Graph-theoretical Descriptors Research Articles

Graph theoretical descriptors differentiate d-Mannose isomers in the principal component proposed feature space: A computational approach

The intricate nature of carbohydrates, particularly monosaccharides, stems from the existence of several chiral centers within their tertiary structures. Predicting and characterizing the molecular geometries and electrostatic landscapes of these substances is difficult due to their complex electrical properties. Moreover, these structures can display a substantial degree of conformational flexibility due to the presence of many rotatable bonds. Moreover, identifying and distinguishing between D and L enantiomers of monosaccharides presents a significant analytical obstacle since there is a need for empirically measurable properties that can distinguish them. This work uses Principal Component Analysis (PCA) to explore the chemical information included in 3D descriptors in order to comprehend the conformational space of d-Mannose stereoisomers. The isomers may be discriminated by utilizing 3D matrix-based indices, geometrical descriptors, and RDF descriptors. The isomers can be distinguished by descriptors, such as the Harary-like index from the reciprocal squared geometrical matrix (H_RG), Harary-like index from Coulomb matrix (H_Coulomb), Wiener-like index from Coulomb matrix (Wi_Coulomb), Wiener-like index from geometrical matrix (Wi_G), Graph energy from Coulomb matrix (SpAbs_Coulomb), Spectral absolute deviation from Coulomb matrix (SpAD_Coulomb), and Spectral positive sum from Coulomb matrix (SpPos_Coulomb). Among these descriptors, the first two, H_RG and H_Coulomb, perform the best in differentiation among the 3D-Matrix-Based Descriptors (3D-MBD) class. The results obtained from this study provide a significant chemical insight into the structural characteristics of the compounds inside the graph theoretical framework. These findings are likely to serve as the basis for developing new methods for analytical experiments.

Structure-property modeling of coumarins and coumarin-related compounds in pharmacotherapy of cancer by employing graphical topological indices.

Coumarins, a subgroup of colorless and crystalline oxygenated heterocyclic compounds originally discovered in the plant Dipteryx odorata, were the subject of a recent study investigating their quantitative structure-activity relationship (QSAR) in cancer pharmacotherapy. This study utilized graph theoretical molecular descriptors, also known as topological indices, as a numerical representation method for the chemical structures embedded in molecular graphs. These descriptors, derived from molecular graphs, play a pivotal role in quantitative structure-property relationship (QSPR) analysis. In this paper, intercorrelation between the Balban index, connective eccentric index, eccentricity connectivity index, harmonic index, hyper Zagreb index, first path Zagreb index, second path Zagreb index, Randic index, sum connectivity index, graph energy and Laplacian energy is studied on the set of molecular graphs of coumarins. It is found that the pairs of degree-based indices are highly intercorrelated. The use of these molecular descriptors in structure-boiling point modeling was analyzed. Finally, the curve-linear regression between considered molecular descriptors with physicochemical properties of coumarins and coumarin-related compounds is obtained.

Mixed Metric Dimension of Certain Carbon Nanocone Networks

Chemical graph theory is an extension of mathematical chemistry that explores chemical phenomena and entities using the conceptual frameworks of graph theory. Chemical graph theory, in particular, uses chemical graphs to represent molecular structures. This chemical graph represents bonds and atoms, respectively, as edges and vertices. Cheminformatics predominantly uses chemical graphs as an essential data type for depicting chemical structures. The computable features of graphs lay the groundwork for (quantitative) structure-property and structure-activity predictions, which is an important component of cheminformatics. The physical and chemical characteristics of chemical compounds are thus reflected in these graphs, which can subsequently be reduced to graph-theoretical descriptors or indices. One of the most extensively researched distance-based graph descriptors is the resolving set W (metric dimension), which differentiates each pair of distinct vertices in every connected simple graph. The mixed metric dimension, which is the most significant variant of metric dimension, is determined for the complex molecular graph of a one-pentagonal carbon nanocone (PCN) in this manuscript. We show that only three distinct non-neighboring vertices (least possible requirement) from PCN can be adopted to uniquely identify all of the edges as well as vertices in PCN.

Curvilinear regression analysis of benzenoid hydrocarbons and computation of some reduced reverse degree based topological indices for hyaluronic acid-paclitaxel conjugates

Graph theoretical molecular descriptors alias topological indices are a convenient means for expressing in numerical form the chemical structure encoded in a molecular graph. The structure descriptors derived from molecular graphs are widely used in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies. The reason for introducing new indices is to obtain predictions of target properties of considered molecules that are better than the predictions obtained using already known indices. In this paper, we apply the reduced reverse degree based indices introduced in 2021 by Vignesh et al. In the QSPR analysis, we first compute the reduced reverse degree based indices for a family of benzenoid hydrocarbon molecules and then we obtain the correlation with the Physico-chemical properties of the considered molecules. We show that all the properties taken into consideration for the benzenoid hydrocarbons can be very effectively predicted by the reduced reverse degree based indices. Also, we have compared the predictive capability of reduced reverse degree based topological descriptors against 16 existing degree based indices. Further, we compute the defined reduced reverse degree based topological indices for Hyaluronic Acid-Paclitaxel Conjugates (HAP)_{n}, n ge 1.

