A Metric on the Space of Reduced Phylogenetic Networks

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1 1 A Metic on the Space of Reduced Phylogenetic Netwoks Luay Nakhleh Abstact Phylogenetic netwoks ae leaf-labeled, ooted, acyclic, diected gaphs, that ae used to model eticulate evolutionay histoies. Seveal measues fo quantifying the topological dissimilaity between two phylogenetic netwoks have been devised, each of which was poven to be a metic on cetain esticted classes of phylogenetic netwoks. A biologically-motivated class of phylogenetic netwoks, namely educed phylogenetic netwoks, was ecently intoduced. None of the existing measues is a metic on the space of educed phylogenetic netwoks. In this pape, we povide a metic on the space of educed phylogenetic netwoks that is computable in time polynomial in the size of the netwoks. Index Tems phylogeny, phylogenetic netwok, indistinguishability, educed phylogenetic netwok, metic. I. INTRODUCTION Phylogenetic tees model the vetical tansmission of genetic mateial fom ancestos to descendants. When eticulate evolutionay events, such as hoizontal gene tansfe and hybid speciation, occu, the evolutionay histoy of the set of oganisms is moe appopiately modeled as a special ooted, diected, acylic gaph, called phylogenetic netwok. Measuing the distance between a pai of phylogenies plays an impotant ole in a vaiety of tasks, including clusteing, eo estimation of econstuction methods, and detection of species/gene tee inconguities. A distance measue, o metic, m, on a space S satisfies thee popeties fo all a, b, c S: P1 m(a, b) = 0 if and only if a = b, P2 m(a, b) = m(b, a), and P3 m(a, b) + m(b, c) m(a, c). In addition to these thee popeties, in phylogenetics it is desied that the measue povides infomation about the similaity of the two evolutionay scenaios epesented by the two phylogenies being compaed. Fo example, viewing each banch of Depatment of Compute Science, Rice Univesity. nakhleh@cs.ice.edu a phylogenetic tee as indicative of a bipatition of the set of taxa at the leaves, the Robinson- Foulds measue quantifies the aveage numbe of bipatitions that the two tees disagee on [1]. As anothe example, the subtee pune and egaft (SPR) measue quantifies an edit distance that is coelated with the numbe of eticulate evolutionay events equied to econcile the two tees being compaed [2]. To illustate a metic that is of little o no utility in phylogenetic compaison, conside the measue m on the space of all phylogenetic tees, whee m(p 1, P 2 ) = 0 if the two phylogenies P 1 and P 2 ae isomophic (whee the bijection also espects the leaf labeling), and m(p 1, P 2 ) = 1 othewise. While it is clealy a metic on the space of phylogenetic tees, such a binay measue tells whethe two phylogenetic tees ae identical (up to isomophism) o not, but does not quantify the degee of similaity/dissimilaity between them. The liteatue on geneal labeled gaphs contains metics that can be applied to phylogenetic netwoks, e.g., [3], [], but the esults they yield do not necessaily povide insight into the similaity o dissimilaity of the evolutionay histoy scenaios povided by a pai of phylogenetic netwoks. To addess this issue, seveal measues have been intoduced to quantify the dissimilaity between a pai of phylogenetic netwok topologies, each of which is metic on a esticted class of phylogenetic netwoks [5] [12]. In 200, Moet et al. intoduced the concepts of phylogenetic netwok indistinguishability and educed phylogenetic netwoks to account fo issues elated to econstuctibility of phylogenetic netwoks [6]. In that pape, a dissimilaity measue was intoduced, which was late shown not to satisfy some of the popeties P1 P3, even on educed phylogenetic netwoks [8]. In this pape, we eview the concepts of phylogenetic netwok eduction and indistinguishability, and povide a novel metic on

