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The authors have declared that no competing interests exist.

Conceived and designed the experiments: ER PM DS. Performed the experiments: ER. Analyzed the data: ER PM DS. Wrote the paper: ER PM DS.

Current address: Computational Biology Center, Memorial Sloan-Kettering Cancer Center, New York, New York, United States of America.

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,

Cellular metabolism is composed of a complex network of biochemical reactions that convert environmental nutrients into biosynthetic building blocks and energetic currency. Genome-scale mathematical models of metabolic networks focus largely on trying to predict the rates – or fluxes - of these reactions. By assuming that the concentrations of intracellular metabolites are at steady-state (flux balance), and invoking optimality, these constraint-based methods for modeling metabolism have offered abundant insight into how metabolic flux is routed through the cell. Here we ask how cellular growth would respond to deviations from steady state (flux imbalance) of every possible intracellular metabolite. This question can be addressed through a sensitivity analysis inherent to linear optimization theory, known as duality. We show how some features of metabolite concentrations, such as their growth-limitation and their transient response, are captured by this sensitivity analysis. Our results suggest that, in addition to predicting fluxes, stoichiometric models offer a valuable route towards probing the metabolites themselves and their relevance to growth dynamics.

Cells endure relentless variations in intra- and extra-cellular conditions. These perturbations propagate through the cell's metabolic and regulatory networks, leading to a diverse range of interdependent, transient responses in the abundance of metabolites, transcripts, and proteins

One approach to this question is to use genome-scale, constraint-based models of metabolism (such as flux balance analysis, FBA

Here, we show that some features of the behavior of intracellular metabolites are shaped by the interplay between the stoichiometric architecture of the metabolic network and the nutrient limitations imposed by environmental conditions, as well as the key role of metabolism as the conduit for allocating cellular resources towards growth. This link between structure and function of metabolism is hidden in a largely unappreciated aspect of the solution to flux balance models, namely the dual solution to the associated linear programming (LP) problem

Flux balance analysis is a method for computing expected reaction rates in complex metabolic networks, and has been described in detail elsewhere ^{LB}^{UB}

For intracellular reactions, the right-hand-side coefficients _{i}_{i}

In fact, every LP calculation can be reformulated in terms of a complementary problem known as the dual problem _{i}_{i}

In analogy with the interpretation of shadow prices in economics and in line with prior work on shadow prices in constraint-based metabolic modeling _{growth}_{i}

Consider the FBA problem with one metabolite and two reactions, formulated as:

To explore the connection between shadow prices and growth limitation, we analyzed previously collected experimental data studying the relationship between intracellular metabolite abundances and growth-limitation in

Boer and colleagues

Metabolites exhibiting

We compared the growth-limitation measurements for each metabolite identified as significantly growth-limiting or non-growth-limiting/overflow in ^{−5}, Pearson ^{−5}) and phosphate limitation (Spearman ^{−5}, Pearson ^{−5}, 7 metabolites).

In agreement with

We repeated the statistical analyses above for two “lumped” datasets containing data from (i) all three natural nutrient limitation conditions (glucose, nitrogen, and phosphate limitation; Spearman ^{−8}, Pearson ^{−5}) and (ii) all three nutrient limitation conditions, together with auxotrophies (Spearman ^{−13}, Pearson ^{−8}) and only cytosolic metabolites (MCC 0.81, p-value = 1×10^{−7}).

