Genome-wide compatible SNP intervals and their properties

Transcription

1 Genome-wde compatble SNP ntervals and ther propertes Jeremy Wang Dept. of Computer Scence Unversty of North Carolna Chapel Hll, NC 27599, USA Fernando Pardo-Manual de Vllena Dept. of Genetcs Unversty of North Carolna Chapel Hll, NC 27599, USA Kyle J. Moore Dept. of Computer Scence Unversty of North Carolna Chapel Hll, NC 27599, USA We Wang Dept. of Computer Scence Unversty of North Carolna Chapel Hll, NC 27599, USA Q Zhang Dept. of Computer Scence Unversty of North Carolna Chapel Hll, NC 27599, USA eonard McMllan Dept. of Computer Scence Unversty of North Carolna Chapel Hll, NC 27599, USA ABSTACT Intraspecfc genomes can be subdvded nto blocks wth lmted dversty. Understandng the dstrbuton and structure of these blocks wll help to unravel many bologcal problems ncludng the dentfcaton of genes assocated wth complex dseases, fndng the ancestral orgns of a gven populaton, and localzng regons of hstorcal recombnaton, gene converson, and homoplasy. We present methods for parttonng a genomnto blocks for whch there are no apparent recombnatons, thus provdng parsmonous sets of compatble genomntervals based on the four-gamete test. Our contrbuton s a thorough analyss of the problem of dvdng a genomnto compatblntervals, n terms of ts computatonal complexty, and by provdng an achevable lower-bound on the mnmal number of ntervals requred to cover an entre data set. In general, such mnmal nterval parttons are not unque. However, wdentfy propertes that are common to every possble soluton. We also defne the noton of an nterval set that acheves thnterval lower-bound, yet maxmzes nterval overlap. We demonstrate algorthms for parttonng both haplotype data from nbred mce as well as outbred heterozygous genotype data usng extensons of the standard four-gamete test. These methods allow our algorthms to be appled to a wde range of genomc data sets. 1. INTODUCTION The local block-structure of genotypes wthn a populaton sheds lght on many bologcal questons [10]. Genotype blocks are central to quantfyng and localzng recombnatons (both recent and hstorcal) [37, 36, 40], are wdely used to dentfy nformatve marker sets [46], and are buldng blocks for constructng genetc maps [35]. Genotype-block structure also underles many genomewde assocaton methods[45], provdes bochemcal evdence for selecton [18], and offers a tool for ascertanng ancestral orgns [9]. The task of decomposng a genomnto meanngful blocks, however, has proven to bll-defned, nconsstent, and often ambguous [31, 34]. In part, the problem resdes n the ad hoc defnton of what consttutes a genotype block. Genotype blocks are often defned to serve a specfc purpose. Examples nclude the mnmum number of taggng SNPs suffcent to capturnformatve genotypes [32, 46], ntervals of SNPs that exceed a gven threshold of nkage Dsequlbrum (D) [33], and maxmal regons whose genotype dversty falls below a threshold [10]. Parttonng genotypes nto blocks supportng perfect phylogenes [37, 17], and, the related, selecton of blocks lackng evdence for recombnaton [41] are also used to construct Ancestral ecombnaton Graphs (AGs). We propose unambguous defntons for haplotype and genotype blocks and effcent methods for computng them. Where ambguty s unavodable, we have uncovered propertes common to all solutons. Our haplotype block defnton drectly supports, and has been used for, assocaton mappng [30], constructon of genetc maps [48], and determnng ancestral orgns wthn local genomc regons [49]. Our proposed genotype blocks can be used n much the same way. Dense genotype data sets that are homozygous at every allele are avalable for many nbred mammal [15] and plant [6, 28] models commonly used for assocaton mappng. However, haplotype data s not drectly avalable for usn human studes. In our approach, t s unnecessary to phase such data sets, yet, we can stll dentfy blocks mportant for explorng the local dversty structures [3, 24, 27, 29], and ancestral orgns [44]. ke others [17, 41, 40], our blocks are chosen for ther lack of hstorcal recombnaton evdence. Our methods can be used as an alternatve to other block methods such as thosn [13, 47, 16]. Partcularly, the genotype block methods wntroduce may be used to nform phasng [25] and to extend these methods to unphased genotype data. Block assocaton methods such as Blossoc [26] and QBlossoc [2], whch utlze small regons that admt perfect phylogenes could potentally beneft from our methods. We defne our blocks n terms of SNP compatblty accordng to the Four-Gamete Test (FGT) [22]. The FGT s of nterest because of ts close relaton to perfect phylogeny [23]. Specfcally, a neces-

2 (a) C(h) Uber (b) C(g) Uber Fgure 1: Example data sets and Uber-cover. The data sets used n ths fgure wll be used as runnng examples. The lower porton of the fgurs the data matrx. The columns correspond to SNPs, and the rows correspond to haplotypes (a) and genotypes (b) derved by parng adjacent haplotypes. Blue and yellow boxes represent homozygous alleles normalzed such that the frst sample s homozygous blue. Green boxes represent heterozygous alleles. The large trangles above the data matrces are the compatblty matrces, whch show the compatblty of each par of SNPs. Gray boxes mply that a par of SNPs defntely does not volate the four-gamete rule. ed boxes mply that a par of SNPs creates four gametes. Green boxes mply that a par must bn-phase to be compatble. Orange boxes mply that a par must be out-of-phase to be compatble. Blue boxes mply that a par must be ether n-phase or out-of-phase to be compatble. sary and suffcent condton for a perfect phylogeny s that all pars of SNPs satsfy the FGT [21]. For unphased genotype