D. Sankoff, Minimal mutation trees of sequences, SIAM J. Appl. Math. 28 (1) (1975) 35--42.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....which minimize the cost of the tree, defined as P (S i ;S j )2T d(S i ; S j ) When T is a star, the problem is called a Steiner problem, and the optimal sequence for the center is called the Steiner sequence for the leaves. The first exact algorithm for tree alignment was proposed by Sankoff in [18], and is based on dynamic programming. Later Altschul and Lipman [1] introduced some bounding rules to reduce the size of the dynamic programming lattice. Due to the prohibitive worst case complexity of exact methods, approximation algorithms for this problem were devised, by Jiang, Lawler and ....

D. Sankoff, Minimal mutation trees of sequences, SIAM J. Applied Math. 28(1) (1975) 35--42

....such as searching for highly conserved subregions among a set of biological sequences and inferring the evolutionary history of a family of sequences [5, 24] Many versions have been proposed [1, 5, 24] Tree alignment is one of the most famous versions. It was first proposed by Sankoff in [14]. For tree alignment, we are given k sequences and a tree Tree of k leaves, each of which is labeled with a unique given sequences. The goal is to construct a sequence for each internal node in Tree such that the cost of the tree is minimized. The cost of an edge in a tree is defined as the edit ....

....given sequences and T be the given topology. Recall that, for fixed topology alignment we use d[i 1 ; i 2 ; i k ] to indicate the cost of the alignment for subsequences s j [1; i j ] s (j = 1; 2; k) where s j [1; i j ] is the subsequence of s j containing the first i j characters [14, 16]. Our algorithm for the new problem is similar. We consider different configurations of the present column in the alignment. For the recombination nodes in T , we can choose one of the parents in the construction of the present column. After we make a choice, the topology becomes a binary tree. ....

D. Sankoff, Minimal mutation trees of sequences, SIAM J. Applied Math. 28(1975), 35-42.

....1 Introduction Multiple sequence alignment (MSA) is important in functional, structural and evolutionary studies of sequence data. Much research has focussed on the formal study of MSA as an optimization problem and several optimization criteria have been discussed at length in the literature [8, 34, 42, 44, 58, 65, 75, 85]. In addition, many software tools for constructing MSA s are available, mostly based on heuristics although some use exact or branch and bound techniques (see [16, 54] for surveys. The concept of an evolutionary tree is a widely used model for MSA, where the tree encodes the historical ....

....sum of the costs of all unordered pairs in the column d SP (a 1j : a kj ) X p q ffi(a pj ; a qj ) where ffi(x; y) is the cost associated with aligning two symbols x and y in a pairwise alignment. This definition is mathematically natural but not biologically intuitive. Tree alignment [65, 66] is based on the assumption that the residues in the columns of the multiple sequence alignment share an evolutionary history and that this history can be expressed as a tree. Under this model, a column is scored by computing the cost of the underlying tree. The score of the tree expresses the ....

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D. Sankoff. Minimal mutation trees of sequences. Journal of Applied Mathematics, 28:443--453, 1975.

....of the problem is to ask for a set of sequences to assign to the inner nodes of this phylogenetic tree such that the sum of the alignment distances along all edges of the tree is minimized. This version of the problem is called Tree Alignment Problem and has been introduced by David Sankoff [11]. However, frequently the phylogenetic tree is not known and should be derived from the set of sequences, too. The resulting problem is called Generalized Tree Alignment Problem: Given a set of sequences find a tree such that with suitably chosen ancestral sequences at the inner nodes the sum of ....

D. Sankoff. Minimal Mutation Trees of sequences. SIAM Journal of Applied Mathematics 28:35--42, 1975.

....4. Enlarging an existing candidate set guarantees to improve the final score, and including all conceivable sequences as candidates yields 2 an optimal assignment. This has been proved for the Candidate Heuristic scheme by Sankoff and Rousseau [3] for metric distances between sequences. Sankoff [4] worked this out for the Hamming metric on sequences. While, in certain settings sequence can be discarded from the candidate set without sacrificing optimality, in the case of sequences, this is not easily possible. On the other hand, it is practically infeasible to include all conceivable ....

