Quantum Bio-Informatics

PREFACE

CONTENTS

No abstract received.

We discuss the characteristic of biological systems and our view to approach the quantum bio-informatics.

Some aspects of time operators including their spectral properties are reviewed.

The entropic chaos degree (ECD) of the two dynamics, the rotation map and the log-linear map, is studied. The ECD is an important quantity to discuss observation of chaos in dynamical systems, and it is related to “observable adaptivity”, one aspect of the idea of adaptive dynamics. In this paper, the typical characteristic of the ECD is demonstrated in the case of the rotations and Log-linear dynamics.

We formulate a variational principle for finding the time-optimal quantum evolution of mixed states governed by the master equation, when the Hamiltonian H and the Lindblad operators L_j are subject to certain constraints. We show that the problem can be reduced to solving first a fundamental equation (the “quantum brachistochrone”) for H(t), which can be written down once the constraints are specified, and then solving the constraints and the master equation for the L_j(t)s and the density operator ρ(t). As an application of our formalism, we analytically solve a simple one qubit model where the optimal Lindblad operators correspond either to a continuous Markovian measurement or to a decoherence process by the environment.

One of the main activities of the brain is the recognition of signals. As it was pointed out in [23, 26] the procedure of recognition can be described as follows: There is a set of complex signals stored in the memory. Choosing one of these signals may be interpreted as generating a hypothesis concerning an “expexted view of the world”. Then the brain compares a signal arising from our senses with the signal chosen from the memory. That changes the state of both signals in such a manner that after the procedure the signals coincide in a certain sense. Furthermore, measurements of that procedure like EEG or MEG are based on the fact that recognition of signals causes a certain loss of excited neurons, i.e. the neurons change their state from “excited” to “nonexcited”. For that reason a statistical model of the recognition process should reflect both – the change of the signals and the loss of excited neurons. According to the general conception of quantum theory the procedure of recognition should be described by operators acting on a certain Hilbert space. In the present paper which is based on [7] we describe in detail the activity in different parts of the brain by using so-called beam splitters well-known in quantum optics. Application of such a beam splitter may be interpreted as an exchange of the support (excited neurons) of the input signals and the signal chosen from the memory, a procedure which was mentioned in [26]. Recognition takes place if one of the outputs of the splitting procedure is collapsing. One can show (cf. Theorem 3.3) that for sufficiently high intensities of the signals this operator equals approximately the operator of projection onto the vacuum state in the considered region of the brain. In the present paper we want to give some overview of the basic ideas, structures and notions of the proposed model of the recognition process. Most of the proofs are omitted and will be given in some forthcoming papers [5, 6]. In this series also the procedures of creation of signals from the memory, accumulation and transformation of input signals, and measurements like EEG and MEG will be treated in detail.

It is an old and important open problem in Mathematical Quantum Field Theory to find a rigorous realization of certain heuristic path integral expressions appearing in the study of Non-Abelian Chern-Simons models. In particular, one is interested in the path integral expressions for the so-called “Wilson loop observables”. In the present paper we consider a closely related problem, namely the question whether it is possible to find a rigorous realization of the modified path integral expressions for the Wilson loop observables which arise in the special case M = Σ × S¹ after “torus gauge fixing” has been applied. Here M is the (compact 3-dimensional) base manifold of the Chern-Simons model considered. We expect that this modified problem can indeed be solved and that white noise analysis will play a key role here, cf.^{17, 18} In the present paper we will briefly sketch the approach in^17,18 with an emphasis on those points related to white noise analysis.

No abstract received.

No abstract received.

We have studied the quantum algorithm for several years to solve NP complete problem in polynomial times. Moreover we have discussed a computational complexity of it in terms of Turing machine. In this paper, we review some of our results and explain some new Turing machine and its language classes following our recent works^1,2.

The aim of this paper is to discuss some problems connected with reconstructibility of trajectories of a class of stochastic equations. In particular we address Cauchy-like problems for the so-called generalized birth-and-death processes. This type of processes are a simple, natural formal framework for modelling a vast variety of biological processes such as population dynamics, genome evolution and somatic evolution of cancers. We describe how empirical data, e.g., mean values of some attributes of systems in questions, can be used to find trajectories of these systems.

Non-Markovian reduced dynamics of an open system is investigated. In the case the initial state of the reservoir is the vacuum state, an approximation is introduced which makes possible to construct a reduced dynamics which is completely positive.

Belavkin and Ohya gave the notion of generalized entanglement which was studied in terms of Hilbert-Schmidt operator. In this article we review their wider definition of quantum entanglement and report the recently established equivalence between PPT condition and B-O entanglement condition on their general scheme.

I will discuss the following four (1)-(4) below from both mathematical and philosophical views: (1) What is (or do we mean) the understanding of the existence? (2) We propose “Adaptive dynamics” to understand the existence. (3) The adaptive dynamics can be used to describe chaos. (4) The adaptive dynamics is applied to the SAT Quantum Algorithm to solve the NP complete problem.

The mutual relation between quantum Micro and classical Macro is clarified by a unified formulation of instruments describing measurement processes and the associated amplification processes, from which some perspective towards a description of emergence processes of spacetime structure is suggested.

