Limits...
Flux imbalance analysis and the sensitivity of cellular growth to changes in metabolite pools.

Reznik E, Mehta P, Segrè D - PLoS Comput. Biol. (2013)

Bottom Line: In particular, we explore the biological significance of shadow prices in a constraint-based method for integrating gene expression data with a stoichiometric model.In this case, shadow prices point to metabolites that should rise or drop in concentration in order to increase consistency between flux predictions and gene expression data.In general, these results suggest that the sensitivity of metabolic optima to violations of the steady state constraints carries biologically significant information on the processes that control intracellular metabolites in the cell.

View Article: PubMed Central - PubMed

Affiliation: Department of Biomedical Engineering, Boston University, Boston, Massachusetts, United States of America.

ABSTRACT
Stoichiometric models of metabolism, such as flux balance analysis (FBA), are classically applied to predicting steady state rates - or fluxes - of metabolic reactions in genome-scale metabolic networks. Here we revisit the central assumption of FBA, i.e. that intracellular metabolites are at steady state, and show that deviations from flux balance (i.e. flux imbalances) are informative of some features of in vivo metabolite concentrations. Mathematically, the sensitivity of FBA to these flux imbalances is captured by a native feature of linear optimization, the dual problem, and its corresponding variables, known as shadow prices. First, using recently published data on chemostat growth of Saccharomyces cerevisae under different nutrient limitations, we show that shadow prices anticorrelate with experimentally measured degrees of growth limitation of intracellular metabolites. We next hypothesize that metabolites which are limiting for growth (and thus have very negative shadow price) cannot vary dramatically in an uncontrolled way, and must respond rapidly to perturbations. Using a collection of published datasets monitoring the time-dependent metabolomic response of Escherichia coli to carbon and nitrogen perturbations, we test this hypothesis and find that metabolites with negative shadow price indeed show lower temporal variation following a perturbation than metabolites with zero shadow price. Finally, we illustrate the broader applicability of flux imbalance analysis to other constraint-based methods. In particular, we explore the biological significance of shadow prices in a constraint-based method for integrating gene expression data with a stoichiometric model. In this case, shadow prices point to metabolites that should rise or drop in concentration in order to increase consistency between flux predictions and gene expression data. In general, these results suggest that the sensitivity of metabolic optima to violations of the steady state constraints carries biologically significant information on the processes that control intracellular metabolites in the cell.

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In a constraint-based method that integrates gene expression (GIMME/TEAM), shadow prices predict the direction of changes in metabolite abundance.(A) Schematic of the GIMME/TEAM algorithm. Enzymes whose constituent genes show very low expression (red) are penalized. Then, a flux distribution is identified with the lowest total penalty (in this case, the alternative pathway with high expression, colored in green). (B) Schematic of the interpretation of shadow prices in TEAM. Consider a situation in which, at steady-state, a reaction with low gene expression (red, high penalty) is inferred by the model to carry a high flux, leading to a high penalty. When the metabolite is allowed to deviate from steady-state by lowering the flux through the highly penalized reaction, the penalty predicted by TEAM falls. The shadow price λM for this metabolite, whose concentration is predicted to be decreasing, is thus positive. (C) Shadow prices predicted by TEAM and observed changes in metabolite abundance are significantly negatively correlated. A threshold of θ = 0.88 was used, although other values of θ yielded similar results (SI Figure S1). Changes in metabolite abundance were calculated using measurements between hours 10 and 11 in [33] where acetate was observed to be secreted from the cell [32]. Expression data used as input to TEAM is taken from hour 35 of [32]. Both time points correspond to the same phase in the metabolic cycle of yeast, during the end of the oxidative and beginning of the reductive/building phase.
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pcbi-1003195-g005: In a constraint-based method that integrates gene expression (GIMME/TEAM), shadow prices predict the direction of changes in metabolite abundance.(A) Schematic of the GIMME/TEAM algorithm. Enzymes whose constituent genes show very low expression (red) are penalized. Then, a flux distribution is identified with the lowest total penalty (in this case, the alternative pathway with high expression, colored in green). (B) Schematic of the interpretation of shadow prices in TEAM. Consider a situation in which, at steady-state, a reaction with low gene expression (red, high penalty) is inferred by the model to carry a high flux, leading to a high penalty. When the metabolite is allowed to deviate from steady-state by lowering the flux through the highly penalized reaction, the penalty predicted by TEAM falls. The shadow price λM for this metabolite, whose concentration is predicted to be decreasing, is thus positive. (C) Shadow prices predicted by TEAM and observed changes in metabolite abundance are significantly negatively correlated. A threshold of θ = 0.88 was used, although other values of θ yielded similar results (SI Figure S1). Changes in metabolite abundance were calculated using measurements between hours 10 and 11 in [33] where acetate was observed to be secreted from the cell [32]. Expression data used as input to TEAM is taken from hour 35 of [32]. Both time points correspond to the same phase in the metabolic cycle of yeast, during the end of the oxidative and beginning of the reductive/building phase.

