Hirohumi Hirayama

Linear systems analysis of activating processes of complement system as a defense mechanism.

The complement system is an important element of the host defense mechanism, although its kinetics and characteristics as a system are still unclear. We have investigated its temporal changes and system properties from the view point of system engineering. The temporal changes of sequential activating processes of the system were expressed by 26 non-linear differential equations using reported values of rate constants and serum concentration for each component. The intermediate products in the activating processes increased parabolically while the membrane attack component as the final product, increased linearly. The little change in inactive precursors afforded validity for system linearization. Linear systems analysis revealed that the system which was insensitive to the changes in rate constants was unstable. The system became stable when the feed-back input from the final product was set to operate on the first step of the activating processes. Seven uncontrollable variables were insensitive to changes in rate constants or system optimization that minimized the changes in concentrations of components in the complement system. The singular values of the complement system were reduced and the impulse responses of the system were improved when the system was optimized. When stronger minimization was imposed on the changes of concentration of the components in the complement system, the singular values were reduced more, the magnitude of the impulse responses was depressed further and the responses terminated earlier than those when the elements in the weighting matrix of concentration of the components were set to be unity. By this potent minimization, the influences of changes in rate constants on the singular values were diminished. The present theoretical analysis is presented to evaluate the ability of defense mechanism of complement system.

A method to evaluate economical carrier-mediated transport across the biological membrane by the optimal control principle

We propose an optimal control strategy for carrier-mediated transport across biological membranes in an attempt to evaluate the functions quantitatively and to create an artificial membrane. The transport system was described by the substrate, unloaded and loaded forms of the carrier where the binding sites were facing to the outside and inside of the membrane with the corresponding control inputs. The temporal behavior of the transport was expressed by a linear four-states model employing the conservation law. We assigned the state variables for the concentrations of the loaded and unloaded carriers on both sides of the unit membrane area. Two control inputs were set on each individual state variable so as to describe the producing and converting processes. The cost function to evaluate the performance of the transport involved the temporal static concentration changes in the loaded and unloaded carriers and the control inputs for driving the system. Minimizing this cost function resulted in a smooth and non wasteful transport with the least energy consumption. The relative magnitude of minimizing these quantities was characterized by the weighting coefficients and we defined that the optimal transport state is achieved when this cost function has been minimized. We utilized reported experimental data of Na/glucose cotransport for the initial condition and rate constants. Since transport by the carrier is a recycling process, we set a rigorous terminal condition as the target state for the optimally controlled transport. The optimized system equations and co-state equations were solved numerically as a multiple points boundary value problem. The influences of a given weighting coefficient were observed not only on the time course of its proper variable but extended to those of other variables. The changes in the time course could be explained by the compensatory action of the optimized control input so as to prevent excessive increase or decrease of the materials. Finally we showed the successful simulation of experimental data by the present method. The present method is available for evaluating the function of biological transport and for creating an artificial membrane.

Optimal control of active transport across a biological membrane

We propose an optimal control principle for active transport across a biological membrane. The modeling of the membrane is based on Hill and Kedems thermodynamic model. The performance function used to evaluate the optimality of the transport involved the rate of time-dependent changes in the concentration of particles in all the membrane layers as the state variables, and the number of receptor sites on the membrane as the control input. We decided that the optimal transport state is achieved when this cost function has been minimized under the constraints of the system equations characterizing the membrane modeling. The changes in the number of particles in the membrane layer evoked by changes in the kinetic parameters can be explained by the compensatory action of the optimal control strategy in order to prevent excessive decrease or increase of the molecular particles in all the membrane layers. The changes in the number of receptors in the paths of some physiological states can be explained by the optimal control modeling of the membrane transport. This model will be made available to create and evaluate an artificial membrane.

Statistical micromechanical method for dynamic kinetic binary interaction among moderate dense gene-regulating bio-molecules

A statistical thermodynamical approach has been introduced to calculate shear viscosity and thermal flux for interactions among the gene-regulating biomolecular protein particles that operate in moderate density rigid-sphere fluids. Starting from the modified Boltzmann equation with the help of linear perturbation theory, the coefficients of distribution function were determined. On the basis of transport theory, we introduced computational forms of diffusion constant, mass flux, shear viscosity, and thermal flux. We examined the influences of changes in mass, diameter, and magnitude of the pair correlation function. The present method will be available to evaluate the local physical reaction properties of the gene-regulating particles.

Computation of interaction potential among the polypeptides composing an ion gating channel that are crossing by acute angles

Distances of point charges on helical turns of cylindrical polypeptides that are composing ion selective gate on excitable membranes have been calculated. The geometry of two interacting point charges was analyzed in vertical and cross sectional planes of two cylindrical peptides. The factor 1/r 6 of parallel positioned cylinders increased as the pitch of helical turn increased. This method will be available for analyzing the mechanism of Allosetric change of molecular structures that are composing ion channels on the excitable, biological membranes.

