Downloaded by guest on September 24, 2021 ftelvl fsufln ihndfeetcrmsms(50–52), chromosomes different same within shuffling the made of been by levels have produced the comparisons gametes Finally, of 45–49). across 39, and (37, 44] individual ref. in reviewed www.pnas.org/cgi/doi/10.1073/pnas.1817482116 and flies (37–42), in [e.g., humans sex (35–37), same the mice of differ- (30–34), individuals exist also across can shuffling There in (26). informative ences shuffling of be value concomitantly adaptive 27), could the differ- which of 26, these 26–29), explain (14, 14, (13, to shuffling ences theories in evolutionary has several differences the literature prompting testing male/female large to A studied 17). 23–25) distinguishing (15, also 15, domestication from (1, of sex effects ranging (reviewed genomic of implications species advantages across with evolutionary the out 22–24), carried refs. been in have shuffling interest. considerable wide empirical of much of quantity occurs subject the a that genetic been therefore, has shuffling selfish and is, of importance of production amount invasion gamete harmful total in the The from (11–14). par- and complexes evolving separately 10) been rapidly (9, act further from asites has populations to shuffling protecting selection in Genetic implicated (3–8). natural efficiency mutations long-term allowing distinct the on by improves adaptation it envi- addition, of changing or In novel (2). to and adaptation ronments gametes aids among thus diversity and genetic (1), offspring to leads It populations. ual T recombination independent crossovers. is to of entirely which that almost of (ii due (i contribution that is and the that 1/2 shuffling assortment, reveals of of value level female possible high maximum and this its male to analysis close human our is from of humans data Application such sequencing. DNA to either by positions or crossover advantage determining cytologically for selective methods of a emergent Utilization ful suggests interference. which crossover spaced, of evenly more that fact are the implica- include metric Intrinsic this assortment. of independent tions number from crossover and be position from can and measure contributions This separate production. into gamete decomposed during shuffled are loci a Veller Carl law and second positions Mendel’s crossover account into shuffling takes genetic that genome-wide of measure rigorous A osntsfe rmteelmttos edfieteparame- the define We that limitations. shuffling these genome-wide from ter of suffer assort- measure not rigorous independent does Here, a law). from present second (Mendel’s shuffling we chromosomes homologous consider traditional of Moreover, to ment tip. the criti- fail more toward also causes measures crossover chromosome are a a of chromosome than middle shuffling a the toward along crossover a because crossovers cal: inadequate, of qualitatively positions is the This of meiosis. number average per the crossovers counting measures simply Existing by quantity. this crossing-over occurs by of consider measure hampered that direct been a shuffling of have absence on genetic studies the Such of critically production. amount gamete rely Singh) total during Nadia genetics and the Przeworski evolutionary of Molly by evaluation reviewed in 2018; 11, studies October review Comparative for (sent 2018 27, November Kleckner, Nancy by Contributed 02138 MA Cambridge, University, Harvard 02138; MA Cambridge, eateto raimcadEouinr ilg,HradUiest,Cmrde A02138; MA Cambridge, University, Harvard Biology, Evolutionary and Organismic of Department orsodnl,cmaaiesuis(,1–1 fgenome- of 15–21) (1, studies comparative Correspondingly, r uiggmt rdcini nipratpoesi sex- in process genes important inherited an paternally is and production gamete maternally during of shuffling he stepoaiiyta h lee ttornol chosen randomly two at alleles the that probability the as a,b | ac Kleckner Nancy , crossing-over c eateto oeua n ellrBooy avr nvriy abig,M 23;and 02138; MA Cambridge, University, Harvard Biology, Cellular and Molecular of Department | sex | c,1 meiosis n atnA Nowak A. Martin and , r slre hncrossovers when larger is 0tmsgetrthan greater times ∼30 r sealdb power- by enabled is Arabidopsis ) a,b,d (43); r in ) aeit con hfigcue yI.Afut esr of measure fourth A measures IA. above by supposed caused the this shuffling beyond of account shuffling None into to take minimum. contribute num- required the that structurally is COs frequency” to CO of chromosomes “excess each ber its the Since for (55), CO (15). properly one number segregate least COs at chromosome of requires haploid number usually the the bivalent distance is (54). of map gametes used excess of sometimes the 1% in is being in that shuffled cM measure are that Another 1 loci CO either cM, linked average measured two 50 is between length as by of Map multiplied prophase number data. frequency the sequence meiotic from simply during or is cytologically occur frequency CO that only fre- length. COs CO considered map specifically, have more or and shuffling quency crossing-over of of contribution measures the used widely most respectively. shuffling, chromo- chromosomal” homologous of (IA), assortment somes crossovers independent by by both and caused (COs) is Shuffling shuffling. genome-wide of sex new as used be 53). to (26, or chromosomes (13) complexes to genetic susceptible selfish most harboring are chromosomes which for implications with 1073/pnas.1817482116/-/DCSupplemental. y at online information supporting contains article This 1 aadpsto:MTA oefralcluain saalbeo iHb( GitHub on available is calculations all for code MATLAB deposition: Data the under Published interest.y of conflict no declare authors Oregon.y The of University N.S., paper.y and and the University; wrote N.K., Columbia M.A.N. C.V., M.P., and Reviewers: research; N.K., C.V., performed and C.V. data; analyzed research; M.A.N. designed C.V. contributions: Author github.com/cveller/rbar † * owo orsodnesol eadesd mi:[email protected]. y Email: addressed. be should correspondence whom To h nmiuu em“eei hfig o ut“hfig)throughout. “shuffling”) use just we (or confusion, shuffling” avoid “genetic To biology. term molecular unambiguous molecules—in the DNA break- of meaning—the rejoining different and a has age recombination but literature, genetics population below. hfigcue yete rsigoe rI srfre oa rcmiain nthe in “recombination” as to referred is IA or