Institute of Fundamental Sciences

An approximate nested tandem repeat (NTR) in a string T is a complex repetitive structure consisting of many approximate copies of two substrings x and X (''motifs'') interspersed with one another. NTRs fall into a class of repetitive structures broadly known as subrepeats. NTRs have been found in real DNA sequences and are expected to be important in evolutionary biology, both in understanding evolution of the ribosomal DNA (where NTRs can occur), and as a potential marker in population genetic and phylogenetic studies. This article describes an alignment algorithm for the verification phase of the software tool NTRFinder developed for database searches for NTRs. When the search algorithm has located a subsequence containing a possible NTR, with motifs X and x, a verification step aligns this subsequence against an exact NTR built from the templates X and x, to determine whether the subsequence contains an approximate NTR and its extent. This article describes an algorithm to solve this alignment problem in O(jTj(jXj + jxj)) space and time. The algorithm is based on Fischetti et al.'s wrap-around dynamic programming.

An approximate nested tandem repeat (NTR) in a string T is a complex repetitive structure consisting of many approximate copies of two substrings x and X (''motifs'') interspersed with one another. NTRs fall into a class of repetitive structures broadly known as subrepeats. NTRs have been found in real DNA sequences and are expected to be important in evolutionary biology, both in understanding evolution of the ribosomal DNA (where NTRs can occur), and as a potential marker in population genetic and phylogenetic studies. This article describes an alignment algorithm for the verification phase of the software tool NTRFinder developed for database searches for NTRs. When the search algorithm has located a subsequence containing a possible NTR, with motifs X and x, a verification step aligns this subsequence against an exact NTR built from the templates X and x, to determine whether the subsequence contains an approximate NTR and its extent. This article describes an algorithm to solve this alignment problem in O(jTj(jXj + jxj)) space and time. The algorithm is based on Fischetti et al.'s wrap-around dynamic programming.

In many applications of tandem repeats the outcome depends critically on the choice of boundaries (beginning and end) of the repeated motif: for example, different choices of pattern boundaries can lead to different duplication history trees. However, the best choice of boundaries or parsing of the tandem repeat is often ambiguous, as the flanking regions before and after the tandem repeat often contain partial approximate copies of the motif, making it difficult to determine where the tandem repeat (and hence the motif) begins and ends. We define the parsing problem for tandem repeats to be the problem of discriminating among the possible choices of parsing. In this paper we propose and compare three heuristic methods for solving the parsing problem, under the assumption that the parsing is fixed throughout the duplication history of the tandem repeat. The three methods are PAIR, which minimises the number of pairs of common adjacent mutations which span a boundary; VAR, which minimises the total number of variants of the motif; and MST, which minimises the length of the minimum spanning tree connecting the variants, where the weight of each edge is the Hamming distance of the pair of variants. We test the methods on simulated data over a range of motif lengths and relative rates of substitutions to duplications, and show that all three perform better than choosing the parsing arbitrarily. Of the three MST typically performs the best, followed by VAR then PAIR.

1. ABSTRACT 1.1. Motivation: We introduce the algorithm NTRFinder to find a complex repetitive structure in DNA we call a nested tandem repeat (NTR). An NTR is a recurrence of two distinct tandem motifs interspersed with each other. We propose that nested tandem repeats can be used as phylogenetic and population markers. A major issue is the parsing problem, which is the problem of finding the optimal boundaries of the respective motifs.

1. ABSTRACT 1.1. Motivation: We introduce the algorithm NTRFinder to find a complex repetitive structure in DNA we call a nested tandem repeat (NTR). An NTR is a recurrence of two distinct tandem motifs interspersed with each other. We propose that nested tandem repeats can be used as phylogenetic and population markers. A major issue is the parsing problem, which is the problem of fi nding the optimal boundaries of the respective motifs. There has been little discus sion about determining the best boundaries of the repeated motifs in tandem repeats. For nested tandem repeats, parsing is a significant issue. 1.2. Results: An algorithm for finding Nested Tandem Repeats and a criterio n for solving the parsing problem are introduced. We have tested our algorithm on both real and simulated data, and present some real nested tandem repeats of interest. We discuss how one of these may assist in determining the cultivation prehistory of the ancient staple food crop taro (Colocasia esculen...

In many applications of tandem repeats the outcome depends critically on the choice of boundaries (beginning and end) of the repeated motif: for example, different choices of pattern boundaries can lead to different duplication history trees. However, the best choice of boundaries or parsing of the tandem repeat is often ambiguous, as the flanking regions before and after the tandem repeat often contain partial approximate copies of the motif, making it difficult to determine where the tandem repeat (and hence the motif) begins and ends. We define the parsing problem for tandem repeats to be the problem of discriminating among the possible choices of parsing. In this paper we propose and compare three heuristic methods for solving the parsing problem, under the assumption that the parsing is fixed throughout the duplication history of the tandem repeat. The three methods are PAIR, which minimises the number of pairs of common adjacent mutations which span a boundary; VAR, which minimises the total number of variants of the motif; and MST, which minimises the length of the minimum spanning tree connecting the variants, where the weight of each edge is the Hamming distance of the pair of variants. We test the methods on simulated data over a range of motif lengths and relative rates of substitutions to duplications, and show that all three perform better than choosing the parsing arbitrarily. Of the three MST typically performs the best, followed by VAR then PAIR.