competition between the sperm of a single male can ......2013/05/03 · joseph and kirkpatrick...

Competition Between the Sperm of a Single Male can Increase theEvolutionary Rate of Haploid Expressed Genes

Kiyoshi Ezawa∗ and Hideki Innan∗,1

∗ The Graduate University for Advanced Studies, Hayama, Kanagawa 240-0193, Japan

A Research Article Submitted to Genetics, May 6, 2013

ABSTRACT

The population genetic behavior of mutations in sperm genes is theoretically investigated. We modeledthe processes at two levels. One is the standard population genetic process, in which the population allelefrequencies change generation by generation, depending on the difference in selective advantages. Theother is the sperm competition during each genetic transmission from a generation to the next generation.For the sperm competition process, we formulate the situation where a huge number of sperm with allelesA and B, produced by a single heterozygous male, compete to fertilize a single egg. This “minimal model”demonstrates that a very slight difference in sperm performance amounts to quite a large difference betweenthe alleles’ winning probabilities. By incorporating this effect of paternity-sharing sperm competitioninto the standard population genetic process, we show that fierce sperm competition can enhance thefixation probability of a mutation with a very small phenotypic effect at the single-sperm level, suggestinga contribution of sperm competition to rapid amino acid substitutions in haploid-expressed sperm genes.Considering recent genome-wide demonstrations that a substantial fraction of the mammalian sperm genesare haploid-expressed, our model could provide a potential explanation of rapid evolution of sperm geneswith a wide variety of functions (as long as they are expressed in the haploid phase). Another advantage ofour model is that it is applicable to a wide range of species, irrespective of whether the species is externallyfertilizing, polygamous or monogamous. The theoretical result was applied to mammalian data to estimatethe selection intensity on non-synonymous mutations in sperm genes.

For sexual organisms, reproduction is an essential processthat allows an individual’s genomic information to sur-vive beyond its lifetime. Years ago, it was thought thatthe functional constraints on genes involved in reproduc-tion should be as strong as those on functionally importantgenes such as histones, etc. (e.g., Miyata and Yasunaga1980; Li 1997), hence it was predicted that such genesshould evolve much more slowly than average. Therefore,it was a surprise when the first molecular evolutionaryanalyses on reproduction-related genes (or proteins) re-vealed their faster than normal evolutionary rates (see e.g.,Swanson et al. 2001; Swanson and Vacquier 2002a,b).Since then, analyses of additional reproductive genes inadditional species continue to support the initial observa-tion that reproductive genes evolve more rapidly than thegenomic average (e.g., Cutter and Ward 2005; Clarket al. 2006; Ramm et al. 2008; Turner and Hoekstra2008; Clark et al. 2009; Findlay and Swanson 2010;Wong 2011). A common and particularly typical pat-

1Corresponding author: The Graduate University for AdvancedStudies, Hayama, Kanagawa 240-0193, Japan.E-mail: innan [email protected].

tern for reproductive genes is a higher ratio, often denotedas dN/dS (= ω), of the number of non-synonymous nu-cleotide substitutions per non-synonymous site (dN) tothe number of synonymous nucleotide substitutions persynonymous site (dS). This pattern seems to be partic-ularly remarkable among “sperm genes”, namely, geneswhose protein products are found in sperm (e.g., Wyck-off et al. 2000; Torgerson et al. 2002; Swanson et al.2003; Nielsen et al. 2005; Artieri et al. 2008; Doruset al. 2010). When dN/dS is computed for the entirecoding region of a gene, we call it a “gene-wide” dN/dSvalue. Sperm genes usually show higher gene-wide dN/dSvalues than the average over all genes in the genome. Fur-thermore, sperm genes commonly have local regions (ordomains) whose dN/dS values significantly exceed 1.

There are a variety of potential explanations for thisobservation. Some of them are not suitable to explainthe general trend that a wide variety of sperm genes ex-hibit high dN/dS values in various species. For exam-ple, (i) relaxation of selective constraints could accountfor elevated gene-wide dN/dS values (Swanson and Vac-quier 2002a), but does not explain the common observa-

Genetics: Early Online, published on May 11, 2013 as 10.1534/genetics.113.152066

Copyright 2013.

2 K. Ezawa and H. Innan

tion that many genes have local regions with dN/dS � 1(e.g., Ramm et al. 2008; Dorus et al. 2010; Wong 2011).(ii) Defense against pathogens could explain the elevateddN/dS values in reproductive genes (e.g., Vacquier et al.1997), but this applies only to genes that are involved inbattles against pathogens. (iii) Reinforcement of repro-ductive incompatibility in a speciation event can also ac-celerate the evolution of sperm genes (Dobzhansky 1940;Howard 1993), but this works only on special occasionswhere two closely related sympatric species co-exist.

Most other models and hypotheses invoke either post-copulatory sexual selection, namely selection on reproduc-tive genes after mating (reviewed e.g., in Birkhead andPizzari 2002; Swanson and Vacquier 2002b; Clarket al. 2006; Turner and Hoekstra 2008), or sexualconflict (e.g., Rice and Holland 1997; Frank 2000;Gavrilets 2000; Chapman et al. 2003; Hayashi et al.2007). So far, these models mainly focused on selectionand/or competition at the individual level; when theyconsider a competition among sperm, it is almost alwaysamong sperm produced by different males. In these mod-els, competition among sperm from a single male play onlya secondary role, if any.