crossing-over either by caused discussed Shuffling as contribution negligible a makes but shuffling causes also conversion Gene ihvlei agl u oidpnetassortment. this independent that to and due one-half largely of is value value high possible maximum its to close htarnol hsnpi flc hfe hi lee na in alleles their shuffles loci of Measuring pair gamete. chosen randomly shuf- features: a these total that account of into measure takes a that the develop fling importantly, We crossovers. and, of assortment positions independent take to account failing crossovers, into of shuffling number genetic average of the count amount simply total chromosomes. the homologous of measures over of Traditional crossing assortment both of by independent shuffling gametes the and in DNA is paternal organisms and sexual maternal in process important An Significance uniaiecmaioslk hs eur rprmeasure proper a require these like comparisons Quantitative ∗ hc opie“nrcrmsml n “inter- and “intrachromosomal” comprise which NSlicense.y PNAS b rga o vltoayDnmc,HradUniversity, Harvard Dynamics, Evolutionary for Program ). y r nhmn,w n htttlsufln is shuffling total that find we humans, in d eateto Mathematics, of Department † www.pnas.org/lookup/suppl/doi:10. 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EVOLUTION aggregate shuffling, which does take into account IA, is Dar- incorporates CO position in the measurement of genome-wide lington’s “recombination index” (RI) (1, 56, 57), defined as the shuffling has been developed and implemented. sum of the haploid chromosome number and CO frequency. The Here, we present a simple intuitive measure of the genome- rationale for this measure derives from the fact that, given no wide level of shuffling. We define r¯ as the probability that a chromatid interference in meiosis (54, 58), two linked loci sep- randomly chosen pair of loci shuffles their alleles in meiosis, arated by one or more COs shuffle their alleles with probability taking into account CO number and position as well as the con- 1/2 in the formation of a gamete, as if the loci were on sepa- tribution of IA. We have chosen the name “r¯” to echo classic rate chromosomes. The RI is, therefore, the average number of population genetics terminology, where the parameter r for a “freely recombining” segments per meiosis. given pair of loci is the probability that they shuffle their alleles Importantly, none of these existing measures take into account in a gamete. Our parameter r¯ is simply this quantity averaged the specific positions of COs on the chromosomes. Intuitively, across all locus pairs. however, CO position is a critical parameter. For example, a CO r¯ is the most direct metric of the primary effect of genetic at the far tip of a pair of homologous chromosomes does little shuffling, the diversification of gametes, and thus the transmis- work in shuffling the genetic material of those chromosomes, sion of new genetic combinations to offspring (1). r¯ should be while a CO in the middle causes much shuffling (Fig. 1). Addi- used when such diversification is the quantity of interest: for tionally, two COs close together may cancel each other’s effect example, in judging the ability of a population to respond to a (except for the few loci that are between them), and thus result sudden change (or continual fluctuations) in its environment (2). in less allelic shuffling than two COs spaced farther apart. Genetic shuffling also has indirect effects, such as the gradual Therefore, existing measures do not actually define the total reduction of allelic correlations (linkage disequilibrium) within genome-wide amount of shuffling, instead serving only as prox- populations over time (6, 7) and the prevention of invasion of ies for this critical parameter. This is not a trivial concern. There selfish genetic complexes (11). Here, CO positions and IA again is significant heterogeneity in the chromosomal positioning of matter, so standard metrics that simply count COs remain inade- COs at all levels of comparison—between species (16, 59, 60), quate. Therefore, r¯ is likely a better measure in these cases as populations within species (61–65), the sexes (14, 26, 66–69), well, although we suggest (Discussion) modifications of r¯ that individuals (42, 70–73), and different chromosomes (50, 69, 74, more directly measure these quantities of interest. 75), all of which will impact the level of shuffling that these COs In this work, we define the quantity r¯, develop it mathemat- cause. Moreover, there can be major differences in chromosome ically and statistically, and document its intrinsic implications. number and size across species (1, 76, 77), which will seriously We show how it can be decomposed into separate components influence the total amount of shuffling due to IA. deriving from COs and from IA. We then discuss the approaches The fact that the positions of COs matter for the total amount now available to allow chromosome-specific and/or genome-wide of shuffling has been recognized for many years (1, 66, 76, 78–80). measurement of CO positions. With these developments in hand, The need for a measure of total shuffling that accounts for CO we present an application of r¯ to quantitative evaluation of positions has also previously been recognized. Indeed, Burt et al. shuffling in human males and females. (27) give an explicit formula, identical to Eq. 3 below, for “the proportion of the genome which recombines” (in the population Derivation of ¯r genetics sense†). Colombo (81) also gives an explicit character- r¯ is the probability that the alleles at two randomly chosen loci ization of what we call r¯ phrased as a generalization of the RI. are shuffled in the production of a gamete. It can be calculated Finally, Haag et al. (82), after noting that terminal COs cause with knowledge of CO positions on each chromosome in con- little shuffling and that map length is, therefore, an imperfect junction with knowledge of the fraction of the total genomic measure of genome-wide shuffling, suggest a better measure to length (in base pairs) accounted for by each chromosome (chro- be “the average likelihood that a CO occurs between two ran- mosome lengths influence shuffling caused by both IA and COs). domly chosen genes.” However, no mathematical expression that In what follows, we always assume that there are sufficiently many loci so that the difference between sampling them with or without replacement is negligible.

