Reconciliation and local gene tree rearrangement can be of mutual profit

Definition 1

 (Reconciliation model). Consider a gene tree G, a species tree S with a time function θ _S , and its subdivision S^′ with a time function θ_S′.

Let α be a function that maps each node u of G onto an ordered sequence of nodes of S^′, denoted α(u). For u ∈ V(G), let ℓ denote the length of α(u) and let α _i (u) be its i^th element (where 1 ≤ i ≤ ℓ). α is said to be a reconciliation between G and S^′ if and only if exactly one of the following atomic events occurs for each couple of nodes u of G and α _i (u) of S^′ (where α _i (u) is denoted x):

child. ( event)

1. 6.

   x is not artificial and α _i+1(u) ∈ {x ₁,x ₂}. ($\mathbb{S}\mathbb{L}$ event)

    
2. 7.

   α _i+1(u) = x ^′ is such that x ^′ ≠ x and ${\theta }_{S^{\prime }}\left(x^{\prime }\right)={\theta }_{S^{\prime }}\left(x\right)$. ($\mathbb{T}\mathbb{L}$ event)

    

The combinatorial events mentioned above ( , , , , , $\mathbb{T}\mathbb{L}$, $\mathbb{S}\mathbb{L}$) are those defined in [2]. See Figure 2 for an illustration of these events and Figure 3 for an example of reconciliation according to this model.

Note that among these events, $\mathbb{T}\mathbb{L}$ and $\mathbb{S}\mathbb{L}$ are in fact a combination of two independent biological events. However, the fact that a loss is always taken into account jointly with another event allows to obtain a recursive algorithm and is done without loss of generality, i.e. does not reduce the power of the model [2].

Given a gene tree G and species tree S, there is an infinite number of possible reconciliations. Discrete evolutionary models compare them by counting the number of events they respectively induce. As different types of event can have different expectancies (e.g. are thought to be more frequent than and ), reconciliation models allow for a specific cost to be given to each kind of event. The cost of a reconciliation is then the sum of the costs of the individual events it induces. In that setting, the parsimony approach is then to prefer a reconciliation of lower cost. This is formalized in the following definition.

Definition 2.

where δ, τ, and λ respectively denote the cost of , , and events, while d, t, and l denote the number of the corresponding events in α. Moreover, a $\mathbb{T}\mathbb{L}$ event is atomic and costs (τ + λ), while a $\mathbb{S}\mathbb{L}$ event just costs λ. Indeed, speciation events are most of the time considered as having a null cost, but the model easily accommodates for non-null costs if necessary.

over all reconciliations α between G and S^′.

Note that several distinct alternative reconciliations can have an optimal reconciliation cost.

Lemma 1 (Consecutive $\mathbb{T}\mathbb{L}$ events)

Consider a gene tree G, the subdivision S^′ of a species tree, and a reconciliation α of optimal cost C(G,S^′) = c(α). For any node u of G, if α _i (u) corresponds to a $\mathbb{T}\mathbb{L}$ event, then α_i + 1(u) does not.

This results from the observation that two $\mathbb{T}\mathbb{L}$ in a row can be replaced by single $\mathbb{T}\mathbb{L}$, leading to a reconciliation of lesser cost.

Definition 3 (Reconciliation cost matrix).

Consider a gene tree G and the subdivision S^′ of a species tree S with a time function θ_S′. Let $c:V\left(G\right)\times V\left(S^{\prime }\right)\to \mathbb{R}$ denote the cost matrix recursively defined as follows for a node u of G and a vertex x of S^′: $c_{\overline{\mathbb{T}\mathbb{L}}}(u,x)=\text{min}\left\{c_{\mathbb{E}}\right(u,x):\mathbb{E}\in \{\mathbb{C},\mathbb{S},\mathbb{D},\mathbb{T},\varnothing ,\mathbb{S}\mathbb{L}\left\}\right\}$ and $c(u,x)=\text{min}\left\{c_{\mathbb{T}\mathbb{L}}\right(u,x),c_{\overline{\mathbb{T}\mathbb{L}}}(u,x\left)\right\}$, where the costs $c_{\mathbb{E}}(u,x)$ for all events x $\mathbb{E}\in \{\mathbb{C},\mathbb{S},\mathbb{D},\mathbb{T},\varnothing ,\mathbb{S}\mathbb{L},\mathbb{T}\mathbb{L}\}$ are defined below

If the above constraints for an event $\mathbb{E}\in \{\mathbb{C},\mathbb{S},\mathbb{D},\mathbb{T},\varnothing ,\mathbb{S}\mathbb{L},\mathbb{T}\mathbb{L}\}$ on node u and vertex x are not respected, the corresponding cost $c_{\mathbb{E}}(u,x)$ is set to ∞.

