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Genetics, Vol. 176, 513-526, May 2007, Copyright © 2007
doi:10.1534/genetics.106.056150
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,1
,**
* Laboratory of Social Genome Sciences, Department of Medical Genome Sciences, Graduate School of Frontier Science and Institute of Medical Science and Graduate Program of Biophysics and Biochemistry, Graduate School of Science, University of Tokyo, Tokyo 108-8639, Japan, Laboratory for Language Development, RIKEN Brain Science Institute, Saitama 351-0198, Japan, Department of Biology, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan and ** Evolution and Ecology Program, International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria
1 Corresponding author: Laboratory of Social Genome Sciences, Department of Medical Genome Sciences, Graduate School of Frontier Science and Institute of Medical Science, University of Tokyo, 4-6-1 Shirokanedai, Minato-ku, Tokyo 108-8639, Japan.
E-mail: ikobaya{at}ims.u-tokyo.ac.jp
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ABSTRACT |
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 TOP ABSTRACT MODELSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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the ratio of the burst size 
under damage to host cell physiology induced by an unrepaired double-strand break to the default burst size 
It was not until this effect was taken into account that the evolutionary advantage of DNA repair became apparent.
The necessity to repair damage on the genome using undamaged genetic material as a template has been considered to be an immediate factor responsible for the origin of sex (BERNSTEIN et al. 1984; LONG and MICHOD 1995; MICHOD and LONG 1995; MICHOD 1998). Recombination genes may have arisen in the first instance because of their role in repair, and this may have remained their major function until today. Indeed, many experiments have demonstrated that homologous recombination is stimulated by damage to DNA. Transformation frequencies in Bacillus subtilis increased with increasing levels of DNA damage when the cultures are given homologous DNA (MICHOD and WOJCIECHOWSKI 1994). A DNA double-strand break is repaired by copying homologous DNA, with and without associated crossing over, in Escherichia coli by lambdoid bacteriophages (KOBAYASHI and TAKAHASHI 1988; TAKAHASHI and KOBAYASHI 1990).
However, the repair hypothesis does not readily explain the origin and maintenance of sex in eukaryotes, which is defined as meiotic crossing over built in the haploid–diploid cycle (MAYNARD SMITH 1988; BARTON and CHARLESWORTH 1998). Previous studies of the evolution of the haploid–diploid cycle showed that the origin and maintenance of this cycle could be solely explained by faster removal of recurrent deleterious mutations in haploids and greater resistance to genetic damage in diploids (KONDRASHOV and CROW 1991; MAYNARD SMITH and SZATHMARY 1995; CAVALIER-SMITH 2002; SANTOS et al. 2003). The necessity of repair was not revealed. Furthermore, it is obvious that double-strand repair does not require meiosis and syngamy of sexual reproduction in eukaryotes at all. Indeed, the most popular hypotheses for the evolution of sex in eukaryotes ascribe the advantage of sex to accelerated adaptation to ever-changing environments, which likely result from antagonistic interactions with other organisms, or to efficient elimination of deleterious mutations. A thorough review of this subject has been carried out by KONDRASHOV (1993).
The molecular mechanisms underlying meiotic recombination may provide some clue as to this issue. Meiotic recombination in yeast is initiated by the formation of a double-strand break in one of the numerous sites along the chromosome (KROGH and SYMINGTON 2004). It is repaired by copying a sister chromatid or a homologous chromosome, which may result in gene conversion. This break repair (gene conversion) is often accompanied by crossing over of the flanking sequences. This led to the hypothesis that the advantage of meiotic recombination is in the elimination of "nonself" sequences from the genome (TAKAHASHI et al. 1997; KOBAYASHI 1998). Similarly, the advantage of sex is hypothesized to be defense against selfish genetic elements (WELCH and MESELSON 2000). The repair hypothesis is strongly related to these hypotheses.
It can be imagined that the costs of sex in the prokaryotes that lack the haploid–diploid cycle are much smaller than those in the eukaryotes, although the machinery for natural transformation appears to be somewhat costly. Therefore, the repair hypothesis can more adequately explain the evolution of sex in the prokaryotes (often called the origin of sex) than in the eukaryotes, although it is not obvious why DNA double-strand break repair has to be often accompanied by crossing over of the flanking sequences (KUSANO et al. 1994) because crossing over has still a potential to break apart favorable combinations of genes (SHIELDS 1988). However, there are also observations and arguments that question experimental evidence of the repair hypothesis for prokaryotes. One of the observations is that transformation with a small part of the Haemophilus influenzae chromosome was as effective in increasing survival as with the whole chromosomal DNA (MONGOLD 1992). This result was not predicted by the repair hypothesis because the DNA fragment supplied would be able to patch <1% of the possible sites of damage in the H. influenzae genome. The above-mentioned experiments with B. subtilis may not have been sufficiently sensitive to detect such modest differences in bacterial survival (REDFIELD 2001).
