RNA motif classification is important for understanding structure/function connections and buildi... more RNA motif classification is important for understanding structure/function connections and building phylogenetic relationships. Using our coarse-grained RNA-As-Graphs (RAG) representations, we identify recurrent dual graph motifs in experimentally solved RNA structures based on an improved search algorithm that finds and ranks independent RNA substructures. Our expanded list of 183 existing dual graph motifs reveals five common motifs found in transfer RNA, riboswitch, and ribosomal 5S RNA components. Moreover, we identify three motifs for available viral frameshifting RNA elements, suggesting a correlation between viral structural complexity and frameshifting efficiency. We further partition the RNA substructures into 1844 distinct submotifs, with pseudoknots and junctions retained intact. Common modules are internal loops and three-way junctions, and three submotifs are associated with riboswitches that bind nucleotides, ions, and signaling molecules. Together, our library of exis...
Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined ... more Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In this paper we present a linear-time algorithm to partition dual graphs into topological components called blocks and determine whether each block contains a pseudoknot or not. We show that a block contains a pseudoknot if and only if the block has a vertex of degree 3 or more; this characterization allows us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Applications to RNA design can be envisioned since modular building blocks with intact pseudoknots can be combined to form new constructs.
Given an undirected graph G = (V,E), two distinguished vertices s and t of G, and a diameter boun... more Given an undirected graph G = (V,E), two distinguished vertices s and t of G, and a diameter bound D, a D-s, t-path is a path between s and t composed of at most D edges. An edge e is called D-irrelevant if does not belong to any D-s, t-path of G. In this paper we study the problem of efficiently detecting D-irrelevant edges and also study the computational complexity of diameter-related problems in graphs. Detection and subsequent deletion of D-irrelevant edges have been shown to be fundamental in reducing the computational effort to evaluate the Source-to-terminal Diameter-Constrained reliability of a graph G, R{s,t}(G,D), which is defined as the probability that at least a path between s and t, with at most D edges, survives after deletion of the failed edges (under the assumption that edges fail independently and nodes are perfectly reliable). Among other results, we present sufficient conditions to efficiently recognize irrelevant edges and we present computational results illu...
Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined ... more Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In previous works, a linear-time algorithm was introduced to partition dual graphs into maximally connected components called blocks and determine whether each block contains a pseudoknot or not. As pseudoknots can not be contained into two different blocks, this characterization allow us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Moreover we have extended the partitioning algorithm by classifying a pseudoknot as either recursive or non-recursive in order to continue with our research in the development of a library of building blocks for RNA design by fragment assembly. In this paper we present a methodology that uses our previous results and classify pseudoknots into the classical H,K,L, and M types, based upon a novel representation of RNA secondary structures as dua...
Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined ... more Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In this paper we present a linear-time algorithm to partition dual graphs into maximal topological components called blocks and determine whether each block contains a pseudoknot or not. We show that a block contains a pseudoknot if and only if the block has a vertex of degree 3 or more; this characterization allows us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Applications to RNA design can be envisioned since modular building blocks with intact pseudoknots can be combined to form new constructs.
An introduction into the usage of graph or network theory tools for the study of RNA molecules is... more An introduction into the usage of graph or network theory tools for the study of RNA molecules is presented. By using vertices and edges to define RNA secondary structures as tree and dual graphs, we can enumerate, predict, and design RNA topologies. Graph connectivity and associated Laplacian eigenvalues relate to biological properties of RNA and help understand RNA motifs as well as build, by computational design, various RNA target structures. Importantly, graph theoretical representations of RNAs reduce drastically the conformational space size and therefore simplify modeling and prediction tasks. Ongoing challenges remain regarding general RNA design, representation of RNA pseudoknots, and tertiary structure prediction. Thus, developments in network theory may help advance RNA biology.