An important question in the above analysis, and in the calculation of shadow prices in general, is whether the possible alternative optima in the FBA optimization problem could give rise to degenerate shadow prices, and hence ambiguity in the comparison with experimental data. As described in detail in the _{i}

Our results so far indicate, in line with our intuition and with prior work on duality in FBA

Given the metabolite-specific associations between shadow prices and growth-limitation, we decided to investigate whether shadow prices could also aid in understanding other features of intracellular metabolites. In particular, we reasoned that if a metabolite is truly growth-limiting, then its concentration in the cell should be tightly controlled. If, in contrast, a growth-limiting metabolite's concentration is allowed to fluctuate or vary uncontrollably, this temporal variability would eventually propagate to growth rate and have potentially deleterious consequences. Our reasoning was further bolstered by recent studies of the metabolic response of

Based on our reasoning and on the two studies in

The results of our analysis are shown in

The height of each bar represents the number of individual metabolites that fall within a bin. Boundaries between the blue and red regions in each panel correspond to the mean values of shadow prices and temporal variation, respectively. We expect that metabolites with negative shadow prices should have small temporal variation, while metabolites with large temporal variation should have small or zero shadow prices (gray regions). Furthermore, metabolites should not exhibit large temporal variation if they have large negative shadow prices (red region). Bars tend not to fall in the red regions (as quantified statistically, see reported p-values,

Despite these statistically significant correlations, a number of outliers (

To further corroborate our findings, we tested whether the differential dynamic behavior of mutant knockout strains could be captured through our analysis. We used additional metabolite time series available for the wild type and two knockout strains (ΔGOGAT and ΔGDH) of

Changes in shadow price between the wild-type strain and two knockout strains in

Second, we tried to recapitulate the primary qualitative finding of the knockout study from

Thus, the shadow prices associated with individual intracellular metabolites provide information not only about the extent to which each metabolite is limiting for growth, but also about its overall temporal variation following a perturbation. Importantly, in the current framework, shadow prices do not provide quantitative predictions about the speed at which metabolites respond, or the new steady-state concentrations they reach. Hence, shadow price analysis should not be treated as a substitute to explicit predictions from kinetics. While our results seem to hold across different experiments in

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 _{RMF}

(_{M}

As depicted in _{i} deviates from zero

To test whether TEAM's shadow prices indeed could predict changes in intracellular metabolite abundances, we re-analyzed a transcriptomics ^{−6}; Pearson ^{−4}; ^{−8}). Interestingly, among the incorrect predictions, many were for amino acids (methionine, ornithine, proline). The failure of TEAM's shadow prices to predict changes in abundance for these compounds suggests that inconsistency with gene-expression data in pathways utilizing these metabolites may not be due to flux imbalances, and may instead indicate that other regulatory mechanisms are at play.

Using the same sensitivity analysis developed in

Constraint-based stoichiometric models of metabolism have become a widely used approach for characterizing and predicting cellular metabolic states

The results we have presented may seem at first glance surprising. How can a steady state solution convey information about the dynamical changes of metabolite pools? The answer is that flux balance models are not simply steady state solutions to a dynamical system. Rather, they use constraints and optimality to predict how a cell should allocate its resources for maximal efficiency, given the underlying network architecture. It would be tempting to make the leap of inferring that the architecture itself truly constrains the dynamics, independent of parameters and regulation. Rather, we suggest that the stoichiometric architecture may dictate how regulation should evolve to guarantee robustness to temporary variations in the intracellular milieu. If the cell cannot allow itself to accumulate or deplete certain metabolites, without incurring a substantial penalty to growth, then the response to variations in these metabolite pools should be swift. This suggests that quick allosteric and post-translational metabolite-induced regulatory feedback should control the stability of these pools

A subtle but potentially important aspect of shadow prices and their biological interpretation in metabolic network models is the fact that they are defined only over a certain range, as dictated by the structure of the feasible space. These ranges capture how large a perturbation can be before the genome-scale optimal flux distribution changes sharply (

The sensitivity of cells to variations in specific metabolite pools suggests a novel, metabolite-centric route towards the computational prediction of drug targets,

Another prospect for future studies will be to evaluate whether shadow prices may shed light on the interplay between evolution, regulation, and the sub-optimal behavior of cells. While most stoichiometric models still use maximization of growth as the central objective, a number of studies have suggested specific applications of alternative objectives. These include the Minimization Of Metabolic Adjustment