data, we defne the noton of optmstc and pessmstc compatblty based on f a regon possbly or necessarly passes the FGT. We partton the genomnto a set of potentally overlappng, maxmal compatble ntervals, each of whch admts a perfect phylogeny, and whose unon covers the full data set. We address the queston of what s the fewest number of such ntervals requred, and we also dentfy suspect SNPs whose removal reduces the overall complexty of the block structure (perhaps ndcatng genotypng errors, homoplasy, or gene conversons). Our contrbuton s an analyss of the problem of dvdng a genome nto compatblntervals based on genotypes and ts complexty. We provde an achevable lower-bound on the number of such ntervals. Whln general there are numerous ways of dvdng a genomnto a mnmum number of compatblntervals (a fact overlooked by others [26, 40, 42]), we also dentfy non-overlappng core subntervals common to all vald solutons. We also defne an nterval set that acheves thnterval lower-bound, yet maxmzes the block overlap, thus mnmzng the number of perfect phylogeny trees, whle provdng the rchest possble set of SNPs to each tree. 2. EATED WOK There are three common approaches for parttonng haplotypes nto blocks. The frst employs D measures [16, 33] and assgns blocks to regons wth hgh parwse D wthn, and low D between, blocks. A second class assgns blocks to regons of low sequence dversty [32]. astly, there are approaches that look for drect evdence of recombnaton, by ether applyng the FGT [22] and defnng blocks as regons free of apparent recombnaton or homoplasy, or durng the constructon of AGs, denotng supportng regons component subtrees [37]. Schwartz et al. [34] performed an analyss of approaches and concluded that the block assgnments of varous methods dffered markedly. Of these methods, the block boundares of the FGT were better correlated to both the D and dversty-based methods than these two methods were to each other. Our approach parttons the genomnto blocks satsfyng the FGT. Ths s not new. The semnal work of Hudson and Kaplan provdes a sketch of a greedy algorthm that processes SNPs n sequence order lookng for runs of compatblntervals that are broken at ponts of ncompatblty. Ths method appears to be wdely used [26, 34, 40, 42]. A dsconcertng feature of ths approach s that one arrves at a dfferent nterval set f the genoms scanned n the reverse order (Fg. 4). Alternatve sets of compatblty ntervals arse when the regon s grown maxmally around each SNP [26]. Moreover, there appears to be many other possble parttons, beggng the queston of whch block sets have the fewest ntervals, and, of these sets, whch mnmzes haplotype dversty. In our model, each block s compatble wth a perfect phylogeny (a sde effect of the FGT) and overlaps between adjacent ntervals are allowed. We extend our methods to unphased genotype data. ttle work has been done on parttonng genotype data wthout frst phasng, however, there has been consderable work on the related topc of phasng by perfect phylogeny [1, 4, 14, 19, 12]. Such methods determne f a gven genotype block admts a perfect phylogeny. Our

3 (a) Flaggng SNPs Fgure 2: Uber ntervals wth flaggng SNPs hghlghted n red. Incompatbltes between flaggng SNPs of adjacent maxmal compatblntervals are hghlghted wth green crcles. contrbuton s to apply the basc nsghts of these methods to extend the noton of a haplotype "scan" to the genotype case. Smlar work has been done [11] n whch local phylogenes are bult over unphased genotype data to nform assocaton mappng, however ths work does not take full advantage of compatble blocks, employng a sngle-marker approach rather than a global block structure. Past attempts at usng perfect phylogeny to analyze genotypes assume they are gven a regon whch admts a perfect phylogeny or does so wthn an error model. Most prevous work ([12, 19, 14, 38, 1]) determnes f the gven data set does n fact admt a perfect phylogeny, and then solves the Perfect Phylogeny Haplotypng problem (PPH) ([19]) for the gven nstance. ecent extensons allow data to fall wthn some error model and subsequently handles cases where the data does not ft a perfect phylogeny. Error models nclude Mssng Data (MD) and Character-removal (C), and the algorthms attan a global perfect phylogeny whle dealng wth erroneous pont cases ([21, 20]). No prevous approach consders the possblty of dfferent PPH solutons as determned by the choce of block partton. Whle we do not propose haplotypng by perfect phylogeny, we use related technques to partton the genomnto blocks whch satsfy a perfect phylogeny that could, n practce, then be haplotyped usng any one of several prevous algorthms. Introducng a genome-wde approach to perfect phylogeny rather than flterng out data as n [21, 20] consders many bologcal factors prevously overlooked. The noton of recombnaton-free blocks n the genoms welldocumented n humans and mce, as well as other speces [16, 47, 40, 30, 28]. In many cases, regons of the genome on ether sde of a recombnaton pont should realstcally admt dfferent phylogenes based on hybrdzaton between subspeces. Smply removng presumed erroneous data and forcng regons separated by hstorcal recombnaton nto a global phylogeny gnores ther bologcal relevance and produces a msleadng soluton. Our method of parttonng allows for bologcally meanngful, though lmted, regons wth whch to perform further analyses. 