Partial DNA Assembly: A Rate-Distortion Perspective

Transcription

1 Partial DN embly: Rate-Ditortion Perpective Ilan Shomorony 1, ovinda M. Kamath 2, Fei Xia 3, homa. Courtade 1, and David N. e 2 1 Univerity of California, Berkeley, US, 2 Stanford Univerity, Stanford, US, 3 inghua Univerity, China. ilan.homorony@berkeley.edu, gkamath@tanford.edu, f12@mail.tinghua.edu.cn, courtade@berkeley.edu, dnte@tanford.edu. btract Earlier formulation of the DN aembly problem were all in the contet of perfect aembly; i.e., given a et of read from a long genome equence, i it poible to perfectly recontruct the original equence? In practice, however, it i very often the cae that the read data i not ufficiently rich to permit unambiguou recontruction of the original equence. While a natural generalization of the perfect aembly formulation to thee cae would be to conider a rate-ditortion framework, partial aemblie are uually repreented in term of an aembly graph, making the definition of a ditortion meaure challenging. In thi work, we introduce a ditortion function for aembly graph that can be undertood a the logarithm of the number of Eulerian cycle in the aembly graph, each of which correpond to a candidate aembly that could have generated the oberved read. We alo introduce an algorithm for the contruction of an aembly graph and analyze it performance on real genome. I. INRODUCION he cot of DN equencing ha been falling at a rate eceeding Moore law. he dominant technology, called hotgun equencing involve obtaining a large number of fragment called read from random location on the DN equence. hi technology come in two flavor: Short-read technologie, which generate read typically horter than 2 bae pair (bp), with error rate around 1%, with ubtitution being the primary form of error. Long-read technologie, which generate read of length around, bp, with error rate around 15%, with inertion and deletion being the primary form of error. he read obtained from either technology are then merged to each other baed on region of overlap uing an aembly algorithm to obtain an etimate of the DN equence. During the lat decade, everal uch algorithm were developed firt aimed at hort-read equencing technologie and, more recently, focued on long-read technologie. While thee approache attained varied degree of ucce in the aembly of many genome, very few of them are known to provide any kind of performance guarantee. theoretical framework to ae the performance of variou algorithm relative to the fundamental limit for DN aembly wa propoed in [1]. However, thi framework focue on perfect aembly; i.e., when the goal i to recontruct the whole genome perfectly. In particular, a critical read length `crit, defined a a function of the repeat pattern of a given genome [1, 2], i proved to be a fundamental lower bound for perfect aembly, and hown to be achievable by their propoed algorithm. Nonethele, for real genome, `crit can be very large, and read data for many practical DN equencing project doe not meet the information-theoretic lower bound from [1, 2], rendering the tak of perfect aembly fundamentally impoible. hi make the derivation of a theoretic framework to compare algorithm in term of partial aembly of paramount importance. he firt tep in contructing a theoretical framework for partial aembly i to elect an appropriate meaure of the quality of a partial aembly. Meauring the quality of partial aemblie i a challenging problem. In practice, a metric that i often ued i the N5. o decribe N5, recall that a contig i an unambiguou equence in a genome that an aembly return. he N5 of an aembly i then defined a the larget length ` uch that the um of the length of contig at leat ` long account for at leat half of the um of the length of all contig returned. hi i practically a very popular metric, becaue it doe not require knowledge of the ground truth genome (which i uually not known in practice) to compute. However, the fact that N5 doe not depend upon the ground truth genome make it mathematically unatifying. For eample, an algorithm could jut output a random tring over ={, C,, } of length 1 trillion and obtain an N5 of 1 trillion, depite the fact that the output i unrelated to the target genome. nother dicouraging apect of N5 i that it cannot capture what i known about