....be decomposed into columns and thus, the candidate sets assigned to inner nodes in the Candidate Heuristic consist of whole sequences instead of single letters. Optimal Algorithms. The Tree alignment problem was proved NP hard by Wang and Jiang [11] The dynamic programming algorithm of Sankoff [4] solves the problem for unit mutation and indel costs, at the expense of run time that is exponential in the number of input sequences. Subsequent extensions include arbitrary mutation costs [8] and the simultaneous prediction of the inner sequences with respect to their three dimensional RNA ....

D. Sankoff, Minimal mutation trees of sequences, SIAM J. Appl. Math., 28 (1975), 35--42.

....phylogenetic trees, one objective that has been proposed and studied in the literature before [1, 3, 4, 5] is the sum of the costs of the tree edges. Most of this work was performed while the authors were at the Department of Computer Science, University of California at Davis. Sankoff [2] introduced the problem and proposed an algorithm to compute the optimal tree alignment using dynamic programming. Sankoff, Cedergren and Laplame [3] gave local improvement algorithms that start with a heuristic alignment labeling internal nodes with leaf sequences and improves these alignments by ....

....algorithm, completing the proof of Theorem 2. Note that this argument specialized to a star gives a performance ratio of two. Dynamic programming While exact algorithms for alignment via a known tree to minimize the total cost of the edges in the tree have been designed using dynamic programming [2], it is not obvious how such an algorithm can be designed for our bottleneck objective. We first sketch an algorithm for a star tree when indels are not permitted in computing the edit distance. More precisely, we are given a star tree with k sequences each of length exactly n and we wish to ....

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D. Sankoff, "Minimal mutation trees of sequences," SIAM J. Appl. Math., 28(1), 35-42, (1975).

....for a new alignment. One of the earliest formalizations of multiple sequence alignment essentially defines it as the problem of reconstructing ancient predecessors to contemporary sequences when the topology (i.e. the branching pattern) of a tree describing the evolution of the sequences is given [28]. The edit distances between the sequences at the nodes of this tree define the length of the edges. The problem is to choose the ancient sequences such that the overall length of the tree is minimized. This formulation is called tree alignment [2] It is one of two commonly used formulations of ....

....perform fairly well [19] Solving the parsimony problem by the Fitch Hartigan algorithm, i.e. given tree topology and alignment, finding the assignments for the inner nodes that make the tree as short as possible, is easy. Without the given alignment the problem has been introduced by Sankoff [28] in one of the earliest papers on multiple sequence alignment. He assumes that a tree topology and a set of unaligned sequences are given. The problem is to assign sequences to the inner nodes such that the sum of the implied edge lengths is minimal. Note that given a tree with all nodes annotated ....

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D. Sankoff. Minimal mutation trees of sequences. SIAM Journal of Applied Mathematics, 28:35--42, 1975.

....interior nodes, can be unrooted or rooted if the evolutionary origin is known, and usually has internal nodes of degree 3. Over the past few decades, many different objective criteria and algorithms for reconstructing phylogenies have been developed, including (not exhaustively) parsimony [6, 9, 28], compatibility [22] distance [10, 27] and maximum likelihood [6, 7, 2] The outcomes of these methods usually depend on the data and the amount of computational resources applied. As a result, in practice they often lead to different trees on the same set of species [18] It is thus of interest ....

D. Sankoff, Minimal mutation trees of sequences, SIAM J. Appl. Math., 28(1975) 35-42.

....interior nodes, can be unrooted or rooted if the evolutionary origin is known, and usually has internal nodes of degree 3. Over the past few decades, many different objective criteria and algorithms for reconstructing phylogenies have been developed, including (not exhaustively) parsimony [6, 9, 27], compatibility [22] distance [10, 26] and maximum likelihood [6, 7, 2] The outcomes of these methods usually depend on the data and the amount of computational resources applied. As a result, in practice they often lead to different trees 2 on the same set of species [18] It is thus of ....

D. Sankoff, Minimal mutation trees of sequences, SIAM J. Appl. Math., 28(1975) 35-42.

....a new alignment. One of the earliest formalizations of multiple sequence alignment essentially defines it as the problem of reconstructing ancient predecessors to contemporary sequences when the topology (i.e. the branching pattern) of a tree describing the evolution of the sequences is given [24]. The edit distances between the sequences at the nodes of this tree define the length of the edges. The problem is to choose the ancient sequences such that the overall length of the tree is minimized. This formulation is called tree alignment [2] It is one of 2 commonly used formulations of ....

