On the history of the connectivity index: from the connectivity index to the exact solution of the protein alignment problem.

We briefly review the history of the connectivity index from 1975 to date. We hope to throw some light on why this unique, by its design, graph theoretical molecular descriptor continues to be of interest in QSAR, having wide use in applications in structure-property and structure-activity studies. We will elaborate on its generalizations and the insights it offered on applications in Multiple Regression Analysis (MRA). Going beyond the connectivity index we will outline several related developments in the development of molecular descriptors used in MRA, including molecular ID numbers (1986), the variable connectivity index (1991), orthogonal regression (1991), irrelevance of co-linearity of descriptors (1997), anti-connectivity (2006), and high discriminatory descriptors characterizing molecular similarity (2015). We will comment on beauty in QSAR and recent progress in searching for similarity of DNA, proteins and the proteome. This review reports on several results which are little known to the structure-property-activity community, the significance of which may surprise those unfamiliar with the application of discrete mathematics to chemistry. It tells the reader many unknown stories about the connectivity index, which may help the reader to better understand the meaning of this index. Readers are not required to be familiar with graph theory.

Anthraquinone profile, antioxidant and antimicrobial activity of bark extracts of Rhamnus alaternus, R. fallax, R. intermedia and R. pumila.

The quantity of phenols, as well as antioxidant and antimicrobial activities, were investigated in bark of Rhamnus alaternus L., R. fallax Boiss., R. intermedia Steud. et Hochst., and R. pumila Turra from natural stands in Croatia. The most abundant anthraquinones in the investigated extracts were chrysophanol in R. alaternus (3.14 mg/g), emodin in R. pumila (0.339 mg/g), and physcion in R. fallax (2.70 mg/g) and R. intermedia (0.285 mg/g). The species exhibiting the highest antioxidant activity were R. fallax and R. pumila. A positive correlation was observed between total phenolic and flavonoid levels of the extracts and antioxidant activity in some of the assays. All species showed antimicrobial activity against Staphylococcus aureus, Pseudomonas aeruginosa, Escherichia coli, Candida albicans, Aspergillus niger and Microsporum gypseum with minimal inhibitory concentrations equal to or below 2.500 mg/mL. The results indicate that the investigated Rhamnus species are a source of anthraquinones and other phenols, which act as multifunctional antioxidants with antimicrobial activity.

On graphical representation of trans-membrane proteins.

In the first part of this paper, we present a novel graphical representation of proteins, which starts with constructing a map of a protein that is obtained from a matrix, the elements of which record the adjacencies of pairs of amino acids in the primary structure of a protein. Starting with the novel protein map, one interprets its matrix elements as vertices of a graph, which are labelled in sequential order as in the protein sequence. The nearest vertices are connected to the nearest neighbour which has a smaller label. In the second part of this paper, we describe the construction of protein binary codes that can serve as protein descriptors. This novel graphical representation of proteins is illustrated on segments of trans-membrane proteins, which are embedded in the membrane.

Very efficient search for protein alignment--VESPA.

A novel approach to the problem of protein alignments is described, which in comparison with existing approaches is visibly more efficient. This approach is based on superposition of amino acid adjacency matrices of a pair of proteins, which have been modified to record the sequential order of amino acids. As a result, one obtains simultaneously all segments of the two proteins which are shifted relative to one another by one or more positions in either directions, without need of a prior exhaustive search for an alignment that included unproductive directions and unknown displacement steps.

Spectral representation of reduced protein models.

We consider a spectrum-like two-dimensional graphical representation of proteins based on a reduced protein model in which 20 amino acids are grouped into five classes. This particular grouping of amino acids was suggested by Riddle and co-workers in 1997. The graphical representation is based on depicting sequentially the amino acids on five horizontal lines at equal separations. One-letter codes, B, O, U, X and Y, to which numerical values 1 to 5 have been assigned, are suggested as labels for the fictional amino acids that represent all the amino acids within each group. The approach is illustrated on ND6 proteins of eight species having from 168 to 175 amino acids. While visual inspection of the novel spectral graphical representations of proteins may reveal local similarities and dissimilarities of protein sequences, arithmetic manipulations of spectra offer an elegant route to graphic visualization of the degree of similarity for selected pairs of proteins.

