December 2001
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11 Reads
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3 Citations
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Publications (58)
June 2001
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6 Reads
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4 Citations
We formulate the GENERALIZED PATTERN MATCHING problem, a natural extension of string searching capturing regularities across scale. The special case of UNAVOIDABILITY TESTING is ob- tained for pure generalized patterns by fixing an appropriate family of text strings - the Zimin words. We investigate the complexity of this restricted decision problem. Although the efficiency of standard string searching is well-known, determining the occurrence of generalized pat- terns in Zimin words does not appear so tractable. We provide an expo- nential lower bound on any algorithmic decision procedure relying exclu- sively on the equivalent deletion sequence characterization of unavoid- able patterns. We also demonstrate that the four other known necessary conditions are not sufficient to decide pattern unavoidability.
5 Reads
3 Reads
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1 Citation
20 Reads
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12 Citations
Motivated by the base pairing of RNA sequences, and as part of our collaboration with A. Condon and H. H. Hoos, we defined the local move operation on two unobstructed edges in a plane tree given here. We now consider the graph G n induced by this operation on T n , the set of plane trees with n edges. We provide a series of results showing that G n is a connected, n-partite graph of diameter n − 1 with disjoint sets whose cardinalities are enumerated by the Narayana numbers. Our partition of G n depends on orienting the edges in a plane tree, and differs from the known Narayana decomposition of ordered trees according to the number of leaves by N. Dershowitz and S. Zaks. We then consider the partial ordering induced by an antisymmetric restriction of our local move operation. After a theorem characterizing the new relation, we prove that the poset (T n ,) is a lattice, which is complemented but not distributive. Finally, in a joint result with S. Fomin, we show that (T n ,) is isomorphic to the lattice of noncrossing partitions.
7 Reads
The fundamental problem confronting attempts to design RNA sequences with specific base pairings in their secondary structures is preventing the formation of alternate foldings with lower minimal free energy. Under the given design strategy, it is possible to analyze the potential reconfigurations through their abstract graphical representation. In this context, we define a new relation, the local move, on plane trees representing RNA secondary structures. We then prove that this induces an intriguing new combinatorial structure on the equivalence class of plane trees with ¡ edges. Specifically, it is shown to be a multipartite graph whose disjoint sets are enumerated by the Narayana numbers.
8 Reads
8 Reads
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1 Citation
We analyze the extent to which RNA sequences with specified base pairings of the secondary struc- ture can be generated under the simplest possible encoding scheme. The potential and limitations of this approach have important implications for the possibility of an efficient deterministic solution to the RNA secondary structure design problem. Furthermore, the analysis relies on interesting new results regarding strings encoding plane trees, bounds on RNA loop energies, and the degree to which alternate configurations are related.
Citations (30)
... Let T n denote the set of plane trees with n edges, and T ∈ T n . Motivated by the molecular biology problem of RNA folding [7], label the boundary of T with [1, 2n] in increasing order counter-clockwise from the root. Let (i, j) denote the edge in T with indices i < j on the left and right sides. ...
- Citing Article
- Full-text available
March 2023
... SHAPE reactivity values were used as constraints to model the vRNA secondary structure using RNAStructure (version 6.4) [36] with the default values of −0.6 kcal/mol and 1.8 kcal/mol [37] for intercept and slope, respectively, and secondary structure were using VARNA (version 3.93) [38]. Secondary structures predicted with RNAStructure v6.4 were compared using RNAStructViz v2.14.18 [39]. ...
- Citing Article
April 2021
Bioinformatics
... We provide here a mathematical motivation for generating alternative predictions based on a parametric analysis of RNA branching Drellich et al. (2017); Barrera-Cruz et al. (2018); Poznanović et al. (2020Poznanović et al. ( , 2021. Using methods from geometric combinatorics Drellich et al. (2017), it is possible to identify all optimal predictions under any parameterization of the entropic branching penalty. ...
- Citing Article
- Full-text available
March 2021
... Our DMS-MaP (Dimethyl Sulfate-Mutational Profiling) experiments on this 87nt construct in Figure 1C indicate low (black), medium (yellow), and high (red) nucleotides based on their reactivity against DMS (Mustoe et al. 2019 Dey, et al. 2021). Both DMS and SHAPE reactivities can be used as a pseudofree energy term in thermodynamic folding algorithms to significantly improve structure prediction (Deigan et al. 2009;Greenwood and Heitsch 2020) Our two DMS-MaP biological replicates in supplementary Figure S1 are quantitatively reproducible (R 2 >0.85). When used for structure prediction with SHAPEknots (Hajdin et al. 2013), we obtained three conformations, shown in Figures 1C, 1D and 1E. ...
- Citing Article
October 2020
Bulletin of Mathematical Biology
... We provide here a mathematical motivation for generating alternative predictions based on a parametric analysis of RNA branching Drellich et al. (2017); Barrera-Cruz et al. (2018); Poznanović et al. (2020Poznanović et al. ( , 2021. Using methods from geometric combinatorics Drellich et al. (2017), it is possible to identify all optimal predictions under any parameterization of the entropic branching penalty. ...
- Citing Article
February 2020
Journal of Structural Biology
... We provide here a mathematical motivation for generating alternative predictions based on a parametric analysis of RNA branching Drellich et al. (2017); Barrera-Cruz et al. (2018); Poznanović et al. (2020Poznanović et al. ( , 2021. Using methods from geometric combinatorics Drellich et al. (2017), it is possible to identify all optimal predictions under any parameterization of the entropic branching penalty. ...
- Citing Article
August 2017
SIAM Journal on Applied Algebra and Geometry
... 1 several implementations of NNTM including RNAstructure [5], the ViennaRNA Package [6], UNAFold [7], and GTfold [8]. Although very useful, the physics-rooted NNTM approach is prone to accuracy errors in its predictions [9,10]. Accuracy can be improved by incorporating additional information, such as highthroughput chemical probing data that further constrain the thermodynamic model [11,12,13]. ...
- Citing Article
June 2017
Biophysical Journal
... We provide here a mathematical motivation for generating alternative predictions based on a parametric analysis of RNA branching Drellich et al. (2017); Barrera-Cruz et al. (2018); Poznanović et al. (2020Poznanović et al. ( , 2021. Using methods from geometric combinatorics Drellich et al. (2017), it is possible to identify all optimal predictions under any parameterization of the entropic branching penalty. ...
- Citing Chapter
January 2017
... Under the energy minimization hypothesis, an RNA sequence will fold into its configuration only when the loop region energies are minimized and their stacked pairs are maximized. Various algorithms and combinatorial models are developed to solve the RNA secondary structure design problem [96,97,174]. An example of RNA secondary structure is demonstrated in Figure 2 (C). ...
- Citing Chapter
October 2014
Natural Computing Series
... A secondary structure for the given RNA sequence is called suboptimal if its ∆G value is close to the MFE score under the current NNTM evaluation. It has long been recognized that prediction accuracy improves when these alternative structures are also considered Mathews (2006); Rogers and Heitsch (2016). For instance, the three different NNTM parameterizations ; Mathews et al. (1999Mathews et al. ( , 2004 have always reported a "best suboptimal" accuracy which can be understood as an indicator of the approximation quality. ...
- Citing Article
March 2016
WIREs RNA