Dependence of the biodegradability of carboxymethylcellulose on its supermolecular structure and molecular parameters

It has been established that the resorption of carboxymethylcellulose (CMC) on its implantation into a living organism depends on its molecular mass, degree of substitution, and supermolecular structure. Its mechanism includes both hydrolytic breakdown of the main chain and macrophagal pinocytosis. It has been shown that CMC is absorbed completely and is excreted from the organism without accumulation.A. S. Sadykov Institute of Bioorganic Chemistry, Academy of Sciences of the Republic of Uzbekistan, Tashkent, fax (3712) 62 70 71. Translated from Khimiya Prirodnykh Soedinenii, No. 2, pp. 306–312, March–April, 1995. Original article submitted November 16, 1994.     

Chirality of toroidal molecular graphs

Symmetry properties of a class of toroidal molecular graphs, arising as covers of certain bipartite cubic Cayley graphs of dihedral groups, are studied. Although these symmetries make all vertices and all edges indistinguishable, they imply intrinsic chirality.     

On subspectral acyclic molecular graphs

Based on the contraction and expansion of graphs, the subspectrality of acyclic molecular graphs is treated in a systematic way. The graphs containing eigenvalues 0, t 1 and ± 2 are discussed in detail, with emphasis on how to detect and construct the concealed species which can neither be recognized by symmetry considerations nor by the Heilbronner procedure. As a consequence, graphs and their counts have been given for species with numbers of vertices less than 16.Also known as Yuan-sun Kiang.     

Reversible peptide folding: Dependence on molecular force field used

Temperature‐dependent nuclear magnetic resonance (NMR) and CD spectra of methanol solutions of a β‐heptapeptide have been interpreted in such a way that the secondary structure, a 3₁₄‐helix, is assumed to be stable in a temperature range of between 298 and 393 K. This is in contrast to the results of a 50‐ns molecular dynamics simulation using the GROMOS 96 force field, which found a melting temperature of about 340 K. This discrepancy is addressed by further computational studies using the OPLS‐AA force field. The conformational energetics of N‐formyl‐3‐aminobutanamide in vacuo are obtained using ab initio and density functional quantum‐mechanical calculations at the HF/6‐31G*, B3LYP/6‐31G*, and B3LYP/6‐311+G* levels of theory. The results permit development of torsional parameters for the OPLS‐AA force field that reproduce the conformational energetics of the monomer. By varying the development procedure, three parameter sets are obtained that focus on reproducing either low‐energy or high‐energy conformations. These parameter sets are tested by simulating the reversible folding of the β‐heptapeptide in methanol. The melting temperature of the helix formed (>360 K) is found to be higher than the one obtained from simulations using the GROMOS 96 force field (∼340 K). Differences in the potential energy functions of the latter two force fields are evaluated and point to the origins of the difference in stability. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 774–787, 2000     

On walks in molecular graphs.

Walks in molecular graphs and their counts for a long time have found applications in theoretical chemistry. These are based on the fact that the (i, j)-entry of the kth power of the adjacency matrix is equal to the number of walks starting at vertex i, ending at vertex j, and having length k. In recent papers (refs 13, 18, 19) the numbers of all walks of length k, called molecular walk counts, mwc(k), and their sum from k = 1 to k = n - 1, called total walk count, twc, were proposed as quantities suitable for QSPR studies and capable of measuring the complexity of organic molecules. We now establish a few general properties of mwc's and twc among which are the linear dependence between the mwc's and linear correlations between the mwc's and twc, the spectral decomposition of mwc's, and various connections between the walk counts and the eigenvalues and eigenvectors of the molecular graph. We also characterize the graphs possessing minimal and maximal walk counts.     

Dependence of the mobility of molecular probes in cellulose acetate solutions on complexation and the solution structure

The spin echo technique with a magnetic field pulse gradient was used to measure the translational diffusion coefficients D of molecular probes in solutions of cellulose acetates in dimethyl sulfoxide. A linear dependence of coefficient D on the extent of substitution of the cellulose acetate was established and the sensitivity of D to the supermolecular structure of the polymer solution was demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 2, pp. 470–472, February, 1990.     

Graphical shapes: Seeing graphs of chemical curves and molecular surfaces

Interior seeing graphs of Jordan curves and two-dimensional closed surfaces are proposed for the characterisation or the shapes of chemical curves and surfaces. Some of the relevant properties of seeing graphs, in particular, of seeing trees, are described. The proposed applications of seeing graphs provide a molecular shape characterization by converting the continuum problem of a molecular contour surface, for example, Mat of an isodensity contour of molecular electronic charge distribution, into a discrete problem of a graph. The dependence of seeing graphs on be charge density contour value, a continuous parameter, is analysed. The family of seeing graphs occurring; within the chemically accessible range of charge density contour values is proposed for a computer-based analysis of molecular similarity. The general method is illustrated with the example of a detailed study on the family of seeing graphs of the electronic charge density of the ethene molecule.     

