Seeing the Results of a Mutation With a Vertex Weighted Hierarchical Graph

We represent the protein structure of scTIM with a graph-theoretic model. We construct a hierarchical graph with three layers - a top level, a midlevel and a bottom level. The top level graph is a representation of the protein in which its vertices each represent a substructure of the protein. In turn, each substructure of the protein is represented by a graph whose vertices are amino acids. Finally, each amino acid is represented as a graph where the vertices are atoms. We use this representation to model the effects of a mutation on the protein. Methods: There are 19 vertices (substructures) in the top level graph and thus there are 19 distinct graphs at the midlevel. The vertices of each of the 19 graphs at the midlevel represent amino acids. Each amino acid is represented by a graph where the vertices are atoms in the residue structure. All edges are determined by proximity in the protein\u27s 3D structure. The vertices in the bottom level are labelled by the corresponding molecular mass of the atom that it represents. We use graph-theoretic measures that incorporate vertex weights to assign graph based attributes to the amino acid graphs. The attributes of the corresponding amino acids are used as vertex weights for the substructure graphs at the midlevel. Graph-theoretic measures based on vertex weighted graphs are subsequently calculated for each of the midlevel graphs. Finally, the vertices of the top level graph are weighted with attributes of the corresponding substructure graph in the midlevel. Results: We can visualize which mutations are more influential than others by using properties such as vertex size to correspond with an increase or decrease in a graph-theoretic measure. Global graph-theoretic measures such as the number of triangles or the number of spanning trees can change as the result. Hence this method provides a way to visualize these global changes resulting from a small, seemingly inconsequential local change. Conclusions: This modelling method provides a novel approach to the visualization of protein structures and the consequences of amino acid deletions, insertions or substitutions and provides a new way to gain insight on the consequences of diseases caused by genetic mutations

A Predictive Model for Secondary RNA Structure Using Graph Theory and a Neural Network

Background: Determining the secondary structure of RNA from the primary structure is a challenging computational problem. A number of algorithms have been developed to predict the secondary structure from the primary structure. It is agreed that there is still room for improvement in each of these approaches. In this work we build a predictive model for secondary RNA structure using a graph-theoretic tree representation of secondary RNA structure. We model the bonding of two RNA secondary structures to form a larger secondary structure with a graph operation we call merge. We consider all combinatorial possibilities using all possible tree inputs, both those that are RNA-like in structure and those that are not. The resulting data from each tree merge operation is represented by a vector. We use these vectors as input values for a neural network and train the network to recognize a tree as RNA-like or not, based on the merge data vector. The network estimates the probability of a tree being RNA-like.Results: The network correctly assigned a high probability of RNA-likeness to trees previously identified as RNA-like and a low probability of RNA-likeness to those classified as not RNA-like. We then used the neural network to predict the RNA-likeness of the unclassified trees.Conclusions: There are a number of secondary RNA structure prediction algorithms available online. These programs are based on finding the secondary structure with the lowest total free energy. In this work, we create a predictive tool for secondary RNA structures using graph-theoretic values as input for a neural network. The use of a graph operation to theoretically describe the bonding of secondary RNA is novel and is an entirely different approach to the prediction of secondary RNA structures. Our method correctly predicted trees to be RNA-like or not RNA-like for all known cases. In addition, our results convey a measure of likelihood that a tree is RNA-like or not RNA-like. Given that the majority of secondary RNA folding algorithms return more than one possible outcome, our method provides a means of determining the best or most likely structures among all of the possible outcomes

Total irredundance in graphs

AbstractA set S of vertices in a graph G is called a total irredundant set if, for each vertex v in G,v or one of its neighbors has no neighbor in S−{v}. We investigate the minimum and maximum cardinalities of maximal total irredundant sets

A predictive model for secondary RNA structure using graph theory and a neural network

Background: Determining the secondary structure of RNA from the primary structure is a challenging computational problem. A number of algorithms have been developed to predict the secondary structure from the primary structure. It is agreed that there is still room for improvement in each of these approaches. In this work we build a predictive model for secondary RNA structure using a graph-theoretic tree representation of secondary RNA structure. We model the bonding of two RNA secondary structures to form a larger secondary structure with a graph operation we call merge. We consider all combinatorial possibilities using all possible tree inputs, both those that are RNA-like in structure and those that are not. The resulting data from each tree merge operation is represented by a vector. We use these vectors as input values for a neural network and train the network to recognize a tree as RNA-like or not, based on the merge data vector. The network estimates the probability of a tree being RNA-like.Results: The network correctly assigned a high probability of RNA-likeness to trees previously identified as RNA-like and a low probability of RNA-likeness to those classified as not RNA-like. We then used the neural network to predict the RNA-likeness of the unclassified trees.Conclusions: There are a number of secondary RNA structure prediction algorithms available online. These programs are based on finding the secondary structure with the lowest total free energy. In this work, we create a predictive tool for secondary RNA structures using graph-theoretic values as input for a neural network. The use of a graph operation to theoretically describe the bonding of secondary RNA is novel and is an entirely different approach to the prediction of secondary RNA structures. Our method correctly predicted trees to be RNA-like or not RNA-like for all known cases. In addition, our results convey a measure of likelihood that a tree is RNA-like or not RNA-like. Given that the majority of secondary RNA folding algorithms return more than one possible outcome, our method provides a means of determining the best or most likely structures among all of the possible outcomes

A Neighborhood Condition Which Implies the Existence of a Class of D‐chromatic Subgraphs

A graph G of order n satisfies the neighborhood condition NCk \u3e s if any collection of k independent vertices is collectively adjacent to more than s vertices. Given a family H of graphs, the decomposition class β(H) is the family of graphs B with the property that for some H ∈ H of chromatic number d, H contains B as an induced subgraph and l̀V(H) − V(B)ǹ is (d − 2) colorable. Let H be a family of d‐chromatic graphs, B an element of β(H) such that B contains an s‐matching as an induced subgraph. Thus the cardinality of one of the partite sets of B is s + r for some integer r ≥ 0. We show that if t is a fixed positive integer, G is a graph of sufficiently large order n that satisfies the neighborhood condition (Formula Presented.) then G contains B + K(d ‐ 2; t) as a subgraph

Vertex-Weighted Graphs and Their Applications

In our recent work in computational biology, our approach to modeling protein structures requires that each vertex be weighted by a vector of weights. This motivates the results we present here. We show that many graphical invariants can be generalized to vertex weights by replacing the maximum(minimum) cardinality of a vertex set with a sum over the vertex weights. We also generalize the Laplacian of a vertex-weighted graph to include vector-weighted vertices, including generalizing some spectral results

A Neighborhood Condition Which Implies the Existence of a Complete Multipartite Subgraph

Given a graph G and uε{lunate}V(G), the neighborhood N(u)={uε{lunate}V(G)|uvε{lunate}E(G)}. We define NCk(G)=min|∪N(ui)| where the minimum is taken over all k independent sets {u1...uk} of vertices in V(G). We shall show that if G is a graph of order n that satisfies the neighborhood condition NCk(G) \u3e d-2 d-1n+cn1- 1 rfor some real number c=c(m, d, k, r) then for sufficiently large n, G contains at least one copy of a K(r,m...md-1) where mi=m for each i and r≥m. When r=1, 2 or 3, this result is best possible