2 2 the space of educed phylogenetic netwoks. II. INDISTINGUISHABILITY AND REDUCED PHYLOGENETIC NETWORKS In this section we eview the concepts of indistinguishability and educed phylogenetic netwoks, which wee fist pesented in [6]. We begin with geneal phylogenetic netwoks. Definition 1: A phylogenetic X -netwok, o X - netwok fo shot, N is an odeed pai (G, f), whee 1 G = (V, E) is a diected, acyclic gaph (DAG) with V being the union of fou paiwisedisjoint sets {}, V L, V T, and V N, whee indeg() = 0 ( is the oot of N); v V L, indeg(v) = 1 and outdeg(v) = 0 (V L ae the leaves of N); v V T, indeg(v) = 1 and outdeg(v) 2 (V T ae the tee-nodes of N); and, v V N, indeg(v) 2 and outdeg(v) 1 (V N ae the netwok-nodes of N), and E V V ae the netwok s edges. f : V L X is the leaf-labeling function, which is a bijection fom V L to X. We also use the notation L(N) fo the set of leaves of phylogenetic netwok N. Notice that the definition implies that the undiected gaph undelying a phylogenetic netwok is connected. Fig. 1 shows two phylogenetic X -netwoks, whee X = {1, 2, 3, }. When the context is clea, we may omit the labeling function f, and wite N = (V, E). Futhe, when the set of taxa X labeling the leaves of a phylogenetic X -netwok is clea fom the context, we may omit the set desciption and efe to it simply as a phylogenetic netwok. In [6], Moet et al. discussed the issue of indistinguishability among phylogenetic netwoks fom a econstuction standpoint, and agued fo the need of a measue that sepaates phylogenetic netwoks up to indistinguishability, o, in othe wods, a measue that is a metic on the space of all educed phylogenetic netwoks. In this subsection, we biefly eview the concepts of distinguishability and educed (phylogenetic) netwoks, and in Section III we pesent a metic on the space of all educed netwoks that is computable in polynomial time in the size of the netwoks. Moet et al. intoduced the concept of indistinguishability to account fo what we tem hee as soft in-polytomy. In the context of (ooted) phylogenetic tees, a (soft) polytomy is epesented by a node that has moe than two childen, and indicates the lack of phylogenetic signal to esolve, o efine, the evolutionay elationship among these childen. Fig. 2 shows a soft polytomy at node, along with the thee possible efinements of that polytomy. Even though the tue evolutionay histoy is one of the thee possible efinements, due to lack of phylogenetic signal the only scenaio that might be econstuctible is the polytomy Fig. 2. A soft polytomy (top) and its thee possible efinements into binay tees (bottom) N 1 N 2 Fig. 1. Two phylogenetic X -netwoks, with X = {1, 2, 3, }. 1 We use indeg and outdeg to denote the in-degee (numbe of paents) and out-degee (numbe of childen) of a node, espectively. In the case of phylogenetic netwoks, a lack of phylogenetic signal may also esult in nodes with moe than two paents. To illustate, conside the scenaio in Fig. 3. Fou genes, o makes, fom five taxa 1, 2, 3,, and x, yield gene tees that diffe in the placement of x, such that in each of the tees it is a sibling of a diffeent taxon. If this diffeence is due to, say, hybid speciation, then, x is clealy a hybid. Nonetheless, due to lack of phylogenetic signal (in

3 3 a b c d 1 x x 2 3 x x a b c d y x Fig. 3. A phylogenetic netwok with a soft in-polytomy at node y (bottom) esulting fom the diffeent placements of x in the fou diffeent gene tees (top). This netwok with in-polytomy can be efined into nine diffeent netwoks in which each node has at most two paents, by adding two nodes x 1 and x 2, and efining node y in all nine possible ways, as descibed in Table. I. this case, it is massive extinction events o vey spase taxon sampling), the phylogenetic netwok that econciles these fou gene tees is one in which node y has indegee, as shown at the bottom of Fig. 3. This phylogenetic netwok may be efined TABLE I THE POSSIBLE REFINEMENTS OF NODE y IN THE PHYLOGENETIC NETWORK IN FIG. 3, WHICH RESULT IN NETWORKS IN WHICH EACH NODES HAS AT MOST TWO PARENTS. THE PHYLOGENETIC NETWORK RESULTING FROM REFINEMENT (1) IS SHOWN IN FIG.. Refinement Paents of x 1 Paents of x 2 Paents of y 1 a, b c, d x 1, x 2 2 a, c b, d x 1, x 2 3 a, d b, c x 1, x 2 a, b x 1, c x 2, d 5 a, b x 1, d x 2, c 6 a, c x 1, b x 2, d 7 a, c x 1, d x 2, b 8 a, d x 1, b x 2, c 9 a, d x 1, c x 2, b into nine diffeent phylogenetic netwoks in which each node has at most two paents, as descibed in Table I. Fig. shows the netwok esulting fom efinement (1) in the table. Howeve, in the absence of any additional infomation, such as divegence times, selecting one of these efinements ove the othes is abitay. Using the teminology of [6], all these nine netwoks ae indistinguishable fom a econstuction point of view, even though they ae not isomophic. Hence, Moet et al. intoduced the concept of educing these netwoks so as to eliminate the distinction among abitay efinements that ae not suppoted by the data. In this case, the netwok in Fig. 3 is the educed vesion of all these nine netwoks. In any of the netwoks esulting fom the efinements descibed in Table I, the set {x 1, x 2 } of nodes is efeed to in [6] as a maximal convegent set. The netwok eduction pocedue basically a b c d x 1 x 2 y x Fig.. Refinement (1) in Table I of the educed phylogenetic netwok in Fig. 3. entails identifying maximal convegent sets in the netwok, and fo each such set, connecting its paent nodes diectly to the maximal subtees, o clades, eachable fom it and eliminating all peviously existing paths fom the nodes in the set to these clades. We now eview the fomal definitions, as given in [6], of the thee concepts of a maximal

