FBA shadow price analysis in knockout strains.

<p>Changes in shadow price between the wild-type strain and two knockout strains in <i>E. coli</i> following nitrogen upshift. In comparison to the wild-type, only the ΔGOGAT contains any metabolites which are substantially more growth-limiting.</p

Shadow prices correlate with temporal variation in metabolite abundance in <i>E.</i><i>coli</i>.

<p>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, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003195#pcbi.1003195.s004" target="_blank">Table S2</a>) highlighting the capacity of shadow prices to capture features of metabolite dynamics. Subplots correspond to different experimental conditions: (<b>A</b>) nitrogen upshift (<b>B</b>) glucose starvation, (<b>C</b>) acetate limitation, and (<b>D</b>) glycerol limitation.</p

Shadow prices anticorrelate with experimental measurements of growth limitation.

<p>Metabolites exhibiting were experimentally determined to be growth-limiting. Growth-limitation and shadow prices in FBA are significantly anticorrelated under all nutrient limitations from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003195#pcbi.1003195-Boer1" target="_blank">[21]</a>. To make the data more comparable across different nutrient limitations, the data is plotted on a log scale. All points to the left of the grey bar have a shadow price of zero. All correlations for this data (calculated using a linear scale, not the log scale depicted in the Figure) are reported in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003195#pcbi.1003195.s003" target="_blank">Table S1</a>. Abbreviations: 6PDG, 6-phospho-d-gluconate; ADE, Adenosine; ALA, Alanine; ARG, Arginine; ATP, ATP; CHO, Choline; CTP, CTP; CYD, Cytidine; CYT, Cytosine; DHAP, Dihydroxyacetone-Phosphate; DOG, Deoxyguanosine; DS7P, D-sedoheptulose-7-phosphate; F16P, Fructose-1,6-bisphosphate; GLN, Glutamine; GLU, Glutamate; GUA, Guanosine; HIS, Histidine; INO, Inosine; LEU, Leucine/isoleucine; LYS, Lysine; NAD, NAD+; NAG1P, N-acetyl-glucosamine-1-phosphate; NIC, Nicotinate; ORN, Ornithine; PHP, Phenylpyruvate; PYR, Pyruvate; RIBP, Ribose-phosphate; SER, Serine; SUC, Sucrose; THR, Threonine; TRE, Trehalose; TRP, Tryptophan; UDPG, UDP-D-glucose; UTP, UTP. For clarity, only cytosolic metabolites from the metabolic model are plotted.</p

Distributions of alternative metrics for correct and randomized orders of colonization.

<p>(A) Similar to what shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077617#pone-0077617-g001" target="_blank">Fig. 1B</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077617#pone-0077617-g002" target="_blank">Fig. 2</a>, we computed inter-species distance between organisms along paths that respect (red) or do not respect (blue) the layered order of colonization of the Kolenbrander map. Here, however, as opposed to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077617#pone-0077617-g002" target="_blank">Fig. 2</a>, we compute the Jaccard distance between two species based on their profiles of non-enzyme genes (as identifiable through KEGG KO numbers). (B) The correct and incorrect orders of colonization are compared based on an information metric, rather than on the Jaccard distance. In walking along a colonization order path from one organism to the next, we compute (in Nats) the amount of information added due to the presence of previously absent enzymes. The added information for each pair of adjacent organisms is summed to form the added information score, along paths that respect (red) or do not respect (blue) the layered order of colonization. The purple distribution is obtained by computing the added information scores for orders of colonization that reflect the layered structure, but walk through it in reverse order (i.e. from the outer layer downwards towards the salivary pellicle).</p

Metabolic pathway enrichment across layers.

<p>Based on the enzyme content of the different species found in different layers of the biofilm (with layers labeled from 1 to 4, see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077617#pone-0077617-g001" target="_blank">Fig. 1</a>), one can estimate whether any given layer is enriched for specific metabolic functions. Enzyme and pathway enrichments for each layer are computed based on a standard GSEA algorithm. Black boxes in the pathways-by-layers matrix denote enrichment of a particular KEGG pathway in a given layer. The pathway abbreviations are as follows: APM-Arginine and proline metabolism, BAOLN-Biosynthesis of alkaloids derived from ornithine lysine and nicotinic acid, CFP-Carbon fixation pathways in prokaryotes, GDM-Glyoxylate and dicarboxylate metabolism, GST-Glycine, serine and threonine metabolism, NM-Nitrogen metabolism, PCM-Porphyrin and chlorophyll metabolism, PGI-Pentose and glucuronate interconversions, PPM-Propanoate metabolism, PYM-Pyruvate metabolism, TBB-Terpenoid backbone biosynthesis, and TCA-Tricarboxylic acid cycle.</p

Expected synergy between metabolic networks as a function of metabolic distance.