In this work, to examine the validity of the repair hypothesis, we focus on sex in bacteriophages in the form of DNA double-strand break repair by gene conversion. A major role of the homologous recombination machinery carried by DNA bacteriophages is suggested to be repair of DNA double-strand breaks made by restriction-modification systems through the double-strand break repair mechanism (TAKAHASHI et al. 1997; KOBAYASHI 1998). Attack by a cellular restriction enzyme on invading DNA of several bacteriophages initiates recombinational repair by gene conversion if there is homologous DNA. Because several restriction-modification systems behave as selfish mobile elements, such as transposons and bacteriophages (NAITO et al. 1995; KOBAYASHI 1998, 2004), there is an aspect of biological interaction in this mode of homologous recombination. We model the interaction between a bacteriophage and a restriction-modification system in a bacterium as antagonistic coevolution and explore conditions for sexual (recombination/repair-proficient) phages to evolve by numerical simulations.
As is already suggested by the repair hypothesis, sex in DNA bacteriophages has a cooperative (altruistic) aspect. A repair enzyme of a sexual (recombination/repair-proficient) phage is able to repair not only a sexual but also an asexual bacteriophage genome if there is a homologous template chromosome for repair. Namely, the DNA repair enzyme can equally act in cis and in trans, providing an equal opportunity of repair to asexual (recombination/repair-defective) phages. In this case, it can be imagined that evolution of sexual (recombination/repair-proficient) phages is not easy even if the cost of sex is small. Competition between sexual (recombination/repair-proficient) and asexual (recombination/repair-defective) phages in the phage population will become apparent and the former can be viewed as altruistic while the latter can be viewed as selfish.
Our simulation revealed that the sexual (recombination repair) allele is able to evolve only under specific conditions of induced damage to the host cell physiology due to an unrepaired double-strand break. It was not until this effect was taken into account that the evolutionary advantage of DNA repair became apparent.
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MODELS |
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Our model is illustrated in Figure 1, and all the symbols used are explained in Table 1. A bacterial cell either carries a restriction enzyme that can attack a sensitive bacteriophage genome (a+) or does not carry it (a–). Each bacteriophage genome has two loci. The first locus (A) harbors either a restriction-sensitive site (A–) or a restriction-resistant site (A+). The second locus (Rec) of the bacteriophage harbors either a sexual (recombination/repair-proficient) allele (Rec+) or an asexual (recombination/repair-defective) allele (Rec–).
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). The relative proportion of a particular combination of bacteria genotype and infecting bacteriophage genotype(s) is assumed to be given by the product of their frequencies (x, 1 – x, and yij's). Inevitable attack of the restriction enzyme on the restriction site of an invading bacteriophage genome can initiate recombinational repair of the restriction break by gene conversion if there is a co-infecting phage genome and if at least one of them is recombination/repair proficient. The probability of successful repair is denoted by r (r < 1) when one of the two co-infecting phages is "Rec+" and the other is "Rec–" (Figure 1A). When the co-infecting phages are both Rec+, the probability of repair increases to 2r because the amount of Rec enzyme in the host cell is doubled (Figure 1B). If repair succeeds, the "A–" allele of the restricted phage genome is changed to "A+" by gene conversion. Our model assumes that a template chromosome for recombinational repair is supplied only by a co-infecting phage. This assumption of frequent multiple infection is based on the abundance of bacteriophage particles in natural environments (BERGH et al. 1989; WALDOR et al. 2005). We assume that repair cannot occur in single infection because there is no template chromosome for repair in the bacterial cell.
