Gene tree reconciliation is a well-studied method for resolving topological inconsistencies between a gene tree and a trusted species tree 234567. Inconsistencies are resolved by invoking gene-duplication and loss events that reconcile the gene tree to be consistent with the actual species tree. Such events do not only reconcile gene trees, but also lay foundation for a variety of evolutionary applications including ortholog/paralog annotation of genes, locating episodes of gene-duplications in species trees 8910, reconstructing domain decompositions 11, and species supertree construction 8121314.

A major problem in the application of gene tree reconciliation is its high sensitivity to error-prone gene trees. Even seemingly insignificant errors can largely mislead the reconciliation process and, typically undetected, infer incorrect phylogenies (e.g., 715). Errors in gene trees are often topological errors and rooting errors. Topological error results in an incorrect topology of the gene tree that can be caused by the inference process (e.g. noise in the underlying sequence data) or the inference method itself (e.g. heuristic results). This problem has been addressed for rooted gene trees by 'correcting the error'; that is, editing the given tree such that the number of invoked gene-duplications and losses is minimized 1617. However, most inference methods used in practice return only unrooted gene trees (e.g. parsimony and maximum likelihood based methods) that have to be rooted for the gene tree reconciliation process. Rooting error is a wrongly chosen root in an unrooted gene tree. Whereas rooting can be typically achieved in species trees by outgroup analysis, this approach may not be possible for gene trees if there is a history of gene duplication and loss 7. Other rooting approaches like midpoint rooting or molecular clock rooting assume a constant rate of evolution that is often unrealistic. However, rooting problems can be bypassed by identifying roots that minimize the invoked number of gene duplications and losses 716171819.

In summary, even small topological error or a slightly misplaced root can incorrectly identify enormous numbers of gene duplications and losses, and therefore largely mislead the reconciliation process. Therefore, gene tree reconciliation requires gene trees that are free of error and correctly rooted at the same time 5. However, as previous work has incorporated topological error-correction only separately from correctly rooting gene trees into the reconciliation process 1618, this process can still be misled.

We address the problem of reconciling erroneous and unrooted gene trees by error-correcting and rooting them at the same time. Solving this problem efficiently is a crucial step towards making gene tree reconciliation more robust, and thus to improve on the accuracy of applications that rely on gene tree reconciliation like the construction of gene-duplication supertrees. We introduce the problem and design an efficient algorithm that facilitates a much more precise gene tree reconciliation, even for large-scale data sets. Our algorithm detects and corrects errors in unrooted gene trees, and thus we avoid the biologists' difficulty and uncertainty of handling erroneous gene trees and correctly rooting them. The presented experimental results suggest that our novel reconciliation algorithms can identify and correct topological error in unrooted input gene trees, and at the same time root them optimally.

Our algorithm is designed to search for the correct and rooted tree of a given unrooted tree in local search neighborhoods of the given tree. The size of these neighborhoods is described by a positive integer k that allows to fine-tune the search. While in theory k can be large it is assumed that gene trees have only small topological error, which typically can be captured by small values of k. For a fixed but freely choosable integer k the runtime of our algorithm is O(l^k + max(n, m)), where n and m is the size of the gene tree and species tree respectively, and l is the number of edges in the gene tree that potentially contain an error (such edges will be called weak). Thus, for a small error, which is expressed by k = 1, our algorithm runs in linear time. Our experiments show that error-correction runs of the algorithm for k = 3 are still possible even for trees with large number of weak edges (e.g., l = 200) on a standard workstation configuration.

Further, we address the problem of constructing rooted supertrees by reconciling unrooted and erroneous gene trees with assigned weak edges, a key problem in illuminating the role and effect of gene duplication and loss in shaping the evolution of organisms. We introduce the problem and develop an effective local search heuristic that makes the construction of more accurate supertrees possible and allows a much better postulation of gene duplication histories. Our experimental results demonstrate that our approach is effective in identifying gene duplication histories given erroneous gene trees and producing more accurate supertrees under gene tree reconciliation.

We introduce the fundamentals of the classical duplication-loss model. Our definitions are mostly adopted from 18. For a more detailed introduction to the duplication-loss model we refer the interested reader to 251020.

