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A graph-theory algorithm for rapid protein side-chain prediction.

Fast and accurate side-chain conformation prediction is important for homology modeling, ab initio protein structure prediction, and protein design applications. Many methods have been presented, although only a few computer programs are publicly available. The SCWRL program is one such method and is widely used because of its speed, accuracy, and ease of use. A new algorithm for SCWRL is presented that uses results from graph theory to solve the combinatorial problem encountered in the side-chain prediction problem. In this method, side chains are represented as vertices in an undirected graph. Any two residues that have rotamers with nonzero interaction energies are considered to have an edge in the graph. The resulting graph can be partitioned into connected subgraphs with no edges between them. These subgraphs can in turn be broken into biconnected components, which are graphs that cannot be disconnected by removal of a single vertex. The combinatorial problem is reduced to finding the minimum energy of these small biconnected components and combining the results to identify the global minimum energy conformation. This algorithm is able to complete predictions on a set of 180 proteins with 34342 side chains in <7 min of computer time. The total chi(1) and chi(1 + 2) dihedral angle accuracies are 82.6% and 73.7% using a simple energy function based on the backbone-dependent rotamer library and a linear repulsive steric energy. The new algorithm will allow for use of SCWRL in more demanding applications such as sequence design and ab initio structure prediction, as well addition of a more complex energy function and conformational flexibility, leading to increased accuracy.

Algorithms↗

Exploring the repertoire of RNA secondary motifs using graph theory; implications for RNA design.

Understanding the structural repertoire of RNA is crucial for RNA genomics research. Yet current methods for finding novel RNAs are limited to small or known RNA families. To expand known RNA structural motifs, we develop a two-dimensional graphical representation approach for describing and estimating the size of RNA's secondary structural repertoire, including naturally occurring and other possible RNA motifs. We employ tree graphs to describe RNA tree motifs and more general (dual) graphs to describe both RNA tree and pseudoknot motifs. Our estimates of RNA's structural space are vastly smaller than the nucleotide sequence space, suggesting a new avenue for finding novel RNAs. Specifically our survey shows that known RNA trees and pseudoknots represent only a small subset of all possible motifs, implying that some of the 'missing' motifs may represent novel RNAs. To help pinpoint RNA-like motifs, we show that the motifs of existing functional RNAs are clustered in a narrow range of topological characteristics. We also illustrate the applications of our approach to the design of novel RNAs and automated comparison of RNA structures; we report several occurrences of RNA motifs within larger RNAs. Thus, our graph theory approach to RNA structures has implications for RNA genomics, structure analysis and design.

Algorithms↗

Predicting calcium-binding sites in proteins - a graph theory and geometry approach.

Identifying calcium-binding sites in proteins is one of the first steps towards predicting and understanding the role of calcium in biological systems for protein structure and function studies. Due to the complexity and irregularity of calcium-binding sites, a fast and accurate method for predicting and identifying calcium-binding protein is needed. Here we report our development of a new fast algorithm (GG) to detect calcium-binding sites. The GG algorithm uses a graph theory algorithm to find oxygen clusters of the protein and a geometric algorithm to identify the center of these clusters. A cluster of four or more oxygen atoms has a high potential for calcium binding. High performance with about 90% site sensitivity and 80% site selectivity has been obtained for three datasets containing a total of 123 proteins. The results suggest that a sphere of a certain size with four or more oxygen atoms on the surface and without other atoms inside is necessary and sufficient for quickly identifying the majority of the calcium-binding sites with high accuracy. Our finding opens a new avenue to visualize and analyze calcium-binding sites in proteins facilitating the prediction of functions from structural genomic information.

Algorithms↗

Molecular fragments associated with non-genotoxic carcinogens, as detected using a software program based on graph theory: their usefulness to predict carcinogenicity.