On the contrary, in real life, it is obvious that numer-ous sperm compete with each other even when a femalemates with only a single male during a reproductive pe-riod (see e.g., Parker and Begon 1993; Manning andChamberlain 1994). For a sperm to successfully fusewith an egg, it has to win a fierce competition with mil-lions to billions of all the others, to be the only one “win-ner”; the remaining 99.999...% of the sperm are destinednot to be involved in fertilization. This process is quitecomplicated and involves many factors. For example, therate of success depends on how fast it can swim in theright direction and how efficiently it can fuse with the egg.The former process may involve chemotaxis, and the lat-ter may involve egg-sperm compatibility. Therefore, anykind of selection on sperm performance may potentiallyincrease dN/dS values of a wide variety of sperm genes,irrespective of whether the species is externally fertilizing,polygamous or monogamous.

The main goal of this study is to examine the effects ofsuch competition among paternity-sharing sperm, which,as mentioned above, have almost always been neglectedthus far. For this purpose, we here provide a “minimalmodel” of sperm gene population genetics that incorpo-rates the intrinsic feature of the fertilization process. Tobe more specific, our minimal model focuses on the com-petition among sperm introduced by a single mating eventwith a single male. Even in this case, sperm can have dif-ferent genotypes; for a single sperm gene, there would betwo alleles if the male is heterozygous. With this mini-mal model, we demonstrate that even a very tiny pheno-typic effect of a mutation at the level of a single spermcan amount to a substantial difference between allelic fit-nesses at the level of a single inheritance, through fierce

competition among millions to billions of sperm per each.This result implies that mutations with very weak effectsat the molecular level can be a potential explanation ofthe widely observed high dN/dS ratios of sperm genes.

Because our theory applies only when each allele im-pacts solely (or preferentially) haploid sperm genomes car-rying it, the generality of our theory largely depends onhow many sperm genes have haploid expression, namely,are expressed during the haploid phase of the sperm de-velopment. It used to be thought that such haploid ex-pression should be very rare because the haploid phasespans only late stages of the sperm development, duringwhich both DNA and cytoplasm are getting compactified(see e.g., Steger 1999). However, recent genome-wide ex-pression analyses estimated that about several hundredsof sperm genes are haploid expressed in mammal (see e.g.,Joseph and Kirkpatrick 2004). This number is compa-rable to that of genes examined in each of other proteome-scale analyses of mammalian sperm (see e.g., Good andNachman 2005; Dorus et al. 2010). Furthermore, asGood and Nachman (2005) showed, it is among spermgenes expressed after the onset of meiosis, but not amongthose expressed before it, that high dN/dS regions werefound significantly more frequently than the genome aver-age. These results indicate that haploid expression seemsto be indeed quite common among sperm genes. If so,the model we propose here could provide an importantexplanation of rapid evolution of sperm genes with a widevariety of functions, as long as they are haploid expressed.

Theory

Modeling sperm-competition process: In the pro-cess of fertilization, selection works in an essentially dif-ferent way from that assumed in the standard populationgenetics, in which what usually matters is the number ofoffspring that mature enough to produce next offspring.As illustrated in Figure 1A, suppose an autosomal locuswith two alleles, A and B, that have different life spandistributions. Assuming all else (such as fertility) beingequal, selection should favor B because it has a higherpossibility to mature and produce offspring.

In contrast, in the fertilization process, millions to bil-lions of sperm compete for fertilizing an egg (or a hand-ful of eggs), and selection should act on the fertilizingability of sperm. In Figure 1B, we again consider twoalleles at a single autosomal locus, A and B, that havedifferent “velocities” in the direction to the egg. Wesuppose that sperm swim toward an egg, perhaps in re-sponse to pheromone-like chemoattractants. How fast asperm can reach and fertilize the egg is determined by thetraits such as responsiveness to pheromone-like chemoat-tractants, swimming speed, ability to overcome obstaclessurrounding the egg, and compatibility with the egg mem-brane. Here, in order to measure the performance of a

Population Genetics Model of Sperm Genes 3

A!

B!

FIGURE 1.– Illustration on how selection works in thestandard population genetic framework (A) and in spermcompetition (B). See text for details.

sperm, we consider the total time from ejaculation till thecompletion of fertilization. Then, we define the “velocity”of the sperm, or the level of performance in fertilizing theegg, as the reciprocal of the total time, so that a spermwith a larger velocity is more likely to win the race forfertilization.

Sperm competition is usually extremely fierce, whereonly one (or a handful of) winner(s) among a huge num-ber of competitors can pass on the genome(s) to the nextgeneration, and the remaining 99.999...% of the sperm areeliminated. In this situation, the important factor is theright tail of each velocity distribution (red or blue shadein Figure 1b), to which the “fastest” sperm with each al-lele likely belongs. Given the distributions in Figure 1b,selection should favor A because it has a better chance tohave the “winner” sperm among all competitors.

This intuitive expectation can be mathematically ex-pressed as follows. Let x be the velocity as defined above.fZ(x) denotes the probability density function (PDF) ofx for a single sperm with allele Z (= A or B), and letXZ (Z = A or B) be a random variable following thePDF fZ(x). Then, the probability, P [X

Z > x], that asperm with allele Z has a velocity larger than x is givenby: P [XZ > x] =

∫ +∞x

dξfZ(ξ) (for Z = A, B).

