MCVEAN (2002) showed that predictions for r 2,a

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1 oyright Ó 2008 by the Genetics Society of America DOI: /genetics Note The Influence of Gene onversion on Linkage Diseuilibrium Around a Selective Swee Danielle A. Jones 1 and John Wakeley Deartment of Organismic and Evolutionary Biology, Harvard University, ambridge, Massachusetts Manuscrit received June 4, 2008 Acceted for ublication July 17, 2008 ABSTRAT In a 2007 article, McVean studied the effect of recombination on linkage diseuilibrium (LD) between two neutral loci located near a third locus that has undergone a selective swee. The results demonstrated that two loci on the same side of a selected locus might show substantial LD, whereas the exected LD for two loci on oosite sides of a selected locus is zero. In this article, we extend McVean s model to include gene conversion. We show that one of the conclusions is strongly affected by gene conversion: when gene conversion is resent, there may be substantial LD between two loci on oosite sides of a selective swee. MVEAN (2002) showed that redictions for r 2,a commonly used measure of LD introduced by Hill and Robertson (1968), deend on the correlations in coalescence times for a air of loci, which in turn deend on the recombination rate between the loci. In alying this result to LD near a locus that has undergone a selective swee, McVean (2007) develoed a new model that features two neutral loci artially linked to the selected locus. He assumed that recombination could occur between each air of loci and that the swee had a articularly simle structure: a star tree. Figure 1 deicts the model, in which a samle at the two neutral loci is taken at the resent time 0, just after the swee has finished. The swee is assumed to have occurred uickly and to have begun at time t M in the ast, measured in units of 2N e generations, where N e is the (diloid) coalescent effective oulation size (Sjödin et al. 2005). On the basis of the work of Maynard Smith and Haigh (1974) and others (Kalan et al. 1989; Stehan et al. 1992; Durrett and Schweinsberg 2004), McVean (2007) used t M ¼ 0.1, and we adot this value below. McVean (2007) tested the validity of this aroximate model by comaring its redictions to the results of fully stochastic simulations of a swee and found them to be largely accurate. McVean (2007) allowed for recombination (recirocal exchange of genetic material as in a single crossover event) but not for gene conversion (nonrecirocal 1 orresonding author: Biolabs 4092, 16 Divinity Ave., ambridge, MA djones@fas.harvard.edu exchange of short tracks of genetic material). However, there is a growing body of evidence for the imortance of gene conversion in shaing genetic variation in humans (Frisse et al. 2001; Jeffreys and May 2004; Padhukasahasram et al. 2004; hen et al. 2007; Gay et al. 2007) and models that do not feature gene conversion, therefore, do not comletely cature the biological causes, and exectations, of genetic variability. Our aim here is to incororate gene conversion into the model and to ask whether this changes the results. We focus on the case in which variation at the two neutral loci is due to mutations that occurred during the neutral hase shown in Figure 1B. In this case, the two neutral loci can be olymorhic only if they do not coalesce along with the selected allele at time t M in Figure 1B. Without recombination or gene conversion, resent-day samles at the two neutral loci will always remain linked to the selected allele and will certainly coalesce with the selected allele. Recombination and gene conversion allow the loci to escae the swee with some robability and to coalesce during the neutral hase, where they might also exerience mutations. To make a rediction for r 2 secifically s 2 d of Ohta and Kimura (1971) it is necessary to comute the exectation of the roduct of the coalescence times at two loci for each of the three samle configurations (A, B, and ) in Figure 1A (McVean 2002). Briefly, in the three-locus model of McVean (2007), for each of these three samling configurations, we must comute the robability that the two neutral loci are in configuration A, B, or at the start of the neutral hase looking back (i.e., time t M ), at which oint all chromosomes in the Genetics 180: (October 2008)

