Comparison of standard transformed reaction Gibbs energy estimates based on component contributions, to estimates based on previously available data.

<p>Comparison of standard transformed reaction Gibbs energy estimates based on component contributions, to estimates based on previously available data.</p

Distribution of the fractions of reaction vectors (black)

<p>in iAF1260 (<i>E. coli</i>) and Recon 1 (human), that were in the range of , and were thus evaluated with reactant contribution (RC). For a reaction , this fraction was calculated as . Passive and facilitated diffusion reactions, where the reactants undergo no chemical changes, are not included in the figure. 9.4% of all evaluated reactions in iAF1260 were fully evaluated using only reactant contributions. These reactions carried approximately half of the total flux (red) in 312 predicted flux distributions. The 8.3% of evaluated reactions in Recon 1 that were fully evaluated with reactant contributions, carried close to a third of the total flux in 97 predicted flux distributions.</p

Cumulative distributions for the cross-validation results.

<p>The CDF of the absolute-value residuals for both group contribution (, pink) and component contribution (, purple). The reactions were separated to ones which are (A) linearly-dependent on the set of all other reactions ( is in the range of , the stoichiometric matrix of all reactions except ), and (B) to ones which are linearly-independent (and thus component contribution uses group decompositions for at least part of the reaction). We found an 80% reduction in the median for the former set and no significant change for the latter (p-value = ).</p

pH and electrical potential in each compartment of the human reconstruction Recon 1.

<p>Electrical potential in each compartment is relative to electrical potential in the cytosol. Temperature was set to 310.15 K (37°C), and ionic strength was assumed to be 0.15 M <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003098#pcbi.1003098-Alberty1" target="_blank">[14]</a> in all compartments. Taken from <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003098#pcbi.1003098-Haraldsdttir1" target="_blank">[10]</a>.</p

The development of Gibbs energy estimation frameworks.

<p>The coverage is calculated as the percent of the relevant reactions in the KEGG database (i.e. reactions that have full chemical descriptions and are chemically balanced). The median residual (in absolute values) is calculated using leave-one-out cross-validation over the set of reactions that are within the scope of each method. Note that the reason component contribution has a higher median absolute residual than RC is only due to its higher coverage of reactions (for reactions covered by RC, the component contribution method gives the exact same predictions). *The residual value for Alberty's method is not based on cross-validation since it is a result of manual curation of multiple data sources – a process that we cannot readily repeat.</p

Metabolic enzyme cost explains variable trade-offs between microbial growth rate and yield

<div><p>Microbes may maximize the number of daughter cells per time or per amount of nutrients consumed. These two strategies correspond, respectively, to the use of enzyme-efficient or substrate-efficient metabolic pathways. In reality, fast growth is often associated with wasteful, yield-inefficient metabolism, and a general thermodynamic trade-off between growth rate and biomass yield has been proposed to explain this. We studied growth rate/yield trade-offs by using a novel modeling framework, Enzyme-Flux Cost Minimization (EFCM) and by assuming that the growth rate depends directly on the enzyme investment per rate of biomass production. In a comprehensive mathematical model of core metabolism in <i>E. coli</i>, we screened all elementary flux modes leading to cell synthesis, characterized them by the growth rates and yields they provide, and studied the shape of the resulting rate/yield Pareto front. By varying the model parameters, we found that the rate/yield trade-off is not universal, but depends on metabolic kinetics and environmental conditions. A prominent trade-off emerges under oxygen-limited growth, where yield-inefficient pathways support a 2-to-3 times higher growth rate than yield-efficient pathways. EFCM can be widely used to predict optimal metabolic states and growth rates under varying nutrient levels, perturbations of enzyme parameters, and single or multiple gene knockouts.</p></div

Uptake and secretion fluxes across EFMs.

<p>(a) Oxygen uptake (scaled by glucose uptake). Flux values are shown by colors in the rate/yield spectrum (same points as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.g002" target="_blank">Fig 2b</a>). The EFMs with the highest growth rates consume intermediate levels of oxygen. The other diagrams show <b>(b)</b> acetate secretion, <b>(c)</b> lactate secretion and <b>(d)</b> succinate secretion, each scaled by glucose uptake. Acetate secretion and <i>O</i>₂ uptake versus biomass yield are shown in Figure 9 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.s001" target="_blank">S1 Text</a>.</p

Metabolic strategies in <i>E. coli</i> metabolism.

<p><b>(a)</b> Network model of core carbon metabolism in <i>E. coli</i>. Each Elementary Flux Mode (EFM) represents a steady metabolic flux mode in the network, scaled to a unit biomass flux. Reaction fluxes defined by the EFM <i>max-gr</i> are shown by colors. In our reference conditions—i.e. high extracellular glucose and oxygen concentrations—this EFM allows for the highest growth rate among all EFMs. Some of the cofactors in the model are not shown. <b>(b)</b> Statistics of biomass-producing EFMs. <b>(c)</b> Spectrum of growth rates and yields achieved by the EFMs. The labeled focal EFMs are described in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.t001" target="_blank">Table 1</a>, and their flux maps are given in Figures 25-30 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.s001" target="_blank">S1 Text</a>. Pareto-optimal EFMs are marked by squares; the Pareto front is shown by a black line. The plot reveals a positive correlation between growth rate and yield, despite the inevitably negative correlation among Pareto-optimal EFMs. See Figure 24 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.s001" target="_blank">S1 Text</a> for a detailed view of the Pareto front and how it was sampled.</p

Predicted protein investments.

<p>(a) Predicted protein demands for the EFM <i>max-gr</i> at reference conditions. (b) Predicted protein demand for the EFM <i>max-gr</i> at varying glucose levels and reference oxygen level. The y-axis shows relative protein demands (normalized to a sum of 1). The dashed line indicates the reference glucose level (100 mM) corresponding to the pie chart in panel (a).</p

Rate/yield trade-offs and calculation of growth-optimal fluxes.

<p>(a) Rate/yield spectrum of Elementary Flux Modes (EFMs) (schematic drawing). In the scatter plot, EFMs are represented by points indicating biomass yield and maximal achievable growth rate in a given simulation scenario. Pareto-optimal EFMs are marked by red squares. The set of Pareto-optimal flux modes (black lines) contains also non-elementary flux modes. An EFM may be Pareto-optimal when compared to other EFMs, but not when compared to all possible flux modes (e.g. the EFM below the Pareto front marked by a the pink square). Growth rate and yield are positively correlated in the entire point cloud, but the points along the Pareto front show a negative correlation, indicating a trade-off. (b) Enzyme cost of metabolic fluxes. The space of stationary flux distributions is spanned by three EFMs (hypothetical example). The flux modes, scaled to unit biomass production, form a triangle. To compute the enzyme cost of a flux mode, we determine the optimal enzyme and metabolite levels. To do so, we minimize the enzymatic cost on the metabolite polytope (inset graphics) by solving a convex optimality problem called Enzyme Cost Minimization (ECM). (c) Calculation of optimal flux modes. The enzymatic cost is a concave function on the flux polytope, and its optimal points must be polytope vertices. In models without flux bounds, these vertices are EFMs and optimal flux modes can be found by screening all EFMs and choosing the one with the minimal cost.</p