Now, let us consider the situation where NA sperm withallele A and NB sperm with allele B compete to fertilizean egg. Suppose that all NA +NB sperm start swimmingfor the egg at the same time, and that there will be onlya single sperm to win. If interactions between sperm arenegligible, the distribution of the velocity of the “fastest”

among NZ sperm with allele Z (=A or B) is given by the(cumulative) probability:

P [max{XZ1 , ..., XZNZ} < x]

=(P [XZ < x]

)NZ=(1− P [XZ > x]

)NZ, (1)

because the maximum among XZ1 , ..., XZNZ

is less than x

if and only if all of XZ1 , ..., XZNZ

are less than x.The winner of the competition will have allele A if the

fastest sperm with allele A outperforms the fastest withallele B, and vise versa. Therefore, the probability thatthe winner has allele A is given by:

P [winner = A | NA, NB ]

=

∫ +∞0

dxd

dxP [max{XA1 , ..., XANA} < x]×

×P [max{XB1 , ..., XBNB} < x]

=

∫ +∞0

dx NAfA(x)(1− P [XA > x]

)NA−1 ××(1− P [XB > x]

)NB= 1−

∫ +∞0

dx(1− P [XA > x]

)NA ××NBfB(x)

(1− P [XB > x]

)NB−1. (2)

The last equation could be derived via partial integration.It should be noted that, as P [winner = A | NA, NB ] +P [winner = B | NA, NB ] = 1, the second term of theright hand side of (2) is the probability that the winnerhas allele B. The “master equation”, Eq. 2, will providethe basis for the theory of sperm competition in generalsituations. When the mutation is exactly neutral, fA(x) ≡fB(x), this master equation can be easily integrated togive: P [winner = A | NA, NB ] = NANA+NB , which faithfullyreproduces the expectation in an exactly neutral situation.

Here, let us define δf(x) as the difference between fA(x)and fB(x), that is, δf(x) ≡ fA(x)− fB(x). In the follow-ing, we set f(x) ≡ fB(x), although the result is essentiallythe same even if we set f(x) ≡ fA(x). Suppose δf(x)is small (

∫ +∞−∞ dx|δf(x)| � 1), then the master equation

Eq. 2 can be approximated up to O(δf) as:

P [winner = A | NA A′s & NB B′s ]

≈ NANA +NB

{1 +

NBNA +NB

ψA

}≈ NA

NA +NB(1− ψA)≈ NA(1 + ψA)NA(1 + ψA) +NB

, (3)

where ψA is given by:

ψA ≈ (NA +NB)×

×∫ +∞0

dx exp (−(NA +NB)P [Xf > x]) δf(x)

(4)


when NA + NB � 1 (For the derivation of (4) and theequation for a wide range of NA and NB , refer to Note 1of supporting information, File S1.) It is very interestingto note that the approximate equation [3] demonstratesthat the sperm-competition process we model here can bedescribed similarly to the inter-individual competition inthe standard population genetics, that is, allele A has aselective advantage, or “competitive advantage”, ψA overallele B in the “population” of competing sperm. This se-lective advantage can be easily incorporated into the basicframework of population genetics as we will show in thenext section.

It is known that in mammals “intercellular bridges” con-nect four haploid spermatids produced by the meiosis of asingle diploid male germline cell (spermatocyte) (e.g. Rus-sell et al. 1990; de Rooji and Russell 2000; Yoshida2010). Such intercellular bridges are believed to trans-port gene products among the connected spermatids, andtherefore could violate our theory’s key assumption thatthe performance of each sperm is determined by (the geneproduct of) its own allele. However, the theory works aslong as intercellular bridges are imperfect so that there issome difference in the composition of allelic gene productsbetween sperm with different alleles. The only requiredadjustment is that the PDF of velocity should take ac-count of the effect of intercellular bridges.

In Figure 2, we consider some typical cases where a mu-tation changes the PDF of the velocity (f(x)): (i) the PDFis exponential and the mutation from B to A increases themean (Figure 2A); (ii) the PDF is a normal distributionand the mutation increases the mean (Figure 2B); (iii) thePDF is a normal distribution and the mutation increasesthe variance without changing the mean (Figure 2C); and(iv) the PDF has a finite maximum and the mutation in-creases the maximum (Figure 2D). As an example of case(iv), we consider a power-law function with the exponentα and the maximum xM :

f(x) =

{α+1xM

(1− xxM )α for 0 < x < xM ,

0 for x ≥ xM .(5)

In all cases, the mutation is adaptive because the win-ning probability of A should be larger than that of B.These cases should cover quite a wide variety of changesin the PDF of the velocity, because what really matters insperm competition is the change in the right tail (or end)of the PDF. Regarding the right tail, case (iv) may bethe most natural because the sperm velocity should neverreach infinity in nature. Other cases are considered be-cause exponential and normal distributions are very pop-ular functions. Each of the four cases specifies f(x) andδf(x), with which we can calculate ψA as a function ofthe numbers, NA and NB , of sperm with alleles A andB, respectively, by using Equation (4). Here, we assumeNA = NB ≡ NhSp and investigate the enhancing effect ofthe number of sperm on the competitive advantage, ψA,

by defining

R[ψA](NhSp) ≡ ψA(NhSp)/ψA(NhSp = 1). (6)

R[ψA](NhSp) represents how much the competitive advan-tage is enhanced in comparison with the extreme case withNhSp = 1, where only one sperm with allele A competesagainst only one with allele B. It should be noted thatR[ψA](NhSp) does not depend on the magnitude of thePDF difference δf(x) (as long as |δf(x)| � 1), becausethe scale factor of δf(x) in the denominator cancels outthat in the numerator. Figure 3 shows the dependence ofR[ψA](NhSp) on NhSp for cases considered here.

