Quantum Bio-Informatics IV

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• QP–DYN algorithms: general scheme

• Dynamical systems underlying the QP–DYN algorithms

• From sequences of vectors to sequences of bits

• Modifications of the dynamical law

• Orbit jump function

• Machine truncation

• Use of multiple dynamical systems

• Attacks to the 1–matrix algorithm

• Attacks to the 2–matrix algorithm

• References

Microarray analysis is a high-throughput method for analyzing expression levels of multiple genes, therefore the microarray have been regarded by many investigators as a powerful method. Treating a huge amount of data and judgment of differentially expressed genes require appropriate statistical analysis. When the microarray analysis suggests there are co-expressed genes under a specific condition, there is high possibility that the common transcriptional factors (TFs) control them. It is also difficult to identify the TFs involved in co-expression through only biochemical experiments. In view of cis-element pattern related to co expressed genes might be one of the solutions to infer the gene expression mechanism clearly.

So far, we have constructed Cis-Module database in order to specify cis-element location and distribution on genome. Using this database and rat microarray data, we have investigated the TFs network related to co-expression of genes. If we could also extract the human genes that are orthologous to co-expressed gene in rat, it will allow us to compare their cis-elements and TFs and to consider difference of gene expression profiles between rat and human. It will be very useful to find out attention to drug discovery targeting gene expression mechanism.

This article is a summary of our talk in QBIC2010. We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group in a Hilbert space with equipped with the strong operator topology. More precisely, for any strongly closed subgroup G of the unitary group in a finite von Neumann algebra , we show that the set of all generators of strongly continuous one-parameter subgroups of G forms a complete topological Lie algebra with respect to the strong resolvent topology. We also characterize the algebra of all densely defined closed operators affiliated with from the viewpoint of a tensor category.

We review some results on determining a general form of time operators of a Hamiltonian with purely discrete spectrum.

Recently a few authors pointed to a possibility to apply the mathematical formalism of quantum mechanics to cognitive psychology, in particular, to games of the Prisoners Dilemma (PD) type.^6_18 In this paper, we discuss the problem of rationality in game theory and point out that the quantum uncertainty is similar to the uncertainty of knowledge, which a player feels subjectively in his decision-making.

An elementary algebraic approach to unified quantum information theory is given. An operational meaning of entanglement as specifically quantum encoding is disclosed. The general relative entropy as information divergence is introduced and three most important types. of relative information, namely, Araki-Umegaki (A-type) and of Belavkin-Staszewski (B-type) and the thermodynamical (C-type) are shown. The true quantum entropy different from the von Neumann semiclassical entropy is introduced and the proper quantum conditional entropy is shown. The general quantum mutual information via entanglement is defined and the corresponding types of quantum channel capacities as the supremum via the generalized encodings are formulated. The additivity problem for quantum logarithmic capacities for the products of arbitrary quantum channels under the appropriate constraints on encodings is discussed. It is proved that the true quantum capacity, which is achieved on the standard entanglement as the optimal quantum encoding, reclaims the additivity property of the logarithmic quantum channel capacities via the entanglement on the products of quantum input states. This earlier obtained by V. P. B. result for quantum logarithmic information of A-type is extended to any type of quantum information

We analyze a local approach to the non-Markovian evolution of open quantum systems. It turns out that any dynamical map representing evolution of such a system may be described either by non-local master equation with memory kernel or equivalently by equation which is local in time. The price one pays for the local approach is that the corresponding generator might be highly singular and it keeps the memory about the starting point 't₀'. Remarkably, singularities of generator may lead to interesting physical phenomena like revival of coherence or sudden death and revival of entanglement.

We explain the relation between the quantum statistical model of the recognition process developed in the last years in a series of papers (cf. [7, 9, 10, 11, 12]) and a certain process of self-collapses. This process will be modeled by a classical homogenous Markov chain.

In many applications, for example cryptography and Monte Carlo simulation, there is need for random numbers. Any procedure, algorithm or device which is intended to produce such is called a random number generator (RNG). What makes a good RNG? This paper gives an overview on empirical testing of the statistical properties of the sequences produced by RNGs and special software packages designed for that purpose. We also present the results of applying a particular test suite—TestU01— to a family of RNGs currently being developed at the Centro Interdipartimentale Vito Volterra (CIVV), Roma, Italy.

In the previous paper,¹ we showed the significant improvement of sequence alignment can be done by considering the entanglement between two consecutive pairs of residues, in which we introduced a new measure defined on compound systems of two sequences which taking entanglement. In this paper, we explain the advantage of our new measure compared with the "sum of pairs" measure as the measure often-used, and we also show the performance evaluation of our algorithm against other six alignment methods. Our alignment is available at our webpage "http://www.rs.noda.tus.ac.jp/%7Eohya-m/".

