that the asexual bdelloid rotifers are
tetraploid (Mark Welch et al. 2008) and that gene
conversion occurs between gene copies (Hur et al. 2008;
Mark Welch et al. 2008). Our model supports the
verbal claim that gene conversion between homologous
gene copies might aid in DNA damage repair and
prevent the genomic degradation that is expected to
accompany strict asexual reproduction. Unlike the Y
chromosome scenario, crossovers between homologous, tetraploid chromosomes will tend to avoid deleterious chromosomal aberrations. The relative rate of
nonhomologous crossovers is an empirical question
that may be difficult to assess, given the likely association
between chromosome abnormalities and embryonic
death, which will lead to a pronounced bias toward
‘‘normal’’ chromosomes. On the other hand, crossing
over between homologous chromatids is likely to
generate copy number polymorphism, which adds a
level of complexity to the evolutionary dynamics of
autosomal gene duplicates or gene families. This may
lead to different evolutionary consequences of crossing
over and gene conversion in asexual lineages compared
to the results that we report for the Y chromosome and
represents an interesting avenue for future theoretical
research.
We are grateful to Roman Arguello, Clement Chow, Margarida
Cardoso-Moreira, Qixin He, Lacey Knowles, Amanda Larracuente,
Rich Meisel, Nadia Singh, and two anonymous reviewers for discussion
286
T. Connallon and A. G. Clark
and comments that substantially improved the quality of the manuscript and to Sarah Otto for comments about the eigenvalue-selectioncoefficient approximation and for sharing an unpublished manuscript. This work was supported by National Institutes of Health grant
GM64590 to A.G.C. and A. B. Carvalho.
LITERATURE CITED
Alföldi, J. E., 2008 Sequence of mouse Y chromosome. Ph.D. Dissertation, MIT, Cambridge, MA.
Bachtrog, D., 2006 A dynamic view of sex chromosome evolution.
Curr. Opin. Genet. Dev. 16: 578–585.
Bachtrog, D., 2008 The temporal dynamics of processes underlying Y chromosome degeneration. Genetics 179: 1513–1525.
Bengtsson, B. O., 1986 Biased conversion as the primary function
of recombination. Genet. Res. 47: 77–80.
Bengtsson, B. O., 1990 The effect of biased conversion on the mutation load. Genet. Res. 55: 183–187.
Carvalho, A. B., L. B. Koerich and A. G. Clark, 2009 Origin and
evolution of Y chromosomes: Drosophila tales. Trends Genet. 25:
270–277.
Charlesworth, B., 2003 The organization and evolution of the human Y chromosome. Genome Biol. 4: 226.
Charlesworth, B., and D. Charlesworth, 2000 The degeneration of Y chromosomes. Philos. Trans. Biol. Sci. 355: 1563–1572.
Clark, A. G., 1994 Invasion and maintenance of a gene duplication.
Proc. Natl. Acad. Sci. USA 91: 2950–2954.
Engelstadter, J., 2008 Muller’s ratchet and the degeneration of Y
chromosomes: a simulation study. Genetics 180: 957–967.
Geraldes, J., T. Rambo, R. Wing, N. Ferrand and M. W. Nachman,
2010 Extensive gene conversion drives the concerted evolution
of paralogous copies of the SRY gene in European rabbits. Mol.
Biol. Evol. (in press).
Gordo, I., and B. Charlesworth, 2000 The degeneration of asexual haploid populations and the speed of Muller’s ratchet. Genetics 154: 1379–1387.
Haigh, J., 1978 Accumulation of deleterious genes in a population—Muller’s ratchet. Theor. Popul. Biol. 14: 251–267.
Haldane, J. B. S., 1937 The effect of variation on fitness. Am. Nat.
71: 337–349.
Heinritz, W., D. Kotzot, S. Heinze, A. Kujat, W. J. Eleemann et al.,
2005 Molecular and cytogenetic characterization of a nonmosaic isodicentric Y chromosome in a patient with Klinefelter
syndrome. Am. J. Med. Genet. A 132A: 198–201.
Hughes, J. F., H. Skaletsky, T. Pyntikova, T. A. Graves, S. K. M.
van Daalen et al., 2010 Chimpanzee and human Y chromosomes are remarkably divergent in structure and gene content.
Nature 463: 536–539.