2 1252 D. A. Jones and J. Wakeley reresent configurations. The exected roduct of coalescence times at the two neutral loci, samled at resent when all chromosomes ossess the selected allele (denoted by the subscrit S), are the averages over the three ancestral configurations. The exected coalescence time for two chromosomes, i and j, samled at locus X is written as t x ij and for chromosomes k and l, at locus Y, it is written as t y kl. For configuration A, we have E S ½t x ij ty ijš ¼f AA E W ½t x ij ty ijš 1 f AB E W ½t x ij ty ikš 1 f A E W ½t x ij ty klš; Figure 1. This is an adatation of Figure 1 of McVean (2007). (A) The three configurations are A, two neutral loci are samled from two chromosomes; B, two neutral loci are samled from three chromosomes; and, two neutral loci are samled from four chromosomes. (B) The selection event occurs as a raid selective swee during which only crossingover events can occur. During the neutral hase, coalescent events can also occur. oulation have the ancestral tye, or wild tye, at the selected locus. There are nine such robabilities in total, one for each air of configurations. These nine robabilities are denoted using f, with subscrits to which is Euation 9 in McVean (2007). The robabilities f AA, f AB, and so on, deend on t M and on the rates of recombination and gene conversion. The exected values on the right-hand side above are those exected during the neutral hase (W stands for wild-tye allele) and are given in Euation 10 of McVean (2007). The redictions about LD deend strongly on the relative osition of the selected locus comared to the neutral loci. McVean considered two cases: (1) the selected locus is located halfway between the two neutral loci (NSN) and (2) the selected locus is located on one side of the two neutral loci (SNN). All of the derivations described above are done searately for these two cases. onsidering only recombination, McVean redicted substantial LD for SNN, because in this case both neutral loci can escae the swee yet remain linked to each other at the beginning of the neutral hase. For the NSN case, the model without gene conversion redicts no LD between the two neutral loci. As McVean Figure 2. There are four cases: SNN without gene conversion and with gene conversion and NSN without gene conversion and with gene conversion. In all cases, during the selective swee hase, there are two arameters that describe the robability of escae via recombination: x ¼ 1 e ðrx=2þtm and y ¼ 1 e ðry=2þtm. During the neutral hase, the robability of recombination is R x ¼ 4N e r and R y ¼ 4N e r, where r is the er generation robability of recombination. The recombination distance between the two neutral loci is ket constant regardless of whether they are in the SNN or the NSN case. Thus for NSN, R x ¼ R y ¼ 1 2 4N er. When gene conversion is added to the model for both SNN and NSN, the gene conversion is restricted to the individual loci. The overall rate of gene conversion is k X ¼ k S ¼ k Y ¼ 4N e g for each locus, where g is the er generation robability of gene conversion.

3 Note 1253 Figure 3. This is an adatation of Figure 2 of McVean (2007) but, imortantly, it includes five novel transitions from configuration A to configurations A and B at the start of the neutral hase that are not ossible when only recombination is resent. The transitions reresented corresond to the transition robabilities resent in Table A1 and are in the same order excluding the five cases where the transition robability is zero as in Table A1. Also shown in brackets is the notation used for each configuration (McVean 2007). The notation for configuration A is [i S j S, i S j S ] which indicates that two chromosomes, i and j, were samled at locus X and locus Y, and both chromosomes ossess the selected allele. The samling configurations at locus X and locus Y are searated by a comma. notes, this reflects the symmetric nature of the recombination rocess for the NSN case: the robabilities of each of the three configurations at the beginning of the neutral hase (A, B, or ) are the same for each of the three samle configurations, so that s 2 d ¼ 0 (see Euation 14 of McVean 2007). This symmetry breaks down when gene conversion is included in the model because gene conversion at the selected locus allows both neutral loci to escae the swee yet remain linked at the beginning of the neutral hase. Figure 2 gives a grahical reresentation of the model for SNN and NSN, with and without gene conversion. We assume that when gene conversion occurs, it coies a tract length of m nucleotides, which is greater than the size of each