The following sections are included:

• Prologue

• An i.i.d. random variables as a representation of parameter set

• Poisson noise

• Calculus of Poisson noise functionals

• References

We analyze a measure of quantum entanglement called degree of entanglement(DEN). It is shown how DEN behaves for well known classes of bipartite states. Moreover, we compare DEN for quantum states having the same marginals. Contrary to naive expectation it is shown that separable state might possesses stronger correlation (measured by DEN) than an entangled state.

Several scientific and technical problems can be described by a stochastic partial differential equation. The solution of the equation could be considered as the limit of a suitable discrete particle model. The existence of such a kind of approximation was discussed in ⁵. A completely discrete particle model, which is constructed to simulate by computer, is considered in ³. In this paper we give proofs of some lemmas which are used to prove the main theorem in ³.

There exists a quantum algorithm with chaos dynamics solving an NP-complete problem in polynomial time, called OMV SAT algorithm. The language class EXPTIME is larger class than NP, there is no classical algorithm to solve it in polynomial time. In this paper we propose a quantum algorithm for one of the problems in EXPTIME, Pebble Game, and compare the computational complexity of it with the classical one. We show that a quantum algorithm with Oracle solves it in polynomial time while a classical algorithm with same Oracle does in exponential time.

The aim of this paper is to discuss the relationship between some problems of the identification of quantum states by geometric methods used in stroboscopic tomography and, on the other hand, by algebraic approach typical for the quantum generalization of classical sufficient statistics. Some examples of necessary and sufficient conditions which must be fulfilled by generators of algebras in order to estimate states of quantum systems are discussed.

The notion of concurrence and its role as a measure of quantum correlations for both pure and mixed states is recalled. However, for mixed states concurrence is hard to compute and some estimations of it are necessary. It has been demonstrated how to use entanglement witnesses in this procedure. In particular, it has been shown that each entanglement witness detecting given bipartite entangled state provides an estimation of its concurrence. The results are illustrated by several examples.

We discuss the conjecture on quantum-like (QL) processing of information in the brain. It is not based on the physical quantum brain (e.g., Penrose) – quantum physical carriers of information. In our approach the brain created the QL representation (QLR) of information in Hilbert space. It uses quantum information rules in decision making. The existence of such QLR was (at least preliminary) confirmed by experimental data from cognitive psychology. The violation of the law of total probability in these experiments is an important sign of nonclassicality of data. In so called "constructive wave function approach" such data can be represented by complex amplitudes. We presented ^1,2 the QL model of decision making. In this paper we speculate on a possible physical realization of QLR in the brain: a classical wave model producing QLR . It is based on variety of time scales in the brain. Each pair of scales (fine – the background fluctuations of electromagnetic field and rough – the cognitive image scale) induces the QL representation. The background field plays the crucial role in creation of "superstrong QL correlations" in the brain.

The relation between two methods which construct the density operator on composite system is shown. One of them is called an entanglement mapping and another one is called a quantum conditional probability operator. On the base of this relation we discuss the quantum correlation by means of some types of quantum entropy.

The method of transforming a hamiltonian of a composite quantum system into an entanglement witness, explored in an earlier paper¹, is extended to the multipartite case. The witness can be made not only to detect general entanglement but also to discriminate among its various multipartite types. Partial knowledge of the lowest part of energy spectrum is sufficient for approximate witness construction. As an example, we carry out analytic calculations with the hamiltonian of a 1-dimensional Heisenberg XXZ model.

In the present paper we first describe quantum quadratic operators (q.q.o) acting on the algebra of 2 × 2 matrices . Moreover, we provide necessary conditions for q.q.o. with Haar to satisfy Kadison-Schwarz condition. By means of such a description we give an example of q.q.o. which is not the Kadision-Schwarz operator.

In the present paper we continue our investigations started in [Accardi L., Ohno, H., Mukhamedov, F., Quantum Markov fields on graphs, Inf. Dim. Analysis, Quantum Probab. Related Topics (accepted) arxiv: 0911.1667]. In [Accardi L., Mukhamedov, F., Saburov M. On Quantum Markov chains on Cayley tree and associated chains with XY-model arxiv: 1004.3623] we provided a construction of forward and backward Quantum Markov Chains (QMC) defined on the Cayley tree, and established uniqueness of QMC associated with XY-model on a Cayley tree order 2. In the present paper we study the same model on a Cayley tree order 3. Surprisingly in this case, we establish a phase transition (i.e. existence of two distinct quantum Markov chains) for the considered model on the Cayley tree order 3.