Dependence of the critical temperature on molecular parameters

At the critical temperature the surface tension between coexisting liquid and vapor phases must be zero, and the repulsive contributions associated with cavity formation must exactly counterbalance those from interactions of a molecule in the cavity and the bulk. An expression for the critical temperature of pure fluids in terms of the parameters of scaled particle theory (SPT) has been obtained, and the calculated critical temperatures are compared with experimental data for a range of pure fluids. These include noble and diatomic gases, short and medium length hydrocarbons, aromatic compounds, halogenated compounds, oxygen-containing compounds, and water. Considering the simplicity of this approach, a remarkably good correlation between calculated and experimental values is found for most of these fluids.     

Dependence of polymer specific volume on molecular characteristics

Linear and branched bisphenol A polycarbonate (PC) samples were characterized by their average molecular weights, $\text{M}{}_{\mathrm{<mtext>n</mtext>}}$ and $\text{M}{}_{\mathrm{<mtext>w</mtext>}}$, polydispersity degree $\text{q =}\text{M}{}_{\mathrm{<mtext>w</mtext>}}\text{/}\text{M}{}_{\mathrm{<mtext>n</mtext>}}$, and branching degree g_v. The weight fraction of microgel was also determined for branched samples. The samples were amorphized and densities were measured at 23°C to obtain the values of specific volume, v_sp. The dependence of v_sp on molecular characteristics is described by the multivariable power function $\text{Δv}{}_{\mathrm{<mtext>sp</mtext>}}\text{=}\text{A}{}_{\mathrm{sp}}\text{M}{}_{\mathrm{<mtext>x</mtext>}}{}^{\mathrm{<mtext>a</mtext>}}\text{q}{}^{\mathrm{<mtext>apx</mtext>}}\text{g}{}_{\mathrm{<mtext>v</mtext>}}{}^{\mathrm{<mtext>ab</mtext>}}$, where Δv_sp = v_sp ? v_sp,∞, and A_sp, a, a_px and a_b are constants. It has been confirmed that a = ?1, a_pn = 0 and a_pw = 1. It has also been found that the branching exponent a_b significantly depends on microgel content. The relationships found for PC should, in principle, be valid for other polymers. Examples based on literature data are given for linear polyethylene and polydimethylsiloxane.     

On the centrality of vertices of molecular graphs

For acyclic systems the center of a graph has been known to be either a single vertex of two adjacent vertices, that is, an edge. It has not been quite clear how to extend the concept of graph center to polycyclic systems. Several approaches to the graph center of molecular graphs of polycyclic graphs have been proposed in the literature. In most cases alternative approaches, however, while being apparently equally plausible, gave the same results for many molecules, but occasionally they differ in their characterization of molecular center. In order to reduce the number of vertices that would qualify as forming the center of the graph, a hierarchy of rules have been considered in the search for graph centers. We reconsidered the problem of “the center of a graph” by using a novel concept of graph theory, the vertex “weights,” defined by counting the number of pairs of vertices at the same distance from the vertex considered. This approach gives often the same results for graph centers of acyclic graphs as the standard definition of graph center based on vertex eccentricities. However, in some cases when two nonequivalent vertices have been found as graph center, the novel approach can discriminate between the two. The same approach applies to cyclic graphs without additional rules to locate the vertex or vertices forming the center of polycyclic graphs, vertices referred to as central vertices of a graph. In addition, the novel vertex “weights,” in the case of acyclic, cyclic, and polycyclic graphs can be interpreted as vertex centralities, a measure for how close or distant vertices are from the center or central vertices of the graph. Besides illustrating the centralities of a number of smaller polycyclic graphs, we also report on several acyclic graphs showing the same centrality values of their vertices. © 2013 Wiley Periodicals, Inc.     

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.     

Quest for molecular graphs with maximal energy: a computer experiment

If lambda(1), lambda(2),..., lambda(n) are the eigenvalues of a graph G, then the energy of G is defined as E(G) = the absolute value of lambda(1) + the absolute value of lambda(2) +.... + the absolute value of lambda(n). If G is a molecular graph, representing a conjugated hydrocarbon, then E(G) is closely related to the respective total pi-electron energy. It is not known which molecular graph with n vertices has maximal energy. With the exception of m = n - 1 and m = n, it is not known which molecular graph with n vertices and m edges has maximal energy. To come closer to the solution of this problem, and continuing an earlier study (J. Chem. Inf. Comput. Sci. 1999, 39, 984-996, ref 7), we performed a Monte Carlo-type construction of molecular (n,m)-graphs, recording those with the largest (not necessarily maximal possible) energy. The results of our search indicate that for even n the maximal-energy molecular graphs might be those possessing as many as possible six-membered cycles; for odd n such graphs seem to prefer both six- and five-membered cycles.     

