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Due to their sessile lifestyle, plants are exposed to a large set of environmental cues. In order to cope with changes in environmental conditions a multitude of complex strategies to regulate metabolism has evolved. The complexity is mainly attributed to interlaced regulatory circuits between genes, proteins and metabolites and a high degree of cellular compartmentalization. The genetic model plant

The performance and distribution of plants is significantly affected by several environmental factors, like for example temperature, drought and soil salinity. Due to their sessile lifestyle, plants have to cope with spontaneous or seasonal changes in these environmental factors in order to survive. To a certain degree, numerous plant species are able to respond to such changes by re-adjustment of their metabolism—a process that is in general termed as acclimation. It comprises a multitude of biochemical and physiological changes, ultimately leading to an increase in the capacity of the plants to cope with environmental stress. One prominent example, which was amongst others intensely studied in the model plant

The analysis of complex metabolic processes involved in acclimation of plant metabolism to environmental stress significantly benefits from the availability of genetically distinct natural populations of the model plant

Besides such complex plant-environment interactions, latest developments in bioanalytical research comprising shotgun and next-generation genome sequencing as well as molecular analysis using OMICS technologies have driven the need for computer-assisted analysis and modeling of biological data. Systems biology has evolved in a research field focusing on the system wide understanding of biological networks, like for example the cellular metabolism in a photosynthetically active plant cell. In a systems biology approach, network elements, such as genes, proteins or metabolites, are considered as interacting components rather than isolated entities in order to deepen the comprehensive understanding of the organization of a complex biological system. A promising way to analyze such complex biological and biochemical networks is formal representation by mathematical models enabling their computer based handling and making biological data accessible to theoretical methods originating from applied mathematics and systems theory. Numerous mathematical approaches to model plant metabolic networks have been suggested and discussed, both relying on and emphasizing the importance of the iterative processes of model development, simulation and validation by experimental data [

A main focus of mathematical modeling in biochemistry and plant science has been on the construction of kinetic models where metabolic states are simulated based on the knowledge about network topology, stoichiometry, rate equations and kinetic parameters. Typically, a system of ordinary differential equations (ODEs) is used to describe the time-dependent changes in state variables, _{M}) or inhibitory constants (K_{i}) are lacking. This enforces the application of parameter estimation to calculate parameter values which are either completely unknown or can be estimated within numerical bounds based on published data on a different condition or organism. Indeed, such assumptions cause uncertainties, which have to be discussed carefully when interpreting the model output. However, although there might be several uncertainties with respect to regulatory instances involved in every single reaction of metabolism, numerous studies have proven kinetic modeling to be a promising approach to comprehensively analyze complex processes in plant biology. An overview of applications is given by Schallau and Junker [

In contrast to kinetic modeling, the approach of structural modeling is based on the idea of constructing models without kinetic information. This modeling approach refers only to the stoichiometry of the reactions within the system which is summarized in the stoichiometric matrix

where

Focusing now again on the complexity of plant-environment interactions and the variability of stress responses in natural accessions of

The mathematical analysis of plant metabolism first of all relies on the representation by a model, which is constructed based on information on biochemical pathways and the interaction of pathway components gained from numerous previous experimental studies. Typically, the first step of model construction consists of a graphical representation of the pathway or network of interest. In case of a kinetic model of metabolism, this may result in a map, which connects nodes by lines. Every node now represents a metabolite and every line describes a metabolite interconversion, _{M}, and the maximum enzyme activity v_{max}. This process is crucial for the successful modeling approach as all further steps of mathematical analysis rely on these assumptions: if the interaction between two network components is described by equations or parameters which do not agree with confirmed experimental results, validation of simulation results by experimental data is not reliable anymore and the model becomes unfeasible. Although a vast number of metabolic interactions have intensively been characterized and many underlying laws of interaction are well known, like for example the Michaelis-Menten kinetics, deriving the most realistic model structure of a metabolic network becomes difficult when assumptions about simplification have to be made. This is frequently the case for kinetic models, based on systems of ODEs, which are intended to provide an insight into the dynamics of metabolism. These dynamics are predominantly nonlinear and model systems are often characterized by a high-dimensional parameter space. Kinetic parameters, characterizing substrate affinity (K_{M}) or inhibition (K_{i}), are often not directly accessible to experimental measurements. In addition to the everlasting question how results of

