Elementary vectors and conformal sums in polyhedral geometry and their relevance for metabolic pathway analysis

A fundamental result in metabolic pathway analysis states that every flux mode can be decomposed into a sum of elementary modes. However, only a decomposition without cancelations is biochemically meaningful, since a reversible reaction cannot have different directions in the contributing elementary modes. This essential requirement has been largely overlooked by the metabolic pathway community. Indeed, every flux mode can be decomposed into elementary modes without cancelations. The result is an immediate consequence of a theorem by Rockafellar which states that every element of a linear subspace is a conformal sum (a sum without cancelations) of elementary vectors (support-minimal vectors). In this work, we extend the theorem, first to "subspace cones" and then to general polyhedral cones and polyhedra. Thereby, we refine Minkowski's and Carath\'eodory's theorems, two fundamental results in polyhedral geometry. We note that, in general, elementary vectors need not be support-minimal, in fact, they are conformally non-decomposable and form a unique minimal set of conformal generators. Our treatment is mathematically rigorous, but suitable for systems biologists, since we give self-contained proofs for our results and use concepts motivated by metabolic pathway analysis. In particular, we study cones defined by linear subspaces and nonnegativity conditions - like the flux cone - and use them to analyze general polyhedral cones and polyhedra. Finally, we review applications of elementary vectors and conformal sums in metabolic pathway analysis

Constraint-based Analysis of Substructures of Metabolic Networks

Constraint-based methods (CBMs) are promising tools for the analysis of metabolic networks, as they do not require detailed knowledge of the biochemical reactions. Some of these methods only need information about the stoichiometric coefficients of the reactions and their reversibility types, i.e., constraints for steady-state conditions. Nevertheless, CBMs have their own limitations. For example, these methods may be sensitive to missing information in the models. Additionally, they may be slow for the analysis of genome-scale metabolic models. As a result, some studies prefer to consider substructures of networks, instead of complete models. Some other studies have focused on better implementations of the CBMs. In Chapter 2, the sensitivity of flux coupling analysis (FCA) to missing reactions is studied. Genome-scale metabolic reconstructions are comprehensive, yet incomplete, models of real-world metabolic networks. While FCA has proved an appropriate method for analyzing metabolic relationships and for detecting functionally related reactions in such models, little is known about the impact of missing reactions on the accuracy of FCA. Note that having missing reactions is equivalent to deleting reactions, or to deleting columns from the stoichiometric matrix. Based on an alternative characterization of flux coupling relations using elementary flux modes, we study the changes that flux coupling relations may undergo due to missing reactions. In particular, we show that two uncoupled reactions in a metabolic network may be detected as directionally, partially or fully coupled in an incomplete version of the same network. Even a single missing reaction can cause significant changes in flux coupling relations. In case of two consecutive E. coli genome-scale networks, many fully-coupled reaction pairs in the incomplete network become directionally coupled or even uncoupled in the more complete reconstruction. In this context, we found gene expression correlation values being significantly higher for the pairs that remained fully coupled than for the uncoupled or directionally coupled pairs. Our study clearly suggests that FCA results are indeed sensitive to missing reactions. Since the currently available genome-scale metabolic models are incomplete, we advise to use FCA results with care. In Chapter 3, a different, but related problem is considered. Due to the large size of genome-scale metabolic networks, some studies suggest to analyze subsystems, instead of original genome-scale models. Note that analysis of a subsystem is equivalent to deletion of some rows from the stoichiometric matrix, or identically, assuming some internal metabolites to be external. We show mathematically that analysis of a subsystem instead of the original model can lead the flux coupling relations to undergo certain changes. In particular, a pair of (fully, partially or directionally) coupled reactions may be detected as uncoupled in the chosen subsystem. Interestingly, this behavior is the opposite of the flux coupling changes that may happen due to the existence of missing reactions, or equivalently, deletion of reactions. We also show that analysis of organelle subsystems has relatively little influence on the results of FCA, and therefore, many of these subsystems may be studied independent of the rest of the network. In Chapter 4, we introduce a rapid FCA method, which is appropriate for genome-scale networks. Previously, several approaches for FCA have been proposed in the literature, namely flux coupling finder algorithm, FCA based on minimal metabolic behaviors, and FCA based on elementary flux patterns. To the best of our knowledge none of these methods are available as a freely available software. Here, we introduce a new FCA algorithm FFCA (Feasibility-based Flux Coupling Analysis). This method is based on checking the feasibility of a system of linear inequalities. We show on a set of benchmarks that for genome-scale networks FFCA is faster than other existing FCA methods. Using FFCA, flux coupling analysis of genome-scale networks of S. cerevisiae and E. coli can be performed in a few hours on a normal PC. A corresponding software tool is freely available for non-commercial use. In Chapter 5, we introduce a new concept which can be useful in the analysis of fluxes in network substructures. Analysis of elementary modes (EMs) is proven to be a powerful CBM in the study of metabolic networks. However, enumeration of EMs is a hard computational task. Additionally, due to their large numbers, one cannot simply use them as an input for subsequent analyses. One possibility is to restrict the analysis to a subset of interesting reactions, rather than the whole network. However, analysis of an isolated subnetwork can result in finding incorrect EMs, i.e. the ones which are not part of any steady-state flux distribution in the original network. The ideal set of vectors to describe the usage of reactions in a subnetwork would be the set of all EMs projected onto the subset of interesting reactions. Recently, the concept of “elementary flux patterns” (EFPs) has been proposed. Each EFP is a subset of the support (i.e. non-zero elements) of at least one EM. In the present work, we introduce the concept of ProCEMs (Projected Cone Elementary Modes). The ProCEM set can be computed by projecting the flux cone onto the lower-dimensional subspace and enumerating the extreme rays of the projected cone. In contrast to EFPs, ProCEMs are not merely a set of reactions, but from the mathematical point of view they are projected EMs. We additionally prove that the set of EFPs is included in the set of ProCEM supports. Finally, ProCEMs and EFPs are compared in the study of substructures in biological networks

Elementary approaches to microbial growth rate maximisation

This thesis, called Elementary approaches to microbial growth rate maximisation, reports on a theoretical search for principles underlying single cell growth, in particular for microbial species that are selected for fast growth rates. First, the optimally growing cell is characterised in terms of its elementary modes. We prove an extremum principle: a cell that maximises a metabolic rate uses few Elementary Flux Modes (EFMs, the minimal pathways that support steady-state metabolism). The number of active EFMs is bounded by the number of growth-limiting constraints. Later, this extremum principle is extended in a theory that explicitly accounts for self-fabrication. For this, we had to define the elementary modes that underlie balanced self-fabrication: minimal self-supporting sets of expressed enzymes that we call Elementary Growth Modes (EGMs). It turns out that many of the results for EFMs can be extended to their more general self-fabrication analogue. Where the above extremum principles tell us that few elementary modes are used by a rate-maximising cell, it does not tell us how the cell can find them. Therefore, we also search for an elementary adaptation method. It turns out that stochastic phenotype switching with growth rate dependent switching rates provides an adaptation mechanism that is often competitive with more conventional regulatory-circuitry based mechanisms. The derived theory is applied in two ways. First, the extremum principles are used to review the mathematical fundaments of all optimisation-based explanations of overflow metabolism. Second, a computational tool is presented that enumerates Elementary Conversion Modes. These elementary modes can be computed for larger networks than EFMs and EGMs, and still provide an overview of the metabolic capabilities of an organism