Recently,we presented a new rigorous few-body equation with the charged particles systems. On the top of that in this paper, we are concerned with the calculation of the proton-proton phase shift in momentum space with a rigorous long range treatment of the Coulomb potential and the Reid soft core nuclear potential. The method is based on the two-potential theory with a special boundary condition at a finite screening range. The results are in a very good agreement with those of the r-space calculations using the Schrödinger equation. The presented method is applicable to few-body problems. The use of the traditional Coulomb renormalization methods in the three-charged particles systems are alerted.

The estimation of the density matrix of a k-level quantum system is studied when the parametrization is given by the real and imaginary part of the entries and they are estimated by a finite number of independent measurements. For pure or nearly pure states, a simple and provable correct constraining algorithm is proposed that yields an asymptotically unbiased estimate.

The quality of the estimate is characterized by the mean quadratic error (MQE) matrix. The determinant of the averaged MQE matrix is minimal for mutually unbiased measurements. For 2-label systems, an adaptive optimal two-step-measurement scheme is proposed that uses a set of initialization measurements to determine the number of measurements in each of the measurement directions that is applied in the second step. An improved version of the algorithm uses the rotation of the measurement axes in the second step, too.

Local density-of-states (LDOS) modulation in Bi₂Sr₂CuO_6+δ is studied by scanning tunneling microscopy and spectroscopy at 4.2 K. Two dimensional modulation aligned with the Cu-O-Cu bond with a periodicity of about five lattice constants is observed in optimally doped samples prepared by the partial substitution of lanthanide ions for Sr ions. At the crest of the LDOS modulation, the new gap feature about 10 meV coexisting with the superconducting gap is found. This new gap feature is thought to be the key to understand the origin of the LDOS modulation.

In the study of unitary representation of infinite symmetric group S(∞) that appears in the investigation of Poisson noise, we have studied an irreducible factor of a unitary representation of S(n) and applied the Bochner's method on constructing the representation of S(∞) by taking the projective limit. In the limit we are given a space of quadratic Hida distributions. Thus we can see the adjoint of the Lévy Laplacian which naturally comes from Gaussian noise. In this sense, we can see a duality between two noises, Gaussian and Poisson.

Dissipative quantum dynamics is newly formulated in order to study small systems such as quantum stochastic resonators which show stochastic resonance even in a single spin system coupled with stochastic environment obeying Arrhenius-type relaxation. This formulation contains two terms, namely a dynamical relaxation term appearing in Zubarev's formulation of non-equilibrium infinite systems and a quasi-equilibrium relaxation term intrinsic to finite systems. The present formulation has the great merit that it is effectively applicable to small finite dissipative quantum systems which are related to quantum information and other mesoscopic devices.

The observation of the readout of a superconducting flux qubit is reviewed. The qubit is an aluminum superconductor loop surrounded by a dc-SQUID for readout. Parametric control of a flux qubit has been achieved by using two-frequency microwave pulses. We have observed Rabi oscillations stemming from parametric transitions between the qubit states when the sum of the two microwave frequencies or the difference between them matches the qubit Larmor frequency. We also have observed the coherent exchange of a single energy quantum between a flux qubit and a superconducting LC circuit acting as a quantum harmonic oscillator. The exchange of an energy quantum is known as the vacuum Rabi oscillation.

A challenging task for biologists and bioinformatic scientists is the analysis of gene expression mediated by cis-regulatory elements (CREs). Regulatory DNA sequences harbor essential information to control specific gene expression and integrate information derived from signaling cascades. By the means of microarrays, the abundance of several thousand transcripts can be monitored simultaneously. This data can be used to model the flow of information from incoming signals via CREs that orchestrate gene expression, which may converge on downstream CREs. In the last few years, many large, directly comparable microarray datasets have been performed in the AtGenExpress project with the plant model organism Arabidopsis thaliana. Each of these datasets constitutes an invaluable information resource for the study of plant developmental processes, physiological responses or interaction with its environment. Moreover, there is an increasing number of multidimensional expression datasets, for which suitable analysis programs that can keep track of all dimensions are still missing. Additional limitations of microarray analysis include overcoming a “smoothing effect” on the relative gene expression when a large number of expression profile datasets are combined for comparison. Ultimately, the investigation of CREs possibly involved in regulating transcription is best aided by using specific gene clusters and determining linkage between gene expression, CRE position, orientation and number.

The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983. Recently, several quantum mutual type measure were introduced in order to solve a sort of quantum coding theorem. In this paper, we compare with these mutual entropy-type measures by calculating for the attenuation channel.

A parametric statistical model is called singular if its Fisher information matrix is singular. In general, a statistical model which has hierarchical structure or hidden variables is singular. In singular statistical models, the log likelihood function can not be represented by any quadratic form of the parameter, resulting that the conventional statistical asymptotic theory does not hold. In this paper, we propose the standard form of the log likelihood function in singular statistical models, and show that a new statistical theory is established based on the standard form.