3. PEIMINAIES Throughout ths paper we assume a data set of m SNPs spannng n haplotypes (or genotypes) that are represented as a bnary data (a) C(p) Max Fgure 3: Pessmstc Max-k cover over the example set of genotypes. In the compatblty matrx, red ndcates that the correspondng SNP par s possbly ncompatble, gray ndcates that the par s defntely compatble. matrx S or ternary matrx S g where each column corresponds to a SNP, and each row s a haplotype or genotype (Fg. 1). Alleles 0 and 1 represent alternatve homozygous alleles and 2 represents heterozygous alleles. A compatblnterval over a set of haplotypes s a sequence of contguous SNPs over S for whch there are no volatons of the FGT between any SNP par. A compatblnterval, I x = [b x, e x], ncludes all SNPs between the startng SNP s bx and endng SNP, s ex. Fg. 1(a) shows a data set of 16 haplotypes and 44 SNPs, together wth eght compatblntervals, I 1 through I 8. Each nterval covers a consecutve set of SNPs. For example, I 3 covers from s 8 to s 26. The trangular matrx above the SNP matrx s the parwse compatblty matrx. If two SNPs exhbt four gametes, the correspondng matrx element s marked ncompatble (red). Darkened trangles ndcate sub-matrces correspondng to SNP pars n the compatblntervals. Note that no trangles enclose red elements. Compatblntervals over genotypes (S g) are less straghtforward due to ambgutes caused by heterozygous alleles. We defne the noton of optmstc and pessmstc compatblty, whether genotypes are possbly or nessesarly compatble, respectvely. esolvng genotypntervals requres more consderatons when performng the FGT. Pars of SNPs are evaluated to determne whch gametes phasng could produce. In cases of homozygous-homozygous and homozygous-heterozygous pars, the possble gametes are trvally determned. For example, the 0-0 produces only the 0-0 gamete and 0-2 produces the 0-0 and 0-1 gametes. Ambguty s caused only by the 2-2 case when there exst heterozygous alleles n the same sample at two dfferent loc. These cases can produce two dfferent sets of gametes, ether 0-0 and 1-1, whch we call n phase gametes, or 0-1 and 1-0, whch we call out of phase gametes. There are three compatblty cases f 2-2 parngs exst. The ambguous cases must bn phasn order to be compatble (f one of the 0-1 or 1-0 gametes are not present), must be out of phase (f one of the 0-0 or 1-1 gametes are not present), or truly ambguous when all 2-2 pars must smply be produce the same set of gametes snct s always possble to produce four gametes wth opposte phasngs of two 2-2 pars. The optmstc algorthm forms a graph wth a vertex representng each locus and an edge representng a phasng (n phase, out of phase, or ambguous) between two vertces. An nterval s optmstcally compatblff there exsts a bpartton of the graph nto

4 two sets A and B such that no edge wthn A s out of phase, no edge wthn B s out of phase, no edge between sets A and B s n phase, and all ambguous edges are unquely resolvable to as ether n phase or out of phase. We use an algorthm smlar to [14] to partton the genomnto blocks of genotypes whch admt a perfect phylogeny. Smlar to the haplotype "scan", wntroduce SNPs one-by-one and test whether the resultng nterval s nternally compatble. For haplotypes, ths s accomplshed by parwse comparsons of prevous SNPs wth every newly ntroduced SNP. For the genotype case, we defne two nterval types. For optmstc ntervals, we use thdea of what Eskn et al [14] refer to as equal and unequal resoluton to create a bpartte graph for each proposed nterval. We "scan" SNPs as descrbed n the haplotype case, addng these SNPs as vertces to the graph untl the graph s no longer realzeable, thus endng an nterval. Pessmstc ntervals are unambguously compatble regardless of the choce of phasng. When the scan s performed, an nterval s ended as soon as t reaches a SNP whch s possbly ncompatble wth any prevous SNP n the nterval. Ths s equvalent to consderng all non-gray ponts as ncompatble (red) and performng a haplotype scan to produce the pessmstc genotypntervals (see Fg. 3). Fg. 1(b) shows a data set of 8 genotypes and 57 SNPs, together wth four of ts optmstc compatblntervals. It remans true that no nterval may enclose an ncompatble SNP par. However, unlke the haplotype case, ntervals are not necessarly bounded by red elements. As descrbed, SNPs may bmplctly ncompatble wth a gven nterval f ther addton forms an unrealzeable graph. A compatblnterval s maxmal f t cannot be extended n ether drecton. All ntervals n Fg. 1(a) (I 1, I 3, I 4, I 6, and I 8) are maxmal, snce further extenson ncludes one or morncompatble SNP pars. We denote the set of all maxmal compatblntervals as C Uber. Throughout, we wll denote a cover over a genome genercally by C. A cover of a set of haplotypes wll be represented by C(h). An optmstc cover of a set of genotypes wll be represented by C(g) and a pessmstc cover by C(p). The darkened trangles n Fg. 1(a) depct C(h) Uber. The two SNPs adjacent to a maxmal compatblnterval, s bx 1 and s ex+1, are the flaggng SNPs of the nterval (Fg. 2). Note that flaggng SNPs arncompatble wth at least one SNP of the maxmal compatblnterval that t flanks. A cover, C x,y, s an ordered set of ntervals, C x,y= {I 1, I 2,..., I c}, where +1, and every SNP n the range [x, y] s covered by somnterval n C but no SNP outsde [x, y] s covered by any nterval n C x,y. C x,y also satsfes +1, snce otherwse +1 s a fully contaned subset of. We call C 1,m a complete cover of S, and C s ts cardnalty. We wll frequently refer to a cover, C, where C = c, as a c-nterval cover, or smply as a c-cover. In addton, we wll refer to specal nstances of complete covers by usng descrptve subscrpts, n whch case a range from [1, m] s mpled. For example, n Fg. 1(a), {I 1, I 3, I 4} s a C(h) 1,29 cover, {I 1, I 3, I 4, I 6, I 8} s a complete 5-cover. In ths paper, we are partcularly nterested n complete k-covers, where k s a reachable lower bound on the number of ntervals for the gven SNP set. In the followng sectons we provde an effectve method for fndng mnmum-length complete