The value c(u,x) is the optimal cost when mapping gene node u to node x in S^′. The optimal cost for reconciling G with S^′, denoted C(G,S^′), is then $\mathit{\text{mi}}{n}_{x\in V\left(S^{\prime }\right)}\left(c\right(r\left(G\right),x)$.

The algorithm of Doyon et al. [2], called Mowgli, fills the dynamic programming cost matrix $V\left(S^{\prime }\right)\times V\left(G\right)\to {\mathbb{R}}^{+}$ by two embedded loops: one loop visits all species nodes of S^′ in time order (e.g. according to the ${\theta }_{S^{\prime }}$ partial order, while the other loop visits nodes of the gene tree G in postorder. Due to an optimization in precomputing the best receiver edge for transfer events of nodes u at a given time, this algorithm has O(|S|².|G|) time complexity.

The problem considered in this paper is the following:

MOST PARSIMONIOUS RECONCILIATION GENE TREE (MPR-GT)

INPUT:

OUTPUT: a gene tree G^′ such that both $\mathcal{ℒ}\left(G\right)=\mathcal{ℒ}\left(G^{\ast }\right)$ and $\mathit{\text{st}}{r}_t\left(G\right)\subseteq E\left(G^{\ast }\right)$, and such that C(G^∗,S^′) is minimum among all such trees.

Combinatorial optimization

We now present results that take advantage of the way the dynamic programming matrix is computed (Definition 3) to avoid recomputing from scratch the cost matrix associated to a gene tree G^′ obtained by an NNI edit operation from a gene tree G. Consider the gene tree G of Figure 4, the NNI operation applied on edge (w,v) that swaps the two subtrees G _b  and G _c , and the resulting gene tree denoted G^′. We can observe that despite the global architecture of G and G^′ differs, the local architectures of subtrees $G_b,G_c,G_d,G_{a_0}$, $\dots G_{a_k}$ remain unchanged. Hence, any cost that differs between the matrices C(G,S^′) and C(G^′,S^′) (see Definition 3) is located in a column (i.e. node of the gene tree) associated to an ancestor of v (including v itself). For each of those nodes, there are two cases: (i) the node belongs to the NNI edge and its two children have subtree that have been modified (e.g. nodes w and v); (ii) the node is a strict ancestor of the NNI edge (w,v) and has exactly one child with a subtree that has been modified (e.g. g _k ,…,g₀).

Lemma 2 below indicates which columns of the cost matrix don’t need to be recomputed.

Algorithm 1 M o w g l i N N I ( G , c ): seeking a gene tree G ^′  of minimum reconciliation cost, starting from a gene tree G and the precomputed matrix reconciliation cost $c:V\left(G\right)\times V\left(S^{\prime }\right)\to \mathbb{R}$ , where S ^′  is the subdivided species tree.

Simulated gene trees and evolutionary histories

A phylogeny of 37 proteobacteria was used as a reference species tree (denoted S) [8]. Along this tree, we simulated the evolutionary history (denoted R _{T r u e} ) of 1000 gene families (G _{T r u e} ), each containing from 10 to 100 genes, according to a birth and death process [35]. Birth events can be one of three kinds of evolutionary events, i.e. speciation, duplication, and horizontal gene transfer. During the simulation process along the species tree, a speciation occurs every time a gene lineage reaches an internal node of the species tree, leading to a split in two gene lineages. A birth event happening strictly between two nodes of the species tree can only correspond to a gene duplication or a horizontal gene transfer event. A birth is decided to be duplication or a transfer according to the input rates of these events.