Involvement of group I metabotropic glutamate receptors and glutamate transporters in the slow excitatory synaptic transmission in the spinal cord dorsal horn.

Our experiments demonstrate a novel role for group I metabotropic glutamate receptor (mGluR) subtypes 1 and 5 in generating a long-lasting synaptic excitation in the substantia gelatinosa (SG) and deep dorsal horn (DH) neurons of the rat spinal cord. In the present study we have investigated a slow excitatory postsynaptic current (EPSC), elicited by a brief high intensity (at Adelta/C fiber strength) and high frequency (20 or 100 Hz) stimulation of primary afferent fibers (PAFs) using whole-cell patch-clamp recordings from neurons located in the DH (laminae II-V) in spinal cord slices of young rats and wild-type and gene-targeted mice lacking mGluR1 subtype. The results shown here suggest that the activation of both mGluR1 and mGluR5 along with NK1 receptors, may be involved in the generation of the slow EPSC in the spinal cord DH. Inhibition of glial and neuronal glutamate transporters by DL-threo-beta-benzyloxyaspartate (TBOA) enhanced the group I mGluR-dependent slow EPSC about eightfold. Therefore, we conclude, that glutamate transporters strongly influence the group I mGluR activation by PAFs possibly at sensory synapses in the DH. Overall these data indicate that stimulus trains can generate a sustained and widespread glutamate signal that can further elicit prolonged EPSCs predominantly mediated by the group I mGluRs. These slow excitatory synaptic currents may have important functional implications for DH cell firing and synaptic plasticity of sensory transmission, including nociception.

On novel representation of proteins based on amino acid adjacency matrix.

A novel characterization of proteins is presented based on selected properties of recently introduced 20 x 20 amino acid adjacency matrix of proteins in which matrix elements count the occurrence of all 400 possible pair-wise adjacencies obtained by reading protein primary sequence from the left to the right. In particular we consider the characterization based on the sum and the difference of the rows and the corresponding columns, which characterize proteins by a pair of 20-component vectors. The approach is illustrated on a set of ND6 proteins of eight species.

Representation of proteins as walks in 20-D space.

A novel representation of proteins was introduced. It is independent of arbitrary decisions with respect to the choice of labels to be assigned to the 20 natural amino acids. The approach is based on an assignment of 20 unit vectors in 20-dimensional vector space to the 20 natural amino acids. Proteins are then represented by a walk, that is, a sequence of steps in the 20-dimensional space analogous to a walk in the (x, y) plane in the case of binary strings. A straightforward numerical characterization of proteins is obtained from the distance matrix associated with the walk representing the protein in 20-dimensional space combining the information on the Euclidean distance between various amino acids in protein sequence. The Line Distance matrix offers additional numerical characterization of proteins, while the lengths of steps of the walk in 20-D space allow construction of a "protein profile," which represents distribution of average lengths of the steps and their powers.

Novel numerical and graphical representation of DNA sequences and proteins.

We have introduced novel numerical and graphical representations of DNA, which offer a simple and unique characterization of DNA sequences. The numerical representation of a DNA sequence is given as a sequence of real numbers derived from a unique graphical representation of the standard genetic code. There is no loss of information on the primary structure of a DNA sequence associated with this numerical representation. The novel representations are illustrated with the coding sequences of the first exon of beta-globin gene of half a dozen species in addition to human. The method can be extended to proteins as is exemplified by humanin, a 24-aa peptide that has recently been identified as a specific inhibitor of neuronal cell death induced by familial Alzheimer's disease mutant genes.

Validation of counter propagation neural network models for predictive toxicology according to the OECD principles: a case study.