covers for a gven SNP set. Thus, establshng k as a tght lower bound. In general, thers no one unque k-cover for a gven data set. We provde several algorthms that generate varous k-covers n tme lnear to the number of genotypes. In addton, we examne features whch are common to all k- covers of a gven data set. We then present a lnear-tme algorthm for fndng a cover composed entrely of maxmal compatblntervals from C Uber, where C Uber k. Fnally, we present an algorthm for fndng the k-cover wth maxmal overlap, the Maxmalk-Cover (C Max). The cover C Max s of partcular nterest snct leads to the constructon of a parsmonous set of perfect phylogeny trees where each ncorporates maxmal nformaton (.e. the maxmum number of SNPs per tree). Fnally, we present an algorthm for fndng crtcal SNPs n S whose removal reduces, C Max, from k to k 1 or smaller, usng a number of tests that are proportonal to C Uber rather than m. 4. A OWE BOUND ON THE NUMBE OF INTEVAS IN A COMPETE COVE 4.1 eft-to-ght and ght-to-eft Covers We frst defne two non-overlappng covers, the eft-to-ght cover (C(h) ), and thght-to-eft cover (C(h) ). A smple greedy algorthm, Scan, whose pseudocods gven n Appendx B, fnds C(h) over a set of haplotypes and t has been prevously descrbed n [40]. It begns at the leftmost SNP (s 1), and ether extends or termnates the current actvnterval as t consders each SNP n sequence order. Scan performs FGTs of the canddate SNP aganst those SNPs already n the actvnterval. If four gametes occur, the actvnterval s closed, and a new nterval begns from the canddate, otherwse the SNP s added to the actvnterval. Ths contnues untl the last SNP s reached, thus closng the fnal nterval (see Fg. 4(a)). The run-tme of Scan depends on the number of SNPs, m, and the number of the FGTs performed for each SNP. Snce the maxmum number of dstnct compatble SNP patterns (SDPs) that can be mutually compatble among n haplotypes s 2n 3 [38], the FGT requres only O(n) operatons per SNP, assumng a constanttme overhead for each FGT. Therefore, the complexty for Scan s O(mn), and thus s lnear n the number of genotypes. A smlar greedy ght-to-eft scan algorthm (Scan) generatng C(h) can be defned va straghtforward modfcatons to (Scan). kewse C(h) can be generated by merely reversng thnput sequence, applyng Scan, and adjustng thndces of the resultng ntervals, ncludng ther startng and endng postons. Note that a cover s nterval ndces are assgned accordng to the sequence order, regardless of the scannng drecton. We defne a smlar noton over a set of genotypes. As descrbed n Secton 3, the only dfferencs the manner n whch the FGT s performed. We fnd an optmstc left-to-rght (C(g) ) and rghtto-left (C(g) ) cover by closng an nterval only when the subsequent SNP wll be defntely and unambguously ncompatble wth a SNP n thnterval (regardless of the phasng chosen). kewse, a pessmstc nterval (C(p) and C(p) ) s closed off f there exsts a phasng of the genotype set for whch the next SNP wll be ncompatble wth SNPs already n thnterval. The run-tme of the pessmstc genotype scan s also O(mn). ke the haplotype case, thers a lmt on the number of dstnct SNPs that can be compatble among n genotypes whch s lnear n n. Smlarly, the adjusted FGT requres only O(n) operatons per SNP to determnf there exsts any possblncompatblty. The run-tme of the optmstc genotype scan s more complex. Eskn et al ([14]) propose an alogorthm wth O(nm 2 ) complexty to determnf a sngle regon admts a perfect phylogeny. We use a

5 (a) C and C (b) C Uber wth cores (c) C Max Fgure 4: Shown abovs the progresson of covers. (a) depcts C (green) and C (orange) and ther overlap (cores) outlned n red. (b) shows C Uber and the cores from (a). In (c), each C Max nterval (a subset of C Uber ) encloses exactly one core. smlar algorthm, addng SNPs ncrementally. Snce each nterval s bounded and we scan lnearly across the genome, ths allows for an O(nm 2 ) algorthm to partton the entre genome. 4.2 Propertes of C and C Our frst theorem states, "The covers C and C have the same number of ntervals k, and k s the mnmal number of ntervals possble for any complete cover." A proof of ths and an assocated emma 1 are provded n Appendx A. Moreover, certan core subntervals are common to all complete k-covers of a gven SNP set. We defne thntersectons of correspondng ntervals from C and C as cores (Fg. 4). Accordng to emma 1, e e and b b (Fg. 10(a), Appendx), therefore Cor = [b, e ]. emma 2 and a detaled proof of ths clam arncluded n Appendx A. Theorem 2 states, "For any complete k-cover C 1,m= {I 1,..., I k }, thth nterval contans the entrth core: Cor = Cor, and, t does not contan any part of another core Core j = φ, 1 j k, j. Ths s due to thnterleavng of the non-overlappng ntervals of C and C. Cor s necessarly compatble wth both C and C and cannot be extended beyond the outsde boundary of ether. Therefore, no part of any two cores may be a part of the samnterval. A k-cover must contan k ntervals each contanng one core exclusvely. Ths leads to two corollares. The frst s that any nterval that does not contan an entre cors not part of any complete k-cover and the second s that thth cors only contaned wthn thth nterval of any k-cover. Cores have several nterestng propertes worth notng. All SNPs n a core are compatble, snce each cors an ntersecton of two compatble regons and adjacent cores must contan at least one par of ncompatble SNPs. Proofs of these propertes are gven n Appendx A. 5. MAXIMA-K-COVE AGOITHM Frst wntroduce UberScan, whch generates the set of all the maxmal compatblntervals, C Uber. UberScan (Appendx B), s smlar to the Scan. Whenever a compatblnterval ends at SNP s, nstead of startng the next nterval from s +1 as Scan does, UberScan fnds the nearest SNP s j (j < + 1) that s ncompatble wth s +1, and the followng SNP, s j+1, begns the next nterval. Note that s +1 s a flaggng SNP of the prevous maxmal compatblnterval and s j s a flaggng SNP of the next maxmal compatblnterval. UberScan s a smple modfcaton of Scan wth added bookkeepng to track of thndex of the last occurrence of each unque SNP pattern 1. A smlar analyss of Scan shows that UberScan also takes O(mn) tme. UberScan generates C Uber, contanng all maxmal ntervals of S, and generally, C Uber k. C Uber contans all canddates for the Maxmal-kcover, C Max, snce a cover wth maxmal overlap must be composed of maxmal ntervals. 