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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Shadow prices in FBA capture the sensitivity of growth to flux imbalances.Consider the FBA problem with one metabolite and two reactions, formulated as: , ; ; . The solid red line indicates the feasible solution space, and the red dot indicates the optimal solution. When the flux balance condition is relaxed and the outgoing flux from M is allowed to increase, the feasible space moves to the right (dashed blue line) and the optimal solution increases. Since the objective function increases as the right-hand-side of the flux balance constraint decreases, the metabolite has a negative shadow price. In general for intracellular metabolites, negative shadow prices correspond to growth-limiting metabolites.
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pcbi-1003195-g001: Shadow prices in FBA capture the sensitivity of growth to flux imbalances.Consider the FBA problem with one metabolite and two reactions, formulated as: , ; ; . The solid red line indicates the feasible solution space, and the red dot indicates the optimal solution. When the flux balance condition is relaxed and the outgoing flux from M is allowed to increase, the feasible space moves to the right (dashed blue line) and the optimal solution increases. Since the objective function increases as the right-hand-side of the flux balance constraint decreases, the metabolite has a negative shadow price. In general for intracellular metabolites, negative shadow prices correspond to growth-limiting metabolites.

Mentions: In analogy with the interpretation of shadow prices in economics and in line with prior work on shadow prices in constraint-based metabolic modeling [16]–[19], FBA's shadow prices estimate the value of each metabolite to the global molecular budget of a growing cell (Figure 1). The interpretation of shadow prices is particularly interesting in the case of the canonical FBA objective function, i.e. maximization of the biomass flux (Z = vgrowth). In this case, a shadow price corresponds to the change in the biomass flux when one of the intracellular metabolites deviates from steady state. Importantly, if a metabolite has a negative shadow price, this means that allowing additional outflow from this metabolite (so that bi<0) will increase the maximal value of the biomass flux, implying that this metabolite is limiting for the biomass objective (Figure 1). In the remainder of this article, we test the hypothesis that shadow prices correlate with the magnitude of growth-limitation of a metabolite using experimental data, and explore the broader implications of shadow prices in modeling genome-scale metabolism.


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

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

Shadow prices in FBA capture the sensitivity of growth to flux imbalances.Consider the FBA problem with one metabolite and two reactions, formulated as: , ; ; . The solid red line indicates the feasible solution space, and the red dot indicates the optimal solution. When the flux balance condition is relaxed and the outgoing flux from M is allowed to increase, the feasible space moves to the right (dashed blue line) and the optimal solution increases. Since the objective function increases as the right-hand-side of the flux balance constraint decreases, the metabolite has a negative shadow price. In general for intracellular metabolites, negative shadow prices correspond to growth-limiting metabolites.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1003195-g001: Shadow prices in FBA capture the sensitivity of growth to flux imbalances.Consider the FBA problem with one metabolite and two reactions, formulated as: , ; ; . The solid red line indicates the feasible solution space, and the red dot indicates the optimal solution. When the flux balance condition is relaxed and the outgoing flux from M is allowed to increase, the feasible space moves to the right (dashed blue line) and the optimal solution increases. Since the objective function increases as the right-hand-side of the flux balance constraint decreases, the metabolite has a negative shadow price. In general for intracellular metabolites, negative shadow prices correspond to growth-limiting metabolites.
Mentions: In analogy with the interpretation of shadow prices in economics and in line with prior work on shadow prices in constraint-based metabolic modeling [16]–[19], FBA's shadow prices estimate the value of each metabolite to the global molecular budget of a growing cell (Figure 1). The interpretation of shadow prices is particularly interesting in the case of the canonical FBA objective function, i.e. maximization of the biomass flux (Z = vgrowth). In this case, a shadow price corresponds to the change in the biomass flux when one of the intracellular metabolites deviates from steady state. Importantly, if a metabolite has a negative shadow price, this means that allowing additional outflow from this metabolite (so that bi<0) will increase the maximal value of the biomass flux, implying that this metabolite is limiting for the biomass objective (Figure 1). In the remainder of this article, we test the hypothesis that shadow prices correlate with the magnitude of growth-limitation of a metabolite using experimental data, and explore the broader implications of shadow prices in modeling genome-scale metabolism.

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