....perform fairly well [17] Solving the parsimony problem by the Fitch Hartigan algorithm, i.e. given tree topology and alignment, finding the assignments for the inner nodes that make the tree as short as possible, is easy. Without the given alignment the problem has been introduced by Sankoff [24] in one of the earliest papers on multiple sequence alignment. He assumes that a tree topology and a set of unaligned sequences are given. The problem is to assign sequences to the inner nodes such that the sum of the implied edge lengths is minimal. Note that given a tree with all nodes annotated ....

[Article contains additional citation context not shown here]

D. Sankoff. Minimal mutation trees of sequences. SIAM Journal of Applied Mathematics, 28:35--42, 1975.

....nodes, is either unrooted or rooted (if the evolutionary origin is known) and usually all of whose internal nodes have degree 3. Over the past few decades, many different objective criteria and algorithms for reconstructing phylogenies have been developed, including (not exhaustively) parsimony [8, 11, 31], compatibility [25] distance [12, 30] and maximum likelihood [1, 8, 9] The outcomes of these methods usually depend on the data and the amount of computational resources applied. As a result, in practice they often lead to different trees on the same set of species [24] It is thus of interest ....

D. Sankoff, Minimal mutation trees of sequences, SIAM J. Appl. Math., 28(1975) 35-42.

....internal nodes, can be unrooted or rooted if the evolutionary origin is known, and usually has internal nodes of degree 3. Over the past few decades, many different objective criteria and algorithms for reconstructing phylogenies have been developed, including (not exhaustively) parsimony [6, 9, 22], compatibility [17] distance [10, 21] and maximum likelihood [6, 7, 1] The outcomes of these methods usually depend on the data and the amount of computational resources applied. As a result, in practice they often lead to different trees on the same set of species [16] It is thus of interest ....

D. Sankoff, Minimal mutation trees of sequences, SIAM J. Appl. Math., 28(1975) 35-42.

....internal nodes, can be unrooted or rooted if the evolutionary origin is known, and usually has internal nodes of degree 3. Over the past few decades, many different objective criteria and algorithms for reconstructing phylogenies have been developed, including (not exhaustively) parsimony [4, 7, 20], compatibility [15] distance [8, 19] and maximum likelihood [4, 5, 1] The outcomes of these methods usually depend on the data and the amount of computational resources applied. As a result, in practice they often lead to different trees on the same set of species [14] It is thus of interest ....

D. Sankoff, Minimal mutation trees of sequences, SIAM J. Appl. Math., 28(1975) 35-42.

....P 1 [j] P k [j] is the sum of the costs of all unordered pairs in the column d SP ( P 1 [j] P k [j] X p q ffi( P p [j] P q [j] for some binary cost function ffi(x; y) This definition is mathematically natural but not biologically intuitive. Tree alignment [30, 31] is based on the assumption that the residues in the columns of the multiple sequence alignment share an evolutionary history and that this history can be expressed as a single tree for all columns. Under this model, a column is scored by computing the cost of the underlying tree. The score of the ....

....on the order of 100 base pairs in length. An optimal alignment of k species can be obtained using the obvious dynamic program in O(2 k N k ) evaluations of the cost function d using O(N k ) space. An exact tree alignment algorithm for a given tree topology has been presented by Sankoff [30]. This requires O(M(2N) k ) steps where M is the number of internal nodes, k is the number sequences and N is the maximum length over all sequences. Approximation algorithms for both SP and tree alignment include those in [4, 15, 25, 26] Notably, a polynomial time approximation scheme, ....

D. Sankoff. Minimal mutation trees of sequences. Journal of Applied Mathematics, 28:443--453, 1975.

....of degree 3. Reconstructing the correct evolutionary tree for a set of species is one of the fundamental yet difficult problems in evolutionary genetics. Over the past few decades, many approaches for reconstructing evolutionary trees have been developed, including (not exhaustively) parsimony [12, 15, 39], compatibility [32] distance [16, 38] maximum likelihood [12, 13, 3] The outcomes of these methods usually depend on the data and the amount of computational resources applied. As a result, in practice they often lead to different trees on the same set of species [28] It is thus of interest ....