5.1 Fndng the Maxmal-k-cover by fndng the longest path n a k-partte graph A Maxmal-k-cover, C Max, s of partcular nterest as t covers the entre SNP set usng the fewest, k, maxmal ntervals. Whle, C Max s not necessarly unque, alternate solutons are generally smlar. Next we provde a fast graph algorthm to compute all Maxmal-k-covers. We consder only those maxmal ntervals n C Uber that entrely enclose a sngle core and no part of a second core, as defned by C and C (4.2). Accordng to Theorem 2, thesntervals are the canddates for Maxmal-k-covers. Next, we organze the canddates nto k groups accordng to the cort contans. Each core s contaned wthn at least one maxmal nterval, thus, no group s empty. We then examne the overlap between groups. A canddate nterval n group can only overlap canddates from groups 1 or + 1, snce any two canddates enclosng non-adjacent cores (say Core x and Core y) mply that at least one of them contans part of the core between Core x and Core y, contradctng ts qualfcaton as a canddate. The Maxmal-k-cover problem s solved by recastng t as fndng the longest path n a drected k-partte graph. Specfcally, each canddate maxmal nterval s mapped to a node and each group as a part, wth part contanng all the canddates coverng Cor. An edge connects nodes correspondng to overlappng canddates. Each edge s weght s the amount of overlap between the two ntervals. The edgs drected towards the canddate that contans the next corn the sequence. As shown prevously edges only exst between adjacent parts. Fnally, we add a source wth edges to all nodes n part 1 and a snk wth edges from all the nodes n part k, both types of edges have weght 0. Fndng a C Max soluton corresponds to fndng the longest path n ths drected graph wth k + 1 edges from source to snk. Note that the greedy approach of takng the largest nterval that encloses each core does not always yeld a 1 ecall that the maxmum number of dstnct SNP patterns wthn a compatblnterval s 2n 3, or O(n).

6 possblnterpretatons (phasngs) of a set of genotypes as haplotypes. We defne two approaches for determnng compatblty among genotypes wthout explctly phasng. The optmstc method determnes ntervals for whch there mght exst a phasng such that thnterval s four-gamete compatble. The pessmstc method determnes ntervals by choosng phasngs such that the current SNP s ncompatble wth the current nterval wherever possble. (a) Fgure 5: The dstrbuton of cover szes for genotypes resultng from all parngs of a set of haplotypes. For a dstrbutons, data was smulated usng thnfnte stes model and recombnaton. The source haplotypes cover sze (the "ground truth") s represented by the sold black lne. For b dstrbutons, a contrved haplotype set was made by phasng a pessmstc genotype result of a. The source haplotypes cover szs the dashed black lne. correct answer, as shown n the fourth core of our runnng example (compare Fg. 2 (a) and Fg. 4 (c)). The problem s a sngle-source shortest path problem for a weghted DAG, except that we search for longest path (maxmzng nstead of mnmzng the sum of weght on the path) wth a constrant on the number of steps. The constrant can bgnored snce all edges lead from one part to the next, thus any path from source to snk wll have k + 1 steps. The problem can be solved usng dynamc programmng and requres only Θ( V + E ) tme, where V s the set of nodes, and E s the set of edges [8]. 5.2 Crtcal SNPs A crtcal SNP s any SNP whose removal reduces k, the mnmum number of ntervals requred to cover the gven SNP set. To check whether a SNP s crtcal, one could smply remove each SNP and recalculate k by ether a Scan or an Scan. Ths nave approach requres m scans, and takes O(nm 2 ) tme. However, t s unnecessary to test every SNP. In fact, only flaggng SNPs of maxmal compatblntervals need to be consdered. A flaggng SNP boundng an nterval on one sde prevents thnterval from growng toward an adjacent nterval on that sde, therefore a flaggng SNP must be removed to allow any nterval to grow, a necessary condton of reducng k. A complete proof s gven n Appendx A. Snce each maxmal compatblnterval has two flaggng SNPs, one on each sde of thnterval, the total number of flaggng SNPs s O( C Uber ). The runnng-tme for computng crtcal SNPs s O(nm C Uber ). 