We assembled 390 chemicals with a structure non-alerting to DNA-reactivity (145 carcinogens and 245 non-carcinogens) for which rodent carcinogenicity data were available. These non-alerting chemicals were defined by the absence in their molecules of DNA-reactive (directly or after metabolic activation) alerting structures, as described by Ashby and coworkers (Mutat. Res., 204 (1988) 17-115; Mutat. Res., 223 (1989) 73-103; Mutat. Res., 257 (1991) 209-227; Mutat. Res., 286 (1993) 3-74). Using our software program based on graph theory we analyzed the compounds in order to estimate the program's ability to predict nonalerting carcinogens. Our software fragmented the structural formula of the chemicals into all possible fragments of contiguous atoms with size between 2 and 8 (non-hydrogen) atoms and learned about statistically significant fragments from a training set of chemicals. These fragments were used to predict carcinogenicity or lack thereof in a verification set of compounds. For 390 runs of the software program we used (n - 1) of the chemicals as a training set, to predict the excluded chemical at each run (as a test set). Using two different probability thresholds to select significant fragments (P = 0.05 and P = 0.125 1-tailed according to binomial distribution), we performed two analyses: in the better one (P = 0.05) 19% of the molecules tested lacked significant fragments, for the remaining 81% the observed level of accuracy of the prediction was 66.0% against an expected level of accuracy of 51.7%. The difference was highly significant (P < 0.0001). We also examined the more significant activating fragments (biophores) and discussed at length both their biological plausibility and the working hypothesis that additional alerting structures for carcinogenicity (not only those related to genotoxicity) can be detected using this type of SAR approach. This new class of alerting structures could identify subfamilies of congeneric analogs active through mechanisms of receptor mediated carcinogenesis.

Carcinogens↗

Use of graph theory to identify patterns of deprivation and high morbidity and mortality in public health data sets.

OBJECTIVE: An important part of public health is identifying patterns of poor health and deprivation. Specific patterns of poor health may be associated with features of the geographic environment where contamination or pollution may be occurring. For example, there may be clusters of poor health surrounding nuclear power stations, whereas major roads or rivers may be associated with areas of poor health alongside the feature in chains. Current methods are limited in their capacity to search for complex patterns in geographic data sets. The objective of this study was to determine whether graph theory could be used to identify patterns of geographic areas that have high levels of deprivation, morbidity, and mortality in a public health database. The geographic areas used in the study were enumeration districts (EDs), which are the lowest level of census geography in England and Wales, representing on average 200 households in the 1991 census. More specifically, the study aimed to identify chains of EDs with high deprivation, morbidity, and mortality that might be adjacent to specific types of geographic features, i.e., rivers or major roads. DESIGN: The maximum common subgraph (MCS) algorithm was used to search for seven query patterns of deprivation and poor health within the Trent region. Query pattern 1 represented a linear chain of five EDs and query patterns 2 to 7 represented the possible clusters of the five EDs. To identify chains of EDs with high deprivation, morbidity, and mortality, the results from the query patterns 2 to 7 were used to remove patterns (option 1) and EDs (option 2) from the results of query pattern 1. MEASUREMENTS: Data on the Townsend Material Deprivation Index, standardized long-term limiting illness and standardized all-cause mortality rates were used for the 10,665 EDs within the Trent region. RESULTS: The MCS algorithm retrieved a range of patterns and EDs from the database for the queries. Query pattern 1 identified 3,838 patterns containing a total of 195 EDs. When the patterns retrieved using query patterns 2 to 7 were removed from the 3,838 patterns using option 1, 1,704 patterns remained containing 161 EDs. When the EDs retrieved using query patterns 2 to 7 were removed from the 195 EDs identified by query pattern 1 using option 2, 12 EDs remained. The MCS algorithm was therefore able to reduce the numbers of patterns and EDs to allow manual examination for chains of EDs and for that which might be associated with them. CONCLUSION: The study demonstrates the potential of the MCS algorithm for searching for specific patterns of need. This method has potential for identifying such patterns in relation to local geographic features for public health.

Algorithms↗

A graph theory model of the semantic structure of attitudes.