Mentions: So far, we have corroborated the notion that shadow prices are indicative of growth limitation, and demonstrated that shadow prices are even more broadly related to metabolite dynamic variability. As described above and in the Methods, shadow prices are dependent on the underlying stoichiometric model, and the specific environmental conditions. Correspondingly, in the analysis shown up to now, we have explored the relevance of shadow prices across different conditions (different nutrient limitations and genetic modifications). There is, however, a third feature that shadow prices crucially depend on, i.e. the specific objective function used in the FBA optimization. Does the analysis of shadow prices have a meaning and an application for stoichiometric problems with radically different objective functions, or is it biologically interpretable only for the growth maximization objective? To answer this question, we decided to explore the significance of shadow prices in a recently proposed optimization problem aimed at identifying genome-scale fluxes that minimize the inconsistency relative to a given set of gene expression data. This approach, pioneered with the GIMME algorithm [30], and recently re-elaborated in the time-dependent TEAM method [31], is a way of integrating gene expression data with stoichiometric models of metabolism, in order to obtain better predictions and understanding of cellular physiology. Instead of maximizing growth, GIMME and TEAM minimize the conflict between gene expression data and flux predictions using a penalty score (see Methods). In particular, fluxes whose corresponding gene(s) exhibit low expression are penalized (Figure 5A) in proportion to how much lower the gene expression is, relative to a given gene-specific threshold (see [31] and Methods). The cumulative penalty obtained from all these costs (termed the Inconsistency Score, IS) is minimized across the entire metabolic network [30], [31]. This problem can be solved again using linear programming, in analogy to the FBA problem illustrated above:(4)(5)(6)(7)where c is a vector of reaction penalties, and the reaction flux vRMF is a required metabolic functionality (RMF), some minimal, user-defined metabolic behavior which the model must reproduce (for example, growth at a minimal rate or the secretion of a metabolite). One of the reasons the RMF constraint is imposed is to avoid the trivial solution v = 0.


Flux imbalance analysis and the sensitivity of cellular growth to changes in metabolite pools.

Reznik E, Mehta P, Segrè D - PLoS Comput. Biol. (2013)

In a constraint-based method that integrates gene expression (GIMME/TEAM), shadow prices predict the direction of changes in metabolite abundance.(A) Schematic of the GIMME/TEAM algorithm. Enzymes whose constituent genes show very low expression (red) are penalized. Then, a flux distribution is identified with the lowest total penalty (in this case, the alternative pathway with high expression, colored in green). (B) Schematic of the interpretation of shadow prices in TEAM. Consider a situation in which, at steady-state, a reaction with low gene expression (red, high penalty) is inferred by the model to carry a high flux, leading to a high penalty. When the metabolite is allowed to deviate from steady-state by lowering the flux through the highly penalized reaction, the penalty predicted by TEAM falls. The shadow price λM for this metabolite, whose concentration is predicted to be decreasing, is thus positive. (C) Shadow prices predicted by TEAM and observed changes in metabolite abundance are significantly negatively correlated. A threshold of θ = 0.88 was used, although other values of θ yielded similar results (SI Figure S1). Changes in metabolite abundance were calculated using measurements between hours 10 and 11 in [33] where acetate was observed to be secreted from the cell [32]. Expression data used as input to TEAM is taken from hour 35 of [32]. Both time points correspond to the same phase in the metabolic cycle of yeast, during the end of the oxidative and beginning of the reductive/building phase.
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC3757068&req=5