6. POPETIES OF GENOTYPE INTEVA COVES Determnng four-gamete compatblty and compatblntervals over unphased genotype data s ambguous n that there are many We compared the haplotype and genotypntervals on three data sets. Frst, we created smulated genotype data usng a smple nfnte-stes model of mutaton wth cross-over recombnaton - ths served as a data source devod of confoundng factors such as expermental error and homoplasy. In the second, we used real data from F1 crosses between sogenc mouse strans. astly, we used two populatons of HapMap data. 6.1 elatng Genotype and Haplotype Covers. In smulated data, one can explore relatonshps between the compatble ntervals of genotypes and the compatblntervals of ther "source" haplotypes. Such smulated data can be used as ground truth data for a set of genotypes. The number of ntervals n a pessmstc genotype cover s greater than or equal to the number of ground truth ntervals. Thus, the number of ntervals n an optmstc and pessmstc genotype scan are lower and upper bounds on the true number of ntervals. Proof of these clams are provded by Theorems 4 and 5 of Appendx A. Fg. 5(a) represents the dstrbuton of the set of all possble "parngs" of a fxed haplotype set nto genotypes. For a dstrbutons, the "ground truth", or the number of ntervals requred to form a cover usng the source haplotypes, s 5. Notce that the covers resultng from every optmstc genotype scans fall closer to the "ground truth" than those from every pessmstc scans. From our expermental results, we observe ths s a common case. In contrast, b dstrbutons represent the same plot for a contrved, nonbology based, haplotype data set. These represent the dstrbuton of the cover sze of all genotypes that can be formed by parngs of the haplotype set that acheves one of the pessmstc covers from a (ths can always be acheved, as dscussed n 6.2). Specfcally, the "true" cover sze for ths contrved set s 23. Notce that the dstrbuton s dfferent from the bology-based model. In partcular, the optmstc and pessmstc dstrbutons are closer together and the "ground truth" s nearer the pessmstc estmatons. 6.2 Achevng Genotype Covers by Phasng In many crcumstances, t s useful to determnf a partcular genotype cover or nterval set s achevable by phasng. Pessmstc genotype covers can always be acheved (see Algorthm 2 gven n Appendx B). However, there does not always exst a phasng to accomplsh a gven optmstc genotypnterval, n practce many canddate covers can be proposed, and t s trval to verfy canddate ntervals usng exstng PPH methods ([19, 14, 1, 4]). 7. EXPEIMENTS AND ESUTS We tested the performance of our algorthms on real datasets. The frst s based on 8 nbred mouse strans and selected F1 crosses between these strans usng a newly developed genome-wde genotypng platform ([44]). The second and thrd are human datasets from the Internatonal HapMap Project [7]. The frst HapMap dataset descrbes a populaton of Utah resdents from northern and western Europe (CEU). The second s a Yoruba populaton from Ibadan, Ngera (YI). The mouse genome contans 20 chromosomes (chromosome 1-19 and chromosome X). The number of SNPs per chro-

7 (a) (b) C(h) Max (c) C(g) Max Fgure 6: (a) shows the k-partte graph used to fnd C(h) Max for the runnng example. The node represents thnterval, and edges connect ntervals that overlap, wth weght representng the number of shared SNPs. The longest path (bold) computed from source to snk corresponds to C Max. It contans ntervals I 1, I 3, I 4, I 6, and I 8 from C Uber, wth a total overlap of 17. (b) s the Maxmumk-cover of the set of haplotypes, C(h) Max, contanng I 1, I 3, I 4, I 6, and I 8. (c) s the optmstc Maxmum-k-cover of the set of genotypes, C(g) Max. mosome vares from 15K to 50K and ncludes strans: 129SvlmJ, A/J, C57B/6J, CAST/EJ, NOD/tJ, NZO/HItJ, PWK/PhJ, and WSB/EJ along wth 37 F1 crosses. Thnbred founder mouse strans were used n the haplotype analyss, whle the F1 crosses were used n the genotype analyses. Wth ths mouse data set, we are able to ascertan an approxmate ground truth, sans genotypng errors, wth real-world rather than smulated data. Only fully nformatve SNPs among the 8 strans were used n our analyss, reducng the 600K total SNPs to 340K. We found compatblntervals for 23 human chromosomes from the phased HapMap data (Chromosome 1-22 and Chromosome X). The CEU dataset has 348 haplotypes (174 ndvduals) wth 34K - 222K SNPs per chromosome and the YI dataset, has 348 haplotypes (174 ndvduals) wth 38K - 252K SNPs per chromosome. Snce the phasng method used by HapMap also mputes mssng data, all data was used to evaluate the haplotype methods. The phased CEU data set has 2.6M SNPs and YI has 2.9M SNPs. Our algorthms wermplemented n Python 2.6 and experments were performed on a 2.67GHz Intel Cor7 processor wth 8.0GB of AM. 7.1 un-tmes The performance of nterval scannng algorthms (Scan, Scan and UberScan) s lnear n the number of genotypes, whch enables us to compute C, C, and C Uber effcently. The run-tmes of all three scans and the Max-k-cover algorthm was recorded for all the data sets. Fg. 7(a) gves the tmes for calculatng the haplotype nterval covers on the mouse data set (nbred founders) and genotype covers for the F1 crosses. Fg. 7(b) shows run tmes for all covers over the HapMap CEU data set (Ys smlar). As shown, the run-tmes for haplotype and pessmstc genotype scans are lnear n the number of genotypes. Optmstc genotype scans are quadratc n the number of SNPs n the largest nterval, whch s not bounded by the number of genotypes as n the other scans, but s far smaller than the sze of the genome. Snce the Scan and Scan are symmetrc procedures, they have smlar run-tmes. UberScan nvolves more bookkeepng and, thus, has hgher tmes than the other scans. The optmstc genotype scan tmes are much hgher than the haplotype and pessmstc scans due to the computatonally ntensve graph algorthm whch must be performed at each step. un-tmes for all scans vary addtonally across chromosomes dependng on propertes of the data such as nterval sze. The k-partte graph component of the Max-k-cover algorthm takes thntervals of C Uber as nput. Fg. 8(b) shows the run-tme of the Max-k-cover algorthm as a functon of the number of ntervals C Uber. The run-tme of the Max-k-cover algorthm has a lnear relatonshp wth the cardnalty of C Uber. 7.2 Interval and Core Statstcs We also collected varous statstcs over the C Max ntervals and cores, ncludng nterval lengths n terms of SNPs and genomc poston, and the number of dstnct haplotypes. Thnterval