The semantic structure underlying the attitudes of pretreatment and posttreatment drug addicts was modeled using a network analysis of free word associations. Measures of graph theoretic properties were used to assess structural differences in the associative networks of the two populations. These measures modeled the information processes of associative networks proposed in the spreading activation theory of semantic processing. As expected based on graph theory, the structure of the associative networks of posttreatment subjects was more dense, less constrained, and more hierarchically organized by the self concept. In a test of the network model, the subjects' evaluations of concepts in the associative network were found to be a function of their evaluations of semantically similar concepts. Although preliminary and limited, the results suggest that graph theory may provide a broad mathematical foundation for diverse models of cognitive systems.

Attitude↗

Constructing a useful tool for characterizing amino acid conformers by means of quantum chemical and graph theory indices.

The aim of this work is to construct a tool to assist in the prediction of peptidic properties resulting from the exchange of two amino acids in a proteic chain. In the past others have used experimental properties for this purpose. However, the nature of these data sets severely limits their access to important properties pertaining to secondary structure, and hence the indices used cannot characterize different backbone conformers like alpha helix and beta strands, or side-chain conformations like gauche +, gauche - and trans. In this study we explore the importance of backbone and side-chain angles with regard to conformer similarity measured with theoretical properties calculated in an ab initio manner. For each of the 20 genetically encoded amino acids, we studied five conformers that correspond to alpha helical and beta strand structures, with three different side chain conformations for each, defined solely by their angles phi, psi and chi1. This methodology allowed each of the 108 conformers to be represented by a mathematical object without ambiguity. The peptidic chain was emulated using two capping models to simulate the effect of nearest neighbors. These are OHC-Xaa-NH2 and Ala-Xaa-Ala, where Xaa is the conformer of interest. We then calculated 40 ab initio quantum chemical and graph theory indices for each backbone-side-chain conformer to obtain a characterization and classification scheme. We found that: (1) while backbone structure is very important to conformer similarity, side-chain conformations do not cluster together in a top-level manner; (2) amino acids with pi electrons group together independent of backbone conformation.

Amino Acids↗

Topology and graph theory applied to cortical anatomy may help explain working memory capacity for three or four simultaneous items.

Cognitive experimentation suggests that at any single instant only three or four items ("chunks") are simultaneously prominent as a working memory (WM) trace, if we disregard the rehearsal component of WM. The reason for small WM capacity may concern combinatorial manageability. How might the neural representations of these few coactive chunks occupy a spatially distributed set of areas of the sheet-like cortex, while providing both order and flexibility to associate items in WM? Each attribute of each simultaneously active WM item must have broad access to the representational facilities of the cortical sheet, comprising tens of thousands of modular "cortical columns." The two hypothesized neural levels of WM during any moment of cognition comprise (a) "binding" together of many distributed attribute representations within each respective WM chunk, and (b) combinatorial play among three or four WM chunk-representations. Anatomical and functional evidence of cortical unity through its depth suggests that cortex may be viewed as essentially planar in its distribution of activations. Thus, a moment's WM is hypothesized here to reside in myriad activated cortical planar "patches," each subdivided into up to four amoeboid "subpatches." Two different lines of topological reasoning suggest orderly associations of such representations. (1) The four-color principle of map topology, and the related K(4) is planar theorem of graph theory, imply that if a small cortical area is dynamically subdivided into no more than four, discretely bounded planar subareas, then each such segment has ample free access to each of the others. (2) A hypothetical alternative to such associative adjacency of simultaneously active cortical representations of chunk-attributes is associative overlap, whereby, in dense cortical neuropil, activated subpatches behave like Venn diagrams of intersecting sets. As the number of Venn-like coactive subpatches within a patch increases, maintaining ad hoc associativity among all combinations requires exponentially proliferating intersections. Beyond four, serpentine subpatch shapes are required, which could easily lead to pathologies of omission or commission. As hypothesized by many researchers, the binding of the widely distributed cortical modules that represent a given chunk may involve synchrony or coherence of a single EEG frequency. Elsewhere, I have conjectured that such a binding frequency for a single chunk may bear a harmonic relationship with the additional EEG frequencies that are simultaneously binding the other WM chunks. Other possible mechanisms of binding have also been hypothesized. Whatever the mechanism, the many attributes of a moment's complement of three or four WM chunks must generally have an accidental relationship with the spatial distribution of the cortical feature analyzers that must be activated to represent those attributes. Therefore, the cortex may need, and have, comprehensive anatomical connections of each of its modules for representing an attribute (or of small redundant module groupings) with every other. If such whole-part cortico-cortical connections are somehow exploited not only to fully represent each cognitive chunk in its bound-together attributes, but also to bring the major business of intensive WM information processing down to the level of local circuits, in the sorts of topological patterning hypothesized here, there may be two adaptive results: (1) Time and other economies would be achieved in the reduction of activity in distant cortico-cortical connections to lower-energy global orchestration, or binding processes. (2) The piecemeal local topological limit to four subpatches would be writ large, across the entire cortex, preventing an unconstrained combinatorial explosion of associations among all attributes of all three or four simultaneously active chunks. Such hypothetical convergence to foci in local subpatch interactions might take place primarily in association cortex, and/or it might involve temporary shifts in response properties in some cortical subpat might involve temporary shifts in response properties in some cortical subpatches. Quantitative studies of the densely packed cortical fine structure, by Braitenberg and Schüz, and others, seem potentially consistent with this vision of cortical function in cognition.

