Quantum Bio-Informatics III

The following sections are included:

• PREFACE

• CONTENTS

The different notions of stochastic independences, introduced in quantum probability open new fascinating possibilities to deepen our intuition on what a composite system.

In the present note we propose a general mathematical definition of composite system which emphasizes the fact that the naive idea, that a physical system is composed of a multiplicity of sub-systems, can be substantiated by a multiplicity of inequivalent mathematical models.

This wealth of possibilities can considerably enrich the present approach to the theory of open systems, with potential implications for the theory of measurement and the theory of complex systems, such as biological or economical ones.

The standard approach to composite system strongly privileges the tensor product construction and the corresponding notion of stochastic independence. But there are a multiplicity of other possibilities whose mathematical and physical investigation is only at the beginning. In particular, to any notion of statistical independence it is canonically associated a corresponding notion of entanglement.

The following sections are included:

• Introduction

• Main results

• Applications

• References

In many models of perfect quantum teleportation, maximal entangled states are postulated between Alice and Bob. Recently, Kossakowski and Ohya proposed a new scheme of perfect teleportation where a non-maximal entangled state is used ³. In this paper, we introduce a concrete teleportation model of K-O scheme, called a m-level teleportation. A non-maximal entangled state in this model is realizable in optical techniques. Moreover, we propose another teleportation model designed by using a mathematical formalism of K-O theory. In the second model, a new operation is added to a conventional teleportation process, which is called a filtering and implemented before Alice's measurement. We show effects of the filtering for success probabilities of teleportation.

A generalisation of the concepts utility and information value is given for the non-commutative case. In particular, states and utility operators are considered in dual linear spaces equipped with pre-orders, generated by a wedge of utility operators. It is shown that solutions to the information value problem give rise to an isotone Galois connection between the pre-ordered spaces. A particular form of this connection depends on the choice of a functional representing information resource. The properties of information resource are discussed from the point of information value theory, and an example is presented that generalises several known forms of classical and quantum information. Potential areas of application of information value in cognitive science, biology and quantum theory are discussed.

We provide partial classification of entanglement witnesses and positive maps which is based on a family of spectral conditions. This construction generalizes the entanglement witness corresponding to the celebrated Choi map which is positive but not completely positive.

We analyze a class of multipartite states invariant under the action of unitary groups. Our class generalizes well known classes of bipartite states: Werner and isotropic states. We perform a detailed analysis of separability conditions for generalized Werner-like and isotropic-like multipartite states. It is shown that necessary and sufficient separability conditions generalize well known PPT criterion for multipartite case.

One of the main activities of the brain is the recognition of signals. A first attempt to explain the process of recognition in terms of quantum statistics was given in [6]. Subsequently, details of the mathematical model were presented in a (still incomplete) series of papers (cf. [7, 2, 5, 10]). In the present note we want to give a general view of the principal ideas of this approach. We will introduce the basic spaces and justify the choice of spaces and operations. Further, we bring the model face to face with basic postulates any statistical model of the recognition process should fulfill. These postulates are in accordance with the opinion widely accepted in psychology and neurology.

The identity of white noise B(t) has been given, so that we proceed to the space that is spanned by the B(t)'s. The system consists of continuously many independent variables and, in fact, spans a separable space. The system is total in that space. Harmonic analysis by introducing transformation groups will be discussed briefly. Together with other notions we can prepare for the analysis of nonlinear functionals of white noise.

The purpose of this report is to give a detailed interpretations to white noise variables Ḃ(t), t ∈ R¹ and their functionals, including the idea of renormalization. By doing so, we can see an infinite dimensional analysis and some applications follow naturally

In ² clustering representations of the position distribution of the ideal Bose gas were considered. In principle that gives rise to possibilities concerning simulations of the system of positions of the particles. But one has to take into account that in case of low temperature the clusters are very large and their origins are far from a fixed bounded volume. For that reason we will consider some estimations of the influence of these clusters on the behaviour of the subsystem of particles located in a fixed bounded volume.

A lifting is a continuous map from a system to a compound system, and it is useful to describe several physical processes. Ohya and Accardi introduced the notion of it and characterized liftings¹. In this paper, we define a generalized quantum mutual entropy through liftings satisfying some conditions. In some cases, it satisfies the Shannon inequality and some cases not.