Additionally, upon availability of comprehensive data on intracellular metabolite concentrations at multiple time steps, one could envisage implementing stoichiometric models that use explicit flux imbalances (rates of accumulation/depletion) as inputs to the constraint-based model. For example, our shadow price analysis with TEAM is readily extendible to cases where the rate of accumulation/depletion is known for one subset of metabolites, but unknown for another set (_{i}

Finally, while the notion of flux imbalance analysis is not the first to bridge between the worlds of stoichiometry and metabolic dynamics

We offer here a simple derivation of the dual problem to flux balance analysis. We begin by posing the primal FBA problem^{LB}^{UB}

We implemented a number of measures to ensure that each shadow price used in our calculations was accurate and meaningful. In particular, we validated that the shadow prices obtained directly from the LP solver could not take on different values depending on whether a metabolite was accumulating or depleting (

In addition to the primal solution (optimal fluxes), the Gurobi LP solver provides the corresponding dual solution to the FBA problem. The dual solution contains (i) the shadow price value relative to each metabolite steady-state constraint and (ii) the upper (^{+}^{−}^{+}−G^{−}_{range} = 10^{−6} were discarded. Other tested values of _{range} in the range 10^{−3} to 10^{−6} led to qualitatively identical results.

Second, we ensured that alternate optimal solutions ^{+} (the change in the objective function when the right-hand-side of a constraint is ^{−} (the change in the objective function when the right-hand-side of a constraint is _{test}

In order to facilitate the implementation of degeneracy checking of shadow prices, we have provided the pseudocode below:

CHECK_DEGENERACY(S,LowerBound,UpperBound,Objective)

1 #

2 [Flux SP SPUpRange SPDownRange OptVal] =

3 p = 0.5 #

4 #

5

6

7 RHSConstraintsPlus = RHSConstraints #

8 RHSConstraintsPlus(i) = SPUpRange(i)*p #

9

10 [FluxPlus SPPlus SPUpRangePlus SPDownRangePlus OptValPlus] =

11

12

13 RHSConstraintsMinus = RHSConstraints;

14 RHSConstraintsMinus(i) = SPDownRange(i)*p;

15

16 [FluxMinus SPMinus SPUpRangeMinus SPDownRangeMinus OptValMinus] =

17

18 #

19 SPPlus(i) = (OptValPlus – OptVal)/SPPlusRange(i)*p

20 SPMinus(i) = (OptValMinus – OptVal)/SPMinusRange(i)*p

21

22 return ERROR #

23

24

In this work, all optimization problems were solved using the Gurobi optimization software

For all simulations relating to

In order to calculate temporal variation of metabolite, we use the coefficient of variation (CV):

In the main text, we show that metabolites with large negative shadow prices exhibit little temporal variation, and metabolites with large temporal variation should exhibit small (or zero) shadow price. To further corroborate the significance of the anticorrelation between shadow prices and temporal variation illustrated in

For each experiment, the vector of shadow prices (_{T}_{S}_{original}_{S}_{T}

We generated 10^{5} random permutations of _{i}_{S}_{T}_{i}<p_{original}

The penalty vector _{i}

First, we describe how we assign a penalty to each gene _{g}

Once the gene expression penalty thresholds have been calculated, the penalty for each gene _{g}_{g}_{g}

An essential part of TEAM's formulation is a user-defined required metabolic functionality (RMF). The RMF is a metabolic behavior (such as growth or the secretion of a metabolite) that TEAM must reproduce. It was observed in _{RMF,min}

Because the metabolomics and transcriptomics measurements were obtained from two distinct experiments in which the periods of the cycles were significantly different (∼8 hours vs. ∼12 hours, respectively), we used dissolved oxygen measurements (DO) (which the authors of

In order to validate whether the results using TEAM were dependent on our choice of penalty threshold

As shown in

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We are grateful to William Harcombe, Yannis Paschalidis, Arion Stettner and members of the Segrè lab for helpful feedback on the manuscript.