In all cases, the sperm competitive advantage, ψA, in-creases as the half-number of sperm, NhSp, increases (Fig-ure 3; For derivation, refer to Notes 2 and 3 of supportinginformation, File S1). For cases (i) and (ii), ψA increasesroughly proportionally to the logarithm or its power of thetotal sperm number (cyan and green lines in Figure 3). In

case (iv), ψA is roughly proportional to (NhSp)1

α+1 (red,blue, and purple lines in Figure 3), where α is the ex-ponent used in Equation 5. The slope is determined byα. The quantitative difference among cases (i), (ii) and(iv) depends on how the allelic difference in the PDF dis-tributes along the velocity axis, especially near the right-tail. Thus, regardless of the PDFs of velocity, Figure 3shows that the efficacy of selection through a sperm com-petition increases dramatically with the number of mutu-ally competing sperm.

Case (iii) is unique among the four cases because thePDFs for the two alleles have the same mean, while there isa difference in the shape of the right tail, which is the partthat really matters. The result for this case is not shownin Figure 3 because the competitive advantage is zero ina one-on-one competition. Nevertheless, the competitiveadvantage is positive when NhSp > 1, and it increasesroughly proportionally to ln(NhSp) when NhSp � 1; itsasymptotic behavior is quite similar to that of case (i). Ina sense, this case (iii) eloquently demonstrates the unique-ness of sperm competition; A mutant allele could gain abig advantage in a competition among numerous spermeven if it has on average no advantage in a one-on-onecompetition between two sperms with different alleles.Fixation probability of a mutation: In the previoussection, we showed that the advantage of allele A over Bin sperm competition can be described by a single param-eter, ψA, which can be readily incorporated into the basicpopulation genetic framework. We are here interested inhow the frequencies of alleles A and B change, from whichwe can derive the fixation probability of a mutant allele.We assume that A and B are selectively neutral except insperm competition. In other words, it is assumed that thephenotypic effect at the sperm level determines the fate ofthe mutation.

Let us consider the expected frequency of allele A atgeneration t + 1 conditional on the frequency at gener-ation t. We ignore recurrent mutations between them.


A! B!

C! D!

Allele B!Allele A!

Allele A!

Allele B!

Allele A!

Allele B!

Allele B!

Allele A!

FIGURE 2.– Illustration of typical patterns of mutational changes in the probability density function of the velocity.Panels A, B, C, and D correspond to cases (i), (ii), (iii), and (iv), respectively, which are detailed in the text.

1 !

10 !

100 !

1,000 !

10,000 !

100,000 !

1,000,000 !

1.0E+00! 2.0E+00! 3.0E+00! 4.0E+00! 5.0E+00! 6.0E+00! 7.0E+00! 8.0E+00! 9.0E+00! 1.0E+01! 1.1E+01!

Com

petit

ive A

dvan

tage

Enh

ance

men

t!

NhSp (= NA = NB)!1! 103! 106! 109!

(i)!

(ii)!

(iv) ! = 3!

(iv) ! = 2!

(iv) ! = 1!

FIGURE 3.– Enhancement, R[ψA](NhSp), of sperm competitive advantage (ψA) for a range of the half-sperm number(NhSp) as compared to that in one-on-one competition. The cyan and green lines represent case (i) and case (ii)respectively. All other lines represent case (iv). The cases with α = 1, 2, and 3 are colored red, blue, and purple,respectively.


Here we focus on only “successful competition”, whereeggs are successfully fertilized by sperm, because unsuc-cessful competition do not contribute to the populationgenetic process. This should be reasonable under the as-sumption that the overall probability of successful fertil-izations is independent of paternal and maternal geno-types. Let P (P ) [Z | Z1Z2] denote the probability that asperm with allele Z (= A or B) wins a successful compe-tition among sperm ejaculated by a male individual withgenotype Z1Z2 (Z1, Z2 = A or B). Obviously, for homozy-gous males, we have: P (P ) [A | AA] = P (P ) [B | BB] = 1,and P (P ) [A | BB] = P (P ) [B | AA] = 0. For heterozygousmales, we can use Eq. 3. We also assume that there areas many sperm with allele A as those with allele B, i.e.NA = NB . Note that this simplified assumption shouldnot be used for mutations that directly change the num-bers of active sperm with different alleles, such as muta-tions on poisson-antidote type genes and apoptosis relatedgenes (e.g., Da Fonseca et al. 2010).