4 convegent set, a educed netwok, and netwok indistinguishability. Definition 2: Let N = (V, E) be a phylogenetic X -netwok. A set U V of intenal nodes is convegent if (1) U 2, and (2) evey leaf eachable fom some node in U is eachable fom all nodes in U. If thee exists no convegent set U V such that U U, we say that U is a maximal convegent set. The eduction pocedue of [6] poceeds as follows, when applied to phylogenetic X -netwok N = (V, E): 1) Fo each maximal subtee (o, clade) t (that includes no netwok nodes) of leaves X X, ooted at node t, ceate a new unique node h t and a new edge (p t, h t ), whee p t is the paent of t, delete the edge (p t, t ), and emove the subtee t. The node h t becomes a symbolic leaf that epesents the clade t. Let the esulting netwok be N. 2) Repeat the following two steps on N until no change occus: a) Fo each convegent set U with leaf-set L U, emove all vetices and edges on the paths fom a vetex in U to a leaf in L U, including all vetices in U and excluding vetices in L U. Fo any edge (x, v) fo which v is in the deleted set, eplace it by a set of edges {(x, l) : l L U is eachable fom v}. b) Fo each node w in the netwok, with indeg(w) = outdeg(w) = 1, say with edges (u, w) and (w, v), eplace these two edges with a single edge (u, v), emove node w, and emove any duplicate edges. Repeat until no such node exists. 3) Reattach to each symbolic leaf h t the clade t by its oot t. Definition 3: Let N = (V, E, f) be a phylogenetic X-netwok. Its educed vesion, denoted by R(N), is the netwok obtained fom N by application of the eduction pocedue. In Fig., the only clade t found in Step (1) of the eduction pocedue is the one that contains one leaf x. Its oot is t = x and its paent is p t = y. A new node h t is added, with the new edge (y, h t ), and the clade (x), along with the edge fom y to it, is emoved. Then, nodes x 1, x 2 and y ae emoved along with all edges incident with them. Fou new edges (a, h t ), (b, h t ), (c, h t ), (d, h t ) ae then added. Finally, the clade (x) is eattached to node h t, esulting in the netwok shown in Fig. 3 (with the node h t labeled y). We ae now in a position to define netwok indistinguishability. Definition : Two phylogenetic netwoks N 1 and N 2 ae indistinguishable if thei educed vesions R(N 1 ) and R(N 2 ) ae isomophic. Fo example, all nine phylogenetic netwoks esulting fom the node efinements descibed in Table I ae (paiwise) indistinguishable. To detemine if two phylogenetic netwoks ae indistinguishable, one can fist educe them, and then compae thei educed vesions. This equies a metic fo compaing educed phylogenetic netwoks, and we develop and pesent such a metic in the next section. III. A METRIC ON THE SPACE OF REDUCED PHYLOGENETIC NETWORKS The esults of Cadona et al. [8] show that the tipatition-based measue intoduced in [6] is not a metic on the space of educed netwoks. In this section, we intoduce a metic on the space of educed phylogenetic X -netwoks that is computable in time polynomial in the size of the netwoks. We begin with the notion of node equivalence. Definition 5: Given a phylogenetic X -netwok N = ((V, E), f), we say that two nodes u, v V ae equivalent, denoted by u v, if u, v V L and f(u) = f(v), o Node u has k childen u 1, u 2,..., u k, node v has k childen v 1, v 2,..., v k, and u i v i fo 1 i k. Since in this pape we ae concened with compaing netwoks with identical leaf-sets, the notion of node equivalence can be extended to nodes fom two diffeent netwoks, as established in the following equivalence mapping. Definition 6: Let N 1 = ((V 1, E 1 ), f 1 ) and N 2 = ((V 2, E 2 ), f 2 ) be two phylogenetic X -netwoks. We define the equivalence mapping between N 1 and N 2, h : V 1 2 V 2, so that v h(u), fo u V 1 and v V 2, if: u L(N 1 ), v L(N 2 ), and f 1 (u) = f 2 (v), o Node u has k childen u 1, u 2,..., u k, node v has k childen v 1, v 2,..., v k, and v i h(u i ) fo 1 i k. We have the following theoem.

