Abstract
A glycolytic model system consisting of the enzymes phosphofructokinase (EC 2.7.1.11) and pyruvate kinase (EC 2.7.1.40) is analyzed when subject to periodic substrate addition. The calculations are performed by using detailed rate laws that have been derived for the enzymes of Escherichia coli. Due to linear relationships between the metabolite concentrations, the numerical solutions can be displayed inside a trapezium, so that the concentrations of four different metabolites are indicated along the trapezium edges. The analysis reveals a rich variety of time patterns, corresponding to different periodic, quasiperiodic, and chaotic attractors. These patterns undergo complex hysteresis loops when bifurcation parameters are slowly changed—for example, by modulating the input amplitude. By using this technique up to four attractors coexisting in phase space are found. The time patterns corresponding to coexisting attractors can be switched into one another by triggering the system with short substrate pulses. Furthermore, conditions exist at which the triggering is autonomous—i.e., self-sustained (intermittent) switchings occur. The time between these switchings can be set externally by the value of the input amplitude. For conditions in which the periods of the oscillations are in the order of minutes, the self-sustained switching—which modulates these oscillations—can be in the order of hours.
Keywords: biorhythms, entrainment, hysteresis, quasiperiodicity, chaos
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