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.

An optimal control theory has been applied to a biological compartment system to show a method to analyze the control principle of biological system represented by compartments. Present theory has been proposed to afford a theoretical back ground and validity for the strategy of drug administration or control of the anesthetic agent in practical medicine. The instantaneous change of the concentration of a given material within a biological system has been expressed by differential equations. Each compartment has been set to be transferred a material from all other compartments and conversely each compartment sends it to all other compartments. The control input was restricted to be one kind. The performance function involved the deviation from the target value, the rate of change in concentration and the amount of the control variables. The biological system was defined to operate optimally only when the performance function has been minimized during a given time period. By the optimal control theory of Pontoriagin, above biological problem has been converted to a mathematical problem and was solved numerically by multiple shooting method. The calculated trajectory of the optimal control has been asymmetric parabolic one with the maximum at its initiation and the minimum at the middle of total reaction time. This pattern has been consistent with that of probable transient change of the concentration of anesthetic agent when it has been inhalated under the most up to date "Rapid Inhalation Induction" method. The optimal trasient change of the concentration at each compartment has beeb affected by the difference in time dependent nature and the magnitude of the transfer rate. Present theory afforded a method to analyze the control strategy of biological system expressed by compartments model and showed an availability for actual clinical medicine. The optimal control principle must be a most adequate one to describe the Homeostasis in biological system.