Unifying structural descriptors for biological and bioinspired nanoscale complexes.

Biomimetic nanoparticles are known to serve as nanoscale adjuvants, enzyme mimics and amyloid fibrillation inhibitors. Their further development requires better understanding of their interactions with proteins. The abundant knowledge about protein-protein interactions can serve as a guide for designing protein-nanoparticle assemblies, but the chemical and biological inputs used in computational packages for protein-protein interactions are not applicable to inorganic nanoparticles. Analysing chemical, geometrical and graph-theoretical descriptors for protein complexes, we found that geometrical and graph-theoretical descriptors are uniformly applicable to biological and inorganic nanostructures and can predict interaction sites in protein pairs with accuracy >80% and classification probability ~90%. We extended the machine-learning algorithms trained on protein-protein interactions to inorganic nanoparticles and found a nearly exact match between experimental and predicted interaction sites with proteins. These findings can be extended to other organic and inorganic nanoparticles to predict their assemblies with biomolecules and other chemical structures forming lock-and-key complexes.

A look inside the black box: Using graph-theoretical descriptors to interpret a Continuous-Filter Convolutional Neural Network (CF-CNN) trained on the global and local minimum energy structures of neutral water clusters.

We describe a method for the post-hoc interpretation of a neural network (NN) trained on the global and local minima of neutral water clusters. We use the structures recently reported in a newly published database containing over 5 × 106 unique water cluster networks (H2O)N of size N = 3-30. The structural properties were first characterized using chemical descriptors derived from graph theory, identifying important trends in topology, connectivity, and polygon structure of the networks associated with the various minima. The code to generate the molecular graphs and compute the descriptors is available at https://github.com/exalearn/molecular-graph-descriptors, and the graphs are available alongside the original database at https://sites.uw.edu/wdbase/. A Continuous-Filter Convolutional Neural Network (CF-CNN) was trained on a subset of 500 000 networks to predict the potential energy, yielding a mean absolute error of 0.002 ± 0.002 kcal/mol per water molecule. Clusters of sizes not included in the training set exhibited errors of the same magnitude, indicating that the CF-CNN protocol accurately predicts energies of networks for both smaller and larger sizes than those used during training. The graph-theoretical descriptors were further employed to interpret the predictive power of the CF-CNN. Topological measures, such as the Wiener index, the average shortest path length, and the similarity index, suggested that all networks from the test set were within the range of values as the ones from the training set. The graph analysis suggests that larger errors appear when the mean degree and the number of polygons in the cluster lie further from the mean of the training set. This indicates that the structural space, and not just the chemical space, is an important factor to consider when designing training sets, as predictive errors can result when the structural composition is sufficiently different from the bulk of those in the training set. To this end, the developed descriptors are quite effective in explaining the results of the CF-CNN (a.k.a. the "black box") model.

A Graph Theoretical Interpretation of Different Types of Energies of Elementary Particles, Atoms and Molecules

The present work illustrates a predictive method, based on graph theory, for different types of energy of subatomic particles, atoms and molecules, to be specific, the mass defect of the first thirteen elements of the periodic table, the rotational and vibrational energies of simple molecules (such as , H2, FH and CO) as well as the electronic energy of both atoms and molecules (conjugated alkenes). It is shown that such a diverse group of energies can be expressed as a function of few simple graph-theoretical descriptors, resulting from assigning graphs to every wave function. Since these descriptors are closely related to the topology of the graph, it makes sense to wonder about the meaning of such relation between energy and topology and suggests points of view helping to formulate novel hypotheses about this relation.

Ion aggregation in high salt solutions. IV. Graph-theoretical analyses of ion aggregate structure and water hydrogen bonding network.