For the analysis of square contingency tables, various models of symmetry and asymmetry are proposed by many statisticians. This paper (1) reviews various models and (2) using models, compares four kinds of data on unaided distance vision of (i) men in Britain, (ii) women in Britain, (iii) students in a university of Japan, and (iv) pupils in elementary schools in Tokyo, Japan. This paper also (3) proposes the use of ratio parameter in the conditional symmetry model for comparing the degree of asymmetry in several tables.

No abstract received.

Based on the definition of helical geometry introduced by Chothia et al. [J. Mol. Biol., 145, 215 (1981)], we have developed various restraint potentials such as the helix-helix distance, crossing angle, and hinge angle of two helices, and tilt angle of a helix. Using these restraint potentials, these corresponding geometries can be maintained around their target values during molecular dynamics (MD) simulations. A series of assessments show that calculated restraint forces are numerically accurate. Since the restraint forces are only exerted on atoms which define helical principal axis, each helix can freely rotate along its helical axis, depending on helix-helix (or helix-environment) interactions. Such a restraint potential enables us to characterize these interactions at atomic level by sampling their conformational space around helical reaction coordinate with (restraint) force-dependent fluctuations. These restraint potentials of helical geometries are promising to significantly improve one's ability to understand biologically important issues.

In the Post-genome era, the interests on the genome sequences seem to shift the analysis of regulatory regions such as findings of cis-regulatory elements or repeat elements which arc not “genes”. However, there are some difficulties for the systematic analysis of cis-regulatory elements by bio-chemical methods. In order to overcome the limitation of biochemical methods, it is clear to need the help of computational approach. At this point in time, many researchers have been proposing the mathematical ideas to compare the sets of cis-regulatory. One of our interests is to find the common rule laid on the sequence pattern of cis-elements or the sophisticated evaluation measure of the complexity of cis-elements. In this paper, we report some our approaches based on information theories to find the relationship of recognition between cis-elements and the corresponding transcription factor.

No abstract received.

In research for life we first need to align the sequences in order to compare several different genes or amino acid sequences. When the number of sequences being compared becomes too large, such alignment takes a very long time. Therefore, we have made an attempt to establish this alignment using quantum algorithms (e.g.,[5,6]). We discuss one of such algorithms here. In future, we plan to use our findings in research on classification and change in living organisms such as HIV, and to link it to the introduction of markers for observing changes in disease progression (see [16-22] for trials along this line). We in this paper explain some of our trials by means of coding theory and entropic chaos degree.

It is well aware that mankind is now facing critical issues of global warming and shortage of energy resources, which will become apparent in the near future. It would be only a solution to overcome these catastrophes to learn detailed mechanisms of sophisticated and efficient phenomena in the nature like photosynthesis and then to apply their principles to productions for social activities and to reduction of energy consumption in human life.

Here, photosynthesis is chosen as an example to investigate such sophisticated phenomena, since photosynthesis has been well studied with both experimental and theoretical approaches for a long time. Especially, charge transfer in photosynthetic reaction center initiated by absorbed solar energy is suitable to learn how efficient biological systems are. If the mechanism is understood in detail, another kind of solar battery would be invented based on its novel principles. But, although many of interesting and amazing features of the charge transfer processes have been revealed at molecular level from experimental studies especially using X-ray structure analyses, the reason for the efficient electron transfer in the photosynthetic reaction center is still kept unknown.

Since molecular simulation has reached the stage of being capable that chemical properties of ordinary but rather small molecules are predicted with enough accuracy. Thus, molecular simulation is strongly expected to be one way to study the mechanism of the charge separation in the reaction center. In the present paper, one theoretical framework for studying the charge transfer is presented and technical problems for parallel computing are briefly described.

We reviewed the origin and evolution of the two pairs of immune genes, (MHC-B and MHC-C) and (MICA and MICB) in man, chimpanzee and rhesus monkey based mainly on our previous work. Since those genes were well known to have been subject to strong natural selection in evolution, they themselves were not suitable for our study. We thus took another approach to use fragmented and nonfunctional LINEs that had coevolved with the two pairs in the same genomic fragments. Our results showed that MHC-B and MHC-C duplicated about 22 Mry (million years) ago, and MICA and MICB duplicated about 14 Myr ago. Interestingly, rhesus monkey was found not to have either pair but many repeats similar to MHC-B. Therefore, we estimated the divergence time of the monkey, and found that it diverged out from a common ancestor of man and chimpanzee about 30 Myr ago. The divergence time was consistent with the duplication times of the two pairs of immune genes. Based on our results we would predict that orangutan and gorilla also have the two pairs, because the both primate species are considered to have diverged less than 14 Myr ago.

It is well known that Gibbs states and the Gaussian distribution are characterized by the maximum entropy principle. In this paper we discuss probability distributions which maximize generalized entropies including Rényi's and Tsal-lis's.

Thanks to the many large-scale genome sequencing projects, thousands of primary sequences have been determined and the number of uncharacterized sequences will continue to grow. One of the major goals of genome science is to create a living system in a computer using such sequence information. To this goal, many researchers are developing several algorithms to predict folding structures of proteins from their amino acid sequences, to predict functions from sequence information or structures, and finally to simulate living systems in a computer. In this review, we describe our trials toward this goal.