Computational theory of reinforcement learning has been helpful in understanding the neural mechanisms of learning. In the application of reinforcement learning to autonomous robots, the designs of reward functions turned out to be crucial. Here we report our new research project on the design and evolution of rewards that enable self-preservation and self-reproduction of mobile robots.

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Miniaturization of chemical or biochemical devices provide a number of advantageous features for microscale reactions and analyses. A PDMS (polydimethylsiloxane) microfluidic device, for instance, which has microchannels for electrophoretic separation can be easily fabricated through molding process. And sealing of those channels doesn’t need any elaborate bonding processes which are usually required for glass chips. We’ve been working on PDMS-based microfluidic devices for biomedical applications, which is named ‘Microbiochemical Systems’. Using this technology, novel setups for in vitro biological experiments can be easily realized. In this paper, we present examples of those devices which could be used for advanced biological experiments and the detailed microfabrication processes are also described.

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This paper presents a new reinforcement learning method for multi-agent systems. On difficult problem in reinforcement learning for multi-agent environments is how to determine the agents that receive rewards for cooperative behavior. We need to solve this problem in order to prevent social dilemma and thereby enhance effectiveness of cooperation among agents. A manually tailored reward assignment strategy may work in some cases, but there are always new environments where there is no such(manually tailored) strategy. We aim at developing reinforcement learning agents that behave cooperatively without the reward distributed algorithm by introducing the generalized reciprocity. Our experimental results show that our reciprocal Q-learning achieves better performance than that of traditional ones.

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In this paper, autonomous learning of reward distribution in multi-agent reinforcement learning was applied to a 4-player game named “not100”. In this game, more shrewd strategy to cooperate with the other agents is required for each agent than the other tasks that the learning was applied previously. The reward distribution ratio after learning was varied among simulation runs. However, the validity of the average reward distribution ratio was examined in some ways. The three agents with higher probability of win after learning were stronger when the distribution ratio was learned than some cases when the agents learned with fixed distribution ratio.

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This paper presents a new framework of multiagent reinforcement learning to acquire cooperative behaviors by generating and coodinating each learning goal interactively among agents. Reinforcement learning (RL) is the major learning mechanism for the agent to adapt itself to various situations of an unknown environment flexibly. However, in amultiagent system environment that has mutual dependency among agents, it is difficult for a human to setup suitable learning goals for each agent, besides the existent framework of RL that aims for egoistic optimality of each agent is inadequate. Therefore, it requires the active and interactive learning mechanism to generate and coordinate each learning goal among agents. To realize this, first we propose to treat each learning goal as reinforcement signal (RS) that can be communicated among agents. Second we introduce motivation rules to integrate RS communicated among agents into areward value for RL of an agent. Then we define cooperative rewards as learning goals with mutual dependency. Learning experiments for two agents with or without learning received RSs as sub-reward are performed. Experimental results show that our proposed method with sub-reward learning is converged better to cooperative behaviors.

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This paper presents a method for simultaneous learning in multiagent environment to emerge the cooperative behaviors. Each agent has two action value functions: one is for action execution based on the policy updated in the previous stage, and the other one is for learning based on the episodes experienced by the execution function. This makes all agents behave based on the fixed policies, by which the non-Markovian problem can be avoided except for the update periods that depends on the learning progress of each agent. In order to avoid the local maxima due to such asynchronous renewal of action value functions, optimistic action values are given as initial ones, that helps the exploration process not to be treapped in the local maxima. The experimental results applied to one of the cooperative task in dynamic, multiagent environment, RoboCup, is shown and a discussion is given.

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Many attempts have been reported that make reinfocement learning agents synthesize coordinated behavior needed to accomplish their common goal effectively. In most of them, however, all the agents are supposed to make their decisions synchronously at regular time intervals, while such assumption does not hold in real-world multi-agent environments in general. We have proposed a design scheme for an autonomous agent based on neural network representation of the decision policy and its optimization with real-coded GA, which is applicable to multi-agent domains where individual agents make their decisions asynchronously. Through its application to the multi-agent seesaw balancing problem, we shows the effectiveness of the proposed idea. In addition, we propose a co-evolutionary generation alternation model, which allows us to apply the scheme to competitive domains.