Comparability graphs and molecular properties: IV. Generalizations and applications

The development of a recently proposed method for calculating molecular properties is outlined. The approach is based on the idea of constructing optimized compound samples for structure—property or structure-activity correlations by means of the so—called comparability graphs (CG) of isomeric compounds. A dynamic comparability principle is devised, proceeding from a series of standard molecular rearrangements described in graph—theoretical terms as rules on molecular branching and cyclicity. An extension of the approach is presented for both the construction of CG's and their combination for variable numbers of atoms. The method is applied to various physico-chemical properties, which are thus divided into three groups according to the degree to which they are conditioned by molecular topology. The Wiener topological index is shown to produce a highly linear correlation with the alkane critical densities and volumes, as well as with their heats and entropies of vaporization.Dedicated to the memory of Professor Oskar E. Polansky, a pioneer of chemical graph theory     

Fast construction of assembly trees for molecular graphs

A number of modeling and simulation algorithms using internal coordinates rely on hierarchical representations of molecular systems. Given the potentially complex topologies of molecular systems, though, automatically generating such hierarchical decompositions may be difficult. In this article, we present a fast general algorithm for the complete construction of a hierarchical representation of a molecular system. This two-step algorithm treats the input molecular system as a graph in which vertices represent atoms or pseudo-atoms, and edges represent covalent bonds. The first step contracts all cycles in the input graph. The second step builds an assembly tree from the reduced graph. We analyze the complexity of this algorithm and show that the first step is linear in the number of edges in the input graph, whereas the second one is linear in the number of edges in the graph without cycles, but dependent on the branching factor of the molecular graph. We demonstrate the performance of our algorithm on a set of specifically tailored difficult cases as well as on a large subset of molecular graphs extracted from the protein data bank. In particular, we experimentally show that both steps behave linearly in the number of edges in the input graph (the branching factor is fixed for the second step). Finally, we demonstrate an application of our hierarchy construction algorithm to adaptive torsion-angle molecular mechanics.     

Common vertex matrix: A novel characterization of molecular graphs by counting

We present a novel matrix representation of graphs based on the count of equal‐distance common vertices to each pair of vertices in a graph. The element (i, j) of this matrix is defined as the number of vertices at the same distance from vertices (i, j). As illustrated on smaller alkanes, these novel matrices are very sensitive to molecular branching and the distribution of vertices in a graph. In particular, we show that ordered row sums of these novel matrices can facilitate solving graph isomorphism for acyclic graphs. This has been illustrated on all undecane isomers C₁₁H₂₄ having the same path counts (total of 25 molecules), on pair of graphs on 18 vertices having the same distance degree sequences (Slater's graphs), as well as two graphs on 21 vertices having identical several topological indices derived from information on distances between vertices. © 2013 Wiley Periodicals, Inc.     

Study of the origin of subspectrality in molecular graphs

Summary The subspectral origin of three families of molecules based on cyclobutadiene, benzene, and cyclooctatetraene is discussed. The graph theoretical decomposition of the fourfold cyclooctatetraene molecular graphs is presented in explicit form and has expedited the computation of their respective eigenvalues. The cyclic automorphism approach of Davidson is clarified and merged with the author's methodology leading to a more comprehensive procedure for rapidly determining the characteristic polynomial and eigenvalues of chemically significant molecular graphs. The graph theoretical determination of the characteristic polynomials and eigenvalues of two sixfold coronene-related molecular families is presented.     

Van der Waals surface graphs and molecular shape

A van der Waals surface graph is the graph defined on a van der Waals surface by the intersections of the atomic van der Waals spheres. A van der Waals shape graph has a vertex for each atom with a visible face on the van der Waals surface, and edges between vertices representing atoms with adjacent faces on the van der Waals surface. These are discrete invariants of three‐dimensional molecular shape. Some basic properties of van der Waals surface graphs are studied, including their relationship with the Voronoi diagram of the atom centres, and a class of molecular embeddings is identified for which the dual of the van der Waals surface graph coincides with the van der Waals shape graph. This revised version was published online in July 2006 with corrections to the Cover Date.     

Generation of molecular graphs for QSAR studies: an approach based on supergraphs

A new algorithm for generation of substituted derivatives of a given structure and its software implementation are described. The program has deterministic and stochastic generation modes and efficiently supports various types of structural constraints. The problem of exhaustive and irredundant generation is discussed, and a new algorithm of the complete rejection of isomorphic molecular graphs is proposed. The main application of the generator is QSAR studies; however, applications in combinatorial chemistry are also possible.