Tree alignment

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In computational phylogenetics, tree alignment is the problem of producing a multiple sequence alignment, which can be used to analyze a set of sequences with an evolutionary relationship using a fixed tree. Essentially, tree alignment is an algorithm for optimizing phylogenetic tree by calculating the edit distance to achieve the minimum value. To be specific, phylogenetic tree shows how an evolutionary relationship between different species and taxa joined together are assumed to have the same ancestor.

Video Tree alignment

Definition

Formally, tree alignment is the following optimization problem.

Input: A set ${\displaystyle S}$ of sequences, a phylogenetic tree ${\displaystyle T}$ leaf-labeled by ${\displaystyle S}$ and an edit distance function ${\displaystyle d}$ between sequences,

Output: A labeling of the internal vertices of ${\displaystyle T}$ such that ${\displaystyle \Sigma _{e\in T}d(e)}$ is minimized, where ${\displaystyle d(e)}$ is the edit distance between the endpoints of ${\displaystyle e}$.

The task is NP-hard.

Maps Tree alignment

Background

Sequence alignment

In bioinformatics, the basic method of information processing is to contrast the sequence data. It is significant for when biologists use it to discover the function, structure and evolutionary information in biological sequences. From the sequence assembly, the phylogenetic analysis, the haplotype comparison, and the prediction of RNA structure are all based on sequence alignment, so the efficiency of sequence alignment (especially multiple sequence alignment) will directly affect the efficacy of solving these problems. Therefore, to design a rational and efficient sequence alignment algorithm becomes a very important branch of research in the field of bioinformatics.

Generally, sequence alignment means constructing a string from two or more given strings with the greatest similarity by adding, deleting letters, or adding a space for each string. The multiple sequence alignment problem is generally based on pairwise sequence alignment and currently, for a pairwise sequence alignment problem, biologists can use a dynamic programming approach to obtain its optimal solution. However, the multiple sequence alignment problem is still one of the more challenging problems in bioinformatics, because finding the optimal solution for multiple sequence alignment has been proved as a NP-complete problem so that only an approximate optimal solution can be obtained.

Edit distance

Edit distance measures the minimum operation number of character insertions, deletions, and substitutions that are required to transform one sequence u to the other sequence v when being operated on a pair of strings. The calculation of edit distance can be based on dynamic programming, and the equation is in O(|u|*|v|) time, where |u| and |v| are the lengths of u and v. Edit distance is the basic principle in computational biology, thus an efficient estimation of edit distance is essential. There are some functions to calculate edit distance, including "symmetrization" used for functions of hereditary properties. Because there are a series of functions being used to calculate edit distance, different functions may result in different results. Finding an optimal edit distance function seems essential for further explanation.

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The problem of tree alignment

Tree alignment problem is a NP-hard problem when we restrict its scoring mode and alphabet size, and it can be found as an algorithm, which is used to find the optimized solution. However, there is an exponential relationship between its efficiency and the number of sequences, which means that when the number of the sequence is very large, the computation time required to get results is an enormous figure, which is unacceptable. Using star alignment is faster than tree alignment to get the approximate optimized solution. However, whatever the degree of multiple-sequence similarity is, the time complexity of star alignment has a proportional relationship with the square of the sequence number and the square of the sequence average length. As usual, the sequence in MSA is so long that it is also inefficient or even unacceptable. Therefore, the challenge of reducing the time complexity to linear is one of the core issues in Tree alignment.

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Combinatorial optimization strategy

Combinatorial optimization is a good strategy to solve MSA problem. The idea of combinatorial optimization strategy is to transform the multiple sequence alignment into pair sequence alignment to solve this problem. Depending on its transformation strategy, the combinatorial optimization strategy can be divided into the tree alignment algorithm and the star alignment algorithm. For a given multi-sequence set ${\displaystyle S}$ ={${\displaystyle s_{1}}$,...,${\displaystyle s_{n}}$ }, find an evolutionary tree which has n leaf nodes and establishing one to one relationship between this evolutionary tree and the set S. By assigning the sequence to the internal nodes of the evolutionary tree we calculate the total score of each edge, and the sum of all edges' score is the score of the evolutionary tree. The aim of tree alignment is to find an assigned sequence, which can obtain a maximum score, and get the final matching result by the evolutionary tree and its nodes' assigned sequence. Star alignment can be seen as a special case of the tree alignment. When we use star alignment, the evolutionary tree has only one internal node and n leaf nodes. The sequence, which is assigned to the internal node, is called core sequence.

The keyword tree theory and the Aho-Corasick search algorithm

When we use combinatorial optimization strategy to transform the multiple sequence alignment into pair sequence alignment, the main problem is changed from "How to improve the efficiency of multiple sequence alignment" to "How to improve the efficiency of pairwise sequence alignment". The Keyword Tree Theory and Aho-Corasick search algorithm is an efficient approach to solve the pairwise sequence alignment problem. The aim of combining the keyword tree theory and Aho-Corasick search algorithm is to solve this kind of problem: for a given long string T and a short strings set ${\displaystyle P}$={${\displaystyle p_{1}}$,${\displaystyle p_{2}}$,... ,${\displaystyle p_{z}}$} (z?N,z>1), find the location of all ${\displaystyle P_{i}}$ in the T. We use keyword tree produced by set ${\displaystyle P}$, and then search in the T with this keyword tree by Aho-Corasick search algorithm. The total time complexity of using this method to find all ${\displaystyle P_{i}}$'s location in the T is O(m+n+k), where m=|T| (the length of T), n=?|${\displaystyle P_{i}}$| (the sum of all ${\displaystyle P_{i}}$'s length) and k means the sum of occurrence for all ${\displaystyle P_{i}}$ in the T.