The death of a gene lineage corresponds to a loss event, which happens according to an input loss rate. The species tree was scaled to the height of 500 million years (Mya). The speciation rate is determined by the topology and the height of the species tree. Each of the 1000 gene families was generated with different event rates, the loss rate being randomly chosen in the range [0.0010–0.0018] events/gene per million year. The ratio between the sum of duplication and transfer rates and the loss rate was randomly chosen in the range [0.5 - 1.1], the duplication rate being [70% - 100%] of the mentionned sum. This birth and death process first output a complete gene tree G ^o , then the “true” gene tree G _{T r u e}  was obtained from G ^o  by pruning extinct subtrees. The “true” evolutionary events to be recovered by the reconciliation programs are those appearing in G _{T r u e} . We denote R _{T r u e}  the history composed by these events. We only considered gene families containing at most ten duplication and transfer events in their true evolution. In particular for the transfer events, this constraint allowed us to limit the number of cases where the true evolution contains a sequence of consecutive transfers where non-transferred genes are lost (i.e. a sequence of$\mathbb{T}\mathbb{L}$ events). Such a piece of history can hardly be recovered by reconciliation methods as it left no trace at all in the gene tree.

Starting from G _True , a further step of the simulation protocol allows to obtain estimates of both this gene tree and the events composing its history (see Figure 6).

The length of the edges in G _True  were first converted from duration to number of substitutions per site by a module taken from [36]. Next, we simulated the evolution of 1500 - 3000 bp DNA sequences along this tree under the GTR model, thanks to the Seq-Gen program [37]. The alignment, composed of one sequence per extant gene, was then given as input to RAxML [38] to obtain a maximum likelihood estimate of the gene tree, denoted G _ML  (also called initial gene tree below). Mowgli[2] and Ranger-DTL-D [3] were then used to infer a most parsimonious evolutionary history R _ML , resp. R _R , between this initial gene tree and the reference species tree S. Separately, MowgliNNI was used to search for an alternative gene tree topology (G _NNI ) of lower reconciliation cost, along with one of its most parsimonious evolutionary history (R _NNI ). The elementary cost considered for each event kind (with being , or ) was computed as follows:

$\mathit{\text{Cos}}{t}_{\mathbb{E}}=\left\{\begin{array}{ll}\text{log}\left(\right.\frac{{\mathbb{D}}_{R_{\mathit{\text{True}}}}+{\mathbb{T}}_{R_{\mathit{\text{True}}}}+{\mathbb{L}}_{R_{\mathit{\text{True}}}}}{{\mathbb{E}}_{R_{\mathit{\text{True}}}}}\left)\right.& \text{if}{\mathbb{E}}_{R_{\mathit{\text{True}}}}\ne 0\\ \text{log}\left(\right.\frac{{\mathbb{D}}_{R_{\mathit{\text{True}}}}+{\mathbb{T}}_{R_{\mathit{\text{True}}}}+{\mathbb{L}}_{R_{\mathit{\text{True}}}}}{0.1}\left)\right.& \text{otherwise}\end{array}\right.$

(1)

where${\mathbb{E}}_{R_{\mathit{\text{True}}}}$ stands for the number of events of the corresponding kind in R _True .

Measuring the accuracy

First, we estimated the improvement in the accuracy of the gene tree’s topology, as measured by the Robinson-Foulds (RF) distance [39] between the true gene tree (G _True ) and the inferred gene tree (G _ML ). As a second measurement of the accuracy of inferred reconciliations we compared the positions of , and events predicted by MowgliNNI and Mowgli with those present in the true history. This is achieved by studying the proportion of true positive (TP), false positive (FP) and false negative (FN) separately for duplications, transfers and losses [2]. True negatives (TN) were not studied as their number is considerably large (if even finite) and hard to determine. An event of R _True  is declared as correctly predicted when it concerns the right part of the gene tree (node or edge) placed in the correct branch or node of the species tree (see [2] for more details). Incidentally, both the receiver and the donor edge of the species tree have to be correctly indicated for a predicted transfer event to be declared as correct.

MowgliNNI provides more accurate inferences

We explored the ability of MowgliNNI to improve the set of G _ML  trees using six different bootstrap values as threshold for defining weak edges, i.e. 20, 40, 60, 80, 90, and 95. The G _ML  trees were inferred from relatively long sequences, they thus contained a large proportion of high bootstrap values, e.g. more than 63% edges had a bootstrap value ≥ 80. Though this left only a moderate number of edges in each gene tree to be considered by MowgliNNI for rearrangement, the method was still able to improve their quality (see below).

