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Complex networks have been successfully employed to represent different levels of biological systems, ranging from gene regulation to protein-protein interactions and metabolism. Network-based research has mainly focused on identifying unifying structural properties, such as small average path length, large clustering coefficient, heavy-tail degree distribution and hierarchical organization, viewed as requirements for efficient and robust system architectures. However, for biological networks, it is unclear to what extent these properties reflect the evolutionary history of the represented systems. Here, we show that the salient structural properties of six metabolic networks from all kingdoms of life may be inherently related to the evolution and functional organization of metabolism by employing network randomization under mass balance constraints. Contrary to the results from the common Markov-chain switching algorithm, our findings suggest the evolutionary importance of the small-world hypothesis as a fundamental design principle of complex networks. The approach may help us to determine the biologically meaningful properties that result from evolutionary pressure imposed on metabolism, such as the global impact of local reaction knockouts. Moreover, the approach can be applied to test to what extent novel structural properties can be used to draw biologically meaningful hypothesis or predictions from structure alone.

Background: Reconstruction of genome-scale metabolic networks has resulted in models capable of reproducing experimentally observed biomass yield/growth rates and predicting the effect of alterations in metabolism for biotechnological applications. The existing studies rely on modifying the metabolic network of an investigated organism by removing or inserting reactions taken either from evolutionary similar organisms or from databases of biochemical reactions (e.g., KEGG). A potential disadvantage of these knowledge-driven approaches is that the result is biased towards known reactions, as such approaches do not account for the possibility of including novel enzymes, together with the reactions they catalyze.
Results: Here, we explore the alternative of increasing biomass yield in three model organisms, namely Bacillus subtilis, Escherichia coil, and Hordeum vulgare, by applying small, chemically feasible network modifications. We use the predicted and experimentally confirmed growth rates of the wild-type networks as reference values and determine the effect of inserting mass-balanced, thermodynamically feasible reactions on predictions of growth rate by using flux balance analysis.
Conclusions: While many replacements of existing reactions naturally lead to a decrease or complete loss of biomass production ability, in all three investigated organisms we find feasible modifications which facilitate a significant increase in this biological function. We focus on modifications with feasible chemical properties and a significant increase in biomass yield. The results demonstrate that small modifications are sufficient to substantially alter biomass yield in the three organisms. The method can be used to predict the effect of targeted modifications on the yield of any set of metabolites (e.g., ethanol), thus providing a computational framework for synthetic metabolic engineering.

Kinetic modelling of complex metabolic networks - a central goal of computational systems biology - is currently hampered by the lack of reliable rate equations for the majority of the underlying biochemical reactions and membrane transporters. On the basis of biochemically substantiated evidence that metabolic control is exerted by a narrow set of key regulatory enzymes, we propose here a hybrid modelling approach in which only the central regulatory enzymes are described by detailed mechanistic rate equations, and the majority of enzymes are approximated by simplified (nonmechanistic) rate equations (e.g. mass action, LinLog, Michaelis-Menten and power law) capturing only a few basic kinetic features and hence containing only a small number of parameters to be experimentally determined. To check the reliability of this approach, we have applied it to two different metabolic networks, the energy and redox metabolism of red blood cells, and the purine metabolism of hepatocytes, using in both cases available comprehensive mechanistic models as reference standards. Identification of the central regulatory enzymes was performed by employing only information on network topology and the metabolic data for a single reference state of the network [Grimbs S, Selbig J, Bulik S, Holzhutter HG & Steuer R (2007) Mol Syst Biol3, 146, doi:10.1038/msb4100186]. Calculations of stationary and temporary states under various physiological challenges demonstrate the good performance of the hybrid models. We propose the hybrid modelling approach as a means to speed up the development of reliable kinetic models for complex metabolic networks.