Undamaged or repaired phage genomes survive and give rise to progeny. We designate the number of progeny as burst size, which is defined as the number of virus particles released per cell (WEINBAUER 2004). As illustrated in Figure 1, we assume that the burst size decreases when a double-strand break of one of the co-infecting phages remains unrepaired. This assumption is based on the experimental evidence that a single unrepaired double-strand break on a plasmid molecule or a yeast artificial chromosome induces lethality to a cell (BENNETT et al. 1993, 1996). We thus introduce another parameter of burst size under induction of damage to the host cell physiology 
which is less than or equal to default burst size Two examples of 
and 
are illustrated in Figure 1. The influence of this parameter is apparent only when co-infection results in survival of one of the infecting phages and death of the other phage with an unrepaired double-strand break. When single infection occurs or co-infection leads to the survival of both phages, any damage is not induced and, therefore, the distinction between and 
is unnecessary. Note that if 
= 1.0, the total burst size is equal to the default burst size whether the repair succeeds and leads to the survival of both restriction-sensitive and -resistant phages or it fails and leaves only resistant phage. In the case of successful repair, the two resulting phage genotypes are assumed to give the same number of progeny because there is an upper limit of intracellular resources available in a host cell and they equally share the resources.
There are four genotypes (A+ Rec+, A+ Rec–, A– Rec+, and A– Rec–) in the phage population and two genotypes [restriction positive (a+) and negative (a–)] in the bacterial population. Phages are sampled randomly from the phage population, with the multiplicity of infection (MOI) from 0 to 2, and allowed to infect one of the two genotypes of bacteria. When no infection occurs in a bacterial cell (MOI = 0), or when the restriction-positive bacterial cell is infected by sensitive phage(s), the bacterial cell multiplies.
After single infection either by a Rec+ or by a Rec– bacteriophage, the phage will kill a restriction-negative bacterial cell and produce progeny. On the other hand, a restriction-positive cell will always prevent the growth of a restriction-sensitive phage, but will always yield to a restriction-resistant phage.
Co-infecting phage pairs can be classified into three cases (Rec+ and Rec+ infection, Rec+ and Rec– infection, and Rec– and Rec– infection), each of which is further divided into their allelic states at the restriction locus (A+ or A–). For each combination, the phages experience three possible events (restriction, repair, and burst).
We assume that there is a cost of carriage of a restriction-modification system on a+ bacterium, 
which is realized as a reduced growth rate. The relative fecundity of a+ bacterium to that of a– bacterium depends on the cost of restriction modification as 
Also assumed are the metabolic cost 
for restriction resistance on A+ phage and that 
for recombination/repair capacity on Rec+ phage, both represented by a reduced burst size (the relative fecundity, see Table 1). The relative fecundity of A+ phage is expressed as 
and that of Rec+ phage as 
If a phage carries both a restriction-resistant site and a Rec allele in its genome, the relative fecundity is given by 
We compile a mating table that contains all the infection patterns, their probability of occurrence, and the number of progeny from each pattern. Part of the mating table is shown in Table 2. Note that all the patterns in Table 2 are those for restriction-positive (a+) bacteria. Other patterns for restriction-negative (a–) bacteria are not included because they are trivial, in the sense that all the infecting phages survive and thus the genotype of their progeny always remains the same as that of their parents. The number of phage progeny from an infected bacterium depends on the relative burst size, which is when both of the co-infecting phages (or the singly infecting phage) survive(s) and when one of the co-infecting phages survives in the presence of an unrepaired double-strand break of another phage's chromosome. The expected number of phage progeny is assumed to be given by the product of the relative burst size, the relative fecundity depending on the metabolic costs of restriction-resistance and recombination/repair-proficient alleles, and probabilities of each infection and repair. The number of progeny of the host bacterial genotype in the next generation is represented similarly.
From the mating table, we can write down the following equations. The frequency x of bacteria that have restriction-modification genes changes between generations as
 | (1) |
is the frequency of restriction-sensitive phages (with 
the frequency of restriction-resistant phages). The phage genotype frequencies in the next generation are expressed as
 | (2) |
is the mean fitness of phage, which is given by the sum of right-hand sides of the above equations. Strongly antagonistic interaction between bacteria and phage genotypes represented by frequency-dependent genotypic fitness easily destabilizes an equilibrium of the coupled genotypic dynamics (1)–(2), which, unless the costs of restriction modification in bacteria and restriction resistance in phage are too large, show complex limit cycles of large amplitudes. Even when the phage population is monomorphic with respect to its recombination/repair locus, the coupled dynamics of restriction-negative and -positive bacteria and restriction-sensitive and -resistant phages exhibit limit cycles. With periodic oscillation in genotypic frequencies of bacteria and phage populations, obtaining the analytical "invasion criteria," the sign of the long-term marginal logarithmic growth rate of Rec+ carrying phage introduced into the resident Rec– population, becomes difficult.