locus and less than the distance between the loci. Thus, gene conversion acts on single loci indeendently. During the neutral hase, on the coalescent timescale, recombination events occur with rates R x and R y, and gene conversion events occur at rates k s, k x, and k y. Following McVean (2007), during the selection hase with duration t M there are two robabilities of escae via recombination: x ¼ 1 e tmrx=2 and y ¼ 1 e tmry=2. To these we add three robabilities of escae by gene conversion: g s ¼ 1 e tmks=2, g x ¼ 1 e tmkx=2, g y ¼ 1 e tmky=2. Tables A1 A6 in the aendix give all the terms needed to comute the robabilities f AA, f AB, and so on, for both SNN and NSN. We follow McVean (2007) in comuting transitions between the resent and the start of the neutral hase, using the intermediate configuration at the end of the selection hase; our Tables A1 A3 corresond to each of the three columns of Aendix A in McVean (2007) and our Tables A4 A6 corresond to each of the three columns of Aendix B in McVean (2007). Again, the key difference between the models is that in the NSN case gene conversion allows resent-day configuration A to remain in configuration A at the start of the neutral hase, whereas this is not ossible by recombination alone. Figure 3 shows the configurations that can be reached from samling configuration A at the resent; it is analogous to Figure 2inMcVean (2007) and shows five of the six novel con-

4 1254 D. A. Jones and J. Wakeley Figure 4. Exected LD for the NSN case. The y-axis is the amount of LD as redicted by s 2 D. The x-axis is the distance, in base airs, between the two neutral loci starting from a distance of 2 3 tract length, m, between them and increasing to 10,000 b. In this examle, m ¼ 300 b. The distance between either neutral locus and the selected site must be at least a tract length, so that any given gene conversion event converts only one locus; the two neutral loci are searated by a minimum of two tract lengths, 600 b. For this distance range, R Neutral ¼ nr 1 2mk. (A) f ¼ 0, there is no gene conversion. (B) f ¼ 1, there is the same amount of gene conversion as there is recombination. () f ¼ 5, there is 5 times as much gene conversion as recombination. This is a reasonable ratio for human data. (D) f ¼ 15, there is 15 times as much gene conversion as recombination. figurations (the bottom five transitions encomassed in a box) that can be reached when gene conversion is included in the model. Following McVean (2007), to generate redictions alicable to molecular data, we assume that the rates of recombination, R x and R y, deend linearly on the distance between the loci. For examle, if locus X is n nucleotides away from locus S, then R x ¼ nr, where r is the rate of recombination between adjacent nucleotides on the coalescent timescale. In addition, we assume that each locus is a single-nucleotide site, so that in the neutral hase all three loci have the same rate of conversion: k s ¼ k x ¼ k y ¼ mk, where k is the rate of initiation of a gene conversion event between two adjacent nucleotides on a coalescent timescale. Finally, we include a arameter, f ¼ k/r, which is the ratio of gene conversion to recombination. Note that, with this arameterization, the er generation robabilities of recombination and gene conversion in Figure 2 are given by r ¼ nr=4n e and g ¼ mk=4n e. To redict the likely effect of gene conversion on LD in human oulations, we substituted lausible genetic arameters from humans: r /b (Frisse et al. 2001); the ratio of gene conversion to recombination, f, has been estimated to be between 1.5 and 14 (Frisse et al. 2001; Jeffreys and May 2004; Padhukasahasram et al. 2004; hen et al. 2007; Gay et al. 2007); and tyical gene conversion tract lengths range from 50 to 500 b Figure 5. Exected LD for the SNN case. The y-axis is the amount of LD as redicted by s 2 D. The x-axis is the distance, in base airs, between the two loci starting from a distance of at least two tract lengths between them, to allow for comarison to the NSN case, and increasing to 10,000 b. In this examle, m ¼ 300 b; therefore, the starting distance between the two neutral loci is 600 b. For this distance range, R Neutral ¼ nr 1 2mk. (A) f ¼ 0, there is no gene conversion. (B) f ¼ 1, there is the same amount of gene conversion as there is recombination. () f ¼ 5, there is 5 times as much gene conversion as recombination. (D) f ¼ 15, there is 15 times as much gene conversion as recombination.