Then, assuming a very small mutational effect (i.e.,δf(x) is small), we have:

P (P ) [A | AB] = 12

(1 +

ψA2

), (7)

P (P ) [B | AB] = 12

(1− ψA

2

). (8)

Then, through some calculations detailed in Note 4 ofsupporting information, File S1, we get a recursion rela-tion:

pt+1(A) = pt(A) +ψA4pt(A)(1− pt(A)) , (9)

andpt+1(B) = 1− pt+1(A). (10)

Here,

pt(Z) ≡1

2

{p(P )t (Z) + p

(M)t (Z)

}is the gender-averaged frequency of allele Z (= A or B) at

the t-th generation. The variables p(P )t (Z) and p

(M)t (Z)

are the frequencies of allele Z transmitted paternally andmaternally, respectively. As long as we use this equation,our theory does not have to assume equal population sizesof males and females.

We can incorporate the above deterministic recursionequation directly into the standard population genetic the-ory, and obtain the fixation probability by taking geneticdrift into account. To do this, it is sufficient to notice thatour recursion equation, Eq. 9, is equivalent to the standarddeterministic recursion equation of the allele frequency:

pt+1(A) = pt(A) + s pt(A)(1− pt(A)) ,

when allele A has a selective advantage of s (� 1) overallele B. Therefore, the diffusion theory framework asunfolded in section 8.8.3 of Crow and Kimura (1970)

applies also to the current case, if s is replaced by ψA/4.Thus, we have the fixation probability u(p) of allele Awhen its initial frequency is p:

u(p) =1− exp(−NeψA p)1− exp(−NeψA)

, (11)

where Ne is the effective population size. The fixationprobability of a single mutation with initial frequency p =1/(2N) is given by

u(1/2N) =ψANe/(2N)

1− exp(−NeψA), (12)

where N is the actual population size. Or, more simply,if Ne = N , equation 11 is approximated as:

u(1/2N) =ψA/2

1− exp(−NψA). (13)

This in turn reduces to u(p) ≈ ψA/2 when NψA � 1.It should be noted that, in this equation, we set the

initial frequency of a mutant following the standard treat-ment in population genetic theory, that is, a single haploidmutant with frequency p = 1/(2N) arises in the popula-tion at rate 2Nµ, where µ is the mutation rate per gen-eration per haploid locus. It should not be unreasonableto assume that this also works in sperm genes. There aremany cell divisions in a single male individual, and a mu-tation that occurred in an early stage would result in alarge number of sperm having the mutation. Mutationsat later stages would occur more frequently but generatefewer mutant sperm each. The probability that the winneris one of such mutant sperm may be affected by selectionon the mutation, but this effect should be very small ina single generation. If so, we can approximate the mu-tation process in a single (male) individual as a randomprocess along a “genealogy” of competing selectively neu-tral sperm that originates from a single zygote. Then, µ(i.e., the per-generation mutation rate) should be definedas the expected number of mutations from the top (i.e.,root) to any tip (i.e., leaf) of the “genealogy”. This situa-tion is analogous to the typical treatment in the standardtheory of population genetics.

It would be intriguing to compare Equation 13 and thefixation probability of a normal adaptive mutation withselective coefficient s, i.e., u(1/2N) = 2s1−exp(−4Ns) . The

two equations are identical when s = ψA/4, or the ef-fect of the sperm competitive advantage is 1/4 as large asthat of the standard additive selective advantage. Thereare two factors, each contributing 1/2 independently, thatare multiplied together to give this ratio of 1/4. Thefirst factor comes from the assumption that the trans-mission of alleles through females is selectively neutral.The other factor comes from the fact that competitionamong paternity-sharing sperm cause a selective bias onlythrough heterozygous males.


Impact of paternity-sharing sperm competition ondN/dS ratios: Now that we established the popula-tion genetics theory of sperm sharing the paternity, wecan estimate their impact on the dN/dS ratios of spermgenes. First, assuming that synonymous mutations do notchange the fertilization efficiency of sperm at all, the fix-ation probability of a synonymous mutation in a diploidpopulation of size N is given by

P [fixed | synonymous] = 12N

,

according to the neutral theory (Kimura 1968, 1983).

Next we consider non-synonymous sites. According totheir impacts on the sperm fertilization efficiency, mu-tations on such sites will be classified into three cate-gories, namely, those that are competitively (a) disad-vantageous, (b) neutral, and (c) advantageous. Then,roughly speaking, three parameters will govern the av-erage fixation probability of non-synonymous mutations:(i) the proportion, pN , of quite strictly neutral mutationsamong all non-synonymous mutations; (ii) the proportion,pCA, of competitively advantageous (CA) mutations; and(iii) the sperm competitive advantage (ψA) that followsa PDF, FCA(ψA), over competitively advantageous muta-tions. Note that because of the inflation in the competitive(dis)advantage deduced above, even slightly disadvanta-geous mutations will be immediately removed from thepopulation. Therefore, the proportion of competitivelydisadvantageous mutations, pCD (= 1 − pN − pCA), con-tributes only negligibly little, if any, to the dN/dS-ratio.

Given these three key parameters, applying Equation 13will provide the average fixation probability:

P [fixed | non− synonymous]

= pCA

∫ +∞0

dψA FCA(ψA)ψA/2

1− exp(−NψA)+pN2N

,

(14)

This is a theoretically rigorous expression, but the func-tional form of FCA(ψA) is usually unknown. In such acase, it is more convenient and intuitive to define ψA asthe “mean” of ψA, via the following equation:

ψA/2

1− exp(−NψA)≡∫ +∞0

dψA FCA(ψA)ψA/2

1− exp(−NψA).

