We obtain the characteristic polynomials and a number of spectral-based indices such as the Riemann-Zeta functional indices and spectral entropies of n-dimensional hypercubes using recursive Hadamard transforms. The computed numerical results are constructed for up to 23-dimensional hypercubes. While the graph energies exhibit a J-curve as a function of the dimension of the n-cubes, the spectra-based entropies exhibit a linear dependence on the dimension. We have also provided structural interpretations for the coefficients of the characteristic polynomials of n-cubes and obtain expressions for the integer sequences formed by the spectral-based Riemann-Zeta functions.

It is shown that the extent of deviation of a molecular shape from spherical can be characterized by comparing the distribution of the circular variances, a measure originally proposed to quantify angular spread, of the vectors from each atom to the rest of the molecule to the circular variance of a collection of atoms filling the unit sphere. Different measures for quantifying the difference between distribution are proposed and compared.

We discuss the chemical synthesis of topological links, in particular higher order links which have the Brunnian property (namely that removal of any one component unlinks the entire system). Furthermore, we suggest how to obtain both two dimensional and three dimensional objects (surfaces and solids, respectively) which also have this Brunnian property.

Biochemistry has many examples of linear chain polymers, i.e., molecules formed from a sequence of units from a finite set of possibilities; examples include proteins, RNA, single-stranded DNA, and paired DNA. In the field of mass spectrometry, it is useful to consider the idea of weighted alphabets, with a word inheriting weight from its letters. We describe the distribution of the mass of these words in terms of a simple recurrence relation, the general solution to that relation, and a canonical form that explicitly describes both the exponential form of this distribution and its periodic features, thus explaining a wave pattern that has been observed in protein mass databases. Further, we show that a pure exponential term dominates the distribution and that there is exactly one such purely exponential term. Finally, we illustrate the use of this theorem by describing a formula for the integer mass distribution of peptides and we compare our theoretical results with mass distributions of human and yeast peptides.

An analytic approach to the modeling of stop-flow amperometric measurements of cellular metabolism with thin glucose oxidase and lactate oxidase electrodes would provide a mechanistic understanding of the various factors that affect the measured signals. We divide the problem into two parts: (1) analytic formulas that provide the boundary conditions for the substrate and the hydrogen peroxide at the outer surface of the enzyme electrode layers and the electrode current expressed through these boundary conditions, and (2) a simple diffusion problem in the liquid compartment with the provided boundary conditions, which can be solved analytically or numerically, depending on the geometry of the compartment. The current in an amperometric stop-flow measurement of cellular glucose or lactate consumption/excretion is obtained analytically for two geometries, corresponding to devices developed at the Vanderbilt Institute for Integrative Biosystems Research and Education: a multianalyte nanophysiometer with effective one-dimensional diffusion and a multianalyte microphysiometer, for which plentiful data for metabolic changes in cells are available. The data are calibrated and fitted with the obtained time dependences to extract several cellular fluxes. We conclude that the analytical approach is applicable to a wide variety of measurement geometries and flow protocols.

We expand on the work of Hosoya to describe a generalization of continued fractions called "tree expressions." Each rooted tree will be shown to correspond to a unique tree expression which can be evaluated as a rational number (not necessarily in lowest terms) whose numerator is equal to the Hosoya index of the entire tree and whose denominator is equal to the tree with the root deleted. In the development, we use Z(G) to define a natural candidate ζ(G, v) for a "vertex topological index" which is a value applied to each vertex of a graph, rather than a value assigned to the graph overall. Finally, we generalize the notion of tree expression to "labeled tree expressions" that correspond to labeled trees and show that such expressions can be evaluated as quotients of determinants of matrices that resemble adjacency matrices.

In this article, we propose two well-defined distance metrics of biological sequences based on a universal complexity profile. To illustrate our metrics, phylogenetic trees of 18 Eutherian mammals from comparison of their mtDNA sequences and 24 coronaviruses using the whole genomes are constructed. The resulting monophyletic clusters agree well with the established taxonomic groups.

New approaches aiming at a detailed similarity/dissimilarity analysis of DNA sequences are formulated. Several corrections that enrich the information which may be derived from the alignment methods are proposed. The corrections take into account the distributions along the sequences of the aligned bases (neglected in the standard alignment methods). As a consequence, different aspects of similarity, as for example asymmetry of the gene structure, may be studied either using new similarity measures associated with four-component spectral representation of the DNA sequences or using alignment methods with corrections introduced in this paper. The corrections to the alignment methods and the statistical distribution moment-based descriptors derived from the four-component spectral representation of the DNA sequences are applied to similarity/dissimilarity studies of -globin gene across species. The studies are supplemented by detailed similarity studies for histones H1 and H4 coding sequences. The data are described according to the latest version of the EMBL database. The work is supplemented by a concise review of the state-of-art graphical representations of DNA sequences.

