Attributing uncertainties to the input space elevates the trustworthiness and explainability of machine learning applications. This paper proposes a novel method called Smoothness Constrained Attribution (SCA), which uses the uncertainty propagation mechanism to propagate the output uncertainty back to the input space. This input uncertainty attribution relies solely on test-time data, the trained uncertainty-aware Machine Learning (ML) model, and assumes a smooth input space, resulting in an efficient and simple system. SCA is compared to existing input Uncertainty Attribution Mechanisms (iUCAMs) based on eXplainable Artificial Intelligence (XAI) and an oracle reference using heteroscedastic noise in different synthetic datasets. These evaluations demonstrate the robustness and improvements of SCA compared to existing methods.

Aspergillus fumigatus is an important human fungal pathogen and its conidia are constantly inhaled by humans. In immunocompromised individuals, conidia can grow out as hyphae that damage lung epithelium. The resulting invasive aspergillosis is associated with devastating mortality rates. Since infection is a race between the innate immune system and the outgrowth of A. fumigatus conidia, we use dynamic optimization to obtain insight into the recruitment and depletion of alveolar macrophages and neutrophils. Using this model, we obtain key insights into major determinants of infection outcome on host and pathogen side. On the pathogen side, we predict in silico and confirm in vitro that germination speed is an important virulence trait of fungal pathogens due to the vulnerability of conidia against host defense. On the host side, we found that epithelial cells, which have been underappreciated, play a role in fungal clearance and are potent mediators of cytokine release. Both predictions were confirmed by in vitro experiments on established cell lines as well as primary lung cells. Further, our model affirms the importance of neutrophils in invasive aspergillosis and underlines that the role of macrophages remains elusive. We expect that our model will contribute to improvement of treatment protocols by focusing on the critical components of immune response to fungi but also fungal virulence traits.

Elementary flux modes (EFMs) emerged as a formal concept to describe metabolic pathways and have become an established tool for constraint-based modeling and metabolic network analysis. EFMs are characteristic (support-minimal) vectors of the flux cone that contains all feasible steady-state flux vectors of a given metabolic network. EFMs account for (homogeneous) linear constraints arising from reaction irreversibilities and the assumption of steady state; however, other (inhomogeneous) linear constraints, such as minimal and maximal reaction rates frequently used by other constraint-based techniques (such as flux balance analysis [FBA]), cannot be directly integrated. These additional constraints further restrict the space of feasible flux vectors and turn the flux cone into a general flux polyhedron in which the concept of EFMs is not directly applicable anymore. For this reason, there has been a conceptual gap between EFM-based (pathway) analysis methods and linear optimization (FBA) techniques, as they operate on different geometric objects. One approach to overcome these limitations was proposed ten years ago and is based on the concept of elementary flux vectors (EFVs). Only recently has the community started to recognize the potential of EFVs for metabolic network analysis. In fact, EFVs exactly represent the conceptual development required to generalize the idea of EFMs from flux cones to flux polyhedra. This work aims to present a concise theoretical and practical introduction to EFVs that is accessible to a broad audience. We highlight the close relationship between EFMs and EFVs and demonstrate that almost all applications of EFMs (in flux cones) are possible for EFVs (in flux polyhedra) as well. In fact, certain properties can only be studied with EFVs. Thus, we conclude that EFVs provide a powerful and unifying framework for constraint-based modeling of metabolic networks.

Bacterial communities are taxonomically highly diverse, yet the mechanisms that maintain this diversity remain poorly understood. We hypothesized that an obligate and mutual exchange of metabolites, as is very common among bacterial cells, could stabilize different genotypes within microbial communities. To test this, we developed a cellular automaton to model interactions among six empirically characterized genotypes that differ in their ability and propensity to produce amino acids. By systematically varying intrinsic (i.e. benefit-to-cost ratio) and extrinsic parameters (i.e. metabolite diffusion level, environmental amino acid availability), we show that obligate cross-feeding of essential metabolites is selected for under a broad range of conditions. In spatially structured environments, positive assortment among cross-feeders resulted in the formation of cooperative clusters, which limited exploitation by non-producing auxotrophs, yet allowed them to persist at the clusters' periphery. Strikingly, cross-feeding helped to maintain genotypic diversity within populations, while amino acid supplementation to the environment decoupled obligate interactions and favored auxotrophic cells that saved amino acid production costs over metabolically autonomous prototrophs. Together, our results suggest that spatially structured environments and limited nutrient availabilities should facilitate the evolution of metabolic interactions, which can help to maintain genotypic diversity within natural microbial populations.