We therefore numerically explore conditions that allow the sexual (recombination/repair-proficient) allele to evolve. The procedure is summarized in Figure 2, which is equivalent to the iteration of the recursion (1)–(2) except for the process of mutation described below. After all the combinations are computed on the basis of the mating table, the progeny of each phage/bacterial genotype is summed up to yield the fitness (i.e., expected number of progeny) for all genotypes in each generation. Selection and mutation then operates, resulting in frequency change for each genotype. Mutation is assumed to occur only at the restriction locus of the bacteriophages, which enables us to eliminate the persistence of the repair/recombination allele by mutation–selection balance, because we are interested in the adaptive evolution of the repair/recombination allele. This evolutionary process for one generation is repeated for thousands of generations.
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(and 
). We then summarize how the condition for the evolution of a sexual allele depends on these parameters. |
RESULTS |
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although it always gave sustained cycles of genotypes for our choices of parameters. The dependence of the advantage of the sex allele on is summarized in Figure 3. Apparently, evolution of the repair allele becomes possible when is small and its cost is small.
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is large, the evolutionary dynamics show victory of Rec– phages over Rec+ phages (Figure 4A). Rec+ phages continue to decrease in frequency and become extinct even when the initial frequency is very high (99%). The recombination/repair-proficient allele cannot evolve under this condition. The intuitive reason for the failure of the sex allele is clear: the damaged sensitive genomes of Rec– phage can be repaired by co-infecting Rec+'s enzymes and templates. This implies that Rec– phage can enjoy the advantage of "free repairs" equally efficiently as altruistic Rec+ phage does, yet without paying any cost.
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Detailed dynamical interaction between bacteria restriction-positive genotypes and bacteriophage restriction-resistance genotypes is presented in Figure 4C. Among Rec– phages, the relative frequency of each allele at the restriction-site locus [A– (restriction sensitive) or A+ (restriction resistant)] shows sustained oscillation. This is also true of Rec+ phages (data not shown). In the bacteria, the relative frequency of each allele [a– (restriction negative) or a+ (restriction positive)] alternates in conjunction with the cycles of phage genotypes. These results represent a continuous coevolutionary force acting both on the phage genome and on the restriction-modification system in bacteria. When a prevalent genotype of bacteria is a+ (restriction positive), A+ (restriction-resistant) phages survive and spread. Once A+ (restriction-resistant) phages become prevalent, however, any restriction enzyme of bacteria is no longer effective while its cost still exists. Then a– (restriction-negative) bacteria increase their frequency in the bacterial population. Once a– (restriction-negative) bacteria become prevalent, however, A– (restriction-sensitive) phages in turn have an advantage because resistance of the phage genome is no longer useful and becomes costly. The prevalence of A– (restriction-sensitive) phages makes a+ (restriction-positive) bacteria advantageous and the dynamics return to the former state. Sustained cycles of phage and bacteria genotypes are thus produced. The Rec+ modifier allele in phage, however, consistently decreases as its ability to repair the cleaved restriction-sensitive site benefits co-infecting Rec– phages equally as well as themselves when is large (equal to or only slightly <1).
From these results we were unable to find a definitive evolutionary advantage of the double-strand break repair. However, the results change dramatically as decreases relative to 
as summarized in Figure 5.
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The evolutionary trajectory reveals the Rec+ allele can remarkably increase in frequency by its building up of positive linkage disequilibrium to restriction-resistant sites, as shown in Figure 5B. The sustained cycles of bacteria and phage genotypes then drive the frequency of Rec+ alleles to fixation. The increase of the Rec+ allele occurs in the phase of cycle in which both a+ (restriction-positive) bacteria and A+ (restriction-resistant) phages are prevalent. This indicates that the double-strand break repair between Rec+ phages has a definitive evolutionary advantage when a+ bacteria predominate. For sufficiently small the mutually altruistic repair in Rec+/Rec+ infections produces an advantage for Rec+ by their larger contribution of progeny to the next generation than that in Rec–/Rec– infections, which can overcome the Rec–'s advantage of free repairs in Rec–/Rec+ heterologous co-infections. This at the same time generates a positive correlation between the Rec+ allele and the restriction-resistant allele. The larger contribution of Rec+/Rec+ homologous infection is due to its prevention of induced damage to the host cell physiology by the unrepaired double-strand break in the Rec+/Rec+ infections, as parameterized in the model as (
).