the relative poition between the contig. In particular, contig are typically etracted from an aembly graph, which i baically a graph with the contig a vertice, and an edge from contig u to contig v if contig v come after one copy of contig u. N5 doe not account for any of the tructural information contained in uch a graph. In thi manucript, we introduce a ditortion metric that attempt to capture how good an aembly graph i. Roughly peaking, thi meaure coincide with the logarithm of the number of Eulerian cycle in the aembly graph. Intuitively, every Eulerian cycle correpond to a ditinct aembly and all of them eplain the data equally well; a uch, the ditortion repreent the miing information till needed for perfect aembly. o the bet of our knowledge, thi i the firt work in thi direction. With the yardtick defined, we then eek an aembly algorithm whoe performance can be characterized in term of the propoed ditortion meaure. While de Bruijn graphbaed algorithm [3] are better undertood from a theoretical tandpoint [1, 3], and would contitute better candidate for the ditortion analyi, they are not very relevant in the contet of aembly from long-read technologie. hi i due to their high enitivity to read error, which prevent them from leveraging the potential of long-read technologie (all of which have error rate above % and will continue to have for the foreeeable future [4, Section 1, 3]). In contrat, overlap-baed aembly approache (e.g., tring graph [5]) are better uited to longread, high-error equencing. hi cla of algorithm relie on the identification of long overlap between read, which i inherently robut to error [4], and ha been recently hown to attain the ame theoretical performance a de Bruijn graphbaed approache in term of perfect aembly [6]. In thi /16/$ IEEE 1799

2 work, we propoe a new overlap-baed aembly algorithm and introduce technique to provide theoretical guarantee in term of the new ditortion meaure. II. PROBLEM SEIN Let be a tring of ` ymbol from the alphabet = {, C,, }. We let = n be the length of the tring, and [i] be it ith ymbol. ubtring of i a contiguou interval of the ymbol in, and i denoted a [i : j], ([i], [i + 1],...,[j]). ubtring of the form [1 : `] i called a prefi (or an `-prefi) of. Similarly, a ubtring of the form [ ` +1: ] i a uffi (or an `-uffi) of. We ay that tring and y have an overlap of length ` if the `-uffi of and the `-prefi of y are equal. We let y denote the concatenation of and y. We aume that there eit an unknown target DN equence of length = which we wih to aemble from a et of N read R. hroughout the paper, we will make two implifying aumption about the et of read: (1) ll read in R have length L. (2) he read in R are error-free. he firt aumption i made to implify the epoition of the reult. he econd aumption i motivated by the eitence of overlapping tool (uch a Dligner [4]), which can efficiently identify ignificant matche between read at error rate of over 15%. he algorithm decribed in thi manucript can be adapted to work with the approimate matche found by uch tool, eentially by treating them a eact matche. For eae of epoition, we will aume that i a circular equence of length ; i.e., [t + ] =[t] for any t. hi way we will avoid edge effect and a read 2Rcan correpond to any ubtring [t : t + L 1], for t =1,...,. We will ue the tandard Poion ampling model for hotgun equencing. hi mean that each of the N read i drawn independently and uniformly at random from the et of length-l ubtring of, {[t : t + L 1] : t =1,...,}.. Repeat and Bridging repeat of length ` in i a ubtring 2 ` appearing at ditinct poition t 1 and t 2 in ; i.e., [t 1 : t 1 + ` 1] = [t 2 : t 2 + ` 1] =, that i maimal; i.e., [t 1 1] 6= [t 2 1] and [t 1 + `] 6= [t 2 + `]. repeat i bridged if there i a read that etend beyond one copy of the repeat in both direction, a hown in Fig. 1. repeat i doubly-bridged if both copie are bridged. Similarly, a triple repeat of length ` i a r 1 r 2 r 3 Fig. 1. Illutration of a bridged repeat in ; Illutration of a triple repeat all-bridged by read r 1, r 2 and r 3. ubtring that appear at three ditinct location in (poibly overlapping); i.e., [t 1 : t 1 + ` 1] = [t 2 : t 2 + ` 1] = [t 3 : t 3 + ` 1] = for ditinct t 1, t 2 and t 3 (modulo, given the circular DN aumption), and i maimal (that