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According to the progress of several genome projects, we have more than 60 genomic sequences. For a large fraction of the gene products, however, no functional information have been identified. Functional annotation is usually done by comparative sequence analysis, but sequence similarities does not always infer their functional relationship. Since structural information of proteins are more tightly coupled with their molecular functions than their amino-acid sequences, some trials to identify protein function from their structural information have been emerging. In this tutorial session, we will introduce such trials and related databases on three dimensional structure of proteins. For the purpose, here we describe an observation of protein structure-function relationship for S-adenosyl-L-methionine dependent methyltransferases superfamily as an example.

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At the late stage of recombinational repair in prokaryotes, RuvA, RuvB, and RuvC proteins process the Holliday junction through formation of two types of complexes, which catalyze branch migration (RuvAB) and resolution (RuvABC resolvasome), respectively. We determined the crystal structures of all three protein components. The RuvA-Holliday junction complex structure revealed that two base pairs at the crossover point are disrupted, implying the intermediate of base pair rearrangements. The crystal structure of the E. coli RuvC dimer allowed us to build a Holliday junction model bound to RuvC. The structure of the Thermus thermophilus RuvB protomer revealed the RuvB architecture, classified into the AAA+ family, and the environments of the ATP or ADP binding site. The X-ray structure of the RuvA-RuvB complex has revealed that two RuvA tetramers form the symmetric and closed octameric shell, where four RuvA domain IIIs spring out in the two opposite directions to be individually caught by a single RuvB. This structural study, including electron microscopy, led to the model of the RuvA-RuvB-Holliday junction complex, which implies that the functional scheme with a fixed RuvA-RuvB interaction may be preferable to that with their rotational interaction.

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ATP synthase is a rotary machine found next to the bacterial flagella motor in the biological world. This enzyme is composed of two rotary motors, Fo and F1 connected by a common shaft. Its diameter and height are 10nm and 15nm respectively, i.e., this is the smallest rotary motor in the world. We developed the experimental systems to visualize the rotary motion of the F1-motor driven by ATP, and revealed its marvelous performance.

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We propose a new recognition method using independent components based on a subspace method. Independent components are calculated by the ensemble learning that can reduce redundant basis functions. The proposed method has some modules for each class and a combiner. A module consists of feature extraction and classification. Features are extracted by the ensemble learning and classification is made on k nearest neighbors. Outputs of modules are combined in the combiner and a category is decided by a majority rule. Simulations are performed for hand-written characters to evaluate the proposed method. Results show the effectiveness of the proposed method.

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This paper presents a vision system that is possible to detect human face in wide view. The system has an advantage that it is capable to simultaneously satisfy incompatible requirements of wide view and high resolution. The authors have studied human symbiotic and collaborative robot. As a part of our effort to realize such robot, the authors have been developing face recognition system. During the development, we have faced a trade-off between wide view and high resolution. When the resolution is increased, i.e. image is closed up, so that a face can be stably detected, field of view becomes narrower. On the contrary, when the filed of view is widened, then the resolution becomes lower and the face detection becomes fail. To resolve this trade-off, the authors introduce camera array of more than one cameras. The camera array enables to satisfy the above mentioned incompatible requirements simultaneously. The authors have already implemented a working prototype system of the camera array based upon the above idea. And we have conducted experiments using the prototype. The experimental results demonstrate feasibility and effectiveness of the idea behind the system. In our experiment, the field of view is widened at 1.7 times. And the execution time is reduced to one third.

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This paper presents a method of visually steerable sound beam forming. The method is a combination of face tracking by motion image processing and beam forming by speaker array. Direction towards a target person can be obtained by the face tracking in real-time. By continuously updating the sound beam direction with the result of the tracking, the system is possible to keep transmitting sounds selectively towards the target person, even if he or she moves around. To detect and track the target person who moves in wide area, it is necessary to simultaneously satisfy conflicting requirements of wide view and high resolution that is enough for the detection. In this work, the authors solve the problem by utilizing camera array and succeed to realize the visually steerable sound beam forming in wide area. Experimental results prove the feasibility and effectiveness of the method.

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In this paper the authors propose a novel sound localization technique named spot forming. We have already succeeded in realizing sound beam forming by speaker array of one straight line. Using the beam forming, we can transmit sound towards desired direction. The spot forming proposed in this paper is a natural extension of the beam forming. It uses orthogonal two lines of speaker array. By arranging speakers in the shape of two orthogonal straight lines, it is expected that spot-like small area of higher sound pressure level can be formed locally around the point where two sound beams cross each other. This paper describes not only speculative discussion nor simulation but also experiments using actually working speaker array system. Experimental results support feasibility of the proposed idea.