The sustained cycle of each genotype [A– (restriction sensitive) or A+ (restriction positive)] among Rec+ phages is similar (Figure 5C) to the case when Rec+ decreases (Figure 4C). Thus apparently same coevolutionary cycles of bacteria restriction-modification genotypes and phage restriction-resistance genotype have quite different effects on the fate of the sexual allele in phage—they can drive the costly sexual allele to fixation if is sufficiently <1, but fail to do so if is large.
The conditions for the evolution of the recombination/repair-proficient allele also critically depend on the probability of co-infection and the probability r of successful repair in the presence of Rec+ phage. The results of extensive simulations shown in Figure 6 demonstrate that the higher the values of these two parameters, the more likely the evolution of the repair allele. These results indicate that considerable co-infection and repair are necessary for evolution of the double-strand break repair, even when is small.
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DISCUSSION |
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TOPABSTRACTMODELSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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the ratio of the burst size b1 under induced damage to the host cell physiology to the default burst size b0. It was only when this effect of the induced damage was taken into account that the evolutionary advantage of the double-strand break repair became apparent. The validity of the repair hypothesis for the origin of sex is, therefore, confirmed under a limited condition.
Under the condition where is high, the repair allele did not increase at all. Namely, double-strand break repair did not show any evolutionary advantage under this condition. This seems counterintuitive, because progeny of the Rec+ phage indeed increase by repair of its genome in A– Rec+/A+ Rec– infection and the cost of repair is assumed to be not very large (Figure 3). In our model, however, the DNA repair enzyme acts equally in cis and in trans, providing an equal opportunity of repair to Rec– phages. In A– Rec–/A+ Rec+ infection (Figure 1A), once-restricted Rec– phage is repaired by an enzyme from Rec+ phage, resulting in a decrease in Rec+ progeny. Therefore, the benefit of repair for the Rec+ phages is completely counterbalanced by that of the Rec– phages. In addition, even in A– Rec+/A+ Rec+ infection, where both infecting phages are Rec+ (Figure 1B), double-strand break repair confers no advantage for Rec+ because repairing the genome does not change the total burst size. For example, in Figure 1B, 100 progeny of Rec+ result from repair failure, while 50 plus 50 progeny of Rec+ result from repair success. This number is the same as that for progeny of Rec– phage in A– Rec–/A+ Rec– infection without any repair. This is why double-strand break repair confers no selective advantage under the condition 
In contrast, the repair allele did increase from a very low initial frequency when 
As in the case of close to 1, the benefit of repair of Rec+ in A– Rec+/A+ Rec– infection is counterbalanced by that of Rec– phages. However, the fitness difference between Rec+ and Rec– phages is generated when A– Rec+/A+ Rec+ and A– Rec–/A+ Rec– infections are compared. In A+ Rec+/A– Rec+ infection, where both the infecting phages are Rec+, the total progeny of Rec+ increases by repair success. For example, 50 progeny of Rec+ result from repair failure, while 50 plus 50 progeny of Rec+ result from repair success (see Figure 1B). In contrast, in A– Rec–/A+ Rec– infection where no repair occurs, the number of progeny of surviving Rec– phage decreases to (50) because a remaining double-strand break of one of the co-infecting phage genomes induces damage to the host cell physiology. This represents the definitive advantage of repair for Rec+ phages under this condition (see the APPENDIX for further analytical explanations).
The disadvantage of A– Rec–/A+ Rec– infection means Rec– is recessive deleterious. Therefore, Rec– phages decreased slowly as they became rare in the population because the probability of Rec–/Rec– co-infection became lower. After the initial 10,000 generations, Rec– phages decreased in frequency from 99 to 
17%. After the next 10,000 generations, however, Rec– phages did not become extinct and decreased more and more slowly (<0.1%).
In our model, there is no fitness difference between Rec+ and Rec– phages when single infection or infection to a– bacteria occurs. This is also true of co-infection in which both phages are A– or A+. We therefore examined our results in the above explanations by focusing on A+/A– co-infection.
