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[Mathematical model of adaptation of the energy metabolism of a cell. Calculation of the influence of ATP on the activity and concentration of the initiator stage enzyme].

A simple kinetic model was constructed to study the adaptation of cell energy metabolism to a varying loading. In this model the initiatory step of energy metabolism, in which the initial substrate S is activated at the expense of ATP molecule energy, is catalyzed by an oligomeric enzyme E dissociable at high ATP concentration to monomers E1. It is assumed that the steady state level of monomers E1 in the cell is maintained by constitutive synthesis of E1 molecules, which balances their continuous hydrolysis by proteases. The properties of the kinetic model were studied using a mathematical model which is a system of nonlinear differential equations describing the change with time of the total enzyme E concentration and the concentration of ATP. The main isoclines of this system can intersect in one, two or three points. The mathematical analysis shows that the kinetic model considered exhibits adaptive properties. A sharp increase of the ATPase activity in the model initiates a transient process which leads to a rise in the total enzyme E concentration and in the efficiency of energy metabolism. As a result, the concentration of ATP drops only slightly. The establishment of a new level of the enzyme E concentration may proceed in the oscillatory fashion.