5 Note 1255 (Jeffreys and May 2004). To simlify our analysis, we assume a fixed tract length of 300 b (Jeffreys and Neumann 2002). Reeating our analysis with a 50-b tract length led to slightly higher levels of LD (results not shown). As Figures 4 and 5 show, adding gene conversion affects LD in both the NSN and the SNN case. In the SNN case, the effect of gene conversion is similar to the effect of recombination, so that increasing f decreases LD (Figure 5, B D). Adding gene conversion in the SNN case creates more oortunities for the two neutral loci to escae the swee indeendently, so LD between them is reduced. In the NSN case, gene conversion increases LD (Figure 4, B D). In this case, gene conversion has a ualitatively different effect. Gene conversion events at the middle (S) locus allow the two neutral loci to escae the swee together. Present-day samles of this tye may then be samles of an ancestral halotye that would otherwise have been lost during the swee. Looking forward in time, in the NSN case gene conversion can reserve the reexisting correlated genealogical structure between the outer loci. By incororating gene conversion into the threelocus model of McVean (2007), we have shown that LD is exected between two loci on oosite sides of a selected locus that has undergone a swee. Although we have focused only on a single air of neutral loci, our results have imlications for genomic scans for selective swees using extended halotye homozygosity (Sabeti et al. 2002), integrated extended halotye homozygosity (Voight et al. 2006), and long-range halotyes (Sabeti et al. 2007). In articular, gene conversion at the selected site will cause some fraction of resent-day chromosomes to show the selected allele but while sitting on an ancestral halotye. Using t M ¼ 0.1 as in McVean (2007), and assuming a tract length of m ¼ 300 and a ratio of gene conversion to recombination of f ¼ 5, the robability of samling such a chromosome is 1 e Then, among 100 chromosomes that all ossess the selected allele, we would exect to see about four of these aberrant halotyes, and the chance that all 100 chromosomes would show the classic, recombination-only swee attern would be Thus, it is ossible that many selected loci have been missed in the recent genomic scans for selection. LITERATURE ITED hen, J.-M., D. N. oer, N.huzhanova,.Ferec and G. P. Patrinos, 2007 Gene conversion: mechanisms, evolution and human disease. Nature 8: Durrett, R., and J. Schweinsberg, 2004 Aroximating selective swees. Theor. Poul. Biol. 66: Frisse, L., R. R. Hudson, A. Bartoszewicz, J. D. Wall, J. Donfack et al., 2001 Gene conversion and different oulation histories may exlain the contrast between olymorhism and linkage diseuilibrium. Am. J. Hum. Genet. 69: Gay, J., S. Myers and G. McVean, 2007 Estimating meiotic gene conversion rates from oulation genetic data. Genetics 177: Hill, W. G., and A. Robertson, 1968 Linkage diseuilibrium in finite oulations. Theor. Al. Genet. 38: Jeffreys, A. J., and. May, 2004 Intense and highly localized gene conversion activity in human meiotic crossover hot sots. Nat. Genet. 36: Jeffreys, A. J., and R. Neumann, 2002 Recirocal crossover asymmetry and meiotic drive in human recombination hot sot. Nat. Genet. 