Models of Hopfield-type quaternion neural networks and their energy functions

Recently models of neural networks that can directly deal with complex numbers, complex-valued neural networks, have been proposed and several studies on their abilities of information processing have been done. Furthermore models of neural networks that can deal with quaternion numbers, which is the extension of complex numbers, have also been proposed. However they are all multilayer quaternion neural networks. This paper proposes models of fully connected recurrent quaternion neural networks, Hopfield-type quaternion neural networks. Since quaternion numbers are non-commutative on multiplication, some different models can be considered. We investigate dynamics of these proposed models from the point of view of the existence of an energy function and derive their conditions for existence.     

New 3D graphical representation of DNA sequence based on dual nucleotides

We introduce a 3D graphical representation of DNA sequences based on the pairs of dual nucleotides (DNs). Based on this representation, we consider some mathematical invariants and construct two 16-component vectors associated with these invariants. The vectors are used to characterize and compare the complete coding sequence part of beta globin gene of nine different species. The examination of similarities/dissimilarities illustrates the utility of the approach.     

DNA motif representation with nucleotide dependency

The problem of discovering novel motifs of binding sites is important to the understanding of gene regulatory networks. Motifs are generally represented by matrices (position weight matrix (PWM) or position specific scoring matrix (PSSM) or strings. However, these representations cannot model biological binding sites well because they fail to capture nucleotide interdependence. It has been pointed out by many researchers that the nucleotides of the DNA binding site cannot be treated independently, e.g. the binding sites of zinc finger in proteins. In this paper, a new representation called Scored Position Specific Pattern (SPSP), which is a generalization of the matrix and string representations, is introduced which takes into consideration the dependent occurrences of neighboring nucleotides. Even though the problem of discovering the optimal motif in SPSP representation is proved to be NP-hard, we introduce a heuristic algorithm called SPSP-Finder, which can effectively find optimal motifs in most simulated cases and some real cases for which existing popular motif finding software, such as Weeder, MEME and AlignACE, fail.     

Similarity analysis for DNA sequences based on chaos game representation. Case study: The albumin

Using chaos game representation we introduce a novel and straightforward method for identifying similarities/dissimilarities between DNA sequences of the same type, from different organisms. A matrix is associated to each CGR pattern and the similarities result from the comparison between the matrices of the sequences of interest. Three different methods of analysis of the resulting difference matrix are considered: a 3-dimensional representation giving both local and global information, a numerical characterization by defining an n-letter word similarity measure and a statistical evaluation. The method is illustrated by implementation to the study of albumin nucleotides sequences from eight mammal species taking as reference the human albumin.     

Conversion of nucleotides sequences into genomic signals

An original tetrahedral representation of the Genetic Code (GC) that better describes its structure, degeneration and evolution trends is defined. The possibility to reduce the dimension of the representation by projecting the GC tetrahedron on an adequately oriented plane is also analyzed, leading to some equivalent complex representations of the GC. On these bases, optimal symbolic-to-digital mappings of the linear, nucleic acid strands into real or complex genomic signals are derived at nucleotide, codon and amino acid levels. By converting the sequences of nucleotides and polypeptides into digital genomic signals, this approach offers the possibility to use a large variety of signal processing methods for their handling and analysis. It is also shown that some essential features of the nucleotide sequences can be better extracted using this representation. Specifically, the paper reports for the first time the existence of a global helicoidal wrapping of the complex representations of the bases along DNA sequences, a large scale trend of genomic signals. New tools for genomic signal analysis, including the use of phase, aggregated phase, unwrapped phase, sequence path, stem representation of components'relative frequencies, as well as analysis of the transitions are introduced at the nucleotide, codon and amino acid levels, and in a multiresolution approach.     

PNN-curve: a new 2D graphical representation of DNA sequences and its application

We introduce a novel 2D graphical representation of DNA sequences based on the pairs of the neighboring nucleotides (PNNs). Then we get the PNNs' distributions and obtain a y-M. The construction of the PNN-curve has some important advantages (1) It avoids loss of information and the PNN-curve standing for DNA sequences does not overlap or intersect with itself. (2) The novel 2D representation is more sensitive. The utility of this method can be illustrated by the examination of similarities/dissimilarities among the coding sequences of the first exon of beta-globin gene of eleven different species in Table 2.     

Interpolating three-dimensional kinematic data using quaternion splines and hermite curves

Kinematic interpolation is an important tool in biomechanics. The purpose of this work is to describe a method for interpolating three-dimensional kinematic data, minimizing error while maintaining ease of calculation. This method uses cubic quaternion and hermite interpolation to fill gaps between kinematic data points. Data sets with a small number of samples were extracted from a larger data set and used to validate the technique. Two additional types of interpolation were applied and then compared to the cubic quaternion interpolation. Displacement errors below 2% using the cubic quaternion method were achieved using 4% of the total samples, representing a decrease in error over the other algorithms.     

Quaternion-based discriminant analysis method for color face recognition

Pattern recognition techniques have been used to automatically recognize the objects, personal identities, predict the function of protein, the category of the cancer, identify lesion, perform product inspection, and so on. In this paper we propose a novel quaternion-based discriminant method. This method represents and classifies color images in a simple and mathematically tractable way. The proposed method is suitable for a large variety of real-world applications such as color face recognition and classification of the ground target shown in multispectrum remote images. This method first uses the quaternion number to denote the pixel in the color image and exploits a quaternion vector to represent the color image. This method then uses the linear discriminant analysis algorithm to transform the quaternion vector into a lower-dimensional quaternion vector and classifies it in this space. The experimental results show that the proposed method can obtain a very high accuracy for color face recognition.     