5 5 Theoem 1: The equivalence of nodes as given in Definition 5 is an equivalence elation. The poof is staightfowad and follows fom the popeties of set equality. Obsevation 1: Let N = ((V, E), f) be a phylogenetic X -netwok, and let u, v V be two nodes whee u v. Then the set {u, v} is convegent. We have the following Lemma. Lemma 1: Let N = ((V, E), f) be a educed phylogenetic X -netwok. If {u, v} V is a convegent set, then u v. Poof: Based on the phylogenetic netwok eduction pocedue, all convegent sets in a phylogenetic netwok ae eliminated, except fo one special type of convegent sets: a set of two nodes {u, v}, whee u is a netwok node and v is the only child of u. In this case, u and v ae not equivalent, since no child of v is equivalent to v. Given two phylogenetic netwoks N 1 = (V 1, E 1 ) and N 2 = (V 2, E 2 ), and a node v 1 V 1, we call the set h(v 1 ) v 1 s mates in N 2, whee h is the equivalence mapping, as given in Definition 6. Notice that h(v 1 ) is empty when v 1 has no equivalent nodes in N 2. Futhe, while in phylogenetic tees we always have h(v 1 ) 1, it may be the case in geneal phylogenetic netwoks that h(v 1 ) > 1 fo some nodes. Since all nodes in h(v 1 ) ae paiwise equivalent, we use h(v 1 ) to denote an abitay node in the set, and NIL when the set is empty. Assume that V 1 = {v 1, v 2,..., v p }. Then the unique nodes of N 1, denoted by U(N 1 ), is the set {v i : j < i, v j v i, 1 i p}. We define U(N 2 ) similaly. Futhe, fo each node v i V 1, we define κ N1 (v i ) = {v V 1 : v v i }, and define κ N2 (u i ) similaly, fo each node u i V 2. We define κ(nil) = 0 fo any netwok N. When the context is clea, we dop the subscipt of κ. We ae now in position to define the measue on pais of phylogenetic X -netwoks. Definition 7: Let N 1 = (V 1, E 1 ) and N 2 = (V 2, E 2 ) be two phylogenetic X -netwoks. Then, m(n 1, N 2 ) equals ( 1 2 v U(N 1 ) max{0, κ(v) κ(v )}+ ) u U(N 2 ) max{0, κ(u) κ(u )}, (1) whee v (u ) is a node in N 2 (N 1 ) that is equivalent to v (u), and if no such equivalent node exists, then v (u ) is NIL. The ationale behind the measue m is that it oughly quantifies the numbe of ooted subnetwoks that ae in one but not both of the netwoks. In the special case whee the two netwoks ae two ooted tees T 1 and T 2, then m(t 1, T 2 ) yields half the symmetic diffeence of thei sets of ooted subtees, whee two subtees fom T 1 and T 2 ae equal if they ae isomophic with espect to the leaf labels. Futhe, like the Robinson- Foulds metic [1], this measue is vey sensitive to small petubations in cetain cases. Fo example, even though the only diffeence between the two netwoks in Fig. 1 is the oientation of a single edge, which is the edge between the paents of leaves 2 and 3, we have m(n 1, N 2 ) = 5. One of the most commonly used distance metics fo compaing phylogenetic tees, namely the RF distance, has a simila popety. Fo example, fo the two tees T 1 = (a, (b, (c, (d, (e, f))))) and T 2 = (f, (a, (b, (c, (d, e))))), whose edit distance is 1, due to the diffeent placement of the leaf f, have RF distance of 3, when consideed unooted, and when consideed ooted (and this effect can be futhe damatized by consideing lage such catepilla tees). We now establish popeties of the measue m. Lemma 2: If m(n 1, N 2 ) = 0 fo two educed phylogenetic netwoks N 1 = (V 1, E 1 ) and N 2 = (V 2, E 2 ), then: 1) V 1 = V 2. 