The aim of this paper is to describe new possibilities related to some applications of frames and fusion frames in stroboscopic tomography of classical and quantum open systems. We assume that the evolutions of systems in question are given by appropriate semigroups on the sets of states and that information about system states is given by expectation values of fixed observables (physical quantities) at some time instants t₁,…, t_r. Natural ways in which frames and fusion frames appear in stroboscopic tomography can be used to reduce the influence of noise and loss of data in determination of states. Overcompletness of frames turns out to be very useful in comparison to the methods based on orthonormal sets of vectors

The purpose of this paper is to show how useful are the notions of frame and fusion frame in the problems of trajectory reconstruction in open quantum systems. In particular, we discuss some problems of so-called stroboscopic quantum tomography. Frames are redundant sets of vectors in a Hilbert space, which assure a natural representation of each vector in the space, but may have infinitely many different representations for any given vector. For a given quantum system represented in a Hilbert space , the question of minimal number η of observables Q₁,…, Q_η, whose expectation values at some instants t₁,…, t_s determine the statistical state of the system is discussed. We assume that the time evolution of the system in question is governed by a semigroup of linear transformations with generator 𝕃.

We analyze the new equation of motion for the damped oscillator. It differs from the standard one by a damping term which is nonlocal in time and hence it gives rise to a system with memory. Both classical and quantum analysis is performed. The characteristic feature of this nonlocal system is that it breaks local composition low for the classical Hamiltonian dynamics and the corresponding quantum propagator.

We give an elementary introduction to the problems of quantization of dissipative systems considering the simplest but still not fully understood model of damped harmonic oscillator. We recall some old and introduce some new methods of quantization of the model starting from the classical damped harmonic oscillator equation. Our methods are based on Hamiltonian formulation of dynamical systems adopted to the case of dissipation and loss of energy. Interestingly, in our approach we can interpret the poles of the analytical continuation of the energy spectrum as resonant states, well-known in atomic physics.

The aim of this review (oriented to biologists looking for applications of QM) is to provide a detailed comparative analysis of classical (Kolmogorovian) and quantum (Dirac-von Neumann) models. We will stress differences in the definition of conditional probability and as a consequence in the structures of matrices of transition probabilities, especially the condition of double stochasticity which arises naturally in QM. One of the most fundamental differences between two models is deformation of the classical formula of total probability (FTP) which plays an important role in statistics and decision making. An additional term appears in the QM-version of FTP – so called interference term. Finally, we discuss Bell's inequality and show that the common viewpoint that its violation induces either nonlocality or "death of realism" has not been completely justified. For us it is merely a sign of non-Kolmogorovianity of probabilistic data collected in a few experiments with incompatible setups of measurement devices.

Recently systems of p-adic numbers were applied to genetics. These systems are fundamentally hierarchical. Geometrically they are represented by homogeneous trees (here p is the number of branches leaving each vertex). It was shown that p-adics provides a natural number theoretic representation of the genetic code. In our previous paper we constructed a representation of the genetic code on so called 2-adic plane (so p = 2 and the dimension of the 2-adic space equals to two as well). In the present paper we investigate degeneracy of the genetic code with the help of 2-adic analysis and show that there exists a linear condition on the coordinates of the 2-adic plane of codons which separates the 2-adic plane in the domains of strong and weak degeneracy of the genetic code. We discuss the relation of this result and the Rumer's symmetry of the genetic code. We extend Rumer's argument that degeneracy of the genetic code related to the energy of binding of the codons with the complementary triples of nucleotides (anticodons). In the 2-adic framework this argument became especially natural.

The following sections are included:

• Introduction

• Why C* -algebras?

• Structure of positive maps versus PPT states

• Decomposition theory for entanglement maps

• Acknowledgments

• References

We describe a constructive method of transforming the Hamiltonian of a system of interacting quantum spins into an entanglement witness. The resulting observable can be used as a simple tool for the detection of entanglement. We verify the efficiency of this detection on a class of analytically solvable 1-dimensional Heisenberg models.

The present paper is a brief, elementary introduction to the theory of entangled states in quantum systems, based on the lecture given by the author at the Suwa Satellite Conference of QBIC09, held at Tokyo University of Science in Suwa, March 6–8, 2009. We focus here on the mathematical problem of entanglement detection. The duality between positive maps and entanglement witnesses is emphasized. One particular class of experimentally relevant entanglement witnesses and the method of their construction is further discussed in another paper appearing in this volume.¹ An interested reader may find much more information about recent developments in the entanglement theory and applications in a review paper [2].