lengths shown n Fgure 9 llustrate the prevalence of many long blocks (suggestve of conserved regons) punctuated by smaller ntervals ndcatve of hot spots. Numbers of dstnct haplotypes (Fg. 9(b)) ndcate the relatve dversty wthn each block. Note that the maxmum number of dstnct haplotypes s Mn(n, s + 1), where s s the number of unque SNPs n thnterval. A large percentage of Max-k ntervals n the optmstc genotype cover of the human CEU data set contan only a sngle SNP. By defnton, these are Crtcal SNPs - removng one reduces the total number of ntervals k by at least one. Such one-snp ntervals, whch arncompatble wth SNPs mmedately adjacent, are not lkely to bnformatve regardng recombnatons and probably represent other bologcal or expermental artfacts such as genotype errors, homoplasy, or gene conversons. Ths explans the prevalence of ntervals wth two unque haplotypes as shown n Fg. 9(b), and the large number of sngle-snp ntervals, as shown n Fg. 9(a). Fg. 9(a) also shows the dstrbuton of nterval szes n SNPs of the Max-k cover after the SNPs makng up these sngle- SNP ntervals have been removed. Notce that the averagnterval szs greater for all three cover types after removng these data. The relatonshp between the haplotype covers and the optmstc and pessmstc genotype covers s llustrated n Fgures 8 and 9. Fg. 8(a) shows the number of ntervals n a k-cover of the CEU human data set for each chromosome. These covers demonstrate Theorems 4 and 5 (Appendx A) that C(g) C(h) C(p). Thus, the genotype scans serve as effectve upper and lower bounds on the "ground truth".

8 (a) Mouse cover run-tmes (b) HapMap CEU cover run-tmes Fgure 7: un-tmes to calculate covers over real data sets. (a) shows C, C, and C Uber run-tmes versus the number of sze of chromosome n SNPs for haplotype, optmstc genotype, and pessmstc genotype covers over the mouse data set. (b) smlarly dpcts run-tmes of the HapMap CEU populaton. 8. DISCUSSION AND FUTUE WOK By provdng an effectve means of parttonng haplotypes and genotypes nto meanngful blocks on a genome-wde scale, we have enabled several new areas for exploraton. An obvous applcaton of FGT compatblntervals s to construct local perfect phylogeny trees, n an effort to fnd sets of frequently recurrng and compatble trees [43]. Ths queston s of partcular mportancn model organsms such as laboratory mce that are thought to derve from small set of founders (.e., fancy mce) [44], and n communtes where there are ongong efforts to generate new model populatons for systems bology [5]. Our method s effectvely used n Yang et al (2010, n preparaton) to dentfy meanngful blocks over whch the hstorcal subspecfc orgn of laboratory mce can be analyzed. Moreover, one can precsely defne the core of such trees and quantfy ther varablty. Both local phylogenetc trees derved from compatble SNP ntervals and the lmted haplotype dversty of compatble SNP ntervals can bncorporated nto dsease assocaton studes, as has been recently demonstrated [39, 26, 30]. Wth our ntroducton of a method for fndng compatblntervals over outbred populatons, t may be possble to gan the same benefts workng wth less controlled populatons, ncludng the human genome. The prevalence of sngle-snp cores n our results also suggests new methods for cleanng data. There are several possble sources for these small local features ncludng genotypng errors, gene conversons, and homoplasy. In addton to the obvous benefts of elmnatng putatve errors from a gven data set, the other two sources for sngle-snp cores are of great nterest to bologsts, but are not well-characterzed. One would expect that a systematc greedy reducton of an nterval set from k to k 1 or k 2 ntervals would expose larger scale structure and phylogenetc trees wth mproved support. 9. EFEENCES [1] Vneet Bafna, Dan Gusfeld, Guseppe anca, and Shbu Yooseph. Haplotypng as perfect phylogeny: A drect approach. Journal of Computatonal Bology, 10(3-4), [2] S. Besenbacher, T. Malund, and M.H. Scherup. ocal phylogeny mappng of quanttatve trats: Hgher accuracy and better rankng than sngle-marker assocaton n genomewde scans. Genetcs, 181: , [3] E. Buckler and M. Gore. An arabdopss haplotype map takes root. Nature Genetcs, 39(9): , [4] en Hua Chung and Dan Gusfeld. Perfect phylogeny haplotyper: haplotypnferral usng a tree model. Bonformatcs Applcaton Note, 19(6), [5] G. A. et al. Churchll. The collaboratve cross, a communty resource for the genetc analyss of complex trats. Nature Genetcs, 36: , [6].M. Clark, G. Schwekert, C. Toomajan, S. Ossowsk, G. Zeller, P. Shnn, N. Warthmann, T.T. Hu, G. Fu, D.A. Hnds, H. Chen, K.A. Frazer, D.H. Huson, B. Scholkopf, M. Nordborg, G. atsch, J.. Ecker, and D. Wegel. Common sequence polymorphsms shapng genetc dversty n arabdopss thalana. Scence, 317: , [7] The Internatonal Hapmap Consortum. A haplotype map of the human genome. Nature, 437: , [8] T.H. Cormen, C.E. eserson, and.. vest. Introducton to Algorthms. MIT Press, [9] E. Cuppen. Haplotype-based genetcs n mce and rats. TENDS n Genetcs, 21(6), [10] M.J. Daly, J.D. oux, S.F. Schaffner, T.J. Hudson, and E.S. ander. Hgh-resoluton haplotype structurn the human genome. Nature Genetcs, 29: , [11] Z. Dng, T. Malund, and Y.S. Song. Effcent whole-genome assocaton mappng usng local phylogenes for unphased genotype data. Bonformatcs, 24(19): , [12] Zhhong Dng, Vladmr Flkov, and Dan Gusfeld. A lnear-tme algorthm for the perfect phylogeny haplotypng (pph) problem. ECOMB, [13] E. Eskn, E. Halpern, and.m. Karp. arge scale reconstructon of haplotypes from genotype data. ECOMB, [14] Eleazar Eskn, Eran Halpern, and chard M Karp. Effcent reconstructon of haplotype structure va perfect phylogeny. Journal of bonformatcs and computatonal bology., 2003.