Mowgli and Ranger-DTL-D showed a similar accuracy in inferring duplications and transfers (Figure 7), though Ranger-DTL-D proposed reconciliations with higher costs in 13% of the cases. As both methods reconciled the same trees and used the same elementary costs for the events, one might wonder why they did not always obtain the same reconciliation costs. This results from different factors among which the most important is that Ranger-DTL-D relies on a less general reconciliation model than Mowgli (e.g. not ensuring time consistency and not allowing gene loss in the donor branch after a transfer), but which on the other hand allows it to run at greater speed. As Mowgli and Ranger-DTL-D performed similarly, in the following we just report results obtained with Mowgli.

MowgliNNI progressively reduced the number of predicted duplications, transfers and losses as the threshold increased. At threshold 0 (where MowgliNNI = Mowgli, 5510 duplications, 2494 transfers and 12190 losses were predicted on the whole dataset; going to threshold 80, these numbers dropped to 4602 duplications, 1676 transfers and 8133 losses, i.e. values that are much closer to the 4535 duplications and 8260 losses contained in the true reconciliations.

Figure 8(a) shows that, no matter the threshold value, the false positive (FP) of MowgliNNI are always less than that of Mowgli both in terms of RF distance and evolutionary events (duplications, transfers and losses). This means that the G _NNI  trees are closer to the true ones than the initial G _ML  trees inferred from sequences only. Similarly, the evolutionary events inferred from the G _NNI  trees are more accurate. As the threshold increases from 0 to 80, R _NNI  contains less and less FP events, hence widening the gap in accuracy between Mowgli and MowgliNNI. As increasing the threshold results in reconsidering a larger number of G _ML  edges for NNI operations, this means that the species tree examined through the reconciliation lens is a good guide tree for correcting wrong edges of the gene trees.

The average number of false positive events of the R _NNI reconciliations decreases as the threshold increases (Figure 8(b)). However, as in Doyon et al. [2], the average number of FP transfers is quite high compared to that of duplications and losses. This can be explained by several reasons. First, a transfer is judged incorrect as soon as (i) it does not depart or end in the same edges of the species tree as the corresponding true transfer, or (ii) it does not concern the same edge in the gene tree. Overall, there is an additional constraint w.r.t. duplications and loss events, leading on average to more incorrect events. This point is all the more sensitive that several most parsimonious reconciliations (MPR) are obtained in a number of cases, while we just accounted for one of them for each gene family. Hence, event error rates we report are pessimistic, and especially for transfers due to the stringent conditions for judging a transfer as correct. Note that the multiplicity of MPRs does not affect the RF error terms. Last, incorrect gene trees lead to incorrect event inferences, but the latter are very sensitive to only small errors in gene trees. The event FP error grows almost exponentially when the RF distance between the initial and the true tree increases from 0 to 10% (Figure 9). Figures 8 and 10 show that transfers are more influenced by this factor, as a result of more stringent conditions for being correct.

Influence of the sequence length parameter

Robustness of reconciliations to imprecision in the event costs

Room for future improvement

To measure how much of the achievable improvement over the G _{M L}  trees was realized by MowgliNNI, we studied the distribution of reconciliation costs of all possible gene trees for several cases involving a computationally manageable number of species. The shape of the distribution together with the relative position of the costs obtained for G _True , G _ML  and G _NNI  within those distributions gives information on how much improvement could be achieved in the future by more sophisticated methods (e.g., relying on SPR moves). We report here on two cases representative of our observations: two true gene trees$G_{\mathit{\text{True}}}^A,G_{\mathit{\text{True}}}^B$ of 8 taxa were generated from the species tree S of 37 proteobacteria according to the protocol described in Figure 6. Their two associated histories A and B were used as starting points to obtain both sequence alignments and reconciliations costs (according to Equation 1). This time, 50 sequence alignments were generated from each of the two gene trees. A maximum likelihood tree was obtained from each of the 100 alignments, with bootstrap supports associated to its edges. These trees were then submitted for improvement to MowgliNNI, applying a threshold 50 to specify weak edges, and relying on event costs corresponding to histories A and B respectively. All reconciliations were performed with respect to the species tree S.

