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How-often-that-high graphs of serum cholesterol. Findings from the Scottish Heart Health and Scottish MONICA studies.

The Scottish Heart Health and Scottish MONICA studies included measurements of serum total cholesterol in 10,450 representative men and women aged 25-64 years recruited across Scotland in 1984-86. The results were typical of Britain as a whole. A new graph, called the HOTH graph ("how often that high?"), shows the percentage of the population with serum total cholesterol at or above any given value. Median (equal to mean) cholesterol levels in men were 5.5, 6.0, 6.3, and 6.2 mmol/l in successive 10-year age groups 25-64, whilst equivalent values in women were 5.2, 5.5, 6.4, and 7.2 mmol/l. By comparison with other countries these are high, but the percentage of the population above specific cutpoints is disproportionate, varying considerably by age and sex. Application of imported cholesterol management algorithms, based on global cut-points, would lead to an overwhelming caseload of patients needing intensive lipid investigation and management, dominated by older women, and incurring great costs. 35% of the population 25-64 years old would be at or above 6.5 mmol/l and 11% at or above 7.8 mmol/l. A population diet and multiple-risk-factor strategy would be more feasible and rational than a one-dimensional cholesterol cut-point approach.

Adult↗

Checking the projection display of multivariate data with colored graphs.

Projection methods such as principal component analysis (PCA), nonlinear mapping (NLM), and the self-organizing map (SOM) are valuable algorithms for visualizing multidimensional data in a two-dimensional plane. Unfortunately, the reduction of the dimensionality involves distortions. In an attempt to graphically localize the distortions of the projected data, we suggest superposing colored graphs onto the 2D plots. The color of the edges of these graphs encodes the original high-dimensional distances between the connected points. The method is applied to a cluster analysis of 37 biologically active compounds and 471 molecules represented by a structural 3D descriptor.

Computer Graphics↗

A reciprocal graph to plot the reciprocal serum creatinine over time.

In patients who have a linear decline in renal function over time, plotting the reciprocal serum creatinine versus time has been found to be useful in monitoring renal disease progression and predicting the start of dialysis. Unfortunately, producing such plots is cumbersome because of the inherent difficulties of plotting a reciprocal number on standard graph paper. This technical note presents a graph in which the y-axis is represented as a reciprocal axis. In this manner, one is able to directly plot the serum creatinine over time and must not rely on plotting the reciprocal value. This approach may make the plotting of such data easier for the nephrologist and make this clinical tool more useful.

Creatinine↗

Robustness of MetaNet graph models: predicting control of urea production in humans.

Urea production in human liver was described by a MetaNet graph, a flowchart-like representation of metabolic pathways that includes parameters for the kinetic constants of the constituent enzymes. Formal operations on the graph facilitate the identification of ligand-binding equilibria that participate in feedback regulation in the network of biochemical reactions. The state of the biochemical network is specified by the concentrations of the intermediates. At any particular time, the influence of an identified locus of regulation is proportional to the respective fractional saturation of the corresponding binding site. Enzymes that make or consume the feedback chemicals share in the control of the strength of the feedback signal in proportion to their fractional saturation. This model predicts control of urea production by the processes that deliver amino groups to the urea cycle enzymes more than by the cycle enzymes themselves. Mitochondrial membrane transport processes are important for transmission of information through the network, but irreversible enzymes and processes far from equilibrium control the strength of the feedback signal. Systematic variation of the parameter values by amounts comparable to the expected variability of their measured values indicated a high probability of invariance in the identities of the predicted control points. The properties of the model are consistent with those of error-tolerant scale-free networks. These results demonstrate the robustness of a MetaNet model's predictions with respect to uncertainties in the values of its parameters.

Computer Simulation↗

Comparison of chemical clustering methods using graph- and fingerprint-based similarity measures.

This paper compares several published methods for clustering chemical structures, using both graph- and fingerprint-based similarity measures. The clusterings from each method were compared to determine the degree of cluster overlap. Each method was also evaluated on how well it grouped structures into clusters possessing a non-trivial substructural commonality. The methods which employ adjustable parameters were tested to determine the stability of each parameter for datasets of varying size and composition. Our experiments suggest that both graph- and fingerprint-based similarity measures can be used effectively for generating chemical clusterings; it is also suggested that the CAST and Yin-Chen methods, suggested recently for the clustering of gene expression patterns, may also prove effective for the clustering of 2D chemical structures.

Algorithms↗

Application of a novel graph-theoretic folding degree index to the study of steroid-DB3 antibody binding affinity.

A novel folding degree index, together with other macromolecular descriptors, is used to study steroid-DB3 antibody interactions. This index is based on graph spectral moments of a matrix representing the dihedral angles of a protein backbone. The causes influencing the different order of binding affinity of steroids to DB3 antibody are identified. It is shown that the changes in the chain compactness of the DB3 antibody with respect to its center of mass (radius of gyration) is compensated by a change in the folding degree index in the contrary sense. In fact, the increment in compactness of chain L and the lower increment in the folding degree index of chain H are able to explain the variations in affinity for DB3 of the steroids studied. Consequently, the highest binding affinities are reached by increasing the compactness of chain L in DB3 at the same time that producing the smallest increment in the folding degree of chain H. This study shows the possibilities of application for the graph-theoretic folding degree index in studying drug-protein interactions.