(15)

Now, taking the ratio of the two fixation probabilitiesand assuming that the mutation rate is identical at syn-onymous and non-synonymous sites, we get a formula forthe average dN/dS-ratio:

(dN/dS) =P [fixed | non-synonymous]P [fixed | synonymous]

= pCA (dN/dS)|CA + pN . (16)

Here (dN/dS)|CA is the average contribution to thedN/dS ratio from competitively advantageous mutations:

(dN/dS)|CA ≡P [fixed | non-synonymous, CA]

P [fixed | synonymous]

=N ψA

1− exp(−N ψA). (17)

If the enhancement effect R[ψA](N) is very large, sayit exceeds 100, we may be able to assume that there arevirtually no quite strictly neutral mutations, i.e. pN � 1,and that almost all non-synonymous mutations could bemore or less selected. In such a case, Equation (16) couldbe simplified as:

(dN/dS) ≈ pCA (dN/dS)|CA . (18)

Thus, the average dN/dS value is now approximately de-termined by two parameters, i.e., the proportion of com-petitively advantageous mutations (pCA) and the “mean”competitive advantage of such mutations (ψA).

The two colored curves in Figure 4 schematically illus-trate the expected dN/dS ratio averaged over CA mu-tations ((dN/dS)|CA) as functions of the winning proba-bility, computed by Equation (17), with N = 104 (greenline), and = 105 (cyan line). In all cases, (dN/dS)|CAmonotonically increases as the winning probability in-creases, and becomes asymptotically proportional to thecompetitive advantage: (dN/dS)|CA ≈ NψA. Hence thegradient of the curve depends almost linearly on the pop-ulation size N .Application to data: Equation (16), or its approxima-tion (18), could be used to estimate the intensity of selec-tion on sperm genes or on any subregions within them. Asan example, we use mammalian sperm protein genes, formany of which rapid amino acid subsitiutions have beendemonstrated (Wyckoff et al. 2000; Torgerson et al.2002; Swanson et al. 2003). We use the data in Torg-erson et al. (2002) and in Swanson et al. (2003). Intheir ML analyses, the proportion of codon sites with ele-vated amino-acid substitution rates and the dN/dS valueaveraged over such sites were inferred for each gene. Bothstudies compared two models of codon evolution underthe framework of Yang et al. (2000). One is a purifyingselection model known as “M7”, in which the expecteddN/dS is assumed to be beta-distributed between 0 and1, and the other is a composite selection model known as“M8” that accommodates an additional class of sites withdN/dS > 1 on top of the classes of negatively selectedsites as incorporated in M7. It was found that M8 fittedsignificantly better than M7 for some sperm genes (fourgenes in Torgerson et al. (2002) and six genes in Swan-son et al. (2003)), strongly indicating that these genesare likely involved in determining sperm performance. Inthe class of positively selected sites, estimates of (dN/dS)ranged from 4.0 to 6.9 for the former four genes and from


Torgenson et al. 2002!

Swanson et al. 2003!

0!

2!

4!

6!

8!

10!

50.000%! 50.005%! 50.010%! 50.015%! 50.020%! 50.025%!

Aver

age

dN/d

S ov

er C

A sit

es!

Winning Probability of Mutant!

N=10,000! N=100,000!

FIGURE 4.– Average dN/dS as a function of the winningprobability of a mutant allele. The colored curves showthe dependence of (dN/dS)|CA on the winning probabil-ity ( 12 (1 + ψA/2)) of the mutant allele A, computed fromEquation (17). The diploid population size (N) is set tobe 10+4 (green line) and 10+5 (cyan line). The red andblue shades represent the ranges of such dN/dS values ob-tained by Torgerson et al. (Torgerson et al. 2002) andby Swanson et al. (Swanson et al. 2003), respectively.

1.3 to 3.9 for the latter six genes (the red shade and theblue shade, respectively, in Figure 4).

To estimate the mean winning probability,

P(P )

[A | AB] = 12 (1 + ψA/2), corresponding tothe above intervals, we use Equation (18). The propor-tion of advantageous mutations (pCA) is usually unknownalthough it may not be very large as most of mutationsshould be disadvantageous. Therefore, we convention-ally assume pCA = 1, which should provide a possiblelower-bound of estimates of ψA. Because this assumptionof pCA = 1 means that almost all non-synonymousmutations on the sites of interest are competitivelyadvantageous, it obviously causes an underestimation ofψA. A caveat in understanding this result is that thetheory might overestimate the role of positive selectionin the haploid phase because it assumes neutrality ofmutations in all other phases.

