Eigenvalues of Jacobian Matrices Report on Steps of Metabolic Reprogramming in a Complex Plant-Environment Interaction

ABSTRACT

Mathematical modeling of biochemical systems aims at improving the knowledge about complex regulatory networks. The experimental high-throughput measurement of levels of biochemical components, like metabolites and proteins, has become an integral part for characterization of biological systems. Yet, strategies of mathematical modeling to functionally integrate resulting data sets is still challenging. In plant biology, regulatory strategies that determine the metabolic output of metabolism as a response to changes in environmental conditions are hardly traceable by intuition. Mathematical modeling has been shown to be a promising approach to address such problems of plant-environment interaction promoting the comprehensive understanding of plant biochemistry and physiology. In this context, we recently published an inversely calculated solution for first-order partial derivatives, i.e. the Jacobian matrix, from experimental high-throughput data of a plant biochemical model system. Here, we present a biomathematical strategy, comprising 1) the inverse calculation of a biochemical Jacobian; 2) the characterization of the associated eigenvalues and 3) the interpretation of the results with respect to biochemical regulation. Deriving the real parts of eigenvalues provides information about the stability of solutions of inverse calculations. We found that shifts of the eigenvalue real part distributions occur together with metabolic shifts induced by short-term and long-term exposure to low temperature. This indicates the suitability of mathematical Jacobian characterization for recognizing perturbations in the metabolic homeostasis of plant metabolism. Together with our previously published results on inverse Jacobian calculation this represents a comprehensive strategy of mathematical modeling for the analysis of complex biochemical systems and plant-environment interactions from the molecular to the ecosystems level.

Cite this paper

T. Nägele and W. Weckwerth, "Eigenvalues of Jacobian Matrices Report on Steps of Metabolic Reprogramming in a Complex Plant-Environment Interaction,"*Applied Mathematics*, Vol. 4 No. 8, 2013, pp. 44-49. doi: 10.4236/am.2013.48A007.

T. Nägele and W. Weckwerth, "Eigenvalues of Jacobian Matrices Report on Steps of Metabolic Reprogramming in a Complex Plant-Environment Interaction,"

References

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[15] S. Henkel, T. Nagele, I. Hormiller, T. Sauter, O. Sawodny, M. Ederer and A. G. Heyer, “A Systems Biology Ap proach to Analyse Leaf Carbohydrate Metabolism in Ara bidopsis thaliana,” EURASIP Journal on Bioinformatics & Systems Biology, 2011, p. 2. doi:10.1186/1687-4153-2011-2

[16] T. Nagele and A. G. Heyer, “Approximating Subcellular Organisation of Carbohydrate Metabolism during Cold Acclimation in Different Natural Accessions of Arabi dopsis thaliana,” New Phytologist, Vol. 198, No. 3, 2013, pp. 777-787. doi:10.1111/nph.12201

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[1] J. A. Morgan and D. Rhodes, “Mathematical Modeling of Plant Metabolic Pathways,” Metabolic Engineering, Vol. 4, No. 1, 2002, pp. 80-89. doi:10.1006/mben.2001.0211

[2] W. Weckwerth, “Green Systems Biology—From Single Genomes, Proteomes and Metabolomes to Ecosystems Re search and Biotechnology,” Journal of Proteomics, Vol. 75, No. 1, 2011, pp. 284-305. doi:10.1016/j.jprot.2011.07.010

[3] G. D. Farquhar, S. V. Caemmerer and J. A. Berry, “A Biochemical Model of Photosynthetic CO2 Assimilation in Leaves of C3 Species,” Planta, Vol. 149, No. 1, 1980, pp. 78-90. doi:10.1007/BF00386

231

[4] G. D. Farquhar, “Models Describing the Kinetics of Ribu lose Biphosphate Carboxylase-Oxygenase,” Archives of Biochemistry and Biophysics, Vol. 193, No. 2, 1979, pp. 456-468. doi:10.1016/0003-9861(79)90052-3

[5] C. J. Bernacchi, J. E. Bagley, S. P. Serbin, U. M. Ruiz Vera, D. M. Rosenthal and A. Vanloocke, “Modeling C3 Photosynthesis from the Chloroplast to the Ecosystem,” Plant, Cell and Environment, 2013. doi:10.1111/pce.12118

[6] W. Cramer, D. W. Kicklighter, A. Bondeau, B. Moore, G. Churkina, B. Nemry, A. Ruimy, A. L. Schloss and P. P. N. M. Intercompariso, “Comparing Global Models of Ter restrial Net Primary Productivity (NPP): Overview and Key Results,” Global Change Biology, Vol. 5, 1999, pp. 1-15. doi:10.1046/j.1365-2486.1999.00009.x

[7] H. Doerfler, D. Lyon, T. Nagele, X. Sun, L. Fragner, F. Hadacek, V. Egelhofer and W. Weckwerth, “Granger Cau sality in Integrated GC-MS and LC-MS Metabolomics Data Reveals the Interface of Primary and Secondary Metabolism,” Metabolomics, Vol. 9, No. 3, 2013, pp. 564-574.

[8] X. Sun and W. Weckwerth, “COVAIN: A Toolbox for Uni and Multivariate Statistics, Time-Series and Correla tion Network Analysis and Inverse Estimation of the Dif ferential Jacobian from Metabolomics Covariance Data,” Metabolomics, Vol. 8, No. 1, 2012, pp. 81-93. doi:10.1007/s11306-012-0399-3

[9] I. Thiele and B. O. Palsson, “A Protocol for Generating a High-Quality Genome-Scale Metabolic Reconstruction,” Nature Protocols, Vol. 5, 2010, pp. 93-121. doi:10.1038/nprot.2009.203

[10] W. Weckwerth, “Unpredictability of Metabolism—The Key Role of Metabolomics Science in Combination with Next-Generation Genome Sequencing,” Analytical and Bioanalytical Chemistry, Vol. 400, No. 7, 2011, pp. 1967-1978. doi:10.1007/s00216-011-4948-9

[11] R. Steuer, J. Kurths, O. Fiehn and W. Weckwerth, “Ob serving and Interpreting Correlations in Metabolomic Net works,” Bioinformatics, Vol. 19, No. 8, 2003, pp. 1019-1026. doi:10.1093/bioin

formatics/btg120

[12] N. G. van Kampen, “Stochastic Processes in Physics and Chemistry,” Elsevier, Amsterdam, 1992.

[13] E. Reznik and D. Segrè, “On the Stability of Metabolic Cycles,” Journal of Theoretical Biology, Vol. 266, No. 4, 2010, pp. 536-549. doi:10.1016/j.jtbi.2010.07.023

[14] C. Eck, H. Garcke and P. Knabner, “Mathematische Mo dellierung,” Springer, Berlin Heidelberg, 2008.

[15] S. Henkel, T. Nagele, I. Hormiller, T. Sauter, O. Sawodny, M. Ederer and A. G. Heyer, “A Systems Biology Ap proach to Analyse Leaf Carbohydrate Metabolism in Ara bidopsis thaliana,” EURASIP Journal on Bioinformatics & Systems Biology, 2011, p. 2. doi:10.1186/1687-4153-2011-2

[16] T. Nagele and A. G. Heyer, “Approximating Subcellular Organisation of Carbohydrate Metabolism during Cold Acclimation in Different Natural Accessions of Arabi dopsis thaliana,” New Phytologist, Vol. 198, No. 3, 2013, pp. 777-787. doi:10.1111/nph.12201

[17] R. M. May, “Stability and Complexity in Model Ecosys tems,” 2nd Edition, Princeton University Press, Princeton, 1974.