Figure 11 shows the distributions of reconciliation costs C(S,G) obtained for all possible binary trees G having the same leaves as$G_{\mathit{\text{True}}}^A$ and$G_{\mathit{\text{True}}}^B$ respectively. The first observation is that though the same species tree was used in both cases, these distributions vary significantly in range and shape depending on the gene tree leaf set. In the case of history B, the number of trees with reconciliation costs falling in a given range varies sporadically, whereas the number of trees in a given range almost follows a normal (or beta) distribution for history A.

History A involved 2 duplications, no transfer and 7 losses and was correctly recovered by the parsimonious reconciliation of Mowgli from$G_{\mathit{\text{True}}}^A$. However, History B (involving 2 duplications, 1 transfer and 5 losses) was incorrectly recovered from$G_{\mathit{\text{True}}}^B$, the achieved$C(G_{\mathit{\text{True}}}^B,S)=3.98$ cost being less than the 5.75 cost for the real history. Though the real cost is in the left part of the distribution, it is not the minimum point of the distribution, showing that parsimony can sometimes be misleading when followed to its extreme.

Nevertheless, in both cases, the true gene tree is among the ones having the minimum reconciliation costs: it is precisely the one leading to the minimum cost for history A and among the nine best trees for history B. On these examples (and other cases not shown), parsimony can be considered as a very good guide towards the correct gene tree, even if the reconciliation from this correct tree can underestimate the number of real events (as discussed above).

For both histories A and B, MowgliNNI proposed a gene tree G _NNI  whose reconciliation cost was on average closer from that of the true gene tree – and from the real cost – than the cost obtained from the maximum likelihood tree.

Conclusion on simulated datasets

In summary, MowgliNNI successfully uses the reconciliation cost as additional information to resolve the uncertain parts of gene trees inferred from sequences only. Though the gene tree resolutions are partly guided by reconciliations with the species tree, they are not attracted away from the true gene trees, but are closer to them than the initial gene trees. As a result, MowgliNNI infers gene events more accurately, which is of prior importance to distinguish orthologs from paralogs and xenologs [14].

A decrease in the number of inferred events and reconciliation costs

MowgliNNI allowed to change the gene tree, hence to lower the reconciliation cost, in 24% of the ≈30,000 families. This gain is non-negligible and has a real importance as changing the gene tree topology has an important impact on the inferred events (as already shown on simulated datasets and discussed below). In turn, these inferred events may serve to predict the function of new sequences on the basis of their orthology relationships with annotated sequences, orthology following from the chosen reconciliation. Among previous reconciliation studies that allowed to modify the gene trees, Berglund-Sonnhammer et al. report that 10% of their families were improved [21] when allowing rearrangements on weak edges under the DL model, while Chaudhary et al. improved all their gene trees in a pure D model when rearranging gene trees with Subtree Prune and Regraft (SPR) operations [24]. Note that the heterogeneity of models and datasets used in these studies limit the comparison of their results, but we cite them for completeness.

For gene families with a lower reconciliation cost (24% of all families), we counted the number of events of each kind ($\mathbb{D},\mathbb{T},\mathbb{L}$) inferred by Mowgli and MowgliNNI. As a rule, MowgliNNI led to a decrease in the number of events in inferred evolutionary histories. In particular, the number of transfers is reduced in 88.3% of these gene families, the number of losses being reduced in 59.9%, while the number of duplications is almost the same (decrease in 5.2% of the families). These results obtained in the DTL model echo those of Durand et al. reporting that in the DL model gene tree rearrangements substantially reduce the number of events needed to explain the data [19]. The differences in reductions we observed among the kind of events can be explained by the costs – estimated from [18, 41] – that we used for the events (τ = 3,δ = 3.5,λ = 1). Given those costs, it is usually more parsimonious to explain the conflicts between a gene and the species tree by a combination of and rather than a combination of , and . Thus, when MowgliNNI infers a gene tree closer to the species tree, it mostly removes the need for artificial transfers (and losses to a lesser extent), while not altering that much the number of duplications.

To give a precise example of how MowgliNNI proposes a more parsimonious reconciliation on a modified gene tree than Mowgli on the initial gene tree, Figure 12 details the particular case of family HBG040981, (a putative tocopherol cyclase). In this case, MowgliNNI proposes a gene history resorting to less events by rearranging an edge with a very small support. The gene tree modified by MowgliNNI leads to a reconciliation having one transfer and one loss less than the reconciliation performed from the initial gene tree. On the whole, the reconciliation cost goes from 9.5 down to 5.5.

