Balancing biomass reaction stoichiometry and measured fluxes in flux balance analysis.

MOTIVATION: Flux balance analysis (FBA) is widely recognized as an important method for studying metabolic networks. When incorporating flux measurements of certain reactions into an FBA problem, it is possible that the underlying linear program may become infeasible, e.g. due to measurement or modeling inaccuracies. Furthermore, while the biomass reaction is of central importance in FBA models, its stoichiometry is often a rough estimate and a source of high uncertainty. RESULTS: In this work, we present a method that allows modifications to the biomass reaction stoichiometry as a means to (i) render the FBA problem feasible and (ii) improve the accuracy of the model by corrections in the biomass composition. Optionally, the adjustment of the biomass composition can be used in conjunction with a previously introduced approach for balancing inconsistent fluxes to obtain a feasible FBA system. We demonstrate the value of our approach by analyzing realistic flux measurements of E.coli. In particular, we find that the growth-associated maintenance (GAM) demand of ATP, which is typically integrated with the biomass reaction, is likely overestimated in recent genome-scale models, at least for certain growth conditions. In light of these findings, we discuss issues related to the determination and inclusion of GAM values in constraint-based models. Overall, our method can uncover potential errors and suggest adjustments in the assumed biomass composition in FBA models based on inconsistencies between the model and measured fluxes. AVAILABILITY AND IMPLEMENTATION: The developed method has been implemented in our software tool CNApy available from https://github.com/cnapy-org/CNApy.

StrainDesign: a comprehensive Python package for computational design of metabolic networks.

SUMMARY: Various constraint-based optimization approaches have been developed for the computational analysis and design of metabolic networks. Herein, we present StrainDesign, a comprehensive Python package that builds upon the COBRApy toolbox and integrates the most popular metabolic design algorithms, including nested strain optimization methods such as OptKnock, RobustKnock and OptCouple as well as the more general minimal cut sets approach. The optimization approaches are embedded in individual modules, which can also be combined for setting up more elaborate strain design problems. Advanced features, such as the efficient integration of GPR rules and the possibility to consider gene and reaction additions or regulatory interventions, have been generalized and are available for all modules. The package uses state-of-the-art preprocessing methods, supports multiple solvers and provides a number of enhanced tools for analyzing computed intervention strategies including 2D and 3D plots of user-selected metabolic fluxes or yields. Furthermore, a user-friendly interface for the StrainDesign package has been implemented in the GUI-based metabolic modeling software CNApy. StrainDesign provides thus a unique and rich framework for computational strain design in Python, uniting many algorithmic developments in the field and allowing modular extension in the future. AVAILABILITY AND IMPLEMENTATION: The StrainDesign package can be retrieved from PyPi, Anaconda and GitHub (https://github.com/klamt-lab/straindesign) and is also part of the latest CNApy package.

Analyzing and Resolving Infeasibility in Flux Balance Analysis of Metabolic Networks.

Flux balance analysis (FBA) is a key method for the constraint-based analysis of metabolic networks. A technical problem may occur in FBA when known (e.g., measured) fluxes of certain reactions are integrated into an FBA scenario rendering the underlying linear program (LP) infeasible, for example, due to inconsistencies between some of the measured fluxes causing a violation of the steady-state or other constraints. Here, we present and compare two methods, one based on an LP and one on a quadratic program (QP), to find minimal corrections for the given flux values so that the FBA problem becomes feasible. We provide a general guide on how to treat infeasible FBA systems in practice and discuss relevant examples of potentially infeasible scenarios in core and genome-scale metabolic models. Finally, we also highlight and clarify the relationships to classical metabolic flux analysis, where solely algebraic approaches are used to compute unknown metabolic rates from measured fluxes and to balance infeasible flux scenarios.

CNApy: a CellNetAnalyzer GUI in Python for analyzing and designing metabolic networks.

