Overload breakdown in models for photosynthesis

2017

Patterns of Dynamics

Self Cite

This paper compares a number of mathematical models for the Calvin cycle of photosynthesis and presents theorems on the existence and stability of steady states of these models. Results on five-variable models in the literature are surveyed. Next a number of larger models related to one introduced by Pettersson and Ryde-Pettersson are discussed. The mathematical nature of this model is clarified, showing that it is naturally defined as a system of differential-algebraic equations. It is proved that there are choices of parameters for which this model admits more than one positive steady state. This is done by analysing the limit where the storage of sugars from the cycle as starch is shut down. There is also a discussion of the minimal models for the cycle due to Hahn.

Section: The Pettersson Model and Modifications Of Itmentioning

confidence: 82%

Section: The Five-species Modelsmentioning

confidence: 99%

Section: The Pettersson Model and Modifications Of Itmentioning

confidence: 99%

See 1 more Smart Citation

A Calvin Bestiary

2017

Patterns of Dynamics

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“…This kind of behaviour can be ruled out if the model admits a suitable conservation law. This is, for instance, the case in a model of [16] whose mathematical properties were studied in [13]. In the model of Hahn with photorespiration boundedness of solutions is obtained without there being a conservation law.…”

Section: Discussionmentioning

confidence: 99%

“…This means intuitively that the production of sugar by the cycle cannot meet the demand for export from the chloroplast. In [13] it was shown that there are solutions of the model of [16] which admit this phenomenon while in a modification of the model due to Poolman [17] these solutions are eliminated. The model of [17] does not include photorespiration but does include the mobilization of glucose from starch.…”

Section: Discussionmentioning

confidence: 99%

The minimal model of Hahn for the Calvin cycle

Obeid

Mathematical Biosciences and Engineering

2019

There are many models of the Calvin cycle of photosynthesis in the literature. When investigating the dynamics of these models one strategy is to look at the simplest possible models in order to get the most detailed insights. We investigate a minimal model of the Calvin cycle introduced by Hahn while he was pursuing this strategy. In a variant of the model not including photorespiration it is shown that there exists exactly one positive steady state and that this steady state is unstable. For generic initial data either all concentrations tend to infinity at lates times or all concentrations tend to zero at late times. In a variant including photorespiration it is shown that for suitable values of the parameters of the model there exist two positive steady states, one stable and one unstable. For generic initial data either the solution tends to the stable steady state at late times or all concentrations tend to zero at late times. Thus we obtain rigorous proofs of mathematical statements which together confirm the intuitive idea proposed by Hahn that photorespiration can stabilize the operation of the Calvin cycle. In the case that the concentrations tend to infinity we derive formulae for the leading order asymptotics using the Poincaré compactification.

Stability of stationary solutions in models of the Calvin cycle

Disselnkötter

Nonlinear Analysis: Real World Applications

2017

Self Cite

In this paper results are obtained concerning the number of positive stationary solutions in simple models of the Calvin cycle of photosynthesis and the stability of these solutions. It is proved that there are open sets of parameters in the model of Zhu et. al. [15] for which there exist two positive stationary solutions. There are never more than two isolated positive stationary solutions but under certain explicit special conditions on the parameters there is a whole continuum of positive stationary solutions. It is also shown that in the set of parameter values for which two isolated positive stationary solutions exist there is an open subset where one of the solutions is asymptotically stable and the other is unstable. In related models derived from the work of [4], for which it was known that more that one positive stationary solution exists, it is proved that there are parameter values for which one of these solutions is asymptotically stable and the other unstable. A key technical aspect of the proofs is to exploit the fact that there is a bifurcation where the centre manifold is one-dimensional.