Alkanes with the First Three Maximal/Minimal Modified First Zagreb Connection Indices.

The modified first Zagreb connection index ( Z C 1 * ) is a molecular descriptor, which was initially appeared within a formula of the total electron energy of alternant hydrocarbons in 1972. In a recent paper [A. Ali, N. Trinajstic, A novel/old modification of the first Zagreb index, Mol. Inform. 37 (2018) 1800008], it was observed that the molecular descriptor Z C 1 * correlates well with the entropy and acentric factor of octane isomers. In this article, the molecules with the first three maximal Z C 1 * values as well as the first three minimal Z C 1 * values are determined from the family of all alkanes with n carbon atoms, for n ≥ 6 . This extends the main results of the aforementioned paper.

A Novel/Old Modification of the First Zagreb Index.

In the seminal paper [I. Gutman, N. Trinajstic, Chem. Phys. Lett. 1972, 17, 535-538], it was shown that total electron energy (Eπ ) of any alternant hydrocarbon depends on the sum of the squares of the degrees of the corresponding molecular graph. Nowadays, this sum is known as the first Zagreb index. In the same paper, another molecular descriptor was proved to influence Eπ , but that descriptor was never restudied explicitly. We call this descriptor as modified first Zagreb connection index and denote it by ZC1* . In this paper, chemical applicability of the molecular descriptor ZC1* is tested for the octane isomers. Some basic properties of ZC1* are also established here. Furthermore, the alkanes with maximum and minimum ZC1* values are determined from the class of all alkanes having fixed number of carbon atoms.

The sum-connectivity index--an additive variant of the Randic connectivity index.

This review discusses structure-property modeling applications of a novel variant of the Randic connectivity index that is called the sum-connectivity index. We compare published one-descriptor quantitative structure-property relationship (QSPR) models obtained with the new sum-connectivity index and with the Randic connectivity index, called here the product-connectivity index. Additionally, the efficiency of both variants of connectivity indices in QSPR modeling is tested on five datasets of alkanes and two datasets of polycyclic hydrocarbons. Several physicochemical properties of alkanes (i.e. boiling and melting points, retention index, molar volume, molar refraction, heat of vaporization, standard Gibbs energy of formation, critical temperature, critical pressure, surface tension, density) and π- electronic energies of two sets of polycyclic hydrocarbons were correlated with the product- and sum-connectivity indices. A comparison of these QSPR models shows that both variants of connectivity indices are equivalent, and only slightly (but not significantly) better results are obtained with the sum-connectivity index. Inter-correlations between the product- and sum-connectivity indices are mostly linear with a slope very close to 1.0 for alkanes, and with a slope more different from 1.0 (0.88) for polycyclic compounds. The comparative analysis presented here supports the use of the sumconnectivity index in QSPR/QSAR studies together with the product-connectivity index. Further studies on larger and more heterogeneous datasets should test the sum-connectivity index in QSPR/QSAR models.

Master connectivity index and master connectivity polynomial.

We give the standard expressions of the Zagreb indices, Randic indices and their variants. Then we present the master connectivity index and show how this index can generate all connectivity indices of both varieties. We also present the master connectivity polynomial and show the relationship between this polynomial and the master connectivity index. Because of this relationship, the master connectivity polynomial can also be used to generate connectivity indices.

On sum-connectivity matrix and sum-connectivity energy of (molecular) graphs.

If G is a (molecular) graph with n vertices, and di is the degree of its i-th vertex, then the sum-connectivity matrix of G is the n × n matrix whose (i, j) -entry is equal to 1/√di + dj if the i-th and the j-th vertices are adjacent and 0 otherwise. The sum-connectivity energy of a graph G is defined as the sum of the absolute values of the eigenvalues of the sum-connectivity matrix. Some properties including upper and lower bounds for the eigenvalues of the sum-connectivity matrix and the sum-connectivity energy are established, and the extremal cases are characterized.

Bond dissociation enthalpies calculated by the PM3 method confirm activity cliffs in radical scavenging of flavonoids.

Radical scavenging potency of flavonoids is associated with activity cliffs, i.e., small chemical modifications on flavonoid core can have a significant effect on activity. The presence or absence of the 3',4'-diOH and/or 3-OH group may serve as an activity switch for radical scavenging. The physicochemical background of such an indicator variable, defined previously (Amic et al. (2003) Croat Chem Acta 76:55-61), is confirmed by computation of bond dissociation enthalpies and selecting the minimal of all values relating to flavonoid OH groups. Bond dissociation enthalpies for hydrogen abstraction from OH groups for 29 flavonoids were calculated by the PM3 method. Minimal bond dissociation enthalpy values were obtained for OH groups attached to C-3, C-3' and C-4' positions, and they correspond to the previously introduced indicator variable. Taking into account some driving forces of the radical scavenging mechanism, it is possible to relate structural characteristics of flavonoids to their radical scavenging potency as well as to develop reliable structure-activity models.