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11 10. APPENDICES 10.1 Appendx A EMMA 1. Any c-nterval cover coverng the range [1, e c], wth e c m, satsfes e c e c, where e c s the endpont of the c th nterval of C over the same sequence. POOF. By nducton, a sngle-nterval cover must end at, or short of, e 1 (an overage would ndcate Scan closed thnterval prematurely). Assume that the emma holds for an -nterval cover, thus e. Ths mples for any ( + 1)-nterval cover +1 b +1. We now prove +1 e +1. Snce there exsts a SNP, s, n the ( + 1)th nterval of C (.e. wthn [b +1, e +1]) where s s ncompatble wth s e +1 +1, [b+1, e+1] cannot be a compatble nterval f +1 > e +1. Therefore, we have +1 e +1. A symmetrc emma for C states that any C-nterval cover coverng the range [b 1, m], wth b 1 1, satsfes b c b c, where b c s the start of the c th nterval of C over the same sequence. core Core j, t s suffcent to prove < b +1 and > e 1. Snce {I 1,..., } s an -cover coverng the range [1, ], accordng to emma 1, we have e < b +1. Smlarly, we can prove > e 1. POPETY 1. All SNPs n a core are compatble, snce each cors an ntersecton of two compatble regons POPETY 2. Adjacent cores must contan at least one par of ncompatble SNPs POOF. Ths can be demonstrated by contradcton. Assume that two adjacent cores, cor and cor+1, are compatble. By defnton, cor = [b, e ]. Snce cor = [b, e ], cor s compatble wth [e + 1, e ]. Smlarly, cor+1 s compatble wth [e + 1, e ]. Therefore, cor+1 [e + 1, e ] cor+1 = [b, e +1] s a compatblnterval contanng both cores and contradctng Theorem 2 (Fg. 13(a)). THEOEM 1. The covers C and C have the same number of ntervals k, and k s the mnmal number of ntervals possble for any complete cover. POOF. Assume C =. Accordng to emma 1, for any c-nterval cover, C, wth c < and startng from the left-most SNP, the end of the cover s range e c <= b c < b = m. Therefore, C cannot be a complete cover f C < C, mplyng that a complete cover must have at least C ntervals. A smlar concluson can be drawn for C usng the symmetrc verson of emma 1. Therefore, we have C = C = k, and k s the lower bound on the number of ntervals for a complete cover. EMMA 2. For all = 1... k, Cor. POOF. By nducton. Assume C = {I1,..., Ik }, and C = {I1,..., Ik }. By defnton Cor s [b, e ], therefore, t s suffcent to prove b e. Ths s trvally the case for = 1; b 1 e 1. Assume that b e holds for, we now provt holds for +1. From b e, we know b < b +1. If b +1 e +1 does not hold, mplyng e +1 e, then we know +1, and SNP ( ) must be compatble wth all SNPs n +1 (Fg. 11(a)). s e s e s outsde and adjacent to I +1. By defnton of thght-to- eft cover, SNP s e must bncompatble wth at least one SNP n +1, whch results n a contradcton. THEOEM 2. For any complete k-cover C 1,m= {I 1,..., I k }, thth nterval contans the entrth core: Cor = Cor, and, t does not contan any part of another core Core j = φ, 1 j k, j. POOF. Snce Cor = [b, e ], to prove thth nterval = [, ] contans Cor, t s suffcent to prove b and e. Snce {I 1,..., 1} s an ( 1)-cover coverng the range [1, 1] startng from the leftmost SNP, accordng to emma 1, we have 1 e 1, whch means b. Smlarly, we can prove e (see Fg. 12(a)). To prove Core j = φ for any other THEOEM 3. Crtcal SNPs are flaggng SNPs of C Uber ntervals. POOF. Consder two adjacent maxmal compatblntervals, Uber and +1 Uber. Ther flaggng SNPs s e Uber +1 and s b Uber +1 1 must bncompatble (see Fg. 2(b) for an example). In fact, ths ncompatblty makes t nfeasble for e Uber to be larger and b Uber +1 to be smaller. Wthout removng at least one of these two flaggng SNPs, t s mpossble for ether of these two ntervals to grow towards each other. Snce C Max C Uber, a necessary condton of reducton n k s that at least onnterval n C Uber grows n sze. Therefore, beng a flaggng SNP s a necessary condton for beng a crtcal SNP. THEOEM 4. C(g) C(h) POOF. By contradcton. Assume C(h) exsts such that C(h) < C(g). There must exst somncompatblty n S g not n S. By defnton, t must bmpossble to phase the genotypes to make ths compatble. Therefore, S must not be a true phasng of S g. Contradcton by defnton. THEOEM 5. C(h) C(p) POOF. By contradcton. Assume C(h) exsts such that C(p) < C(h). There must exst somncompatblty n S not n S g. By defnton, t must bmpossble to phase the genotypes to make ths ncompatble. Therefore, S must not be a true phasng of S g. Contradcton by defnton. COOAY 1. C(g) C(h) C(p) 10.2 Appendx B