MOTIVATION: The analysis of high-throughput molecular data in the context of metabolic pathways is essential to uncover their underlying functional structure. Among different metabolic pathway concepts in systems biology, elementary flux modes (EFMs) hold a predominant place, as they naturally capture the complexity and plasticity of cellular metabolism and go beyond predefined metabolic maps. However, their use to interpret high-throughput data has been limited so far, mainly because their computation in genome-scale metabolic networks has been unfeasible. To face this issue, different optimization-based techniques have been recently introduced and their application to human metabolism is promising. RESULTS: In this article, we exploit and generalize the K-shortest EFM algorithm to determine a subset of EFMs in a human genome-scale metabolic network. This subset of EFMs involves a wide number of reported human metabolic pathways, as well as potential novel routes, and constitutes a valuable database where high-throughput data can be mapped and contextualized from a metabolic perspective. To illustrate this, we took expression data of 10 healthy human tissues from a previous study and predicted their characteristic EFMs based on enrichment analysis. We used a multivariate hypergeometric test and showed that it leads to more biologically meaningful results than standard hypergeometric. Finally, a biological discussion on the characteristic EFMs obtained in liver is conducted, finding a high level of agreement when compared with the literature.

The dynamics of circadian rhythms needs to be adapted to day length changes between summer and winter. It has been observed experimentally, however, that the dynamics of individual neurons of the suprachiasmatic nucleus (SCN) does not change as the seasons change. Rather, the seasonal adaptation of the circadian clock is hypothesized to be a consequence of changes in the intercellular dynamics, which leads to a phase distribution of electrical activity of SCN neurons that is narrower in winter and broader during summer. Yet to understand this complex intercellular dynamics, a more thorough understanding of the impact of the network structure formed by the SCN neurons is needed. To that effect, we propose a mathematical model for the dynamics of the SCN neuronal architecture in which the structure of the network plays a pivotal role. Using our model we show that the fraction of long-range cell-to-cell connections and the seasonal changes in the daily rhythms may be tightly related. In particular, simulations of the proposed mathematical model indicate that the fraction of long-range connections between the cells adjusts the phase distribution and consequently the length of the behavioral activity as follows: dense long-range connections during winter lead to a narrow activity phase, while rare long-range connections during summer lead to a broad activity phase. Our model is also able to account for the experimental observations indicating a larger light-induced phase-shift of the circadian clock during winter, which we show to be a consequence of higher synchronization between neurons. Our model thus provides evidence that the variations in the seasonal dynamics of circadian clocks can in part also be understood and regulated by the plasticity of the SCN network structure.

MOTIVATION: The reconstruction of metabolic networks at the genome scale has allowed the analysis of metabolic pathways at an unprecedented level of complexity. Elementary flux modes (EFMs) are an appropriate concept for such analysis. However, their number grows in a combinatorial fashion as the size of the metabolic network increases, which renders the application of EFMs approach to large metabolic networks difficult. Novel methods are expected to deal with such complexity. RESULTS: In this article, we present a novel optimization-based method for determining a minimal generating set of EFMs, i.e. a convex basis. We show that a subset of elements of this convex basis can be effectively computed even in large metabolic networks. Our method was applied to examine the structure of pathways producing lysine in Escherichia coli. We obtained a more varied and informative set of pathways in comparison with existing methods. In addition, an alternative pathway to produce lysine was identified using a detour via propionyl-CoA, which shows the predictive power of our novel approach. AVAILABILITY: The source code in C++ is available upon request.

The question whether fatty acids can be converted into glucose in humans has a long standing tradition in biochemistry, and the expected answer is "No". Using recent advances in Systems Biology in the form of large-scale metabolic reconstructions, we reassessed this question by performing a global investigation of a genome-scale human metabolic network, which had been reconstructed on the basis of experimental results. By elementary flux pattern analysis, we found numerous pathways on which gluconeogenesis from fatty acids is feasible in humans. On these pathways, four moles of acetyl-CoA are converted into one mole of glucose and two moles of CO₂. Analyzing the detected pathways in detail we found that their energetic requirements potentially limit their capacity. This study has many other biochemical implications: effect of starvation, sports physiology, practically carbohydrate-free diets of inuit, as well as survival of hibernating animals and embryos of egg-laying animals. Moreover, the energetic loss associated to the usage of gluconeogenesis from fatty acids can help explain the efficiency of carbohydrate reduced and ketogenic diets such as the Atkins diet.