AB Formulas for r. In the ideal situation, CO positions are defined for individual gametes, and the parental origins of the chromo- some segments delimited by these COs are known. In this case, the proportion p of the gamete’s genome that is paternal can be determined, and the probability that the alleles of a randomly chosen pair of loci were shuffled during formation of the gamete (equivalently, the proportion of locus pairs at which the alle- les are shuffled), either by crossing-over or by IA, is simply the product of the probabilities that one locus is of paternal origin (probability p) and the other is of maternal origin (probability 1 − p) in each of two possible combinations:

r¯ = 2p(1 − p). [1]

Such data can emerge directly from sequencing of single gam- etes (Fig. 2B) or of a diploid offspring in which the haploid contribution from a single gamete can be identified (Fig. 2C). Fig. 1. The position of a CO affects the amount of genetic shuffling that In some analyses, while CO positions for a gamete are known, it causes. The figure shows the number of shuffled locus pairs that result the specific parental identities of the chromosome segments that from crossing-over between two chromatids in a one-step meiosis. The chro- these COs delimit are not known, because the gamete producer mosome is arbitrarily divided into eight loci. (A) A CO at the tip of the has been sequenced but its parents have not. In this case, we can chromosome between the seventh and eighth loci causes 7 of 28 locus pairs nonetheless calculate an expected value of r¯ for the gamete. The to be shuffled in each resulting gamete. (B) A CO in the middle of the chro- parental origin of segments will alternate across COs on each mosome between the fourth and fifth loci causes 16 of 28 locus pairs to be chromosome in the gamete, and therefore, we can still define, shuffled in each resulting gamete. The central CO thus causes more shuffling for any given chromosome k, the proportions that originate from than the terminal CO. the two different parents (pk and 1 − pk ). However, it is not

2 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1817482116 Veller et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 oiaesufldi eutn aeei / utpidb the by multiplied 1/2 segments: of separate is pair on are gamete chosen loci resulting randomly two the a a that at in probability meiosis alleles shuffled at the are COs that loci of probability configuration the interference the chromatid I, given no Therefore, assumes 57)]. [this (54, alle- 1/2 seg- their probability shuffle different they with on segments, les different situated is on be situated which indeed to of are need probability production loci the the two in ments, alleles the its gamete, shuffle a to of pair locus selected domly oa eoelnt is length segment genome that suppose total and order, (arbitrary) some of into number bivalents haploid Burt the the by divide If given 2A). (Fig. formulas (81) is the chromosomes Colombo and following (27) case al. this et in calculated be 1 elre al. et Veller their way. about quantitative assertions rigorous a previous shuffling), in evaluate contributions (interchromosomal to relative IA us allowing from thereby deriving component and a shuffling) (intrachromosomal COs from deriving component (Discussion). of species 86) of number 85, a formulation the (1, Our frequency for increase CO to way than is rather effective chromosomes shuffling IA, most genome-wide is increase the species to sexual that in correspondingly, shuffling of or, source predominant of the Components that Intrachromosomal and Inter- in especially diversity, of measure (84). used ecology commonly a (83), index E[¯ for value expected An I. prophase meiotic of stage chromosomes bivalent that along at positions CO define to allows somes gamete. the in realized actually value the Eq. by so multiplied probabilistic, chromosome), is that chromosomes, same term different the probability on on the are are is loci term chosen 1 second randomly The two the origin. parental ent k Eq. mosomes in term first The proportion a are there is chromosome fraction and If shuffled a set alle- Law. been haploid the the Second have in Mendel’s chromosomes that chromosomes with separate probability consistent on the loci one-half, that two address at We assuming sets parents. les which by different next or ambiguity same the the this to from chromosome are one segments of from know to possible ‡ fpsil atrso hoai novmn nterslto fCsa eoi and I meiosis at COs of thereafter. distribution resolution chromatids the the of to in patterns respect involvement segregation with chromatid time of this patterns but possible along value of expected COs its observe calculate we we therefore, when I, Similarly, meiosis to at assortments. respect chromosomes possible with bivalent expectation these its of calculate distribution we parental Therefore, the possible segments. the various of assortments the different of for origins then If different known, being “parameter”). value a not realized is are its it with origins here, paternal (i.e., and known maternal precisely the shuffled—is been have alleles of which value realized the segments, chromosome ie aeesqec n nweg fteptra n aenloiiso all of origins maternal and paternal the of knowledge and sequence gamete a Given − − r ial,ctlgclaayi