We report some of our recent tunneling results on various high-T_C superconductors, including cuprate superconductors and a newly discovered iron-based pnictide superconductor. After the introduction of Section 1, in Section 2 we explain the experimental details, and in Section 3, we show the characteristic features of the superconducting gap on electron-doped cuprate superconductor, subsequently we report the two kinds of superconducting gap on an iron-based pnictide superconductor NdFeAsO_1-y with T_C~51K.

The Josephson bifurcation amplifier (JBA) readout process of a superconducting qubit is analyzed quantum mechanically. We examined the dynamics of the density operator of a probe (a driven nonlinear oscillator) and a qubit coupled system during the measurement process. We have observed the qubit-probe (JBA) entangled state and it is divided into two separable states at the moment of the JBA transition begins. To readout the measurement result, however, we must wait until the two JBA states are macroscopically well separated.

Various versions of "independence" are actively inverstigated in quantum probability. In the context of relativistic QFT, we show here that the physical origin of "independence" can be sought in the asymptotic condition through which asymptotic fields and states exhibiting the independence emerge from the non-independent interacting Heisenberg fields in a kind of "central limit". From the algebraic viewpoint, this condition is equivalent to the on-shell condition to pick up free one-particle modes, which also reduces to Einstein's famous formula E = mc². A scenario to reconstruct interacting Heisenberg fields as Micro-objects from these "independent" =free Macro-objects intertwined by an S-matrix as a measurable quantity is formulated according to the Micro-Macro Duality associated with a new notion of a cocycle of K-T operators.

The Markov property of states of algebras of the canonical commutation relation is studied and in the case of Gaussian states several equivalent properties are obtained. The detailed description is given in terms of a block matrix. The relation to classical multivariate Gaussian Markov triplets is also described. The minimizer of relative entropy with respect to a Gaussian markov state has the Markov property.

The RNA-Crypto System (shortly RCS) is a symmetric key algorithm to cipher data. The idea for this new algorithm starts from the observation of nature. In particular from the observation of RNA behavior and some of its properties. In particular the RNA sequences have some sections called Introns. Introns, derived from the term "intragenic regions", are non-coding sections of precursor mRNA (pre–mRNA) or other RNAs, that are removed (spliced out of the RNA) before the mature RNA is formed. Once the introns have been spliced out of a pre–mRNA, the resulting mRNA sequence is ready to be translated into a protein. The corresponding parts of a gene are known as introns as well. The nature and the role of Introns in the pre–mRNA is not clear and it is under ponderous researches by Biologists but, in our case, we will use the presence of Introns in the RNA-Crypto System output as a strong method to add chaotic non coding information and an unnecessary behaviour in the access to the secret key to code the messages. In the RNA-Crypto System algorithm the introns are sections of the ciphered message with non-coding information as well as in the precursor mRNA.

We first recall the canonical representation theory of Gaussian processes (Hida²) which plays an important role in our study. For a given multiple Markov Gaussian process in a restricted sense, a new multiple Gaussian process is obtained using the differential operator in a sense of duality. We introduce a new concept to uniformly multiple Markov Gaussian process.

New approaches to protein functional inference based on protein structure and evolution are described. First, FINDSITE, a threading based approach to protein function prediction, is summarized. Then, the results of large scale benchmarking of ligand binding site prediction, ligand screening, including applications to HIV protease, and GO molecular functional inference are presented. A key advantage of FINDSITE is its ability to use low resolution, predicted structures as well as high resolution experimental structures. Then, an extension of FINDSITE to ligand screening in GPCRs using predicted GPCR structures, FINDSITE/QDOCKX, is presented. This is a particularly difficult case as there are few experimentally solved GPCR structures. Thus, we first train on a subset of known binding ligands for a set of GPCRs; this is then followed by benchmarking against a large ligand library. For the virtual ligand screening of a number of Dopamine receptors, encouraging results are seen, with significant enrichment in identified ligands over those found in the training set. Thus, FINDSITE and its extensions represent a powerful approach to the successful prediction of a variety of molecular functions.

The following sections are included:

• Chernoff's formula.

• Quantum dynamics and diffusion on [0, ∞).

• Feynman formulae for quantum dynamics and diffusion of particle with position-dependent mass.

• Diffusion on graphs.

• Feynman formulae for equations on Riemannian manifolds.

• Concluding remarks.

• References

We introduce Poisson analysis as a means to study processes of particle configurations in the continuum and present the free Kawasaki dynamics as illustration.