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This paper presents spatial distribution of sound pressure level by multiple axes speaker array. The authors have already implemented an actually working speaker array system of one straight line. And we have succeeded in realizing sound beam forming. Using this beam forming effect by the speaker array, we can transmit sound towards desired direction. In this paper, the authors extend the array by arranging speakers in the shape of two or three orthogonal straight lines and examine the effect of the extension by simulation. According to our simulation results, it is expected that the multiple axes speaker array can locally form small area of higher sound pressure level at the desired point on the 2 dimensional plane or in the 3 dimensional space instead of the desired direction.

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We review key topics for developments since 1990s in cross-disciplinary fields between statistical mechanics and information sciences bridged by probabilistic models.

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It is shown that the demodulation problem of the code-division multiple-access (CDMA) communication channel can be mapped to a spin glass model in statistical mechanics. This correspondence allows us to apply various tools in spin glass theory to analyzing the performance of the CDMA demodulation. Results of the replica analysis of the CDMA demodulation problem are shown.

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In this paper, Bayesian networks and their applications to intelligent information systems are introduced in the context of probabilistic information processing researches in AI.

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Probabilistic models that are applicable to probabilistic inference are often given as graphical representations. Moreover, it is known that the probabilistic models for probabilistic inference are sometimes equivalent to familiar statistical-mechanical models. A cluster variation method is one of familiar statistical-mechanical techniques for calculating some statistical quantities in massive probabilistic models. The algorithms constructed by applying the cluster variation method for probabilistic models with loopy graphical representations is equivalent to belief propagation algorithms. In this paper, the review of the application of a cluster variation method to probabilistic inference is given and a new method for calculating correlation functions between any pairs of nodes is proposed by combining the cluster variation method with a linear response theory.

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Markov-chain Monte Carlo (MCMC) methods have successfully been used in various fields including statistical physics, statistical sciences, optimization problems and so on. The ideas of MCMC have been developed into the simulated annealing method for optimization problems and the extended ensemble method for simulating complex systems. We give a brief introduction on MCMC method, in particular, from viewpoint of statistical physics. Subsequently, an application of MCMC to the N queens problem which is a kind of constraint-satisfaction problem is presented. We show our numerical results on the number of solutions of the N queens problem for large N where exact enumeration is hard to perform.

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We review recent developments in analytical investigations of finite-dimensional spin glasses. A particular emphasis is put on the identification of the exact location of the multicritical point of the ±J Ising model on the square lattice. Duality arguments are introduced for this purpose, and arguments leading to a conjecture on the exact location are explained. Implications of the result on information processing, error-correcting codes in particular, are discussed both under classical and quantum situations.

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Recently, neurophysiology has made rapid progress, showing that particular neurons produce various periodic patterns such as walking motions, biological rhythms, and so on. Accordingly, it is strongly desirable to realize artificial neural oscillators which generate periodic oscillations. We have already proposed synthesis method of neural oscillators with specified periods and shapes by learning of neural networks. However, it does not consider the degree of stability of them. It is well known the stability of limit cycles can be estimated by checking eigenvalues of Jacobian of Poincare map defined on them. This paper presents a learning method that can assign desired eigenvalues to the Jacobian of Poincare map. Using this method, we can realize neural oscillators generating limit cycles with specified degree of stability.

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This paper is concerned with oscillation phenomena in a coupled oscillator system with Belousov-Zhabotinsky (BZ) reaction, which is a typical type of coupled chemical oscillator system. Synchronization phenomena emerged in coupled plural BZ reactions was observed, and the observed patterns were compared with analytical results based on the symmetric Hopf bifurcation theory used to analyze spatio-temporal patterns in systems with geometrical symmetry. Tuning rules of the coupling parameters were obtained, that control the desired transition between plural synchronized oscillation patterns.

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This paper presents a new method for the description of algorithms. We took algorithms as “the process which minimize some values under some constraints,” so we propose that algorithms can be written as an optimization problem, in stead of procedural ways. As a basic research for this method, we apply this method for some examples. Precisely, we show that some algorithms such as sort, basis transform, etc. can be described with LMI format (one of the optimization problem), and that we can obtain the answers by solving LMI.

