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<Paper uid="W06-1508">
  <Title>Stochastic Multiple Context-Free Grammar for RNA Pseudoknot Modeling</Title>
  <Section position="3" start_page="0" end_page="57" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> Non-coding RNAs fold into characteristic structures determined by interactions between mostly Watson-Crick complementary base pairs. Such a base paired structure is called the secondary structure. Pseudoknot (Figure 1 (a)) is one of the typical substructures found in the secondary structures of several RNAs, including rRNAs, tmRNAs and viral RNAs. An alternative graphic representation of a pseudoknot is arc depiction where arcs connect base pairs (Figure 1 (b)). It has been recognized that pseudoknots play an important role in RNA functions such as ribosomal frameshifting and regulation of translation.</Paragraph>
    <Paragraph position="1"> Many attempts have so far been made at modeling RNA secondary structure by formal grammars. In a grammatical approach, secondary structure prediction can be viewed as parsing problem. However, there may be many different derivation trees for an input sequence. Thus, it is necessary to have a method of extracting biologically realistic  derivation trees among them. One solution to this problem is to extend a grammar to a probabilistic model and find the most likely derivation tree, and another is to take free energy minimization into account. Eddy and Durbin (1994), and Sakakibara et al. (1994) modeled RNA secondary structure without pseudoknots by using stochastic context-free grammars (stochastic CFGs or SCFGs). For pseudoknotted structure (Figure 1 (a)), however, another approach has to be taken since a single CFG cannot represent crossing dependencies of base pairs in pseudoknots (Figure 1 (b)) for the lack of generative power. Brown and Wilson (1996) proposed a model based on intersections of SCFGs to describe RNA pseudoknots. Cai et al. (2003) introduced a model based on parallel communication grammar systems using a single CFG synchronized with a number of regular grammars.</Paragraph>
    <Paragraph position="2"> Akutsu (2000) provided dynamic programming algorithms for RNA pseudoknot prediction without using grammars. On the other hand, several grammars have been proposed where the grammar itself can fully describe pseudoknots. Rivas and Eddy (1999, 2000) provided a dynamic programming  algorithm for predicting RNA secondary structure including pseudoknots, and introduced a new class of grammars called RNA pseudoknot grammars (RPGs) for deriving sequences with gap. Uemura et al. (1999) defined specific subclasses of tree adjoining grammars (TAGs) named SL-TAGs and extended SL-TAGs (ESL-TAGs) respectively, and predicted RNA pseudoknots by using parsing algorithm of ESL-TAG. Matsui et al. (2005) proposed pair stochastic tree adjoining grammars (PSTAGs) based on ESL-TAGs and tree automata for aligning and predicting pseudoknots, which showed good prediction accuracy. These grammars have generative power stronger than CFGs and polynomial time algorithms for parsing problem. null In our previous work (Kato et al., 2005), we identified RPGs, SL-TAGs and ESL-TAGs as subclasses of multiple context-free grammars (MCFGs) (Kasami et al., 1988; Seki et al., 1991), which can model RNA pseudoknots, and showed a candidate subclass of the minimum grammars for representing pseudoknots. The generative power of MCFGs is stronger than that of CFGs and MCFGs have a polynomial time parsing algorithm like the CYK (Cocke-Younger-Kasami) algorithm for CFGs. In this paper, we extend the above candidate subclass of MCFGs to a probabilistic model called a stochastic MCFG (SM-CFG). We present a polynomial time parsing algorithm for finding the most probable derivation tree, which is applicable to RNA pseudoknot prediction. In addition, we mention a probability parameter estimation method based on the EM (expectation maximization) algorithm. Finally, we show some experimental results on pseudoknot prediction for three RNA families using SMCFG algorithm, which show good prediction accuracy.</Paragraph>
  </Section>
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