i, all three copie can not be etended in a direction). triple repeat i aid to be bridged if at leat one of it copie i bridged. It i aid to be all-bridged if all of it copie are bridged, a r illutrated in Fig. 1, and all-unbridged if none of it copie are bridged. III. DISORION MERIC FOR SSEMBLY RPHS o motivate our notion of ditortion for aembly graph, let u firt conider an idealized etting in which all read of length L from are given. We call thi enemble of read the L-mer compoition of, defined a the multiet C L () ={[i : i + L 1] : 1 apple i apple }. (1) We will let C L () repreent the upport of C L (); i.e., C L () without the copy count. he L-mer compoition C L () doe not, in general, determine unambiguouly. In particular, any equence with C L () =C L () i inditinguihable from when only C L () i oberved. hu, one require at leat log { : C L () =C L ()} (2) additional bit to determine unambiguouly from all other equence having the ame L-mer compoition. hi uncertainty characterization can be tranlated to the language of equence graph through the notion of a k-mer graph 1 of, B k () [3]. Definition 1. he multigraph B k () ha C k 1 () a it node et and, for, y 2 C k 1 (), we place m edge from to y if the k-mer y[k 1] ha multiplicity m in C k (). It i eay to ee that B k () i an Eulerian graph for any k, and every with C k () =C k () correpond to a ditinct Eulerian cycle in B k (). Furthermore, two Eulerian cycle that are ditinct up to edge multiplicitie correpond to ditinct equence. herefore, a natural meaure of the ditortion of B k () a an aembly graph of would be D k (), log ec(b k ()), (3) where ec() i the number of Eulerian cycle in, ditinct up to edge multiplicity. We point out that the number of Eulerian cycle in a k-mer graph ha been previouly ued in the related contet of DN-baed torage channel [7]. illutrated in Fig. 2, D k () can be computed for real Dk() lcrit k Fig. 2. D k () a a function of k, when i the genome of E. coli 536. Notice that D k () reache zero when k = `crit () [1]. genome, and can be interpreted a a lower bound on how good an aembly from read of length k can be. In the actual etting for the aembly problem, however, one doe not have acce to the entire L-mer compoition of, nor can be epected to perfectly contruct B k () for ome k < L (other than for mall value of k). Hence, when defining a ditortion metric for aembly graph, one mut conider a larger cla of graph than B k (). In thi work, we will conider the following: 1 In the aembly literature, uch graph are ometime referred to a de Bruijn graph. However, ince our propoed algorithm i not a de Bruijn graph baed algorithm in the uual ene of [3], we avoid the terminology. 18

3 C (c) C C (d) C where ec() i the number of Eulerian cycle in that are ditinct up to edge multiplicitie, and D 1 () i the ditortion achieved by the 1-mer graph of, B 1 (). We note that if contain all of {, C,, }, then D 1 () would be the ditortion achieved by the graph, hown in Figure 3. It i not difficult to ee that the ditortion of any ufficient equence graph i at mot D 1 (). Fig. 3 how the computation of thi ditortion in a toy eample. Fig. 3. he trivial equence graph i alway ufficient. (b,c,d) n eample of the ditortion computed for the aembly of a cyclic equence = C i hown. If the graph in i returned by an aembly algorithm, then a the graph i not a ufficient h equence i graph with repect to, the ditortion i computed to be log = If the 7 2,1,2,2 equence graph of order k =1in (c) i returned, then [] i a hown in (d). he ditortion i D(, ) =a there i eactly one Eulerian cycle in [] (modulo difference in travering edge between the ame two vertice). Definition 2. equence graph =(V,E, ) of order k i a directed multigraph where each edge e 2 E i labeled with a k-mer (e) 2 k, and each node v 2 V i labeled with a (k 1)-mer (v) 2 k 1 atifying the property that if (u, v) =, then (u) =[1 : k 1] and (v) =[2 : k]. Notice that any path p =(v 1,...,v`) on a equence graph of order k naturally define a length-(` 2+k) tring t(p), (v 1,v 2 )[1]... (v` 2,v` 1 )[1] (v` 1,v`). If a path p =(v 1,...,v`) end in a node with out-degree zero, it will be called a graph uffi, and if it tart in a node with in-degree zero, it will be called a graph prefi. Definition 3. Chinee Potman cycle in a equence graph, i a