Animals↗

Application of graph theory to analysis of Langerhans cell types on the basis of their cell nuclei ultrastructure.

Studies were performed aimed at estimating similarities and differences in ultrastructure of Langerhans cells of albino mouse epidermis. Graph theory-based analysis of surface and circumference of nucleus cross-section, heterochromatin and interchromatin area permitted to advance a view that Langerhans cells constitute a uniform cell population, as far as morphology of cell nucleus is concerned.

Animals↗

Random walks and chemical graph theory.

Simple random walks probabilistically grown step by step on a graph are distinguished from walk enumerations and associated equipoise random walks. Substructure characteristics and graph invariants correspondingly defined for the two types of random walks are then also distinct, though there often are analogous relations. It is noted that the connectivity index as well as some resistance-distance-related invariants make natural appearances among the invariants defined from the simple random walks.

Journal Article↗

[An attempt to use graph theory for the description of demographic events].

"The possibility is discussed of using graphs to describe a sequence of demographic events such as births, marriages and deaths. Among all sequences of demographic events those with perturbations are singled out. The following perturbations are defined: childlessness, premature termination of childbearing caused by the death of the female or termination of marriage due to divorce or death of spouse, remarriage of females and pre-marital births." (SUMMARY IN ENG, RUS)

Demography↗

Integration of graph theory and quantum chemistry for structure-activity relationships.

The objective of this article is to outline both graph-theoretically based and quantum chemically based structural indices of potential use in quantitative structure activity correlations. We consider graph-theoretical indices such as the connectivity index, topological index, Wiener index and molecular ID indices. Several structural and geometry-dependent indices can be derived from semiempirical and ab initio quantum calculations based on the charge densities, overlap matrices, frontier orbitals, molecular hardness, free valence, density matrices, quantum spectral difference indices, quantum spectral indices and bond matrices. Finally, the use of electrostatic potentials and charge densities for the prediction of reactive sites will be discussed.

Butadienes↗

Graph theory for fused cubic clusters of water dodecamer.

The stable structures of the fused cubic water cluster (H2O)12 are examined using graph theoretical techniques and ab initio calculations. The calculations are obtained by scanning the symmetry of digraph structures of hydrogen-bond network spanning 12 oxygen atom vertexes. Using the Pólya theorem the cycle index expressions for 12 vertexes and 20 edges of a cuboid in point-group symmetry D(4h) are developed. A total of 91 energy-allowed fused cubic structures are obtained, which are classified by 8 point-group symmetries: 1 D(2h), 2 S4, 5 C4, 1 D2, 11 C2, 10 C(i), 1 C(s), and 60 C1. An energy level diagram of the structures reveals 14 bands that correspond to 14 unique two-colored graphs derived from the distributions of four free hydrogens of the cluster.