Let ℐ be the set of species consisting of N > 0 elements. The unrooted gene tree is an undirected acyclic graph in which each node has degree 3 (internal nodes) or 1 (leaves), and the leaves are labeled by the elements from ℐ. A species tree is a rooted binary tree with N leaves uniquely labeled by the elements from ℐ. In some cases, a node of a tree will be referred by "cluster" of labels of its subtree leaves. For instance, a species tree (a, (b, c)) has 5 nodes denoted by: a, b, c, bc and abc. A rooted gene tree is a rooted binary tree with leaves labeled by the elements from ℐ. The internal nodes of a tree T we denote by int(T).

Let S = ⟨ V S , E S ⟩ be a species tree. can be viewed as an upper semilattice with + a binary least upper bound operation and ⊤ the top element, that is, the root. In particular for a, b ∈ V S , a <b means that a and b are on the same path from the root, with b being closer to the root than a. We define the comparability predicate D(a, b) = 1, if a ≤ b or b ≤ a and D(a, b) = 0, when a and b are incomparable. The distance function ρ(a, b) is used to denote the number of edges on the unique (non-directed) path connecting a and b.

We call distinct nodes a, b∈VSsiblings when a + b is a parent of a and b. For a, b∈VS let Sb(a, b) be the set of nodes defined by the following recurrent rule: (i) Sb(a, b) = ∅ if a = b or a and b are siblings, (ii) Sb(a, b) = {c} ∪ Sb(a + c, b), if a <b or a + c <a + b; here c is the sibling of a, and (iii) Sb(a, b) = Sb(b, a) otherwise.

By L(a, b) we denote the number of elements in Sb(a, b). Observe that L(a, b) = ρ(a, b) - 2 · (1 - D(a, b)). Let M : V G → V S be the least common ancestor (lca) mapping, from rooted into that preserves the labeling of the leaves. Formally, if v is a leaf in then M(v) is the node in labeled by the label of v. If v is internal node in with two children a, b, then M(v) = M(a) + M(b). An example is depicted in Figure 1.

In this general setting let us assume that we are given a cost function ξ : V G × V S → R which for all nodes v ∈ V G , a ∈ V S assigns a real ξ(v, a) representing a contribution to node a which comes from v when reconciling with . Having ξ we can define k ( v ) = ∑ a ξ v ,   a to be a total contribution from v in the reconciliation of with . We call κ a contribution function. Finally, σ = ∑ v k v is the total cost of reconciliation of with .

Now we present examples of cost functions that are used in the duplication model. We assume that if v is an internal node in then w₁ and w₂ are its children. The Duplication cost function is defined as follows: ξ^D(v, a) = 1 if v ∈ int ( G ) and M(v) = M(w_i) = a for some i, and ξ^D(v, a) = 0 otherwise. The Loss cost function: ξ^L(v, a) = 1 if v∈int(G) and a ∈ Sb (M(w₁), M(w₂)), and ξ^L(v, a) = 0 otherwise. It can be proved that if v∈int(G) then κ^D(v) = D(M(w₁), M(w₂)) and κ^L (v) = L(M(w₁), M(w₂)) (in both cases 0 if v is a leaf).

The Duplication cost function is defined as follows: ξ^D(v, a) = 1 if v∈int(G) and M(v) = M(w_i) = a for some i, and ξ^D(v, a) = 0 otherwise. Loss cost function: ξ^L(v, a) = 1 if v∈int(G) and a ∈ Sb(M(w₁), M(w₂)), and ξ^L(v, a) = 0 otherwise. It can be proved that if v∈int(G) then κ^D(v) = D(M(w₁), M(w₂)) and κ^L(v) = L(M(w₁), M(w₂)) (in both cases 0 if v is a leaf).

Observe that a node v ∈ V G is called a duplication 413 if κ^D(v) = 1. Moreover, κ^L(v) = l(v), where l(v) is the number of gene losses associated to v. It can be proved that σ^D and σ^L are the minimal number of gene duplications and gene losses (respectively) required to reconcile (or to embed) with . Please refer to 18 for more details. The example of an embedding is depicted in Figure 1.

Here we highlight some results from 18 that are used for the design of our algorithm. From now on, we assume that G = 〈 V G , E G   〉 is an unrooted gene tree. We define a rooting of by selecting an edge e ∈ E G on which the root is to be placed. Such a rooted tree will be denoted by G e , where v_* is a new node defining the root. To distinguish between rootings of , the symbols defined in previous section for rooted gene trees will be extended by inserting index e. Please observe, that the mapping of the root of Ge is independent of e. Without loss of generality the following is assumed: (A1) and have at least one internal node and (A2) M_e(v_*)=⊤; that is, the root of every rooting is mapped into the root of (we may always consider the subtree of the species tree rooted in M_e(v_*) with no change of the cost).

