A biochemical model with two instability mechanisms

In the two-variable models studied for glycolytic oscillations and birhythmicity, periodic behaviour originates from a unique instability mechanism based on the autocatalytic regulation of an allosteric enzyme by its reaction product. The question arises as to what happens when two instabiUty-generating mechanisms are present and coupled within the same system can new modes of dynamic behaviour arise from such an interaction  

An example of such a situation was considered at the end of the preceding chapter the system with two oscillatory isozymes (fig. 3.23) contains two instability mechanisms coupled in parallel. Compared with the model based on a single product-activated enzyme, new behavioural modes may be observed, such as birhythmicity, hard excitation and multiple oscillatory domains as a function of a control parameter. The modes of dynamic behaviour in that model remain, however, limited, because it contains only two variables. For complex oscillations such as bursting or chaos to occur, it is necessary that the system contain at least three variables. 

The coupling in series of two enzyme reactions with autocatalytic regulation (fig. 4.1) permits the construction of a three-variable biochemical prototype containing two instability-generating mechanisms (Decroly, 1987a,b Decroly Goldbeter, 1982). As in the model for glycolytic oscillations, the substrate S of the first enzyme is introduced at a constant rate into the system this substrate is transformed by enzyme Ej into product Pi, which serves as substrate for a second enzyme E2 that transforms Pj into P2. The two allosteric enzymes are both activated by their reaction product Pj and P2 are thus positive effectors of enzymes Ej and E2, respectively. 

Each of the two enzymes thus behaves as phosphofructokinase in the model considered for glycolytic oscillations (chapter 2). To limit the study to temporal organization phenomena, the system is considered here as spatially homogeneous, as in the case of experiments on glycolytic oscillations (Hess et ai, 1969). In the case where the kinetics of the two enzymes obeys the concerted allosteric model (Monod et al, 1965), the time evolution of the model is governed by the kinetic equations (4.1), which take the form of three nonlinear, ordinary differential equations  

For simplicity, we consider that the rate of enzyme Ej depends in a linear manner on the concentration j8 of its substrate, which is the same 

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