Ions in high salt solutions form a variety of ion aggregates, from ion pairs to clusters and networks. Their influences on water hydrogen bonding (H-bonding) network structures have long been of great interest. Recently, we have shown that the morphological structures of ion aggregates can be analyzed by using a spectral graph analysis theory, where each ion cluster or ion network is represented by a properly defined graph with edges and vertices. Here, to further examine the network properties of ion aggregates and water H-bonding networks in high salt solutions, we consider a few representative graph-theoretical descriptors: clustering coefficient, minimum path length, global efficiency, and degree distribution of ion aggregates. From the molecular dynamics trajectories, these graph theoretical properties of ion aggregates and water structures in NaCl and kosmotropic solutions are calculated and shown to be strongly dependent on the two types of ion aggregate structures, i.e., ion cluster and ion network. Ion clusters in high NaCl solutions exhibit typical behaviors of scale free network. The corresponding graph theoretical properties of ion networks in high KSCN solutions are notably different from those of NaCl ion clusters and furthermore they are very similar to those of water hydrogen-bonding network. The present graph-theoretical analysis results indicate that the high solubility limits of KSCN and other ion-network-forming salts might originate from their ability to form a large scale morphological network that can be intertwined with co-existing water H-bonding network. Furthermore, it is shown that the graph-theoretical properties of water H-bonding network structures do not strongly depend on the nature of dissolved ions nor on the morphological structures of ion aggregates, indicating that water's H-bonding interaction and network-forming capability are highly robust. We anticipate that the present graph-theoretical analysis results of high salt solutions would provide important information on the Hofmeister ion effects on water structure.

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.

Chemistry Explained by Topology: An Alternative Approach

Molecular topology can be considered an application of graph theory in which the molecular structure is characterized through a set of graph-theoretical descriptors called topological indices. Molecular topology has found applications in many different fields, particularly in biology, chemistry, and pharmacology. The first topological index was introduced by H. Wiener in 1947 [1]. Although its very first application was the prediction of the boiling points of the alkanes, the Wiener index has demonstrated since then a predictive capability far beyond that. Along with the Wiener index, in this paper we focus on a few pioneering topological indices, just to illustrate the connection between physicochemical properties and molecular connectivity.

Use of Structure Codes (Counts) for Computing Topological Indices of Carbon Nanotubes: Sadhana (Sd) Index of Phenylenes and its Hexagonal Squeezes

Structural codes vis-a-vis structural counts, like polynomials of a molecular graph, are important in computing graph-theoretical descriptors which are commonly known as topological indices. These indices are most important for characterizing carbon nanotubes (CNTs). In this paper we have computed Sadhana index (Sd) for phenylenes and their hexagonal squeezes using structural codes (counts). Sadhana index is a very simple W-Sz-PItype topological index obtained by summing the number of edges on both sides of the elementary cuts of benzenoid graphs. It has the similar discriminating power as that of the Weiner (W)-, Szeged (Sz)-, and PI-indices.

The Graph-Theoretical Descriptors for Carbon Nanotubes (CNTs): Sadhana (Sd) Index of Phenylene (PH) and Its Hexagonal Squeeze (HS)

The Graph-Theoretical Descriptors for Carbon Nanotubes (CNTs): Sadhana (Sd) Index of Phenylene (PH) and Its Hexagonal Squeeze (HS)

Use of structure codes (counts) for computing Sadhana (Sd) index of phenylenes and its hexagonal squeezes for QSAR studies

Structural codes vis-a-vis structural counts, like polynomials of a molecular graph, are important in computing graph-theoretical descriptors which are commonly known as topological indices. These indices are most important for characterizing carbon nanotubes (CNTs). In this paper we have computed Sadhana index (Sd) for phenylenes and their hexagonal squeezes using structural codes (counts). Sadhana index is a very simple W-Sz-PI-type topological index obtained by summing the number of edges on both sides of the elementary cuts of benzenoid graphs. It has the similar discriminating power as that of the Weiner (W)-, Szeged (Sz)-, and PI-indices. The methodology is useful for computing Sadhana index in drug designing.

Predicting the vapour pressure of chemicals from structure: a comparison of graph theoretic versus quantum chemical descriptors

In this paper a set of graph theoretic molecular descriptors was used to predict the normal vapour pressure of a collection of 121 chlorinated organic chemicals. The easily calculated topological descriptors resulted in a robust quantitative structure–property relationship (QSPR) model with q2 of 0.988, which is comparable to a model published previously developed using the computationally expensive density functional theory (DFT) method at the B3LYP level (Becke three-parameter exchange, Lee–Yang–Parr correlation). The addition of computer-intensive quantum chemical descriptors, including polarizability, to the set of topological descriptors did not improve the predictive ability of the model.

True prediction of lowest observed adverse effect levels

A database of structurally heterogeneous chemical structures with their experimental values of Lowest Observed Adverse Effect Levels (LOAELs) was modeled using graph theoretical descriptors. Variable selection for multiple linear regression (MLR) and linear discriminant analysis (LDA) was accomplished by the Internal Test Set (ITS) method in order to achieve true predicted LOAEL values. The results obtained can be considered good if we take in count the structural diversity of the training set.