Antioxidant QSAR modeling as exemplified on polyphenols.

Methodology for deriving quantitative structure-activity relationship (QSAR) models based on computed molecular descriptors, representing numerically structural features of polyphenols, and applicable to the antioxidant activity of polyphenols is delineated. The application of this methodology is illustrated on a data set of 100 polyphenols. Prior to the computation of molecular descriptors, molecular structures are coded in the SMILES form, a computer-acceptable version of structure, and then converted to the 3D form by the CORINA program. Using 3D structures, molecular descriptors can be calculated by one of several programs developed (we used the DRAGON program in this study). Finally, using computer program for selection of most important descriptors in the model, a two-descriptor model is selected and its use is illustrated.

SAR and QSAR of the antioxidant activity of flavonoids.

Flavonoids are a group of naturally occurring phytochemicals abundantly present in fruits, vegetables, and beverages such as wine and tea. In the past two decades, flavonoids have gained enormous interest because of their beneficial health effects such as anti-inflammatory, cardio-protective and anticancer activities. These findings have contributed to the dramatic increase in the consumption and use of dietary supplements containing high concentrations of plant flavonoids. The pharmacological effect of flavonoids is mainly due to their antioxidant activity and their inhibition of certain enzymes. In spite of abundant data, structural requirements and mechanisms underlying these effects have not been fully understood. This review presents the current knowledge about structure-activity relationships (SARs) and quantitative structure-activity relationships (QSARs) of the antioxidant activity of flavonoids. SAR and QSAR can provide useful tools for revealing the nature of flavonoid antioxidant action. They may also help in the design of new and efficient flavonoids, which could be used as potential therapeutic agents.

Toxicity of aliphatic ethers: a comparative study.

The CROMRsel procedure was used to model the toxicity of aliphatic ethers against mice. The best model obtained is based on three molecular descriptors and is a better model than other QSAR models from the literature. The only comparable model is one by Ren, based on four descriptors.

On reformulated Zagreb indices.

Zagreb indices were reformulated in terms of the edge-degrees instead of the vertex-degrees as the original Zagreb indices. Three types of Zagreb indices were considered: original, modified and variable Zagreb indices. It is found that the optimum exponent of the variable reformulated Zagreb M2 index (v = -1/2) is identical with the exponent of the vertex-connectivity index, which is the most used topological index in QSPR and QSAR. The close relationship between the graph and its line graph is used to relate the original and reformulated indices.

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.

Use of variable selection in modeling the secondary structural content of proteins from their composition of amino acid residues.

The possibility of prediction of protein secondary structure content from composition of their amino acid residues can help in bridging the gap between proteins of known primary sequence having an unknown secondary structure. Almost all recently published models for understanding the relationship between composition (frequency of occurrence) of amino acid residues and secondary structure content of proteins involved composition of all 20 amino acid residues. However, it is well-known that many amino acid residues are mutually similar according to their physicochemical properties (hydrophobicity, hydrophilicity, charge, size, etc.). Because of that, we were motivated to investigate the possibility of reduction of the total number of terms (frequencies of amino acid residues) in the models for describing the relation between the composition of amino acid residues and the percentage of residues belonging to alpha, beta, and coil secondary structure. For this purpose, the CROMRsel algorithm (J. Chem. Inf. Comput. Sci. 1999, 39, 121-132) for selection of a small subset of the most important variables/descriptors into the multiregression (MR) models, i.e., frequency of occurrence of amino acid residues in proteins, was used. Analysis was performed on a data set containing 475 proteins, taken from Proteins 1996, 25, 157-168. A complete data set was partitioned into a 317-protein training set and 158-protein test set. The best possible linear models containing I=1, ..., 20 frequencies were selected among all 20 frequencies of occurrence of amino acid residues on the 317-protein training set, and were used for performing prediction of the corresponding percentage of secondary structure content on the 158-protein test set. For the 317-protein data set the best selected concise models for the alpha, beta, and coil secondary structure contain only 9, 5, and 8 frequencies, respectively. Selected concise models are of the same or better fitted, cross-validated, and predictive statistical parameters than the models containing all 20 frequencies. Additionally, for each I (I=1, ...., 20) 30 the best possible random models were selected. In each case, the best possible real models are much better than each of the best possible random models, showing clearly that there is no risk of a chance correlation (what one could expect due to the application of an exhaustive search for the best model having I frequencies among all 20!/I!(20-I)! possible models). Finally, the best selected models on the complete 475-protein data set for the alpha, beta, and coil secondary structure contain only 7, 4, and 7 frequencies of amino acid residues, respectively. These models are much simpler and have better fitted and cross-validated errors than the corresponding models from the literature, that were obtained without using a procedure for selection of the most important frequencies of amino acid residues in proteins.

On use of the variable Zagreb vM2 index in QSPR: boiling points of benzenoid hydrocarbons.