Identifying and analyzing RNA pseudoknots based on graph-theoretical properties of dual graphs: a... more Identifying and analyzing RNA pseudoknots based on graph-theoretical properties of dual graphs: a partitioning approach
Let G = (V,E) be a graph with a distinguished set of terminal vertices K ⊆ V. We define the K-dia... more Let G = (V,E) be a graph with a distinguished set of terminal vertices K ⊆ V. We define the K-diameter of G as the maximum distance between any pair of vertices of K. If the edges fail randomly and independently with known probabilities (vertices are always operational), the diameter-constrained K-terminal reliability of G, RK(G,D), is defined as the probability that surviving edges span a subgraph whose K-diameter does not exceed D. In general, the computational complexity of evaluating RK(G,D) is NP-hard, as this measure subsumes the classical K-terminal reliability RK(G), known to belong to this complexity class. In this note, we show that even though for two terminal vertices s and t and D = 2, R{s,t}(G,D) can be determined in polynomial time, the problem of calculating R{s,t}(G,D) for fixed values ofD, D ≥ 3, is NP-hard. We also generalize this result for any fixed number of terminal vertices. Although it is very unlikely that general efficient algorithms exist, we present a re...
Dual graphs have been applied to model RNA secondary structures. The purpose of the paper is two-... more Dual graphs have been applied to model RNA secondary structures. The purpose of the paper is two-fold: we present new graph-theoretic properties of dual graphs to validate the further analysis and classification of RNAs using these topological representations; we also present a linear-time algorithm to partition dual graphs into topological components called {\it blocks} and determine if each block contains a {\it pseudoknot} or not. We show that a block contains a pseudoknot if and only if the block has a vertex of degree $3$ or more; this characterization allows us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Even though non-topological techniques to detect and classify pseudoknots have been efficiently applied, structural properties of dual graphs provide a unique perspective for the further analysis of RNAs. Applications to RNA design can be envisioned since modular building...
IAENG international journal of computer science, 2017
Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined ... more Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In this paper we present a linear-time algorithm to partition dual graphs into maximal topological components called blocks and determine whether each block contains a pseudoknot or not. We show that a block contains a pseudoknot if and only if the block has a vertex of degree 3 or more; this characterization allows us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Applications to RNA design can be envisioned since modular building blocks with intact pseudoknots can be combined to form new constructs.
Ab stract— In this paper we present a new network reliability measure that is particularly useful... more Ab stract— In this paper we present a new network reliability measure that is particularly useful to evaluate performance objectives of wireless sensor networks. A communication network can be modeled as directed graph G = (V, E), composed of a set of nodes V, and a set of directed links E. Given that the links of the network underlying graph fail independently with known probabilities (nodes are perfectly reliable), and given a set K of terminal nodes (or participating nodes) and a distinguished terminal node s of K, the Kterminal-to-sink reliability measure, RK,s(G), is the probability of the event that the surviving links span a sub-digraph of G such that for each node u of K, there exists an operational directed path from u to s. In this paper we study a combinatorial property of graphs called the domination invariant which has been applied to efficiently compute the reliability of communication networks. Moreover we model wireless networks as random digraphs using current resul...
We are concerned with undirected graphs G= (V,E) with distinguished set of vertices K ⊆ V, |K| ≥ ... more We are concerned with undirected graphs G= (V,E) with distinguished set of vertices K ⊆ V, |K| ≥ 2, called terminal vertices. A K-Steiner tree T of G is a minimal tree to the number of edges, containing all the vertices of K. The K-edge connectivity of a connected graph G with terminal vertices K, and denoted as λK (G), is the minimum number of edges whose removal disconnect at least two vertices of K in G. In this talk, we will investigate the relationship between the maximum number of edge-disjoint K-Steiner trees and the K-edge-connectivity of a graph G. This problem, known as a Steiner Tree Packing Problem (STPP), it has attracted considerable attention from researchers in different areas because of its wide applicability as for example in the design of VLSI circuits. In 2003, the EGRES combinatorial group of the Hungarian Academy of Sciences conjected that any graph G= (V,E) with arbitary set of terminal vertices K ⊆ V, |K| ≥ 2, contains at least ⌊ λk (G)/2⌋ edge-disjoint K-Ste...