The OECD has proposed five principles for validation of QSAR models used for regulatory purposes. Here we present a case study investigating how these principles can be applied to models based on Kohonen and counter propagation neural networks. The study is based on a counter propagation network model that has been built using toxicity data in fish fathead minnow for 541 compounds. The study demonstrates that most, if not all, of the OECD criteria may be met when modeling using this neural network approach.

2-D graphical representation of proteins based on virtual genetic code.

We consider a novel 2-D graphical representation of proteins in which individual nucleic acids are represented as "spots" within a square frame distributed according to specific construction rules. The resulting "images" of proteins can not only serve to facilitate visual comparison of similarities and dissimilarities between lengthy protein sequences, but also offer a way for mathematical characterization of protein sequences, analogous to similar considerations for lengthy DNA sequences. Basically the approach is based on the concept of virtual genetic code, which is a hypothetical string of RNA nucleic acid bases, A, C, U and G, which generates reported protein sequences, without the knowledge of the actual genetic code that produces the protein.

Highly compact 2D graphical representation of DNA sequences.

Most 2D graphical representations of primary DNA sequences, while offering visual geometrical patterns for depicting sequences, do require considerable space if enough details of such representations are to be visible. In this contribution, we consider a highly compact graphical representation of DNA, which allows visual inspection and numerical characterization of DNA sequences having a large number of nucleic acid bases. The approach is illustrated on the DNA sequences of the first exon of human beta-globin. The same graphical approach not only allows one to depict differences in composition within a single DNA, but makes possible graphical representation of protein sequences, which have hitherto evaded similar 2D visual representations.

Peripheral inflamation-induced increase of AMPA-mediated currents and Ca2+ transients in the presence of cyclothiazide in the rat substantia gelatinosa neurons.

This study employing a rodent model of acute pain investigated the influence of carrageenan-induced inflammation on the ability of S-alpha-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptor activation to induce membrane currents and rises in cytosolic free calcium concentration ([Ca2+]i) in the rat substantia gelatinosa (SG) neurons using simultaneous whole-cell patch-clamp recording and fura-2 calcium imaging in spinal cord slices of L4-L5 segments. The novel finding of this study is that carrageenan-induced inflammation, in the presence of cyclothiazide, an inhibitor of AMPA receptor desensitization, produces a sustained facilitation of the AMPA-mediated membrane current and rises in [Ca2+]i in both the soma and proximal dendrites of SG neurons recorded on the injected side 3 h after the induction of inflammation. These results suggest that in carrageenan-inflamed rats AMPA receptors undergo some alterations that influence AMPA receptors desensitization and/or sensitivity to cyclothiazide.

Novel matrix invariants for characterization of changes of proteomics maps.

Previous studies on mathematical characterization of proteomics maps by sets of map invariants were based on the construction of a set of distance-related matrices obtained by matrix multiplication of a single matrix by itself. Here we consider an alternative characterization of proteomics maps based on a set of matrices characterizing local features of an embedded zigzag curve over the map. It is shown that novel invariants can well characterize proteomics maps. Advantages of the novel approach are discussed.

On graphical and numerical characterization of proteomics maps.

We outlined a mathematical approach suitable for characterization of experimental data given by 2-D densitograms. In particular we consider numerical characterization of proteomics maps. The basis of our approach is to order "spots" of a 2-D map and assign them unique labels (that in general will depend on the criteria used for ordering). In this way a map is "translated" into a sequence. In the next step one associates with the generated sequence a geometrical path and views such a path as a mathematical object that needs characterization. We have ordered spots representing proteins in 2-D gel plates according to their relative intensities which results in a zigzag path that produces a complicated "fingerprint" pattern. Mathematical characterization of zigzag pattern follows similar mathematical characterizations of embedded patterns based on matrices, the elements of which are given as quotients of Euclidean distance between spots and the distance along the zigzag path. The leading eigenvalue of constructed matrices is taken to represent characterization of the original 2-D map. Comparison of different 2-D maps (simulated by using random generator) allows one to construct partial order, which although qualitative in nature gives some insight into perturbation induced by foreign agents to the proteome of the control cell.

