Kinetic constraints for formation of steady states in biochemical networks

The constraint-based analysis has emerged as a useful tool for analysis of biochemical networks. This work introduces the concept of kinetic constraints. It is shown that maximal reaction rates are appropriate constraints only for isolated enzymatic reactions. For biochemical networks, it is revealed that constraints for formation of a steady state require specific relationships between maximal reaction rates of all enzymes. The constraints for a branched network are significantly different from those for a cyclic network. Moreover, the constraints do not require Michaelis-Menten constants for most enzymes, and they only require the constants for the enzymes at the branching or cyclic point. Reversibility of reactions at system boundary or branching point may significantly impact on kinetic constraints. When enzymes are regulated, regulations may impose severe kinetic constraints for the formation of steady states. As the complexity of a network increases, kinetic constraints become more severe. In addition, it is demonstrated that kinetic constraints for networks with co-regulation can be analyzed using the approach. In general, co-regulation enhances the constraints and therefore larger fluctuations in fluxes can be accommodated in the networks with co-regulation. As a first example of the application, we derive the kinetic constraints for an actual network that describes sucrose accumulation in the sugar cane culm, and confirm their validity using numerical simulations.     

Systemic metabolic reactions are obtained by singular value decomposition of genome-scale stoichiometric matrices

Genome-scale metabolic networks can be reconstructed. The systemic biochemical properties of these networks can now be studied. Here, genome-scale reconstructed metabolic networks were analysed using singular value decomposition (SVD). All the individual biochemical conversions contained in a reconstructed metabolic network are described by a stoichiometric matrix (S). SVD of S led to the definition of the underlying modes that characterize the overall biochemical conversions that take place in a network and rank-ordered their importance. The modes were shown to correspond to systemic biochemical reactions and they could be used to identify the groups and clusters of individual biochemical reactions that drive them. Comparative analysis of the Escherichia coli, Haemophilus influenzae, and Helicobacter pylori genome-scale metabolic networks showed that the four dominant modes in all three networks correspond to: (1) the conversion of ATP to ADP, (2) redox metabolism of NADP, (3) proton-motive force, and (4) inorganic phosphate metabolism. The sets of individual metabolic reactions deriving these systemic conversions, however, differed among the three organisms. Thus, we can now define systemic metabolic reactions, or eigen-reactions, for the study of systems biology of metabolism and have a basis for comparing the overall properties of genome-specific metabolic networks.     

Sequence-based Network Completion Reveals the Integrality of Missing Reactions in Metabolic Networks

Genome-scale metabolic models are central in connecting genotypes to metabolic phenotypes. However, even for well studied organisms, such as Escherichia coli, draft networks do not contain a complete biochemical network. Missing reactions are referred to as gaps. These gaps need to be filled to enable functional analysis, and gap-filling choices influence model predictions. To investigate whether functional networks existed where all gap-filling reactions were supported by sequence similarity to annotated enzymes, four draft networks were supplemented with all reactions from the Model SEED database for which minimal sequence similarity was found in their genomes. Quadratic programming revealed that the number of reactions that could partake in a gap-filling solution was vast: 3,270 in the case of E. coli, where 72% of the metabolites in the draft network could connect a gap-filling solution. Nonetheless, no network could be completed without the inclusion of orphaned enzymes, suggesting that parts of the biochemistry integral to biomass precursor formation are uncharacterized. However, many gap-filling reactions were well determined, and the resulting networks showed improved prediction of gene essentiality compared with networks generated through canonical gap filling. In addition, gene essentiality predictions that were sensitive to poorly determined gap-filling reactions were of poor quality, suggesting that damage to the network structure resulting from the inclusion of erroneous gap-filling reactions may be predictable.     

Candidate states of Helicobacter pylori's genome-scale metabolic network upon application of "loop law" thermodynamic constraints

Constraint-based modeling has proven to be a useful tool in the analysis of biochemical networks. To date, most studies in this field have focused on the use of linear constraints, resulting from mass balance and capacity constraints, which lead to the definition of convex solution spaces. One additional constraint arising out of thermodynamics is known as the "loop law" for reaction fluxes, which states that the net flux around a closed biochemical loop must be zero because no net thermodynamic driving force exists. The imposition of the loop-law can lead to nonconvex solution spaces making the analysis of the consequences of its imposition challenging. A four-step approach is developed here to apply the loop-law to study metabolic network properties: 1), determine linear equality constraints that are necessary (but not necessarily sufficient) for thermodynamic feasibility; 2), tighten V(max) and V(min) constraints to enclose the remaining nonconvex space; 3), uniformly sample the convex space that encloses the nonconvex space using standard Monte Carlo techniques; and 4), eliminate from the resulting set all solutions that violate the loop-law, leaving a subset of steady-state solutions. This subset of solutions represents a uniform random sample of the space that is defined by the additional imposition of the loop-law. This approach is used to evaluate the effect of imposing the loop-law on predicted candidate states of the genome-scale metabolic network of Helicobacter pylori.     

