A non-linear optimization procedure to estimate distances and instantaneous substitution rate matrices under the GTR model
A non-linear optimization procedure to estimate distances and instantaneous substitution rate matrices under the GTR model
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DOI：
10.1093/bioinformatics/btk001
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发表时间：
2006-03-15
期刊：
影响因子：
5.8
通讯作者：
Milinkovitch, MC
中科院分区：
文献类型：
--
作者：
Catanzaro, D;Pesenti, R;Milinkovitch, MC
动机：一般时间可逆（GTR）模型是最流行的核苷酸取代模型之一，因为它在数学易处理性和生物现实性之间构成了良好的权衡。然而，当它被应用于推断进化距离和/或瞬时速率矩阵，GTR模型似乎更容易不适用于更严格的时间可逆模型。虽然它以前已经注意到，棘手的原因是由计算的对数的矩阵特征在于负特征值的不可能性所造成的，这个问题还没有得到进一步的调查。结果：在这里，我们正式的数学条件的特征，并讨论其生物学解释，这导致GTR模型的不适用性。我们调查之间的关系，一方面，负特征值的发生，另一方面，序列长度和序列的分歧。然后，我们提出了一个可能的重新制定以前的程序在一个非线性优化问题。我们分析研究我们的方法估计的进化距离和转移概率矩阵的效果。最后，我们提供了我们提出的解决方案的优点的分析。最后给出了一个算例。
Motivation: The general-time-reversible (GTR) model is one of the most popular models of nucleotide substitution because it constitutes a good trade-off between mathematical tractability and biological reality. However, when it is applied for inferring evolutionary distances and/or instantaneous rate matrices, the GTR model seems more prone to inapplicability than more restrictive time-reversible models. Although it has been previously noted that the causes for intractability are caused by the impossibility of computing the logarithm of a matrix characterised by negative eigenvalues, the issue has not been investigated further.Results: Here, we formally characterize the mathematical conditions, and discuss their biological interpretation, which lead to the inapplicability of the GTR model. We investigate the relations between, on one hand, the occurrence of negative eigenvalues and, on the other hand, both sequence length and sequence divergence. We then propose a possible re-formulation of previous procedures in terms of a non-linear optimization problem. We analytically investigate the effect of our approach on the estimated evolutionary distances and transition probability matrix. Finally, we provide an analysis on the goodness of the solution we propose. A numerical example is discussed.