Hur, J. H., K. Van Doninck, M. L. Mandigo and M. Meselson,
2008 Degenerate tetraploidy was established before bdelloid
rotifer families diverged. Mol. Biol. Evol. 26: 375–383.
Innan, H., and F. A. Kondrashov, 2010 The evolution of gene duplications: classifying and distinguishing between models. Nat.
Rev. Genet. 11: 97–108.
Kimura, M., 1957 Some problems of stochastic processes in genetics. Ann. Math. Stat. 28: 882–901.
Kimura, M., 1962 On the probability of fixation of mutant genes in
a population. Genetics 47: 713–719.
Kimura, M., and T. Maruyama, 1966 Mutational load with epistatic
gene interactions in fitness. Genetics 54: 1337–1351.
Koerich, L. B., X. Wang, A. G. Clark and A. B. Carvalho,
2008 Low conservation of gene content in the Drosophila Y
chromosome. Nature 456: 949–951.
Kondrashov, A. S., 1988 Deleterious mutations and the evolution
of sexual reproduction. Nature 336: 435–440.
Kondrashov, A. S., and J. F. Crow, 1988 King’s formula for the mutation load with epistasis. Genetics 120: 853–856.
Krsticevic, F. J., H. L. Santos, S. Januario, C. G. Schrago and
A. B. Carvalho, 2009 Functional copies of the Mst77F gene
on the Y chromosome of Drosophila melanogaster. Genetics 184:
295–307.
Lange, J., H. Skaletsky, S. K. M. van Daalen, S. L. Embry, C. M.
Korver et al., 2009 Isodicentric Y chromosomes and sex disorders as byproducts of homologous recombination that maintains
palindromes. Cell 138: 855–869.
Mano, S., and H. Innan, 2008 The evolutionary rate of duplicate
genes under concerted evolution. Genetics 180: 493–505.
Marais, G., 2003 Biased gene conversion: implications for genome
and sex evolution. Trends Genet. 19: 330–338.
Marais, G., P. R. A. Campos and I. Gordo, 2010 Can intra-Y gene
conversion oppose the degeneration of the human Y chromosome?: A simulation study. Genome Biol. Evol. 2: 347–357.
Mark Welch, D. B., J. L. Mark Welch and M. Meselson,
2008 Evidence for degenerate tetraploidy in bdelloid rotifers.
Proc. Natl. Acad. Sci. USA 105: 5145–5149.
Murphy, W. J., A. J. P. Wilkerson, T. Raudsepp, R. Agarwala,
A. A. Schaffer et al., 2006 Novel gene acquisition on carnivore
Y chromosomes. PLoS Genet. 2: e43.
Noordam, M. J., and S. Repping, 2006 The human Y chromosome:
a masculine chromosome. Curr. Opin. Genet. Dev. 16: 225–232.
Ohta, T., 1989 The mutational load of a multigene family with uniform members. Genet. Res. 53: 141–145.
Orr, H. A., and Y. Kim, 1998 An adaptive hypothesis for the evolution of the Y chromosome. Genetics 150: 1693–1698.
Otto, S. P., and D. Bourguet, 1999 Balanced polymorphisms and
the evolution of dominance. Am. Nat. 153: 561–574.
Otto, S. P., and T. Day, 2007 A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution. Princeton University Press, Princeton, NJ.
Otto, S. P., and D. B. Goldstein, 1992 Recombination and the evolution of diploidy. Genetics 131: 745–751.
Otto, S. P., and P. Yong, 2002 The evolution of gene duplicates.
Homol. Eff. 46: 451–483.
Peck, J. R., 1994 A ruby in the rubbish: beneficial mutations, deleterious mutations and the evolution of sex. Genetics 137: 597–606.
R Development Core Team, 2005 R: A Language and Environment
for Statistical Computing, reference index version 2.2.1. R Foundation for Statistical Computing, Vienna.
Repping, S., H. Skaletsky, J. Lange, S. Silber, F. van der Veen et al.,
2002 Recombination between palindromes P5 and P1 on the
human Y chromosome causes massive deletions and spermatogenic failure. Am. J. Hum. Genet. 71: 906–922.
Rice, W. R., 1987 Genetic hitchhiking and the evolution of reduced
genetic activity on the Y sex chromosome. Genetics 116: 161–167.
