Fig 1.
A new approach to calculating metabolic fluxes.
The state of the art in measuring metabolic fluxes (13C MFA) involves using extracellular metabolite concentration data (top left) and 13C experimental data (bottom left) to find the fluxes that best fit the data (optimization approach) for a core metabolic model. The extracellular metabolite concentration data is converted into exchange fluxes, and the 13C experimental data involves the metabolite labeling patterns or Mass Distribution Vector, MDV, or Mass Isotopomer Distribution, MID (e.g., for cytosolic ribose-5-phosphate, r5p_c, among others) after labeled glucose is metabolically transformed. However, core metabolite models only represent a small fraction of all possible reactions and involve simplifications that can have an inordinate impact on the calculated fluxes. Genome-scale models can be systematically derived from the organism genome and represent a comprehensive description of metabolism, but also display more degrees of freedom (reactions) than measurements. This mismatch results in several flux profiles being compatible with the experimental data, which are badly represented by a single flux solution, even if coupled with a confidence interval. Whereas the optimization approach (top right, where each point in the x axis is a different flux, and the y axis its value and confidence interval) only provides a best estimate for the flux profile (e.g., vCS for citrate synthase, cs) and a confidence interval, BayFlux uses Bayesian Inference and Monte Carlo sampling to provide the full distribution of fluxes compatible with the experimental data (bottom right, where the x axis is the flux value, and the y axis represents P(vCS): the probability of the flux being vCS).
Fig 2.
Graphical illustration of Artificial centering Metropolis sampling (AcMet) behind the BayFlux software package.
The AcMet algorithm is used to sample the phase space and find the probability for each flux profile (see Eq (1), and Algorithm 1). Each frame illustrates a step in the AcMet algorithm, shown in only two dimensions for simplicity. The black outline represents the feasible flux phase space (a polytope), as determined by the genome scale model stoichiometric matrix. 1. Center identification. Initial ‘edge points’ are identified on the edges of the flux space by minimizing and maximizing each reaction. A running average of all samples is maintained as the ‘center’ and a series of samples are taken, always moving the current point in a direction determined by the current center and one of the edge points. Once a direction is determined, a sample is chosen from the uniform distribution within the allowable bounds, and all samples are accepted. Sufficient samples are collected to obtain a stable center. 2. Metropolis sampling. Once a stable center is identified, the center is locked, and all previous samples are discarded. New proposed samples are collected in the same manner as step 1., but without updating the center. 3. Reject low probability samples. Samples are accepted or rejected probabilistically based on the ratio of the likelihood of the data given the new sample, divided by the likelihood of the data given the current sample, L(data|new sample)/L(data|current sample). All higher likelihood samples are accepted. 4. If a sample is rejected, back up a step. If a sample is rejected, it is discarded, and the sampler is moved back to the previous sample location, and records an additional sample at the previous location. 5. If a sample is accepted, continue. If a sample is accepted, it is recorded and more samples are collected, just as in step 1, but without updating the center. 6. Halt and report posterior probability. After a sufficient number of samples are collected, they are used to describe the posterior probability distribution. See Materials and methods for further details.
Fig 3.
Flux profiles for core metabolic models obtained through BayFlux (sampling, in blue) and 13CFLUX2 (optimization, in red) are similar.
The best sample (e.g. highest posterior probability) from BayFlux (for ten million samples) is here compared with the best fit obtained from 13CFLUX2. Results for best fits and samples are similar: while there are differences for some TCA cycle fluxes (e.g. tca3, tca4), the credible intervals for these fluxes overlap with the 13CFLUX2 best fit and its confidence interval (Fig 4 and C in S1 Text), indicating that the difference is not significant given the current data. In general, BayFlux credible intervals overlap with the 13CFLUX2 best fit and its confidence interval for all different inputs (Figs B in S1 Text and C in S1 Text). All the fluxes are in units of mmol/gDW/h. Reaction names correspond to Core Metabolic Model 1 (see Materials and methods).
Fig 4.
Fluxes obtained from BayFlux using a flux sampling approach are compatible with the optimization results from 13CFLUX2, but offer more information.
