Parameter Balancing in Kinetic Models of Cell Metabolism

Kinetic modeling of metabolic pathways has become a major field of systems biology. It combines structural information about metabolic pathways with quantitative enzymatic rate laws. Some of the kinetic constants needed for a model could be collected from ever-growing literature and public web resources, but they are often incomplete, incompatible, or simply not available. We address this lack of information by parameter balancing, a method to complete given sets of kinetic constants. Based on Bayesian parameter estimation, it exploits the thermodynamic dependencies among different biochemical quantities to guess realistic model parameters from available kinetic data. Our algorithm accounts for varying measurement conditions in the input data (pH value and temperature). It can process kinetic constants and state-dependent quantities such as metabolite concentrations or chemical potentials, and uses prior distributions and data augmentation to keep the estimated quantities within plausible ranges. An online service and free software for parameter balancing with models provided in SBML format (Systems Biology Markup Language) is accessible at www.semanticsbml.org. We demonstrate its practical use with a small model of the phosphofructokinase reaction and discuss its possible applications and limitations. In the future, parameter balancing could become an important routine step in the kinetic modeling of large metabolic networks

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

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

Growth rates achieved with two variants of glycolysis.

<p>(a) Glucose- and oxygen-dependent growth rates predicted for wild-type <i>E. coli</i>. Same data as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.g004" target="_blank">Fig 4(c)</a>, but shown as a heatmap. <i>E. coli</i> can employ two variants of glycolysis: the Embden-Meyerhof-Parnas (EMP) pathway, which is common also to eukaryotes, and the Entner-Doudoroff (ED) pathway, which provides a lower ATP yield at a much lower enzyme demand [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.ref021" target="_blank">21</a>]. (b) A simulated ED knockout strain that must use the EMP pathway. The heatmap shows the relative growth advantage of the wild-type strain (i.e. of reintroducing the ED pathway to the cell). The ED pathway provides its highest advantage at low oxygen and medium to low glucose levels. (c) Growth advantage provided by the EMP pathway. The advantage is highest at glucose concentrations below 10 μM. (d) Comparison between the two knockout strains. Blue areas indicate conditions where ED is more favorable, and red areas indicate conditions where EMP would be favored. The dark blue region at low oxygen and medium glucose levels may correspond to the environment of bacteria such as <i>Z. mobilis</i>, which uses the ED pathway exclusively [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.ref050" target="_blank">50</a>]. The same data are shown as Monod surface plots in Figure 21 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.s001" target="_blank">S1 Text</a>.</p

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

Focal EFMs representing different growth strategies.

<p>Metabolic fluxes are given in carbon moles (or O₂ moles) per carbon moles of glucose uptake. Growth rates are given for reference conditions [glucose] = 100 mM, and [O₂] = 0.21 mM. For more details, see Table 10 in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006010#pcbi.1006010.s001" target="_blank">S1 Text</a>. Abbreviations: * <i>max-gr</i>: maximum growth rate; <i>max-yield</i>: maximum yield; <i>pareto</i>: a Pareto optimal EFM with higher growth rate than max-yield, and higher yield than max-gr; <i>ana-lac</i>: anaerobic lactate fermentation; <i>aero-ace</i>: aerobic acetate fermentation; <i>exp</i>: experimentally measured flux distribution.</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

Enzyme demand in a metabolic pathway.

<p>(a) Pathway with reversible Michaelis-Menten kinetics (equilibrium constants, catalytic constants, and <i>K</i>_M values are set to values of 1, [A] and [B] denote the variable concentrations of intermediates A and B in mM). The external metabolite levels [X] and [Y] are fixed. Plots (b)-(d) show the enzyme demand of reactions 1, 2, and 3 at given flux <i>v</i> = 1 according to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005167#pcbi.1005167.e004" target="_blank">Eq (2)</a>. Grey regions represent infeasible metabolite profiles. At the edges of the feasible region (where A and B are close to chemical equilibrium), the thermodynamic driving force goes to zero. Since small forces must be compensated by high enzyme levels, edges of the feasible region are always dark blue. For example, in reaction 1 (panel (b)), enzyme demand increases with the level of A (x-axis) and goes to infinity as the mass-action ratio [<i>A</i>]/[<i>X</i>] approaches the equilibrium constant (where the driving force vanishes). (e) Total enzyme demand, obtained by summing all enzyme levels. The metabolite polytope—the intersection of feasible regions for all reactions—is a triangle, and enzyme demand is a convex function on this triangle. The point of minimum total enzyme demand defines the optimal metabolite levels and optimal enzyme levels. (f) As the <i>k</i>_cat value of the first reaction is lowered by a factor of 5, states close to the triangle edge of reaction 1 become more expensive and the optimum point is shifted away from the edge. (g) The same model with a physiological upper bound on the concentration [A]. The bound defines a new triangle edge. Since this edge is not caused by thermodynamics, it can contain an optimum point, in which driving forces are far from zero and enzyme costs are kept low.</p

Mathematical symbols used.

<p>The fitness unit Darwin (D) is a proxy for the different fitness units used in cell models. Reaction must be orientated in such a way that all fluxes are positive. To define metabolite log-concentrations, we use the standard concentration <i>c</i>_<i>σ</i> = 1 mM. For a more comprehensive list of mathematical symbols used in ECM, see Table C in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005167#pcbi.1005167.s001" target="_blank">S1 Text</a>.</p