Two approaches for metabolic pathway analysis?

Metabolic pathway analysis is becoming increasingly important for assessing inherent network properties in (reconstructed) biochemical reaction networks. Of the two most promising concepts for pathway analysis, one relies on elementary flux modes and the other on extreme pathways. These concepts are closely related because extreme pathways are a subset of elementary modes. Here, the common features, differences and applicability of these concepts are discussed. Assessing metabolic systems by the set of extreme pathways can, in general, give misleading results owing to the exclusion of possibly important routes. However, in certain network topologies, the sets of elementary modes and extreme pathways coincide. This is quite often the case in realistic applications. In our opinion, the unification of both approaches into one common framework for metabolic pathway analysis is necessary and achievable.     

Detection of elementary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering

Rational metabolic engineering requires powerful theoretical methods such as pathway analysis, in which the topology of metabolic networks is considered. All metabolic capabilities in steady states are composed of elementary flux modes, which are minimal sets of enzymes that can each generate valid steady states. The modes of the fructose-2,6-bisphosphate cycle, the combined tricarboxylic-acid-glyoxylate-shunt system and tryptophan synthesis are used here for illustration. This approach can be used for many biotechnological applications such as increasing the yield of a product, channelling a product into desired pathways and in functional reconstruction from genomic data.     

Minimal cut sets in biochemical reaction networks

MOTIVATION: Structural studies of metabolic networks yield deeper insight into topology, functionality and capabilities of the metabolisms of different organisms. Here, we address the analysis of potential failure modes in metabolic networks whose occurrence will render the network structurally incapable of performing certain functions. Such studies will help to identify crucial parts in the network structure and to find suitable targets for repressing undesired metabolic functions. RESULTS: We introduce the concept of minimal cut sets for biochemical networks. A minimal cut set (MCS) is a minimal (irreducible) set of reactions in the network whose inactivation will definitely lead to a failure in certain network functions. We present an algorithm which enables the computation of the MCSs in a given network related to user-defined objective reactions. This algorithm operates on elementary modes. A number of potential applications are outlined, including network verifications, phenotype predictions, assessing structural robustness and fragility, metabolic flux analysis and target identification in drug discovery. Applications are illustrated by the MCSs in the central metabolism of Escherichia coli for growth on different substrates. AVAILABILITY: Computation and analysis of MCSs is an additional feature of the FluxAnalyzer (freely available for academic users upon request, special contracts for industrial companies; see web page below). Supplementary information: http://www.mpi-magdeburg.mpg.de/projects/fluxanalyzer     

Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems

The elucidation of organism-scale metabolic networks necessitates the development of integrative methods to analyze and interpret the systemic properties of cellular metabolism. A shift in emphasis from single metabolic reactions to systemically defined pathways is one consequence of such an integrative analysis of metabolic systems. The constraints of systemic stoichiometry, and limited thermodynamics have led to the definition of the flux space within the context of convex analysis. The flux space of the metabolic system, containing all allowable flux distributions, is constrained to a convex polyhedral cone in a high-dimensional space. From metabolic pathway analysis, the edges of the high-dimensional flux cone are vectors that correspond to systemically defined "extreme pathways" spanning the capabilities of the system. The addition of maximum flux capacities of individual metabolic reactions serves to further constrain the flux space and has led to the development of flux balance analysis using linear optimization to calculate optimal flux distributions. Here we provide the precise theoretical connections between pathway analysis and flux balance analysis allowing for their combined application to study integrated metabolic function. Shifts in metabolic behavior are calculated using linear optimization and are then interpreted using the extreme pathways to demonstrate the concept of pathway utilization. Changes to the reaction network, such as the removal of a reaction, can lead to the generation of suboptimal phenotypes that can be directly attributed to the loss of pathway function and capabilities. Optimal growth phenotypes are calculated as a function of environmental variables, such as the availability of substrate and oxygen, leading to the definition of phenotypic phase planes. It is illustrated how optimality properties of the computed flux distributions can be interpreted in terms of the extreme pathways. Together these developments are applied to an example network and to core metabolism of Escherichia coli demonstrating the connections between the extreme pathways, optimal flux distributions, and phenotypic phase planes. The consequences of changing environmental and internal conditions of the network are examined for growth on glucose and succinate in the face of a variety of gene deletions. The convergence of the calculation of optimal phenotypes through linear programming and the definition of extreme pathways establishes a different perspective for the understanding of how a defined metabolic network is best used under different environmental and internal conditions or, in other words, a pathway basis for the interpretation of the metabolic reaction norm.     

