The steady-state behavior of a circular and linear array of three cells containing a substrate-inhibited-like kinetics catalyzed by immobilized thylakoids is studied. The photobiochemical reaction used to model the system is based on previous studies concerning a single and a two-cell system. In a general model all the cells in the array are considered to be continuously fed by the substrate and under diffusional relation with each others. Several models are then considered (circular and linear arrangements) depending upon the presence or the absence of these previous characteristics on each cell. The behavior of the various configurations is studied as a function of both the external substrate input concentration a0, and the ratio between the transport terms λ. The results given by bifurcation analysis and limit point continuation allows to determine three domains of stable stationary behavior: I, monostability; II, bistability and multistability; III, multistability and occurrence of dissipative structures. The existence of domain III is strictly dependent on the existence of a topological and functional symmetry in the arrangement. The experimental occurrence of both stable symmetric and asymmetric steady states in a circular and linear array of cells is also illustrated.
The dynamics of the fructose 6-phosphate/fructose-1,6-bisphosphate cycle operating in an open and homogeneous system reconstituted from purified enzymes was extensively studied. In addition to 6-phosphofructokinase and fructose-l,6-bisphosphatase, pyruvate kinase, adenylate kinase and glucose-6-phosphate isomerase were involved. In that multi-enzyme system, the main source of non-linearity is the reciprocal effect of AMP on the activities of 6-phosphofructokinase and fructose-I ,6-bisphosphatase. Depending upon the experimental parameter values, stable attractors, various types of multiple states and sustained oscillations were shown to occur.In the present report we show that irreversible transitions are also likely to occur for realistic operating conditions. Two parameters of the system, that is the adenylate energy charge of the influx and the fructose-1,6-bisphosphatase maximal activity, are potential candidates to provoke such irreversible transitions from one steady state to the other: (a) when varying the maximal activity of fructose-l,6-bisphosphatase, the system can jump irreversibly from a low to a high stable steady state, and (b) when the adenykdte energy charge of the influx is the changing parameter, irreversible transitions occur from a high stable steady state to a stable oscillatory state (limit cycle motion). This behavior can be predicted by constructing the loci of limit points and Hopf bifurcation points.
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