Revealing unique properties of the ribosome using a network based analysis

The ribosome is a complex molecular machine that offers many potential sites for functional interference, therefore representing a major target for antibacterial drugs. The growing number of high-resolution structures of ribosomes from different organisms, in free form and in complex with various ligands, provides unique data for structural and comparative analyses of RNA structures. We model the ribosome structure as a network, where nucleotides are represented as nodes and intermolecular interactions as edges. As shown previously for proteins, we found that the major functional sites of the ribosome exhibit significantly high centrality measures. Specifically, we demonstrate that mutations that strongly affect ribosome function and assembly can be distinguished from mild mutations based on their network properties. Furthermore, we observed that closeness centrality of the rRNA nucleotides is highly conserved in the bacteria, suggesting the network representation as a comparative tool for the ribosome analysis. Finally, we suggest a global topology perspective to characterize functional sites and to reveal the unique properties of the ribosome.     

A cut-off based approach for gene expression analysis of formalin-fixed and paraffin-embedded tissue samples

Microarray analysis of formalin-fixed and paraffin-embedded (FFPE) tissue seems to be of importance for the detection of molecular marker sets in prostate cancer (PC). The compromised RNA integrity of FFPE tissue results in a high degree of variability at the probe level of microarray data as shown by degradation plot. We tested methods that reduce the variability by including all probes within 300 nucleotides, within 600 nucleotides, or up to a calculated breakpoint with reference to the 3'-end. Accepted PC pathways such as the Wnt signaling pathway could be observed to be significantly regulated within FFPE microarray datasets. The best representation of PC gene expression, as well as better comparability to meta-analysis and fresh-frozen microarray data, could be obtained with a 600-nucleotide cutoff. Beyond the specific impact for PC microarray data analysis we propose a cutoff of 600 nucleotides for samples for which the integrity of the RNA cannot be guaranteed.     

Quaternion-based definition of protein secondary structure straightness and its relationship to Ramachandran angles

We describe here definitions of "local helical axis" and "straightness" that are developed using a simple quaternion-based analysis of protein structure without resort to least-squares fitting. As part of this analysis, it is shown how quaternion differences can be visualized to depict accurately the local helical axis relating any two adjacent amino acid residues in standard, nonidealized proteins. Three different options for the definition of amino acid residue orientation in terms of quaternion frames are described. Two of these, the "C(α) frame" and the "P frame," are shown to be correlated strongly with a simple approximate measure derived solely from Ramachandran angles. The relationship between quaternion-based straightness and recognized DSSP-derived secondary structure motifs is discussed.     

Classification analysis of dual nucleotides using dimension reduction

We introduce a new approach to investigate the dual nucleotides compositions of 11 Gram-positive and 12 Gram-negative eubacteria recently studied by Sorimachi and Okayasu. The approach firstly obtains a 16-dimension vector set of dual nucleotides by PN-curve from the complete genome of organism. Each vector of the set corresponds to a single gene of genome. Then we reduce the 16-dimension vector set to 2-dimension by principal components analysis (PCA). The reduction avoids possible loss of information averaging all 16-dimension vectors. Then we suggest a 2D graphical representation based on the 2-dimension vector to investigate the classification patters among different organisms.     

Rook’s tour representation of the genetic code

Disconnected recurrences of the stop signal, serine and arginine appear in the original representation of the genetic code, and of the stop signal, arginine, serine and leucine in the codon ring representation. To achieve connectedness along with structural continuity, arook’s tour representation is presented here. On the basis of structural similarities and disparities in their side groups, each of the 20 amino acids is associated with a domain comprised of from one to six contiguous squares on the chess board. As the rook moves on the chess board, it reaches all 64 squares in the ordering of the codon numbers, which prescribe the codons by a simple formula based on the position and size of the nucleotides in a triplet. Recurrences of the stop signal, arginine and serine occur naturally on the tour as the rook enters each of the latter domains for the second time. A mathematical equivalent of the rook’s tour may enter as a programming device in the implementation of the code by the RNAs.     

Reorientational Correlation Functions,Quaternions and Wigner Rotation Matrices

Abstract

The main results of this paper are tabulations of Wigner rotation matrix elements in terms of quaternion parameters and direction cosines. These will be useful for the calculation of orientational probability functions or reorientational correlation functions from simulations of molecular systems using programs based either on quaternions or constraints. We review the relationship between quaternions and the L = ½ Wigner rotation matrix, give some useful properties of quaternions and Wigner rotation matrices and show how the Lth Wigner rotation matrix can be constructed from the quaternion parameters without first obtaining the Euler angles.     

RNA aptamers to the adenosine moiety of S-adenosyl methionine: structural inferences from variations on a theme and the reproducibility of SELEX.

We used in vitro selection (SELEX) to isolate RNA 'aptamers' to S-adenosyl methionine (SAM). Individual aptamer sequences conform to the structural element noted previously for adenosine binding in selections for aptamers to ATP and NAD+. When we compare the patterns of sequence conservation among 65 adenosine-binding sequences to the published structure of the adenosine aptamer, we find that the most highly conserved nucleotides contact the bound adenosine directly, and that one conserved nucleotide outside the binding pocket is in position to stabilize nucleotides within the binding pocket. The aptamer's ability to bind diverse adenosine-containing cofactors is easily understood in terms of its mode of binding, which leaves the 5'position exposed to solvent. We propose that aptamers that bind their targets away from the reactive moiety may be particularly well suited for catalysis. Finally, we estimate that one sequence in 10(11) may be able to form this structural motif, and that there may be many other adenosine-binding motifs that have escaped detection because of their lower representation in the starting random pools.