Prompted by the recent growing evidence of oscillatory behavior involving MAPK cascades we present a systematic approach of analyzing models and elucidating the nature of biochemical oscillations based on reaction network theory. In particular, we formulate a minimal biochemically consistent mass action subnetwork of the Huang-Ferrell model of the MAPK signalling that provides an oscillatory response when a parameter controlling the activation of the top-tier kinase is varied. Such dynamics are either intertwined with or separated from the earlier found bistable/hysteretic behavior in this model. Using the theory of stability of stoichiometric networks, we reduce the original MAPK model, convert kinetic to convex parameters and examine those properties of the minimal subnetwork that underlie the oscillatory dynamics. We also use the methods of classification of chemical oscillatory networks to explain the rhythmic behavior in physicochemical terms, i.e., we identify of the role of individual biochemical species in positive and negative feedback loops and describe their coordinated action leading to oscillations. Our approach provides an insight into dynamics without the necessity of knowing rate coefficients and thus is useful prior the statistical evaluation of parameters.

Synchronizations of oscillatory regimes of the Belousov−Zhabotinskii (BZ) reaction in a circular array of three identical CSTRs coupled via symmetric passive diffusion/convection mass transfer were studied experimentally. Stability of symmetric and asymmetric phase-shifted oscillatory regimes with respect to variations of the coupling strength among the reaction cells was examined. The all-in-phase regime was found to be the only regime stable over the entire range of coupling strength values. Phase-shifted oscillatory regimes were found to be stable only within a narrow interval of very low coupling strength values. Spontaneous transitions of the phase-shifted regimes to the synchronized mode due to stochastic fluctuations of the coupling strength were observed. Numerical simulations with the four-variable Oregonator based model of the BZ reaction qualitatively confirmed the experimental findings. Propagation of an excitable response to periodic pulsed stimulations in a linear three-array of coupled chemical excitators (Belousov−Zhabotinskii reaction) was studied in dependence on the coupling strength, on the excitability level of the reaction mixture, and on the period and amplitude of pulse stimulation. Regimes of complete and partial propagation of the excitable response and the regimes of partial and complete propagation failure were observed. Numerical simulations predict qualitatively well excitatory regimes observed in experiments.

Early experimental observations of chaotic behavior arising via the period-doubling route for the CO catalytic oxidation both on Pt(110) and Ptgamma-Al(2)O(3) porous catalyst were reported more than 15 years ago. Recently, a detailed kinetic reaction scheme including over 20 reaction steps was proposed for the catalytic CO oxidation, NO(x) reduction, and hydrocarbon oxidation taking place in a three-way catalyst (TWC) converter, the most common reactor for detoxification of automobile exhaust gases. This reactor is typically operated with periodic variation of inlet oxygen concentration. For an unforced lumped model, we report results of the stoichiometric network analysis of a CO reaction subnetwork determining feedback loops, which cause the oscillations within certain regions of parameters in bifurcation diagrams constructed by numerical continuation techniques. For a forced system, numerical simulations of the CO oxidation reveal the existence of a period-doubling route to chaos. The dependence of the rotation number on the amplitude and period of forcing shows a typical bifurcation structure of Arnold tongues ordered according to Farey sequences, and positive Lyapunov exponents for sufficiently large forcing amplitudes indicate the presence of chaotic dynamics. Multiple periodic and aperiodic time courses of outlet concentrations were also found in simulations using the lumped model with the full TWC kinetics. Numerical solutions of the distributed model in two geometric coordinates with the CO oxidation subnetwork consisting of several tens of nonlinear partial differential equations show oscillations of the outlet reactor concentrations and, in the presence of forcing, multiple periodic and aperiodic oscillations. Spatiotemporal concentration patterns illustrate the complexity of processes within the reactor.

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