Dynamical Properties of Models for the Calvin Cycle

Springer Science and Business Media LLC - Tập 26 - Trang 673-705 - 2014
Alan D. Rendall1, Juan J. L. Velázquez2
1Institute for Mathematics, University of Mainz, Mainz, Germany
2Institute for Applied Mathematics, University of Bonn, Bonn, Germany

Tóm tắt

Modelling the Calvin cycle of photosynthesis leads to various systems of ordinary differential equations and reaction-diffusion equations. They differ by the choice of chemical substances included in the model, the choices of stoichiometric coefficients and chemical kinetics and whether or not diffusion is taken into account. This paper studies the long-time behaviour of solutions of several of these systems, concentrating on the ODE case. In some examples it is shown that there exist two positive stationary solutions. In several cases it is shown that there exist solutions where the concentrations of all substrates tend to zero at late times and others (runaway solutions) where the concentrations of all substrates increase without limit. In another case, where the concentration of ATP is explicitly included, runaway solutions are ruled out.

Tài liệu tham khảo

Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Watson, J.D.: Molecular Biology of the Cell. Garland Science, New York (2002) Anderson, D.A.: Boundedness of trajectories for weakly reversible, single linkage class reaction systems. J. Math. Chem. 49, 2275–2290 (2011) Anderson, D.A.: A proof of the global attractor conjecture in the single linkage class case. SIAM J. Appl. Math. 71, 1487–1508 (2011) Arnold, A., Nikoloski, Z.: A quantitative comparison of Calvin–Benson cycle models. Trends Plant Sci. 16, 676–682 (2011) Conradi, C., Flockerzi, D., Raisch, J., Stelling, J.: Subnetwork analysis reveals dynamical features of complex (bio)chemical networks. Proc. Natl. Acad. Sci. USA 104, 19175–19180 (2007) Ellison, P., Feinberg, M.: How catalytic mechanisms reveal themselves in multiple steady-state data. J. Mol. Catal. A 154, 155–167 (2000) Feinberg, M.: Lectures on Chemical Reaction Networks. http://www.chbmeng.ohio-state.edu/~feinberg/research/ (1980) Feinberg, M.: Chemical reaction network structure and the stability of complex isothermal reactors. II Multiple steady states for networks of deficiency one. Chem. Eng. Sci. 43, 1–25 (1988) Grimbs, S., Arnold, A., Koseska, A., Kurths, J., Selbig, J., Nikoloski, Z.: Spatiotemporal dynamics of the Calvin cycle: multistationarity and symmetry breaking instabilities. Biosystems 103, 212–223 (2011) Hartshorne, R.: Algebraic geometry. Springer, Berlin (1977) Heineken, F.G., Tsuchiya, H.M., Aris, R.: On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics. Math. Biosci. 1, 95–113 (1967) Hek, G.: Geometric singular perturbation theory in biological practice. J. Math. Biol. 60, 347–386 (2010) Hirsch, M.W.: Systems of differential equations which are competitive or cooperative II: convergence almost everywhere. SIAM J. Math. Anal. 16, 432–439 (1985) Horn, F., Jackson, R.: General mass action kinetics. Arch. Rat. Mech. Anal. 47, 81–116 (1972) Huang, C.-Y.F., Ferrell, J.E.: Ultrasensitivity in the mitogen-activated protein kinase cascade. Proc. Natl. Acad. Sci. USA 93, 10078–10083 (1996) Jablonsky, J., Bauwe, H., Wolkenhauer, O.: Modeling the Calvin–Benson cycle. BMC Syst. Biol. 5, 185 (2011) Lei, H.-B., Wang, X., Wang, R., Chen, L., Zhang, J.-F.: A parameter condition for ruling out multiple equilibria of the photosynthetic carbon metabolism. Asian J. Control 13, 611–624 (2011) Mallet-Paret, J., Smith, H.L.: The Poincaré–Bendixson theorem for monotone cyclic feedback systems. J. Dyn. Diff. Eq. 2, 367–421 (1990) Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000) Pettersson, G., Ryde-Pettersson, U.: A mathematical model of the Calvin photosynthesis cycle. Eur. J. Biochem. 175, 661–672 (1988) Poolman, M.G., Olcer, H., Lloyd, J.C., Raines, C.A., Fell, D.: Computer modelling and experimental evidence for two steady states in the photosynthetic Calvin cycle. Eur. J. Biochem. 268, 2810–2816 (2001) Rendall, A.D.: Mathematics of the NFAT signalling pathway. SIAM J. Appl. Dyn. Sys. 11, 988–1006 (2012) Wang, L., Sontag, E.D.: On the number of steady states in a multiple futile cycle. J. Math. Biol. 57, 29–52 (2008) Zhu, X.-G., de Sturler, E., Long, S.P.: Optimizing the distribution of resources between enzymes of carbon metabolism can dramatically increase photosynthetic rate: a numerical simulation using an evolutionary algorithm. Plant Physiol. 145, 513–526 (2007) Zhu, X.-G., Alba, R., de Sturler, E.: A simple model of the Calvin cycle has only one physiologically feasible steady state under the same external conditions. Nonlin. Anal. RWA 10, 1490–1499 (2009)