SUMMARY: Constraint-based reconstruction and analysis (COBRA) is a widely used modeling framework for analyzing and designing metabolic networks. Here, we present CNApy, an open-source cross-platform desktop application written in Python, which offers a state-of-the-art graphical front-end for the intuitive analysis of metabolic networks with COBRA methods. While the basic look-and-feel of CNApy is similar to the user interface of the MATLAB toolbox CellNetAnalyzer, it provides various enhanced features by using components of the powerful Qt library. CNApy supports a number of standard and advanced COBRA techniques and further functionalities can be easily embedded in its GUI facilitating modular extension in the future. AVAILABILITY AND IMPLEMENTATION: CNApy can be installed via conda and its source code is freely available at https://github.com/cnapy-org/CNApy under the Apache 2 license.

Speeding up the core algorithm for the dual calculation of minimal cut sets in large metabolic networks.

BACKGROUND: The concept of minimal cut sets (MCS) has become an important mathematical framework for analyzing and (re)designing metabolic networks. However, the calculation of MCS in genome-scale metabolic models is a complex computational problem. The development of duality-based algorithms in the last years allowed the enumeration of thousands of MCS in genome-scale networks by solving mixed-integer linear problems (MILP). A recent advancement in this field was the introduction of the MCS2 approach. In contrast to the Farkas-lemma-based dual system used in earlier studies, the MCS2 approach employs a more condensed representation of the dual system based on the nullspace of the stoichiometric matrix, which, due to its reduced dimension, holds promise to further enhance MCS computations. RESULTS: In this work, we introduce several new variants and modifications of duality-based MCS algorithms and benchmark their effects on the overall performance. As one major result, we generalize the original MCS2 approach (which was limited to blocking the operation of certain target reactions) to the most general case of MCS computations with arbitrary target and desired regions. Building upon these developments, we introduce a new MILP variant which allows maximal flexibility in the formulation of MCS problems and fully leverages the reduced size of the nullspace-based dual system. With a comprehensive set of benchmarks, we show that the MILP with the nullspace-based dual system outperforms the MILP with the Farkas-lemma-based dual system speeding up MCS computation with an averaged factor of approximately 2.5. We furthermore present several simplifications in the formulation of constraints, mainly related to binary variables, which further enhance the performance of MCS-related MILP. However, the benchmarks also reveal that some highly condensed formulations of constraints, especially on reversible reactions, may lead to worse behavior when compared to variants with a larger number of (more explicit) constraints and involved variables. CONCLUSIONS: Our results further enhance the algorithmic toolbox for MCS calculations and are of general importance for theoretical developments as well as for practical applications of the MCS framework.

An extended and generalized framework for the calculation of metabolic intervention strategies based on minimal cut sets.

The concept of minimal cut sets (MCS) provides a flexible framework for analyzing properties of metabolic networks and for computing metabolic intervention strategies. In particular, it has been used to support the targeted design of microbial strains for bio-based production processes. Herein we present a number of major extensions that generalize the existing MCS approach and broaden its scope for applications in metabolic engineering. We first introduce a modified approach to integrate gene-protein-reaction associations (GPR) in the metabolic network structure for the computation of gene-based intervention strategies. In particular, we present a set of novel compression rules for GPR associations, which effectively speedup the computation of gene-based MCS by a factor of up to one order of magnitude. These rules are not specific for MCS and as well applicable to other computational strain design methods. Second, we enhance the MCS framework by allowing the definition of multiple target (undesired) and multiple protected (desired) regions. This enables precise tailoring of the metabolic solution space of the designed strain with unlimited flexibility. Together with further generalizations such as individual cost factors for each intervention, direct combinations of reaction/gene deletions and additions as well as the possibility to search for substrate co-feeding strategies, the scope of the MCS framework could be broadly extended. We demonstrate the applicability and performance benefits of the described developments by computing (gene-based) Escherichia coli strain designs for the bio-based production of 2,3-butanediol, a chemical, that has recently received much attention in the field of metabolic engineering. With our extended framework, we could identify promising strain designs that were formerly unpredictable, including those based on substrate co-feeding.

MoVE identifies metabolic valves to switch between phenotypic states.