31: Kalan, N. L., R. R. Hudson and. H. Langley, 1989 The hitchhiking effect revisited. Genetics 123: Maynard Smith, J., and J. Haigh, 1974 The hitchhiking effect of a favourable gene. Genet. Res. 23: McVean, G., 2002 A genealogical interretation of linkage diseuilibrium. Genetics 16: McVean, G., 2007 The structure of linkage diseuilibrium around a selective swee. Genetics 175: Ohta, T., and M. Kimura, 1971 Linkage diseuilibrium between two segregating nucleotide sites under steady flux of mutations in a finite oulation. Genetics 68: Padhukasahasram, B., P. Marjoram and M. Nordborg, 2004 Estimating the rate of gene conversion on human chromosome 21. Am. J. Hum. Genet. 75: Sabeti, P.., D. E. Reich, J. M. Higgins, H. Z. P. Levine, D. J. Richter et al., 2002 Detecting recent ositive selection in the human genome from halotye structure. Nature 419: Sabeti, P.., P. Varilly, B.Fry, J.Lohmueller, E.Hostetter et al., 2007 Genome-wide detection and characterization of ositive selection in human oulations. Nature 449: Sjödin, P., I. Kaj, S. Krone, M. Lascoux and M. Nordborg, 2005 On the meaning and existence of an effective oulation size. Genetics 169: Stehan, W., T. H. E. Wiehe and M. W. Lenz, 1992 The effect of strongly selected substitutions on neutral olymorhisms: analytical results based on diffusion theory. Theor. Poul. Biol. 41: Voight, B. F., S. Kudaravalli, X. Wen and J. K. Pritchard, 2006 A ma of recent ositive selection in the human genome. PLoS Biol. 4(3): e72. ommunicating editor: N. Takahata APPENDIX Notation The transition euations used in Tables A1 A6 are comlicated by the addition of gene conversion events. In an attemt to simlify the euations used to build the tables, a new notation, which incororates the notation of McVean (2007), is used. The new notation is outlined and comared to McVean s below. All of the symbols used refer to escae robabilities during the selection hase that result from either recombination or gene conversion. For NSN (Tables A1 A3): McVean (2007): x is used to indicate that no recombination event has occurred between locus X and locus S. x is used to indicate that a recombination event has occurred between locus X and locus S. y is used to indicate that no recombination event has occurred between locus S and locus Y. y is used to indicate that a recombination event has occurred between locus S and locus Y.

6 1256 D. A. Jones and J. Wakeley TABLE A1 Transition robabilities for NSN for the starting configuration A to the four states, A, B,, or O, resent at the beginning of the neutral hase onfiguration at the end of selection hase Probability given starting configuration [i S j S, i S j S ] onfiguration at the start of the neutral hase [i S j S, i S j S ] ( [i S j S, i S k W ] 2( y *)(*(1 g x s) [i S j W, i S k S ] 2( y *)(*(1 g x s) [i S j W, i S k W ] 2( y *)(* x * y x * * g y s 1 x * * g y s) B [i S j S, k W l W ] (( y *))2 O [i W j W, k S l S ] ( [i S j W, k S l W ] 2( y *)(*(1 g x s) [i S j W, k W l W ] 2( y *)(* x * y x * y * g s 1 x * * g y s) [i W j W, k S l W ] 2( x * y * g s 1 x * * g y s)( y *) [i W j W, k W l W ] ( x * * g y s 1 x * * g y s) 2 [i S j S, i S k S ] 0 O [i S j S, k S l W ] 0 O [i S j W, k S l S ] 0 O [i S j S, k S l S ] 0 O [i S j W, i S j W ] 2( y *)(* g x s y *) A [i W j W, i W k S ] 2( y *)(*(1 g x s) [i W j W, i W j