Macromolecular Substances↗

An algorithm based on graph theory for the assembly of contigs in physical mapping of DNA.

An algorithm is described for mapping DNA contigs based on an interval graph (IG) representation. In general terms, the input to the algorithm is a set of binary overlapping relations among finite intervals spread along a real line, from which the algorithm generates sets of ordered overlapping fragments spanning that line. The implications of a more general case of the IG, called a probe interval graph (PIG), in which only a subset of cosmids are used as probes, are also discussed. In the specific case of cosmids hybridizing to regions of a YAC, the algorithm takes cross-hybridization information using the cosmids as probes, and orders them along the YAC; if gaps exist due to insufficient coverage of cosmid contigs along the length of the YAC, repetitive use of the algorithm generates sets of ordered overlapping fragments. Both the IG and the PIG can expose problems caused by false overlaps, such as hybridizations due to repetitive elements. The algorithm, has been coded in C; CPU time is essentially linear with respect to the number of cosmids analyzed. Results are presented for the application of a PIG to cosmid contig assembly along a human chromosome 13-specific YAC. An alignment of 67 cosmids spanning a YAC took 0.28 seconds of CPU time on a Convex 220 computer.

Algorithms↗

Application of graph theory to detect disconnected structures in a crystallographic database: copper oxide perovskites as a case study

Every crystal structure can be described as a graph with atoms as vertices and bonds as edges. Although such a graph loses the space arrangement of atoms and symmetry elements, it can mathematically represent the connectivity between atoms. This topological approach was used to develop a new method for detecting disconnected structures, in which individual atoms or structural fragments are located too far from each other, forming impossibly large gaps. Approximately 2300 perovskite-related crystal structures have been extracted from the Inorganic Crystal Structure Database (in 1999) and the maximum disconnecting distances, and the relations between them and the ionic radii of elements, have been analysed. Several disconnected structures, which are erroneous by our definition, have been revealed. Conventional tests for crystallographic data checking did not detect those entries.

Journal Article↗

A study of chemical systems using signal flow graph theory.

Photochemistry of giant planets and their satellites is characterized by numerous reactions involving a lot of chemical species. In the present paper, chemical systems are modeled by signal flow graphs. Such a technique evaluates the transmission of any input into the system (solar flux, electrons ... ) and gives access to the identification of the most important mechanisms in the chemical system. This method is applied to the production of hydrocarbons in the atmospheres of giant planets. In particular, the production of C2H6 in the atmosphere of Neptune from the photodissociation of CH4 is investigated. Different pathways of dissociation of CH4 are possible from L alpha radiation. A chemical system containing 14 species and 30 reactions including these different pathways of dissociation is integrated. The main mechanism of production of C2H6 is identified and evaluated for each model of dissociation. The importance of various reaction pathways as a function of time is presented.

Ethane↗

A study of chemical systems using signal flow graph theory: application to Neptune.

Photochemistry of giant planets and their satellites is characterized by numerous reactions involving many chemical species. In the present paper, chemical systems are modeled by signal flow graphs. Such a technique evaluates the transmission of any input into the system (solar flux, electrons...) and gives access to the identification of the most important mechanisms in the chemical system. For a given chemical system, we first evaluate rate coefficients. Then, in order to obtain concentrations of each compound, we integrate the set of continuity equations by Gear's method. Gear's method is chosen rather than another classical method because it is recommended for a system of stiff equations due to the existence of greatly differing time constants. Finally, the technique of signal flow graphs is used. This method is applied to the production of hydrocarbons in the atmospheres of giant planets. In particular, the production of C2H6 in the atmosphere of Neptune from the photodissociation of CH4 is investigated. Different paths of dissociation of CH4 are possible from L alpha radiations. A chemical system containing 14 species and 30 reactions including these different paths of dissociation is integrated. The main mechanism of production of C2H6 is identified and evaluated for each model of dissociation. The importance of various reaction paths as a function of time is discussed.

Atmosphere↗