Metabolism is highly regulated, allowing for robust and complex behavior. This behavior can often be achieved by controlling a small number of important metabolic reactions, or metabolic valves. Here, we present a method to identify the location of such valves: the metabolic valve enumerator (MoVE). MoVE uses a metabolic model to identify genetic intervention strategies which decouple two desired phenotypes. We apply this method to identify valves which can decouple growth and production to systematically improve the rate and yield of biochemical production processes. We apply this algorithm to the production of diverse compounds and obtained solutions for over 70% of our targets, identifying a small number of highly represented valves to achieve near maximal growth and production. MoVE offers a systematic approach to identify metabolic valves using metabolic models, providing insight into the architecture of metabolic networks and accelerating the widespread implementation of dynamic flux redirection in diverse systems.

OptMDFpathway: Identification of metabolic pathways with maximal thermodynamic driving force and its application for analyzing the endogenous CO2 fixation potential of Escherichia coli.

Constraint-based modeling techniques have become a standard tool for the in silico analysis of metabolic networks. To further improve their accuracy, recent methodological developments focused on integration of thermodynamic information in metabolic models to assess the feasibility of flux distributions by thermodynamic driving forces. Here we present OptMDFpathway, a method that extends the recently proposed framework of Max-min Driving Force (MDF) for thermodynamic pathway analysis. Given a metabolic network model, OptMDFpathway identifies both the optimal MDF for a desired phenotypic behavior as well as the respective pathway itself that supports the optimal driving force. OptMDFpathway is formulated as a mixed-integer linear program and is applicable to genome-scale metabolic networks. As an important theoretical result, we also show that there exists always at least one elementary mode in the network that reaches the maximal MDF. We employed our new approach to systematically identify all substrate-product combinations in Escherichia coli where product synthesis allows for concomitant net CO2 assimilation via thermodynamically feasible pathways. Although biomass synthesis cannot be coupled to net CO2 fixation in E. coli we found that as many as 145 of the 949 cytosolic carbon metabolites contained in the genome-scale model iJO1366 enable net CO2 incorporation along thermodynamically feasible pathways with glycerol as substrate and 34 with glucose. The most promising products in terms of carbon assimilation yield and thermodynamic driving forces are orotate, aspartate and the C4-metabolites of the tricarboxylic acid cycle. We also identified thermodynamic bottlenecks frequently limiting the maximal driving force of the CO2-fixing pathways. Our results indicate that heterotrophic organisms like E. coli hold a possibly underestimated potential for CO2 assimilation which may complement existing biotechnological approaches for capturing CO2. Furthermore, we envision that the developed OptMDFpathway approach can be used for many other applications within the framework of constrained-based modeling and for rational design of metabolic networks.

Growth-coupled overproduction is feasible for almost all metabolites in five major production organisms.

Computational modelling of metabolic networks has become an established procedure in the metabolic engineering of production strains. One key principle that is frequently used to guide the rational design of microbial cell factories is the stoichiometric coupling of growth and product synthesis, which makes production of the desired compound obligatory for growth. Here we show that the coupling of growth and production is feasible under appropriate conditions for almost all metabolites in genome-scale metabolic models of five major production organisms. These organisms comprise eukaryotes and prokaryotes as well as heterotrophic and photoautotrophic organisms, which shows that growth coupling as a strain design principle has a wide applicability. The feasibility of coupling is proven by calculating appropriate reaction knockouts, which enforce the coupling behaviour. The study presented here is the most comprehensive computational investigation of growth-coupled production so far and its results are of fundamental importance for rational metabolic engineering.

Use of CellNetAnalyzer in biotechnology and metabolic engineering.

Mathematical models of the cellular metabolism have become an essential tool for the optimization of biotechnological processes. They help to obtain a systemic understanding of the metabolic processes in the used microorganisms and to find suitable genetic modifications maximizing the production performance. In particular, methods of stoichiometric and constraint-based modeling are frequently used in the context of metabolic and bioprocess engineering. Since metabolic networks can be complex and comprise hundreds or even thousands of metabolites and reactions, dedicated software tools are required for an efficient analysis. One such software suite is CellNetAnalyzer, a MATLAB package providing, among others, various methods for analyzing stoichiometric and constraint-based metabolic models. CellNetAnalyzer can be used via command-line based operations or via a graphical user interface with embedded network visualizations. Herein we will present key functionalities of CellNetAnalyzer for applications in biotechnology and metabolic engineering and thereby review constraint-based modeling techniques such as metabolic flux analysis, flux balance analysis, flux variability analysis, metabolic pathway analysis (elementary flux modes) and methods for computational strain design.

