Improvement of the stoichiometric network analysis for determination of instability conditions of complex nonlinear reaction systems

Abstract

Stoichiometric network analysis (SNA), a known method for analyzing complex reaction systems including biochemical ones, is improved and applied to a nonlinear process studied far from equilibrium in a continuously fed, well stirred tank reactor (CSTR). A particular attention is focused on the determination of the narrow range of the control parameter values where the main steady state is unstable and where different dynamic states can be simulated numerically. The instability region, the most important feature of nonlinear reaction systems, is calculated as a function of the SNA parameters (current rates and reciprocal concentrations of intermediate species in the steady state) and simplified by retaining only the dominant terms. Since the number of the current rates is usually larger than the number of linearly independent equations to be used for their calculation, it is shown here that the current rates can be replaced with a smaller number of reaction rates at the steady state. These rates are related to the experimental data in a simple manner. The instability conditions is also written as a function of dimensionless parameters derived from the SNA. This general approach is applied to a model of the Bray-Liebhafsky (BL) reaction having seven reactions without direct autocatalysis or autoinhibition, studied under CSTR conditions. Since the model has six intermediate species, it would be very difficult to analyze its instability condition by the conventional procedure, where a sixth order characteristics equation would have to be solved. On the other hand, the instability condition, obtained easily by the improved SNA, locates correctly the oscillatory region using numerical integration. Other dynamic states found earlier with a larger model of the BL reaction, such as mixed-mode oscillations, period doubling and chaos, are also obtained within the theoretically predicted oscillatory region. Thus, besides the general advantages of the improved stoichiometric network analysis as a method appropriate for the examination of complex nonlinear reactions, we show that the various mentioned dynamic states can be obtained by a very simple variant of the model of the BL reaction realized under CSTR conditions.