fmitcpcyeechromo- pachytene meiotic of analysis cytological Finally, utpidb h rbblt ht fs,te r fdiffer- of are they so, if that, probability the by multiplied ] p P sdfie yEq. by defined as , k k n so h te,then other, the of is =1 E[¯ k L L r k 2 k httornol hsnlc r nchromosome on are loci chosen randomly two that = ] p ie,1 (i.e., ftegnm’ eghadi edtriethat determine we if and length genome’s the of k n X k fchromosome of =1 n n hr r oa of total a are there and r ¯ 2p E[¯ − 2 losu opriinttlsufln noa into shuffling total partition to us allows r k h rbblt htarno aro loci of pair random a that probability the stepoaiiysme vralchro- all over summed probability the is l = ] (1 i spootoa oteGini–Simpson the to proportional is 3, n with , − ¯ r + 1 2 narsliggmt saanarno variable; random a again is gamete resulting a in 2 p I

k sa xetto of expectation an is )L emns ae h emnsin segments the Label segments. 1 l 1 − k k 2 + ¯ r bigtepooto flcspisat pairs locus of proportion the —being + so n aetloii and origin parental one of is n X 1 i · · · =1 +I − 1 2 l P +

i 2 ‡ ! I 1 l i n ¯ r n =1 +I . − O,te hs COs these then COs, +I eoe admvariable, random a becomes r . X k l ti fe argued often is It 1 = =1 n i 2 hssecond This 1/2. ftetoloci two the If . L o n ran- any For . k i r k 2 ¯ sfato of fraction ’s consfor accounts ! ahrthan rather . r ¯ can [2] [3] n L is length genomic total of length tion genomic total of fractions with is chromosomes lent of number haploid the that pose Eq. as same the is which Eq. in term first the then is retain partition we therefore, and 4 which), is which and origin parental we proportion then a delimit, the calculate they not still that but can segments complement the haploid of origin a parental in specific realizations CO know we If i.2. Fig. (B nagmt rhpodcmlmn fa fsrn.I chromo- If is offspring. decomposition an for appropriate of decomposition complement this and haploid some chromosome or present same gamete first a the We in on alleles. are and is their loci component chromosomes shuffle two intrachromosomal that separate the probability on while the are alleles, their loci shuffle chosen randomly two C B A 1 u oeifrainaottescn em h appropriate The term. second the about information lose but and enwpeettedcmoiinfor decomposition the present now We h necrmsmlcmoetof component interchromosomal The + k DNA ingamete ofpaternal proportion total genomiclength of length asafraction segment bivalent chromosome crossover positionon L C k exhibits 2 acltn h xetdvleof value expected the Calculating (A) Calculating ) Paternal gamete + otisaproportion a contains E[¯ r ¯ · · · = i r ’s genomic = ] X | k + =1 n nrcrm component Intrachrom. I k a | X L k

=1 2p n nrcrm component Intrachrom. O,dvdn tit segments into it dividing COs, n ¯ r 1 = k ngmtso fsrn sn Eq. using offspring or gametes in 2p 1 (1 {z − k hn Eq. Then, . − (1 {z p 2. chromosome 1 p k − k fteohr(lhuhntknowing not (although other the of )L p k L chromosome 2 k 2 } )L p k p + k k = k 2 } fchromosome of | X 3 fptra otn,te the then content, paternal of i l + 6=j k spriinda follows: as partitioned is necrm component Interchrom. ,1 l | 2 1 2p k necrm component Interchrom. ¯ + r ,i NSLts Articles Latest PNAS ,

E[ i r ¯ Chromosome . l chromosome 1 (1 k 1 E[¯ ¯ r ,2 stepoaiiythat probability the is ] {z − tmissIuigEq. using I meiosis at , − r + Other gamete Other {z ] X k chromosome 2 p tmissI Sup- I. meiosis at =1 · · · n Expected valueof Expected j i Maternal gamete 1. )L n 1, = L + k n htbiva- that and i k 2 L ob fone of be to ! l k j } . . . } ,I . , k k +1 | , sfrac- ’s I f10 of 3 with , k

1 + [5] [4] 3. r ¯

EVOLUTION n Ik +1 ! n ! 1 X 2 X 2 1 X 2 Properties and Intrinsic Implications of r [¯r] = L − l + 1 − L . [6] E 2 k k,i 2 k Properties. We note three properties of r¯. First, its minimum k=1 i=1 k=1 value is zero, and its maximum value is one-half, the latter rely- | {z } | {z } Intrachrom. component Interchrom. component ing on our assumption of many loci. The maximum value of r¯ in a gamete can occur by chance equal segregation of mater- Averaging r. The above measures can be aggregated to obtain nal and paternal DNA to the gamete. The maximum value of the average value of r¯ across many gametes (or meiocytes). We E[¯r] at meiosis I requires, unrealistically, that every pair of loci denote the average value of a variable x by hxi. To average r¯ either is on separate chromosomes or experiences at least one given sequence data from many gametes and supposing that, for CO between them in every meiosis. The minimum value of r¯ in each gamete, we can distinguish which sequences are paternal a gamete could result from chance segregation of only CO-less and which are maternal, we can take the average value of Eq. 4, chromatids of one parental origin to the gamete. The minimum noting that the Lk are constants: value of E[¯r] at meiosis I requires crossing-over to be absent [as in Drosophila males and Lepidoptera females (76)] and either a n karyotype of one chromosome or a meiotic process that causes X 2 X hr¯i = h2pk (1 − pk )i Lk + h2pi (1 − pj )i Li Lj . [7] multiple chromosomes to segregate as a single linkage group [as k=1 i=6 j in some species in the evening primrose genus, Oenothera (87)]. | {z } | {z } Second, r¯ satisfies the intuitive property that a CO at the Intrachrom. component Interchrom. component tip of a chromosome causes less shuffling than a CO in the middle (some example calculations of r¯ are given in Fig. 3). If segregation is Mendelian, then in a large sample of gametes, This is easily seen in Eq. 3. Consider a bivalent on which a h2p (1 − p )i ≈ 1/2 i j , and therefore, the interchromosomal com- single CO will be placed. This will divide the chromosome 1 1 − Pn L2
