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.     

Simple and fast alignment of metabolic pathways by exploiting local diversity

MOTIVATION: An important tool for analyzing biological networks is the ability to perform homology searches, i.e. given a pattern network one would like to be able to search for occurrences of similar (sub)networks within a set of host networks. In the context of metabolic pathways, Pinter et al. [Bioinformatics, 2005] proposed to solve this computationally hard problem by restricting it to the case where both the pattern and host networks are trees. This restriction, however, severely limits the applicability of their algorithm. RESULTS: We propose a very fast and simple algorithm for the alignment of metabolic pathways that does not restrict the topology of the host or pattern network in any way; instead, our algorithm exploits a natural property of metabolic networks that we call 'local diversity property'. Experiments on a test bed of metabolic pathways from the BioCyc database indicate that our algorithm is much faster than the restricted algorithm of Pinter et al.-the metabolic pathways of two organisms can be aligned in mere seconds-and yet has a wider range of applicability and yields new biological insights. Our ideas can likely be extended to work for the alignment of various types of biological networks other than metabolic pathways. AVAILABILITY: Our algorithm has been implemented in C++ as a user-friendly metabolic pathway alignment tool called METAPAT. The tool runs under Linux or Windows and can be downloaded at http://theinf1.informatik.uni-jena.de/metapat/     

Network methods for describing sample relationships in genomic datasets: application to Huntington’s disease

Background

We consider the possibility of engineering metabolic pathways in a chassis organism in order to synthesize novel target compounds that are heterologous to the chassis. For this purpose, we model metabolic networks through hypergraphs where reactions are represented by hyperarcs. Each hyperarc represents an enzyme-catalyzed reaction that transforms set of substrates compounds into product compounds. We follow a retrosynthetic approach in order to search in the metabolic space (hypergraphs) for pathways (hyperpaths) linking the target compounds to a source set of compounds.

Results

To select the best pathways to engineer, we have developed an objective function that computes the cost of inserting a heterologous pathway in a given chassis organism. In order to find minimum-cost pathways, we propose in this paper two methods based on steady state analysis and network topology that are to the best of our knowledge, the first to enumerate all possible heterologous pathways linking a target compounds to a source set of compounds. In the context of metabolic engineering, the source set is composed of all naturally produced chassis compounds (endogenuous chassis metabolites) and the target set can be any compound of the chemical space. We also provide an algorithm for identifying precursors which can be supplied to the growth media in order to increase the number of ways to synthesize specific target compounds.

Conclusions

We find the topological approach to be faster by several orders of magnitude than the steady state approach. Yet both methods are generally scalable in time with the number of pathways in the metabolic network. Therefore this work provides a powerful tool for pathway enumeration with direct application to biosynthetic pathway design.     

Probabilistic path ranking based on adjacent pairwise coexpression for metabolic transcripts analysis

MOTIVATION: Pathway knowledge in public databases enables us to examine how individual metabolites are connected via chemical reactions and what genes are implicated in those processes. For two given (sets of) compounds, the number of possible paths between them in a metabolic network can be intractably large. It would be informative to rank these paths in order to differentiate between them. RESULTS: Focusing on adjacent pairwise coexpression, we developed an algorithm which, for a specified k, efficiently outputs the top k paths based on a probabilistic scoring mechanism, using a given metabolic network and microarray datasets. Our idea of using adjacent pairwise coexpression is supported by recent studies that local coregulation is predominant in metabolism. We first evaluated this idea by examining to what extent highly correlated gene pairs are adjacent and how often they are consecutive in a metabolic network. We then applied our algorithm to two examples of path ranking: the paths from glucose to pyruvate in the entire metabolic network of yeast and the paths from phenylalanine to sinapyl alcohol in monolignols pathways of arabidopsis under several different microarray conditions, to confirm and discuss the performance analysis of our method.     