W ] ( y *)2 A [i W j S, i W k W ] 2( y *)(*(1 g x s) [i W j W, i W k W ] 2( y *)(* x * y x * * g y s 1 x * * g y s) B [i W j S, i W k S ] 0 A O corresonds to a state where at least one of the two neutral loci has coalesced. There are six new states created by the addition of gene conversion that are not resent when recombination is the only crossingover event otion. These are the six last states. This table corresonds to the transition robabilities in the second column of Aendix A of McVean (2007) and the same notation, exlained in detail in Figure 3, is used. Notation with gene conversion: The terminology is the same as that in McVean (2007) but with the additional consideration that escae from the selection swee can also come from gene conversion. x * ¼ xð1 g x Þ: there is no recombination between locus X and locus S and no gene conversion at locus X. x * ¼ 1 * ¼½ x xð1 g x Þ 1 ð1 x Þg x x g x Š: there is either at least one recombination event between locus X and locus S or at least one gene conversion at locus X. y * ¼ yð1 g y Þ: there is no recombination between locus S and locus Y and no gene conversion at locus Y. y * ¼ 1 * ¼½ y yð1 g y Þ 1 ð1 y Þg y y g y Š: there is either at least one recombination event between locus S and locus Y or at least one gene conversion at locus Y. For SNN (Tables A4 A6): McVean (2007): x and x have the same meaning as their NSN counterarts. y is used to indicate that no recombination event has occurred between locus X and locus Y. y is used to indicate that at least one recombination event has occurred between locus X and locus Y. Notation with gene conversion: x * and x * have the same meaning as their NSN counterarts. y * ¼ yð1 g y Þ: there is no recombination between locus X and locus Y and no gene conversion at locus Y. y * ¼ 1 y * ¼ [ y ð1 g y Þ 1 ð1 y Þg y y g y Š ¼ ½ y 1 g y y g y Š: there is either at least one recombination event between locus X and locus Y or at least one gene conversion at locus Y. There are two additional robabilities resent in the case of SNN: s ¼ (1 x )(1 g s ) is the robability that no recombination event has occurred between locus S and locus X and no gene conversion event has occurred at locus S. s ¼ [ x (1 g s ) 1 (1 x )g s x g s ] ¼ [ x 1 g s x g s ]is the robability that a recombination event between locus S and locus X or a gene conversion event at locus S or both has occurred.

7 Note 1257 TABLE A2 Transition robabilities for NSN for the starting configuration B onfiguration at the end of selection hase Probability given starting configuration [i S j S, i S k S ] onfiguration at start of neutral hase [i S j S, i S j S ] 0 O [i S j S, i S k W ] y * x *(1 g s)( y * y * * g y s) O [i S j W, i S k S ] x *) y *(1 g s) O [i S j W, i S k W ] y *((* 1 x * g x s))( y * g s) B [i S j S, k W l W ] y * * (1 g x s)( y * g s) O [i W j W, k S l S ] x *)*(1 g y s) O [i S j W, k S l W ] ( *(1 g *(* 1 g *)*(1 g *(1 g B x s) y x s x y s) x s) y * * (1 g x s)( y * g s) 1 ( x * * g y s 1 x * * g y s) x *(1 g s) 2 y *) [i S j W, k W l W ] ( y *(* 1 x * g x s)( y * g s) 1 ( 1 [i W j W, k S l W ] x * y s x * y x y y ( *(1 g x s) y *(* 1 x * g x s)( * g y s) 1 ( 1 x * * g y s 1 x * * g y s)( * 1 g x s x *)*(1 g y s) * g 1 * g s) *(1 g s)( * 1 * g s)) [i W j W, k W l W ] ( x * * g y s 1 