Rozen, S., H. Skaletsky, J. Lange, S. Silber, F. van der Veen et al.,
2003 Abundant gene conversion between arms of palindromes
in human and ape Y chromosomes. Nature 423: 873–876.
Skaletsky, H., T. Kuroda-Kawaguchi, P. J. Minx, H. S. Cordum, L.
Hillier et al., 2003 The male-specific region of the human Y
chromosome is a mosaic of discrete sequence classes. Nature
423: 825–837.
Tanaka, K. M., and K. R. Takahasi, 2009 Enhanced fixation and
preservation of a newly arisen duplicate gene by masking deleterious loss-of-function mutations. Genet. Res. 91: 267–280.
Verkaar, E. L. C., C. Zijlstra, E. M. van ’t Veld, K. Boutaga, D. C.
J. Boxtel et al., 2004 Organization and concerted evolution
of the ampliconic Y-chromosomal TSPY genes from cattle.
Genomics 84: 468–474.
Walsh, B., 1985 Interaction of selection and biased gene conversion
in a multigene family. Proc. Natl. Acad. Sci. USA 82: 153–157.
Wilkerson, A. J. P., F. Raudsepp, T. Graves, D. Albracht, W.
Warren et al., 2008 Gene discovery and comparative analysis
of X-degenerate genes from the domestic cat Y chromosome.
Genomics 92: 329–338.
Yong, P., 1998 Theoretical population genetic model of the invasion of an initial duplication. Honours Thesis, Department of Zoology, University of British Columbia, Vancouver, BC, Canada.
Yu, Y.-H., Y.-W. Lin, J.-F. Yu, W. Schempp and P. H. Yen,
2008 Evolution of the DAZ gene and AZFc region on primate
Y chromosomes. BMC Evol. Biol. 8: 96.
Communicating editor: D. Charlesworth
GENETICS
Supporting Information
http://www.genetics.org/cgi/content/full/genetics.110.116756/DC1
Gene Duplication, Gene Conversion and the Evolution
of the Y Chromosome
Tim Connallon and Andrew G. Clark
Copyright Ó 2010 by the Genetics Society of America
DOI: 10.1534/genetics.110.116756
2 SI
T. Connallon and A. G. Clark
FILE S1
I. Invasion of gene duplicates on Y chromosomes that carry an arbitrary number of linked genes.
Y-linked duplicate genes evolve within the genetic background of the entire Y chromosome, which is likely to contain
multiple functional genes, particularly during early stages of sex chromosome evolution. To determine the generality of the single
gene duplication scenario in the main text, we developed a second model to examine the evolutionary dynamics of rare, Y-linked
duplicates on ancestral chromosomes carrying an arbitrary number (n) of single-copy genes.
Consider a rare, Y-linked duplicate on Y chromosome carrying n single-copy genes. By duplicating one of the n single-copy
genes, the individual has n – 1 single-copy genes and a single duplicated pair. Though expanding the number of loci greatly
increases the number of possible genotypes to follow within the population, subsequent calculations can be simplified by making
each gene essential. In other words, fitness drops to zero (s = 1) unless each of the n genes has at least one functional copy.
Given this simplification, there are four relevant genotypic classes within the population: (i) individuals with n functional
singletons and no duplicates, each at frequency xn and with fitness wn = 1 – sh; (ii) those with n + 1 functional genes (n – 1
singleton) at frequency xn1 and with fitness wn1 = 1; (iii) those with n + 1 genes (n – 1 singleton), of which n are functional, at
frequency xn0 and with fitness wn0 = 1 – sh; and (4) a class of sterile individuals, at frequency xs and with fitness ws = 1 – s = 0, that
either lack a functional copy of an essential gene, or carry an abnormal Y chromosome.
In an individual carrying n singletons, the Y chromosome deleterious mutation rate per gamete per generation is U = nu, and
the distribution of mutations across gametes is reasonably modeled as a Poisson variable with mean of nu. However, given that the
diploid, genomic deleterious mutation rate is unlikely to be much greater than one, and Y chromosomes typically represent a tiny
fraction of a genome, the number of new mutations should be close to the Bernoulli distribution: U = nu is probability of one
mutation, and 1 – U represents the probability of zero mutations, per generation. For an individual carrying n + 1 total genes, the
overall mutational target will be slightly increased, and the Y chromosome mutation rate becomes Udup = U(n + 1)/n, per
generation. The presence of gene duplicates introduces an opportunity for gene conversion, which as before, are governed by
recombination rate (d), crossover (c), and conversion bias (b) parameters.
