Vicente Acuña

Modes and cuts in metabolic networks: Complexity and algorithms

Constraint-based approaches recently brought new insight into our understanding of metabolism. By making very simple assumptions such as that the system is at steady-state and some reactions are irreversible, and without requiring kinetic parameters, general properties of the system can be derived. A central concept in this methodology is the notion of an elementary mode (EM for short) which represents a minimal functional subsystem. The computation of EMs still forms a limiting step in metabolic studies and several algorithms have been proposed to address this problem leading to increasingly faster methods. However, although a theoretical upper bound on the number of elementary modes that a network may possess has been established, surprisingly, the complexity of this problem has never been systematically studied. In this paper, we give a systematic overview of the complexity of optimisation problems related to modes. We first establish results regarding network consistency. Most consistency problems are easy, i.e., they can be solved in polynomial time. We then establish the complexity of finding and counting elementary modes. We show in particular that finding one elementary mode is easy but that this task becomes hard when a specific EM (i.e. an EM containing some specified reactions) is sought. We then show that counting the number of elementary modes is musical sharpP-complete. We emphasize that the easy problems can be solved using currently existing software packages. We then analyse the complexity of a closely related task which is the computation of so-called minimum reaction cut sets and we show that this problem is hard. We then present two positive results which both allow to avoid computing EMs as a prior to the computation of reaction cuts. The first one is a polynomial approximation algorithm for finding a minimum reaction cut set. The second one is a test for verifying whether a set of reactions constitutes a reaction cut; this test can be readily included in existing algorithms to improve their performance. Finally, we discuss the complexity of other cut-related problems.

A note on the complexity of finding and enumerating elementary modes

In the context of the study into elementary modes of metabolic networks, we prove two complexity results. Enumerating elementary modes containing a specific reaction is hard in an enumeration complexity sense. The decision problem if there exists an elementary mode containing two specific reactions is NP-complete. The complexity of enumerating all elementary modes remains open.

Graph-Based Analysis of the Metabolic Exchanges between Two Co-Resident Intracellular Symbionts, Baumannia cicadellinicola and Sulcia muelleri, with Their Insect Host, Homalodisca coagulata

Endosymbiotic bacteria from different species can live inside cells of the same eukaryotic organism. Metabolic exchanges occur between host and bacteria but also between different endocytobionts. Since a complete genome annotation is available for both, we built the metabolic network of two endosymbiotic bacteria, Sulcia muelleri and Baumannia cicadellinicola, that live inside specific cells of the sharpshooter Homalodisca coagulata and studied the metabolic exchanges involving transfers of carbon atoms between the three. We automatically determined the set of metabolites potentially exogenously acquired (seeds) for both metabolic networks. We show that the number of seeds needed by both bacteria in the carbon metabolism is extremely reduced. Moreover, only three seeds are common to both metabolic networks, indicating that the complementarity of the two metabolisms is not only manifested in the metabolic capabilities of each bacterium, but also by their different use of the same environment. Furthermore, our results show that the carbon metabolism of S. muelleri may be completely independent of the metabolic network of B. cicadellinicola. On the contrary, the carbon metabolism of the latter appears dependent on the metabolism of S. muelleri, at least for two essential amino acids, threonine and lysine. Next, in order to define which subsets of seeds (precursor sets) are sufficient to produce the metabolites involved in a symbiotic function, we used a graph-based method, PITUFO, that we recently developed. Our results highly refine our knowledge about the complementarity between the metabolisms of the two bacteria and their host. We thus indicate seeds that appear obligatory in the synthesis of metabolites are involved in the symbiotic function. Our results suggest both B. cicadellinicola and S. muelleri may be completely independent of the metabolites provided by the co-resident endocytobiont to produce the carbon backbone of the metabolites provided to the symbiotic system (., thr and lys are only exploited by B. cicadellinicola to produce its proteins).

Enumerating Precursor Sets of Target Metabolites in a Metabolic Network

We present the first exact method based on the topology of a metabolic network to find minimal sets of metabolites (called precursors) sufficient to produce a set of target metabolites. In contrast with previous proposals, our model takes into account self-regenerating metabolites involved in cycles, which may be used to generate target metabolites from potential precursors. We analyse the complexity of the problem and we propose an algorithm to enumerate all minimal precursor sets for a set of target metabolites. The algorithm can be applied to identify a minimal medium necessary for a cell to ensure some metabolic functions. It can be used also to check inconsistencies caused by misannotations in a metabolic network. We present two illustrations of these applications.

Algorithms and complexity of enumerating minimal precursor sets in genome-wide metabolic networks

MOTIVATION In the context of studying whole metabolic networks and their interaction with the environment, the following question arises: given a set of target metabolites T and a set of possible external source metabolites , which are the minimal subsets of that are able to produce all the metabolites in T. Such subsets are called the minimal precursor sets of T. The problem is then whether we can enumerate all of them efficiently. RESULTS We propose a new characterization of precursor sets as the inputs of reaction sets called factories and an efficient algorithm to decide if a set of sources is precursor set of T. We show proofs of hardness for the problems of finding a precursor set of minimum size and of enumerating all minimal precursor sets T. We propose two new algorithms which, despite the hardness of the enumeration problem, allow to enumerate all minimal precursor sets in networks with up to 1000 reactions. AVAILABILITY Source code and datasets used in our benchmarks are freely available for download at http://sites.google.com/site/pitufosoftware/download. CONTACT vicente77@gmail.com, pvmilreu@gmail.com or marie-france.sagot@inria.fr.

