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

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.