Distributions of electrostatic potentials and forces in a DNA molecule by a concentric cylindrical model. II. Effects of the Debye-Huckel parameter

We have developed a mathematical method to compute the three-dimensional distributions of potentials, circumferential forces, and electrostatic free energy in a DNA molecule. A DNA molecule was described by a three-layered concentric cylinder. For geometric reasons, discrete point phosphate charges are arranged helically around the central axis of the DNA in equivocal circumferential steps. The Debye-Huckel theory was applied to linearize the Poisson-Boltzmann equation and obtain analytical solutions. As the value of the Debye-Huckel parameter κ increased, the potentials in the inner region increased, and the reaction potentials and the middle-region potentials decreased. Forces in the inner, middle, and outer regions decreased, while the reaction force increased. The influence of the Debye-Huckel parameter strongly depended on the radial position of the DNA. Changes in the dielectrical constant of the middle region had a significant influence on the electrostatic energy of the DNA. This method will be useful for evaluating the interaction between the physiological ionic circumstances around DNA, the ion adsorption, and the melting process of the DNA double strand.

Method for evaluating thermodynamic and statistical molecular dynamic interactions among the biomolecular particles that participate in gene expression

A mathematical method is introduced for evaluating the biomolecular dynamic interactions in gene expression and the regulation of an artificial life. The theoretical basis was founded on the thermodynamics and statistical molecular dynamics of multicomponent dilute gas systems, which are characterized by the Boltzmann equations and molecular collision integrals. We introduce the mathematical processes for computing shear viscosity, and the thermal conductivity of two interacting biomolecules that have different geometries, number density, and mass, in great detail. The computed normalized shear viscosity, normalized thermal conductivity, self diffusion, and thermal diffusion coefficients showed multimodal complex behaviors as functions of radius, length, mass distribution parameters, number density, and the mass of the second interacting particle. This method and the computed results, in a more generalized version, would give quantitative evaluations of physical collisional interactions among biomolecular particles as the ultimate process of biochemical reactions.

Stochastic diffusion control for gene-regulating protein particles

We introduce a precise analytical method for computing the temporal changes in concentration and fluxes of gene-regulating repressor protein particles. The temporal changes in repressor particle concentration is described by an integral-differential diffusion equation in cylindrical coordinates. The equation consists of the memory-less first-time-arrival probability and the integration of the return probability of the particles to the operator region of the DNA. By using the Laplace transformation, we could derive analytical forms of the temporal changes in concentration and flux in a radial direction, the total flux, and the first-time-arrival probability. We also computed the impulse responses of the first-time-arrival probability of the repressor to the sink. The computed diffusion of the repressor particles decreased rapidly from the onset of the reaction. As the diffusion constant in the medium around the DNA increased, the first-time-arrival probability, the diffusion, and the flux of the particles decreased, while the total flux into the target sink increased. As the chemical factor became predominantly a diffusion factor, the first-time-arrival probability, the diffusion, and the flux of the particles decreased. As the dissociation rate of the particles increased, the flux into the sink increased. The number of dissociated particles was significantly more influenced by the chemical factor than by the diffusion. The first-time-arrival probability oscillated significantly at the onest of the reaction. When this method has been extended, it will be available for predicing genetic expression and creating artificial life.

H2 control strategy for Na ion channels on the neural cellular membrane

The channel gating process of neural cells is the first step of neural information transmission. We have proposed a kinetic model for state transitions for a sodium (Na) ion gating channel under H2 control. The channel state consisted of an open state, three closed but activated states, and four inactivated but not closed states. This modeling was based strictly on molecular biological observations. Three charged amino acid helixes of the specific subunits of the Na channel hole act as activating gates. Another helix of the subunit having membrane voltagesensing properties behaves as an inactivating particle that invades the Na channel hole after membrane depolarization. This particle blocks the free movements of the three activating gates and inactivates the Na channel gating function. In total there are eight channel states, which consist of four inactivated states, three closed states, and one open state. We expressed the transitions among these states by eight linear differential equations using the law of conservation. For the control principle, the channel system is always exposed to a biological mimetic that is a false transmitter and competes for the channel sites with Na ions. Hence, we regarded such biological agencies as noises in the system that disturb the effective transmission of information, i.e., rapid transitions through the channel gating systems. The physiological Na gating is understood to minimize influences from the disturbing noises on the transition of the channels states, and we have proposed the H2 control principle. The computed results of temporal changes in the amount of channel species per unit membrane area showed rapid changes and then termination. This behavior was strongly dependent on the membrane potential. Our modeling could describe the rapid excitation and resetting of the Na ion channel gating function of the neural system. These results strongly reflect the digital nature of the neural system. The present investigation could be used to evaluate the function of neural systems that minimizes the influences of noises on the information transmission process by the transitions of the Na ion channel gating state.

A mathematical method for investigating dynamic behavior of an idiotype network of the immune system: The time minimum optimal control theory

We proposed a mathematical method to investigate an integrated property of an idiotype immune network under the time minimum optimal control. The transient changes of amounts of B cell receptor bound antibodies and immune complex in the network system were expressed by detailed differential equations. The rate constant for binding the second Fab arm of antibody was set as a function of coulombic repulsive force to express the influence of redistribution of electrical charges in the ligand-receptor molecular complex. We proposed time minimum optimal control strategy as an organizing principle for rapid reactions of the immune system. Based on the rigorous mathematical foundations of the optimal control theory, we determined the differential equations for co-state variables for the state variables to compute the time minimum transient changes in the amount of the species. Biological parameters in the immune reactions were utilized from the reported experimental data. Numerical computation disclosed that influence of changes in a rate constant extended to all the species of the network. Changes in a rate constant in a different B cell system reinforced the collaborations among the idiotypes and lead them to set in motion the ejection of the antigen. Simulation of reported experimental data by the present method was successful. There were, however, some inevitable dissociations between reported experimental data and computed results. The present method will be available for evaluating the time minimum reaction of the immune network system.