BACKGROUND: Subtle alternative splicing events involving tandem splice sites separated by a short (2-12 nucleotides) distance are frequent and evolutionarily widespread in eukaryotes, and a major contributor to the complexity of transcriptomes and proteomes. However, these events have been either omitted altogether in databases on alternative splicing, or only the cases of experimentally confirmed alternative splicing have been reported. Thus, a database which covers all confirmed cases of subtle alternative splicing as well as the numerous putative tandem splice sites (which might be confirmed once more transcript data becomes available), and allows to search for tandem splice sites with specific features and download the results, is a valuable resource for targeted experimental studies and large-scale bioinformatics analyses of tandem splice sites. Towards this goal we recently set up TassDB (Tandem Splice Site DataBase, version 1), which stores data about alternative splicing events at tandem splice sites separated by 3 nt in eight species. DESCRIPTION: We have substantially revised and extended TassDB. The currently available version 2 contains extensive information about tandem splice sites separated by 2-12 nt for the human and mouse transcriptomes including data on the conservation of the tandem motifs in five vertebrates. TassDB2 offers a user-friendly interface to search for specific genes or for genes containing tandem splice sites with specific features as well as the possibility to download result datasets. For example, users can search for cases of alternative splicing where the proportion of EST/mRNA evidence supporting the minor isoform exceeds a specific threshold, or where the difference in splice site scores is specified by the user. The predicted impact of each event on the protein is also reported, along with information about being a putative target for the nonsense-mediated decay (NMD) pathway. Links are provided to the UCSC genome browser and other external resources. CONCLUSION: TassDB2, available via http://www.tassdb.info, provides comprehensive resources for researchers interested in both targeted experimental studies and large-scale bioinformatics analyses of short distance tandem splice sites.

MOTIVATION: Elementary flux modes (EFMs) represent a key concept to analyze metabolic networks from a pathway-oriented perspective. In spite of considerable work in this field, the computation of the full set of elementary flux modes in large-scale metabolic networks still constitutes a challenging issue due to its underlying combinatorial complexity. RESULTS: In this article, we illustrate that the full set of EFMs can be enumerated in increasing order of number of reactions via integer linear programming. In this light, we present a novel procedure to efficiently determine the K-shortest EFMs in large-scale metabolic networks. Our method was applied to find the K-shortest EFMs that produce lysine in the genome-scale metabolic networks of Escherichia coli and Corynebacterium glutamicum. A detailed analysis of the biological significance of the K-shortest EFMs was conducted, finding that glucose catabolism, ammonium assimilation, lysine anabolism and cofactor balancing were correctly predicted. The work presented here represents an important step forward in the analysis and computation of EFMs for large-scale metabolic networks, where traditional methods fail for networks of even moderate size. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Abstract Motivation: In recent years, several methods have been proposed for determining metabolic pathways in an automated way based on network topology. The aim of this work is to analyse these methods by tackling a concrete example relevant in biochemistry. It concerns the question whether even-chain fatty acids, being the most important constituents of lipids, can be converted into sugars at steady state. It was proved five decades ago that this conversion using the Krebs cycle is impossible unless the enzymes of the glyoxylate shunt (or alternative bypasses) are present in the system. Using this example, we can compare the various methods in pathway analysis. Results: Elementary modes analysis (EMA) of a set of enzymes corresponding to the Krebs cycle, glycolysis and gluconeogenesis supports the scientific evidence showing that there is no pathway capable of converting acetyl-CoA to glucose at steady state. This conversion is possible after the addition of isocitrate lyase and malate synthase (forming the glyoxylate shunt) to the system. Dealing with the same example, we compare EMA with two tools based on graph theory available online, PathFinding and Pathway Hunter Tool. These automated network generating tools do not succeed in predicting the conversions known from experiment. They sometimes generate unbalanced paths and reveal problems identifying side metabolites that are not responsible for the carbon net flux. This shows that, for metabolic pathway analysis, it is important to consider the topology (including bimolecular reactions) and stoichiometry of metabolic systems, as is done in EMA. Contact: [email protected]; [email protected] Supplementary information: Supplementary data are available at Bioinformatics online.

