Reconciling a non-binary gene tree with a binary species tree can be done efficiently in the absence of horizontal gene transfers, but becomes NP-hard in the presence of gene transfers. Here, we focus on the special case of endosymbiotic gene transfers (EGT), i.e. transfers between the mitochondrial and nuclear genome of the same species. More precisely, given a multifurcated (non-binary) gene tree with leaves labeled 0 or 1 depending on whether the corresponding genes belong to the mitochondrial or nuclear genome of the corresponding species, we investigate the problem of inferring a most parsimonious Duplication, Loss and EGT (DLE) Reconciliation of any binary refinement of the tree. We present a general two-steps method: ignoring the 0–1 labeling of leaves, output a binary resolution minimizing the Duplication and Loss (DL) Reconciliation and then, for such resolution, assign a known number of 0s and 1s to the leaves in a way minimizing EGT events. While the first step corresponds to the well studied non-binary DL-Reconciliation problem, the complexity of the label assignment problem corresponding to the second step is unknown. We show that this problem is NP-complete, even when the tree is restricted to a single polytomy, and even if transfers can occur in only one direction. We present a general algorithm solving each polytomy separately, which is shown optimal for a unitary cost of operation, and a polynomial-time algorithm for solving a polytomy in the special case where genes are specific to a single genome (mitochondrial or nuclear) in all but one species. This work represents the first algorithmic study for reconciliation with endosymbiotic gene transfers in the case of a multifurcated gene tree.

中文翻译：

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内共生基因转移非二叉树协调的复杂性

在没有水平基因转移的情况下，可以有效地协调非二元基因树与二元物种树，但在存在基因转移的情况下就变得NP困难了。在这里，我们重点关注内共生基因转移（EGT）的特殊情况，即同一物种的线粒体和核基因组之间的转移。更准确地说，给定一个多分叉（非二元）基因树，其叶子标记为 0 或 1，具体取决于相应基因是否属于相应物种的线粒体或核基因组，我们研究推断最简约的重复、丢失和EGT (DLE) 树的任何二元细化的协调。我们提出了一种通用的两步方法：忽略叶子的 0-1 标签，输出最小化重复和丢失 (DL) 协调的二进制分辨率，然后，对于这样的分辨率，将已知数量的 0 和 1 分配给叶子一种最小化 EGT 事件的方法。虽然第一步对应于经过充分研究的非二元 DL 协调问题，但第二步对应的标签分配问题的复杂性未知。我们证明这个问题是 NP 完全的，即使树被限制为单一的多分枝，即使转移只能在一个方向上发生。我们提出了一种分别解决每个多项式的通用算法，该算法对于单一操作成本而言是最佳的，并且提出了一种多项式时间算法，用于在基因特定于单个基因组（线粒体或核）的特殊情况下解决多项式时间算法但只有一种。这项工作代表了第一个在多分叉基因树的情况下与内共生基因转移相协调的算法研究。