While the square root of Dirac's is not defined in any standard mathematical formalism, postulating its existence with some further assumptions defines a generalized function called which permits a quasi-classical treatment of simple systems like the H atom or the 1D harmonic oscillator for which accurate quantum mechanical energies were previously reported. The so-defined is neither a traditional function nor a distribution, and it remains to be seen that any consistent mathematical approaches can be set up to deal with it rigorously. A straightforward use of generates several paradoxical situations which are collected here. The help of the scientific community is sought to resolve these paradoxa.

The central idea observes a recursive mapping of [Formula: see text]-body intramolecular interactions to [Formula: see text]-body terms that is consistent with the molecular topology. Iterative application of the line graph transformation is identified as a natural and elegant tool to accomplish the recursion. The procedure readily generalizes to arbitrary [Formula: see text]-body potentials. In particular, the method yields a complete characterization of [Formula: see text]-body interactions. The hierarchical structure of atomic index lists for each interaction order [Formula: see text] is compactly expressed as a directed acyclic graph. A pseudo-code description of the generating algorithm is given. With suitable data structures (e.g., edge lists or adjacency matrices), automatic enumeration and indexing of [Formula: see text]-body interactions can be implemented straightforwardly to handle large bio-molecular systems. Explicit examples are discussed, including a chemically relevant effective potential model of taurocholate bile salt.

Recent studies have found an unusual way of dissociation in formaldehyde. It can be characterized by a hydrogen atom that separates from the molecule, but instead of dissociating immediately it roams around the molecule for a considerable amount of time and extracts another hydrogen atom from the molecule prior to dissociation. This phenomenon has been coined roaming and has since been reported in the dissociation of a number of other molecules. In this paper we investigate roaming in Chesnavich's model. During dissociation the free hydrogen must pass through three phase space bottleneck for the classical motion, that can be shown to exist due to unstable periodic orbits. None of these orbits is associated with saddle points of the potential energy surface and hence related to transition states in the usual sense. We explain how the intricate phase space geometry influences the shape and intersections of invariant manifolds that form separatrices, and establish the impact of these phase space structures on residence times and rotation numbers. Ultimately we use this knowledge to attribute the roaming phenomenon to particular heteroclinic intersections.

We have obtained graph-theoretically based topological indices for the characterization of certain graph theoretical networks of biochemical interest. We have derived certain distance, degree and eccentricity based topological indices for various -level hypertrees and corona product of hypertrees. We have also pointed out errors in a previous study. The validity of our results is supported by computer codes for the respective indices. Several biochemical applications are pointed out.

The present study, which is a continuation of the previous paper, augments a recent work on the use of phylogenetic networks. We develop techniques to characterize the topology of various X-trees and binary trees of biological and phylogenetic interests. We have obtained the results for various -level -trees and phylogenetic networks with variants of Zagreb, Szeged, Padmakar-Ivan, Schultz and Atom Bond Connectivity topological indices.

In this work, a new version of Rényi's divergence is presented. The expression obtained is used as a tool to identify molecules that could share some chemical or structural properties, and a data basis set of 1641 molecules is used in this study. Our results suggest that this new form of Rényi divergence could be a useful tool that will eventually permit complementary studies in which the main goal is to obtain molecules with similar properties.

Given a bi-classification of nucleotides, we can obtain a reduced binary sequence of a primary DNA sequence. This binary sequence will undoubtedly retain some biological information and lose the rest. Here we want to know what kind of and how much biological information an individual binary sequence carries. Three classifications of nucleotides are explored in the present article. Phylogenetic trees are built from these binary sequences by the Neighbor-Joining (NJ) method, with evolutionary distance evaluated on the basis of a symbolic sequence complexity. We find that, for all data sets studied, binary sequences reduced by the purine/pyrimidine classification give reliable phylogeny (almost the same as that from the primary sequences), while the other two result in discrepancies at different levels. Some possible reasons and a simple model of sequence evolutionary are introduced to interpret this phenomenon.

A systematic analysis is presented of the algorithm for converting a virtual-bond chain, defined by the coordinates of the alpha-carbons of a given protein, into a complete polypeptide backbone. An alternative algorithm, based upon the same set of geometric parameters used in the Purisima-Scheraga algorithm but with a different "linkage map" of the algorithmic procedures, is proposed. The global virtual-bond chain geometric constraints are more easily separable from the loal peptide geometric and energetic constraints derived from, for example, the Ramachandran criterion, within the framework of this approach.

A new tool of the classification of DNA sequences is introduced. The method is based on 2D-dynamic graphs and their descriptors. Using the descriptors created by centers of masses, moments of inertia, angles between the axis and the principal axis of inertia of the 2D-dynamic graphs one can obtain classification diagrams in which similar sequences are clustered in separated areas.