On a seven-dimensional representation of RNA secondary structures

In this paper, we proposed a seven-dimensional (7D) representation of ribonucleic acid (RNA) secondary structures. The use of the 7D representation is illustrated by constructing structure invariants. Comparisons with the similarity/dissimilarity results based on 7D representation for a set of RNA 3 secondary structures at the 3′-terminus of different viruses, are considered to illustrate the use of our structure invariants based on the entries in derived sequence matrices restricted to a selected width of a band along the main diagonal.     

On a six-dimensional representation of RNA secondary structures

In this paper, we proposed a 6-D representation of RNA secondary structures. The use of the 6-D representation is illustrated by constructing structure invariants. Comparisons with the similarity/dissimilarity results based on 6-D representation for a set of RNA secondary structures, are considered to illustrate the use of our structure invariants based on the entries in derived sequence matrices restricted to a selected width of a band along the main diagonal.     

On A Six-Dimensional Representation of RNA Secondary Structures

Abstract

In this paper, we proposed a 6-D representation of RNA secondary structures. The use of the 6-D representation is illustrated by constructing structure invariants. Comparisons with the similarity/dissimilarity results based on 6-D representation for a set of RNA secondary structures, are considered to illustrate the use of our structure invariants based on the entries in derived sequence matrices restricted to a selected width of a band along the main diagonal.     

Markov invariants, plethysms, and phylogenetics

We explore model-based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the Log–Det distance measure. We take as our primary tool group representation theory, and show that it provides a general framework for analyzing Markov processes on trees. From this algebraic perspective, the inherent symmetries of these processes become apparent, and focusing on plethysms, we are able to define Markov invariants and give existence proofs. We give an explicit technique for constructing the invariants, valid for any number of character states and taxa. For phylogenetic trees with three and four leaves, we demonstrate that the corresponding Markov invariants can be fruitfully exploited in applied phylogenetic studies.     

Markov invariants and the isotropy subgroup of a quartet tree

The purpose of this article is to show how the isotropy subgroup of leaf permutations on binary trees can be used to systematically identify tree-informative invariants relevant to models of phylogenetic evolution. In the quartet case, we give an explicit construction of the full set of representations and describe their properties. We apply these results directly to Markov invariants, thereby extending previous theoretical results by systematically identifying linear combinations that vanish for a given quartet. We also note that the theory is fully generalizable to arbitrary trees and is equally applicable to the related case of phylogenetic invariants. All results follow from elementary consideration of the representation theory of finite groups.     

An algebraic-combinatorial model for the identification and mapping of biochemical pathways

We develop the mathematical machinery for the construction of an algebraic-combinatorial model using Petri nets to construct an oriented matroid representation of biochemical pathways. For demonstration purposes, we use a model metabolic pathway example from the literature to derive a general biochemical reaction network model. The biomolecular networks define a connectivity matrix that identifies a linear representation of a Petri net. The sub-circuits that span a reaction network are subject to flux conservation laws. The conservation laws correspond to algebraic-combinatorial dual invariants, that are called S-(state) and T-(transition) invariants. Each invariant has an associated minimum support. We show that every minimum support of a Petri net invariant defines a unique signed sub-circuit representation. We prove that the family of signed sub-circuits has an implicit order that defines an oriented matroid. The oriented matroid is then used to identify the feasible sub-circuit pathways that span the biochemical network as the positive cycles in a hyper-digraph.     

3D Modelling of Biological Systems for Biomimetics

1　IntroductionBasedonthereviewofthepreviousworkof 3Dgeometricalmodellingtechniquesandsystemsdevelopedforindustrial,medicalandanimationapplications,thispaperdiscussestheproblemsassociatedwiththeexist ingtechniquesandsystems ,especiallywhenappliedto3Dmodellingof plants ,insectsandanimalsforbiomimeticsresearchanddevelopment .Then ,paperproposessomeareasofresearchinterestsin 3Dmod ellingofplants ,insectsandanimalsforBiomimetics .Toavoidtherepeating ,inthispaper ,biologicalobjectswillbeusedtorep…     

Directed graphs of DNA sequences and their numerical characterization

In this paper we (1) introduce a directed graphical representation of DNA primary sequences; (2) describe a scheme that transforms the directed graph of a DNA sequence into an upper triangular matrix; (3) investigate whether or not the existing matrix-based invariants of DNA sequences are compatible for the upper triangular matrix representation. The utility of our method is illustrated by an examination of the similarity between human and other seven species.     

A 3D Graphical Representation of RNA Secondary Structures

Abstract

In this paper, we proposed a 3-D graphical representation of RNA secondary structures. Based on this representation, we outline an approach by constructing a 3-component vector whose components are the normalized leading eigenvalues of the L/L matrices associated with RNA secondary structure. The examination of similarities/dissimilarities among the secondary structure at the 3′-terminus of different viruses illustrates the utility of the approach.     

Characterization of complex biological systems by matrix invariants.

One direction in exploring similarities among biological sequences (such as DNA, RNA, and proteins), is to associate with such systems ordered sets of sequence invariants. These invariants represent selected properties of mathematical objects, such as matrices, that one can associate with biological sequences. In this article, we are exploring properties of recently introduced Line Distance matrices, and in particular we consider properties of their eigenvalues. We prove that Line Distance matrices of size n have one positive and n - 1 negative eigenvalues. Visual representation of Cauchy's interlacing property for Line Distance matrices is considered. Matlab programs for line distance matrices and examples are available on the following website: www.fmf.uni-lj.si/ approximately jaklicg/ldmatrix.html.     