ACoM: A classification method for elementary flux modes based on motif finding

Elementary flux mode analysis is a powerful tool for the theoretical study of metabolic networks. However, when the networks are complex, the determination of elementary flux modes leads to combinatorial explosion of their number which prevents from drawing simple conclusions from their analysis. To deal with this problem we have developed a method based on the Agglomeration of Common Motifs (ACoM) for classifying elementary flux modes. We applied this algorithm to describe the decomposition into elementary flux modes of the central carbon metabolism in Bacillus subtilis and of the yeast mitochondrial energy metabolism. ACoM helps to give biological meaning to the different elementary flux modes and to the relatedness between reactions. ACoM, which can be viewed as a bi-clustering method, can be of general use for sets of vectors with values 0, +1 or −1.     

Interplay between Constraints,Objectives, and Optimality for Genome-Scale Stoichiometric Models

High-throughput data generation and genome-scale stoichiometric models have greatly facilitated the comprehensive study of metabolic networks. The computation of all feasible metabolic routes with these models, given stoichiometric, thermodynamic, and steady-state constraints, provides important insights into the metabolic capacities of a cell. How the feasible metabolic routes emerge from the interplay between flux constraints, optimality objectives, and the entire metabolic network of a cell is, however, only partially understood. We show how optimal metabolic routes, resulting from flux balance analysis computations, arise out of elementary flux modes, constraints, and optimization objectives. We illustrate our findings with a genome-scale stoichiometric model of Escherichia coli metabolism. In the case of one flux constraint, all feasible optimal flux routes can be derived from elementary flux modes alone. We found up to 120 million of such optimal elementary flux modes. We introduce a new computational method to compute the corner points of the optimal solution space fast and efficiently. Optimal flux routes no longer depend exclusively on elementary flux modes when we impose additional constraints; new optimal metabolic routes arise out of combinations of elementary flux modes. The solution space of feasible metabolic routes shrinks enormously when additional objectives---e.g. those related to pathway expression costs or pathway length---are introduced. In many cases, only a single metabolic route remains that is both feasible and optimal. This paper contributes to reaching a complete topological understanding of the metabolic capacity of organisms in terms of metabolic flux routes, one that is most natural to biochemists and biotechnologists studying and engineering metabolism.     

Calculability analysis in underdetermined metabolic networks illustrated by a model of the central metabolism in purple nonsulfur bacteria

Metabolite balancing has turned out to be a powerful computational tool in metabolic engineering. However, the linear equation systems occurring in this analysis are often underdetermined. If it is difficult or impossible to find the missing constraints, it is nevertheless feasible in some cases to determine the values of a subset of the unknown rates. Here, a procedure for finding out which reaction rates can be uniquely calculated in underdetermined metabolic networks and computing these rates is given. The method is based on the null space to the stoichiometry matrix corresponding to the reactions with unknown rates. It is shown that this method is considerably easier to handle than an algorithm given previously (Van der Heijden et al., 1994a). Furthermore, a useful elementary representation of the null space is presented which is closely related with the elementary flux modes. This unique representation is central to a more general approach to observability/calculability analysis. In particular, it allows one to find, in an easy way, those sets of measurable rates that enable a calculation of a certain unknown rate. Besides, rates which are never calculable by metabolite balancing may be easily detected by this method. The applicability of these methods is illustrated by a model of the central metabolism in purple nonsulfur bacteria. The photoheterotrophic growth of these representatives of anoxygenic photosynthetic bacteria is stoichiometrically analyzed. Interesting metabolic constraints caused by the necessary balancing of NADPH can be detected in a highly underdetermined system. This is, to our knowledge, the first application of stoichiometric analysis to the metabolic network in this bacteria group using metabolite balancing techniques. A new software tool, the FluxAnalyzer, is introduced. It allows quantitative and structural analysis of metabolic networks in a graphical user interface.     

The geometry of the flux cone of a metabolic network

The analysis of metabolic networks has become a major topic in biotechnology in recent years. Applications range from the enhanced production of selected outputs to the prediction of genotype-phenotype relationships. The concepts used are based on the assumption of a pseudo steady-state of the network, so that for each metabolite inputs and outputs are balanced. The stoichiometric network analysis expands the steady state into a combination of nonredundant subnetworks with positive coefficients called extremal currents. Based on the unidirectional representation of the system these subnetworks form a convex cone in the flux-space. A modification of this approach allowing for reversible reactions led to the definition of elementary modes. Extreme pathways are obtained with the same method but splitting up internal reactions into forward and backward rates. In this study, we explore the relationship between these concepts. Due to the combinatorial explosion of the number of elementary modes in large networks, we promote a further set of metabolic routes, which we call the minimal generating set. It is the smallest subset of elementary modes required to describe all steady states of the system. For large-scale networks, the size of this set is of several magnitudes smaller than that of elementary modes and of extreme pathways.     