ponent in Eq. 7 will be close to 2 k=1 k . into two segments of length l1 and l2, where l1 + l2 = L is If we have sequence data from many gametes and can deter- the chromosome’s fraction of total genomic length. The con- mine CO positions but not the parental origin of the sequences tribution of this CO to E[¯r] is seen from Eq. 3 to be 1/2 − 5 2 2 that these COs delimit, then we take the average of Eq. , not- (l + l )/2, which simplifies to l1l2 under the constraint l1 + l2 = L 1 2 ing again that the k are constants (which here means that the L. l1l2 is maximized when l1 = l2 = L/2 (i.e., when the CO is interchromosomal component is constant): placed in the middle of the bivalent). It is minimized when l1 = 0 or l2 = 0 (i.e., when the CO is placed at one of the far ends of the n n ! X 2 1 X 2 bivalent). In general, as the CO is moved farther from the mid- h [¯r]i = h2p (1 − p )i L + 1 − L . [8] E k k k 2 k dle of the bivalent, E[¯r] decreases. This is true regardless of the k=1 k=1 positions of the COs on other chromosomes. Importantly, this | {z } | {z } Intrachrom. component Interchrom. component relationship also holds if, instead of considering where to place a single CO on a whole bivalent, we consider where to place a Given data of CO positions along bivalent chromosomes in new CO on a segment already delimited by two COs (or by a CO many meiocytes, we take the average of Eq. 6: and a chromosome end). In general, if we place a fixed number of COs along a single chromosome, E[¯r] is increased if they are n *Ik +1 +! n ! evenly spaced (SI Appendix). 1 X 2 X 2 1 X 2 h [¯r]i = L − l + 1 − L . [9] Third, r¯ will tend to increase sublinearly with both the num- E 2 k k,i 2 k k=1 i=1 k=1 ber of COs and the number of chromosomes (Figs. 3 D–F). | {z } | {z } For example, increasing the number of COs by some factor will Intrachrom. component Interchrom. component increase the intrachromosomal component of r¯ by a lesser factor (and r¯ as a whole by a yet lesser factor, because its interchromo- Eqs. 7–9 mean that we can estimate the average intrachro- somal component is unaffected). Similarly, doubling the number mosomal contribution to r¯ separately for each chromosome, of chromosomes in the haploid set will cause a less than twofold possibly from different sets of data, and then combine these aver- increase in the interchromosomal component of r¯. That is, there ages into a final measure of r¯. This is useful, because often in are “diminishing returns” to genome-wide shuffling from having sequencing or cytological studies of large numbers of gametes increasingly more COs and more chromosomes. These effects or meiosis I nuclei, it is possible (or desired) to obtain accurate are described in mathematical detail in SI Appendix. measurements for only a subset of the chromosomes in each cell so that the sets of cells from which measurements are taken for Intrinsic Implications. We have noted above that, given some two chromosomes will not overlap. This is the case, for exam- number of COs along a chromosome at meiosis I, E[¯r] is max- ple, in the rich cytological data from human pachytene nuclei imized if they are evenly spaced. This observation carries the in ref. 48. interesting implication that positive CO interference—a classical Unrelated to averaging r¯ across multiple measurements of phenomenon (88, 89) where the presence of a CO at some point individual gametes or meiocytes, an average value for r¯ can also along a bivalent chromosome inhibits the formation of nearby be calculated directly given pairwise average rates of shuffling COs—will tend to increase r¯. It thus suggests a possible selective for all loci. These can be estimated from linkage maps generated advantage for this phenomenon (Discussion). from pooled sequence data (see below). Suppose that we have Also, the general diminishing returns to r¯ of having more COs measured, for each locus pair (i, j ), their rate of shuffling rij . hr¯i (above) carry particular implications for differences in total auto- is then simply the average value of rij across all locus pairs (i, j ). somal shuffling in the two sexes of a species. When the number If Λ is the total number of loci, then of COs and/or their localization do not differ grossly between the ! ! ! sexes, then the amount of shuffling will be similar in both sexes X  Λ X  Λ X  Λ hr¯i = rij = rij + rij , (e.g., for male and female humans as discussed below). 2 2 2 i