Metabolic networks are characterized by complex interactions and regulatory mechanisms between many individual components. These interactions determine whether a steady state is stable to perturbations. Structural kinetic modeling (SKM) is a framework to analyze the stability of metabolic steady states that allows the study of the system Jacobian without requiring detailed knowledge about individual rate equations. Stability criteria can be derived by generating a large number of structural kinetic models (SK-models) with randomly sampled parameter sets and evaluating the resulting Jacobian matrices. Until now, SKM experiments applied univariate tests to detect the network components with the largest influence on stability. In this work, we present an extended SKM approach relying on supervised machine learning to detect patterns of enzyme-metabolite interactions that act together in an orchestrated manner to ensure stability. We demonstrate its application on a detailed SK-model of the Calvin-Benson cycle and connected pathways. The identified stability patterns are highly complex reflecting that changes in dynamic properties depend on concerted interactions between several network components. In total, we find more patterns that reliably ensure stability than patterns ensuring instability. This shows that the design of this system is strongly targeted towards maintaining stability. We also investigate the effect of allosteric regulators revealing that the tendency to stability is significantly increased by including experimentally determined regulatory mechanisms that have not yet been integrated into existing kinetic models.

Structural kinetic modeling (SKM) enables the analysis of dynamical properties of metabolic networks solely based on topological information and experimental data. Current SKM-based experiments are hampered by the time-intensive process of assigning model parameters and choosing appropriate sampling intervals for MonteCarlo experiments. We introduce a toolbox for the automatic and efficient construction and evaluation of structural kinetic models (SK models). Quantitative and qualitative analyses of network stability properties are performed in an automated manner. We illustrate the model building and analysis process in detailed example scripts that provide toolbox implementations of previously published literature models.

This work presents mathematical and computational approaches to cover various aspects of metabolic network modelling, especially regarding the limited availability of detailed kinetic knowledge on reaction rates. It is shown that precise mathematical formulations of problems are needed i) to find appropriate and, if possible, efficient algorithms to solve them, and ii) to determine the quality of the found approximate solutions. Furthermore, some means are introduced to gain insights on dynamic properties of metabolic networks either directly from the network structure or by additionally incorporating steady-state information. Finally, an approach to identify key reactions in a metabolic networks is introduced, which helps to develop simple yet useful kinetic models. The rise of novel techniques renders genome sequencing increasingly fast and cheap. In the near future, this will allow to analyze biological networks not only for species but also for individuals. Hence, automatic reconstruction of metabolic networks provides itself as a means for evaluating this huge amount of experimental data. A mathematical formulation as an optimization problem is presented, taking into account existing knowledge and experimental data as well as the probabilistic predictions of various bioinformatical methods. The reconstructed networks are optimized for having large connected components of high accuracy, hence avoiding fragmentation into small isolated subnetworks. The usefulness of this formalism is exemplified on the reconstruction of the sucrose biosynthesis pathway in Chlamydomonas reinhardtii. The problem is shown to be computationally demanding and therefore necessitates efficient approximation algorithms. The problem of minimal nutrient requirements for genome-scale metabolic networks is analyzed. Given a metabolic network and a set of target metabolites, the inverse scope problem has as it objective determining a minimal set of metabolites that have to be provided in order to produce the target metabolites. These target metabolites might stem from experimental measurements and therefore are known to be produced by the metabolic network under study, or are given as the desired end-products of a biotechological application. The inverse scope problem is shown to be computationally hard to solve. However, I assume that the complexity strongly depends on the number of directed cycles within the metabolic network. This might guide the development of efficient approximation algorithms. Assuming mass-action kinetics, chemical reaction network theory (CRNT) allows for eliciting conclusions about multistability directly from the structure of metabolic networks. Although CRNT is based on mass-action kinetics originally, it is shown how to incorporate further reaction schemes by emulating molecular enzyme mechanisms. CRNT is used to compare several models of the Calvin cycle, which differ in size and level of abstraction. Definite results are obtained for small models, but the available set of theorems and algorithms provided by CRNT can not be applied to larger models due to the computational limitations of the currently available implementations of the provided algorithms. Given the stoichiometry of a metabolic network together with steady-state fluxes and concentrations, structural kinetic modelling allows to analyze the dynamic behavior of the metabolic network, even if the explicit rate equations are not known. In particular, this sampling approach is used to study the stabilizing effects of allosteric regulation in a model of human erythrocytes. Furthermore, the reactions of that model can be ranked according to their impact on stability of the steady state. The most important reactions in that respect are identified as hexokinase, phosphofructokinase and pyruvate kinase, which are known to be highly regulated and almost irreversible. Kinetic modelling approaches using standard rate equations are compared and evaluated against reference models for erythrocytes and hepatocytes. The results from this simplified kinetic models can simulate acceptably the temporal behavior for small changes around a given steady state, but fail to capture important characteristics for larger changes. The aforementioned approach to rank reactions according to their influence on stability is used to identify a small number of key reactions. These reactions are modelled in detail, including knowledge about allosteric regulation, while all other reactions were still described by simplified reaction rates. These so-called hybrid models can capture the characteristics of the reference models significantly better than the simplified models alone. The resulting hybrid models might serve as a good starting point for kinetic modelling of genome-scale metabolic networks, as they provide reasonable results in the absence of experimental data, regarding, for instance, allosteric regulations, for a vast majority of enzymatic reactions.