Keyword tree theory

The keyword tree of the set ${\displaystyle P}$={${\displaystyle p_{1}}$,${\displaystyle p_{2}}$,... ,${\displaystyle p_{z}}$ } (z?N,z>1) is a rooted tree, whose root denoted by K, and this keyword tree satisfies:

(1): Each edge clearly demarcates one letter.

(2): Any two edges separated from the same node are to correspond to different letters.

(3) Each pattern ${\displaystyle P_{i}}$ (i=1,2,...,z) corresponds to a node ${\displaystyle v}$, and the path from the root K to the node ${\displaystyle v}$ can exactly correctly spell the string ${\displaystyle P_{i}}$.

For each leaf node of this K tree, it corresponds to one of certain patterns of set ${\displaystyle P}$.

Also, we use ${\displaystyle L(v)}$ to represent the STRING which is connected from the root node to the node ${\displaystyle v}$. We then use ${\displaystyle Lp(v)}$ to represent the length of the longest suffix (also, this suffix is the prefix of one of patterns in the set ${\displaystyle P}$). Searching this prefix from the root node in the keyword tree, and the last node denoted by ${\displaystyle n_{v}}$ when the search is over. When ${\displaystyle Lp(v)}$=0, ${\displaystyle n_{v}}$ =K. The ordered pair (${\displaystyle v}$, ${\displaystyle n_{v}}$) is called a failure link.

For example, the set ${\displaystyle P}$={potato, tattoo, theater, other}, and the keyword tree is shown on the right. Obviously, in that example if ${\displaystyle L(v)}$=potat, then ${\displaystyle Lp(v)}$=|tat|=3, and the failure link of the node ${\displaystyle v}$ is shown in that figure.

To establish a failure link is the key to improve the time complexity of the Aho-Corasick algorithm. It can be used to reduce the original polynomial time to the linear time for searching. Therefore, the core of keyword tree theory is to find all failure links (which also means finding all ${\displaystyle n_{v}}$) of a keyword tree in the linear time. We assume that we find every ${\displaystyle n_{v}}$ of all nodes ${\displaystyle v}$ whose distance from the root node is less than or equal k, and now we are seeking the ${\displaystyle n_{v}}$ of the node ${\displaystyle v}$ whose distance from the root node is k + 1. Its parent node is ${\displaystyle v'}$, and the letter represented by the node ${\displaystyle v}$ and ${\displaystyle v'}$, is x.

(1): If the next letter of the node ${\displaystyle n_{v}'}$ is x, we set the other node of this edge as ${\displaystyle w}$, and ${\displaystyle n_{v}}$=${\displaystyle w}$.

(2): If all letters is not x by searching all edges between ${\displaystyle n_{v}'}$ and its child nodes, ${\displaystyle L(n_{v})}$ is a suffix of ${\displaystyle L(n_{v}')}$ plus x. Because this suffix matches the STRING beginning with the root node (similar to prefix), we can detect if there is x after ${\displaystyle n_{v}'}$ or not. And if not, continue this process until we find x or find the root node.

Aho-Corasick search algorithm

After establishing all failure links in the keyword tree, we use the Aho-Corasick search algorithm to find the locations of all ${\displaystyle P_{i}}$ (i=1,2,...,z) in the linear time. In this step, the time complexity is O(m+k).

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Other strategies

In MSA, DNA, RNA, and proteins sequences are usually generated and they are assumed to have an evolutionary relationship. By comparing generated maps of RNA, DNA, and sequences from evolutionary families, people can assess conservation of protein, find functional gene domains by comparing differences between evolutionary sequences. Generally, heuristic algorithms and tree alignment graphs are also adopted to solve multiple sequence alignment problems.

Heuristic algorithm

Generally heuristic algorithms rely on the iterative strategy, which is to say that based on a comparison method, optimizing the results of multiple sequence alignment by the iterative process. Davie M. proposed using particle swarm optimization algorithm to solve the multiple sequence alignment problem; Ikeda Takahiro proposed a heuristic algorithm which is based on A* search algorithm; E. Birney first proposed using hidden Markov model to solve the multiple sequence alignment problem; and many other biologists use genetic algorithm to solve it. All these algorithms generally are robust and insensitive to the number of sequences, but they also have shortcomings. For example, the results from particle swarm optimization algorithm is unstable and its merits depend on the selection of random numbers, the runtime of A * search algorithm is too long, and the genetic algorithm is easy to fall into local excellent.

Tree alignment graph

Roughly, tree alignment graph aims to align trees into a graph and finally synthesize them to develop statistics. In biology, tree alignment graphs (TAGs) are used to remove the evolutionary conflicts or overlapping taxa from sets of trees and can be queried to explore uncertainty and conflict. By integrating methods of aligning, synthesizing and analyzing, the TAG aims to solve the conflicting relationships and partial overlapping taxon sets obtained from a wide range of sequence. Also, tree alignment graph serves as a fundamental approach for supertree and grafting exercise, which have been successfully tested to construct supertrees by Berry et al. Because the transformation from trees to a graph contain similar nodes and edges from their source trees, TAGs can also provide extraction of original source trees for further analysis. TAG is a combination of a set of aligning trees, it can store conflicting hypotheses evolutionary relationship and synthesize the source trees to develop evolutionary hypotheses. Therefore, it is a basic method to solve other alignment problems.

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See also

• Generalized tree alignment

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References

Source of the article : Wikipedia