Generating Boolean networks with a prescribed attractor structure

MOTIVATION: Dynamical modeling of gene regulation via network models constitutes a key problem for genomics. The long-run characteristics of a dynamical system are critical and their determination is a primary aspect of system analysis. In the other direction, system synthesis involves constructing a network possessing a given set of properties. This constitutes the inverse problem. Generally, the inverse problem is ill-posed, meaning there will be many networks, or perhaps none, possessing the desired properties. Relative to long-run behavior, we may wish to construct networks possessing a desirable steady-state distribution. This paper addresses the long-run inverse problem pertaining to Boolean networks (BNs). RESULTS: The long-run behavior of a BN is characterized by its attractors. The rest of the state transition diagram is partitioned into level sets, the j-th level set being composed of all states that transition to one of the attractor states in exactly j transitions. We present two algorithms for the attractor inverse problem. The attractors are specified, and the sizes of the predictor sets and the number of levels are constrained. Algorithm complexity and performance are analyzed. The algorithmic solutions have immediate application. Under the assumption that sampling is from the steady state, a basic criterion for checking the validity of a designed network is that there should be concordance between the attractor states of the model and the data states. This criterion can be used to test a design algorithm: randomly select a set of states to be used as data states; generate a BN possessing the selected states as attractors, perhaps with some added requirements such as constraints on the number of predictors and the level structure; apply the design algorithm; and check the concordance between the attractor states of the designed network and the data states. AVAILABILITY: The software and supplementary material is available at http://gsp.tamu.edu/Publications/BNs/bn.htm     

Hierarchy of metabolic compounds based on their synthesising capacity

The concept of scopes is applied to analyse large metabolic networks. Scopes are defined as sets of metabolites that can be synthesised by a metabolic network when it is provided with given seeds (Sets of initial metabolic compounds). Thus, scopes represent synthesising capacities of the seeds in the network. A hierarchy is discussed in the sense that compounds, which are part of the scope of another compound, possess scopes themselves that are subsets of the former scope. This hierarchy is analysed by means of a directed acyclic graph. Using a simple chemical model, it is found that this hierarchy contains specific structures that can, to a large extent, be explained by the chemical composition of the participating compounds. In this way, it represents a new kind of map of metabolic networks, arranging the metabolic compounds according to their chemical capacity.     

The players may change but the game remains: network analyses of ruminal microbiomes suggest taxonomic differences mask functional similarity

By mapping translated metagenomic reads to a microbial metabolic network, we show that ruminal ecosystems that are rather dissimilar in their taxonomy can be considerably more similar at the metabolic network level. Using a new network bi-partition approach for linking the microbial network to a bovine metabolic network, we observe that these ruminal metabolic networks exhibit properties consistent with distinct metabolic communities producing similar outputs from common inputs. For instance, the closer in network space that a microbial reaction is to a reaction found in the host, the lower will be the variability of its enzyme copy number across hosts. Similarly, these microbial enzymes that are nearby to host nodes are also higher in copy number than are more distant enzymes. Collectively, these results demonstrate a widely expected pattern that, to our knowledge, has not been explicitly demonstrated in microbial communities: namely that there can exist different community metabolic networks that have the same metabolic inputs and outputs but differ in their internal structure.     

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.     

Metabolic Pathfinding Using RPAIR Annotation

Metabolic databases contain information about thousands of small molecules and reactions, which can be represented as networks. In the context of metabolic reconstruction, pathways can be inferred by searching optimal paths in such networks. A recurrent problem is the presence of pool metabolites (e.g., water, energy carriers, and cofactors), which are connected to hundreds of reactions, thus establishing irrelevant shortcuts between nodes of the network. One solution to this problem relies on weighted networks to penalize highly connected compounds. A more refined solution takes the chemical structure of reactants into account in order to differentiate between side and main compounds of a reaction. Thanks to an intensive annotation effort at KEGG, decompositions of reactions into reactant pairs (RPAIR) categorized by their role (main, trans, cofac, ligase, and leave) are now available.The goal of this article is to evaluate the impact of RPAIR data on pathfinding in metabolic networks. To this end, we measure the impact of different parameters concerning the construction of the metabolic network: mapping of reactions and reactant pairs onto a graph, use of selected categories of reactant pairs, weighting schemes for compounds and reactions, removal of highly connected metabolites, and reaction directionality. In total, we tested 104 combinations of parameters and identified their optimal values for pathfinding on the basis of 55 reference pathways from three organisms.The best-performing metabolic network combines the biochemical knowledge encoded by KEGG RPAIR with a weighting scheme penalizing highly connected compounds. With this network, we could recover reference pathways from Escherichia coli with an average accuracy of 93% (32 pathways), from Saccharomyces cerevisiae with an average accuracy of 66% (11 pathways), and from humans with an average accuracy of 70% (12 pathways). Our pathfinding approach is available as part of the Network Analysis Tools.     