x * * g y s)( x * 1 * g x s)( y * g s) [i S j S, i S k S ] y * x *(1 g s) 2 [i S j S, k S l W ] y * * (1 g x s) 2 [i S j W, k S l S ] y * * (1 g x s) 2 [i S j S, k S l S ] 0 O [i S j W, i S j W ] 0 A [i W j W, i W k S ] x *) y *(1 g s) B [i W j W, i W j W ] 0 A [i W j S, i W k W ] y * * (1 g x s)( y * g s) B [i W j W, i W k W ] y *(* 1 x * g x s)( y * g s) B [i W j S, i W k S ] y * * (1 g x s) 2 y * A The six new states created by the addition of gene conversion are the last six rows of the table. This corresonds to the transition robabilities in the third column of Aendix A of McVean (2007). onfiguration at the end of selection hase TABLE A3 Transition robabilities for NSN for the starting configuration Probability given starting configuration [i S j S, k S l S ] onfiguration at the start of neutral hase [i S j S, i S j S ] 0 O [i S j S, i S k W ] 0 O [i S j W, i S k S ] 0 O [i S j W, i S k W ] 0 B [i S j S, k W l W ] ( x *(1 g s)( y * g s)) 2 O [i W j W, k S l S ] (( * 1 g s *)*(1 g s)) 2 O [i S j W, k S l W ] x x y 2( x *(1 g s) 2 y *(* 1 x * g x s)( y * g s) 1 x *(1 g s)( y * g s) B ( x * 1 g s x *)*(1 g y s)) x x y x * x y y [i W j W, k W l W ] (( * 1 x * g x s)( * g y s)) 2 [i S j S, i S k S ] 0 O [i S j W, k W l W ] [i W j W, k S l W ] 2 x *(1 g s)( y * g s)( x * 1 * g x s)( y * g s) 2( * 1 g s *) *(1 g s)( * 1 g s)( * 1 * g s) [i S j S, k S l W ] 2 x *(1 g s) 2 y * * (1 g x s)( y * g s) O [i S j W, k S l S ] 2 x *(1 g s) 2 x *)*(1 g y s) O [i S j S, k S l S ] ( x *(1 g s) 2 [i S j W, i S j W ] 0 A [i W j W, i W k S ] 0 B [i W j W, i W j W ] 0 A [i W j S, i W k W ] 0 B [i W j W, i W k W ] 0 B [i W j S, i W k S ] 0 A There are six new states created by the addition of gene conversion; these are described in the last six rows of the table. This corresonds to the transition robabilities in the fourth column of Aendix A of McVean (2007).

8 1258 D. A. Jones and J. Wakeley TABLE A4 Transition robabilities for SNN when the starting configuration is A onfiguration at the end of selection hase Probability given starting configuration [i S j S, i S j S ] onfiguration at the start of neutral hase [i S j S, i S j S ] ( x * y *(1 g s)) 2 O [i S j W, i S j W ] 2 x * * (1 g y s)( x 1 g s x g s )(1 g x ) y * A [i S j S, i S k W ] 2( x * * (1 g y s))( [i S j W, i S k W ] 2 x * * (1 g y s)((g s x x g s )( [i W j W, i W j W ] (( x 1 g s x g s )(1 g x ) y *)2 A [i W j S, i W k W ] 2( x 1 g s x g s )(1 g x ) y * * (1 g x s) y * B [i S j S, k W l W ] ( [i S j W, k W l W ] 2 y *((g s x x g s )( y *) [i W j W, i W k W ] 2( x 1 g s x g s )(1 g x ) y *((g s x x g s )( [i W j W, k W l W ] (((g s x x g s )( y *))2 [i S j S, i S k S ] 0 O [i S j W, i S k S ] 2( x * * (1 g y s))( [i S j S, k S l W ] 0 O [i W j S, i W k S ] 0 A [i S j W, k S l W ] 2 y * y * B [i W j W, i W k S ] 2( x 1 g s x g s )(1 g x ) y * * g s x y * B [i W j W, k S l W ] 2 y *((g s x x g s )( y *) [i S j W, k S l S ] 0 O [i W j W, k S l S ] ( [i S j S, k S l S ] 0 O There are no new states created with the addition of gene conversion but there are new transition robabilities. This table corresonds to the transition robabilities in the second column of Aendix B of McVean (2007). TABLE A5 Transition robabilities for SNN when the starting configuration is B onfiguration at the end of selection