2) v 1 U(N 1 ) κ(v 1) = v 2 U(N 2 ) κ(v 2). Poof: Let h 1 : V 1 V 2 and h 2 : V 2 V 1 be two equivalence mappings, as given by Definition 6. Since m(n 1, N 2 ) = 0, it follows that κ(v 1 ) = κ(h 1 (v 1 )) fo all v 1 V 1 and κ(v 2 ) = κ(h 2 (v 2 )) fo all v 2 V 2. Fom this both esults follow. Lemma 3: If m(n 1, N 2 ) = 0 fo two educed phylogenetic X -netwoks N 1 = ((V 1, E 1 ), f 1 ) and N 2 = ((V 2, E 2 ), f 2 ), then the equivalence mappings h 1 : V 1 V 2 and h 2 : V 2 V 1 ae both isomophisms. Poof: We show the poof fo h 1 ; the poof fo h 2 is identical. By Lemma 2 it follows that V 1 = V 2 and fom Obsevation 1 and Lemma 1 it follows that h(v 1 ) is defined and unique fo each v 1 V 1. We now show that if (u, v) E 1 then (u, v ) E 2, whee u = h 1 (u) and v = h 1 (v). Given that u = h 1 (u) it follows that u and u ae equivalent, which, by the definition of equivalence,

6 6 implies that both u and u have equivalent childen. By Obsevation 1 and Lemma 1, and fom the assumption that m(n 1, N 2 ) = 0, N 2 has exactly one node that is equivalent to v, which must be v. Theefoe, (u, v ) E 2. We now pove that the measue m satisfies popety P1 stated in Section I. Lemma : Let N 1 = (V 1, E 1 ) and N 2 = (V 2, E 2 ) be two educed phylogenetic X -netwoks. Then, N 1 and N 2 ae isomophic if and only if m(n 1, N 2 ) = 0. Poof: Fo the only if diection, let h 1 : V 1 V 2 be the equivalence mapping as given in Definition 6. Mapping h 1 is a bijection, since N 1 and N 2 ae isomophic. Then, based on Definition 7, we have m(n 1, N 2 ) = 0. The if diection follows diectly fom Lemma 3. Fom the definition of the measue, popety P2 follows immediately. Lemma 5: Fo any pai of phylogenetic X - netwoks N 1 and N 2, we have m(n 1, N 2 ) = m(n 2, N 1 ). The measue m(n 1, N 2 ) can be viewed as half the symmetic diffeence of two multisets on the same set of elements, whee the multiplicity of element u in N 1 is κ N1 (u), and similaly fo N 2. Since the symmetic diffeence defines a metic on multisets [13], we have the following esult. Lemma 6: Let N 1, N 2, and N 3 be thee phylogenetic X -netwoks. Then, m(n 1, N 2 )+m(n 2, N 3 ) m(n 1, N 3 ). Fom Lemmas, 5, and 6, we have the following main esult. Theoem 2: The measue m is a metic on the space of educed phylogenetic netwoks. We have poved that the measue m, as given by Definition 7, is a metic on the space of all educed phylogenetic X-netwoks. Finally, it is woth noting that the metic is computable in polynomial time in the size of the netwoks. IV. CONCLUSIONS In this pape, we eviewed the concepts of phylogenetic netwok indistinguishability and eduction, and devised a polynomially-computable metic on the space of all educed phylogenetic netwoks. This fixes the poblem with the measue intoduced in [6], which was late shown not to be metic on the space of educed phylogenetic netwoks [8]. To detemine whethe two phylogenetic X -netwoks ae indistinguishable, the two netwoks ae fist educed, using the eduction pocedue of [6] (eviewed in Section II), and the measue m is applied to the two esulting netwoks. V. ACKNOWLEDGMENTS The autho would like to