The evolutionary dynamics of Rec+ and Rec– were smooth compared to those of A+ and A– alleles as in Figures 4 and 5. In contrast, if a genetic correlation between the modifer (Rec) and the selected (A) locus is close to 100% and the recombination rate between them is close to 0%, frequencies of modifier alleles (Rec+ or Rec–) strongly depend on those of selected alleles (A+ or A–). This corresponds to a situation where a modifier (Rec) locus sits very close to a selected (A) locus and recombination between them does not occur. Although the situation is possible if there are some restriction sites in a genome, our model assumed one restriction site (A) for simplicity. Therefore, the modifier (Rec) locus did not gain an association with a selected (A) locus and frequencies of modifier alleles (Rec+ or Rec–) changed more slowly than those of selected alleles (A+ or A–). The selection coefficient of the modifier locus is the squared order of that of selected locus (ISHII et al. 1989).
The predominance of Rec– phages under a large condition is caused by complementation. Co-infection of a virus supplying a gene product leads to a defective virus gene that is then represented in the progeny, which instead decreases the progeny of the former functional virus (DENNEHY and TURNER 2004; FROISSART et al. 2004; NOVELLA et al. 2004). This is apparently disadvantageous for the functional viruses (Rec+ phages in our model), which can be viewed as altruists, while the defective (Rec–) phages can be viewed as free riders or cheaters (MAYNARD SMITH and SZATHMARY 1995; KELLER 1999; FOSTER et al. 2004). Meanwhile, the condition of small selectively benefits Rec+ progeny on Rec+/Rec+ infection, as shown in Figure 5B. The repair process under competition between co-infecting phages is considered to act as a mechanism that constrains cheaters (TRAVISANO and VELICER 2004). Our model represents one of the mechanisms for constraining cheaters in microbes (FOSTER et al. 2004). Although cheating, cooperation, and sociality in microbes have not been the focus of attention until recently, these are now being pointed out as fundamental issues in evolutionary theory and in pathogenicity control (SMITH 2001; FROISSART et al. 2004; GRIFFIN et al. 2004; TRAVISANO and VELICER 2004).
We assume that repair-defective (Rec–) phages can produce progeny in the absence of a bacterial recombination system (RecBCD). In lambdoid bacteriophages, packaging of the phage genome into a viable phage particle needs a concatemer form, in which phage DNA units are joined together in a head-to-tail manner (FUJISAWA and MORITA 1997). Formation of the concatemer is blocked by the RecBCD DNase of the host E. coli, which degrades nonself DNA but repairs self DNA marked by an ID sequence (HANDA et al. 1997, 2000). Lambda and other bacteriophages produce an inhibitor of RecBCD DNase (SMITH 1983). Therefore, our model corresponds to the RecBCD-negative states.
In reality, even a single infecting phage genome might encounter homologous prophage genomes in the host cell that can serve as a template for repair. Prophages are abundant in the sequenced bacterial genomes. For example, a natural isolate of E. coli carries 18 prophages and phage remnants, among which 13 are related to bacteriophage lambda (HAYASHI et al. 2001). In addition, an infecting bacteriophage may start replication before attack by a certain type of restriction enzyme, which seems to produce a template for repair of a sister chromosome (HANDA and KOBAYASHI 2005). These effects might provide an additional advantage of double-strand break repair in that a single infecting "A– Rec+" bacteriophage is able to revive to some extent after attack by a restriction enzyme, which would selectively benefit Rec+ phages while eliminating cheaters.
The burst size of a once-restricted and repaired bacteriophage could be lower than that of an undamaged phage on the assumption that an infecting bacteriophage would start replication before attack by certain types of restriction enzyme, as explained above. Accordingly, the restriction and repair process would delay replication of the bacteriophage, which could in turn increase progeny of the undamaged coexisting phage. Our simulation does not explicitly include this effect. However, we already confirmed that the effect could not change the result because Rec+ and Rec– phages had equal opportunities to increase their progeny by this effect.
Our model was constructed in the framework of evolutionary game theory: a powerful tool in both social science and evolutionary biology to analyze social problems involving interdependence among several agents (MAYNARD SMITH 1982; NOWAK and SIGMUND 2004). It is now recognized as being applicable to social interactions such as cheating and cooperation in microbes as well (TURNER and CHAO 1999; KERR et al. 2002; NOWAK and SIGMUND 2002; WOLF and ARKIN 2003; PFEIFFER and SCHUSTER 2005; TURNER 2005; WOLF et al. 2005). It has been claimed that one of the most important challenges lying ahead is to model the interaction of strategies encoded in genomic sequences (NOWAK and SIGMUND 2004). Our model represents one of the first examples of such attempts (see also MOCHIZUKI et al. 2006).