cycle that travere every edge at leat once. natural formulation for the genome aembly problem i to identify a Chinee Potman cycle in the contructed equence graph which correpond to the true equence [8, 9]. Definition 4. equence graph i aid to be ufficient (for the aembly of ) if it contain a Chinee Potman cycle c uch that t(c )= (up to cyclic hift). While it i natural to define the goal of the partial aembly problem to be the contruction of a ufficient equence graph, it i typically unreaonable to epect the aembly algorithm to correctly etimate the multiplicitie of all the edge; i.e., the number of time c travere each edge. he reaon i that the length of the genome i not known in advance, and hence neithe the coverage depth (i.e., the average number of read covering a given poition). Other practical iue like uneven coverage and equence pecific biae only add to the difficulty there. hu our ditortion metric hould not penalize incorrect multiplicitie, and thu not require the produced graph to be Eulerian. o define our ditortion metric, we will conider an Eulerian verion of the contructed equence graph. More preciely, if =(V,E, ) i a ufficient equence graph and c i a Chinee Potman cycle in correponding to the equence, we will let [] be the multigraph obtained by etting the multiplicity of edge e to be the number of time c travere e. Definition 5. he ditortion of a equence graph i 8 < log ec([]) if i a ufficient D(, ), equence graph for : D 1 ()+1 otherwie (4) IV. REEDY LORIHM FOR PRIL SSEMBLY In thi ection we decribe an algorithm to aemble a equence graph. We then analyze it performance in term of it ability to produce a ufficient equence graph and the reulting ditortion. he algorithm can be een a a generalization of the greedy algorithm for equence aembly []. In the tandard greedy algorithm, prefie and uffie of read are iteratively merged in order to produce a ingle equence. However, when an incorrect merging occur, it ha no way of detecting and fiing it at lateteration. Our algorithm overcome thi iue by allowing a read prefi/uffi to be merged to the interior of another read, or to a previouly merged prefi/uffi, a illutrated in Fig. 4. we will how, thi additional fleibility i helpful in contructing a ufficient equence graph in the ene of Definition 4, making the algorithm robut from the point of view of partial aembly. Fig. 4. In the greedy merging algorithm, we allow matche between a prefi/uffi of a read and the interior of another read, producing a graph that i not a line, a i the cae with the tandard greedy algorithm []; Initial equence graph for read CC, C, and C for k =3. Notice that a match of ` ymbol correpond to a path of ` k +1edge. Our algorithm will maintain at all time a equence graph in the ene of Definition 2, where each read 2Rcorrepond to a path p i with L k +2 node and L k +1 edge, which correpond to the L k +1 conecutive k-mer of. Initially, all N path will be dijoint component of the graph, a illutrated in Fig. 4.he algorithm then proceed by finding matche between a previouly unued prefi or uffi and any part of another read, and merging the correponding path. he algorithm i termed greedy ince it earche for matche in decreaing order of length. lgorithm 1 reedy merging algorithm 1: Input: Initial equence graph (Fig. 4), and parameter k 2: for ` = L, L 1,L 2,...,k do 3: X { 2 ` : i a current graph prefi or uffi that appean more than one read} 4: for 2 X do 5: Merge the path correponding to from all read that contain the ubtring 6: Output: Reulting equence graph of order k 181

4 # of triple repeat c = N L= riple repeat length ditribution for S. aureu riple repeat length Required coverage depth `crit Read length Fig. 5. Ditribution of triple repeat length on S. aureu, and coverage depth required for the condition in heorem 1 to be achieved with probability.99. he parameter k hould be choen a the minimum overlap we epect adjacent read to have, and can be made large for equencing eperiment with high coverage depth. Fontance, when aembling long read (, bp) with high error rate, a typical choice for the minimum match length k i [4]. We proceed to analyze the ditortion achieved by the equence graph that lgorithm 1 output in two tep. We firt obtain condition for the equence graph to be ufficient, and then characterize