Whereas the optimization approach only provides the best fit and confidence intervals, BayFlux supplies the probability distribution of all fluxes compatible with the experimental 13C data (Fig 1). Probability densities (blue), best sample (vertical magenta line), and mean (vertical green line) from BayFlux for ten million flux samples are shown vs. 13CFLUX2 best fit with confidence intervals (in orange) for 5 out of 66 fluxes (see Fig 3 for best fits and best samples for a greater number of reactions). Reaction names correspond to Core Metabolic Model 1 (see Materials and methods). The credible intervals for, e.g., fluxes tca3 and tca4 (see Fig 3) overlap with the 13CFLUX2 best fit and confidence intervals. This shows that the difference is not significant, given the current data, and highlights the importance of quantifying flux uncertainty.
Fig 5.
Using genome-scale models produces more biologically meaningful solutions.
The results obtained from BayFlux with a core metabolic model (blue) are compared with those obtained from a genome-scale model (orange). Using a genome-scale model produces a narrower flux distribution (higher certainty posterior probability distributions), as informed by a greater amount of biological knowledge encoded in the genome-scale model. Notice too, that certain reactions display very different averages. For example, GLUDY shows very different averages for the genome-scale and core metabolic models, advising caution in assuming strong inferences from 13C MFA since the results may depend significantly on the model used. Additionally, several of the probability distributions are non-Gaussian, which can only be meaningfully represented as a full distribution rather than a point or interval. We show here only reactions which occur in both models, and which show convergence across 4 repeated BayFlux runs (, Gelman-Rubin statistic [42], see main text). Reaction names correspond to Core Metabolic Model 2 (see Materials and methods).
Fig 6.
Comparison P-13C MOMA and P-13C ROOM predictions with traditional MOMA and ROOM flux predictions for the core metabolic model reactions.
The x axis represent the Euclidean distances for the flux profiles predicted by MOMA, ROOM, P-13C MOMA and P-13C ROOM to the original fluxes calculated by Toya et al. [39] using a core metabolic model (ground truth fluxes, the smaller the Euclidean distance the better the prediction). Since P-13C MOMA and P-13C ROOM yield distributions of predicted flux profiles, the y axis represents the density of distances for these methods. MOMA and ROOM yield a single predicted flux profile, so we plot a single line to represent them. A. Distribution of euclidean distances to Toya et al. pyk5h flux predictions. 23.5% of the P-13C MOMA and 36.5% of the P-13C ROOM prediction distribution were more accurate than the traditional (FBA-based) MOMA and ROOM results, respectively. B. Distribution of euclidean distances to Toya et al. pgi16h flux predictions. 18.5% of the P-13C MOMA and 38.5% of the P-13C ROOM predicted distribution flux profiles were more accurate than the traditional MOMA and ROOM results, respectively.
Fig 7.
Knockout predictions for genome-scale fluxes improve by leveraging BayFlux flux probability distributions.
Here we show the knockout prediction performance for four methods as judged by the distance of the prediction to the experimentally measured flux profile distribution (computed with BayFlux from 13C experimental data). Rather than using single fluxes to determine which method performs better (Fig E in S1 Text), distances between full flux profile distributions comprising all fluxes are calculated through a classical measure of how two probability distributions differ from each other: the multivariate Kullback-Leibler divergence [47] (higher value → worse prediction, lower value → better prediction). The distance between the WT base profile distribution and the KO experimentally observed flux profile distribution is provided for reference. Notice how P-13C MOMA and P-13C ROOM produce smaller distances to the experimental results as compared with MOMA and ROOM, indicating improved predictions. The improvement is particularly pronounced for P-13C MOMA, whereas it is marginal for P-13C ROOM. All distances are shown as relative to the knockout strains (on the left) but flux profiles inhabit a multidimensional space, so similar distances do not mean the distance among them is small (e.g., the fact that the wild type and the P-13C MOMA have a similar distance to the pyk5h knockout does not mean that these two flux distributions are similar to one another).
Fig 8.
The number of samples needed for BayFlux convergence scales approximately linearly with number of reactions in the model.
Shown are the number of samples required to reach convergence across four parallel chains, for three different size models and the fit to a linear model. We define convergence as having at least 80% of reactions with a net flux Gelman-Rubin statistic across 4 parallel chains, and exclude reactions with no sampling variance, e.g. reactions that are fully constrained and have only a single possible flux value (allowing for small amounts of numerical error) [42]. The data used here for the two largest models are the wild type 5 hour data from Toya et. al. [39].
Table 1.
Notation.