On 3-D graphical representation of proteomics maps and their numerical characterization.

We consider numerical characterization of proteomics maps by representing a map as a three-dimensional graphical object based on x, y coordinates of the spots and using their relative abundance as the z coordinate. In our representation the protein spots are first ordered based on their relative abundance and labeled accordingly. In the next step a 3-D path is constructed connecting spots having adjacent labels. Finally a matrix is constructed by assigning to each pairs of labels (i, j) matrix element, the numerical value of which is based on the quotients of the Euclidean distance and the distance along the 3-D zigzag between the two points. The approach has been illustrated on a fragment of a proteomics map and compared with 2-D graphical representation of proteomics maps.

On interpretation of well-known topological indices.

Many topological indices lack an interpretation in terms of simple physicochemical quantities. We have reexamined the structural interpretation of well-known topological indices: the connectivity index (1)chi, the Wiener index W, and the Hosoya topological index Z. We relate the success of various indices in structure-property studies to the degree to which they differentiate contributions from more exposed terminal bonds and more buried interior bonds. When considering bond additive properties of alkanes we find better regressions when greater weights are assigned to terminal CC bonds and lesser weights to internal CC. We suggest here that topological indices be discussed in terms of their partitioning into bond contributions, which for different indices and different bonds will assume different values. With this insight we modified the Wiener index W into a new index W, in which bond contributions are determined using the reciprocal of the product of the number of atoms on each side of a bond. Similarly we modified the Hosoya index Z into Z in which the frequency of occurrence of individual CC bonds in the patterns of disjoint bonds are considered. Novel indices are compared with other indices and they show to yield better regressions for boiling points of octane isomers. This suggests a useful classification of topological indices based on the relative magnitudes of the contributions of terminal and interior bonds. To extend such considerations to other indices one needs to consider partitioning of global molecular indices into bond additive terms. A general scheme for partitioning of molecular descriptors into bond contributions is outlined for indices derived from a selection of matrices associated with molecular graphs.

Characterization of DNA primary sequences based on the average distances between bases.

We outline numerical characterization of DNA primary sequence based on calculation of the average distance between pairs of nucleic acid bases. This leads to a representation of DNA by a condensed 4 x 4 symmetrical matrix, the elements of which give the average separation between pair of bases X, Y in DNA (X, Y = A, C, G, T). As an invariant of choice we consider the leading eigenvalue of the derived 4 x 4 matrix. Additional structurally related invariants were obtained by constructing additional "higher order" 4 x 4 matrices derived from the initial 4 x 4 matrix by raising its elements to higher powers. Suitably normalized leading eigenvalue of these matrices offer a novel characterization of DNA primary sequences, referred to as "DNA profiles". The approach is illustrated on exon 1 of human beta-globin gene.

Novel shape descriptors for molecular graphs.

We report on novel graph theoretical indices which are sensitive to the shapes of molecular graphs. In contrast to the Kier's kappa shape indices which were based on a comparison of a molecular graph with graphs representing the extreme shapes, the linear graph and the "star" graph, the new shape indices are obtained by considering for all atoms the number of paths and the number of walks within a graph and then making the quotients of the number of paths and the number of walks the same length. The new shape indices show much higher discrimination among isomers when compared to the kappa shape indices. We report the new shape indices for smaller alkanes and several cyclic structures and illustrate their use in structure-property correlations. The new indices offer regressions of high quality for diverse physicochemical properties of octanes. They also have lead to a novel classification of physicochemical properties of alkanes.

The variable molecular descriptors based on distance related matrices.

Recently variable molecular connectivity index and variable paths have been tested as molecular descriptors in several structure-property regressions. Here we outline the construction of several variable molecular descriptors, derived from the distance matrix and the "reversed" distance matrix. This includes the variable Balaban J index and the "reversed" Balaban index 1/J as well as a novel index 1/JJ derived from J and 1/J. All the variable descriptors mentioned were constructed by augmenting the distance matrix by replacing the diagonal zeroes with the variables x, y, z,.