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doi: 10.3934/dcdsb.2021268
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Analysis of a model of the Calvin cycle with diffusion of ATP

Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany

* Corresponding author: Alan D. Rendall

Received  July 2021 Revised  September 2021 Early access November 2021

The dynamics of a mathematical model of the Calvin cycle, which is part of photosynthesis, is analysed. Since diffusion of ATP is included in the model a system of reaction-diffusion equations is obtained. It is proved that for a suitable choice of parameters there exist spatially inhomogeneous positive steady states, in fact infinitely many of them. It is also shown that all positive steady states, homogeneous and inhomogeneous, are nonlinearly unstable. The only smooth steady state which could be stable is a trivial one, where all concentrations except that of ATP are zero. It is found that in the spatially homogeneous case there are steady states with the property that the linearization about that state has eigenvalues which are not real, indicating the presence of oscillations. Numerical simulations exhibit solutions for which the concentrations are not monotone functions of time.

Citation: Burcu Gürbüz, Alan D. Rendall. Analysis of a model of the Calvin cycle with diffusion of ATP. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021268
References:
[1]

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, Garland, New York, 2008.

[2]

A. Arnold and Z. Nikoloski, In search for an accurate model of the photosynthetic carbon metabolism, Math. Comp. in Simulation, 96 (2014), 171-194.  doi: 10.1016/j.matcom.2012.03.011.

[3]

S. Cygan, A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of all regular stationary solutions to reaction-diffusion-ODE systems, preprint, arXiv: 2105.05023.

[4]

S. Disselnkötter and A. D. Rendall, Stability of stationary solutions in models of the Calvin cycle, Nonlin. Analysis: RWA, 34 (2017), 481-494.  doi: 10.1016/j.nonrwa.2016.09.017.

[5]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer, Berlin, 1979.

[6]

S. GrimbsA. ArnoldA. KoseskaJ. KurthsJ. Selbig and Z. Nikoloski, Spatiotemporal dynamics of the Calvin cycle: Multistationarity and symmetry breaking instabilities, Biosystems, 103 (2011), 212-223.  doi: 10.1016/j.biosystems.2010.10.015.

[7]

B. D. Hahn, Photosynthesis and photorespiration: Modelling the essentials, J. Theor. Biol., 151 (1991), 123-139.  doi: 10.1016/S0022-5193(05)80147-X.

[8]

J. JablonskyH. Bauwe and O. Wolkenhauer, Modelling the Calvin-Benson cycle, BMC Syst. Biol., 5 (2011), 185. 

[9]

A. Marciniak-CzochraG. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., 99 (2013), 509-543.  doi: 10.1016/j.matpur.2012.09.011.

[10]

A. Marciniak-CzochraG. Karch and K. Suzuki, Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., 74 (2017), 583-618.  doi: 10.1007/s00285-016-1035-z.

[11]

H. Obeid and A. D. Rendall, The minimal model of Hahn for the Calvin cycle, Math. Biosci. Eng., 16 (2019), 2353-2370.  doi: 10.3934/mbe.2019118.

[12]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, Berlin, 1984. doi: 10.1007/978-1-4612-5282-5.

[13]

A. D. Rendall, A Calvin bestiary, in Patterns of Dynamics, 318–337, Springer Proc. Math. Stat., 205, Springer, Cham, (2017). doi: 10.1007/978-3-319-64173-7_18.

[14]

A. D. Rendall and J. J. L. Velázquez, Dynamical properties of models for the Calvin cycle, J. Dyn. Diff. Eq., 26 (2014), 673-705.  doi: 10.1007/s10884-014-9385-y.

[15]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Springer, Berlin, 1984. doi: 10.1007/BFb0099278.

[16]

J. Shatah and W. Strauss, Spectral condition for abstract instability, in Nonlinear PDE's, Dynamics and Continuum Physics, 189–198, Contemp. Math., 255, Amer. Math. Soc., Providence, RI, (2000). doi: 10.1090/conm/255/03982.

[17]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[18]

M. E. Taylor, Partial Differential Equations III. Nonlinear Equations, Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

show all references

References:
[1]

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, Garland, New York, 2008.

[2]

A. Arnold and Z. Nikoloski, In search for an accurate model of the photosynthetic carbon metabolism, Math. Comp. in Simulation, 96 (2014), 171-194.  doi: 10.1016/j.matcom.2012.03.011.

[3]

S. Cygan, A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of all regular stationary solutions to reaction-diffusion-ODE systems, preprint, arXiv: 2105.05023.

[4]

S. Disselnkötter and A. D. Rendall, Stability of stationary solutions in models of the Calvin cycle, Nonlin. Analysis: RWA, 34 (2017), 481-494.  doi: 10.1016/j.nonrwa.2016.09.017.

[5]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer, Berlin, 1979.

[6]

S. GrimbsA. ArnoldA. KoseskaJ. KurthsJ. Selbig and Z. Nikoloski, Spatiotemporal dynamics of the Calvin cycle: Multistationarity and symmetry breaking instabilities, Biosystems, 103 (2011), 212-223.  doi: 10.1016/j.biosystems.2010.10.015.

[7]

B. D. Hahn, Photosynthesis and photorespiration: Modelling the essentials, J. Theor. Biol., 151 (1991), 123-139.  doi: 10.1016/S0022-5193(05)80147-X.

[8]

J. JablonskyH. Bauwe and O. Wolkenhauer, Modelling the Calvin-Benson cycle, BMC Syst. Biol., 5 (2011), 185. 

[9]

A. Marciniak-CzochraG. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., 99 (2013), 509-543.  doi: 10.1016/j.matpur.2012.09.011.

[10]

A. Marciniak-CzochraG. Karch and K. Suzuki, Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., 74 (2017), 583-618.  doi: 10.1007/s00285-016-1035-z.

[11]

H. Obeid and A. D. Rendall, The minimal model of Hahn for the Calvin cycle, Math. Biosci. Eng., 16 (2019), 2353-2370.  doi: 10.3934/mbe.2019118.

[12]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, Berlin, 1984. doi: 10.1007/978-1-4612-5282-5.

[13]

A. D. Rendall, A Calvin bestiary, in Patterns of Dynamics, 318–337, Springer Proc. Math. Stat., 205, Springer, Cham, (2017). doi: 10.1007/978-3-319-64173-7_18.

[14]

A. D. Rendall and J. J. L. Velázquez, Dynamical properties of models for the Calvin cycle, J. Dyn. Diff. Eq., 26 (2014), 673-705.  doi: 10.1007/s10884-014-9385-y.

[15]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Springer, Berlin, 1984. doi: 10.1007/BFb0099278.

[16]

J. Shatah and W. Strauss, Spectral condition for abstract instability, in Nonlinear PDE's, Dynamics and Continuum Physics, 189–198, Contemp. Math., 255, Amer. Math. Soc., Providence, RI, (2000). doi: 10.1090/conm/255/03982.

[17]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[18]

M. E. Taylor, Partial Differential Equations III. Nonlinear Equations, Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

Figure 1.  Non-monotone behaviour near $ P_1 $
Figure 2.  Long-time behaviour of the solution in Fig. 1
Figure 3.  Long-time behaviour of a solution approaching the point $ P_2 $
Figure 4.  Plot in the $ x_{\rm GAP} $-$ x_{\rm Ru5P} $ plane of a solution starting near $ P_2 $
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