4 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1817482116 Veller et al. 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Sequence above derived formulas the into for substituted cytolog- be from measured can (easily spreads) length Therefore, ical axis (micrometers). total length of fractions axis their physical total of cal- fractions for required as of pairs), (base culation length genomic total of segments’ packing fractions chromosome the average approximately considerations, is the these nucleus Given that a constant. within infer chromosomes different we of which ratio from S2 species), Fig. Appendix, diverse (SI axis pachytene physical relative at genomic their lengths relative match closely the chromosomes Second, of 101). vary lengths ref. to in 107 appear reviewed (refs. 111–114; nuclei species, within and of chromosomes variety across and a along in minimally and, directly measured of be analysis for required rel- distances the r inter-CO at of least scale at chromosomes crude nucleus, different atively each Moreover, across within constant 111). and approximately along 110, is ratio 108, axis dis- packing 106, bivalent local (101, genomic pachytene the ratio”) a the “packing along define the local distance Therefore, (the directly physical 107–109). 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EVOLUTION A contains a bivalent of paired X chromosomes, which, in this case, has not been distinguished from the other bivalents (e.g., by FISH). Therefore, the X is included in the calculation for the oocyte, so the interchromosomal components of E[¯r] for the spermatocyte and oocyte are not expected to be equal. Although we have estimated them directly from the single cyto- logical spreads in Fig. 4, they can be calculated exactly from ♂ known genomic lengths of the chromosomes. Substituting the chromosome lengths reported in assembly GRCh38.p11 of the B human reference genome (available online at https://www.ncbi. nlm.nih.gov/grc/human/data?asm=GRCh38.p11) (lengths are listed in Dataset S1) into the first term in Eq. 6, we find that the value for the spermatocyte should be 0.4730, close to the value of 0.4744 calculated in Fig. 4A, while the value for the oocyte should be 0.4744, close to the value of 0.4750 calculated in Fig. 4B. These close matches illustrate the reliability of physical length as a proxy for genomic length in the calculation of E[¯r]. Interestingly, although the female spread exhibits about 50% Fig. 4. Calculation of E[¯r] from CO patterns at meiosis I in humans. Left more COs than the male spread (74 vs. 48), the intrachromo- shows micrographs of immunostained spreads of a pachytene spermatocyte somal component of E[¯r] for the female spread is only about (A) and a pachytene oocyte (B). The red lines show localization of the synap- 30% larger than that of the male spread. This is expected from tonemal complex protein SYCP3 and demarcate the structural axes of the the general arguments given earlier concerning the diminishing chromosomes. The green dots along the axes show foci of the protein MLH1 returns of more COs (Fig. 3 D and F). and demarcate the positions of CO associations along the axes. The white We have carried out measurement of [¯r] using data from arrow in A points to the paired X and Y chromosomes. In B, the blue dots E along the chromosome axes mark the centromeres. The paired X chromo- 824 spermatocytes from 10 male humans (data from ref. 122) somes in B have not been identified and are included in the calculation of (provided by Fei Sun, Nantong University, Nantong, China). 9 E[¯r]. A, Center and B, Center show the traced axes and CO positions from Using Eq. , we calculate each chromosome’s average contribu- the spreads. From these, the lengths of the segments separated by the COs tion to the intrachromosomal component of E[¯r] (SI Appendix, can be measured and converted to fractions of the total physical axis length Table S1). Summing these values, we find that the average intra- (these measurements can be found in Dataset S1). The segments’ fractions of chromosomal component of E[¯r] is 0.0143, similar to the value total physical axis length are approximately equal to their fractions of total calculated for the single spermatocyte in Fig. 4A. Restricting genomic length for reasons described in the text. With these fractions of attention to the 715 cells with data for every chromosome allows genomic length, E[¯r] may be calculated for each spread according to Eq. 3, us to calculate the average and SD of the intrachromosomal com- and its inter- and intrachromosomal components can be determined accord- ponent of E[¯r] for each of the 10 individuals. The SD for each ing to Eq. 6 (A, Right and B, Right). Notice that we do not require individual individual is of order 0.0015 (SI Appendix, Table S2) or about ¯ chromosomes to be identified to calculate E[r] or its intra- and interchro- 10% of the average value. There is some variation among the mosomal components. (A) Modified from ref. 103 by permission of Oxford University Press. (B) Modified from ref. 39. individuals for the average value of the intrachromosomal com- ponent of E[¯r]: the minimum is 0.0130, while the maximum is 0.0153. Interestingly, the minimum and maximum do not come individual’s diploid genome has been haplophased. Haplophas- from the individuals with the smallest and largest average CO ing can be achieved by sequencing an extended pedigree involv- frequencies, respectively (SI Appendix, Table S2). In fact, the ing the individual (40, 73, 116); by sequencing multiple offspring, individual with the largest average value of the intrachromoso- gametes, and/or polar bodies of the individual (42, 52, 95, 97, 117, mal component of E[¯r] has only the 4th largest average number 118); or by isolating individual chromosomes from a somatic cell of COs per spermatocyte. This