hase Probability given starting configuration [i S j S, i S k S ] onfiguration at the start of neutral hase [i S j S, i S j S ] 0 O [i S j W, i S j W ] 0 A [i S j S, i S k W ] ( x * * (1 g y s))( x *(1 g s))( y * xg s 1 x y g y x y 1 x y x y ) O [i S j W, i S k W ] x * * (1 g y s)( x * 1 (1 g x)g s (1 x ))( y * xg s 1 x y g y x y 1 x y x y ) B [i W j W, i W j W ] 0 A [i W j S, i W k W ] ( x 1 g s x g s )(1 g x ) y *(*(1 g x s))( y * xg s 1 x y g y x y 1 x y x y ) B [i S j S, k W l W ] y *( x *(1 g s))( y * xg s 1 x y g y x y 1 x y x y ) O [i S j W, k W l W ] y *(* 1 (1 g x x)g s (1 x ))( y * xg s 1 x y g y x y 1 x y x y ) 1 ((g s x x g s )( y *)(*(1 g x s))( y * xg s 1 x y g y 1 x y 1 x y x y ) [i W j W, i W k W ] ( x 1 g s x g s )(1 g x ) y *(* 1 (1 g x x)g s (1 x )) B ( y * xg s 1 x y g y x y 1 x y x y ) [i W j W, k W l W ] ((g s x x g s )( y * g x)1 x (1 g s )g x y *)(* 1 (1 g x x)g s (1 x ))( y * xg s 1 x y g y x y 1 x y x y ) [i S j S, i S k S ] x * * (1 g y s) x * x(1 g s ) 2 [i S j W, i S k S ] ( x * * (1 g y s))( x * 1 g s [i S j S, k S l W ] y * * x x(1 g s ) 2 [i W j S, i W k S ] ( x 1 g s x g s )(1 g x ) y * x * x(1 g s ) 2 y * A [i S j W, k S l W ] y *(*(1 g x s))( y * xg s 1 x y g y x y 1 x y x y )1 x *(1 g s) B y *)1 ((g s x x g s )( y * g x) 1 x (1 g s )g x y *)* x x(1 g s ) 2 y * [i W j W, i W k S ] ( x 1 g s x g s )(1 g x ) [i W j W, k S l W ] y *(* 1 (1 g x x)g s (1 x ))( y * xg s 1 x y g y x y 1 x y x y )1 ((g s x x g s )( y *)(* 1 g x s y *) [i S j W, k S l S ] y * * x x(1 g s ) 2 [i W j W, k S l S ] [i S j S, k S l S ] 0 O There are no new states created with the addition of gene conversion but there are no new transition robabilities. This corresonds to the transition robabilities in the third column of Aendix B of McVean (2007).

9 Note 1259 TABLE A6 Transition robabilities for SNN when the starting configuration is onfiguration at the end of selection hase Probability given starting configuration [i S j S, k S l S ] onfiguration at the start of neutral hase [i S j S, i S j S ] 0 O [i S j W, i S j W ] 0 A [i S j S, i S k W ] 0 O [i S j W, i S k W ] 0 B [i W j W, i W j W ] 0 A [i W j S, i W k W ] 0 B [i S j S, k W l W ] (( x *(1 g s))( y * xg s 1 x y g y x y 1 x y x y )) 2 O [i S j W, k W l W ] 2( x *(1 g s))( y * xg s 1 x y g y x y 1 x y x y )( x * 1 (1 g x) g s (1 x ))*( y * xg s 1 x y g y x y 1 x y x y ) [i W j W, i W k W ] 0 B [i W j W, k W l W ] ( x * 1 (1 g x)g s (1 x ))( y * xg s 1 x y g y x y 1 x y x y )) 2 [i S j S, i S k S ] 0 O [i S j W, i S k S ] 0 O [i S j S, k S l W ] 2 x * x(1 g s ) 2 y *(*(1 g x s))( y * xg s 1 x y g y x y 1 x y x y ) O [i W j S, i W k S ] 0 A [i S j W, k S l W ] 2( x *(1 g s))( y * xg s 1 x y g y x y 1 x y x y )( x * 1 g s x *) B ( x (1 g s ) y *)1 2* x x(1 g s ) 2 y *(* 1 (1 g x x)g s (1 x )) ( y * xg s 1 x y g y x y 1 x y x y ) [i W j W, i W k S ] 0 B [i W j W, k S l W ] 2( x * 1 g s y *)(* 1 (1 g x x)g s (1 x ))( y * xg s 1 x y g y 1 x y 1 x y x y ) [i S j W, k S l S ] 2 x * x(1 g s ) 2 y * [i W j W, k S l S ] (( x * 1 g s y *))2 O [i S j S, k S l S ] ( x * x(1 g s ) 2 There are no new states created with the addition of gene conversion but there are new transition robabilities. This corresonds to the transition robabilities in the fourth column of Aendix B of McVean (2007).