We show a geometric method to compute the Van der Waals factor 1/r7 between the assemblies of amino acid molecules of the subunits of acetyl choline (abbreviated by Ach: a kind of neuro chemical transmitter) sensitive channel on the post synaptic membrane of the neural system. We induced a analytical geometric formula for the distances between helically arranged ten assemblies of the point amino acid molecules on two interacting membrane perforating poly peptides, M2 helices during channel opening deformation. Detailed geometric parameters have been utilized from reported biophysical measurements. The computed Van der Waals factor decreased rapidly as the slope of the first M2 helix along the central axis of the channel pore has increased. The Van der Waals factor also decreased by an increase in rising angle of the helically arranged amino acids on the M2 helices. The Van der Waals factor increased significantly as the first M2 helix has rotated around the central axis of the channel pore to take an opening position. We discussed the time dependent molecular structural changes of the Ach sensitive channel opening in conjunction with the Allosteric properties of the bio molecules. The molecular mechanism of Ach sensitive channel opening in terms of the Allosteric property may derive from the characteristic helical constitutional nature of the membrane perforating part (M2 helix) of the subunits of the channel molecule.

The controllability and the stability of the blood clotting system are examined with the linear system analysis. The dynamic behavior of the clotting system consisting of a cascade of ten proteolytic reactions of the clotting factors with multiple positive feed back and feed forward loops is represented by the rate equations in a system of non linear ordinary differential equations with 35 variables. The time courses of concentration change in every factor are revealed by numerical integration of the rate equations. Linearization of the rate equations based on the dynamic behavior of the chemical species relevant to the nonlinear terms leads to the linear systems analysis of the clotting system to clarify the essential features of blood coagulation. It follows from the analysis that the clotting system is uncontrollable regardless of changes in any system parameters and control input and that all the chemical species of the system are uncontrollable so that the sequential reactions in the cascade proceed irreversibly, once they are activated. More over by the analysis of the eigen values, the clotting reaction as a total system was shown to be unstable which was insensitive to changes in the system parameters. These characteristic natures of clotting system must be derived in the sequential cascade reaction pattern and the inherent multiple positive feed back and feed forward regulation.

The regulatory mechanism of protein synthesis on a messenger RNA was analyzed from view point of the optimal control and discussed about availability for artificial production of peptide and protein. The transient movements of a ribozome through a messenger RNA with its production of peptide was based on the theory proposed by Gordon (1968). The optimal state of total process was defined as the state at which the time dependent change of each process of peptide synthesis has been minimized during a given time interval. This biological problem was converted into mathematical one by setting state variables and utilizing the optimal control theory with the help of Hamiltonian function. The first process of transition of a ribozome on a messenger RNA showed the largest change and with progress of state, the magnitude of change of each process decreased and became a simpler pattern. The effect of weighting coefficient relating with individual process was not confined only to its proper process but extended to all other processes. Each process was affected from all other processes. These were manifestations of effective and rational control strategies particularly for regulation of the sequential reaction in peptide synthesis. Such results were originated in the operation of the optimal control. By simulating physiological experimental data, it is possible to predict at what process and at what degree, the synthesis is regulated in order to achieve the optimal synthesis state. By analyzing the optimal synthesis process in combination with physiological experimental data, it would be possible to create artificial peptide and protein.

We described short time span idiotype immune network reactions by rigorous mathematical equations. For each idiotype, we described the temporal changes in concentration of (1) single bound antibody, one of its two Fab arms has bound to the complemental receptor site on the B cell. (2) double bound antibody, both of its two Fab arms have bound to the complemental receptor sites on the B cell and (3) an immune complex which is a product of reaction among the antibodies. Stimulation and secretion processes of an antibody in the idiotype network were described by non linear differential equations characterized by the magnitude of cross-linking of the complemental antibody and B cell receptor. The affinity between the mutually complemental antibody and receptor was described by an weighted affinity matrix. The activating process was expressed by an exponential function with threshold. The rate constant for the linkage of the second Fab arm of an antibody was induced from the molecular diffusion process that was modified by the Coulomb repulsive force. By using reported experimental data, we integrated 60 non linear differential equations for the idiotype immune network to obtain the temporal behavior of concentrations of the species in hour span. The concentrations of the idiotype antibody and immune complex changed synchronously. The influence of a change in one rate constant extended to all the members of the idiotype network. The concentrations of the single bound antibody, double bound antibody and immune complex oscillated as functions of the concentration of the free antibody particularly at its low concentration. By comparing to the reported experimental data, the present computational approach seems to realize biological immune network reactions.

We have shown a non-invasive method for estimating transient changes in aortic flow and ventricular volume based on optimal control theory by using successful simulations of reported experimental data. The performance function to evaluate the optimality of the cardiovascular system was proposed based oh physical, fluid mechanical and pathophysiological considerations. It involved the work of the ventricle, the rate of changes in the aortic flow and the ventricular pressure. We determined that the cardiovascular system operates optimally when the performance function has been minimized. The relative magnitudes of the reductions of changes in these terms were expressed by the weighting coefficients. The arterial system was described by the Wind Kessel model using arterial resistance, aortic compliance and aortic valvular resistance. We set boundary conditions and transitional conditions derived from the systolic and diastolic phases of the aortic flow and the arterial pressure. The optimized system equations were converted to 6 linear simultaneous differential equations with 6 boundary conditions. The optimal ventricular pressure and aortic flow rate that minimize the performance function were obtained by solving these differential equations. By alternating the weighting coefficients of the work of ventricle and the rate of change in the ventricular ejection pressure, successful simulations of the ventricular pressures recorded from human subjects and those from isolated canine ventricle were obtained. Once the sets of weighting coefficients had been determined by successful simulations of ventricular pressures, the calculated aortic flow curves and pressure volume loops by the present method coincided with the reported experimental data. The changes in ventricular pressure and aortic flow produced by alternating the weighting coefficients to simulate the reported ventricular pressures and aortic flow curves under the different afterload conditions were consistent with biophysical experimental data. The present method is useful to estimata aortic flow curve and ventricular pressure volume loops non-invasively.