The minimal model of Hahn for the Calvin cycle

Hussein Obeid; Alan D. Rendall; Hussein Obeid; Alan D. Rendall

doi:10.3934/mbe.2019118

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2353-2370. doi: 10.3934/mbe.2019118

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The minimal model of Hahn for the Calvin cycle

Hussein Obeid ,
Alan D. Rendall ^,

Institut für Mathematik, Johannes Gutenberg-Universität, Staudingerweg 9, 55128 Mainz, Germany

Received: 10 December 2018 Accepted: 26 February 2019 Published: 15 March 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 late 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.
- photosynthesis,
- dynamical system,
- steady state,
- stability,
- asymptotic behaviour
Citation: Hussein Obeid, Alan D. Rendall. The minimal model of Hahn for the Calvin cycle[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2353-2370. doi: 10.3934/mbe.2019118

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Abstract

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 late 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.

References

[1]	A. Arnold and Z. Nikoloski, A quantitative comparison of Calvin-Benson cycle models, Trends in Plant Sci., 16 (2011), 676–683.
[2]	A. Arnold and Z. Nikoloski, In search for an accurate model of the photosynthetic carbon metabolism, Math. Comp. in Sim., 96 (2014), 171–194.
[3]	J. Jablonsky, H. Bauwe and O.Wolkenhauer, Modelling the Calvin-Benson cycle, BMC Syst. Biol. 5 (2011), 185.
[4]	A. D. Rendall, A Calvin bestiary, in Patterns of Dynamics (eds. P. Gurevich, J. Hell, B. Sanstede and A. Scheel), Springer, Berlin, 2017.
[5]	B. D. Hahn, Photosynthesis and photorespiration: modelling the essentials, J. Theor. Biol. 151 (1991), 123–139.
[6]	H.-W. Heldt and B. Piechulla, Plant biochemistry, Academic Press, New York, 2011.
[7]	C. Kuehn, Multiple scale dynamics, Springer, Berlin, 2015.
[8]	L. Perko, Differential Equations and Dynamical Systems, Springer, Berlin, 2001.
[9]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative theory of planar differential systems, Springer, Berlin, 2006.
[10]	J. Carr, Applications of centre manifold theory, Springer, Berlin, 1981.
[11]	A. D. Rendall and J. J. L. Velázquez, Dynamical properties of models for the Calvin cycle, J. Dyn. Diff. Eq., 26 (2014), 673–705.
[12]	J. D. Murray, Mathematical Biology, Springer, Berlin, 1989.
[13]	B. D. Hahn, A mathematical model of leaf carbon metabolism, Ann. Botany 54 (1984), 325–339.
[14]	B. D. Hahn, A mathematical model of photorespiration and photosynthesis, Ann. Botany 60 (1987), 157–189.
[15]	M. Banaji and C. Pantea, The inheritance of nondegenerate multistationarity in chemical reaction networks, SIAM J. Appl. Math., 78 (2018), 1105–1130.
[16]	E. Feliu and C. Wiuf, Simplifying biochemical models with intermediate species, J. R. Soc. Interface 10 (2013), 20132484.
[17]	S. Grimbs, A. Arnold, A. Koseska, et al., Spatiotemporal dynamics of the Calvin cycle: multistationarity and symmetry breaking instabilities, Biosystems 103 (2011), 212–223.
[18]	G. Petterson and U. Ryde-Petterson, A mathematical model of the Calvin photosynthesis cycle. Eur. J. Biochem., 175 (1988), 661-672.
[19]	D. Möhring and A. D. Rendall, Overload breakdown in models of photosynthesis, Dyn. Sys. 32 (2017), 234–248.
[20]	M. G. Poolman, Computer modelling applied to the Calvin cycle, PhD Thesis, Oxford Brookes University, 1999.

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