Structural robustness of metabolic networks with respect to multiple knockouts

We present a generalised framework for analysing structural robustness of metabolic networks, based on the concept of elementary flux modes (EFMs). Extending our earlier study on single knockouts [Wilhelm, T., Behre, J., Schuster, S., 2004. Analysis of structural robustness of metabolic networks. IEE Proc. Syst. Biol. 1(1), 114-120], we are now considering the general case of double and multiple knockouts. The robustness measures are based on the ratio of the number of remaining EFMs after knockout vs. the number of EFMs in the unperturbed situation, averaged over all combinations of knockouts. With the help of simple examples we demonstrate that consideration of multiple knockouts yields additional information going beyond single-knockout results. It is proven that the robustness score decreases as the knockout depth increases.We apply our extended framework to metabolic networks representing amino acid anabolism in Escherichia coli and human hepatocytes, and the central metabolism in human erythrocytes. Moreover, in the E. coli model the two subnetworks synthesising amino acids that are essential and those that are non-essential for humans are studied separately. The results are discussed from an evolutionary viewpoint. We find that E. coli has the most robust metabolism of all the cell types studied here. Considering only the subnetwork of the synthesis of non-essential amino acids, E. coli and the human hepatocyte show about the same robustness.     

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.     

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     

Structural Control of Metabolic Flux

Organisms have to continuously adapt to changing environmental conditions or undergo developmental transitions. To meet the accompanying change in metabolic demands, the molecular mechanisms of adaptation involve concerted interactions which ultimately induce a modification of the metabolic state, which is characterized by reaction fluxes and metabolite concentrations. These state transitions are the effect of simultaneously manipulating fluxes through several reactions. While metabolic control analysis has provided a powerful framework for elucidating the principles governing this orchestrated action to understand metabolic control, its applications are restricted by the limited availability of kinetic information. Here, we introduce structural metabolic control as a framework to examine individual reactions'' potential to control metabolic functions, such as biomass production, based on structural modeling. The capability to carry out a metabolic function is determined using flux balance analysis (FBA). We examine structural metabolic control on the example of the central carbon metabolism of Escherichia coli by the recently introduced framework of functional centrality (FC). This framework is based on the Shapley value from cooperative game theory and FBA, and we demonstrate its superior ability to assign “share of control” to individual reactions with respect to metabolic functions and environmental conditions. A comparative analysis of various scenarios illustrates the usefulness of FC and its relations to other structural approaches pertaining to metabolic control. We propose a Monte Carlo algorithm to estimate FCs for large networks, based on the enumeration of elementary flux modes. We further give detailed biological interpretation of FCs for production of lactate and ATP under various respiratory conditions.     

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.     

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.     

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.     

The fractional contributions of elementary modes to the metabolism of Escherichia coli and their estimation from reaction entropies

The metabolism of a cell can be viewed as a weighted sum of elementary modes. Due to the multiplicity of modes the identification of the individual weights represents a non-trivial problem. To enable the determination of weighting factors we have identified and implemented two gene deletions in combination with defined growth conditions that limit the metabolism from 4374 original elementary modes to 24 elementary modes for a non-PHB synthesizing control and 40 modes for a PHB synthesizing strain. These remaining modes can be further grouped into five families that have the same overall stoichiometry. Thus, the complexity of the problem is significantly reduced, and weighting factors for each family of modes could be determined from the measurement of accumulation rates of metabolites. Moreover, it is shown that individual weights are inversely correlated with the entropy generated by the operation of the used pathways defined in elementary modes. This suggests that evolution developed cellular regulatory patterns that permit diversity of pathways while favoring efficient pathways with low entropy generation. Furthermore, such correlation provides a rational way of estimating metabolic fluxes based on the thermodynamic properties of elementary modes. This is demonstrated with an example in which experimentally determined, intracellular fluxes are shown to be highly correlated with fluxes computed based on elementary modes and reaction entropies. The analysis suggests that the set of elementary modes can be interpreted analogous to a metabolic ensemble of quantum states of a macroscopic system.     

Stoichiometric design of metabolic networks: multifunctionality,clusters, optimization,weak and strong robustness

Starting from a limited set of reactions describing changes in the carbon skeleton of biochemical compounds complete sets of metabolic networks are constructed. The networks are characterized by the number and types of participating reactions. Elementary networks are defined by the condition that a specific chemical conversion can be performed by a set of given reactions and that this ability will be lost by elimination of any of these reactions. Groups of networks are identified with respect to their ability to perform a certain number of metabolic conversions in an elementary way which are called the network’s functions. The number of the network functions defines the degree of multifunctionality. Transitions between networks and mutations of networks are defined by exchanges of single reactions. Different mutations exist such as gain or loss of function mutations and neutral mutations. Based on these mutations neighbourhood relations between networks are established which are described in a graph theoretical way. Basic properties of these graphs are determined such as diameter, connectedness, distance distribution of pairs of vertices. A concept is developed to quantify the robustness of networks against changes in their stoichiometry where we distinguish between strong and weak robustness. Evolutionary algorithms are applied to study the development of network populations under constant and time dependent environmental conditions. It is shown that the populations evolve toward clusters of networks performing a common function and which are closely neighboured. Under changing environmental conditions multifunctional networks prove to be optimal and will be selected.     

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.     

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.     

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.     

Application of hybrid cybernetic model in simulating myeloma cell culture co-consuming glucose and glutamine with mixed consumption patterns

A dynamic model called hybrid cybernetic model (HCM) based on structured metabolic network is established for simulating mammalian cell metabolism featured with partially substitutable and partially complementary consumption patterns of two substrates, glucose and glutamine. Benefiting from the application of elementary mode analysis (EMA), the complicated metabolic network is decomposed into elementary modes (EMs) facilitating the employment of the hybrid cybernetic framework to investigate the external and internal flux distribution and the regulation mechanism among them. According to different substrate combination, two groups of EMs are obtained, i.e., EMs associated with glucose uptake and simultaneous uptake of glucose and glutamine. Uptake fluxes through various EMs are coupled together via cybernetic variables to maximize substrate uptake. External fluxes and internal fluxes could be calculated and estimated respectively, by the combination of the stoichiometrics of metabolic networks and fluxes through regulated EMs. The model performance is well validated via three sets of experimental data. Through parameter identification of limited number of experimental data, other external metabolites are precisely predicted. The obtained kinetic parameters of three experimental cultures have similar values, which indicates the robustness of the model. Furthermore, the prediction performance of the model is successfully validated based on identified parameters.     

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.     

Evolution under Fluctuating Environments Explains Observed Robustness in Metabolic Networks

A high level of robustness against gene deletion is observed in many organisms. However, it is still not clear which biochemical features underline this robustness and how these are acquired during evolution. One hypothesis, specific to metabolic networks, is that robustness emerges as a byproduct of selection for biomass production in different environments. To test this hypothesis we performed evolutionary simulations of metabolic networks under stable and fluctuating environments. We find that networks evolved under the latter scenario can better tolerate single gene deletion in specific environments. Such robustness is underlined by an increased number of independent fluxes and multifunctional enzymes in the evolved networks. Observed robustness in networks evolved under fluctuating environments was “apparent,” in the sense that it decreased significantly as we tested effects of gene deletions under all environments experienced during evolution. Furthermore, when we continued evolution of these networks under a stable environment, we found that any robustness they had acquired was completely lost. These findings provide evidence that evolution under fluctuating environments can account for the observed robustness in metabolic networks. Further, they suggest that organisms living under stable environments should display lower robustness in their metabolic networks, and that robustness should decrease upon switching to more stable environments.