First, we transform into a directed graph G ^ = ⟨ V G , E G ^ ⟩ where E G ^ = { ⟨ v , w ⟩ | { v , w } ∈ E G } . In other words each edge 〈v, w〉 in is replaced in G ^ by a pair of directed edges 〈v, w〉 and 〈w, v〉.

Edges in G^ are labeled by nodes of as follows. If v ∈ V G is a leaf labeled by a, then the edge ⟨ v , w ⟩ ∈ E G ^ is labeled by a. When v is an internal node in G^ we assume that 〈w₁, v〉 and 〈w₂, v〉 are labeled by b₁ and b₂, respectively. Then the edge ⟨ v , w 3 ⟩ ∈ E G ^ , such that w₃ ≠ w₁ and w₃ ≠ w₂ is labeled by b₁ + b₂. Such labeling will be used to explore mappings of rootings of . An edge {v, w} in is called asymmetric if exactly one of the labels of 〈v, w〉 and 〈w, v〉 in G^ is equal to ⊤, otherwise it is called symmetric.

Every internal node v, and its neighbors in G^ define a subtree of E G ^ , called a star with a center v, as depicted in Figure 2. The edges 〈v, w_i〉 are called outgoing, while the edges 〈w_i, v〉 are called incoming. We will refer to the undirected edge {v, w_i} as e_i, for i = 1, 2, 3.

The are several types of possible star topologies based on the labeling (for proofs and details see 18): (S1) a star has one incoming edge labeled by ⊤ and two outgoing edges labeled ⊤ and these edges are connected to the three siblings of the center, (S2) a star has exactly two outgoing edges labeled by ⊤, (S3) a star has all outgoing edges and exactly one incoming edgd labeled by ⊤, (S4) a star has all edges labelled by top, and (S5) a star has all outgoing edges and exactly two incoming edges labeled by ⊤. Figure 2 illustrates the star topologies.

In summary stars are basic 'puzzle-like' units that can be used to assemble them into unrooted gene trees. However, not all star compositions represent a gene tree. For instance, there is no gene tree with 3 stars of type S2. It follows from 18 (see Lemma 4) that we need the following additional condition: (C1) if a gene tree has two stars of type S2 then they share a common edge.

Now we overview the main result of 18 (see Theorem 1 for more details). Let be a species tree and be unrooted gene tree. The set of optimal edges, that is, candidates for best rootings, is defined as follows: M i n G = { e ∈ E G | σ e M α , β  is minimal } , where σ e M α , β is the total cost for the weighted mutation cost defined by ξ e M α , β ( v , a ) = α ⋅ ξ e D ( v , a ) + β ⋅ ξ e L ( v , a ) , e is an edge in and α, β are two positive reals. Then (M1) if | M i n G | > 1 , then M i n G consists of all edges present in all stars of type S4 or S5, (M2) if | M i n G | = 1 , then MinG contains exactly one symmetric edge that is present in star of type S2 or S3. From the above statements, (C1) and star topologies we can easily determine MinG. More precisely, the star edges outside MinG are asymmetric and share the same direction. Thus, to find an optimal edge it is sufficient to follow the direction of non ⊤ edges in G^.

Now we summarize the time complexity of this procedure. It follows from 21 that a single lca-query (that, is a + b for nodes a and b in ) can be computed in constant time after an initial preprocessing step requiring O ( | S | ) time. Other structures like G^ with the labeling can be computed in O ( | G | ) time. The same complexity has the procedure of finding an optimal edge in . In summary an optimal edge/rooting and the minimal cost can be computed in linear time. See 18 for more details and other properties.

Now we show that a single NNI operation can be completed in constant time if all structures required for computing the optimal rootings are already constructed. First, let us assume that the following is given: (a) two positive reals α and β, a species tree , (b) lca structure for that allows to answer lca-queries in constant time, (c) an unrooted gene tree , (d) G^ with the labeling of edges, (e) MinG - the set of optimal edges, and (f) σ - the minimal total weighted mutation cost. As observed in the previous section (b),(d)-(f) can be computed in O ( max ( | S | , | G | ) ) . Now we show that (c)-(f) can be computed in constant time after a single NNI operation.

NNI operation (c) and the update of lca-mappings (d).