Animals↗

Estimation of additive, dominance and epistatic variance components using finite locus models implemented with a single-site Gibbs and a descent graph sampler.

In a previous contribution, we implemented a finite locus model (FLM) for estimating additive and dominance genetic variances via a Bayesian method and a single-site Gibbs sampler. We observed a dependency of dominance variance estimates on locus number in the analysis FLM. Here, we extended the FLM to include two-locus epistasis, and implemented the analysis with two genotype samplers (Gibbs and descent graph) and three different priors for genetic effects (uniform and variable across loci, uniform and constant across loci, and normal). Phenotypic data were simulated for two pedigrees with 6300 and 12,300 individuals in closed populations, using several different, non-additive genetic models. Replications of these data were analysed with FLMs differing in the number of loci. Simulation results indicate that the dependency of non-additive genetic variance estimates on locus number persisted in all implementation strategies we investigated. However, this dependency was considerably diminished with normal priors for genetic effects as compared with uniform priors (constant or variable across loci). Descent graph sampling of genotypes modestly improved variance components estimation compared with Gibbs sampling. Moreover, a larger pedigree produced considerably better variance components estimation, suggesting this dependency might originate from data insufficiency. As the FLM represents an appealing alternative to the infinitesimal model for genetic parameter estimation and for inclusion of polygenic background variation in QTL mapping analyses, further improvements are warranted and might be achieved via improvement of the sampler or treatment of the number of loci as an unknown.

Epistasis, Genetic↗

An algorithm for sampling descent graphs in large complex pedigrees efficiently.

No exact method for determining genotypic and identity-by-descent probabilities is available for large complex pedigrees. Approximate methods for such pedigrees cannot be guaranteed to be unbiased. A new method is proposed that uses the Metropolis-Hastings algorithm to sample a Markov chain of descent graphs which fit the pedigree and known genotypes. Unknown genotypes are determined from each descent graph. Genotypic probabilities are estimated as their means. The algorithm is shown to be unbiased for small complex pedigrees and feasible and consistent for moderately large complex pedigrees.

Algorithms↗

Wiener index extension by counting even/odd graph distances.

Chemical structures of organic compounds are characterized numerically by a variety of structural descriptors, one of the earliest and most widely used being the Wiener index W, derived from the interatomic distances in a molecular graph. Extensive use of such structural descriptors or topological indices has been made in drug design, screening of chemical databases, and similarity and diversity assessment. A new set of topological indices is introduced representing a partitioning of the Wiener index based on counts of even and odd molecular graph distances. These new indices are further generalized by weighting exponents which can be optimized during the quantitative structure-activity/-property relationship (QSAR/QSPR) modeling process. These novel topological indices are tested in QSPR models for the boiling temperature, molar heat capacity, standard Gibbs energy of formation, vaporization enthalpy, refractive index, and density of alkanes. In many cases, the even/odd distance indices proposed here give notably improved correlations.

Journal Article↗

Graph valence shells as molecular descriptors.

We have introduced a new simple structural descriptor for molecules that is based on the count of the valence shells for vertices in molecular graphs. The construction of the new descriptor is illustrated on 2,3-dimethylhexane and is reported for the 18 octane isomers. The relationship of the new descriptor to the path numbers of a graph is discussed. It can be seen that the path counts and the count of valence of neighbor shells are related for paths of length two (and shells of range two). There is no appreciable correlation between the count of the longer paths and the count of the corresponding neighbor valence shells at larger separations. Use of the neighbor valence shells as molecular descriptors is illustrated on the boiling point, the entropy, and the density of octanes. An intriguing situation is observed for regressions involving considered properties of n-octane isomers C8H18 in that the paths of length two, three, and four and the shells of the range two, three, and four give identical multivariate regression statistics. An explanation for this somewhat unusual aspect of MRA (multiple regression analysis) is offered.

Journal Article↗

Automation of protein 2D proton NMR assignment by means of fuzzy mathematics and graph theory.

The novel methodology for protein 2D NMR assignment presented in this paper is based upon protein spin coupling graph theory analysis, fuzzy graph pattern recognition, and tree searching. The method required to formalize the whole assignment procedure into a logical system which can be properly processed by computer software is also discussed. Solutions for peak overlaps, spin coupling network overlaps, and details related to the automated assignment of BPTI are reported as well.

Algorithms↗

A new graph descriptor for molecules containing cycles. Application as screening criterion for searching molecular structures within large databases of organic compounds.

The search of molecular structures inside a large database of chemical compounds is a critical step for many computer programs used in several domains of chemistry. During the last years, the size of many chemical databases has dramatically increased, hence in the meantime, search engines needed to be more and more powerful. The speed and the efficiency of screening processes of the chemical compounds are thus essential. Looking forward for algorithms dedicated to structure and substructure search, we have developed a new graph descriptor for structures containing cycles in order to find efficient indexation and classification criteria of molecular structures. This graph descriptor can be used as a screening criteria for structure and substructure search in large databases of organic compounds.