condition under which the reulting ditortion can be upper bounded by D k () for ome k>1. Definition 6. We ay that R k-cover the equence if there i a read tarting in every k-length ubtring of. heorem 1. lgorithm 1 contruct a ufficient equence graph of order k if the et of read R k-cover the equence and every triple repeat i either unbridged or all-bridged. decribed in Section V, given the condition in heorem 1, one can bound the probability that the graph produced by lgorithm 1 i not ufficient. hi bound can then be tranlated into a value of coverage depth c = N L/ for which the reulting equence graph i ufficient with a deired probability 1. hi i illutrated in Fig. 5 for the S. aureu genome from the E dataet [11]. We notice that for value of L that are far from the length of ome triple repeat, a mall coverage depth uffice. We remark that an intereting open quetion i to determine if the non-monotonicity caued by the peak in required coverage near triple repeat length (a hown in Fig. 5) repreent a fundamental barrier or a limitation of the algorithm. Mot eiting algorithm face challenge when there are triple repeat of length that are cloe to the read length. In fact, overlapbaed algorithm alo uffer from imilar problem when there are double repeat of length cloe to L. In addition to the ufficiency property guaranteed by heorem 1, we need a way to characterize the ditortion achieved by the reulting graph. o do o, we will bound the ditortion achieved by aembling read of length L by the quantity D q (), defined in (3), for ome q < L. We begin with a definition. Definition 7. wo repeat [a 1 : a 1 + `], [a 2 : a 2 + `] and [b 1 : b 1 + m], [b 2 : b 2 + m] are aid to be linked if a 2 <b 1 apple a 2 + ` +1. We call a 2 + ` +1 b 1 the link length. illutrated in Fig. 6, linked repeat are potential caue of ambiguity in the equence graph. heorem 2. Suppoe that the et of read R from the equence atifie the following condition: each triple repeat i either all-bridged or all-unbridged, [b 1 : b 1 + m] [a 1 : a 1 + `] [a 2 : a 2 + `] [b 2 : b 2 + m] Fig. 6. Illutration of linked repeat with link length a 2 + ` +1 b 1. If we merge both repeat in the equence graph, the link (red egment) create a path that i not in the true equence. all repeat of length apple q are doubly-bridged, (c) for all pair of linked repeat with link length ` atifying k 1 apple ` apple q, at leat one i doubly-bridged. hen the ufficient equence graph produced by lgorithm 1 ha a ditortion atifying D(, ) apple D q (). Notice that if all repeat in are doubly bridged, the condition in heorem 2 are atified for any q, implying that D(, ) =. he tandard greedy algorithm [], on the other hand, achieve perfect aembly when all repeat are bridged, not necearily doubly bridged [1]. Intuitively, the more tringent requirement of double bridging i the price paid to obtain guarantee in a range of L where the genome i much more repetitive. V. DISORION ON REL ENOME Clearly in practice we cannot verify whether the condition in heorem 1 and 2 are atified, a we do not have acce to the genome being equenced. he purpoe of thee reult i to allow u to compute the rate-ditortion tradeoff achieved by lgorithm 1 on previouly aembled genome. hi provide a framework to analyze the algorithm performance and compare it to the fundamental lower bound (or to other algorithm). For an organim whoe whole genome i known, we can compute repeat tatitic, which can then be ued to numerically compute the number of read N required to guarantee that the condition in heorem 1 hold with probability at leat 1, for ome target error probability > (ee [12] for detail). hi yield the curve in Fig. 5. Similarly, by identifying the ditribution of repeat length and characterizing which pair of repeat are linked, one can compute the probability that condition and (c) in heorem 2 are not atified for a given q. By computing D q () Ditortion Lower bound c=2 c= c=5 l crit Read length L Fig. 7. Ditortion achieved by lgorithm 1 with k =3on S. aureu with probability.99 for different coverage depth c = NL/, compared to the lower bound D L (). ap indicate that the probability of the condition of heorem 2 not being atified i at leat