<p>The synergy is computed as the count of elementary flux modes (pathways) that are feasible for a metabolic network that is the union of two networks with a given Jaccard’s distance from each other, normalized to the count of elementary flux modes of the constituent networks. The count of elementary flux modes can be thought of as an estimate of the number of distinct metabolic tasks that the network can perform, i.e. its versatility. Hence, the graph shows how the versatility of two conjoined networks relative to the constituent networks is maximal for an intermediate Jaccard’s distance between such networks. 100 random paired networks were generated for each of several possible Jaccard’s distances. Bar heights reflect the average normalized increase in the number of elementary flux modes, whereas error bars represent the standard error of the mean.</p

Flux Imbalance Analysis and the Sensitivity of Cellular Growth to Changes in Metabolite Pools

<div><p>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>i.e.</i> that intracellular metabolites are at steady state, and show that deviations from flux balance (<i>i.e.</i> flux imbalances) are informative of some features of <i>in vivo</i> 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 <i>Saccharomyces cerevisae</i> 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 <i>Escherichia coli</i> 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.</p></div

Temporal Expression-based Analysis of Metabolism

<div><p>Metabolic flux is frequently rerouted through cellular metabolism in response to dynamic changes in the intra- and extra-cellular environment. Capturing the mechanisms underlying these metabolic transitions in quantitative and predictive models is a prominent challenge in systems biology. Progress in this regard has been made by integrating high-throughput gene expression data into genome-scale stoichiometric models of metabolism. Here, we extend previous approaches to perform a Temporal Expression-based Analysis of Metabolism (TEAM). We apply TEAM to understanding the complex metabolic dynamics of the respiratorily versatile bacterium <em>Shewanella oneidensis</em> grown under aerobic, lactate-limited conditions. TEAM predicts temporal metabolic flux distributions using time-series gene expression data. Increased predictive power is achieved by supplementing these data with a large reference compendium of gene expression, which allows us to take into account the unique character of the distribution of expression of each individual gene. We further propose a straightforward method for studying the sensitivity of TEAM to changes in its fundamental free threshold parameter <em>θ</em>, and reveal that discrete zones of distinct metabolic behavior arise as this parameter is changed. By comparing the qualitative characteristics of these zones to additional experimental data, we are able to constrain the range of <em>θ</em> to a small, well-defined interval. In parallel, the sensitivity analysis reveals the inherently difficult nature of dynamic metabolic flux modeling: small errors early in the simulation propagate to relatively large changes later in the simulation. We expect that handling such “history-dependent” sensitivities will be a major challenge in the future development of dynamic metabolic-modeling techniques.</p> </div

Metabolic Proximity in the Order of Colonization of a Microbial Community

<div><p>Microbial biofilms are often composed of multiple bacterial species that accumulate by adhering to a surface and to each other. Biofilms can be resistant to antibiotics and physical stresses, posing unresolved challenges in the fight against infectious diseases. It has been suggested that early colonizers of certain biofilms could cause local environmental changes, favoring the aggregation of subsequent organisms. Here we ask whether the enzyme content of different microbes in a well-characterized dental biofilm can be used to predict their order of colonization. We define a metabolic distance between different species, based on the overlap in their enzyme content. We next use this metric to quantify the average metabolic distance between neighboring organisms in the biofilm. We find that this distance is significantly smaller than the one observed for a random choice of prokaryotes, probably reflecting the environmental constraints on metabolic function of the community. More surprisingly, this metabolic metric is able to discriminate between observed and randomized orders of colonization of the biofilm, with the observed orders displaying smaller metabolic distance than randomized ones. By complementing these results with the analysis of individual vs. joint metabolic networks, we find that the tendency towards minimal metabolic distance may be counter-balanced by a propensity to pair organisms with maximal joint potential for synergistic interactions. The trade-off between these two tendencies may create a “sweet spot” of optimal inter-organism distance, with possible broad implications for our understanding of microbial community organization.</p></div

Shadow prices in FBA capture the sensitivity of growth to flux imbalances.

<p>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 <i>M</i> 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 <i>intracellular</i> metabolites, negative shadow prices correspond to growth-limiting metabolites.</p