Following the events order of (i) birth, (ii) selection, (iii) mutation, (iv) recombination, and (v) fertilization, the Y chromosome
recursions are:
x n1 '=
x n1[2Ud(1 c)b + (n U Un)(1 dc)] x n 0 (1 h)(1 U)d(1 c)b
+
[x n1 + (x n 0 + x n )(1 h)]n
x n1 + (x n 0 + x n )(1 h)
x n 0 '=
2x n1U(1 d)
x (1 h)(1 U)(1 d)
+ n0
[x n1 + (x n 0 + x n )(1 h)]n x n1 + (x n 0 + x n )(1 h)
x n '=
x n (1 h)(1 U)
x n1 + (x n 0 + x n )(1 h)
T. Connallon and A. G. Clark
3 SI
x s '= x n1 '+ x n 0 '+ x n '
Stability of the equilibrium xn1 = xn0 = 0,
=
xˆ n = 1 U = 1 xˆ s , and w = (1 U)(1 h)
is governed by the eigenvalue:
2Ud(1 c)b + (n U Un)(1 dc) + (1 h)(1 U)(1 d)n
+
2(1 h)(1 U)n
2
{2Ud(1 c)b + (n U Un)(1 dc) + (1 h)(1 U)(1 d)n} 4(n U Un)(1 dc)(1 d)(1 h)(1 U)n
2(1 h)(1 U)n
When there is no recombination (d = 0), a rare gene duplicate is favored by selection when sh > U/(n – nU). Substituting for
U = nu yields sh > u/(1 – nu). This result differs slightly from the previous model of a duplicate linked to a single essential gene (the
former model predicts that a duplicate invades when sh > u/(1 – u)). Multiple Y-linked genes will therefore decrease opportunities
for positive selection in favor of new duplicates.
When selection is weak (sh 0), recombination can promote selection in favor of the duplicate. For sh = c = 0, the Taylor
series approximation around d = 0 gives a leading eigenvalue of:
= d=0 +
d + O(d 2 ) 1+ d(2b 1)
d d = 0
which is greater than one for b > 0.5, as in the previous model. Numerical simulations of the leading eigenvalue under a broad
range of parameter space show that, as before, the opportunity for positive selection for a new duplicate is greater with
recombination.
4 SI
T. Connallon and A. G. Clark
II. Invasion Probability of Duplicate Genes with Gene Conversion
FIGURE S1.—The probability of fixation for Y-linked duplicate genes. The red line depicts the analytical approximation from
Eq. (2). To facilitate comparison between these results and those of Fig. 2 from the main text, we show the approximation for N =
1000, s = 1, d = 0, and u = 10-5, and present representative simulation results for d > 0 and various combinations of the remaining
parameters (c, b). Circles represent the proportion of duplicate genotypes (out of 100,000 replicate simulations for each data point)
that eventually become fixed within the population.
T. Connallon and A. G. Clark
5 SI
III. Maintenance of Functional Gene Duplicates
FIGURE S2.—Gene conversion and the maintenance of functionally redundant paralogs. Results are presented for two
extremes of selection: gene conversion between paralogs of an essential gene (s = 1) and between paralogs of a nonessential gene (s
= 0.001). In each case, gene conversion is unbiased (b = 0.5) and the mutation rate is u = 10-5. Under essentiality and nonessentiality, fitness is maximized when at least one of the paralog copies is functional (i.e., masking of knockout mutations is
complete: h = 0). Each point represents the fraction of 100 simulation replicates where both copies are maintained as functional
within the population. For each simulation run, the population is initially fixed for two functional Y-linked genes, and then
evolves under mutation, recombination, selection, and genetic drift for 100,000 generations.
6 SI
T. Connallon and A. G. Clark
IV. Frequency of the ‘least loaded class’ under biased gene conversion.
FIGURE S3.—Gene conversion increases the frequency of Y chromosomes haplotypes that carry zero deleterious mutations
(i.e., the “least-loaded” genotypic class). Results use the same parameters as those of Fig. 3 with n = 50, and with the biased gene
conversion parameter (b) permitted to vary.