Aggregation of amyloid β peptide (Aβ) in the brain is the primary element in the pathogenesis of Alzheimer's disease. Aβ is derived from amyloid precursor protein (APP) in the membrane due to the cleavages by β- and γ-secretases. Here, we predict the transmembrane structures of the wild-type and mutant APP in the biological membrane by replica-exchange molecular dynamics simulations. The simulations illustrate large conformational differences between the wild type and mutant APP fragments in the membrane. Dimerization of the wild type occurs due to the Cα-H⋯O hydrogen bonds at the Gly-XXX-Gly motifs between two APP fragments, whereas the mutant dimer is stabilized by the interactions between hydrophobic side chains. We also observe the downward shift of γ-cleavage site in the mutant APP, which may cause the prohibition of Aβ production.

For the analysis of square contingency tables with same row and column classifications, Stuart (1955) considered the marginal homogeneity model which indicates that the row marginal distribution is identical with the column marginal distribution. Tahata, Iwashita and Tomizawa (2006, 2008) proposed the measure of departure from the marginal homogeneity model for square contingency tables with ordered categories. The present paper (1) reviews these measures and (2) compares between several Japanese social mobility data using these measures.

Recently after human genome sequence has been determined almost perfectly, more and more researchers have been studying genes in detail. Therefore, we are sure that accumulated gene information for human will be getting more important in the near future to develop customized medicine and to make gene interactions clear. Among plenty of information, micro array might be one of the most important analysis method for genes because it is the technique that can get big amount of the gene expressions data from one time experiment and also can be used for DNA isolation. To get the novel knowledge from micro array data, we need to enrich statistical tools for its data analysis. So far, many mathematical theories and definition have been proposing. However, many of those proposals are tested with strict conditions or customized to data for specific species. In this paper, we reviewed existing typical statistical methods for micro array analysis and discussed the repeatability of the analysis, construction the guideline with more general procedure. First we analyzed the micro array data for TG rats, with statistical methods of family-wise error rate (FWER) control approach and False Discovery Rate (FDR) control approach. As existing report, no significantly different gene could be detected with FWER control approach. On the other hand, we could find several genes significantly with FDR control approach even q=0.5. To find out the reliability of FDR control approach with micro array conditions, we have analyzed 2 more pieces of data from Gene Expression Omnibus (GEO) public database on the web site with SAM in addition to FWER and FDR control approaches. We could find a certain number of significantly different genes with BH method and SAM in the case of q=0.05. However, we have to note that the number and kinds of detected genes are different when we compare our result with the one from the published paper. Even if the same approach is used to analyze the same micro array data, we might get a different result because the distinct definition for micro array data has not been set yet. It means that from the same data we will get different results depending on researchers. We are afraid that this problem will have a big effect on developing new medicines and to progress the next step, like a 2^nd screening. So, we suggest that we should have certain guidelines to analyze Micro-Array data validly with statistic method and it will surely be helpful for Micro-Array analysis for medical studies in the future.

The time irreversibility problem is the problem of how to explain that there is the reversible microscopic dynamics and the irreversible macroscopic physics. In this paper an attempt is performed of the following solution of the irreversibility problem: a formulation of microscopic dynamics is suggested which is irreversible in time. In this way the contradiction between the reversibility of microscopic dynamics and irreversibility of macroscopic dynamics is avoided since both dynamics in the proposed approach are irreversible.

A widely used notion of microscopic state of the system at a given moment of time as a point in the phase space does not have an immediate physical meaning since arbitrary real numbers are non observable. In the approach presented in this paper the physical meaning is attributed not to an individual trajectory but only to a bunch of trajectories or to the distribution function on the phase space.

The fundamental equation of the microscopic dynamics in the proposed "functional" approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. It is shown that the Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton trajectories are computed. Possible applications to the information and molecular dynamics are mentioned.

Gene expression data not only provide us insights into discrete transcript abundance of specific genes, but contain cryptic information that can not readily be assessed without interpretation. We again used data of the plant Arabidopsis thaliana as our reference organism, yet the analysis presented herein can be performed with any organism with various data sources. Within the cell, information is transduced via different signaling cascades and results in differential gene expression responses. The incoming signals are perceived from upstream signaling components and handed to downstream messengers that further deliver the signals to effector proteins which can directly influence gene expression. In most cases, we can assume that proteins, which are connected to other signaling components within such a regulatory network, exhibit similar expression trajectories. Thus, we extracted a known functional network from literature and demonstrated that it is possible to superimpose microarray expression data onto the pathways. Thereby, we could follow the information flow through time reflected by gene expression changes. This allowed us to predict, whether the upstream signal was transmitted from known elements contained in the network or relayed from outside components. We next conducted the vice versa approach and used large scale microarray expression data to build a co-expression matrix for all genes present on the array. From this, we computed a regulatory network, which allowed us to deduce known and novel signaling pathways.