pcbi-1003195-g005: In a constraint-based method that integrates gene expression (GIMME/TEAM), shadow prices predict the direction of changes in metabolite abundance.(A) Schematic of the GIMME/TEAM algorithm. Enzymes whose constituent genes show very low expression (red) are penalized. Then, a flux distribution is identified with the lowest total penalty (in this case, the alternative pathway with high expression, colored in green). (B) Schematic of the interpretation of shadow prices in TEAM. Consider a situation in which, at steady-state, a reaction with low gene expression (red, high penalty) is inferred by the model to carry a high flux, leading to a high penalty. When the metabolite is allowed to deviate from steady-state by lowering the flux through the highly penalized reaction, the penalty predicted by TEAM falls. The shadow price λM for this metabolite, whose concentration is predicted to be decreasing, is thus positive. (C) Shadow prices predicted by TEAM and observed changes in metabolite abundance are significantly negatively correlated. A threshold of θ = 0.88 was used, although other values of θ yielded similar results (SI Figure S1). Changes in metabolite abundance were calculated using measurements between hours 10 and 11 in [33] where acetate was observed to be secreted from the cell [32]. Expression data used as input to TEAM is taken from hour 35 of [32]. Both time points correspond to the same phase in the metabolic cycle of yeast, during the end of the oxidative and beginning of the reductive/building phase.
Mentions: So far, we have corroborated the notion that shadow prices are indicative of growth limitation, and demonstrated that shadow prices are even more broadly related to metabolite dynamic variability. As described above and in the Methods, shadow prices are dependent on the underlying stoichiometric model, and the specific environmental conditions. Correspondingly, in the analysis shown up to now, we have explored the relevance of shadow prices across different conditions (different nutrient limitations and genetic modifications). There is, however, a third feature that shadow prices crucially depend on, i.e. the specific objective function used in the FBA optimization. Does the analysis of shadow prices have a meaning and an application for stoichiometric problems with radically different objective functions, or is it biologically interpretable only for the growth maximization objective? To answer this question, we decided to explore the significance of shadow prices in a recently proposed optimization problem aimed at identifying genome-scale fluxes that minimize the inconsistency relative to a given set of gene expression data. This approach, pioneered with the GIMME algorithm [30], and recently re-elaborated in the time-dependent TEAM method [31], is a way of integrating gene expression data with stoichiometric models of metabolism, in order to obtain better predictions and understanding of cellular physiology. Instead of maximizing growth, GIMME and TEAM minimize the conflict between gene expression data and flux predictions using a penalty score (see Methods). In particular, fluxes whose corresponding gene(s) exhibit low expression are penalized (Figure 5A) in proportion to how much lower the gene expression is, relative to a given gene-specific threshold (see [31] and Methods). The cumulative penalty obtained from all these costs (termed the Inconsistency Score, IS) is minimized across the entire metabolic network [30], [31]. This problem can be solved again using linear programming, in analogy to the FBA problem illustrated above:(4)(5)(6)(7)where c is a vector of reaction penalties, and the reaction flux vRMF is a required metabolic functionality (RMF), some minimal, user-defined metabolic behavior which the model must reproduce (for example, growth at a minimal rate or the secretion of a metabolite). One of the reasons the RMF constraint is imposed is to avoid the trivial solution v = 0.

Bottom Line: In particular, we explore the biological significance of shadow prices in a constraint-based method for integrating gene expression data with a stoichiometric model.In this case, shadow prices point to metabolites that should rise or drop in concentration in order to increase consistency between flux predictions and gene expression data.In general, these results suggest that the sensitivity of metabolic optima to violations of the steady state constraints carries biologically significant information on the processes that control intracellular metabolites in the cell.

View Article: PubMed Central - PubMed

Affiliation: Department of Biomedical Engineering, Boston University, Boston, Massachusetts, United States of America.

ABSTRACT
Stoichiometric models of metabolism, such as flux balance analysis (FBA), are classically applied to predicting steady state rates - or fluxes - of metabolic reactions in genome-scale metabolic networks. Here we revisit the central assumption of FBA, i.e. that intracellular metabolites are at steady state, and show that deviations from flux balance (i.e. flux imbalances) are informative of some features of in vivo metabolite concentrations. Mathematically, the sensitivity of FBA to these flux imbalances is captured by a native feature of linear optimization, the dual problem, and its corresponding variables, known as shadow prices. First, using recently published data on chemostat growth of Saccharomyces cerevisae under different nutrient limitations, we show that shadow prices anticorrelate with experimentally measured degrees of growth limitation of intracellular metabolites. We next hypothesize that metabolites which are limiting for growth (and thus have very negative shadow price) cannot vary dramatically in an uncontrolled way, and must respond rapidly to perturbations. Using a collection of published datasets monitoring the time-dependent metabolomic response of Escherichia coli to carbon and nitrogen perturbations, we test this hypothesis and find that metabolites with negative shadow price indeed show lower temporal variation following a perturbation than metabolites with zero shadow price. Finally, we illustrate the broader applicability of flux imbalance analysis to other constraint-based methods. In particular, we explore the biological significance of shadow prices in a constraint-based method for integrating gene expression data with a stoichiometric model. In this case, shadow prices point to metabolites that should rise or drop in concentration in order to increase consistency between flux predictions and gene expression data. In general, these results suggest that the sensitivity of metabolic optima to violations of the steady state constraints carries biologically significant information on the processes that control intracellular metabolites in the cell.

Show MeSH
Related in: MedlinePlus