Spatiotemporal dynamics of the Calvin cycle multistationarity and symmetry breaking instabilities
(2011)

The possibility of controlling the Calvin cycle has paramount implications for increasing the production of biomass. Multistationarity, as a dynamical feature of systems, is the first obvious candidate whose control could find biotechnological applications. Here we set out to resolve the debate on the multistationarity of the Calvin cycle. Unlike the existing simulation-based studies, our approach is based on a sound mathematical framework, chemical reaction network theory and algebraic geometry, which results in provable results for the investigated model of the Calvin cycle in which we embed a hierarchy of realistic kinetic laws. Our theoretical findings demonstrate that there is a possibility for multistationarity resulting from two sources, homogeneous and inhomogeneous instabilities, which partially settle the debate on multistability of the Calvin cycle. In addition, our tractable analytical treatment of the bifurcation parameters can be employed in the design of validation experiments.

Motivation: Metabolic engineering aims at modulating the capabilities of metabolic networks by changing the activity of biochemical reactions. The existing constraint-based approaches for metabolic engineering have proven useful, but are limited only to reactions catalogued in various pathway databases.
Results: We consider the alternative of designing synthetic strategies which can be used not only to characterize the maximum theoretically possible product yield but also to engineer networks with optimal conversion capability by using a suitable biochemically feasible reaction called 'stoichiometric capacitance'. In addition, we provide a theoretical solution for decomposing a given stoichiometric capacitance over a set of known enzymatic reactions. We determine the stoichiometric capacitance for genome-scale metabolic networks of 10 organisms from different kingdoms of life and examine its implications for the alterations in flux variability patterns. Our empirical findings suggest that the theoretical capacity of metabolic networks comes at a cost of dramatic system's changes.

Robustness of biochemical systems has become one of the central questions in systems biology although it is notoriously difficult to formally capture its multifaceted nature. Maintenance of normal system function depends not only on the stoichiometry of the underlying interrelated components, but also on the multitude of kinetic parameters. Invariant flux ratios, obtained within flux coupling analysis, as well as invariant complex ratios, derived within chemical reaction network theory, can characterize robust properties of a system at steady state. However, the existing formalisms for the description of these invariants do not provide full characterization as they either only focus on the flux-centric or the concentration-centric view. Here we develop a novel mathematical framework which combines both views and thereby overcomes the limitations of the classical methodologies. Our unified framework will be helpful in analyzing biologically important system properties.

Integration of high-throughput data with functional annotation by graph-theoretic methods has been postulated as promising way to unravel the function of unannotated genes. Here, we first review the existing graph-theoretic approaches for automated gene function annotation and classify them into two categories with respect to their relation to two instances of transductive learning on networks - with dynamic costs and with constant costs - depending on whether or not ontological relationship between functional terms is employed. The determined categories allow to characterize the computational complexity of the existing approaches and establish the relation to classical graph-theoretic problems, such as bisection and multiway cut. In addition, our results point out that the ontological form of the structured functional knowledge does not lower the complexity of the transductive learning with dynamic costs - one of the key problems in modern systems biology. The NP-hardness of automated gene annotation renders the development of heuristic or approximation algorithms a priority for additional research.