Quaternion representation of RNA sequences and tertiary structures

A quaternion representation of nucleotides is proposed, with representation of RNA sequences by vectors whose elements are quaternions. Structure and transition matrices in quaternion representation are defined. Correspondence between diagrammatic technique in complex-number and quaternion representation of nucleotides is delineated.     

Moment invariants as shape recognition technique for comparing protein binding sites

MOTIVATION: An approach for identifying similarities of protein-protein binding sites is presented. The geometric shape of a binding site is described by computing a feature vector based on moment invariants. In order to search for similarities, feature vectors of binding sites are compared. Similar feature vectors indicate binding sites with similar shapes. RESULTS: The approach is validated on a representative set of protein-protein binding sites, extracted from the SCOPPI database. When querying binding sites from a representative set, we search for known similarities among 2819 binding sites. A median area under the ROC curve of 0.98 is observed. For half of the queries, a similar binding site is identified among the first two of 2819 when sorting all binding sites according the proposed similarity measure. Typical examples identified by this method are analyzed and discussed. The nitrogenase iron protein-like SCOP family is clustered hierarchically according to the proposed similarity measure as a case study. AVAILABILITY: Python code is available on request from the authors.     

A group matrix representation relevant to scales of measurement of clinical disease states via stratified vectors

Background

Previously, we applied basic group theory and related concepts to scales of measurement of clinical disease states and clinical findings (including laboratory data). To gain a more concrete comprehension, we here apply the concept of matrix representation, which was not explicitly exploited in our previous work.

Methods

Starting with a set of orthonormal vectors, called the basis, an operator R_j (an N-tuple patient disease state at the j-th session) was expressed as a set of stratified vectors representing plural operations on individual components, so as to satisfy the group matrix representation.

Results

The stratified vectors containing individual unit operations were combined into one-dimensional square matrices [R_j]s. The [R_j]s meet the matrix representation of a group (ring) as a K-algebra. Using the same-sized matrix of stratified vectors, we can also express changes in the plural set of [R_j]s. The method is demonstrated on simple examples.

Conclusions

Despite the incompleteness of our model, the group matrix representation of stratified vectors offers a formal mathematical approach to clinical medicine, aligning it with other branches of natural science.

     

Dimensional Analysis Using Toric Ideals: Primitive Invariants

Classical dimensional analysis in its original form starts by expressing the units for derived quantities, such as force, in terms of power products of basic units etc. This suggests the use of toric ideal theory from algebraic geometry. Within this the Graver basis provides a unique primitive basis in a well-defined sense, which typically has more terms than the standard Buckingham approach. Some textbook examples are revisited and the full set of primitive invariants found. First, a worked example based on convection is introduced to recall the Buckingham method, but using computer algebra to obtain an integer matrix from the initial integer matrix holding the exponents for the derived quantities. The matrix defines the dimensionless variables. But, rather than this integer linear algebra approach it is shown how, by staying with the power product representation, the full set of invariants (dimensionless groups) is obtained directly from the toric ideal defined by . One candidate for the set of invariants is a simple basis of the toric ideal. This, although larger than the rank of , is typically not unique. However, the alternative Graver basis is unique and defines a maximal set of invariants, which are primitive in a simple sense. In addition to the running example four examples are taken from: a windmill, convection, electrodynamics and the hydrogen atom. The method reveals some named invariants. A selection of computer algebra packages is used to show the considerable ease with which both a simple basis and a Graver basis can be found.     

Slicing hyperdimensional oranges: the geometry of phylogenetic estimation

A new view of phylogenetic estimation is presented where data sets, tree evolution models, and estimation methods are placed in a common geometric framework. Each of these objects is placed in a vector space where the character patterns are the basis vectors. This viewpoint allows intuitive understanding of various complex properties of the phylogeneticestimation problem structure. This is illustrated with examples discussing data set combinations, mixture models, consistency, and phylogenetic invariants.     

Bayesian signal extraction from noisy FT NMR spectra

Summary The statistical interpretation of the histogram representation of NMR spectra is described, leading to an estimation of the probability density function of the noise. The white-noise and Gaussian hypotheses are discussed, and a new estimator of the noise standard deviation is derived from the histogram strategy. The Bayesian approach to NMR signal detection is presented. This approach homogeneously combines prior knowledge, obtained from the histogram strategy, together with the posterior information resulting from the test of presence of a set of reference shapes in the neighbourhood of each data point. This scheme leads to a new strategy in the local detection of NMR signals in 2D and 3D spectra, which is illustrated by a complete peak-picking algorithm.     

Singular Value Decomposition of the Radial Distribution Function for Hard Sphere and Square Well Potentials

We compute the singular value decomposition of the radial distribution function for hard sphere, and square well solutions. We find that decomposes into a small set of basis vectors allowing for an extremely accurate representation at all interpolated densities and potential strengths. In addition, we find that the coefficient vectors describing the magnitude of each basis vector are well described by a low-order polynomial. We provide a program to calculate in this compact representation for the investigated parameter range.     

Prediction of protein coarse contact maps

Prediction of topological representations of proteins that are geometrically invariants can contribute towards the solution of fundamental open problems in structural genomics like folding. In this paper we focus on coarse grained protein contact maps, a representation that describes the spatial neighborhood relation between secondary structure elements such as helices, beta sheets, and random coils. Our methodology is based on searching the graph space. The search algorithm is guided by an adaptive evaluation function computed by a specialized noncausal recursive connectionist architecture. The neural network is trained using candidate graphs generated during examples of successful searches. Our results demonstrate the viability of the approach for predicting coarse contact maps.