Definition 1. (Single NNI operation) An NNI operation transforms a gene tree G = ( ( T 1 , T 2 ) , ( T 3 , T 4 ) ) into G ′ = ( ( T 2 , T 3 ) , ( T 1 , T 4 ) ) , where T_i-s are (rooted) subtrees of . The edge that connects the roots of (T₁, T₂) and (T₃, T₄) in is denoted by e₀ and called the center edge. For each i = 1, 2, 3, 4 we assume the following: w_i is the root of T_i, e_i is the edge connecting w_i with e₀ and a_i is the lca-mapping of T_i. Similarly, we define the center edge e 0 ′ and e i ′ in G ′ .

An NNI operation is depicted in Figure 3 with the transformation of G^ into G ′ ^ . The notation will be used from now on. Note that there is a second NNI operation, when is replaced with ((T₁, T₃), (T₂, T₄)). However, it can be easily defined and therefore it is omitted here. Observe that the NNI operation (without updating of lca-mappings) can be performed in constant time for both trees.

The right part of Figure 3 depicts the transformation of G^. Observe that the labels of the incoming and outgoing edges attached to each w_i in G^ do not change during this operation. Lemma 1 follows directly from this observation.

Lemma 1. An NNI operation changes only the labels of the center edge.

We conclude that updating G^ requires only two lca-queries, and therefore can be performed in constant time.

Reconstruction of optimal edges (e). We analyze the changes of the optimal set of edges M i n G . To this end we consider a number of cases depending on the relation between the optimal set of edges and the set of edges, incident to the nodes of the center edge. Let C G = { e i } i = 0 , … , 4 .

For convenience, assume that the NNI operation replaces e_i with e i ′ as indicated in Figure 3. We call two disjoint edges from C G semi-alternating if they share a common node after the NNI operation. In Figure 3 {e₁, e₄} and {e₂, e₃} are semi-alternating. For two edges a and b that are incident to the same node let ⋆(a, b) be the set of three edges defining the unique star that contains a and b.

Lemma 2. Assuming that e_i is replaced by e i ′ after the NNI operation the set of optimal edges does not require additional changes if and only if one of the following conditions is satisfied:

( E Q 1 ) M i n G ∩ C G = ∅ ,

( E Q 2 ) M i n G ⊇ C G and each pair of semi-alternating edges contains at least one symmetric edge,

( E Q 3 ) M i n G consists of only the center edge,

( E Q 4 ) M i n G ∩ C G = { e i } for some i >0 and the center is asymmetric after the NNI operation.

Proof: (EQ1) All edges in C G are asymmetric (2 stars S1). Then, after the NNI operation e 0 ′ is asymmetric and ( C G ′ has 2 stars S1). (EQ2) C G consists of 2 stars of type S4/S5 and at most two asymmetric edges. It follows from EQ2 that the asymmetric edges in CG′ cannot form a star of type other than S5. Together with M1 it follows that CG′ is optimal. (EQ3) By M1 the center is symmetric in . It remains symmetric after NNI. From C1 and M2, M i n G ′ consists of the center edge. (EQ4) Note, that the type of ⋆ ( e i ′ , e 0 ′ ) is S1, S2 or S3.

Lemma 3 (NE1). If M i n G ⊇ C G and there exists a pair {e_i , e_j} of asymmetric semi-alternating edges, then M i n ′ G = M i n G \ C G ∪ ( C G ′ \ { e i ′ , e j ′ } ) .

Proof: The type of ⋆ ( e i ′ , e j ′ ) is S1 or S3 and the other star has type S4 or S5. By M2 e i ′ and e j ′ are not optimal.

Lemma 4 (NE2). If M i n G ∩ C G = { e i } for some i >0 and the center is symmetric after the NNI operation then M i n ′ G = M i n G \ { e i } ∪ ⋆ ( e 0 ′ , e j ′ ) .

Proof: In this case e 0 ′ has two arrows and ⋆ ( e 0 ′ , e i ′ ) is of type S5.

Lemma 5. Assume that M i n G ∩ C G = { e 0 , e i , e j } , where i ≠ 0,

(NE3) If both e_i and e_j are symmetric then M i n G ′ = M i n G \ C G ∪ C G ′ ,

(NE4) If e_j is asymmetric and e0′ is symmetric then M i n G ′ = M i n G \ C G ∪ ⋆ ( e 0 ′ , e i ′ ) .

(NE5) If both e_j and e0′ are asymmetric then M i n G ′ = M i n G \ C G ∪ { e i ′ } .