Binets: Fundamental Building Blocks for Phylogenetic Networks

Phylogenetic networks are a generalization of evolutionary trees that are used by biologists to represent the evolution of organisms which have undergone reticulate evolution. Essentially, a phylogenetic network is a directed acyclic graph having a unique root in which the leaves are labelled by a given set of species. Recently, some approaches have been developed to construct phylogenetic networks from collections of networks on 2- and 3-leaved networks, which are known as binets and trinets, respectively. Here we study in more depth properties of collections of binets, one of the simplest possible types of networks into which a phylogenetic network can be decomposed. More specifically, we show that if a collection of level-1 binets is compatible with some binary network, then it is also compatible with a binary level-1 network. Our proofs are based on useful structural results concerning lowest stable ancestors in networks. In addition, we show that, although the binets do not determine the topology of the network, they do determine the number of reticulations in the network, which is one of its most important parameters. We also consider algorithmic questions concerning binets. We show that deciding whether an arbitrary set of binets is compatible with some network is at least as hard as the well-known graph isomorphism problem. However, if we restrict to level-1 binets, it is possible to decide in polynomial time whether there exists a binary network that displays all the binets. We also show that to find a network that displays a maximum number of the binets is NP-hard, but that there exists a simple polynomial-time 1/3-approximation algorithm for this problem. It is hoped that these results will eventually assist in the development of new methods for constructing phylogenetic networks from collections of smaller networks.     

Data Identification for Improving Gene Network Inference using Computational Algebra

Identification of models of gene regulatory networks is sensitive to the amount of data used as input. Considering the substantial costs in conducting experiments, it is of value to have an estimate of the amount of data required to infer the network structure. To minimize wasted resources, it is also beneficial to know which data are necessary to identify the network. Knowledge of the data and knowledge of the terms in polynomial models are often required a priori in model identification. In applications, it is unlikely that the structure of a polynomial model will be known, which may force data sets to be unnecessarily large in order to identify a model. Furthermore, none of the known results provides any strategy for constructing data sets to uniquely identify a model. We provide a specialization of an existing criterion for deciding when a set of data points identifies a minimal polynomial model when its monomial terms have been specified. Then, we relax the requirement of the knowledge of the monomials and present results for model identification given only the data. Finally, we present a method for constructing data sets that identify minimal polynomial models.     

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     

Generalized concept of minimal cut sets in biochemical networks

Recently, the concept of minimal cut sets has been introduced for studying structural fragility and identifying knock-out strategies in biochemical reaction networks. A minimal cut set (MCS) has been defined as a minimal set of reactions whose removal blocks the operation of a chosen objective reaction. In this report the theoretical framework of MCSs is refined and extended increasing the practical applicability significantly. An MCS is now defined as a minimal (irreducible) set of structural interventions (removal of network elements) repressing a certain functionality specified by a deletion task. A deletion task describes unambiguously the flux patterns (or the functionality) to be repressed. It is shown that the MCSs can be computed from the set of target modes, which comprises all elementary modes that exhibit the functionality to be attacked. Since a deletion task can be specified by several Boolean rules, MCSs can now be determined for a large variety of complex deletion problems and may be utilized for inhibiting very special flux patterns. It is additionally shown that the other way around is also possible: the elementary modes belonging to a certain functionality can be computed from the respective set of MCSs. Therefore, elementary modes and MCSs can be seen as dual representations of network functions and both can be converted into each other. Moreover, there exist a strong relationship to minimal hitting sets known from set theory: the MCSs are the minimal hitting sets of the collection of target modes and vice versa. Another generalization introduced herein is that MCSs need not to be restricted to the removal of reactions they may also contain network nodes. In the light of the extended framework of MCSs, applications for assessing, manipulating, and designing metabolic networks in silico are discussed.     