Contact:[email protected]; [email protected] The main points of criticism by Faust et al. (2009) concerning the work in de Figueiredo et al., (2009a) are the following: the cases presented are biased; different tools should be supplied with the same input networks; the number of study cases should be representative; and tools should be evaluated by neutral assessors. We think that the models in de Figueiredo et al. (2009a) illustrate concrete biological problems that are very well documented. In particular, the conversion of fatty acids into sugars was not so trivial to be answered and is historically relevant (cf. Weinman et al., 1957). Today, many biochemistry textbooks dedicate at least one paragraph explaining why there is no net conversion of acetyl-CoA to glucose via the tricarboxylic acid cycle in vertebrates (cf. Nelson and Cox, 2000; Stryer, 1995; Voet and Voet, 2004). Thus, new tools for metabolic pathway prediction have to be able to answer these problems correctly. Analysing the same network used in (de Figueiredo et al., 2009a) by Path Finding (Croes et al., 2006), paths converting acetyl-CoA into glucose are computed, even though this is impossible for humans in the network studied. Thus, the critique of Faust et al. (2009) does not bring anything new to what was discussed in de Figueiredo et al. (2009a). It is often said that the size of the input network limits the computation of elementary flux modes (EFMs; Faust et al., 2009; Papin et al., 2003, 2004). Indeed, the enumeration of all EFMs in genome-scale models with the existing methods is difficult (Klamt et al., 2007; Schwarz et al., 2005; Terzer and Stelling, 2008; von Kamp and Schuster, 2006). However, there are approaches to compute at least a subset of EFMs in such models (Acuña et al., 2009; de Figueiredo et al., 2009b; Kaleta et al., 2009), for example, the shortest EFMs (de Figueiredo et al., 2009b). Regarding the number of test cases presented in (de Figueiredo et al., 2009a), we are sure they are representative of the issue that is discussed in that article. More cases exist, for example, the conversion of hypoxanthine into ATP in human erythrocytes, for which EFM analysis can be performed at the cell level (Schuster and Kenanov, 2005). Of course, a comparison between tools is preferably made by neutral assessors. However, it is usual in bioinformatics that authors who have established a new tool compare their method with others (Klamt et al., 2007; Urbanczik and Wagner, 2005; Wagner and Urbanczik, 2005). In addition, one co-author (C.K.) has written an article where EFMs are compared with chemical organizations (Kaleta et al., 2006). In the letter by Faust et al. (2009), it is argued that an incorrect definition of internal metabolites in EFM analysis, can generate wrong pathway predictions. The study of any biochemical system requires the definition of the system's boundary (see, e.g. Schilling and Palsson, 1998). Thus, the definition of internal and external metabolites in EFM analysis is nothing more than the definition of the boundary conditions found in many other modelling methods. Faust et al. (2009) say that the steady-state constraint is not always an appropriate assumption. Although, in some cases, this assumption is not justified, in many other cases, it is. Accordingly, it is used in many approaches such as Metabolic Control and Flux Balance Analyses. Faust et al. (2009) cite the work of Teusink et al. (2000) to support their statement. However, that work shows that the experimental system does reach a steady state. Additionally, we do not think that the study summarized in Table 1 of Faust et al. (2009) is exhaustive enough with respect to EFMs (cf. Trinh et al., 2009, for a review). Nevertheless, the communication from Faust et al. (2009) raises an important point concerning the comparison of tools for metabolic pathway prediction. It is suggested to follow a CASP-like protocol to evaluate the methods for metabolic pathway prediction and that this task should be performed by an independent committee. The validation process of some of the new tools for metabolic pathway prediction has been performed using the pathway information present in metabolic pathway databases (Blum and Kohlbacher, 2008; Croes et al., 2006). A CASP-like protocol, to be developed, has to take into account the fact that these databases contain errors (Likić, 2006; Poolman et al., 2006). On the other hand, only very well-documented pathways are stored in these databases. Many pathways are missing due to the lack of information or simply due to the fact that this classification is performed manually. Furthermore, these databases do not contain all the functional modes of a pathway within a metabolic network, e.g. all the five functional modes of the pentose phosphate pathway in conjunction with glycolysis (Schuster et al., 2000; Stryer, 1995). Moreover, EFM analysis has been successful in predicting relevant, hitherto unknown pathways, for example, the catabolic PEP-glyoxylate cycle in Escherichia coli (Schuster et al., 1999), which was later found in experiment (Fischer and Sauer, 2003) and that, to our knowledge, is not present in KEGG nor in MetaCyc (Caspi et al., 2008; Kanehisa et al., 2008). In conclusion, it is a challenge to represent the combinatorial multitude of biochemical pathways in metabolic databases (Sauer, 2006). Funding: PhD grant SFRH/BD/32961/2006 (to L.F.F.); by the following Portuguese entities: Fundaçãao Calouste Gulbenkian, Fundação para a Ciência e a Tecnologia (FCT) and Siemens SA Portugal (to L.F.F.); Biotechnology and Biosciences Research Council (BB/·E00203X/1 to D.A.F.); German Ministry for Research and Education (BMBF) within the Forsys Partner initiative (to C.K.). Conflict of Interest: none declared.