Our one-locus model has been simplified from the gene-for-gene model used by SASAKI (2000), which assumed multilocus and asymmetric gene-for-gene interaction. This simplification enabled us to write down all the interactions between bacteriophages and host bacteria into a simple mating table, even if we also consider a modifier locus (Rec) and co-infection. Multilocus models yield a much greater number of genotypes and of interactions between them, which makes analysis and interpretation difficult. Despite the simplification, our model similarly showed protected genetic polymorphism in the genotype of the host (phage genome in our model) and the parasite (restriction-modification system in bacteria in our model) and produced a sustained cycle of genotype frequencies. This is a robust tendency in many gene-for-gene models, which has been considered to give an advantage to recombination and sexual reproduction, although the cycle itself has not been experimentally proven (see, for example, KORONA and LEVIN 1993). To the best of our knowledge, this work represents the first study examining whether these characteristics of the dynamics enable sex (recombination repair) in bacteriophages to evolve. Because our one-locus models cannot distinguish between gene conversion not associated with flanking crossing over and gene conversion associated with flanking crossing over, whether the dynamics yield a short-term advantage for crossing over remains an unexplored question.
The repair process of our model is assumed to begin only after a double-strand break by a restriction enzyme, which is similar to the "damage-induced sex" proposed by Michod and colleagues (LONG and MICHOD 1995; MICHOD and LONG 1995; MICHOD 1998). However, they assumed different molecular mechanisms, in that gene damage was repaired by cell or protocell fusion with damaged or undamaged partners. These differences lead to different results, especially in a situation in which sexual cells mate with asexual (cheater in our model) cells (MICHOD and LONG 1995).
In this work, we assumed that gene conversion initiated by a restriction break is limited to the broken locus. It would be interesting to consider the case where the gene conversion continues across the Rec site. The biological plausibility of such coconversion between linked loci is well established in meiotic recombination in Ascomycetes (STAHL 1979; PETES et al. 1991). The underlying mechanism could be expansion of the double-strand breakage into a double-strand gap (6:2 segregation of alleles) (SZOSTAK et al. 1983), repair synthesis (5:3 segregation of alleles) (SUN et al. 1991), or mismatch correction on heteroduplex DNA formed from the break site (6:2, 5:3) (NAG et al. 1989; ALLERS and LICHTEN 2001). In the DNA double-strand break repair mediated by bacteriophage lambda recombination machinery, there is evidence for DNA double-strand gap repair (TAKAHASHI and KOBAYASHI 1990). Clustering of exchanges (called high negative interference), which can be explained by mismatch repair on heteroduplex DNA, is observed in the presence of DNA replication (STAHL 1979). A long heteroduplex DNA formed from the cos site (for the molecular ends) was shown to serve as a substrate for mismatch repair (WHITE and FOX 1974; STAHL et al. 1985; LEACH 1996), leading to high negative interference in the presence of block to DNA replication. Such coconversion from A– Rec+/A+ Rec– may lead to A+ Rec–/A+ Rec–. Here the very action of the Rec+ allele leads to its disappearance from the progeny. This cost should be dependent on linkage with the cut locus and the allelic state of the interacting genomes. Therefore, it may be regarded as more similar to the cost of sex that has been analyzed in terms of population genetics (MICHOD and LEVIN 1988) than the simple metabolic type of cost of sex assumed in this work. Understanding its effects would represent another goal in the study of DNA double-strand break repair by gene conversion.
It is conceivable that the repair mechanism used in our model represents one form of bacteriophage adaptation to attack by restriction enzymes (KOBAYASHI 1998), that is, an example of anti-restriction strategies (TOCK and DRYDEN 2005). An advantage of the mechanism, however, becomes obvious only when a remaining double-strand break of one of the co-infecting phage genomes induces damage to the host cell physiology, resulting in a decrease in the burst size, although some additional advantages might exist as well.
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APPENDIX |
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TOPABSTRACTMODELSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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is sufficiently larger than the cost of recombination c, a positive linkage disequilibrium between the restriction-resistance locus and the recombination locus can be built up. The latter fact indicates that the recombination allele increases both by its direct benefit by restoring the loss due to unrepaired break and by the hitchhiking effect on the linkage disequilibrium driven by strong selection for restriction-resistant phage.