Using the topology of metabolic networks to predict viability of mutant strains

Understanding the relationships between the structure (topology) and function of biological networks is a central question of systems biology. The idea that topology is a major determinant of systems function has become an attractive and highly disputed hypothesis. Although structural analysis of interaction networks demonstrates a correlation between the topological properties of a node (protein, gene) in the network and its functional essentiality, the analysis of metabolic networks fails to find such correlations. In contrast, approaches utilizing both the topology and biochemical parameters of metabolic networks, e.g., flux balance analysis, are more successful in predicting phenotypes of knockout strains. We reconcile these seemingly conflicting results by showing that the topology of the metabolic networks of both Escherichia coli and Saccharomyces cerevisiae are, in fact, sufficient to predict the viability of knockout strains with accuracy comparable to flux balance analysis on large, unbiased mutant data sets. This surprising result is obtained by introducing a novel topology-based measure of network transport: synthetic accessibility. We also show that other popular topology-based characteristics such as node degree, graph diameter, and node usage (betweenness) fail to predict the viability of E. coli mutant strains. The success of synthetic accessibility demonstrates its ability to capture the essential properties of the metabolic network, such as the branching of chemical reactions and the directed transport of material from inputs to outputs. Our results strongly support a link between the topology and function of biological networks and, in agreement with recent genetic studies, emphasize the minimal role of flux rerouting in providing robustness of mutant strains.     

Identification of biochemical network modules based on shortest retroactive distances

Modularity analysis offers a route to better understand the organization of cellular biochemical networks as well as to derive practically useful, simplified models of these complex systems. While there is general agreement regarding the qualitative properties of a biochemical module, there is no clear consensus on the quantitative criteria that may be used to systematically derive these modules. In this work, we investigate cyclical interactions as the defining characteristic of a biochemical module. We utilize a round trip distance metric, termed Shortest Retroactive Distance (ShReD), to characterize the retroactive connectivity between any two reactions in a biochemical network and to group together network components that mutually influence each other. We evaluate the metric on two types of networks that feature feedback interactions: (i) epidermal growth factor receptor (EGFR) signaling and (ii) liver metabolism supporting drug transformation. For both networks, the ShReD partitions found hierarchically arranged modules that confirm biological intuition. In addition, the partitions also revealed modules that are less intuitive. In particular, ShReD-based partition of the metabolic network identified a 'redox' module that couples reactions of glucose, pyruvate, lipid and drug metabolism through shared production and consumption of NADPH. Our results suggest that retroactive interactions arising from feedback loops and metabolic cycles significantly contribute to the modularity of biochemical networks. For metabolic networks, cofactors play an important role as allosteric effectors that mediate the retroactive interactions.     

Structure versus time in the evolutionary diversification of avian carotenoid metabolic networks

Historical associations of genes and proteins are thought to delineate pathways available to subsequent evolution; however, the effects of past functional involvements on contemporary evolution are rarely quantified. Here, we examined the extent to which the structure of a carotenoid enzymatic network persists in avian evolution. Specifically, we tested whether the evolution of carotenoid networks was most concordant with phylogenetically structured expansion from core reactions of common ancestors or with subsampling of biochemical pathway modules from an ancestral network. We compared structural and historical associations in 467 carotenoid networks of extant and ancestral species and uncovered the overwhelming effect of pre‐existing metabolic network structure on carotenoid diversification over the last 50 million years of avian evolution. Over evolutionary time, birds repeatedly subsampled and recombined conserved biochemical modules, which likely maintained the overall structure of the carotenoid metabolic network during avian evolution. These findings explain the recurrent convergence of evolutionary distant species in carotenoid metabolism and weak phylogenetic signal in avian carotenoid evolution. Remarkable retention of an ancient metabolic structure throughout extensive and prolonged ecological diversification in avian carotenoid metabolism illustrates a fundamental requirement of organismal evolution – historical continuity of a deterministic network that links past and present functional associations of its components.     