Reconstructible Phylogenetic Networks: Do Not Distinguish the Indistinguishable

Phylogenetic networks represent the evolution of organisms that have undergone reticulate events, such as recombination, hybrid speciation or lateral gene transfer. An important way to interpret a phylogenetic network is in terms of the trees it displays, which represent all the possible histories of the characters carried by the organisms in the network. Interestingly, however, different networks may display exactly the same set of trees, an observation that poses a problem for network reconstruction: from the perspective of many inference methods such networks are indistinguishable. This is true for all methods that evaluate a phylogenetic network solely on the basis of how well the displayed trees fit the available data, including all methods based on input data consisting of clades, triples, quartets, or trees with any number of taxa, and also sequence-based approaches such as popular formalisations of maximum parsimony and maximum likelihood for networks. This identifiability problem is partially solved by accounting for branch lengths, although this merely reduces the frequency of the problem. Here we propose that network inference methods should only attempt to reconstruct what they can uniquely identify. To this end, we introduce a novel definition of what constitutes a uniquely reconstructible network. For any given set of indistinguishable networks, we define a canonical network that, under mild assumptions, is unique and thus representative of the entire set. Given data that underwent reticulate evolution, only the canonical form of the underlying phylogenetic network can be uniquely reconstructed. While on the methodological side this will imply a drastic reduction of the solution space in network inference, for the study of reticulate evolution this is a fundamental limitation that will require an important change of perspective when interpreting phylogenetic networks.     

Thermodynamics-based Metabolite Sensitivity Analysis in metabolic networks

The increasing availability of large metabolomics datasets enhances the need for computational methodologies that can organize the data in a way that can lead to the inference of meaningful relationships. Knowledge of the metabolic state of a cell and how it responds to various stimuli and extracellular conditions can offer significant insight in the regulatory functions and how to manipulate them. Constraint based methods, such as Flux Balance Analysis (FBA) and Thermodynamics-based flux analysis (TFA), are commonly used to estimate the flow of metabolites through genome-wide metabolic networks, making it possible to identify the ranges of flux values that are consistent with the studied physiological and thermodynamic conditions. However, unless key intracellular fluxes and metabolite concentrations are known, constraint-based models lead to underdetermined problem formulations. This lack of information propagates as uncertainty in the estimation of fluxes and basic reaction properties such as the determination of reaction directionalities. Therefore, knowledge of which metabolites, if measured, would contribute the most to reducing this uncertainty can significantly improve our ability to define the internal state of the cell. In the present work we combine constraint based modeling, Design of Experiments (DoE) and Global Sensitivity Analysis (GSA) into the Thermodynamics-based Metabolite Sensitivity Analysis (TMSA) method. TMSA ranks metabolites comprising a metabolic network based on their ability to constrain the gamut of possible solutions to a limited, thermodynamically consistent set of internal states. TMSA is modular and can be applied to a single reaction, a metabolic pathway or an entire metabolic network. This is, to our knowledge, the first attempt to use metabolic modeling in order to provide a significance ranking of metabolites to guide experimental measurements.     