MUFINS: multi-formalism interaction network simulator.

Systems Biology has established numerous approaches for mechanistic modeling of molecular networks in the cell and a legacy of models. The current frontier is the integration of models expressed in different formalisms to address the multi-scale biological system organization challenge. We present MUFINS (MUlti-Formalism Interaction Network Simulator) software, implementing a unique set of approaches for multi-formalism simulation of interaction networks. We extend the constraint-based modeling (CBM) framework by incorporation of linear inhibition constraints, enabling for the first time linear modeling of networks simultaneously describing gene regulation, signaling and whole-cell metabolism at steady state. We present a use case where a logical hypergraph model of a regulatory network is expressed by linear constraints and integrated with a Genome-Scale Metabolic Network (GSMN) of mouse macrophage. We experimentally validate predictions, demonstrating application of our software in an iterative cycle of hypothesis generation, validation and model refinement. MUFINS incorporates an extended version of our Quasi-Steady State Petri Net approach to integrate dynamic models with CBM, which we demonstrate through a dynamic model of cortisol signaling integrated with the human Recon2 GSMN and a model of nutrient dynamics in physiological compartments. Finally, we implement a number of methods for deriving metabolic states from ~omics data, including our new variant of the iMAT congruency approach. We compare our approach with iMAT through the analysis of 262 individual tumor transcriptomes, recovering features of metabolic reprogramming in cancer. The software provides graphics user interface with network visualization, which facilitates use by researchers who are not experienced in coding and mathematical modeling environments.

Genome-scale strain designs based on regulatory minimal cut sets.

MOTIVATION: Stoichiometric and constraint-based methods of computational strain design have become an important tool for rational metabolic engineering. One of those relies on the concept of constrained minimal cut sets (cMCSs). However, as most other techniques, cMCSs may consider only reaction (or gene) knockouts to achieve a desired phenotype. RESULTS: We generalize the cMCSs approach to constrained regulatory MCSs (cRegMCSs), where up/downregulation of reaction rates can be combined along with reaction deletions. We show that flux up/downregulations can virtually be treated as cuts allowing their direct integration into the algorithmic framework of cMCSs. Because of vastly enlarged search spaces in genome-scale networks, we developed strategies to (optionally) preselect suitable candidates for flux regulation and novel algorithmic techniques to further enhance efficiency and speed of cMCSs calculation. We illustrate the cRegMCSs approach by a simple example network and apply it then by identifying strain designs for ethanol production in a genome-scale metabolic model of Escherichia coli. The results clearly show that cRegMCSs combining reaction deletions and flux regulations provide a much larger number of suitable strain designs, many of which are significantly smaller relative to cMCSs involving only knockouts. Furthermore, with cRegMCSs, one may also enable the fine tuning of desired behaviours in a narrower range. The new cRegMCSs approach may thus accelerate the implementation of model-based strain designs for the bio-based production of fuels and chemicals. AVAILABILITY AND IMPLEMENTATION: MATLAB code and the examples can be downloaded at http://www.mpi-magdeburg.mpg.de/projects/cna/etcdownloads.html. CONTACT: krishna.mahadevan@utoronto.ca or klamt@mpi-magdeburg.mpg.de SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Enumeration of smallest intervention strategies in genome-scale metabolic networks.

One ultimate goal of metabolic network modeling is the rational redesign of biochemical networks to optimize the production of certain compounds by cellular systems. Although several constraint-based optimization techniques have been developed for this purpose, methods for systematic enumeration of intervention strategies in genome-scale metabolic networks are still lacking. In principle, Minimal Cut Sets (MCSs; inclusion-minimal combinations of reaction or gene deletions that lead to the fulfilment of a given intervention goal) provide an exhaustive enumeration approach. However, their disadvantage is the combinatorial explosion in larger networks and the requirement to compute first the elementary modes (EMs) which itself is impractical in genome-scale networks. We present MCSEnumerator, a new method for effective enumeration of the smallest MCSs (with fewest interventions) in genome-scale metabolic network models. For this we combine two approaches, namely (i) the mapping of MCSs to EMs in a dual network, and (ii) a modified algorithm by which shortest EMs can be effectively determined in large networks. In this way, we can identify the smallest MCSs by calculating the shortest EMs in the dual network. Realistic application examples demonstrate that our algorithm is able to list thousands of the most efficient intervention strategies in genome-scale networks for various intervention problems. For instance, for the first time we could enumerate all synthetic lethals in E.coli with combinations of up to 5 reactions. We also applied the new algorithm exemplarily to compute strain designs for growth-coupled synthesis of different products (ethanol, fumarate, serine) by E.coli. We found numerous new engineering strategies partially requiring less knockouts and guaranteeing higher product yields (even without the assumption of optimal growth) than reported previously. The strength of the presented approach is that smallest intervention strategies can be quickly calculated and screened with neither network size nor the number of required interventions posing major challenges.

SBML qualitative models: a model representation format and infrastructure to foster interactions between qualitative modelling formalisms and tools.

BACKGROUND: Qualitative frameworks, especially those based on the logical discrete formalism, are increasingly used to model regulatory and signalling networks. A major advantage of these frameworks is that they do not require precise quantitative data, and that they are well-suited for studies of large networks. While numerous groups have developed specific computational tools that provide original methods to analyse qualitative models, a standard format to exchange qualitative models has been missing. RESULTS: We present the Systems Biology Markup Language (SBML) Qualitative Models Package ("qual"), an extension of the SBML Level 3 standard designed for computer representation of qualitative models of biological networks. We demonstrate the interoperability of models via SBML qual through the analysis of a specific signalling network by three independent software tools. Furthermore, the collective effort to define the SBML qual format paved the way for the development of LogicalModel, an open-source model library, which will facilitate the adoption of the format as well as the collaborative development of algorithms to analyse qualitative models. CONCLUSIONS: SBML qual allows the exchange of qualitative models among a number of complementary software tools. SBML qual has the potential to promote collaborative work on the development of novel computational approaches, as well as on the specification and the analysis of comprehensive qualitative models of regulatory and signalling networks.

Minimal cut sets in a metabolic network are elementary modes in a dual network.

MOTIVATION: Elementary modes (EMs) and minimal cut sets (MCSs) provide important techniques for metabolic network modeling. Whereas EMs describe minimal subnetworks that can function in steady state, MCSs are sets of reactions whose removal will disable certain network functions. Effective algorithms were developed for EM computation while calculation of MCSs is typically addressed by indirect methods requiring the computation of EMs as initial step. RESULTS: In this contribution, we provide a method that determines MCSs directly without calculating the EMs. We introduce a duality framework for metabolic networks where the enumeration of MCSs in the original network is reduced to identifying the EMs in a dual network. As a further extension, we propose a generalization of MCSs in metabolic networks by allowing the combination of inhomogeneous constraints on reaction rates. This framework provides a promising tool to open the concept of EMs and MCSs to a wider class of applications. CONTACT: utz-uwe.haus@math.ethz.ch; klamt@mpi-magdeburg.mpg.de SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

An application programming interface for CellNetAnalyzer.

CellNetAnalyzer (CNA) is a MATLAB toolbox providing computational methods for studying structure and function of metabolic and cellular signaling networks. In order to allow non-experts to use these methods easily, CNA provides GUI-based interactive network maps as a means of parameter input and result visualization. However, with the availability of high-throughput data, there is a need to make CNA's functionality also accessible in batch mode for automatic data processing. Furthermore, as some algorithms of CNA are of general relevance for network analysis it would be desirable if they could be called as sub-routines by other applications. For this purpose, we developed an API (application programming interface) for CNA allowing users (i) to access the content of network models in CNA, (ii) to use CNA's network analysis capabilities independent of the GUI, and (iii) to interact with the GUI to facilitate the development of graphical plugins. Here we describe the organization of network projects in CNA and the application of the new API functions to these projects. This includes the creation of network projects from scratch, loading and saving of projects and scenarios, and the application of the actual analysis methods. Furthermore, API functions for the import/export of metabolic models in SBML format and for accessing the GUI are described. Lastly, two example applications demonstrate the use and versatile applicability of CNA's API. CNA is freely available for academic use and can be downloaded from http://www.mpi-magdeburg.mpg.de/projects/cna/cna.html.

Computing combinatorial intervention strategies and failure modes in signaling networks.

The identification of combinatorial intervention strategies and the elucidation of failure modes that may cause aberrant behavior of cellular signaling networks are highly relevant topics in cell biology, medicine, and pharmaceutical industry. We have recently introduced the concept of minimal intervention sets (MISs)--minimal combinations of knock-ins and knock-outs provoking a desired/observed response in certain target nodes--to tackle those problems within a Boolean/logical framework. We first generalize the notion of MISs and then present several techniques for search space reduction facilitating the enumeration of MISs in networks of realistic size. One strategy exploits topological information about network-wide interdependencies between the nodes to discard unfavorable single interventions. A similar technique checks during the algorithm whether all target nodes of an intervention problem can be influenced in appropriate direction (up/down) by the interventions contained in MIS candidates. Another strategy takes lessons from electrical engineering: certain interventions are equivalent with respect to their effect on the target nodes and can therefore be grouped in fault equivalence classes (FECs). FECs resulting from so-called structural equivalence can be easily computed in a preprocessing step, with the advantage that only one representative per class needs to be considered when constructing the MISs in the main algorithm. With intervention problems from realistic networks as benchmarks, we show that these algorithmic improvements may reduce the computation time up to 99%, increasing the applicability of MISs in practice.

Computing paths and cycles in biological interaction graphs.

BACKGROUND: Interaction graphs (signed directed graphs) provide an important qualitative modeling approach for Systems Biology. They enable the analysis of causal relationships in cellular networks and can even be useful for predicting qualitative aspects of systems dynamics. Fundamental issues in the analysis of interaction graphs are the enumeration of paths and cycles (feedback loops) and the calculation of shortest positive/negative paths. These computational problems have been discussed only to a minor extent in the context of Systems Biology and in particular the shortest signed paths problem requires algorithmic developments. RESULTS: We first review algorithms for the enumeration of paths and cycles and show that these algorithms are superior to a recently proposed enumeration approach based on elementary-modes computation. The main part of this work deals with the computation of shortest positive/negative paths, an NP-complete problem for which only very few algorithms are described in the literature. We propose extensions and several new algorithm variants for computing either exact results or approximations. Benchmarks with various concrete biological networks show that exact results can sometimes be obtained in networks with several hundred nodes. A class of even larger graphs can still be treated exactly by a new algorithm combining exhaustive and simple search strategies. For graphs, where the computation of exact solutions becomes time-consuming or infeasible, we devised an approximative algorithm with polynomial complexity. Strikingly, in realistic networks (where a comparison with exact results was possible) this algorithm delivered results that are very close or equal to the exact values. This phenomenon can probably be attributed to the particular topology of cellular signaling and regulatory networks which contain a relatively low number of negative feedback loops. CONCLUSION: The calculation of shortest positive/negative paths and cycles in interaction graphs is an important method for network analysis in Systems Biology. This contribution draws the attention of the community to this important computational problem and provides a number of new algorithms, partially specifically tailored for biological interaction graphs. All algorithms have been implemented in the CellNetAnalyzer framework which can be downloaded for academic use at http://www.mpi-magdeburg.mpg.de/projects/cna/cna.html.

Structural robustness of metabolic networks with respect to multiple knockouts.

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

Understanding the roadmap of metabolism by pathway analysis.

The theoretical investigation of the structure of metabolic systems has recently attracted increasing interest. In this chapter, the basic concepts of metabolic pathway analysis are described and various applications are outlined. In particular, the concepts of nullspace and elementary flux modes are explained. The presentation is illustrated by a simple example from tyrosine metabolism and a system describing lysine production in Corynebacterium glutamicum. The latter system gives rise to 37 elementary modes, 36 of which produce lysine with different molar yields. The examples illustrate that metabolic pathway analysis is a useful tool for better understanding the complex architecture of intracellular metabolism, for determining the pathways on which the molar conversion yield of a substrate-product pair under study is maximal, and for assigning functions to orphan genes (functional genomics). Moreover, problems emerging in the modeling of large networks are discussed. An outlook on current trends in the field concludes the chapter.