Classical dynamical entropy is an important tool to discuss coding theorems in classical information theory. Quantum dynamical entropy was first studied by Connes, Størmer and Emch. Thereafter, there have been many researches to construct or calculate the dynamical entropies for several models.

In this paper, we briefly review two formulations due to (i) Ohya and (ii) Kossakowski, Ohya and Watanabe. Some numerical computations of these entropies are carried for several states.

It is important to look for accurate sequence alignment because biological sequence analysis depends on the quality of alignment. However it is still a difficult computational problem because of not only the computational complexity but also unsatisfactory objective functions for measuring alignment. Here, we show the significant improvement of sequence alignment can be done by introducing a new metric defined on compound systems of two sequences. In the benchmark tests by PREFAB4 and HOMSTRAD, our method gives higher accuracy than other methods such as Needleman-Wunsch, ClustalW2. Especially for the sequences with less than 30% sequence identity, our method improves the alignment accuracy significantly.

Many Gram-negative bacteria initiate infections by injecting effector proteins into host cells through the type III secretion apparatus (TTSA) that is comprised of a basal body, a needle, and a tip. The needle channel is formed by the assembly of a single needle protein. To explore the export mechanisms of MxiH needle protein through the needle of Shigella flexneri, an essential step during needle assembly, we have performed steered molecular dynamics simulations in implicit solvent. Interestingly, the electronegative channel interior creates an energy barrier for MxiH to enter the channel, while the same may facilitate the ejection of the effectors into host cells. Structurally-known basal regions and ATPase underneath the basal region have also such electronegative interior, while effector proteins have considerable electronegative patches on their surfaces. Based on these observations, we propose a repulsive electrostatic mechanism for protein translocation through the TTSA. This mechanism is supported by the suggestion that an ATPase is required for protein translocation through these nanomachines, which may provide the energy to overcome the initial electrostatic energy barrier. A similar mechanism may be applicable to macromolecular channels in other secretion systems or viruses through which proteins or nucleic acids are transported.

In this paper, we discuss possibilities to use single-wall carbon nanotubes (SWCNTs) as building blocks of quantum computing devices, particularly as quantum bits (qubits). To do this, electron transport properties of SWCNT quantum dots (QDs) are described, with experimental data that show artificial atom behaviors of the QD. It is shown that the SWCNT-QD behaves as a one-dimensional artificial atom with simple shell structures and energy scales associated with the artificial atom in the THz frequency range. The simple artificial atom feature would be useful for spin qubits. Another qubit type with normal materials is a charge qubit consisting of the coupled quantum dots. The experimental effort to fabricate it is also described.

Protein-protein interactions play many important roles in biological function. Knowledge of protein-protein complex structure is required for understanding the function. The determination of protein-protein complex structure by experimental studies remains difficult, therefore computational prediction of protein structures by structure modeling and docking studies is valuable method. In addition, MD simulation is also one of the most popular methods for protein structure modeling and characteristics. Here, we attempt to predict protein-protein complex structure and property using some of bioinformatic methods, and we focus botulinum toxin complex as target structure.

Microarrays provide genome-wide gene expression changes. In current analyses, the majority of genes on the array are frequently eliminated for further analysis just in order for computational effort to be affordable. This strategy risks failure to discover whole sets of genes related to a quantitative trait of interest, which is generally controlled by several loci that might be eliminated in current approaches. Here, we describe a high-throughput gene discovery method based on correspondence analysis with a new index for expression ratios [arctan (1/ratio)] and three artificial marker genes. This method allows us to quickly analyze the whole microarray dataset without elimination and discover up/down-regulated genes related to a trait of interest. We employed an example dataset to show the theoretical advantage of this method. We then used the method to identify 88 cancer-related genes from a published microarray data from patients with breast cancer. This method can be easily performed and the result is also visible in three-dimensional viewing software that we have developed. Our method is useful for revaluating the wealth of microarray data available from web-sites.