Telling metabolic stories to explore metabolomics data: A case study on the Yeast response to cadmium exposure

Motivation: The increasing availability of metabolomics data enables to better understand the metabolic processes involved in the immediate response of an organism to environmental changes and stress. The data usually come in the form of a list of metabolites whose concentrations significantly changed under some conditions, and are thus not easy to interpret without being able to precisely visualize how such metabolites are interconnected. Results: We present a method that enables to organize the data from any metabolomics experiment into metabolic stories. Each story corresponds to a possible scenario explaining the flow of matter between the metabolites of interest. These scenarios may then be ranked in different ways depending on which interpretation one wishes to emphasize for the causal link between two affected metabolites: enzyme activation, enzyme inhibition or domino effect on the concentration changes of substrates and products. Equally probable stories under any selected ranking scheme can be further grouped into a single anthology that summarizes, in a unique subnetwork, all equivalently plausible alternative stories. An anthology is simply a union of such stories. We detail an application of the method to the response of yeast to cadmium exposure. We use this system as a proof of concept for our method, and we show that we are able to find a story that reproduces very well the current knowledge about the yeast response to cadmium. We further show that this response is mostly based on enzyme activation. We also provide a framework for exploring the alternative pathways or side effects this local response is expected to have in the rest of the network. We discuss several interpretations for the changes we see, and we suggest hypotheses that could in principle be experimentally tested. Noticeably, our method requires simple input data and could be used in a wide variety of applications. Availability and implementation: The code for the method presented in this article is available at http://gobbolino.gforge.inria.fr. Contact: pvmilreu@gmail.com; vincent.lacroix@univ-lyon1.fr; marie-france.sagot@inria.fr Supplementary information: Supplementary data are available at Bioinformatics online.

Enumerating chemical organisations in consistent metabolic networks: complexity and algorithms

The structural analysis of metabolic networks aims both at understanding the function and the evolution of metabolism. While it is commonly admitted that metabolism is modular, the identification of metabolic modules remains an open topic. Several definitions of what is a module have been proposed. We focus here on the notion of chemical organisations, i.e. sets of molecules which are closed and self-maintaining. We show that finding a reactive organisation is NP-hard even if the network is flux-consistent and that the hardness comes from blocking cycles. We then propose new algorithms for enumerating chemical organisations that are theoretically more efficient than existing approaches.

Enumeration of minimal stoichiometric precursor sets in metabolic networks

BackgroundWhat an organism needs at least from its environment to produce a set of metabolites, e.g. target(s) of interest and/or biomass, has been called a minimal precursor set. Early approaches to enumerate all minimal precursor sets took into account only the topology of the metabolic network (topological precursor sets). Due to cycles and the stoichiometric values of the reactions, it is often not possible to produce the target(s) from a topological precursor set in the sense that there is no feasible flux. Although considering the stoichiometry makes the problem harder, it enables to obtain biologically reasonable precursor sets that we call stoichiometric. Recently a method to enumerate all minimal stoichiometric precursor sets was proposed in the literature. The relationship between topological and stoichiometric precursor sets had however not yet been studied.ResultsSuch relationship between topological and stoichiometric precursor sets is highlighted. We also present two algorithms that enumerate all minimal stoichiometric precursor sets. The first one is of theoretical interest only and is based on the above mentioned relationship. The second approach solves a series of mixed integer linear programming problems. We compared the computed minimal precursor sets to experimentally obtained growth media of several Escherichia coli strains using genome-scale metabolic networks.ConclusionsThe results show that the second approach efficiently enumerates minimal precursor sets taking stoichiometry into account, and allows for broad in silico studies of strains or species interactions that may help to understand e.g. pathotype and niche-specific metabolic capabilities. sasita is written in Java, uses cplex as LP solver and can be downloaded together with all networks and input files used in this paper at http://sasita.gforge.inria.fr/.

Modeling Parsimonious Putative Regulatory Networks: Complexity and Heuristic Approach

A relevant problem in systems biology is the description of the regulatory interactions between genes. It is observed that pairs of genes have significant correlation through several experimental conditions. The question is to find causal relationships that can explain this experimental evidence. A putative regulatory network can be represented by an oriented weighted graph, where vertices represent genes, arcs represent predicted regulatory interactions and the arc weights represent the p-value of the prediction. Given such graph, and experimental evidence of correlation between pairs of vertices, we propose an abstraction and a method to enumerate all parsimonious subgraphs that assign causality relationships compatible with the experimental evidence. When the problem is modeled as the minimization of a global weight function, we show that the enumeration of scenarios is a hard problem. As an heuristic, we model the problem as a set of independent minimization problems, each solvable in polynomial time, which can be combined to explore a relevant subset of the solution space. We present a logic-programming formalization of the model implemented using Answer Set Programming. We show that, when the graph follows patterns that can be found in real organisms, our heuristic finds solutions that are good approximations to the full model. We encoded these approach using Answer Set Programming, applied this to a specific case in the organism E. coli and compared the execution time of each approach.

Coding with variable block maps

In this article we study a special class of sliding block maps that we call variable block maps. We characterize the subsets of finite and infinite sequences that can be obtained as the image of another subset of symbolic sequences by a variable block map. On the other way, we show that the coding process induced by such kind of block maps can be reversed, even with partial knowledge about the variable block maps, and we give an explicit construction of a canonical antecedent.