D. Sankoff, Minimal mutation trees of sequences, SIAM J. Appl. Math., 28, pp. 35-42, 1975.

.... lengths, that are similar in a certain way [20] 21] 31] It is a different approach from the more classical one of optimizing the value of a function of similarity defined between strings in view of obtaining a multiple alignment of the corresponding macromolecules [2] 8] 9] 15] 19] 28] [39] [40] Obtaining a list of common similar words may lead in a second step to such an alignment but it can also represent an end in itself. We stop here at this first end. It is a multiple comparison we are interested in, not a multiple global alignment. We wish to synthetize a series of works [35] ....

D. Sankoff. Minimal mutation trees of sequences. SIAM J. Appl. Math., 28:35--42, 1975.

....the sequences labeling its endpoints, and the objective is to minimize the sum of the edge lengths. For reasons that will be explained shortly, this problem is usually called multiple sequence alignment under a fixed evolutionary tree, and is often abbreviated to simply tree alignment. Sankoff [16] initiated the formal study of tree alignment and gave an exact algorithm for the problem. By observing that for every solution there is a pairwise alignment on each tree edge that achieves the edge length and that any tree of pairwise alignments induces a multiple alignment of the leaf sequences, ....

....P to each element of S. In our application, the metric space is the set of all sequences under edit distance. In this context, a Steiner point for a set of k sequences of length at most n from an alphabet of size s can be computed in O(k2 k n k ) time, given O(ks k 1 ) time preprocessing [16]. R. Ravi and J.D. Kececioglu 4 sequence P that minimizes the sum P 1ik D(P; S i ) where D(x; y) denotes the edit distance between sequences x and y. Sequence P is called a Steiner sequence for the multiset. Approximation algorithm Our algorithm is parameterized by p, the maximum size of a ....

[Article contains additional citation context not shown here]

Sankoff, D. "Minimal mutation trees of sequences." SIAM Journal on Applied Mathematics 28:1, 35--42, 1975.

....we discuss each of them very briefly. See the excellent survey in [20, 72] for more details. Parsimony methods construct phylogenetic trees for the given sequences such that, in some sense, the total number of changes (i.e. base substitutions) or some weighted sum of the changes is minimized. See [18, 22, 62] for some of the papers in this direction. Distance methods [10, 23, 61] try to fit a tree to a matrix of pairwise distances between a set of n species. Entries in the distance matrices are assumed to represent evolutionary distance between species represented by the sequences in the tree, i.e. ....

....s 0 j (i) such that P (p;q)2E (s 0 p (i) s 0 q (i) is minimized. The score (s 0 1 (i) s 0 2 (i) s 0 k (i) of the i th column is thus defined as (s 0 1 (i) s 0 2 (i) s 0 k (i) X (p;q)2E (s 0 p (i) s 0 q (i) This measure has been discussed in [1, 4, 62, 63, 64]. Multiple sequence alignment with tree score is often referred to as tree alignment in the literature. Note that, a tree alignment induces a set of reconstructed sequences, each corresponding to an internal node. Thus, it is convenient to reformulate tree alignment as follows: Given a set X of k ....

D. Sankoff. Minimal mutation trees of sequences, SIAM Journal of Applied Mathematics, 28 (1975), pp. 35-42.

....of protein and of nucleic acid sequences are presented. 1 Introduction The study of molecular evolution, in particular the study of the evolution of DNA and amino acid sequences involves the solution of two complex optimization problems. The first one is the multiple sequence alignment problem ([11, 23], see[3] or [22] for review) the second is the reconstruction of a phylogenetic tree displaying the evolutionary relationships of the sequences (see [6] or [14] for review) Most commonly used approaches that reconstruct a tree rely on multiple sequence alignments as input data. Given a specific ....

D. Sankoff. Minimal mutation trees of sequences. SIAM Journal of Applied Mathematics, 28:35--42, 1975.

....At the low error rates of current practice, our alignment graphs have a regular underlying structure. When no error is present, the sequences are identical, and the alignment 12 This observation, expressed in different language, can be found in many papers. Perhaps the first occurrence is in [29]. 29 graph is a series of columns, each column a complete subgraph. When a rare error is present, its effect on this structure is to displace or delete some edges local to the defect. For such graphs, most edges in a pairwise trace (A;C) will coincide with the trace of A and C induced by ....

Sankoff, David. Minimal mutation trees of sequences. SIAM Journal on Applied Mathematics 28:1, 35--42, 1975.

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D. Sankoff, Minimal mutation trees of sequences, SIAM J. Appl. Math. 28 (1) (1975) 35--42.

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