A decrease in the number of equally most parsimonious reconciliations

In addition to reductions in number of events and hence reconciliation cost, the modified gene trees proposed by MowgliNNI usually reduced the number of alternative MPRs, i.e. equally most parsimonious histories. On a random sample of two dozens modified gene trees, the number of MPRs is reduced in 63% of the cases (by a factor of 18 in the best case), and increased in 21% (by a factor 3 at worst). This echoes similar observations done by other authors.

The improvement in running time due to the optimized version of MowgliNNI

We measured the running time of Mowgli and that of both the naive and optimized versions of MowgliNNI (see Methods), respectively called M o w g l i N N I ⁿ  and MowgliNNI. From the ≈30k families of our dataset, we extracted a random sample of 100 families uniformly spanning from 10 to 80 taxa, and respecting the previously observed proportion of improving / non-improving cases in the reconciliation cost (i.e., 24% and 76% resp.). This sample was used to run the three programs. Figure 13 reports the ratios of MowgliNNI ⁿ  and MowgliNNI running times over those of Mowgli, with respect to the number of weak edges within the input gene tree. Indeed, for each gene tree, the number of tested rearrangements during the optimization search depends on the number of weak edges, which hence strongly impacts MowgliNNI ⁿ  and MowgliNNI running times. Note that the number of weak edges often represents between 10% and 20% of the gene tree edges, but can go up to 70%.

Figure 13 shows that MowgliNNI is 20 (resp. 50 and 80) times faster than MowgliNNI ⁿ , when facing 1–20 (resp. 20–40 and 40–60) weak edges. This shows that the combinatorial optimization proposed in the Methods section is crucial in practice.

Now, when compared to Mowgli, the rearrangements tried by MowgliNNI ⁿ  on weak edges to obtain a better gene tree are done at the price of a relatively small computation time overcost. We also indicate the regression line of MowgliNNI running times with respect to those of Mowgli, plotted against the number of weak edges. Its slope is only 0.01186, meaning that MowgliNNI (the optimized version) is able to take into account the gene tree uncertainties with just a slight increase in the running time.

Transfers in prokaryotic phyla

On our whole dataset of 29,709 homologous gene families, we particularly studied transfers in 5 bacterial and 1 archaeal phyla: Proteobacteria (169 genomes), Actinobacteria (31 genomes), Cyanobacteria (14 genomes), Chlamydiae (7 genomes), Spirochaetes (7 genomes) and Crenarchaeota (10 genomes). We compared our results obtained with Mowgli and MowgliNNI to those of Abby et al. [30] obtained with the Prunier method [29] that infers transfers in mono-copy gene families on another basis than reconciliation.

In order to compare our results to the Abby et al. study, we extracted particular families from HOGENOM v4. For each of the 6 phyla of interest, we collected the list of families having at most one copy of the gene for the genomes of this phylum and separated them into two groups: families having one copy of the gene for each genome of the phylum, so-called “universal families”, and families having a copy of the gene for at least 7 genomes of the phylum, so-called “non-universal families”.

For each phylum, we computed the number of intra-phylum transfers inferred by reconciliations of Mowgli and MowgliNNI for families of the two groups (universal and non-universal). As the number of families we found in several groups among the various phyla varied slightly from those reported by Abby et al. [30] we summarized the findings of both studies in terms of transfer rates, expressed in number of transfers per million year and per family.

Figures 14 and 15 display transfer rates of universal, resp. non-universal families, for the studied phyla ordered by increasing transfer rate. We note that the obtained order on the phyla depends on the profile of the studied families, as in Abby et al. [30]. For both the universal and the non-universal families, the transfer rates obtained through Mowgli and MowgliNNI follow the same trend as that prediced by the Prunier method, i.e. the phyla are ordered in the same way, from the Spirochaetes up to the Actinobacteria in the case of universal families and from the Proteobacteria up to the Crenarchaeota in the case of non-universal families.

Finally, as expected, MowgliNNI reduced the number of inferred transfers compared to Mowgli, leading to transfer rates closer to that inferred by Prunier.

Theorem 1.