Assuming the population size to be N = 104 (a typi-cal effective population size for humans, from Takahata(1993)), we estimated that the winning probability rangedfrom 50.010 % to 50.017 % and from 50.0014 % to 50.010%, corresponding to Torgerson et al. (2002) and Swan-son et al. (2003), respectively. As demonstrated in Fig-ure 3, competition among millions to billions of sperm areexpected to enhance the effect of a mutation greatly, al-

though the degree of enhancement differs depending on thePDF of velocity and the type of its change. If we assumecase (iv) with α = 3 as an example, and given NhSp ∼ 108(an approximate number for humans, from e.g., Manningand Chamberlain (1994)), we predict that the selectiveadvantage at the single-sperm level should be, on average,4.7×10−8 to 5.7×10−7. If these values are directly applied,they are equivalent to the population selection intensity,2Ns, of 9.4×10−4 to 0.011. If we assume a bigger popula-tion size, e.g., N ∼ 106 for mice (Keightley et al. 2005),the obtained range of equivalent 2Ns is unchanged. Thisis because the population size, N , impacts the averagedN/dS-ratio only through the product NψA/4 or Ns (seee.g., Equation 17). In contrast, if we are given a smallernumber of sperm, e.g., NhSp ∼ 106 for mice (Manningand Chamberlain 1994), the single-sperm level selectiveadvantage would be equivalent to 2Ns ∼ 2.8×10−3 - 0.034,which is about 3 times larger than the range inferred forhumans. This results from the positive correlation be-tween NhSp and ψA (Figure 3).

Thus, this results indicates that typical selective ad-vantages of mutations at the single-sperm level should bevery small; if selection acted on these mutations in thestandard fashion (as in Figure 1A), the effect of selec-tion would be negligible and they would behave almostas if they were neutral or nearly neutral. In other words,fierce competition among paternity-sharing sperm enableselection to act quite efficiently on mutations with suchtiny phenotypic effects, which could cause dN/dS valuesto significantly exceed 1 in some regions.

Discussion

In this article, we theoretically examined the populationgenetic behavior of mutations in sperm genes. We mod-eled the processes at two levels. One is the standard pop-ulation genetic process, in which the population allele fre-quencies change generation by generation, depending onthe difference in selective advantages. The other is thesperm competition during each genetic transmission froma generation to the next generation.

For the sperm competition process, we considered a verysimple situation with monogamous mating, so that selec-tion needs to be considered only in matings involving het-erozygous males. In a single mating process involving aheterozygous male, a huge number of sperm with allelesA and B (approximately equal in number) compete to fer-tilize a single egg. Our theory demonstrates that a veryslight difference in sperm performance (i.e., velocity as wedefined) amounts to quite a large difference in the win-ning probability. We found that this probability is givenby a function of an important parameter, ψA, namely thecompetitive advantage of allele A over allele B in a sin-gle mating. We also demonstrated that ψA is much largerthan it would be in a one-on-one competition between a


pair of sperm, one with allele A and the other with B.This suggests that a very small phenotypic difference atthe single sperm level can be enhanced by fierce spermcompetition.

For the generation-by-generation process, the standardpopulation genetic theory can be directly applied withslight modifications. The only difference is that selec-tion works only through heterozygous males. In a simpleone-locus two-allele model with alleles A and B, our the-ory shows that the fixation probability of a newly arisenmutant with A is given by Equation 13 if allele A has acompetitive advantage of ψA over B. This equation indi-cates that four times ψA in our model is equivalent to theselective advantage s in the standard model of additiveselection. Of this reduction to a quarter in the efficacyof selection, a half is due to the neutrality (actually, ab-sence) of the process for females, and another half is dueto selection operating only through heterozygous males.

While the efficacy of selection is reduced to 1/4 forsperm genes in the generation-by-generation process, theenhancement of selection in the sperm competition dur-ing each generation is much larger, potentially increasingadaptive amino acid substitutions in sperm genes. Indeed,as we demonstrated theoretically, the elevated dN/dS val-ues observed in sperm genes in mammals could be ex-plained by mutations whose effects on individual spermare so weak that they would be classified as “neutral” ina normal population genetic framework.

Our minimal model focuses on competition amongsperm that share paternity. These competition are ubiq-uitous and occur regardless of whether the mating systemis monogamous, polygamous, or external. Thus, our re-sult could provide a potentially important explanation ofrapid evolution of sperm genes with a variety of functionsin a wide range of species, as long as they are expressedin the haploid phase. The extent to which our model canbe applied depends on how common haploid expression isamong sperm genes. There are multiple lines of evidencethat haploid expression in sperm is indeed quite commonas mentioned in the Introduction (see e.g., Joseph andKirkpatrick 2004; Good and Nachman 2005; Doruset al. 2010). A notable work is the recent proteome-scaleevolutionary analysis in mouse by Good and Nachman(2005), which showed that sperm genes expressed aftermeiosis tend to have higher dN/dS values than othersperm genes. Their study implies that, in such genes withhaploid expression, the contribution of sperm competitionto the rapid amino acid evolution should be large.

Our model may be well-merged with previous models ofpost-copulatory sexual selection (e.g., Birkhead and Piz-zari 2002; Swanson and Vacquier 2002b; Clark et al.2006; Turner and Hoekstra 2008) or sexual conflict(e.g., Rice and Holland 1997; Frank 2000; Gavrilets2000; Chapman et al. 2003; Hayashi et al. 2007), all ofwhich focus mainly on competition or conflicts among in-dividuals. Our results suggest that competition among

paternity-sharing sperm can boost any mode of post-copulatory sexual selection and/or sexual conflict. Forexample, coupling our model with sexual conflict providesa more powerful explanation of the long-term accelerationof dN/dS in sperm genes. Our model alone does not guar-antee that the acceleration of amino acid substitution lastslong. This is because each sperm gene eventually reachesthe optimum of its fitness landscape, from which no adap-tive mutations are expected. Thus a high dN/dS cannotbe expected as long as the gene’s fitness landscape keepsits shape (or optimum). This indicates that a long-termacceleration may require factors that shift the optimum.