thank Gabiel Cadona, C. Randal Linde, Benad M.E. Moet, Fancesc Rosselló, Gabiel Valiente, Tandy Wanow, and the thee anonymous eviewes fo vey helpful comments on the manuscipt. This wok was suppoted in pat by DOE gant DE-FG02-06ER2573, NSF gant CCF , and gant R01LM0099 fom the National Libay of Medicine. The contents ae solely the esponsibility of the autho and do not necessaily epesent the official views of the DOE, NSF, National Libay of Medicine o the National Institutes of Health. REFERENCES [1] D. Robinson and L. Foulds, Compaison of phylogenetic tees, Mathematical Biosciences, vol. 53, pp , [2] L. Nakhleh, T. Wanow, and C. Linde, Reconstucting eticulate evolution in species theoy and pactice, in Poceedings of the 8th Annual Intenational Confeence on Computational Molecula Biology, 200, pp [3] H. Bunke and K. Sheae, A gaph distance metic based on the maximal common subgaph, Patten Recognition Lettes, vol. 19, pp , [] H. Bunke, X. Jiang, and A. Kandel, On the minimum common supegaph of two gaphs, Computing, vol. 65, pp , [5] L. Nakhleh, J. Sun, T. Wanow, R. Linde, B. Moet, and A. Tholse, Towads the development of computational tools fo evaluating phylogenetic netwok econstuction methods, in Poceedings of the 8th Pacific Symposium on Biocomputing. Wold Scientific Pub., 2003, pp [6] B. Moet, L. Nakhleh, T. Wanow, C. Linde, A. Tholse, A. Padolina, J. Sun, and R. Timme, Phylogenetic netwoks: modeling, econstuctibility, and accuacy, IEEE/ACM Tansactions on Computational Biology and Bioinfomatics, vol. 1, no. 1, pp , 200. [7] M. Baoni, C. Semple, and M. Steel, A famewok fo epesenting eticulate evolution, Annals of Combinatoics, vol. 8, pp , 200. [8] G. Cadona, F. Rosselló, and G. Valiente, Tipatitions do not always disciminate phylogenetic netwoks, Mathematical Biosciences, vol. 211, no. 2, pp , [9] G. Cadona, M. Llabés, F. Rosselló, and G. Valiente, A distance metic fo a class of tee-sibling phylogenetic netwoks, Bioinfomatics, vol. 2, no. 13, pp , [10], Metics fo phylogenetic netwoks I: Genealizations of the obinson-foulds metic, IEEE/ACM Tansactions on Computational Biology and Bioinfomatics, vol. 6, no. 1, pp. 1 16, 2009.

7 7 [11] G. Cadona, F. Rosselló, and G. Valiente, Compaison of tee-child phylogenetic netwoks, IEEE/ACM Tansactions on Computational Biology and Bioinfomatics, 2009, to appea. [12] G. Cadona, M. Llabés, F. Rosselló, and G. Valiente, Metics fo phylogenetic netwoks II: Nodal and tiplets metics, IEEE/ACM Tansactions on Computational Biology and Bioinfomatics, [13] F. Restle, A metic and an odeing on sets, Psychometika, vol. 2, no. 3, pp , Luay Nakhleh is an Assistant Pofesso of Compute Science at Rice Univesity. He eceived his BSc in Compute Science in 1996 fom the Technion Isael Institute of Technology, the Maste s degee in Compute Science fom Texas A&M Univesity in 1998, and the PhD degee in Compute Science fom The Univesity of Texas at Austin. His eseach inteests fall in the geneal aeas of computational biology and bioinfomatics; in paticula, he woks on computational phylogenetics, compaative genomics, and biological netwok analysis. Luay eceived the Roy E. Campbell Faculty Development Awad fom Rice Univesity in May 2006 and the DOE Ealy Caee Awad in August 2006.