Decomposition of metabolic network into functional modules based on the global connectivity structure of reaction graph

MOTIVATION: Metabolic networks are organized in a modular, hierarchical manner. Methods for a rational decomposition of the metabolic network into relatively independent functional subsets are essential to better understand the modularity and organization principle of a large-scale, genome-wide network. Network decomposition is also necessary for functional analysis of metabolism by pathway analysis methods that are often hampered by the problem of combinatorial explosion due to the complexity of metabolic network. Decomposition methods proposed in literature are mainly based on the connection degree of metabolites. To obtain a more reasonable decomposition, the global connectivity structure of metabolic networks should be taken into account. RESULTS: In this work, we use a reaction graph representation of a metabolic network for the identification of its global connectivity structure and for decomposition. A bow-tie connectivity structure similar to that previously discovered for metabolite graph is found also to exist in the reaction graph. Based on this bow-tie structure, a new decomposition method is proposed, which uses a distance definition derived from the path length between two reactions. An hierarchical classification tree is first constructed from the distance matrix among the reactions in the giant strong component of the bow-tie structure. These reactions are then grouped into different subsets based on the hierarchical tree. Reactions in the IN and OUT subsets of the bow-tie structure are subsequently placed in the corresponding subsets according to a 'majority rule'. Compared with the decomposition methods proposed in literature, ours is based on combined properties of the global network structure and local reaction connectivity rather than, primarily, on the connection degree of metabolites. The method is applied to decompose the metabolic network of Escherichia coli. Eleven subsets are obtained. More detailed investigations of the subsets show that reactions in the same subset are really functionally related. The rational decomposition of metabolic networks, and subsequent studies of the subsets, make it more amenable to understand the inherent organization and functionality of metabolic networks at the modular level. SUPPLEMENTARY INFORMATION: http://genome.gbf.de/bioinformatics/     

Ab initio prediction of thermodynamically feasible reaction directions from biochemical network stoichiometry

Analysis of the stoichiometric structure of metabolic networks provides insights into the relationships between structure, function, and regulation of metabolic systems. Based on knowledge of only reaction stoichiometry, certain aspects of network functionality and robustness can be predicted. Current theories focus on breaking a metabolic network down into non-decomposable pathways able to operate in steady state. The physics underlying these theories is based on mass balance and the laws of thermodynamics. However, due to the inherent nonlinearity of the thermodynamic constraints on metabolic fluxes, computational analysis of large-scale biochemical systems can be expensive. In this study, it is shown how the feasible reaction directions may be determined by either computing the allowable ranges under the mass-balance and thermodynamic constraints or by analyzing the stoichiometric structure of the network. The computed reaction directions translate into a set of linear constraints necessary for thermodynamic feasibility. This set of necessary linear constraints is shown to be sufficient to guarantee feasibility in certain cases, thus translating the nonlinear thermodynamic constraints to linear. We show that for a reaction network of 44 internal reactions representing energy metabolism, the computed linear inequality constraints represent necessary and sufficient conditions for thermodynamic feasibility.     

Beyond topology: coevolution of structure and flux in metabolic networks

Interactions between the structure of a metabolic network and its functional properties underlie its evolutionary diversification, but the mechanism by which such interactions arise remains elusive. Particularly unclear is whether metabolic fluxes that determine the concentrations of compounds produced by a metabolic network, are causally linked to a network's structure or emerge independently of it. A direct empirical study of populations where both structural and functional properties vary among individuals’ metabolic networks is required to establish whether changes in structure affect the distribution of metabolic flux. In a population of house finches (Haemorhous mexicanus), we reconstructed full carotenoid metabolic networks for 442 individuals and uncovered 11 structural variants of this network with different compounds and reactions. We examined the consequences of this structural diversity for the concentrations of plumage‐bound carotenoids produced by flux in these networks. We found that concentrations of metabolically derived, but not dietary carotenoids, depended on network structure. Flux was partitioned similarly among compounds in individuals of the same network structure: within each network, compound concentrations were closely correlated. The highest among‐individual variation in flux occurred in networks with the strongest among‐compound correlations, suggesting that changes in the magnitude, but not the distribution of flux, underlie individual differences in compound concentrations on a static network structure. These findings indicate that the distribution of flux in carotenoid metabolism closely follows network structure. Thus, evolutionary diversification and local adaptations in carotenoid metabolism may depend more on the gain or loss of enzymatic reactions than on changes in flux within a network structure.     