Sexual conflict would be one example, which continu-ously shifts the optimum by a “co-evolutionary chase” be-tween the male and female genes (e.g., in Rice and Hol-land 1997; Frank 2000; Gavrilets 2000; Chapmanet al. 2003; Hayashi et al. 2007). An interesting questionin this regard is why high dN/dS values are observed com-monly among sperm genes but quite rarely among female-reproductive genes in many species (e.g. Swanson andVacquier 2002b; Clark et al. 2006). One importantconclusion of our theory is that fierce sperm competitioncould enable even a mutation with a tiny phenotypic ben-efit to be fixed as if it were strongly selected for. Thissuggests that, even to a small shift in the female environ-ment (e.g., caused by only a single amino acid change),a sperm gene could respond via a large number of aminoacid changes, each of which alters the sperm phenotypeonly slightly.

Another potentially important factor would be inter-male competition through different phenotypes of spermor other reproductive apparatuses (such as copulatoryplugs). This could also change the environment in whichsperm compete against one another, and thus could shiftthe optimum of the fitness landscape of each sperm gene.

In this work, we only focused on the effects of mutationson sperm performance and ignored their effects on otherphenotypes. If, however, a sperm gene is pleiotropic, itis obvious that the fitness at the sperm level is not theentire factor to determine the fate of the mutation (seee.g., Crow 2012). Nevertheless, such pleiotropic effectsmight not significantly affect the main conclusions of thisstudy, for two reasons. First, a majority of sperm genesobserved to have high dN/dS values seem to be specificto sperm (e.g., Good and Nachman 2005; Turner andHoekstra 2008). Second, even if a sperm gene is indeedpleiotropic, it is important to notice, again, that fiercesperm competition could enhance the effect of a muta-tion on sperm functions but not its effect on other tissues.Therefore, there should be quite many occasions wheresperm competition contribute to the accelerated dN/dSin sperm genes with haploid expression. This, combinedwith the observation that haploid expressed genes accountfor a substantial fraction of the sperm proteome (e.g.,Joseph and Kirkpatrick 2004; Good and Nachman2005), could be one of the major explanations of the gen-


eral trend that a wide variety of sperm genes show highdN/dS ratios in various taxonomic lineages.

Our theory may explain some previous observationsthat seemed enigmatic. One is on the role of post-copulatory sexual selection among (male) individuals asa potential major cause of the elevated dN/dS values. Ifthis plays the major role, one would expect a positive cor-relation between dN/dS and the intensity of sexual se-lection. However, there was no significant correlation ina molecular evolutionary analysis of male sperm genes inrodents (Ramm et al. 2008). Why can dN/dS be ele-vated in a species with weak sexual selection as much asin species with intense sexual selection? One possible andsimple answer could be that, at least in rodents, competi-tion among paternity-sharing sperm are much more potentthan inter-male sperm competition. The former competi-tion takes place irrespective of whether sexual selection oc-curs at the individual level or not, and our theory predictsthat the former greatly enhances the selective advantageof a mutation. Thus, if the former’s “baseline” effects aremuch larger than the effects influenced by individual-levelsexual selection, the correlation between the protein evo-lution rate and the intensity of sexual selection would berelatively too small to be observed.

It should be noted here that our theory applies exclu-sively to sperm genes, especially haploid expressed ones,but that it does not apply to male reproductive genes lack-ing expression in sperm. Some genes of the latter typeare known to have high dN/dS and show positive corre-lation between dN/dS and the intensity of (inter-male)sexual selection. Among those showing the correlation isa seminal vesicle-derived protein, Svs2, which is a majorcomponent of the copulatory plug in rodents (Ramm et al.2008). Similarly, semenogelins, SEMG1 and SEMG2, thatare known to prevent other males’ sperm from reachingthe oocyte, also showed correlations in the primate lin-eages (e.g., Ramm et al. 2008, and references therein).The rapidly evolving Drosophila male accessory gland pro-teins (e.g., Swanson et al. 2001) also belong to this type.From their functions, it is obvious that these genes playimportant roles in inter-male competition, and that theirrapid evolution is not caused by sperm competition. Itis indicated that inter-male sexual selection is an impor-tant mechanism to increase dN/dS especially of such malereproductive genes not expressed in sperm.

Flies are unique in that their sperm genes do notshow significantly higher dN/dS values than genome av-erage (Dorus et al. 2006). Given that any kind of se-lection works very efficiently through fierce competitionamong sperm sharing the paternity, this unique obser-vation should indicate that such sperm competition maynot be very intensive in flies. Indeed, there are severallines of evidences that files have quite small numbers ofcompeting sperm per egg and little haploid expression(see e.g., Erickson 1990; Manning and Chamberlain1994). This agrees with our prediction that mutational

effects on sperm genes should be enhanced in a mannerpositively correlated with the number of sperm per egg(e.g., Figure 3).

We thank two anonymous reviewers for their helpful com-ments. This work is in part supported by grants from theJapan Society for the Promotion of Science (JSPS) to HI.

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