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 stases 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 pathophysiogical 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.