MOTIVATION: In recent years, several methods have been proposed for determining metabolic pathways in an automated way based on network topology. The aim of this work is to analyse these methods by tackling a concrete example relevant in biochemistry. It concerns the question whether even-chain fatty acids, being the most important constituents of lipids, can be converted into sugars at steady state. It was proved five decades ago that this conversion using the Krebs cycle is impossible unless the enzymes of the glyoxylate shunt (or alternative bypasses) are present in the system. Using this example, we can compare the various methods in pathway analysis. RESULTS: Elementary modes analysis (EMA) of a set of enzymes corresponding to the Krebs cycle, glycolysis and gluconeogenesis supports the scientific evidence showing that there is no pathway capable of converting acetyl-CoA to glucose at steady state. This conversion is possible after the addition of isocitrate lyase and malate synthase (forming the glyoxylate shunt) to the system. Dealing with the same example, we compare EMA with two tools based on graph theory available online, PathFinding and Pathway Hunter Tool. These automated network generating tools do not succeed in predicting the conversions known from experiment. They sometimes generate unbalanced paths and reveal problems identifying side metabolites that are not responsible for the carbon net flux. This shows that, for metabolic pathway analysis, it is important to consider the topology (including bimolecular reactions) and stoichiometry of metabolic systems, as is done in EMA. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

BACKGROUND: Modeling of metabolic networks includes tasks such as network assembly, network overview, calculation of metabolic fluxes and testing the robustness of the network. RESULTS: YANAsquare provides a software framework for rapid network assembly (flexible pathway browser with local or remote operation mode), network overview (visualization routine and YANAsquare editor) and network performance analysis (calculation of flux modes as well as target and robustness tests). YANAsquare comes as an easy-to-setup program package in Java. It is fully compatible and integrates the programs YANA (translation of gene expression values into flux distributions, metabolite network dissection) and Metatool (elementary mode calculation). As application examples we set-up and model the phospholipid network in the phagosome and genome-scale metabolic maps of S.aureus, S.epidermidis and S.saprophyticus as well as test their robustness against enzyme impairment. CONCLUSION: YANAsquare is an application software for rapid setup, visualization and analysis of small, larger and genome-scale metabolic networks.

The local position measurement system LPM is a novel system for fast and accurate measurements of object positions with active transponders. With an accuracy of few centimeters and measurement rates of 1000 measurements per second LPM is by now one of the most advanced systems in this field. For sensor fusion purposes each transponder is equipped with a telemetry channel. In this contribution we discuss the sensor fusion of the position measurements of the LPM system with accelerometers and gyroscopes. The focus of this work is to investigate the impact of additional sensor information on the accuracy of the estimated states based on different state transition matrices of a Kalman filter. The results based on synthetic data are verified by measurements of a test car on a race track.

SUMMARY: Elementary modes analysis is a powerful tool in the constraint-based modeling of metabolic networks. In recent years, new approaches to calculating elementary modes in biochemical reaction networks have been developed. As a consequence, the program Metatool, which is one of the first programs dedicated to this purpose, has been reimplemented in order to make use of these new approaches. The performance of Metatool has been significantly increased and the new version 5.0 can now be run inside the GNU octave or Matlab environments to allow more flexible usage and integration with other tools. AVAILABILITY: The script files and compiled shared libraries can be downloaded from the Metatool website at http://pinguin.biologie.uni-jena.de/bioinformatik/networks/index.html. Metatool consists of script files (m-files) for GNU octave as well as Matlab and shared libraries. The scripts are licensed under the GNU Public License and the use of the shared libraries is free for academic users and testing purposes. Commercial use of Metatool requires a special contract.

