Oscillatory model of crassulacean acid metabolism with a dynamic hysteresis switch

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Abstract
The mechanism of endogenous circadian photosynthesis oscillations of plants performing crassulacean acid metabolism (CAM) is investigated in terms of a nonlinear theoretical model. Unlike previous CAM models containing a discrete element, we use throughout continuous time differential equations which more adequately reflect the CAM dynamics. By incorporating results from a complementary and as well continuous membrane model, a detailed description of the molecular malate transport in and out of the vacuole through the tonoplast membrane is achieved. Our analysis shows that the membrane effectively acts as hysteresis switch regulating the oscillations. It thus provides a molecular basis for the circadian clock. The model shows regular endogenous limit cycle oscillations that are stable for a wide range of temperatures, in a manner that complies well with experimental data. The circadian period length is explained simply in terms of the filling time of the vacuole. The results emphasize the central role of membrane dynamics for the generation of circadian oscillations, and thus have general relevance for the explanation of biological clocks.