5 y y Fig. 8. Subtring of and mapping to: ame egment in ; ditinct egment in correponding to an unbridged repeat. (c) If at iteration L ` the `-uffi of and the `-prefi of are not merged, they mut have been merged to ome read t i and t j in previou iteration. for a range of value of q, which can be done uing the wellknown BES heorem, we can upper bound the ditortion achieved by lgorithm 1 with a deired probability 1. Notice that D L () i alo the minimum ditortion that can be achieved with read of length L, which provide a lower bound to the ditortion that can be achieved by any algorithm. In Fig. 7 we how thee curve computed for S. aureu for different value of the coverage depth c = NL/. We notice that the upper bound curve follow the lower bound cloely but have gap in them, repreenting the range of L where the condition of heorem 1 are not atified with the deired probability, and the achieved ditortion jump to D 1 (). VI. PROOFS OF MIN RESULS Here, we provide proof ketche for the main reult in the paper. he longer verion of thi manucript [12] ha full proof.. Sufficiency of equence graph (heorem 1) Note that each r 2Rinduce a mapping between it ymbol and a egment of length L in, from where r wa ampled. Lemma 1. Suppoe that read and hare ubtring that (i) map to the ame tring in, a in Fig. 8, or (ii) map to two different tring in, which are part of an unbridged repeat, a hown in Fig. 8. t the end of iteration L ` of lgorithm 1, the path correponding to in and are merged. iven Lemma 1, heorem 1 follow immediately becaue, at the end of the algorithm, the overlapping part of any two conecutive read and +1 (which mut be of length at leat k when R k-cover ) mut be merged in the equence graph. Lemma 1 i proved by induction on = L 1,L 2,...,k. he bae cae follow ince, if read and have a matching ubtring of ize L 1, they mut correpond to a graph uffi or prefi in the beginning of the algorithm and will thu be merged, o aume the lemma hold up to = `+1. Suppoe and hare a ubtring with = ` and we are in cae (i). If lgorithm 1 doe not merge the path correponding to in and at iteration L `, it mut be the cae that a longer uffi of and a longer prefi of were merged to other read, ay t i and t j in previou iteration, a illutrated in Fig. 8(c). Now if the ubtring in t i (or t j ) map to the ame egment of a and, it ha an overlap greater than ` with both and and, by the induction hypothei, it i merged to both, cauing the path in and to be merged. Similarly, if the ubtring in t i and t j map to the ame egment of, by the induction hypothei, they have been previouly merged, cauing and to be merged a well. Finally, if the ubtring (c) t i t j in t i and t j map to two other egment in, i a triple repeat. Since i unbridged at the interection of and, by the aumption in heorem 1, it mut be all-unbridged. Hence, t i and t j are part of an unbridged repeat, and by the induction hypothei have been previouly merged to each other. Cae (ii) follow imilarly. B. Ditortion Bound (heorem 2) Firt we how that when the condition in heorem 2 are atified, lgorithm 1 can only merge two path if they either correpond to the ame egment in, or they correpond to an unbridged repeat (or a ubtring of it) in. imple proof by contradiction i ued to how thi. Fig. 9. Cycle equence graph (of order k = 1) for the equence CCCC; Contracted graph U(R) where U(R) correpond to the two pair of repeat hown in red and blue. We then conider the et of repeat in that are not doubly bridged by R, U(R), and define a ufficient equence graph U(R) by taking a cycle graph repreentation of and merging the repeat in U(R), a illutrated in Fig. 9. Uing the previou reult, we how that, the graph returned by lgorithm 1, can be converted to U(R) via node contraction. hi implie that there i a urjection between the et of Eulerian cycle in U(R) [] to the et of Eulerian cycle in []. hi thu give u that, D(, ) apple D( U(R), ). Finally we how that, when the condition of heorem 2 are met, any path of q k edge in U(R) correpond to a q-mer from C q (), the q-mer compoition of. hi directly give u that the equence correponding to any Eulerian cycle in U(R) [] ha q-mer compoition C q (). hi implie that D( U(R), ) apple D q (), which i the deired reult. REFERENCES [1]. Breler, M. Breler, and D. e, Optimal embly for High hroughput Shotgun Sequencing, BMC Bioinformatic, 213. [2] E. 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