Let G = (V;E) be a digraph with a distinguished set of terminal vertices K V and a vertex s 2 K .... more Let G = (V;E) be a digraph with a distinguished set of terminal vertices K V and a vertex s 2 K . We de ne the s;K-diameter of G as the maximum distance between s and any of vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the Diameter-constrained s;K-terminal reliability of G, Rs;K(G;D), is de ned as the probability that surviving arcs span a subgraph whose s;Kdiameter does not exceed D [5, 11]. A graph invariant called the domination of a graphG was introduced by Satyanarayana and Prabhakar [13] to generate the non-canceling terms of the classical reliability expression, Rs;K(G), based on the same reliability model (i.e. arcs fail randomly and independently and where nodes are perfect), and de ned as the probability that the surviving arcs span a subgraph of G with unconstrained nite s;K-diameter. This result allowed the generation of rapid algorithms for the computation of Rs;K(G). In this paper we present a ch...
This paper is intended as a summary of combinatorial and computational properties of a Diameter-c... more This paper is intended as a summary of combinatorial and computational properties of a Diameter-constrained network reliability model. Classical network reliability models are based on the existence of end-to-end paths between network nodes, not taking into account the length of these paths; for many applications this is inadequate because the connection will only be established or attain the required quality if the distance between the connecting nodes does not exceed a given value. An alternative topological reliability model is the Diameter-constrained reliability of a network (DCR); this measure considers not only the underlying topology, but also imposes a bound on the diameter, which is the maximum distance between the nodes of the network. As communication networks can be modeled by undirected as well as directed graphs, we present a synopsis of the results known up until now pertaining the DCR for both topological representations. Moreover we show some important combinatoria...
RNA motif classification is important for understanding structure/function connections and buildi... more RNA motif classification is important for understanding structure/function connections and building phylogenetic relationships. Using our coarse-grained RNA-As-Graphs (RAG) representations, we identify recurrent dual graph motifs in experimentally solved RNA structures based on an improved search algorithm that finds and ranks independent RNA substructures. Our expanded list of 183 existing dual graph motifs reveals five common motifs found in transfer RNA, riboswitch, and ribosomal 5S RNA components. Moreover, we identify three motifs for available viral frameshifting RNA elements, suggesting a correlation between viral structural complexity and frameshifting efficiency. We further partition the RNA substructures into 1844 distinct submotifs, with pseudoknots and junctions retained intact. Common modules are internal loops and three-way junctions, and three submotifs are associated with riboswitches that bind nucleotides, ions, and signaling molecules. Together, our library of exis...
Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined ... more Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In this paper we present a linear-time algorithm to partition dual graphs into topological components called blocks and determine whether each block contains a pseudoknot or not. We show that a block contains a pseudoknot if and only if the block has a vertex of degree 3 or more; this characterization allows us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Applications to RNA design can be envisioned since modular building blocks with intact pseudoknots can be combined to form new constructs.
Given an undirected graph G = (V,E), two distinguished vertices s and t of G, and a diameter boun... more Given an undirected graph G = (V,E), two distinguished vertices s and t of G, and a diameter bound D, a D-s, t-path is a path between s and t composed of at most D edges. An edge e is called D-irrelevant if does not belong to any D-s, t-path of G. In this paper we study the problem of efficiently detecting D-irrelevant edges and also study the computational complexity of diameter-related problems in graphs. Detection and subsequent deletion of D-irrelevant edges have been shown to be fundamental in reducing the computational effort to evaluate the Source-to-terminal Diameter-Constrained reliability of a graph G, R{s,t}(G,D), which is defined as the probability that at least a path between s and t, with at most D edges, survives after deletion of the failed edges (under the assumption that edges fail independently and nodes are perfectly reliable). Among other results, we present sufficient conditions to efficiently recognize irrelevant edges and we present computational results illu...
Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined ... more Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In previous works, a linear-time algorithm was introduced to partition dual graphs into maximally connected components called blocks and determine whether each block contains a pseudoknot or not. As pseudoknots can not be contained into two different blocks, this characterization allow us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Moreover we have extended the partitioning algorithm by classifying a pseudoknot as either recursive or non-recursive in order to continue with our research in the development of a library of building blocks for RNA design by fragment assembly. In this paper we present a methodology that uses our previous results and classify pseudoknots into the classical H,K,L, and M types, based upon a novel representation of RNA secondary structures as dua...
Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined ... more Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In this paper we present a linear-time algorithm to partition dual graphs into maximal topological components called blocks and determine whether each block contains a pseudoknot or not. We show that a block contains a pseudoknot if and only if the block has a vertex of degree 3 or more; this characterization allows us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Applications to RNA design can be envisioned since modular building blocks with intact pseudoknots can be combined to form new constructs.
An introduction into the usage of graph or network theory tools for the study of RNA molecules is... more An introduction into the usage of graph or network theory tools for the study of RNA molecules is presented. By using vertices and edges to define RNA secondary structures as tree and dual graphs, we can enumerate, predict, and design RNA topologies. Graph connectivity and associated Laplacian eigenvalues relate to biological properties of RNA and help understand RNA motifs as well as build, by computational design, various RNA target structures. Importantly, graph theoretical representations of RNAs reduce drastically the conformational space size and therefore simplify modeling and prediction tasks. Ongoing challenges remain regarding general RNA design, representation of RNA pseudoknots, and tertiary structure prediction. Thus, developments in network theory may help advance RNA biology.
Identifying and analyzing RNA pseudoknots based on graph-theoretical properties of dual graphs: a... more Identifying and analyzing RNA pseudoknots based on graph-theoretical properties of dual graphs: a partitioning approach
Let G = (V,E) be a graph with a distinguished set of terminal vertices K ⊆ V. We define the K-dia... more Let G = (V,E) be a graph with a distinguished set of terminal vertices K ⊆ V. We define the K-diameter of G as the maximum distance between any pair of vertices of K. If the edges fail randomly and independently with known probabilities (vertices are always operational), the diameter-constrained K-terminal reliability of G, RK(G,D), is defined as the probability that surviving edges span a subgraph whose K-diameter does not exceed D. In general, the computational complexity of evaluating RK(G,D) is NP-hard, as this measure subsumes the classical K-terminal reliability RK(G), known to belong to this complexity class. In this note, we show that even though for two terminal vertices s and t and D = 2, R{s,t}(G,D) can be determined in polynomial time, the problem of calculating R{s,t}(G,D) for fixed values ofD, D ≥ 3, is NP-hard. We also generalize this result for any fixed number of terminal vertices. Although it is very unlikely that general efficient algorithms exist, we present a re...
Dual graphs have been applied to model RNA secondary structures. The purpose of the paper is two-... more Dual graphs have been applied to model RNA secondary structures. The purpose of the paper is two-fold: we present new graph-theoretic properties of dual graphs to validate the further analysis and classification of RNAs using these topological representations; we also present a linear-time algorithm to partition dual graphs into topological components called {\it blocks} and determine if each block contains a {\it pseudoknot} or not. We show that a block contains a pseudoknot if and only if the block has a vertex of degree $3$ or more; this characterization allows us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Even though non-topological techniques to detect and classify pseudoknots have been efficiently applied, structural properties of dual graphs provide a unique perspective for the further analysis of RNAs. Applications to RNA design can be envisioned since modular building...
IAENG international journal of computer science, 2017
Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined ... more Dual graphs have been applied to model RNA secondary structures with pseudoknots, or intertwined base pairs. In this paper we present a linear-time algorithm to partition dual graphs into maximal topological components called blocks and determine whether each block contains a pseudoknot or not. We show that a block contains a pseudoknot if and only if the block has a vertex of degree 3 or more; this characterization allows us to efficiently isolate smaller RNA fragments and classify them as pseudoknotted or pseudoknot-free regions, while keeping these sub-structures intact. Applications to RNA design can be envisioned since modular building blocks with intact pseudoknots can be combined to form new constructs.