Mass-balanced randomization of metabolic networks

MOTIVATION: Network-centered studies in systems biology attempt to integrate the topological properties of biological networks with experimental data in order to make predictions and posit hypotheses. For any topology-based prediction, it is necessary to first assess the significance of the analyzed property in a biologically meaningful context. Therefore, devising network null models, carefully tailored to the topological and biochemical constraints imposed on the network, remains an important computational problem. RESULTS: We first review the shortcomings of the existing generic sampling scheme-switch randomization-and explain its unsuitability for application to metabolic networks. We then devise a novel polynomial-time algorithm for randomizing metabolic networks under the (bio)chemical constraint of mass balance. The tractability of our method follows from the concept of mass equivalence classes, defined on the representation of compounds in the vector space over chemical elements. We finally demonstrate the uniformity of the proposed method on seven genome-scale metabolic networks, and empirically validate the theoretical findings. The proposed method allows a biologically meaningful estimation of significance for metabolic network properties.     

Thermodynamic calculations for biochemical transport and reaction processes in metabolic networks

Thermodynamic analysis of metabolic networks has recently generated increasing interest for its ability to add constraints on metabolic network operation, and to combine metabolic fluxes and metabolite measurements in a mechanistic manner. Concepts for the calculation of the change in Gibbs energy of biochemical reactions have long been established. However, a concept for incorporation of cross-membrane transport in these calculations is still missing, although the theory for calculating thermodynamic properties of transport processes is long known. Here, we have developed two equivalent equations to calculate the change in Gibbs energy of combined transport and reaction processes based on two different ways of treating biochemical thermodynamics. We illustrate the need for these equations by showing that in some cases there is a significant difference between the proposed correct calculation and using an approximative method. With the developed equations, thermodynamic analysis of metabolic networks spanning over multiple physical compartments can now be correctly described.     

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.     

Thermodynamically based profiling of drug metabolism and drug-drug metabolic interactions: a case study of acetaminophen and ethanol toxic interaction

Drug-drug metabolic interactions can result in unwanted side effects, including reduced drug efficacy and formation of toxic metabolic intermediates. In this work, thermodynamic constraints on non-equilibrium metabolite concentrations are used to reveal the biochemical interactions between the metabolic pathways of ethanol and acetaminophen (N-acetyl-p-aminophenol), two drugs known to interact unfavorably. It is known that many reactions of these pathways are coupled to the central energy metabolic reactions through a number of metabolites and the cellular redox potential. Based on these observations, a metabolic network model has been constructed and a database of thermodynamic properties for all participating metabolites and reactions has been compiled. Constraint-based computational analysis of the feasible metabolite concentrations reveals that the non-toxic pathways for APAP metabolism and the pathway for detoxifying N-acetyl-p-benzoquinoneimine (NAPQI) are inhibited by network interactions with ethanol metabolism. These results point to the potential utility of thermodynamically based profiling of metabolic network interactions in screening of drug candidates and analysis of potential toxicity.     

METABOLIC FLUX IS A DETERMINANT OF THE EVOLUTIONARY RATES OF ENZYME‐ENCODING GENES

Relationships between evolutionary rates and gene properties on a genomic, functional, pathway, or system level are being explored to unravel the principles of the evolutionary process. In particular, functional network properties have been analyzed to recognize the constraints they may impose on the evolutionary fate of genes. Here we took as a case study the core metabolic network in human erythrocytes and we analyzed the relationship between the evolutionary rates of its genes and the metabolic flux distribution throughout it. We found that metabolic flux correlates with the ratio of nonsynonymous to synonymous substitution rates. Genes encoding enzymes that carry high fluxes have been more constrained in their evolution, while purifying selection is more relaxed in genes encoding enzymes carrying low metabolic fluxes. These results demonstrate the importance of considering the dynamical functioning of gene networks when assessing the action of selection on system‐level properties.     

