Impact of stoichiometry representation on simulation of genotype-phenotype relationships in metabolic networks

Abstract

Genome-scale metabolic networks provide a comprehensive structural framework for modeling genotype-phenotype relationships through flux simulations. The solution space for the metabolic flux state of the cell is typically very large and optimization-based approaches are often necessary for predicting the active metabolic state under specific environmental conditions. The objective function to be used in such optimization algorithms is directly linked with the biological hypothesis underlying the model and therefore it is one of the most relevant parameters for successful modeling. Although linear combination of selected fluxes is widely used for formulating metabolic objective functions, we show that the resulting optimization problem is sensitive towards stoichiometry representation of the metabolic network. This undesirable sensitivity leads to different simulation results when using numerically different but biochemically equivalent stoichiometry representations and thereby makes biological interpretation intrinsically subjective and ambiguous. We hereby propose a new method, Minimization of Metabolites Balance (MiMBl), which decouples the artifacts of stoichiometry representation from the formulation of the desired objective functions, by casting objective functions using metabolite turnovers rather than fluxes. By simulating perturbed metabolic networks, we demonstrate that the use of stoichiometry representation independent algorithms is fundamental for unambiguously linking modeling results with biological interpretation. For example, MiMBl allowed us to expand the scope of metabolic modeling in elucidating the mechanistic basis of several genetic interactions in Saccharomyces cerevisiae.

Figure 1. Minimization of overall intracellular flux leads to divergent predictions for flux distribution when using biochemically equivalent stoichiometry representations.

Shown are predicted fluxes through key pathways within the S. cerevisiae central carbon metabolism, using numerically different but biochemically equivalent stoichiometric representation of reaction RPI1 (θ_RPI1, Methods). θ_RPI1 is represented on the x-axis, while fold-change of fluxes relatively to θ = 1 is represented on the y-axis. A representative reaction from each of the pathways was selected to illustrate the flux re-arrangement; FBA1 for glycolysis, ZWF1 for pentose phosphate pathway, CIT1 for tricarboxilic acid cycle and NID1 for oxidative phosphorylation. Note that θ = 1 is an arbitrary reference, as the stoichiometric representation of any reaction is subjective, often scaled to have coefficient of 1 for one of the reactants/products.

Figure 2. MiMBl shows robust simulation results while using alternative stoichiometry representations – illustration using a toy-model.

a) Toy-model: R1 to R7 and A to D represent reactions and metabolites, respectively. In the wild-type, or reference, flux goes from A to D via R5. R6 and R2–R3–R4 are two alternative pathways for flux re-distribution after deletion of R5. b) Flux through reactions R2 (full symbols) and R6 (open symbols) obtained after simulation of minimization of metabolic adjustment with lMoMA (black), quadratic MoMA (qMoMA, gray) and MiMBl (red) using numerically different but biochemically equivalent representations of reaction R6 (given by different scaling factor θ_R6, Methods). c) Formulation of objective functions of minimization of metabolic adjustment for lMoMA, qMoMA and MiMBl (Methods). d) Optimal objective function value (distance) obtained for minimization of metabolic adjustment using lMoMA (black), qMoMA (gray) and MiMBl (red) as function of θ_R6.

Figure 3. Sensitivity of MiMBl towards the use of alternative reference flux distributions.

a) The histogram shows the distribution of variability in the predicted growth of single gene knockout mutants while using 500 different FBA alternative optima as reference flux distributions. b) Case study of YLR377C knockout simulations using different reference flux distributions as input. The predicted growth varies between 50–100% of that of the wild-type.

Figure 4. Understanding genetic interactions by using MiMBl.

a, b) The accuracy of genetic interaction predictions by FBA, lMoMA and MiMBl was assessed by calculating the sensitivity and precision for positive (a) and negative (b) interactions. Sensitivity was calculated as the fraction of experimentally observed interactions captured by the algorithm, while precision was estimated as the fraction of experimentally observed interactions among the predicted interactions. c) Venn diagram showing the overlap of the correctly predicted interactions by FBA, MiMBl and lMoMA. d, e) Distribution of the graph theoretical distances, within the yeast metabolic network, between the interacting genes captured by FBA (d) and MiMBl (e). As MiMBl also captured the majority of FBA predicted interactions, only exclusive MiMBl interactions are shown in (e). f) The S. cerevisiae genetic interactions network correctly predicted by MiMBl and/or FBA (FBA – dashed line, MiMBl – dotted line, both – full line). Positive and negative interactions are distinguished by color (orange and blue, respectively) and the opacity of the edges is inversely proportional to the network distance between the corresponding genes. Gray-filled nodes represent genes that display both positive and negative interactions. Gray areas enclose isoenzymes where at least one of them was found to interact with other genes in the metabolic network.