The steady-state assumption in oscillating and growing systems.

The steady-state assumption, which states that the production and consumption of metabolites inside the cell are balanced, is one of the key aspects that makes an efficient analysis of genome-scale metabolic networks possible. It can be motivated from two different perspectives. In the time-scales perspective, we use the fact that metabolism is much faster than other cellular processes such as gene expression. Hence, the steady-state assumption is derived as a quasi-steady-state approximation of the metabolism that adapts to the changing cellular conditions. In this article we focus on the second perspective, stating that on the long run no metabolite can accumulate or deplete. In contrast to the first perspective it is not immediately clear how this perspective can be captured mathematically and what assumptions are required to obtain the steady-state condition. By presenting a mathematical framework based on the second perspective we demonstrate that the assumption of steady-state also applies to oscillating and growing systems without requiring quasi-steady-state at any time point. However, we also show that the average concentrations may not be compatible with the average fluxes. In summary, we establish a mathematical foundation for the steady-state assumption for long time periods that justifies its successful use in many applications. Furthermore, this mathematical foundation also pinpoints unintuitive effects in the integration of metabolite concentrations using nonlinear constraints into steady-state models for long time periods.

Hierarchical decomposition of metabolic networks using k-modules.

The optimal solutions obtained by flux balance analysis (FBA) are typically not unique. Flux modules have recently been shown to be a very useful tool to simplify and decompose the space of FBA-optimal solutions. Since yield-maximization is sometimes not the primary objective encountered in vivo, we are also interested in understanding the space of sub-optimal solutions. Unfortunately, the flux modules are too restrictive and not suited for this task. We present a generalization, called k-module, which compensates the limited applicability of flux modules to the space of sub-optimal solutions. Intuitively, a k-module is a sub-network with low connectivity to the rest of the network. Recursive application of k-modules yields a hierarchical decomposition of the metabolic network, which is also known as branch decomposition in matroid theory. In particular, decompositions computed by existing methods, like the null-space-based approach, introduced by Poolman et al. [(2007) J. Theor. Biol. 249: , 691-705] can be interpreted as branch decompositions. With k-modules we can now compare alternative decompositions of metabolic networks to the classical sub-systems of glycolysis, tricarboxylic acid (TCA) cycle, etc. They can be used to speed up algorithmic problems [theoretically shown for elementary flux modes (EFM) enumeration] and have the potential to present computational solutions in a more intuitive way independently from the classical sub-systems.

Obstructions to Sampling Qualitative Properties.

BACKGROUND: Sampling methods have proven to be a very efficient and intuitive method to understand properties of complicated spaces that cannot easily be computed using deterministic methods. Therefore, sampling methods became a popular tool in the applied sciences. RESULTS: Here, we show that sampling methods are not an appropriate tool to analyze qualitative properties of complicated spaces unless RP = NP. We illustrate these results on the example of the thermodynamically feasible flux space of genome-scale metabolic networks and show that with artificial centering hit and run (ACHR) not all reactions that can have variable flux rates are sampled with variables flux rates. In particular a uniform sample of the flux space would not sample the flux variabilities completely. CONCLUSION: We conclude that unless theoretical convergence results exist, qualitative results obtained from sampling methods should be considered with caution and if possible double checked using a deterministic method.

Fast flux module detection using matroid theory.

Flux balance analysis (FBA) is one of the most often applied methods on genome-scale metabolic networks. Although FBA uniquely determines the optimal yield, the pathway that achieves this is usually not unique. The analysis of the optimal-yield flux space has been an open challenge. Flux variability analysis is only capturing some properties of the flux space, while elementary mode analysis is intractable due to the enormous number of elementary modes. However, it has been found by Kelk et al. (2012) that the space of optimal-yield fluxes decomposes into flux modules. These decompositions allow a much easier but still comprehensive analysis of the optimal-yield flux space. Using the mathematical definition of module introduced by Müller and Bockmayr (2013b), we discovered useful connections to matroid theory, through which efficient algorithms enable us to compute the decomposition into modules in a few seconds for genome-scale networks. Using that every module can be represented by one reaction that represents its function, in this article, we also present a method that uses this decomposition to visualize the interplay of modules. We expect the new method to replace flux variability analysis in the pipelines for metabolic networks.

Generic flux coupling analysis.

Flux coupling analysis (FCA) has become a useful tool for aiding metabolic reconstructions and guiding genetic manipulations. Originally, it was introduced for constraint-based models of metabolic networks that are based on the steady-state assumption. Recently, we have shown that the steady-state assumption can be replaced by a weaker lattice-theoretic property related to the supports of metabolic fluxes. In this paper, we further extend our approach and develop an efficient algorithm for generic flux coupling analysis that works with any kind of qualitative pathway model. We illustrate our method by thermodynamic flux coupling analysis (tFCA), which allows studying steady-state metabolic models with loop-law thermodynamic constraints. These models do not satisfy the lattice-theoretic properties required in our previous work. For a selection of genome-scale metabolic network reconstructions, we discuss both theoretically and practically, how thermodynamic constraints strengthen the coupling results that can be obtained with classical FCA. A prototype implementation of tFCA is available at http://hoverboard.io/L4FC.