Proof: Note that {e₀, e_i , e_j} must be a star in G ⋅ ( NE 3 ) ⋆ ( e i , e j ) has type S4 or S5. After the transformation the two stars ⋆ ( e 0 ′ , e i ′ ) and ⋆ ( e 0 ′ , e j ′ ) have type S5. Both are optimal in G ′ ⋅ ( NE4 ) ⋆ ( e i , e j ) has type S5. After the transformation ⋆ ( e 0 ′ , e i ′ ) has type S5 and ⋆ ( e 0 ′ , e j ′ ) has type S3. Only the first is optimal in G ′ ⋅ ( NE5 ) ⋆ ( e i , e j ) has type S5 while the other star in CG has type S3. After the transformation only e i ′ remains symmetric in C G ′ therefore it is the only optimal edge in C G ′ .

Computing the optimal cost (f). Observe that from Lemmas 2-5 at least one optimal edge remains optimal after the NNI operation. Therefore, to compute the difference in costs between optimal rootings of and G ′ we start with the cost analysis for the rootings of such edge.

First, we introduce a function for computing the cost differences. Consider three nodes x, y, z of some rooted gene tree such that x and y are siblings and the parent of them (denoted by xy), is a sibling of z. In other words we can denote this subtree by ((x, y), z). Then, the partial contribution of ((x, y), z) to the total weighted mutation cost can be described as follows: ∑ a ∈ S α * ( ξ D ( x y , a ) + ξ D ( x y z , a ) ) + β * ( ξ L ( x y , a ) + ξ L ( x y z , a ) ) . Assume that x, y and z are mapped into a, b and c (from the species tree), respectively. It can be proved from the defnition of ξ^D and ξ^L that the above contribution equals: ϕ(a, b, c) = α * (D(a, b) + D(a + b, c)) + β * (L(a, b) + L(a + b, c)). Now, assume that a single NNI operation changes ((x, y), z)) into (x, (y, z)). It should be clear that the cost difference is given by: Δ₃(a, b, c) = ϕ(c, b, a) - ϕ(a, b, c). Similarly, we can define a cost difference when a single NNI operation changes ((x, y), (z, v)) into ((x, v), (y, z)). Assume, that v is mapped into d. Then, the cost contribution of the first subtree is ϕ'(a, b, c, d) = ϕ(a, b, c + d) + α * (D(c, d) + β * L(c, d). The cost difference is given by: Δ₄(a, b, c, d) = ϕ'(a, d, b, c) - ϕ'(a, b, c, d).

Lemma 6. If the center edge is optimal and remains optimal after the NNI operation then the cost difference equals Δ₄(a₁, a₂, a₃, a₄), where a_i (for i = 1, 2, 3, 4) is the mapping as indicated in Figure 3.

As mentioned the above lemma can be proved by comparing the rootings placed on the center edges in and G′. Lemma 6 gives a solution for cases: EQ2, EQ3, NE1 and NE3. The next lemma gives a solution for the remaining cases.

Lemma 7. If for some i >0 there exists an optimal edge in T_i ∪ {e_i} that remains optimal after the NNI operation (under assumption that e_i is replaced by e i ′ ) then the cost difference is Δ₃(a₄, a₃, a₂) if i = 1, Δ₃(a₃, a_4, a₁) if i = 2, Δ₃(a₂, a₁, a₄) if i = 3 and Δ₃(a₁, a₂, a₃) if i = 4.

Similarly to Lemma 6 we can prove Lemma 7 by comparing the rootings of e_i and e i ′ .

Error correction algorithm. Finally, we can present the algorithm for computing the optimal weighted mutation cost for a given gene tree and its k-NNI neighborhood. See Figure 4 for details. It should be clear that the complexity of this algorithm is O ( | G | k + max ( | G | , | S | ) ) . We write that a gene tree has errors if the optimal cost is computed for one of its NNI variants. Otherwise, we write that a gene tree does not require corrections. Please note that it for a special case of k = 1, this algorithm is linear in time (see also our preliminary article 22).

We present several approaches to problems of error correction and phylogeny reconstruction. Let us assume that σ α , β , k ( S , G ) is the cost computed by algorithm from Figure 4, where α, β > 0, k ≥ 0, is a rooted species tree and is an unrooted gene tree.

Problem 1 (kNNIC). Given a rooted species tree and a set of unrooted gene trees, G compute the total cost ∑ G ∈ G σ α , β , k ( S , G ) .