The variable Zagreb (v)M(2) index is introduced and applied to the structure-boiling point modeling of benzenoid hydrocarbons. The linear model obtained (the standard error of estimate for the fit model S(fit)=6.8 degrees C) is much better than the corresponding model based on the original Zagreb M2 index (S(fit)=16.4 degrees C). Surprisingly,the model based on the variable vertex-connectivity index (S(fit)=6.8 degrees C) is comparable to the model based on (v)M2 index. A comparative study with models based on the vertex-connectivity index, edge-connectivity index and several distance indices favours models based on the variable Zagreb (v)M2 index and variable vertex-connectivity index.However, the multivariate regression with two-, three- and four-descriptors gives improved models, the best being the model with four-descriptors (but (v)M2 index is not among them) with S(fit)=5 degrees C, though the four-descriptor model contaning (v)M2 index is only slightly inferior (S(fit)=5.3 degrees C).

Toward generating simpler QSAR models: nonlinear multivariate regression versus several neural network ensembles and some related methods.

In this study we want to test whether a simple modeling procedure used in the field of QSAR/QSPR can produce simple models that will be, at the same time, as accurate as robust Neural Network Ensemble (NNE) ones. We present results of application of two procedures for generating/selecting simple linear and nonlinear multiregression (MR) models: (1) method for selecting the best possible MR models (named as CROMRsel) and (2) Genetic Function Approximation (GFA) method from the Cerius2 program package. The obtained MR models are strictly compared with several NNE models. For the comparison we selected four QSAR data sets previously studied by NNE (Tetko et al. J. Chem. Inf. Comput. Sci. 1996, 36, 794-803. Kovalishyn et al. J. Chem. Inf. Comput. Sci. 1998, 38, 651-659.): (1) 51 benzodiazepine derivatives, (2) 37 carboquinone derivatives, (3) 74 pyrimidines, and (4) 31 antimycin analogues. These data sets were parameterized with 7, 6, 27, and 53 descriptors, respectively. Modeled properties were anti-pentylenetetrazole activity, antileukemic activity, inhibition constants to dihydrofolate reductase from MB1428 E. coli, and antifilarial activity, respectively. Nonlinearities were introduced into the MR models through 2-fold and/or 3-fold cross-products of initial (linear) descriptors. Then, using the CROMRsel and GFA programs (J. Chem. Inf. Comput. Sci. 1999, 39, 121-132) the sets of I (I < or = 8, in this paper) the best descriptors (according to the fit and leave-one-out correlation coefficients) were selected for multiregression models. Two classes of models were obtained: (1) linear or nonlinear MR models which were generated starting from the complete set of descriptors, and (2) nonlinear MR models which were generated starting from the same set of descriptors that was used in the NNE modeling. In addition, the descriptor selection method from CROMRsel was compared with the GFA method included in the QSAR module of the Cerius2 program. For each data set it has been found that the MR models have better cross-validated statistical parameters than the corresponding NNE models and that CROMRsel selects somewhat better MR models than the GFA method. MR models are also much simpler than NNEs, which is the important surprising fact, and, additionally, express calculated dependencies in a functional form. Moreover, MR models were shown to be better than all other models obtained by different methods on the same data sets ("old" multivariate regressions, functional-link-net models, back-propagation neural networks, genetic algorithm, and partial least squares models). This study also indicated that the robust NNE models cannot generate good models when applied on small data sets, suggesting that it is perhaps better to apply robust methods (like NNE ones) on larger data sets.

Atomic walk counts of negative order.

Atomic walk counts (awc's) of order k (k > or = 1) are the number of all possible walks of length k which start at a specified vertex (atom) i and end at any vertex j separated by m (0 < or = m < or = k) edges from vertex i. The sum of atomic walk counts of order k is the molecular walk count (mwc) of order k. The concept of atomic and molecular walk counts was extended to zero and negative orders by using a backward algorithm based on the usual procedure used to obtain the values of mwc's. The procedure can also be used in cases in which the adjacency matrix A related to the actual structure is singular and therefore A(-1) does not exist. awc's and mwc's of negative order may assume noninteger and even negative values. If matrix A is singular, atomic walk counts of zero order may not be equal to one.

Nanotubes: number of Kekulé structures and aromaticity.

Carbon nanotubes (CNTs) are composed of cylindrical graphite sheets consisting of sp(2) carbons. Due to their structure CNTs are considered to be aromatic systems. In this work the number of Kekulé structures (K) in "armchair" CNTs was estimated by using the transfer matrix technique. All Kekulé structures of the cyclic variants of naphthalene and benzo[c]phenanthrene have been generated and the basic patterns have been obtained. From this information the elements of the transfer matrix were derived. The results obtained indicate that K (and the resonance energy) is greater if tubulenes are extended in the vertical than in the horizontal direction. Tubulenes are therefore more stabile than cyclic strips. An illustration, obtained by using scanning probe microscope, has been attached to affirm the existence of thin CNTs.