The microscopic origin of space(-time) geometry is explained on the basis of an emergence process associated with the condensation of infinite number of microscopic quanta responsible for symmetry breakdown, which implements the basic essence of "Quantum-Classical Correpondense" and of the forcing method in physical and mathematical contexts, respectively. From this viewpoint, the space(-time) dependence of physical quantities arises from the "logical extension" ⁶ to change "constant objects" into "variable objects" by tagging the order parameters associated with the condensation onto "constant objects"; the logical direction here from a value y to a domain variable x (to materialize the basic mechanism behind the Gel'fand isomorphism) is just opposite to that common in the usual definition of a function f : x → f(x) from its domain variable x to a value y = f(x).

The following sections are included:

• Introduction

• Ingredients

• The algorithm

• Early implementations

• The role of the Coding Functions

• Statistical Analysis

• Conclusion

• References

In this paper we discuss how to approximate white noise functionals and the operators in white noise analysis by using variables depending on discrete parameter. We discuss to understand the basic idea and real meaning of approximation of operators.

The following sections are included:

• Introduction

• Symplectic locally convex spaces and Hamilton's equations.

• Liouville's equations with respect to measures.

• Systems of equations with respect to finite-dimensional distributions of probabilities.

• Bogolyubov's systems of equations.

• Wigner measures.

• Generalization of Poincaré's model.

• References

For a two-way contingency table with nominal row and column variables, the measures which describe the proportional reduction in variation (PRV) from the marginal distribution of one variable to the conditional distribution given the other variable are proposed by Goodman and Kruskal (1954), Theil (1970), and Freeman (1987, p. 101). Tomizawa, Seo and Ebi (1997), and Miyamoto, Usui and Tomizawa (2005) proposed the generalization of those measures. Tomizawa, Miyamoto and Yajima (2002), and Yamamoto and Tomizawa (2009) proposed the PRV measures for a nominal-ordinal contingency table and for an ordinal-ordinal contingency table, respectively. The present paper (1) reviews these PRY measures and (2) analyzes and compares between several categorical data using these PRY measures.

The electron reservoir model for the integer quantum Hall effects, the magnetoplasmon dispersion plateaus, and the radiation-induced magnetoresistance oscillations, are briefly reviewed.

The world view underlying traditional science is based on reductionism and determinism when there is an empty space (vacuum) and material points which move along the Newtonian trajectories. This approach may be called "mechanistic" or "Newtonian". Quantum mechanics, in its Copenhagen interpretation, also adopts this world view. However this world view is not satisfactory by at least two reasons. First, there is uncertainty in the derivation of the position and velocity of the material point and second, it can not solve the time irreversibility problem. Moreover, the Newtonian approach is not well suited for applications of mathematics and physics to life science. Recently a new approach to classical mechanics was proposed in which the basic notion is not the trajectory but a probability distribution. In this functional mechanics approach one deals with the mean trajectories and one has corrections to the Newtonian equation of motion.

In this note we consider correspondence between the Newtonian trajectories for an anharmonic oscillator and the averaged trajectories in the functional mechanics and compute the dependence of the characteristic time from the dispersion.

Regulatory non-coding DNA is important to drive gene transcription and thereby influence mRNA and consequently protein abundance. Therefore, biologists and bioinformation scientists aim to extract meaningful information from these sequence regions, in particular upstream regulation regions called promoters, and conclude on regulatory sequence function. While some approaches have been successful for single genes or a single genome, it is an open question whether information on promoter function can readily be transferred between different species. Thus, it is useful for biologists to know more about the general structure and composition of promoters including the occurrence of cisregulatory DNA-elements (CREs) to be able to compare promoter architecture between organisms. To approach this task, we utilized the fully sequenced genomes of the plant model organisms: mouse-ear cress (Arabidopsis thaliana), western balsam poplar (Populus trichocarpa), Sorghum bicolor and rice (Oryza sativa). For the interspecies comparison we made use of quantile-quantile (QQ)-plots of the variances of hexanucleotides or known functional CREs of core-promoter regions. Here, we suggest that the differences in promoter architecture correlate with the sizes of the intergenic space, i.e. regions, in which the promoters are located. In contrast, analysis of CREs is hampered by the general lack of well characterized transcription factor-CRE-relationships.

In a study of several systems, we are interested to examine (1) the dynamics of state change and (2) the complexity of states. Ohya introduced in 1991 (ref.[21]) a general idea so-called a Information Dynamics (ID), which constitutes a theory under a frame of the ID by synthesizing the formalities of the investigations of (1) and (2). There are two kind of complexities in ID. One is a complexity of state describing system itself and another is a transmitted complexity between two systems. Entropies of classical and quantum systems are the example of these complexities. In order to treat a flow of dynamics process, dynamical entropies were introduced in not only classical but also quantum systems.