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Optimization methods by using chaos dynamics interests as a class of global optimization methods by which the global minimum can be obtained without trapping in local minima. The chaos dynamics are classified into discretized models of gradient dynamics and continuous dissipative dynamics with a nonlinear damping term. In this paper, new dissipative chaos dynamics embedded in a simplex are presented, and the chaos dynamics are applied to optimization problems constrained on the simplex. By using the two types of models suggested by Fujita et al. and Tani as nonlinear dissipative models, chaotic trajectories embedded in the simplex are certified through the computational simulations.

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There are two international standards, IEC 60848 and IEC 61131-3, regarding SFC (Sequential Control Chart) and both of them have been revised. IEC 60848 has been drastically revised and its scope has been expanded to cover some original parts of SFC. On the other hand, many new features have been added into IEC 61131-3 to adapt the rapid progress of computer hardware and software in recent years. In this paper, a survey about the revisions of these IEC standards will be given and some technical trends of SFC and its peripheral technology will be introduced.

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SFC(Sequential Function Chart) is standardized in IEC61 13-3, and has many strong point but has not found into almost industrial actual case. In this papers, I presentation the new technology of CAD·TOOL for putting SFC to practical use.

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This report describes an educational approach using a graphical modeling and programming of sequential control by means of net oriented method. While the evolution of programmable sequential controllers (PLCs) and the standardization of programming languages as in IEC and JIS have been promoted, few endeavors to systematize sequential control methodologies were executed. Standing on a point of view of Mark Flow Graph (MFG) for modeling of structures and behaviors of discrete event control systems, the authors propose a programming and demonstration tool with graphical user’s interface for the system control synthesis as well as its application on the engineering education procedure.

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Discrete event systems present themselves in the real world as logic control systems also known as “sequence control” systems. As logic control systems are getting more and more sophisticated and complicated a new problem of lack of human affinity toward system behaviors become evident. This paper describes important aspects discrete event system research should take note of.

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Recently in Japan, we were informed of serious problems occurred on the computer control and network systems in main banks or in air lines after the integration of different systems. And also we are facing at the news that these kind of big issues were found not only in computer systems but also in more public related fields such as milk products contamination or BSE in beef in food industry. As the result, we can not but much concern about the lack of the basic attention and/or the technology for information systems including complex interfaces or procedures. In this paper, I will introduce the latest SP95 technology on the IT/Control interface, and also show the problem we are facing or experienced in actual industrial field as the engineering company and system integrator for ERPⁱ, MESⁱ and Control systems. We hope that the information, based on the experience, allows many uses/suppliers of the systems to avoid or solve the same kind of problems if it be.

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An automation surprise may occur when human operator’s situational recognition differs from that of the automation or he/she fails to understand the intention of the automation. Highly sophisticated automations are spreading to wide variety of application areas. Among them, automobile is one of applications that must be investigated with great care in the design of driver-automation interactions. It is not appropriate to assume that every car driver is trained well so that he/she may have deep understanding of the machine intelligence in his/her car. Automation surprise thus may happen more frequently or widely in cases of automobile than in cases of systems with well-trained operators, such as aircraft. This paper investigates driver-automation interactions in the low-speed range adaptive cruise control (ACC) systems with automatic stopping capability, and shows, through a driving simulator experiment, that an automation surprise may occur in certain circumstances.

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This paper focuses on the driver’s impression to the false alarms of driving support system related to the intelligent transport systems. The hypothesis is proposed that the driver tends to have not so bad impression to false alarms when the driver finds it difficult to detect the phenomena that system needs to alarm, compared with the other situation that driver finds it easy to do that. For proving this hypothesis, experiments of tracking previous vehicle at the same headway distance with correcting the headway by alarm system, which has some false alarms, are carried out. According the analysis of these experiments, our hypothesis is partially verified.

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We measured driving behavior on the real road environment using equipped vehicles. Repeated driving behavioral data for 320 trips from 8 subjects were collected when the vehicle approached to an intersection with a stop sign. The timing of the turn signal operation, onset/offset of pedal operation and other event timing were analyzed and modeled using Baysian Network (BN) model. The BN model was applied to evaluated the timing of the driver’s operational event. The timing of onset of braking was selected to as an index for detecting unusual operational behavior.

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While human ability has not changed within limitations, machine systems have become more complicated these years. Human, therefore, developed automation systems, but this caused human error in some situation. We will need the intent inference method in order to solve these problems because intent inference is an important element of proposing system information optimally. In this paper, we propose intent inference method. Our model contains intent inference using the whole recognition process and we use Bayesian network (BN) as inference engine. We made experiments and extract knowledge that is needed to construct BN and compare this result with simulation result.

