Chaotic patterns in a coupled oscillator-excitator biochemical cell system

Sufficient Conditions for Coordination of Coupled Nonlinear Biochemical Systems: Analysis of a Simple, Representative Example

Under conditions of stress and environmental fluctuations, how do coupled biochemical systems maintain the balance of metabolic fluxes? Using a model derived from a representative glycolysis example, this work derives sufficient coordination conditions for guaranteeing the balance of metabolic fluxes in the coupled systems subjected to any fluctuations. For diffusion-like coupling, it is explicitly shown that the sufficient conditions are the linear summation of individual uncoupled systems. For this case, coupling strength is not a key factor affecting the balance of metabolic fluxes, although it may affect the emergence of certain dynamic patterns. When the sufficient coordination conditions are satisfied, coupled systems always settle onto stable finite states, independently of the nature (type, period or amplitude) of external signals. When they are not, external signals may force the stable states of coupled systems to lose their coordination, resulting in the coupled systems having no stable finite states which is inconsistent with most normal biological functions. Numerical simulations support the analytical results. The generalization of this result for other coupled nonlinear biochemical systems is discussed. Finally, the maintenance of such biological coordination is discussed in the context of genetic engineering and environmental fluctuations.

Nonlinear Dynamics in the Uncatalyzed Bromate–Pyrocatechol Reaction

Spontaneous chemical oscillations are uncovered in the uncatalyzed bromate–pyrocatechol reaction conducted in a stirred batch reactor. Phase diagrams show that the oscillatory behavior is more sensitive to the ratio of [bromate]/[pyrocatechol] than their actual concentrations. The long induction time observed in the system is insensitive to the initial presence of brominated pyrocatechol species. Quenching experiments illustrate that the studied chemical oscillator is both bromide and radical controlled. In addition, this new uncatalyzed bromate oscillator exhibits both light-induced and light-quenched oscillatory behaviors. Preliminary mechanistic characterization with UV/vis absorption method and mass spectrometry suggests the formation of bromo-1,2-benzoquinone through the reaction between pyrocatechol and bromine.

Transient complex oscillations in a closed chemical system with coupled autocatalysis

In this study, hydroquinone was introduced to the classic Belousov-Zhabotinsky (BZ) reaction to build up coupled autocatalytic feedbacks. Various complex dynamical behaviors including successive period-adding bifurcations, irregular oscillations, and frequency modulations were observed in the coupled reaction system. Not only the complexity of oscillations but also the time period during which complex oscillations persist were found to depend greatly on the initial concentration of hydroquinone, which was expected to manifest the coupling strength in the studied system. Dependence of the observed transient complex oscillations on concentrations of ferroin, sulfuric acid, bromate, and malonic acid was also characterized systematically. Numerical simulations with a modified BZ model via incorporating reactions involving hydroquinone and products of hydroquinone qualitatively reproduced the influence of hydroquinone seen in experiments.

Excitable dynamics and threshold sets in nonlinear systems

Following our previous work [J. Zagora et al., Faraday Discuss. 120, 313 (2001)], we present a quantitative definition of a threshold that separates large-amplitude excitatory responses and small-amplitude nonexcitatory responses to a perturbation of an excitable system with a single globally attracting steady state. For systems with two variables, finding the threshold set is formulated as a boundary value problem supplemented by a condition of a maximum separation rate. For this highly nonlinear problem we formulate a numerical method based on the use of multiple shooting and continuation methods. The threshold phenomena are examined by using an example dynamical system with chemical reaction--the bromate-sulfite-ferrocyanide system. In a model of this reaction we find the threshold set, construct a bifurcation diagram and discuss how excitability can vanish. These results are compared with recent experiments. We also discuss relevance of other definitions of the excitability threshold including the concept of nullclines.

Symmetry breaking, bifurcations, quasiperiodicity, and chaos due to electric fields in a coupled cell model

A model for the asymmetric coupling of two oscillatory cells is considered. The coupling between the cells is both through diffusional exchange (symmetric) and through the electromigration of ionic reactant species from one cell to the other (asymmetric) in applied electric fields. The kinetics in each cell are the same and based on the Gray-Scott scheme. Without the electric field, only simple, stable dynamics are seen. The effect of the asymmetry (applying electric fields) is to create a wide variety of stable dynamics, multistability, multiperiodic oscillations, quasiperiodicity and chaos being observed, this complexity in response being more prevalent at weaker coupling rates and at weaker field strengths. The results are obtained using a standard dynamical systems continuation program, though asymptotic results are obtained for strong coupling rates and strong electric fields. These are seen to agree well with the numerically determined values in the appropriate parameter regimes. (c) 2002 American Institute of Physics.

Dynamical properties of chemical systems near Hopf bifurcation points

In this paper, we numerically investigate local properties of dynamical systems close to a Hopf bifurcation instability. We focus on chemical systems and present an approach based on the theory of normal forms for determining numerical estimates of the limit cycle that branches off at the Hopf bifurcation point. For several numerically ill-conditioned examples taken from chemical kinetics, we compare our results with those obtained by using traditional approaches where an approximation of the limit cycle is restricted to the center subspace spanned by critical eigenvectors, and show that inclusion of higher-order terms in the normal form expansion of the limit cycle provides a significant improvement of the limit cycle estimates. This result also provides an accurate initial estimate for subsequent numerical continuation of the limit cycle. (c) 2000 American Institute of Physics.