关于DCJ下三个基因组的两两距离和中位数得分
On pairwise distances and median score of three genomes under DCJ
摘要：
在比较基因组学中，两个基因组之间的重排距离（等于将它们转化为单个基因组所需的最小数量的基因组重排）通常用于测量它们的进化距离。将这一指标推广到三个基因组被称为中位数得分（而得到的基因组被称为中位数基因组）。与两个基因组之间的重排距离可以在线性时间内计算相反，计算三个基因组的中值得分是NP困难的。这激发了对更简单和更快的中值分数近似的追求，其中最自然的似乎是成对距离的一半和，实际上代表了中值分数的下限。在这项工作中，我们研究的关系和三个基因组之间的成对距离和他们的中位数得分双切割和连接（DCJ）重排模型的相互作用。最值得注意的是，我们表明，虽然重排可能会改变成对距离的总和最多2（从而改变下限最多1），即使是在这方面最“强大”的重排，增加下限1（通过移动一个基因组远离其他两个基因组中的每一个），我们称之为强，不一定会影响中位数得分。这一观察结果意味着，这两个措施并不像人们的直觉所暗示的那样相互关联。我们进一步证明，中位数得分达到下界正好在三个基因组，可以从一个单一的基因组与强重排。虽然因子为2/3的成对距离之和代表了中位数得分的上限，但其紧密程度仍不清楚。尽管如此，我们表明，中位数得分和它的下限的差异是不受限制的常数。
英文摘要：
In comparative genomics, the rearrangement distance between two genomes (equal the minimal number of genome rearrangements required to transform them into a single genome) is often used for measuring their evolutionary remoteness. Generalization of this measure to three genomes is known as the median score (while a resulting genome is called median genome). In contrast to the rearrangement distance between two genomes which can be computed in linear time, computing the median score for three genomes is NP-hard. This inspires a quest for simpler and faster approximations for the median score, the most natural of which appears to be the halved sum of pairwise distances which in fact represents a lower bound for the median score. In this work, we study relationship and interplay of pairwise distances between three genomes and their median score under the model of Double-Cut-and-Join (DCJ) rearrangements. Most remarkably we show that while a rearrangement may change the sum of pairwise distances by at most 2 (and thus change the lower bound by at most 1), even the most "powerful" rearrangements in this respect that increase the lower bound by 1 (by moving one genome farther away from each of the other two genomes), which we call strong, do not necessarily affect the median score. This observation implies that the two measures are not as well-correlated as one's intuition may suggest. We further prove that the median score attains the lower bound exactly on the triples of genomes that can be obtained from a single genome with strong rearrangements. While the sum of pairwise distances with the factor 2/3 represents an upper bound for the median score, its tightness remains unclear. Nonetheless, we show that the difference of the median score and its lower bound is not bounded by a constant.
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