Predictability and prediction of lowest observed adverse effect levels in a structurally heterogeneous set of chemicals

A database of chronic lowest observed adverse effect levels (LOAELs) for 234 compounds, previously compiled from different sources (Toxicology Letters 79, 131–143 (1995)), was modelled using graph theoretical descriptors. This study reveals that data are not homogeneous. Only those data originating from the U.S. Environmental Protection Agency (EPA) reports could be well modelled by multilinear regression (MLR) and linear discriminant analysis (LDA). In contrast, data available from the specific procedures of the National Toxicology Program (NTP) database introduced noise and did not render good models either alone, or in combination with the EPA data.

Structural invariants for the prediction of relative toxicities of polychloro dibenzo-p-dioxins and dibenzofurans

Multivariate models are reported that can predict the relative toxicity of compounds with severe environmental impact, namely polychloro dibenzo-p-dioxins (PCDDs) and dibenzofurans (PCDFs). Multiple linear regression analysis (MLR) and partial least square projections of latent variables (PLS) show the usefulness of graph-theoretical descriptors, mainly topological charge indices (TCIs), in these series. The general trends of the group are correctly reproduced and better results are presented than have previously been published. In general, the more toxic compounds exhibit more symmetric molecular structures.

Chemical genomic profiling of biological networks using graph theory and combinations of small molecule perturbations.

Genome-wide measurements of multiple experimental samples yield rich fingerprints for comparison and interpretation. Here, a two-dimensional matrix of the cellular effects of all possible pairwise combinations of 24 small molecules, each with a different structure and bioactivity, was used to profile otherwise isogenic deletion strains of the yeast Saccharomyces cerevisiae. Using principles from graph theory, we derived a discrete model of the data for each strain by encoding the information in the form of a binary adjacency matrix. This matrix was used to construct a graph composed of nodes representing small molecules and edges connecting combinations that inhibited cell cycle progression. Computation of a set of graph theoretic descriptors for each chemical genetic network provided a topological fingerprint that showed genotype-dependent fluctuations. Because the structure of the genetic network determines the structure of the chemical genetic network, multidimensional chemical genomic profiling can be used for the characterization of perturbations in biological networks or the networks themselves. This application of small molecules could be useful for discerning the molecular basis of highly complex biological phenotypes, including those involved in the susceptibility to or etiology of human disease.

What are the limits of applicability for graph theoretic descriptors in QSPR/QSAR? Modeling dipole moments of aromatic compounds with TOPS-MODE descriptors.

The numerous possibilities of using graph theoretic descriptors in QSPR/QSAR are analyzed, and some misunderstandings on the role of this theoretical approach in chemistry are clarified. Principal component analysis is used to obtain a property space for several physicochemical properties of aromatic compounds. It is proved that most of the QSPR applications of the graph-theoretic structure descriptors are concentrated to the description of properties in a very limited region of this property space. Here, we show that graph-theoretic approaches are also applicable to the modeling of physicochemical properties that are far away from this region traditionally studied. The molecular dipole moments of benzene derivatives, mono-, ortho-, meta-, and para-susbtituted, are modeled by using the Topological Sub-Structural Molecular Design (TOPS-MODE) approach. The TOPS-MODE approach used permits to calculate group dipole moments that are given for several substituents. The differences between these group dipoles and those obtained by simple difference between experimental values are analyzed. Some difficulties arising from this traditional way of deriving substituent constants are identified and analyzed.

On characterization of dose variations of 2-D proteomics maps by matrix invariants.

We explore the characterization of 2-D electrophoresis proteomics maps by certain structural invariants derived from matrixes constructed by considering for all pairs of spots in a proteomics maps the shortest (Euclidean) distances and distances measured along zigzag lines connecting protein spots of the neighboring abundance. This paper is a sequel to previous papers in which we outlined the idea of characterizing 2-D proteomics maps by graph-theoretical descriptors. To illustrate the approach, we selected data of Anderson et al. (Anderson, N. L.; Esquer-Blasco, R.; Richardson, F.; Foxworthy, P.; Eacho, P. The effects of peroxisome proliferators on protein abundances in mouse liver. Toxicol. Appl. Pharmacol. 1996, 137, 75-89) on protein abundance in mouse liver under a series of dose of peroxisome proliferator LY1711883. We found strong linear correlation between the experimentally applied doses and the leading eigenvalue of a D/D-type matrix (Randić, M.; Kleiner, A. F.; DeAlba, L. M. Distance/ distance matrices. J. Chem. lnf. Comput. Sci. 1994, 34, 277-286) constructed for the experimental proteomics maps.