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Knowing an operator’s intended plans can significantly improve the effectiveness and reliability of human-machine systems and several works have been carried out on intention inference and its applications. While some progress has been demonstrated, efforts to date have focused on a single operator. In large and complex artifacts such as power plants and aircrafts, however, a team generally operates the system. It is therefore necessary to develop an inference method of team intended plans because team intention is not reducible to mere summation of individual intensions and inference methods for a single user are not directly applicable to such systems. This paper describes a method of team intention inference based on the definition of We-intention and its application: conflict detection. We applied it to the operation of a plant simulator operated by a two-person team, and the strengths and limitations of the method are discussed.

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In this paper, we surveyed Japanese pilots’ attitudes toward automation and other crew members in order to clarify why the pilots have attained aircraft safety. We have found that the pilots’ attitudes toward automation were more variant than those toward other crew members. One possible interpretation is that pilots overcome problems due to automation by coordination among crew members.

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This paper describes a concept of human-machine interface (HMI) design for collaboration of human operators and automated systems. For supporting operators to prevent conflicts with automated systems, intentions (i.e., goals and means) of automated systems are explicitly represented on the interface. In this research, the ways of using information at state comprehension in current states and future states, and the decision task of control actions are identified trough cognitive experiments with a prototype interface. The results show operators can achieve higher goal by using information about intentions of automated systems and they may acquire deep knowledge about the target plant behavior and the automated systems.

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The goal of this study is developing the new software for creating interactive virtual environments of machine-maintenance training using virtual reality technology. The problem of existing virtual reality softwares is that special knowledge is needed in order to create interactive virtual environments where a user can assemble and disassemble virtual apparatus. Therefore this paper presents a new interface in which a user intuitively creates interactive virtual environments by attaching ActCube, which is a rectangular parallelepiped with a peculiar function, to 3D model as like building blocks.

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A part of the surface mount work of a printed circuit board is conducted by people’s hand, and the human error resulting from hand working is an serious problem. So the education of the workers is important in order to prevent such human error, but the present education is not so efficient. In this study, we have been developing an education system for the surface mount work of a printed circuit board utilizing an information technology. In this paper, our progress situation of development is reported, especially focusing on recognition of a worker’s situation.

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We have proposed the expansion-contraction mechanism of collective strategy as a framework of adaptive collective intelligent systems, and applied it to some specific task-oriented problems. In this paper, the collective behavior is formulated and numerically simulated to evaluate the basic property of adaptation process.

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Recently, Multi Agent Systems (MAS) have been studied to control large and complicated systems efficiently. As one of these research, Multi Agent Systems with Symbiotic Learning and Evolution named Masible which introduce symbiotic phenomena among creatures into MAS has been proposed and studied. In this paper, we focus on evolutionary Masbiole and apply it to the tileworld model which is a benchmark example of MAS study. It is clarified from simulations that symbiotic behaviors among agents and symbiotic Pareto solutions had emerged by symbiotic evolution.

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In this Paper, we describe on dynamics of two-layered organization in a poly-agent system. Almost all of models of multi-agent systems with hierarchical structure in previous researches had no features on dynamic changes of organization. This research is toward understanding the dynamics of organizational changes in multi-layered systems, such as human society. As an example domain, we design a simplified model of international relationship focusing on foods and war, where each country involves some villages that can move belonging country and even can be independent country by its own decision. Through some types of evolutionary simulation of poly-agent system constructed from countries and villages, we provide some comments on dynamics of this kind of complex organization.

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This research tackles the possibility of dissolution of a social dilemma, which happens between individual’s action and the social environment. Especially we focus on individual decision-making. In this research we propose a theoretic model based on structural reform approach, and presents some consideration about individual’s spontaneous cooperation action which is taken in not only rational decision-making but also irrational decision-making and the changes of environment.

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The selection of business partners (i.e. partnering) is very important and critical activity in the operation of company in Virtual Enterprise (VE). In this paper, we focus on the negotiation process in VE formulation as a basic research to clarify its effective management. Each enterprise in VE is defined as agent with multi-utilities and agent behavior is modeled with lexicographic model and Maximum Likelihood Hierarchical (MLH) model especially in consumer behavior and N-Person Cooperative Game in the game theory to make a compromise on their multiple objectives. We develop a computer simulation model to form a virtual enterprise through multiple negotiations amongst several potential members in the negotiation domain, and finally clarify the formulation dynamism with the negotiation process.