Let denote the mean fitness of phage, which is given by the sum of the right-hand sides of (2) in the text. If the cost of recombination c is ignored, the mean fitness is given by
 | (A1) |
is the frequency of the restriction-resistant allele in the first locus, 
is the frequency of the recombination-efficient allele in the second locus, and 
is the linkage disequilibrium between the loci. We denote by 
the reduction of phage production when unrepaired double-strand breaks remain in the bacteria cell and by 
the fraction of multiple infection. We also denote, for notational convenience, by s the relative fecundity of restriction-positive phage and by 
the relative fecundity of the recombination-efficient allele.
We can see that in the last term of (A1),
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co-infects with the other phage that results in the mixture of restriction-sensitive and -resistant phage genome in a restriction-positive bacterium. If recombination occurs in such a case with probability 
the double-strand break of the restriction-sensitive genome is repaired and the phages restore the amount 
of reduction in phage production. It is clear from this that the recombination-efficient allele is not a neutral modifier but is directly subject to selection, and the intensity of the direct selection for the recombination allele is proportional to 
and 
In other words, the direct selection for the recombination-efficient allele exists only when there is multiple infection (
), restriction-positive bacteria (
), nonzero recombination (
), and the reduction in phage production in the presence of the unrepaired double-strand break (
).
The second to last term of (A1), on the other hand, represents the contribution, not through recombination, to phage fitness from the sensitive/resistant heterologous infection to the restriction-positive bacterium. This includes the reduction 
of phage particles due to the unrepaired double-strand break. The first two terms of (A1) simply represent the facts that restriction-sensitive phages can multiply only in restriction minus bacterium and restriction-resistant phages can multiply in any bacterium but pay a cost of reduced fecundity (
).
If we assume that the cost 
of recombination is sufficiently small, the changes in the frequency of restriction-resistant allele 
and that of recombination modifier allele 
in one generation are given by
 | (A2) |
 | (A3) |
can be expressed as 
where 
is the mutation rate between restriction-sensitive and -resistant alleles. Formal neglect of the explicit mutation term, however, does not affect the qualitative results we derive in the following, because we need only the protected polymorphism in the restriction-resistance locus (i.e.,
) to establish the results. The expression for the change in the linkage disequilibrium 
is complicated, but its derivation is straightforward from (2) in the text. With combined dynamics ofx, p, q, and D defined here, we establish the following results:
i.e., if there is no reduction in phage production in the presence of the unrepaired double-strand break, the surface 
is locally attracting, indicating that the recombination-efficient allele can never invade the phage population. of the unrepaired double-strand break is sufficiently larger than the cost c of recombination, a positive linkage disequilibrium between the restriction-resistance locus and the recombination locus can be built up.
We first show that if there is no cost of unrepaired double-strand break (
), the surface 
the state with no Rec+ allele and no linkage disequilibrium, is locally attracting. Because bacteria restriction genotypes and phage restriction-resistance genotypes show sustained cycles, the local stability of is determined by examining a linearized system with respect to q and D with time-dependent coefficients [i.e., those depend on the frequency of restriction-positive bacteria, 
and the frequency of restriction-resistant phage,
]. The Jacobian matrix at the surface 
and 
is given, by linearizing (A3) and 
defined in (2) in the text with respect to q and D, as
 | (A4) |
is a function of time whose explicit form does not affect the stability, and
 |
 | (A5) |
 | (A6) |
 | (A7) |
is locally attracting if implying that the Rec+ allele can never invade the population if there is no cost, in terms of phage production, of unrepaired double-strand break in the infected cell.
Second, we show that, if 
i.e., if there is nonzero cost of unrepaired double-strand break, a positive linkage disequilibrium can be built up from locally repelling surface of equilibria 
The change in the linkage disequilibrium is complicated, but it always increases at 
 | (A8) |
and 
Thus as long as both host and parasite are polymorphic in each locus, and there is a cost of unrepaired double-strand break (), a positive linkage disequilibrium is generated. We need to guarantee that nonzero q is maintained to show the buildup of positive linkage disequilibrium near the surface (polymorphisms in the phage restriction-resistance locus and in bacteria are certain). The finite rate of increase of the frequency of recombination-efficient allele Rec+ at is
 | (A9) |
m, and r relative to c is necessary. If this is the case, a positive is maintained, and hence from (A8) a positive linkage disequilibrium is generated. |
ACKNOWLEDGEMENTS |
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TOPABSTRACTMODELSRESULTSDISCUSSIONAPPENDIXACKNOWLEDGEMENTSLITERATURE CITED |
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Results for 
and 
are shown. Other parameter values are listed in 
is fixed, 
and 
(A) Wide view; (B) close-up view. Other parameter values are listed in