Debra J. Knisley

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

BackgroundDetermining 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.ResultsThe 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.ConclusionsThere 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.

Research Article: Predicting protein-protein interactions using graph invariants and a neural network

The PDZ domain of proteins mediates a protein-protein interaction by recognizing the hydrophobic C-terminal tail of the target protein. One of the challenges put forth by the DREAM (Discussions on Reverse Engineering Assessment and Methods) 2009 Challenge consists of predicting a position weight matrix (PWM) that describes the specificity profile of five PDZ domains to their target peptides. We consider the primary structures of each of the five PDZ domains as a numerical sequence derived from graph-theoretic models of each of the individual amino acids in the protein sequence. Using available PDZ domain databases to obtain known targets, the graph-theoretic based numerical sequences are then used to train a neural network to recognize their targets. Given the challenge sequences, the target probabilities are computed and a corresponding position weight matrix is derived. In this work we present our method. The results of our method placed second in the DREAM 2009 challenge.

Hereditary domination and independence parameters

For a graphical property P and a graph G, we say that a subset S of the vertices of G is a P-set if the subgraph induced by S has the property P. Then the P-domination number of G is the minimum cardinality of a dominating P-set and the P-independence number the maximum cardinality of a P-set. We show that several properties of domination, independent domination and acyclic domination hold for arbitrary properties P that are closed under disjoint unions and subgraphs.

Mentoring Interdisciplinary Undergraduate Students via a Team Effort

We describe how a team approach that we developed as a mentoring strategy can be used to recruit, advance, and guide students to be more interested in the interdisciplinary field of mathematical biology, and lead to success in undergraduate research in this field. Students are introduced to research in their first semester via lab rotations. Their participation in the research of four faculty members—two from biology and two from mathematics—gives them a first-hand overview of research in quantitative biology and also some initial experience in research itself. However, one of the primary goals of the lab rotation experience is that of developing teams of students and faculty that combine mathematics and statistics with biology and the life sciences, teams that subsequently mentor undergraduate research in genuine interdisciplinary environments. Thus, the team concept serves not only as a means of establishing interdisciplinary research, but also as a means of incorporating new students into existing research efforts that will then track those students into meaningful research of their own. We report how the team concept is used to support undergraduate research in mathematical biology and what types of team-building strategies have worked for us.

Total irredundance in graphs

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

Classifying Multigraph Models of Secondary RNA Structure Using Graph-Theoretic Descriptors

The prediction of secondary RNA folds from primary sequences continues to be an important area of research given the significance of RNA molecules in biological processes such as gene regulation. To facilitate this effort, graph models of secondary structure have been developed to quantify and thereby characterize the topological properties of the secondary folds. In this work we utilize a multigraph representation of a secondary RNA structure to examine the ability of the existing graph-theoretic descriptors to classify all possible topologies as either RNA-like or not RNA-like. We use more than one hundred descriptors and several different machine learning approaches, including nearest neighbor algorithms, one-class classifiers, and several clustering techniques. We predict that many more topologies will be identified as those representing RNA secondary structures than currently predicted in the RAG (RNA-As-Graphs) database. The results also suggest which descriptors and which algorithms are more informative in classifying and exploring secondary RNA structures.

Inferring gene expression from ribosomal promoter sequences, a crowdsourcing approach

The Gene Promoter Expression Prediction challenge consisted of predicting gene expression from promoter sequences in a previously unknown experimentally generated data set. The challenge was presented to the community in the framework of the sixth Dialogue for Reverse Engineering Assessments and Methods (DREAM6), a community effort to evaluate the status of systems biology modeling methodologies. Nucleotide-specific promoter activity was obtained by measuring fluorescence from promoter sequences fused upstream of a gene for yellow fluorescence protein and inserted in the same genomic site of yeast Saccharomyces cerevisiae. Twenty-one teams submitted results predicting the expression levels of 53 different promoters from yeast ribosomal protein genes. Analysis of participant predictions shows that accurate values for low-expressed and mutated promoters were difficult to obtain, although in the latter case, only when the mutation induced a large change in promoter activity compared to the wild-type sequence. As in previous DREAM challenges, we found that aggregation of participant predictions provided robust results, but did not fare better than the three best algorithms. Finally, this study not only provides a benchmark for the assessment of methods predicting activity of a specific set of promoters from their sequence, but it also shows that the top performing algorithm, which used machine-learning approaches, can be improved by the addition of biological features such as transcription factor binding sites.

Graph-Theoretic Models of Mutations in the Nucleotide Binding Domain 1 of the Cystic Fibrosis Transmembrane Conductance Regulator

Cystic fibrosis is one of the most common inherited diseases and is caused by a mutation in a membrane protein, the cystic fibrosis transmembrane conductance regulator (CFTR). This protein serves as a chloride channel and regulates the viscosity of mucus lining the ducts of a number of organs. Although much has been learned about the consequences of mutations on the energy landscape and the resulting disrupted folding pathway of CFTR, a level of understanding needed to correct the misfolding has not been achieved. The most common mutations of CFTR are located in one of two nucleotide binding domains, namely, the nucleotide binding domain 1 (NBD1). We model NBD1 using a nested graph model. The vertices in the lowest layer each represent an atom in the structure of an amino acid residue, while the vertices in the mid layer each represent the residue. The vertices in the top layer each represent a subdomain of the nucleotide binding domain. We use this model to quantify the effects of a single point mutation on the protein domain. We compare the wildtype structure with eight of the most common mutations. The graph-theoretic model provides insight into how a single point mutation can have such profound structural consequences.

Seeing the results of a mutation with a vertex weighted hierarchical graph

BackgroundWe 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.MethodsThere 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 proteins 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.ResultsWe 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.ConclusionsThis 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 neighborhood condition which implies the existence of a complete multipartite subgraph

Abstract Given a graph G and uϵV ( G ), the neighborhood N ( u )={ uϵV ( G )| uvϵE ( G )}. We define NC k ( G )= min |∪ N ( u i )| where the minimum is taken over all k independent sets { u 1 … u k } of vertices in V ( G ). We shall show that if G is a graph of order n that satisfies the neighborhood condition NC k (G) > d−2 d−1 n+cn 1− 1 r for 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 … m d −1 ) where m i = m for each i and r ⩾ m . When r =1, 2 or 3, this result is best possible.