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Application of the H-infinite control for multi-states conformational transition of calcium ion selective gating channels on excitable cellular membranes

Hirayama H.

Systems Science

2002

Vol. 28, nr 1

107-34

We introduced H infinite control theory for evaluating the noise filtering functions of biological membranes. Calcium ion selective gating channel on excitable cellular membranes was described by four subunits allosteric modeling. The modeling was characterized by cooperative positional changes of an intrinsic voltage sensitive molecule in each of the subunits and concerted conformational changes of all the subunits. We applied the H-infinite control strategy for minimizing the influence of noises as the worst disturbances on the filtering function of the calcium channel. The allosteric cooperative positioning of the S4 and concerted actions of the subunits were approximated by powers of non-dimensional parameter F/sup n/. The temporal changes in amounts of calcium channel states were described by ten differential equations. We induced the corresponding differential equations for observers and accompanied Riccati equations. The transient changes in amounts per unit membrane area of calcium ion channel species were almost parallel. Those of the observers were a variety of patterns. The change in amount of control input for the concerted opening was the smallest while those for inter open states transitions were the largest. The computed temporal changes in the worst disturbances and the worst noises showed characteristic time courses for individual channel species and observers.

Allosteric systems on biological membrane analysis by time minimum optimal control principle

Hirayama H., Okita Y.

Systems Science

2000

Vol. 26, nr 4

121-145

We calculated temporal changes in concentrations of species in biological allosteric system. The allosteric nature was described by co-operativity and concerted actions of identical subunits composing the system. The co-operativity was characterized by accelerated bindings of substrates to the subunits and their structural changes. The concerted mechanism provokes all or non-conformational change of an entire molecule from its inactive to active sate at one time. As the number of bound substrate increases, the concerted conformational transition of all the subunits strongly directs to active state. These two properties save the time to complete activation of the system and full substrate binding. The allosteric actions are frequently observed in the critical emergency conditions such as oxygen binding, immune defense reaction and ionic channel gating on excitable membrane of the bio-signal transmission. Hence, the temporal actions of allosteric systems can be interpreted as the shortest time controlled system. We thus, proposed the time minimum optimal control principle as an organizing strategy for activating and binding processes of the allosteric systems. We presented fifteen non-linear differential equations for describing the temporal changes in concentrations of the species composing a two activators-two substrates allosteric system. Another set of fifteen non-linear differential equations for co-state variables were obtained by partial differentiation of the Hamiltonian of the system based on the minimum optimal control theory. The computed temporal concentrations of species oscillated. A reduction of an allosteric parameter shifted these temporal changes synchronously. When the amounts of substrate and activator were set as time invariant, the oscillations disappeared and the concentration of the species approached a steady level. These computed results are qualitatively consistent with actual biophysical phenomena. The present mathematical description of transient changes in concentrations of species in biological allosteric system will be available for evaluating the effective and economical performance of the allosteric system acting under the critical emergency circumstances.