further highlights the importance and sequencing them separately (94, 119). of CO position for the amount of genetic shuffling and hints that r¯ can also be calculated from linkage maps derived from individuals might differ systematically in their CO positioning in pooled sequencing data [as obtained, for example, from pop- ways that affect the amount of shuffling in their gametes. ulation pedigrees (40) or pooled sperm (120)]. A linkage map gives the map distance (in centimorgans) between pairs of linked Analysis of Gamete Sequence Data. Hou et al. (97) sequenced markers. Using this, we can generate a grid of evenly spaced loci the products of multiple meioses (first polar bodies, second for each chromosome (to ensure even sampling of loci in the cal- polar bodies, and ova) of several females and haplophased these culation of r¯) and impute map distances between these by linear females directly from the sequences, allowing the CO points in interpolation from distances between true markers in the linkage those products to be determined. Fig. 5 is a modified version map (details are in SI Appendix). Thus, we generate a map dis- of figure S2 in ref. 97, showing the CO points along the chro- tance dij between every pair of linked loci (i, j ), which we can mosomes of an egg cell from one of the individuals. Because convert to a rate of shuffling rij using a map function (54) [e.g., relatives of the individual were not sequenced, the paternal and 1 Kosambi’s (121): rij = r(dij ) = 2 tanh (2dij )]. This gives a rate maternal sequences cannot be discerned. Therefore, calculation of shuffling for every linked locus pair, from which we calculate of E[¯r] proceeds according to Eq. 2, with its inter- and intrachro- the average intrachromosomal component of hr¯i using Eq. 10. mosomal components determined according to Eq. 5. Carrying out this calculation for all chromosomes, including the X, yields Measuring r in Humans E[¯r] = 0.4971, with an interchromosomal component of 0.4756 Analysis of Cytological Data. Fig. 4 shows the calculation of E[¯r] (calculated using chromosome lengths as they appear in Fig. 5) from the immunostained pachytene chromosomes of a single and an intrachromosomal component of 0.0215 (all details are human spermatocyte (103) and oocyte (39). For both sperma- in Dataset S2). The interchromosomal contribution is about 22 tocyte and oocyte, E[¯r] is close to its maximum value of 1/2. This times larger than the intrachromosomal contribution. Excluding high level of shuffling is due almost entirely to IA: in both cases, the X, E[¯r] = 0.4965, with an interchromosomal component of the interchromosomal component of E[¯r] is much larger than the 0.4735 and an intrachromosomal component of 0.0230. intrachromosomal component (from COs)—by a factor of about Fan et al. (119) haplophased a male using microfluidic tech- 35 in the male and 27 in the female. niques, and Wang et al. (94) sequenced 91 individual sperm cells The spermatocyte in Fig. 4A contains easily identified, par- from this male using the haplophasing from ref. 119 to determine tially paired X and Y chromosomes, while the oocyte in Fig. 4B the CO points in each sperm’s sequence (given in table S2 in

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EVOLUTION value of the intrachromosomal contribution to E[¯r] is about 15% would be easiest with sequence data, especially in species with higher than this interferenceless value. well-annotated genomes. For species without well-annotated CO interference, first described more than a century ago genomes, proxies of regional gene density, such as euchromatin (88), is a deeply conserved feature of the meiotic program content or guanine–cytosine richness, could be used. Correcting (128, 129). Despite this, its adaptive function, if any, remains for gene density with cytological data would be more compli- unclear (104, 130, 131). One category of hypotheses invokes cated, since it requires knowledge of the identity and orienta- mechanical advantages of spread-out COs either to bivalent sta- tion of the various chromosomes. At a crude resolution, this bility in prophase (132) or to homolog segregation at meiosis can be achieved with chromosome-specific FISH probes (and I (133). However, such ideas are challenged by the fact that if required, centromere localization). Finer-scale identification meiotic segregation proceeds without trouble in organisms that of different genomic regions in cytological spreads could be lack CO interference [e.g., fission yeast and Aspergillus (134)] achieved using high-resolution FISH (147–149). In any case, or that have had interference experimentally reduced or elimi- the usual problems would arise in deciding which parts of the nated (135, 136). This study raises the possibility of a qualitatively genome are “functional” (150, 151). different idea: that CO interference provides an evolutionary advantage via its effects on genetic transmission. By explicitly Taking Heterozygosity into Account. Genetic shuffling of alleles taking into account the positions of COs in quantifying how much at two loci in meiosis is practically relevant only if both loci shuffling they cause, we show that CO interference serves to are heterozygous. A simple modification of r¯ that measures the increase the shuffling caused by a given number of