Yuzo Fukuyama

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.

A new method for evaluation and dietary therapy of congenital: deficiencies of amino acid metabolic enzymes. Linear system analysis and optimization of feedback inputs for the metabolic pathways of lysine, methionine and isoleucine.

We intended to elucidate an integrated mathematical model of amino acid metabolism and we propose a system for optimization treatment of disturbed metabolic states caused by congenital enzyme deficiencies. Our analysis focused on the metabolic pathway starting at asparaginic acid proceeding to isoleucine, methionine and lysine. The rate of change in the concentration of the biochemical species was expressed as 21 linear rate equations. We obtained the rate constants and the magnitude of feedback from reported experimental data. Linear systems analysis revealed that the metabolic system under study was stable but uncontrollable. These properties were insensitive to changes in the magnitude of feedback. To show the effect of optimizing the feedback so that it minimizes the square of the concentration of the species and the control input, we analyzed the impulse response of the species, transient response and the singular value of the system for four cases; (1) at the physiological state without optimizing the feedback, (2) at the physiological state attained after optimizing the feedback, (3) at the pathophysiological state attained with enzyme deficiency states for lysine and methionine metabolism without optimizing the feedback, and (4) at the pathophysiological state attained after optimizing the feedback for enzyme deficiencies. In the enzyme deficient model, the impulse response oscillated and lasted longer than that in the physiological state. These changes appeared even in the species on other branched pathways. The singular value was elevated in the enzyme deficient state. By optimizing the feedback, all the impulse responses in the enzyme deficient state recovered to nearly those in the normal physiological state. Similarly, the transient response and the singular value in the enzyme deficient state recovered to nearly the normal physiological values. We elucidated the numerical value of the feedback gain for this optimization. The present analysis is useful for the evaluation of the integrated properties of amino acid metabolism and the optimization technique is potentially of use for determining a treatment course for congenital metabolic enzyme deficiencies.

Optimization of growth of peptide chain by ribosomes on messenger RNA

To elucidate the integrated functional properties and organization strategy of the peptide elongation process of ribosomes on messenger RNA, we took a theoretical approach by applying a linear system analysis and optimal hypothesis. The basic theoretical model published by Gerst in 1965 was founded on reported experimental data. We have simplified the mechanism of the peptide elongation process by making the following assumptions. (1) When the left-most site (the 5′ terminal) on the messenger RNA is empty, the newly arrived amino-acyl-transfer-RNA (AT) links to the 5′ terminal even though other sites are empty. (2) When all other sites on the ribosomes except the 5′ terminal are empty and the first ribosome is not yet filled, ATs which arrive subsequently continue to bind to the 5′ terminal site of the first ribosome until the first ribosome has been filled. (3) When a given ribosome has been filled by an amount of product equivalent to the product ofp (amino acid residues) andq ribosomes, the filled ribosome advances to the next ribosomal region (q+1). There, the amino acid attainspq+1. After this transition, theqth ribosomal region becomes empty. (4) When the sites on the ribosomes are not all filled, newly arrived AT binds from the right of the total sequence. By applying the mass action law, the elongation process was expressed by 23 rate equations. The rate constants for the binding ribosome, turn over, peptide bond formation and transitional movement of ribosomes were utilized from reported experimental data. The optimal state was defined when the square of the concentration of all the transitional states and control input were minimized. This means that the elongation process is in the most economical state when there are no excessive changes at any steps of the elongation process and the least energy consumption. Linear system analysis showed that system was stable and controllable. The singular value was depressed by system optimization. The impulse response terminated earlier by a smaller peak amplitude in the optimized system than in those without optimization. The present model under optimal control is available to evaluate the functions of peptide synthesis on a messenger RNA, and to produce artificial protein from the standpoint of system optimization.