As exemplified by these approaches of mathematical modeling, realistic output of model simulations can be expected despite a significant simplification of the model structure. This was also proven by many other kinetic modeling approaches (for an overview of several approaches please refers to [

where Z represents the cost function to be minimized, y_{exp} contains experimentally determined state variables (for example metabolite concentrations), y_{pred}(p,t) is the model prediction of state variables depending on estimated parameters p and time t, and W(t) is the weighting matrix containing information about the level of importance of single state variables and determining their influence on the cost function. This optimization problem of minimization of Z is subject to the differential/algebraic equality constraints describing the systems dynamics and additional requirements for system performance. Additionally, the estimation of model parameter p is subject to lower (p^{low}) and upper (p^{up}) bounds:

Due to nonlinearities in objective function and constraints, solving these optimization problems frequently means having to cope with multimodality,

Due to its capability to develop methods for comprehensive analysis of complex data sets and provide strategies of how to solve nonlinear problems, optimization theory represents an essential component for mathematical modeling of plant metabolism and other biological systems. Beyond that, the prediction of metabolism from first principles only becomes possible by application of optimization approaches [

Reconstruction of metabolic networks is based on information about whole genome sequences finally resulting in the stoichiometric matrix

With respect to such comprehensive metabolic network simulations, quantitative measurement of metabolism is necessary to validate the output of such simulations, which can be accomplished applying bioanalytical methods in metabolomics science [

The analysis of metabolomics results on numerous biological replicates under different environmental conditions or with genotypic variation represents a multidimensional task and results in a complex data matrix. The covariance matrix C results from multivariate statistics representing a central result of the experiments [

In this equation, J represents the Jacobian matrix and D is the fluctuation/diffusion matrix. The diagonal entries D_{ii} characterize the magnitude of fluctuations of each metabolite, whereas off-diagonal entries D_{ij} (i≠j) represent the fluctuation of metabolites caused by the interaction between enzymes i and j. The interconnection between metabolic networks and the Jacobian Matrix as well as the fluctuation matrix is described in detail elsewhere [

Here, N is the stoichiometric matrix, r represents the rates for each reaction and M is the metabolite concentration. Based on equations (4) and (5), an approach of inverse calculation of a Jacobian from metabolomics covariance data was recently derived [_{ij}, defining the relative change of two Jacobians J_{a} and J_{b} which are associated with different treatments,

Calculation of the differential Jacobian reveals perturbation sites between two different metabolic states hinting at a significant regulatory event, e.g., the change of enzymatic reaction rates due to environmental perturbations. In principle, using this approach it is possible to conveniently connect a large metabolomics experiment with many samples and thousands of variables directly with the predicted genome-scale metabolic network to calculate biochemical regulation in the investigated biological system (for more detail see [

With regard to currently emerging discussions about world population feeding, global climate change and limited energy resources with fossil fuels, plant biology and biotechnology are central topics of life sciences in the coming decades [

Overview of modeling approaches and their interaction by validation. Data represent results of experiments on the metabolome, proteome, enzyme activities or transcriptome.

Kinetic modeling approaches are limited by lack of kinetic information and stoichiometric modeling approaches are limited by their reference to a steady state. Yet, if a stoichiometric modeling approach delivers information about potential perturbation sites in metabolism, this will enable systematic in-depth analysis, for example by kinetic modeling, promoting a comprehensive understanding of how plant metabolism is composed functionally.

We would like to thank the members of the Department Molecular Systems Biology for fruitful discussions. We would also like to thank the reviewers of this article for their constructive advice to improve its quality and coverage. TN is funded by a Marie Curie ITN project of the European Union, Grant Agreement number 264474.

The authors declare no conflict of interest.

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