COs. This extent of such “effective” shuffling in meiosis is the proportion of finding, therefore, suggests a possible adaptive function of CO locus pairs that shuffle their alleles and are both heterozygous. interference. Again, we are mostly interested in the shuffling of functional elements, in which case estimation of this measure requires Additional Measures of Genetic Shuffling. r¯ is a suitable measure knowledge of the genomic landscape of functional heterozygosity of the total amount of shuffling, linearly aggregating the rates [e.g., nonsynonymous heterozygosity in protein-coding genes as of shuffling between different locus pairs. In this way, it quanti- available for humans (152)] together with knowledge of linkage fies the first-order effect of shuffling: the genetic diversification disequilibrium between nearby loci [e.g., for humans (153)]. of gametes/offspring (1). However, certain population genetic properties exhibit nonlinear dependence on the rates of shuffling Taking Gene Conversion into Account. Another source of shuf- between loci. For example, theoretical studies have determined fling, in addition to COs and IA, is gene conversion (GC) (154, that certain allelic associations can jointly increase in frequency 155): the unidirectional copying of a DNA tract from one chro- only if their constituent loci shuffle their alleles at a rate below matid (the donor) to a homologous chromatid (the acceptor) some critical threshold. This has been shown for coadapted either in mitosis or in meiosis. In SI Appendix, we show how gene complexes exhibiting positive fitness epistasis (137–139), to take both mitotic and meiotic GC into account in measur- “poison–antidote” gamete-killing haplotypes involved in meiotic ing r¯. In meiosis, chromosome pairing involves many points drive (11, 140–142), and associations between sex-determining of interaction between homologous chromatids [∼ 200 across genes and genes with sexually antagonistic fitness effects (53, all chromosomes in mouse spermatocytes (156)], each induced 143–145). Similarly, Hill–Robertson interference is effective only by a double-strand break on one of the chromatids. In most among loci that are tightly linked, shuffling their alleles at a rate organisms, only a minority of these interactions result in COs below a small threshold (which depends on the effective pop- (104, 157, 158), the majority instead resulting in non-COs. Both ulation size and other parameters) (146). In quantifying these outcomes are associated with short tracts of GC [e.g., ∼ 100 various properties at an aggregate level, a more suitable measure bp for non-CO GCs and ∼ 500 bp for CO GCs in mice (98, would be the proportion of locus pairs that shuffle their alleles at 156)]. These tracts, although numerous, are too short to seriously a rate below the critical threshold. Such measures would be infor- affect the total amount of shuffling [for example, the contribu- mative, for example, of which chromosomes are most likely to be tion from meiotic GCs in mammals will be of the order 10,000 co-opted as new sex chromosomes, of how susceptible a species times smaller than the contribution from COs (SI Appendix); is to the invasion of meiotic drive complexes (or within species, in yeast, this figure is smaller—about 200 (SI Appendix)]; how- which chromosomes are most likely to harbor such complexes), ever, they may have important local effects (154, 159). Mitotic or of the average fitness reduction caused by Hill–Robertson GC tracts result from homologous chromosome repair, are interference. Calculating these threshold-based aggregate mea- typically much longer than meiotic GC tracts (often of order sures would require estimates of the average rates of shuffling for 10 kb) (160, 161), and if they occur in germline cell divisions specific locus pairs. High-density linkage maps would, therefore, leading to meiocytes, can lead to shuffling of maternal and be especially promising for such measurements. paternal DNA in gametes (which should especially affect males, who have more germline cell divisions). Too little is presently Taking Gene Density into Account. The calculation of r¯, as we known about the global frequency of these events for us to have defined it, treats all regions of equal genomic length as give a reasonable quantitative estimate of how much shuffling of the same importance. However, the quantity that we seek to they cause. measure is really the amount of shuffling of functional genetic elements—it is largely irrelevant if two functionless pseudogenes ACKNOWLEDGMENTS. We thank Anthony Geneva, David Haig, Arbel shuffle, for example. Correcting the calculation of r¯ to take Harpak, Dan Hartl, Alison Hill, Thomas Lenormand, Pavitra Muralidhar, gene density into account is an important extension, a point Naomi Pierce, Molly Przeworski, Nadia Singh, and Martin White for helpful comments and Colin Donihue for help with figure preparation. C.V. thanks already noted by Burt et al. (27). This is especially so if the Bob Trivers for impressing on him the importance of CO position for the distribution of genes is nonrandom along or across chromo- amount of recombination. The research of N.K. was supported by NIH Grant somes. Correcting for gene densities of different genomic regions GM-044794.

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