Metabolic network properties help assign weights to elementary modes to understand physiological flux distributions

MOTIVATION: Elementary modes (EMs) analysis has been well established. The existing methodologies for assigning weights to EMs cannot be directly applied for large-scale metabolic networks, since the tremendous number of modes would make the computation a time-consuming or even an impossible mission. Therefore, developing more efficient methods to deal with large set of EMs is urgent. RESULT: We develop a method to evaluate the performance of employing a subset of the elementary modes to reconstruct a real flux distribution by using the relative error between the real flux vector and the reconstructed one as an indicator. We have found a power function relationship between the decrease of relative error and the increase of the number of the selecting EMs, and a logarithmic relationship between the increases of the number of non-zero weighted EMs and that of the number of the selecting EMs. Our discoveries show that it is possible to reconstruct a given flux distribution by a selected subset of EMs from a large metabolic network and furthermore, they help us identify the 'governing modes' to represent the cellular metabolism for such a condition.     

Flux modules in metabolic networks

The huge number of elementary flux modes in genome-scale metabolic networks makes analysis based on elementary flux modes intrinsically difficult. However, it has been shown that the elementary flux modes with optimal yield often contain highly redundant information. The set of optimal-yield elementary flux modes can be compressed using modules. Up to now, this compression was only possible by first enumerating the whole set of all optimal-yield elementary flux modes. We present a direct method for computing modules of the thermodynamically constrained optimal flux space of a metabolic network. This method can be used to decompose the set of optimal-yield elementary flux modes in a modular way and to speed up their computation. In addition, it provides a new form of coupling information that is not obtained by classical flux coupling analysis. We illustrate our approach on a set of model organisms.     

Finding minimal generating set for metabolic network with reversible pathways

Elementary flux modes give a mathematical representation of metabolic pathways in metabolic networks satisfying the constraint of non-decomposability. The large cost of their computation shifts attention to computing a minimal generating set which is a conically independent subset of elementary flux modes. When a metabolic network has reversible reactions and also admits a reversible pathway, the minimal generating set is not unique. A theoretical development and computational framework is provided which outline how to compute the minimal generating set in this case. The method is based on combining existing software to compute the minimal generating set for a “pointed cone” together with standard software to compute the Reduced Row Echelon Form.     

Combinatorial complexity of pathway analysis in metabolic networks

Elementary flux mode analysis is a promising approach for a pathway-oriented perspective of metabolic networks. However, in larger networks it is hampered by the combinatorial explosion of possible routes. In this work we give some estimations on the combinatorial complexity including theoretical upper bounds for the number of elementary flux modes in a network of a given size. In a case study, we computed the elementary modes in the central metabolism of Escherichia coli while utilizing four different substrates. Interestingly, although the number of modes occurring in this complex network can exceed half a million, it is still far below the upper bound. Hence, to a certain extent, pathway analysis of central catabolism is feasible to assess network properties such as flexibility and functionality.     

Pathway classification of TCA cycle

The structural analysis of large metabolic networks exhibits a combinatorial explosion of elementary modes. A new method of classification has been developed [called aggregation around common motif (ACoM)], which groups elementary modes into classes with similar substructures. This method is applied to the tricarboxylic acid cycle and metabolite carriers. The analysis of this network evidences a great number of elementary flux modes (204) despite the low number of reactions (23). The ACoM is used to class these elementary modes in a low number of sets (8) with biological meanings.     

Computation of elementary modes: a unifying framework and the new binary approach

Background  

Metabolic pathway analysis has been recognized as a central approach to the structural analysis of metabolic networks. The concept of elementary (flux) modes provides a rigorous formalism to describe and assess pathways and has proven to be valuable for many applications. However, computing elementary modes is a hard computational task. In recent years we assisted in a multiplication of algorithms dedicated to it. We require a summarizing point of view and a continued improvement of the current methods.     

Analyzing molecular reaction networks

Pathways are typically the central concept in the analysis of biochemical reaction networks. A pathway can be interpreted as a chain of enzymatical reactions performing a specific biological function. A common way to study metabolic networks are minimal pathways that can operate at steady state called elementary modes. The theory of chemical organizations has recently been used to decompose biochemical networks into algebraically closed and self-maintaining subnetworks termed organizations. The aim of this paper is to elucidate the relation between these two concepts. Whereas elementary modes represent the boundaries of the potential behavior of the network, organizations define metabolite compositions that are likely to be present in biological feasible situations. Hence, steady state organizations consist of combinations of elementary modes. On the other hand, it is possible to assign a unique (and possibly empty) set of organizations to each elementary mode, indicating the metabolites accompanying the active pathway in a feasible steady state.