Ab stract— In this paper we present a new network reliability measure that is particularly useful... more Ab stract— In this paper we present a new network reliability measure that is particularly useful to evaluate performance objectives of wireless sensor networks. A communication network can be modeled as directed graph G = (V, E), composed of a set of nodes V, and a set of directed links E. Given that the links of the network underlying graph fail independently with known probabilities (nodes are perfectly reliable), and given a set K of terminal nodes (or participating nodes) and a distinguished terminal node s of K, the Kterminal-to-sink reliability measure, RK,s(G), is the probability of the event that the surviving links span a sub-digraph of G such that for each node u of K, there exists an operational directed path from u to s. In this paper we study a combinatorial property of graphs called the domination invariant which has been applied to efficiently compute the reliability of communication networks. Moreover we model wireless networks as random digraphs using current resul...
We are concerned with undirected graphs G= (V,E) with distinguished set of vertices K ⊆ V, |K| ≥ ... more We are concerned with undirected graphs G= (V,E) with distinguished set of vertices K ⊆ V, |K| ≥ 2, called terminal vertices. A K-Steiner tree T of G is a minimal tree to the number of edges, containing all the vertices of K. The K-edge connectivity of a connected graph G with terminal vertices K, and denoted as λK (G), is the minimum number of edges whose removal disconnect at least two vertices of K in G. In this talk, we will investigate the relationship between the maximum number of edge-disjoint K-Steiner trees and the K-edge-connectivity of a graph G. This problem, known as a Steiner Tree Packing Problem (STPP), it has attracted considerable attention from researchers in different areas because of its wide applicability as for example in the design of VLSI circuits. In 2003, the EGRES combinatorial group of the Hungarian Academy of Sciences conjected that any graph G= (V,E) with arbitary set of terminal vertices K ⊆ V, |K| ≥ 2, contains at least ⌊ λk (G)/2⌋ edge-disjoint K-Ste...
Let G = (V;E) be a digraph with a distinguished set of terminal vertices K V and a vertex s 2 K .... more Let G = (V;E) be a digraph with a distinguished set of terminal vertices K V and a vertex s 2 K . We de ne the s;K-diameter of G as the maximum distance between s and any of vertices of K. If the arcs fail randomly and independently with known probabilities (vertices are always operational), the Diameter-constrained s;K-terminal reliability of G, Rs;K(G;D), is de ned as the probability that surviving arcs span a subgraph whose s;Kdiameter does not exceed D [5, 11]. A graph invariant called the domination of a graphG was introduced by Satyanarayana and Prabhakar [13] to generate the non-canceling terms of the classical reliability expression, Rs;K(G), based on the same reliability model (i.e. arcs fail randomly and independently and where nodes are perfect), and de ned as the probability that the surviving arcs span a subgraph of G with unconstrained nite s;K-diameter. This result allowed the generation of rapid algorithms for the computation of Rs;K(G). In this paper we present a ch...
This paper is intended as a summary of combinatorial and computational properties of a Diameter-c... more This paper is intended as a summary of combinatorial and computational properties of a Diameter-constrained network reliability model. Classical network reliability models are based on the existence of end-to-end paths between network nodes, not taking into account the length of these paths; for many applications this is inadequate because the connection will only be established or attain the required quality if the distance between the connecting nodes does not exceed a given value. An alternative topological reliability model is the Diameter-constrained reliability of a network (DCR); this measure considers not only the underlying topology, but also imposes a bound on the diameter, which is the maximum distance between the nodes of the network. As communication networks can be modeled by undirected as well as directed graphs, we present a synopsis of the results known up until now pertaining the DCR for both topological representations. Moreover we show some important combinatoria...
Uploads
Papers