The main purpose of this paper is to compare with these mean entropies and the complexities in ID, we calculate these mean entropies for some simple models to discuss the complexity of information transmission for OOK and PSK modulations.

We propose a local scheme to enhance the security of quantum key distribution in Ekert protocol (E91). Our proposal is a fair sampling test meant to detect an eavesdropping attempt that would use a biased sample to mimic an apparent violation of Bell inequalities. The test is local and non disruptive: it can be unilaterally performed at any time by either Alice or Bob during the production of the key, and together with the Bell inequality test.

The interiors of all living cells are highly crowded with macro molecules, which differs considerably the thermodynamics and kinetics of biological reactions between in vivo and in vitro. For example, the diffusion of green fluorescent protein (GFP) in E. coli is ~10-fold slower than in dilute conditions. In this study, we performed Brownian dynamics (BD) simulations of rigid macromolecules in a crowded environment mimicking the cytosol of E. coli to study the motions of macromolecules. The simulation systems contained 35 70S ribosomes, 750 glycolytic enzymes, 75 GFPs, and 392 tRNAs in a 100 nm × 100 nm × 100 nm simulation box, where the macromolecules were represented by rigid-objects of one bead per amino acid or four beads per nucleotide models. Diffusion tensors of these molecules in dilute solutions were estimated by using a hydrodynamic theory to take into account the diffusion anisotropy of arbitrary shaped objects in the BD simulations. BD simulations of the system where each macromolecule is represented by its Stokes radius were also performed for comparison. Excluded volume effects greatly reduce the mobility of molecules in crowded environments for both molecular-shaped and equivalent sphere systems. Additionally, there were no significant differences in the reduction of diffusivity over the entire range of molecular size between two systems. However, the reduction in diffusion of GFP in these systems was still 4-5 times larger than for the in vivo experiment. We will discuss other plausible factors that might cause the large reduction in diffusion in vivo.

Considering the important issues concerning food, environment, and energy that humans are facing in the 21st century, humans mostly depend on plants. Unlike animals which move from an inappropriate environment, plants do not move, but rapidly sense diverse environmental changes or invasion by other organisms such as pathogens and insects in the place they root, and adapt themselves by changing their own bodies, through which they developed adaptability. Whole genetic information corresponding to the blueprints of many biological systems has recently been analyzed, and comparative genomic studies facilitated tracing strategies of each organism in their evolutional processes. Comparison of factors involved in intracellular signal transduction between animals and plants indicated diversification of different gene sets. Reversible binding of Ca²⁺ to sensor proteins play key roles as a molecular switch both in animals and plants. Molecular mechanisms for signaling network of environmental sensing and adaptation in plants will be discussed with special reference to Ca²⁺ as a key element in information processing.

Networks are a natural and popular mechanism for the representation and investigation of a broad class of systems. But extracting information from a network can present significant challenges. We present NetzCope, a software application for the display and analysis of networks. Its key features include the visualization of networks in two or three dimensions, the organization of vertices to reveal structural similarity, and the detection and visualization of network communities by modularity maximization.

We studied the evolution of HIV-1 (Human Immuno-deficiency Virus Type 1) by means of coding theory and information dynamics. More precisely, (1) we applied various artificial codes to look for the similarity between these codes and the code of HIV-1; (2) the entropic chaos degree was used to describe the evolution of HIV-1.

Many of biological system mediated through protein-protein interactions. Knowledge of protein-protein complex structure is required for understanding the function. The determination of huge size and flexible protein-protein complex structure by experimental studies remains difficult, costly and five-consuming, therefore computational prediction of protein structures by homolog modeling and docking studies is valuable method. In addition, MD simulation is also one of the most powerful methods allowing to see the real dynamics of proteins. Here, we predict protein-protein complex structure of botulinum toxin to analyze its property. These bioinformatics methods are useful to report the relation between the flexibility of backbone structure and the activity.

In the present paper we will discuss two important roles of the interplay of Jahn-Teller physics and Mott physics. One is the small Fermi surface. The "Fermi arcs" observed in ARPES should be one of the edges of small Fermi pockets, based on the Kamimura-Suwa model (K-S model). This prediction is consistent with ARPES results by Tanaka et al. Another is the mechanism of superconductivity in cuprates. This can be explained by the interplay of strong electron-phonon interactions and local AF order. It is shown that the characteristic phase difference of wave functions between up- and down-spin carriers in the presence of the local AF order leads to the superconducting gap of d_x²-y² symmetry even in the phonon-involved mechanism.