 Consider a gene tree G, the subdivision S^′ of a species tree S, an edge (w, v) of G, a gene tree G^′ obtained by an NNI operation on (w,v), and any strict ancestor u of w in G where the unique child of u that is an ancestor of w is u₁ w.l.o.g. (i.e. w ≤ u₁ in both G and G’). If c(u₁ ,x) ≤ ;c^′ (u₁,x) holds for all x ∈ V(S^′), then c (u,x) ≤ c^′ (u,x) holds for all x ∈ V(S^′), and as a corollary C(G,S^′) ≤ C(G^′,S^′).

Proof.

 First remark that an NNI operation performed around the edge (w,v) of G to obtain a modified tree G^′ does not alter the order of the nodes above v, which are then considered below indifferently of the tree they belong.

The proof is done with a recurrence over increasing time t ∈ {0,1,…,h(r(S^′))} for the subset of nodes V _t (S^′) ⊂ V(S^′). Recall that, in S^′ the height of a node u (denoted h(u)) is a valid time function (see Figure 1) and that u₁ is the child of u that is an ancestor of w (whereas u₂ is incomparable with w). □

Base case

For time t = 0, the possible events for the internal node u and any leaf x ∈ V₀(S^′) are , , and$\mathbb{T}\mathbb{L}$ (see the reconciliation model of Definition 1).

For each event$\mathbb{E}\in \{\mathbb{D},\mathbb{T}\}$,$c_{\mathbb{E}}(u,x)$ (resp.$c_{\mathbb{E}}^{\prime }(u,x)$) depends on the costs c(u _i ,y) (resp. c^′(u _i ,y)) over all children u _i  ∈ {u₁,u₂} and vertices y ∈ V _t (S^′) (see Definition 3). Since u₂ (resp. u₁) is incomparable to (resp. an ancestor of) w, Lemma 2 implies that c(u₂,y) = c^′ (u₂,y) and the assumption states that c(u₁,y) ≤ c^′(u₁,y).

That all costs used in the equation of$c_{\mathbb{E}}^{\prime }(u,x)$ are not lower than the corresponding costs in that of$c_{\mathbb{E}}(u,x)$ leads to properties listed in the following remark.

Inductive step

For a height 0 ≤ t < h(S), we now suppose that the expected property c(u,y) ≤ c^′(u,y) holds for all vertices y ∈ V _t (S^′) and prove that it still holds for any vertex x∈V_t+1(S). , , , ,$\mathbb{S}\mathbb{L}$, and$\mathbb{T}\mathbb{L}$ are the possible events for node u and vertex x. Following exactly the same arguments as in the base case, Remark 1 ( and ) and Remark 2 ($\mathbb{T}\mathbb{L}$) still hold for the current time (t+1).

The dependencies of the corresponding cost for , , and$\mathbb{S}\mathbb{L}$ events are as follows:$c_{\mathbb{S}}(u,x)$ depends on the costs c(u _i ,x _i ) for u _i  ∈ {u₁,u₂} and x _i  ∈ {x₁,x₂}, with x _i  ∈ V _t (S^′);$c_{\varnothing }(u,x)$ on c(u,x₁), with x₁ ∈ V _t (S^′); and$c_{\mathbb{S}\mathbb{L}}(u,x)$ on c(u,x _i ) for x _i  ∈ {x₁,x₂}, with x _i  ∈ V _t (S^′). The same dependencies apply for$c_{\mathbb{S}}^{\prime }(u,x)$,$c_{\varnothing }^{\prime }(u,x)$, and$c_{\mathbb{S}\mathbb{L}}^{\prime }(u,x)$. Recall that u₂ (resp. u₁) is incomparable to (resp. an ancestor of) w and that Lemma 2 (resp. the assumption) implies that c(u₂,x _i ) = c^′(u₂,x _i ) (resp. c(u₁,x _i ) ≤ c^′(u₁,x _i )) for each x _i  ∈ {x₁,x₂}. Moreover, the inductive hypothesis states that c(u,x _i ) ≤ c^′(u,x _i ) holds for each child x _i  of x since x _i  ∈ V _t (S^′). For each event$\mathbb{E}\in \{\mathbb{S},\varnothing ,\mathbb{S}\mathbb{L}\}$, that all costs used in the equation of$c_{\mathbb{E}}^{\prime }(u,x)$ are not lower than the corresponding costs used in the equation of$c_{\mathbb{E}}(u,x)$ leads to the following result: