Exact quantification of cellular robustness in genome-scale metabolic networks

Answer Set Programming for Computing Constraints-Based Elementary Flux Modes: Application to Escherichia coli Core Metabolism

Elementary Flux Modes (EFMs) provide a rigorous basis to systematically characterize the steady state, cellular phenotypes, as well as metabolic network robustness and fragility. However, the number of EFMs typically grows exponentially with the size of the metabolic network, leading to excessive computational demands, and unfortunately, a large fraction of these EFMs are not biologically feasible due to system constraints. This combinatorial explosion often prevents the complete analysis of genome-scale metabolic models. Traditionally, EFMs are computed by the double description method, an efficient algorithm based on matrix calculation; however, only a few constraints can be integrated into this computation. They must be monotonic with regard to the set inclusion of the supports; otherwise, they must be treated in post-processing and thus do not save computational time. We present aspefm, a hybrid computational tool based on Answer Set Programming (ASP) and Linear Programming (LP) that permits the computation of EFMs while implementing many different types of constraints. We apply our methodology to the Escherichia coli core model, which contains 226×106 EFMs. In considering transcriptional and environmental regulation, thermodynamic constraints, and resource usage considerations, the solution space is reduced to 1118 EFMs that can be computed directly with aspefm. The solution set, for E. coli growth on O2 gradients spanning fully aerobic to anaerobic, can be further reduced to four optimal EFMs using post-processing and Pareto front analysis.

Environmental flexibility does not explain metabolic robustness

Cells show remarkable resilience against genetic and environmental perturbations. However, its evolutionary origin remains obscure. In order to leverage methods of systems biology for examining cellular robustness, a computationally accessible way of quantification is needed. Here, we present an unbiased metric of structural robustness in genome-scale metabolic models based on concepts prevalent in reliability engineering and fault analysis. The probability of failure (PoF) is defined as the (weighted) portion of all possible combinations of loss-of-function mutations that disable network functionality. It can be exactly determined if all essential reactions, synthetic lethal pairs of reactions, synthetic lethal triplets of reactions etc. are known. In theory, these minimal cut sets (MCSs) can be calculated for any network, but for large models the problem remains computationally intractable. Herein, we demonstrate that even at the genome scale only the lowest-cardinality MCSs are required to efficiently approximate the PoF with reasonable accuracy. Building on an improved theoretical understanding, we analysed the robustness of 489 E. coli, Shigella, Salmonella, and fungal genome-scale metabolic models (GSMMs). In contrast to the popular "congruence theory", which explains the origin of genetic robustness as a byproduct of selection for environmental flexibility, we found no correlation between network robustness and the diversity of growth-supporting environments. On the contrary, our analysis indicates that amino acid synthesis rather than carbon metabolism dominates metabolic robustness.

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.

Computing irreversible minimal cut sets in genome-scale metabolic networks via flux cone projection

MOTIVATION

Minimal cut sets (MCSs) for metabolic networks are sets of reactions which, if they are removed from the network, prevent a target reaction from carrying flux. To compute MCSs different methods exist, which may fail to find sufficiently many MCSs for larger genome-scale networks.

RESULTS

Here we introduce irreversible minimal cut sets (iMCSs). These are MCSs that consist of irreversible reactions only. The advantage of iMCSs is that they can be computed by projecting the flux cone of the metabolic network on the set of irreversible reactions, which usually leads to a smaller cone. Using oriented matroid theory, we show how the projected cone can be computed efficiently and how this can be applied to find iMCSs even in large genome-scale networks.

AVAILABILITY AND IMPLEMENTATION

Software is freely available at https://sourceforge.net/projects/irreversibleminimalcutsets/.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Optimal bioprocess design through a gene regulatory network - Growth kinetic hybrid model: Towards replacing Monod kinetics

Currently, design and optimisation of biotechnological bioprocesses is performed either through exhaustive experimentation and/or with the use of empirical, unstructured growth kinetics models. Whereas, elaborate systems biology approaches have been recently explored, mixed-substrate utilisation is predominantly ignored despite its significance in enhancing bioprocess performance. Herein, bioprocess optimisation for an industrially-relevant bioremediation process involving a mixture of highly toxic substrates, m-xylene and toluene, was achieved through application of a novel experimental-modelling gene regulatory network - growth kinetic (GRN-GK) hybrid framework. The GRN model described the TOL and ortho-cleavage pathways in Pseudomonas putida mt-2 and captured the transcriptional kinetics expression patterns of the promoters. The GRN model informed the formulation of the growth kinetics model replacing the empirical and unstructured Monod kinetics. The GRN-GK framework's predictive capability and potential as a systematic optimal bioprocess design tool, was demonstrated by effectively predicting bioprocess performance, which was in agreement with experimental values, when compared to four commonly used models that deviated significantly from the experimental values. Significantly, a fed-batch biodegradation process was designed and optimised through the model-based control of TOL Pr promoter expression resulting in 61% and 60% enhanced pollutant removal and biomass formation, respectively, compared to the batch process. This provides strong evidence of model-based bioprocess optimisation at the gene level, rendering the GRN-GK framework as a novel and applicable approach to optimal bioprocess design. Finally, model analysis using global sensitivity analysis (GSA) suggests an alternative, systematic approach for model-driven strain modification for synthetic biology and metabolic engineering applications.

Genome-scale biological models for industrial microbial systems

The primary aims and challenges associated with microbial fermentation include achieving faster cell growth, higher productivity, and more robust production processes. Genome-scale biological models, predicting the formation of an interaction among genetic materials, enzymes, and metabolites, constitute a systematic and comprehensive platform to analyze and optimize the microbial growth and production of biological products. Genome-scale biological models can help optimize microbial growth-associated traits by simulating biomass formation, predicting growth rates, and identifying the requirements for cell growth. With regard to microbial product biosynthesis, genome-scale biological models can be used to design product biosynthetic pathways, accelerate production efficiency, and reduce metabolic side effects, leading to improved production performance. The present review discusses the development of microbial genome-scale biological models since their emergence and emphasizes their pertinent application in improving industrial microbial fermentation of biological products.

Designing Optimized Production Hosts by Metabolic Modeling

Many of the complex and expensive production steps in the chemical industry are readily available in living cells. In order to overcome the metabolic limits of these cells, the optimal genetic intervention strategies can be computed by the use of metabolic modeling. Elementary flux mode analysis (EFMA) is an ideal tool for this task, as it does not require defining a cellular objective function. We present two EFMA-based methods to optimize production hosts: (1) the standard approach that can only be used for small and medium scale metabolic networks and (2) the advanced dual system approach that can be utilized to directly compute intervention strategies in a genome-scale metabolic model.

Stoichiometric Representation of Gene-Protein-Reaction Associations Leverages Constraint-Based Analysis from Reaction to Gene-Level Phenotype Prediction

Genome-scale metabolic reconstructions are currently available for hundreds of organisms. Constraint-based modeling enables the analysis of the phenotypic landscape of these organisms, predicting the response to genetic and environmental perturbations. However, since constraint-based models can only describe the metabolic phenotype at the reaction level, understanding the mechanistic link between genotype and phenotype is still hampered by the complexity of gene-protein-reaction associations. We implement a model transformation that enables constraint-based methods to be applied at the gene level by explicitly accounting for the individual fluxes of enzymes (and subunits) encoded by each gene. We show how this can be applied to different kinds of constraint-based analysis: flux distribution prediction, gene essentiality analysis, random flux sampling, elementary mode analysis, transcriptomics data integration, and rational strain design. In each case we demonstrate how this approach can lead to improved phenotype predictions and a deeper understanding of the genotype-to-phenotype link. In particular, we show that a large fraction of reaction-based designs obtained by current strain design methods are not actually feasible, and show how our approach allows using the same methods to obtain feasible gene-based designs. We also show, by extensive comparison with experimental 13C-flux data, how simple reformulations of different simulation methods with gene-wise objective functions result in improved prediction accuracy. The model transformation proposed in this work enables existing constraint-based methods to be used at the gene level without modification. This automatically leverages phenotype analysis from reaction to gene level, improving the biological insight that can be obtained from genome-scale models.

Which sets of elementary flux modes form thermodynamically feasible flux distributions?

Elementary flux modes (EFMs) are non-decomposable steady-state fluxes through metabolic networks. Every possible flux through a network can be described as a superposition of EFMs. The definition of EFMs is based on the stoichiometry of the network, and it has been shown previously that not all EFMs are thermodynamically feasible. These infeasible EFMs cannot contribute to a biologically meaningful flux distribution. In this work, we show that a set of thermodynamically feasible EFMs need not be thermodynamically consistent. We use first principles of thermodynamics to define the feasibility of a flux distribution and present a method to compute the largest thermodynamically consistent sets (LTCSs) of EFMs. An LTCS contains the maximum number of EFMs that can be combined to form a thermodynamically feasible flux distribution. As a case study we analyze all LTCSs found in Escherichia coli when grown on glucose and show that only one LTCS shows the required phenotypical properties. Using our method, we find that in our E. coli model < 10% of all EFMs are thermodynamically relevant.