BACKGROUND: A number of algorithms for steady state analysis of metabolic networks have been developed over the years. Of these, Elementary Mode Analysis (EMA) has proven especially useful. Despite its low user-friendliness, METATOOL as a reliable high-performance implementation of the algorithm has been the instrument of choice up to now. As reported here, the analysis of metabolic networks has been improved by an editor and analyzer of metabolic flux modes. Analysis routines for expression levels and the most central, well connected metabolites and their metabolic connections are of particular interest. RESULTS: YANA features a platform-independent, dedicated toolbox for metabolic networks with a graphical user interface to calculate (integrating METATOOL), edit (including support for the SBML format), visualize, centralize, and compare elementary flux modes. Further, YANA calculates expected flux distributions for a given Elementary Mode (EM) activity pattern and vice versa. Moreover, a dissection algorithm, a centralization algorithm, and an average diameter routine can be used to simplify and analyze complex networks. Proteomics or gene expression data give a rough indication of some individual enzyme activities, whereas the complete flux distribution in the network is often not known. As such data are noisy, YANA features a fast evolutionary algorithm (EA) for the prediction of EM activities with minimum error, including alerts for inconsistent experimental data. We offer the possibility to include further known constraints (e.g. growth constraints) in the EA calculation process. The redox metabolism around glutathione reductase serves as an illustration example. All software and documentation are available for download at http://yana.bioapps.biozentrum.uni-wuerzburg.de. CONCLUSION: A graphical toolbox and an editor for METATOOL as well as a series of additional routines for metabolic network analyses constitute a new user-friendly software for such efforts.

Abstract Motivation: Reconstructing and analyzing the metabolic map of microorganisms is an important challenge in bioinformatics. Pathway analysis of large metabolic networks meets with the problem of combinatorial explosion of pathways. Therefore, appropriate algorithms for an automated decomposition of these networks into smaller subsystems are needed. Results: A decomposition algorithm for metabolic networks based on the local connectivity of metabolites is presented. Interrelations of this algorithm with alternative methods proposed in the literature and the theory of small world networks are discussed. The applicability of our method is illustrated by an analysis of the metabolism of Mycoplasma pneumoniae , which is an organism of considerable medical interest. The decomposition gives rise to 19 subnetworks. Three of these are here discussed in biochemical terms: arginine degradation, the tetrahydrofolate system, and nucleotide metabolism. The interrelations of pathway analysis of biochemical networks with Petri net theory are outlined. Availability: Metatoolis available from ftp://mudshark.brookes.ac.uk/pub/software/ibmpc or http://www.bioinf.mdc-berlin.de/metabolic/. The program Separatorfor decomposing metabolic networks is available from http://www.bioinf.mdc-berlin.de/metabolic/. Supplementary information: http://www.bioinf.mdc-berlin.de/metabolic/metatool/http://www.bork.embl-heidelberg.de/Annot/MP/ (re-annotation of M. pneumoniae genome) Contact: [email protected]; [email protected]; [email protected]; [email protected]

MOTIVATION: To reconstruct metabolic pathways from biochemical and/or genome sequence data, the stoichiometric and thermodynamic feasibility of the pathways has to be tested. This is achieved by characterizing the admissible region of flux distributions in steady state. This region is spanned by what can be called a convex basis. The concept of 'elementary flux modes' provides a mathematical tool to define all metabolic routes that are feasible in a given metabolic network. In addition, we define 'enzyme subsets' to be groups of enzymes that operate together in fixed flux proportions in all steady states of the system. RESULTS: Algorithms for computing the convex basis and elementary modes developed earlier are briefly reviewed. A newly developed algorithm for detecting all enzyme subsets in a given network is presented. All of these algorithms have been implemented in a novel computer program named METATOOL, whose features are outlined here. The algorithms are illustrated by an example taken from sugar metabolism. AVAILABILITY: METATOOL is available from ftp://bmsdarwin.brookes.ac. uk/pub/software/ibmpc/metatool. SUPPLEMENTARY INFORMATION: http://www. biologie.hu-berlin.de/biophysics/Theory/tpfeiffer/metatoo l.html

A refined algorithm together with a computer procedure for determining the complete set of non-negative, steady-state fluxes in biochemical reaction systems of any complexity, with or without some flux rates fixed, is given. It is shown that this set is a convex polyhedron, which may or may not be bounde. The algorithm is illustrated by several examples; one of them concerns intermediary metabolism. A computer code in standard C is presented.