Fig 1.
A schematic view of the various steps towards a large-scale pathway analysis with the proposed ensemble modeling approach. The individual ensembles (Boxes with x-, k-, v-ensemble) are generated (--) and tested(—) separately. A reduced set of its members is then selected for further analysis. Mixed ensembles (Boxes with x/k-, v/x-ensemble) are obtained by generating all possible combinations (⋯‧) of the selected members of the former ensemble partitions. At the end of the process, the amount of models actually processed is the cartesian product of the small subsets of selected models in each ensemble. This subset is but a small fraction of all the possible combinations that have thus been proven inadequate at an early stage without the need for extensive testing or excessive computationally expensive operations.
Fig 2.
A three-step unbranched pathway with negative feedback loop is chosen as the model pathway of this case study. The selected pathway, with the initial precursor concentration x0 and the final product concentration x3, is reversible at every intermediary step. The dependent variable concentrations of the metabolites are named x1 to x3 and an additional variable x4 has been added to reflect variations in product demand.
Fig 3.
Thermodynamic feasibility analysis of the x-ensemble.
The distributions of metabolite concentrations generated using a uniform distribution () and the subset filtered by thermodynamic feasibility for different values of the second equilibrium constant: K2 = 1000 (
), K2 = 1 (
), K2 = 0.05 (
). All other equilibrium constants are fixed at 1.
Fig 4.
Examples of projections of the criterion space.
Each model of the ensemble is represented by a point. The Pareto fronts are calculated for selected goal combinations. Depending on the number of selected goals, 2-dimensional or multi-dimensional Pareto frontiers can be calculated. The datasets are then grouped into the non-efficient datasets () and the Pareto-efficient datasets (
). The utopian point is shown by a red star (
). At the utopian point, both goals reach their optimization goal.
Fig 5.
X-ensemble in criterion and design space.
The upper left figure shows ensembles in the criterion space with a Pareto front for simultaneous minimization of total metabolite concentration and draft. The upper right panel shows the same data in the design space (unfeasible cases not shown). The lower left panel shows a projection of a different two-dimensional criterion space of draft vs. thermodynamic cost. In this case, the Pareto optimal solutions were calcuated using all three criteria simultaneously. On the lower right panel, only the Pareto optimal solutions are shown in design space. In all design space plots, the feasible area is marked as a tetrahedron and the vertex drawn in red corresponds to the x-total vs. draft Pareto front. Datapoints are colored depending on whether they correspond to thermodynamically unfeasible (), feasible (
) or the Pareto-optimal ensembles (
).
Fig 6.
Criterion space projections of supply-driven and demand-driven models.
Projections of different models in 2D criterion space related to response to supply (left) and to demand (right). The bulk of models are marked in blue (), and those models in the intersection of the upper 50% for all three robustness criteria—APS, ISS, and ISD—are superimposed in red (
).
Fig 7.
Analysis of demand-driven pathway designs.
A: Flux vs. product concentration response to an increase in demand. B: Definition of a new criterion, demand performance, that ensures a balanced response both in flux and in product concentration. C: Pareto front for the tradeoff between demand performance and sensitivity. In all cases, the bulk of the models is represented by blue points (), Pareto efficient solutions in red (
), and a Pareto front enriched by convex combinations is shown in yellow (
).
Fig 8.
Analysis of supply-driven pathway designs.
A: Flux vs. product concentration response of the pathway to an increase in supply. B: Pareto front for the tradeoff between supply performance and sensitivity. In all cases, the bulk of the models is represented by blue points (), Pareto efficient solutions in red (
), and a Pareto front enriched by convex combinations is shown in yellow (
).
Fig 9.
Parameter values of the logistic regression model.
The influence of each-value (model parameter) on the respective goal and kinetic order is shown. Coefficients for logistic regressions to classify the the top 50th percentile for each analyzed goal: Intermediate sensitivity to supply (ISS), intermediate sensitivity to demand (ISD), aggregate parameter sensitivity (APS), product supply gain (PSG), flux supply gain (FSG), product demand gain (PDG) and flux demand gain (FDG).
Fig 10.
Left: The synthesized data are shown in the criterion space. The location of designs generated by direct sampling in PCA coordinates () is compared with non-efficient datasets (
). The synthetic data were generated as two independent segments. In both cases, the first principal component was varied as a free parameter, while a minimal set of other PCs were varying according to how they correlated in the corresponding segment of the Pareto front. For the first segment (horizontal trend), PC2 that was varied linearly to the function PC2 = −0.70PC1 − 0.83, while all other PCs were kept constant. For the second segment, two PCs other than PC1 (PC4 and PC7) had to be varied according to the following fucntions: PC4 = 0.95PC1 − 2.07 and PC7 = −0.54PC1 + 1.18. Thus, the segments were obtained through the variation of only two and three PCs respectively. Right: The right panel shows the relevant principal components as bar plots of the coefficients with which each kinetic order contributes to the corresponding PC. The dashed red lines show the average value of all the coefficients in absolute value. Thus, the relevant kinetic orders for each PC are identified as those with coefficients exceeding the dashed lines, such as f2,2 in PC2.
Fig 11.
Dynamic simulation of perturbed pathway designs.
Models of demand-driven pathways are chosen and visualized in the criterion space with the respective Pareto front of ISD vs. demand performance. Simulations are performed using the normalized system with a reference state, setting/fixating the initial steady state concentration at 1. After one second, the demand factor x4 is increased by 100%. Three (1-3) designs are chosen from distinct regions along the Pareto front and one (4) was selected from a region away from the front. The dynamic response of the different models is visualized in for the sum of the intermediate concentrations x1 + x2 (middle panel) and for the product concentration x3 (bottom panel).