Analyzing steady states of dynamics of bio-molecules from the structure of regulatory networks

Regulatory relations between biological molecules constitute complex network systems and realize diverse biological functions through the dynamics of molecular activities. However, we currently have very little understanding of the relationship between the structure of a regulatory network and its dynamical properties. In this paper we introduce a new method, named “linkage logic” to analyze the dynamics of network systems. By this method, we can restrict possible steady states of a given complex network system from the knowledge of regulatory linkages alone. The regulatory linkage simply specifies the list of variables that affect the dynamics of each variable. We formalize two aspects of the linkage logic: the “Principle of Compatibility” determines the upper limit of the diversity of possible steady states of the dynamics realized by a given network; the “Principle of Dependency” determines the possible combinations of states of the system. By combining these two aspects, (i) for a given network, we can identify a cluster of nodes that gives an alternative representation of the steady states of the whole system, (ii) we can reduce a given complex network into a simpler one without loss of the ability to generate the diversity of steady states, (iii) we can examine the consistency between the structure of network and observed set of steady states, and (iv) sometimes we can predict unknown states or unknown regulations from an observed set of steady states alone. We illustrate the method by several applications to an experimentally determined regulatory network for biological functions.     

Expanding Metabolic Networks: Scopes of Compounds, Robustness, and Evolution

A new method for the mathematical analysis of large metabolic networks is presented. Based on the fact that the occurrence of a metabolic reaction generally requires the existence of other reactions providing its substrates, series of metabolic networks are constructed. In each step of the corresponding expansion process those reactions are incorporated whose substrates are made available by the networks of the previous generations. The method is applied to the set of all metabolic reactions included in the KEGG database. Starting with one or more seed compounds, the expansion results in a final network whose compounds define the scope of the seed. Scopes of all metabolic compounds are calculated and it is shown that large parts of cellular metabolism can be considered as the combined scope of simple building blocks. Analyses of various expansion processes reveal crucial metabolites whose incorporation allows for the increase in network complexity. Among these metabolites are common cofactors such as NAD⁺, ATP, and coenzyme A. We demonstrate that the outcome of network expansion is in general very robust against elimination of single or few reactions. There exist, however, crucial reactions whose elimination results in a dramatic reduction of scope sizes. It is hypothesized that the expansion process displays characteristics of the evolution of metabolism such as the temporal order of the emergence of metabolic pathways. [Reviewing Editor : Dr. David Pollock]     

NIBBS-search for fast and accurate prediction of phenotype-biased metabolic systems

Understanding of genotype-phenotype associations is important not only for furthering our knowledge on internal cellular processes, but also essential for providing the foundation necessary for genetic engineering of microorganisms for industrial use (e.g., production of bioenergy or biofuels). However, genotype-phenotype associations alone do not provide enough information to alter an organism's genome to either suppress or exhibit a phenotype. It is important to look at the phenotype-related genes in the context of the genome-scale network to understand how the genes interact with other genes in the organism. Identification of metabolic subsystems involved in the expression of the phenotype is one way of placing the phenotype-related genes in the context of the entire network. A metabolic system refers to a metabolic network subgraph; nodes are compounds and edges labels are the enzymes that catalyze the reaction. The metabolic subsystem could be part of a single metabolic pathway or span parts of multiple pathways. Arguably, comparative genome-scale metabolic network analysis is a promising strategy to identify these phenotype-related metabolic subsystems. Network Instance-Based Biased Subgraph Search (NIBBS) is a graph-theoretic method for genome-scale metabolic network comparative analysis that can identify metabolic systems that are statistically biased toward phenotype-expressing organismal networks. We set up experiments with target phenotypes like hydrogen production, TCA expression, and acid-tolerance. We show via extensive literature search that some of the resulting metabolic subsystems are indeed phenotype-related and formulate hypotheses for other systems in terms of their role in phenotype expression. NIBBS is also orders of magnitude faster than MULE, one of the most efficient maximal frequent subgraph mining algorithms that could be adjusted for this problem. Also, the set of phenotype-biased metabolic systems output by NIBBS comes very close to the set of phenotype-biased subgraphs output by an exact maximally-biased subgraph enumeration algorithm ( MBS-Enum ). The code (NIBBS and the module to visualize the identified subsystems) is available at http://freescience.org/cs/NIBBS.