Inferring meaningful pathways in weighted metabolic networks

An approach is presented for computing meaningful pathways in the network of small molecule metabolism comprising the chemical reactions characterized in all organisms. The metabolic network is described as a weighted graph in which all the compounds are included, but each compound is assigned a weight equal to the number of reactions in which it participates. Path finding is performed in this graph by searching for one or more paths with lowest weight. Performance is evaluated systematically by computing paths between the first and last reactions in annotated metabolic pathways, and comparing the intermediate reactions in the computed pathways to those in the annotated ones. For the sake of comparison, paths are computed also in the un-weighted raw (all compounds and reactions) and filtered (highly connected pool metabolites removed) metabolic graphs, respectively. The correspondence between the computed and annotated pathways is very poor (<30%) in the raw graph; increasing to approximately 65% in the filtered graph; reaching approximately 85% in the weighted graph. Considering the best-matching path among the five lightest paths increases the correspondence to 92%, on average. We then show that the average distance between pairs of metabolites is significantly larger in the weighted graph than in the raw unfiltered graph, suggesting that the small-world properties previously reported for metabolic networks probably result from irrelevant shortcuts through pool metabolites. In addition, we provide evidence that the length of the shortest path in the weighted graph represents a valid measure of the "metabolic distance" between enzymes. We suggest that the success of our simplistic approach is rooted in the high degree of specificity of the reactions in metabolic pathways, presumably reflecting thermodynamic constraints operating in these pathways. We expect our approach to find useful applications in inferring metabolic pathways in newly sequenced genomes.     

Challenges to be faced in the reconstruction of metabolic networks from public databases

In the post-genomic era, the biochemical information for individual compounds, enzymes, reactions to be found within named organisms has become readily available. The well-known KEGG and BioCyc databases provide a comprehensive catalogue for this information and have thereby substantially aided the scientific community. Using these databases, the complement of enzymes present in a given organism can be determined and, in principle, used to reconstruct the metabolic network. However, such reconstructed networks contain numerous properties contradicting biological expectation. The metabolic networks for a number of organisms are reconstructed from KEGG and BioCyc databases, and features of these networks are related to properties of their originating database.     

The evolution of connectivity in metabolic networks

Processes in living cells are the result of interactions between biochemical compounds in highly complex biochemical networks. It is a major challenge in biology to understand causes and consequences of the specific design of these networks. A characteristic design feature of metabolic networks is the presence of hub metabolites such as ATP or NADH that are involved in a high number of reactions. To study the emergence of hub metabolites, we implemented computer simulations of a widely accepted scenario for the evolution of metabolic networks. Our simulations indicate that metabolic networks with a large number of highly specialized enzymes may evolve from a few multifunctional enzymes. During this process, enzymes duplicate and specialize, leading to a loss of biochemical reactions and intermediary metabolites. Complex features of metabolic networks such as the presence of hubs may result from selection of growth rate if essential biochemical mechanisms are considered. Specifically, our simulations indicate that group transfer reactions are essential for the emergence of hubs.     

The metabolic core and catalytic switches are fundamental elements in the self-regulation of the systemic metabolic structure of cells

Background

Experimental observations and numerical studies with dissipative metabolic networks have shown that cellular enzymatic activity self-organizes spontaneously leading to the emergence of a metabolic core formed by a set of enzymatic reactions which are always active under all environmental conditions, while the rest of catalytic processes are only intermittently active. The reactions of the metabolic core are essential for biomass formation and to assure optimal metabolic performance. The on-off catalytic reactions and the metabolic core are essential elements of a Systemic Metabolic Structure which seems to be a key feature common to all cellular organisms.

Methodology/Principal Findings

In order to investigate the functional importance of the metabolic core we have studied different catalytic patterns of a dissipative metabolic network under different external conditions. The emerging biochemical data have been analysed using information-based dynamic tools, such as Pearson''s correlation and Transfer Entropy (which measures effective functionality). Our results show that a functional structure of effective connectivity emerges which is dynamical and characterized by significant variations of bio-molecular information flows.

Conclusions/Significance

We have quantified essential aspects of the metabolic core functionality. The always active enzymatic reactions form a hub –with a high degree of effective connectivity- exhibiting a wide range of functional information values being able to act either as a source or as a sink of bio-molecular causal interactions. Likewise, we have found that the metabolic core is an essential part of an emergent functional structure characterized by catalytic modules and metabolic switches which allow critical transitions in enzymatic activity. Both, the metabolic core and the catalytic switches in which also intermittently-active enzymes are involved seem to be fundamental elements in the self-regulation of the Systemic Metabolic Structure.