Journal Article↗

Computational techniques for vertex partitioning of graphs.

A powerful vertex-partitioning algorithm is developed and applied for vertex partitioning of graphs of chemical and spectroscopic interest. The codes developed on the basis of these algorithms are tested and compared for performance with other methods based on the Morgan algorithm and the principal eigenvector algorithm based on the Givens-Householder method. The newly developed algorithm and codes appear to be more powerful than the Morgan and the principal eigenvector algorithms for vertex partitioning of graphs.

Chemistry, Physical↗

Modeling boiling points of cycloalkanes by means of iterated line graph sequences.

A class of models for predicting boiling points of cycloalkanes is put forward, based on iterated line graphs L(i), i = 1, 2,., of the molecular graph G = L(0). Let m(i) be the number of edges of L(i), i = 0, 1, 2,. The models analyzed are of the form a(0)m(i)()(0) + a(1)m(i)(1) + a(2)m(i)(2) +. + a(k)m(ik) + b. Our optimal QSPR formulas contain m(0), m(1), m(2), m(3), and/or m(4) but never m(5) and m(6). Their precision is as good as or better than the approximations recently reported by Rücker and Rücker (J. Chem. Inf. Comput. Sci. 1999, 39, 788-802).

Journal Article↗

Prediction of aquatic toxicity: use of optimization of correlation weights of local graph invariants.

Quantitative structure-activity relationships (QSARs) were developed for three sets of toxicity data. Chemicals in each set represented a number of narcoses and electrophilic mechanisms of toxic action. A series of quantitative structure-toxicity models correlating toxic potency with a number of optimization of correlation weights of local graph invariants were developed. In the case of the toxicity of a heterogeneous set of benzene derivatives to Tetrahymena pyriformis, the QSARs were based on the Descriptor of Correlation Weights (DCW) using atoms and extended connectivity (EC) graph invariants. The model [log (IGC(50)(-1)) = 0.0813 DCW(a(k),(3)EC(k)) + 2.636; n = 157, r(2) = 0.883, s = 0.27, F = 1170, Pr > F = 0.0001] based on third-order EC of 89 descriptors was observed to be best for the benzene data. However, fits for these data of > 0.800 were achieved ECs with as few as 23 variables. The relationship between the toxicity predicted by this model and experimental toxicity values for the test set [obs. log(IGC(50)(-1))) = 0.991 (pred. (log(IGC(50)(-1))) - 0.012; n = 60, r(2) = 0.863, s = 0.28, F = 372, Pr > F = 0.0001] is excellent. The utility of the approach was demonstrated by the model [log (IGC(50)(-1)) = 0.1744(DCW (a(k), (2)EC) - 3.505; n = 39, r(2) = 0.900, s = 0.35, F = 333, Pr > F = 0.0001] for the toxicity data for T. pyriformis exposed to halo-substituted aliphatic compounds and the model [log (IC(50)(-1)) = 0.1699(DCW (a(k), (2)EC)) - 2.610; n = 66, r(2) = 0.901, s = 0.31, F = 583, Pr > F = 0.0001] for the Vibrio fischeri toxicity data.

Animals↗

Immanants and immanantal polynomials of chemical graphs.

The much-studied determinant and characteristic polynomial and the less well-known permanent and permanental polynomial are special cases of a large class of objects, the immanants and immanantal polynomials. These have received some attention in the mathematical literature, but very little has appeared on their applications to chemical graphs. The present study focuses on these and also generalizes the acyclic or matching polynomial to an equally large class of acyclic immanantal polynomials, generalizes the Sachs theorem to immanantal polynomials, and sets forth relationships between the immanants and other graph properties, namely, Kekulé structure count, number of Hamiltonian cycles, Clar covering polynomial, and Hosoya sextet polynomial.

Journal Article↗

Variable neighborhood search for extremal graphs. 10. Comparison of irregularity indices for chemical trees.

Chemical graphs, as other ones, are regular if all their vertices have the same degree. Otherwise, they are irregular, and it is of interest to measure their irregularity both for descriptive purposes and for QSAR/QSPR studies. Three indices have been proposed in the literature for that purpose: those of Collatz-Sinogowitz, of Albertson, and of Bell's variance of degrees. We study their properties for the case of chemical trees. Structural conjectures are generated with the system AutoGraphiX, and most of them proved later by mathematical means. Analytical expressions for extremal values are obtained, and extremal graphs are characterized for the two last indices.

Journal Article↗

A graph-based genetic algorithm and its application to the multiobjective evolution of median molecules.

In this paper we propose a novel graph-based genetic algorithm for the evolution of novel molecular graphs from a predefined set of elements or molecular fragments with an external objective function. A brief overview of existing genetic algorithm approaches in molecular design is provided followed by a description of our approach. The paper continues to suggest a novel application of this program to the multiobjective evolution of median molecules that are structurally representative of a set of objective molecules. We conclude with a summary of our initial results along with a discussion of a variety of improvements and applications of our approach.

Algorithms↗