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Recently biochemical networks such as genetic networks and protein-protein interaction networks have been focused on analyzing mRNA expression profiles obtained by cDNA microarrays and DNA chips. Various mathematical models including Boolean network, reverse engineering network, Bayesian network and Graph model have been used to model gene regulatory mechanisms in living cells. However the mathematical approaches have clear limitation in the point that they cannot show any biological significance without interpretation by experts. In this sense, Genomic knowledge extracted from literature databases is a key to understand biochemical networks and compensate for the weakness of mathematical approaches.

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We propose a new method for constructing genetic network from gene expression data by using Bayesian networks. We use nonparametric regression for capturing nonlinear relationships between genes and derive a new criterion for choosing the network in general situations. In a theoretical sense, our proposed theory and methodology include previous methods based on Bayes approach. We applied the proposed method to the S. cerevisiae cell cycle data and showed the effectiveness of our method by comparing with previous methods.

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Mammalian cells reproduce by going through the cell cycle. To accurately describe the molecular mechanism of the mammalian cell cycle, we developed a new computational model that makes use of Petri Nets (PNs). The model shows the status of a given cell, such as the concentration of various proteins working at each stage and the activity level of enzymes. It describes the dynamic changes in each cell-cycle-controlling molecule and the topology of activity of the Cdc2-cyclin B kinase (MPF), one of the core networks of cell cycle engines. We found that the model reproduces some previously-reported biological abnormalities in the cell cycle field. The model can also easily reproduce the “knockout” or over-expression of specific gene products related to cell cycle control in living cells. Therefore, this model is a potentially useful means of exploring the relationship between gene functions and diseases.

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Integration for genome and transcriptome informatics is useful for understanding gene transcription regulation networks in view of genome position. In the present study, we have developed integrated analytical tools for genome and transcriptome informatics mainly focused on (1) gene clusterings by self-organizing mapping, (2) statistical determination methods for transcription units and (3) predicting methods for transcription networks in microarray and science literature. (1) Gene clusterings: Analyzing gene-expression profiles such as microarray data is important for understanding gene-expression network in the whole cell system. To attain this purpose, it is important to classify genes in high precision by expression profile. We used butch-learning self-organizing mapping (BL-SOM) for classifying genes based on expression profiles. BL-SOM makes it possible to understanding data distribution in multi-dimensional space. Using this method, we classified genes from log to stationary phases. Translation-related genes such as rps and rpl genes tend to be localized in inherent regions in the SOM map developed. Otherwise a group of stationary dependent genes are localized in the different regions in the translation-related genes. Classification of genes obtained in the present study may play an important roles for understanding function of genes based on expression profiles. (2) Determination of Transcription Units: Some adjacent genes in bacterial genomes are transcribed in a single mRNA molecule (transcription unit) in prokaryotes. Sequence information in the signal sequence, the distance between genes, and co-expression pattern from DNA microarray experiments has been used for predicting transcription units. We have proposed a method for predicting transcription units based on correlation of expression profiles between genes and on genome information such as neighboring relation. (3) Transcription network construction: In E. coli, about 300 genes are classified into transcription factors. We compiled recognition sequences and regulon data for 120 transcription factors involving sigma factors on the basis of literature information. Using these information, we have developed a method for validating microarray data and predicting transcription regulation networks.

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Replication process of double stranded DNA was modeled according to Zimmerman’s polymerization model. The t temporal changes in the probability of occupation for growing single stranded DNA for the definite template site has been computed under the noises based on the H infinite control principle. The theoretical foundation owes to that the disturbance during the replication process can be understood as the noise on the system and the physiological replication is the state of minimum H infinite norm of the transfer function from noise to out put. The present study will be available when more experimental information has been incorporated.

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We introduced a precise analytical and numerical method for computing the temporal changes in concentration and fluxes of gene regulating Repressor protein particles which has been firstly proposed by Berg O, T (1976). The temporal changes in Repressor particle concentration was described by the diffusion equation driven by the memory less first time arrival probability and the integration of return back lifting probability of the particles to the Operator region of DNA. The computed diffusion of the Repressor decreased rapidly from the onset of the reaction.

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