^{1}

^{*}

^{1}

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.

The molecular organization of plant metabolism is of high complexity. This complexity arises from a sophisticated level of subcellular organization and of highly interlaced regulatory circuits interconnecting different levels of biochemical organisation. Mathematical modeling is a powerful way to improve the current understanding of plant cells on a molecular level [_{i} of metabolite concentrations M_{i} with respect to metabolite concentrations (Equation (1)):

The metabolite functions f_{i} are defined by the system of ODEs describing time-dependent changes of metabolite concentrations (Equation (2)):

Here, M(t) is the n-dimensional vector of metabolite concentrations, v is an r-dimensional vector of metabolic fluxes and N describes the m x r stoichiometrix matrix of the biochemical reaction network. If the full stoichiometry of a large biochemical network cannot be resolved experimentally, a metabolic interaction matrix, N_{I}, can be derived by interconnecting the experimentally accessible components. The term indicates that the metabolic flux depends on time-variant metabolite concentrations, i.e. the substrates, products and effectors of the enzymatic reaction, and also on time-variant parameters, e.g. enzyme kinetic parameters or thermodynamic constraints. Hence, to determine the Jacobian matrix of a metabolic system the stoichiometry as well as the timevariant parameters or fluxes must be known. While the method of metabolic reconstruction allows the computational assisted construction of genome-scale stoichiometric matrices [

Here, J represents the Jacobian matrix, C is the covariance matrix derived from the experimental data and D is the so-called fluctuation matrix integrating metabolite fluctuations which can be modelled by a Langevintype equation (Equation (4)):

Time-dependent changes in the matrix of metabolite concentrations Ξ are directly linked to the Gaussian noise function ψ(t). Stationary solutions of the Langevin-type equation are linked to the covariance matrix by a corresponding Fokker-Planck equation and finally result in (Equation (3)) [11,12]. Applying this inverse approach to experimental metabolomics data of leaf material from the genetic model plant Arabidopsis thaliana, we could unravel regulatory instances in plant primary metabolism contributing to the establishment of a new metabolic homeostasis induced by changes in light intensity and temperature [

Model construction and experimental analysis of metabolite content was described previously [^{TM} (V4.3; http://celldesigner.org). The model file is provided on request in Systems Biology Markup Language (SBML). We used this model structure to derive the metabolic interaction matrix, N_{I}, allowing for the calculation of the Jacobian matrix (Equations (1) and (2)).

Singular value decomposition (SVD) was applied for the inverse calculation of the Jacobian matrix [_{2}-ratio of two different conditions a and b (Equation (5)):

In this way we compared the ratio of Jacobian matrices derived from leaf samples of plants which were not exposed (non-acc), 2 days exposed (2d), 8 days exposed (8d), 14 days expose (14d) and 18 days exposed (18d) to 4˚C and increased light intensity. We found a major impact of cold exposure on primary metabolism after 2 days and 14 days at 4˚C pointing to a short-term and longterm effect of cold on plant primary metabolism [

Based on our results of inverse calculations, we derived the eigenvalues of the Jacobian entries as described in the following steps. All of the compared Jacobian matrices represent solutions for a certain steady state, i.e. an infintesimal time unit for which all changes in the systems equations can be assumed to equal zero (Equation (6)):

Metabolite concentration at this steady state, , is related to the metabolite concentration M_{i} in (Equation (2)) by a perturbation term θ_{i}(t) yielding (Equation (7)):

Linearization around this steady state by Taylor expansion discarding all but the first element results in (Equation (8)):

Here, j_{ik} are the elements of the Jacobian matrix J at the considered steady state. Solutions of (Equation (8)) are described by

are constants depending on the initial values of the perturbations. The constants represent the eigenvalues of the Jacobian matrix J describing the behaviour of the system. Solutions for can be derived from the characteristic equation:

Here, I is the m ´ m unit matrix. In general, eigenvalues are complex numbers, composed of a real part and an imaginary part. From (Equation (9)) it becomes obvious that real parts < 0 result in an exponential decay of the perturbation matrix while real parts > 0 induce exponential increase indicating instable system behaviour. The imaginary part of the eigenvalues determine the sinusoidal oscillation which becomes clear when rewriting in polar coordinates instead of Cartesian coordinates.

To analyse whether solutions of inverse calculations may indicate a difference of system stability properties in leaf metabolism when exposed to perturbed environmental conditions, we compared the real parts of eigenvalues. All calculations and graphical representations were performed in MATLAB^{®} (V7.12.0 R2011a). Histograms and scatterplots indicate the presence of positive real parts in all samples following the environmental perturbation (

Inverse problems may often be characterized as ill-conditioned because either solutions do not exist, or solutions are not unique, or solutions are not data-dependent in a uniformly continuous way [

plex natural systems. Even more, characterizing a biochemical system of interest by the Jacobian matrix enables the application of methods from systems theory which is inevitable for the comprehensive and systematic analysis of large metabolic networks [

We would like to thank the members of the Department of Molecular Systems Biology for many fruitful discussions. We thank the EU-Marie-Curie ITN MERIT (GA 2010-264474) for financial support of TN.