Fig 1.
Schematic diagram of information-economic experimental design.
The task is to determine trade-off designs that constitute the Pareto front (blue cycles, dashed blue line) in the objective space. The Pareto front trades-off information-rich, economic designs from sub-optimal, dominated solutions (orange squares, gray area). All solutions located on the Pareto front are considered to be equally good solutions of the MO-ED task.
Fig 2.
Characteristic measurement information of analytical platforms for a C3 metabolite (C1-C2-C3).
The techniques yield specific measurement groups composed of sub-sets or linear combinations of isotopomers as indicated by gray boxes. As an example, 1H-NMR and 13C-NMR measurements for C2, C-IRMS for the total fraction of unlabeled and one-labeled carbon content, MS for the intact precursor ion and MS/MS measurement for the combination of complete precursor and C2-C3 fragment ion, delivering in effect positional information, are shown. For 13C MFA only the carbon backbone of the metabolites and metabolite fractions are relevant. Further details are found in S1 Text.
Fig 3.
Geometrical interpretation of covariance based information criteria.
Fig 4.
Diagram showing the coupling the 13CFLUX2 simulator and the jMetal library.
Fig 5.
Visual elements for the interpretation of MO-ED results in the context of 13C MFA.
A: Chord diagram linking designs and objective (circular node elements) by inlying chords, here for the case of two objectives (right segment) and four input substrate species (left segments). An example is given with three substrates contributing to a design, roughly 25% Substrate1, 0% Substrate2, 50% Substrate3, and 25% Substrate4. The proportions in which the substrate species contribute is indicated by percentages. In addition, the (relative) frequency with which a certain proportion of a substrate species is proposed among the Pareto-optimal solutions is displayed by histograms located at the left outer bands. Information and cost values are scaled to the range of 0–100%. The graphic is created with Circos [54] (www.circos.ca). B: 2D scatter diagram representing the Pareto front with the dominated objective region being grayed-out. The slope of the Pareto front reflects the progressive increase in cost per information gain. The region of the Pareto front in the vicinity of a jump (green arrow) reveals that a higher information value requires the addition of at least one costly input substrate or measurement group which leads to a large cost increase. To the contrary, densely populated flat Pareto fronts indicate that CLE costs can be tuned well. The black bar on the right indicates the overall cost spread. C: Ternary triangles are commonly used in 13C MFA to represent mixture designs with three tracer species. The dashed lines relate the design point (yellow star) on the CD and the Pareto frontier with the mixture composition.
Table 1.
Design parameters for MO-ED case study.
Fig 6.
Ranges of Pareto-optimal D-criterion values ΦD,21 versus costs ΦCosts of the 3D-MO-ED problem for different analytical platforms and platform combinations filtered for solutions with full model dimensionality (p = 21).
Pareto frontiers connect lower left and upper right corners of the boxes (see also the Pareto fronts in Fig 7). Axes are log scaled. See supplementary information for results for p < 21 (S4 Text).
Fig 7.
3D-MO-ED results for GC-MS, LC-MS, LC-MS/MS, and 13C-NMR filtered for models with maximal degree of freedom (p = 21).
Top: Chord diagrams relate tracer compositions with D-criterion values and costs by showing an overlay of all Pareto-optimal tracer designs. Substrate species that contributed less than 1% to a mixture are omitted for clarity. Mid: Scatter plots showing Pareto fronts. Overall costs (black) are itemized into experimental (red) and analytical (blue) parts. Step increases of the fronts in costs are attributed to the addition of isotopically labeled substrate species as indicated. In regions of low D-criterion values, the wriggled characteristics of the graphs originates from frequent switching between input substrates and measurement replicates. Cost axes are log-scaled. Bottom: Substrate clusters. D-criterion, costs and input species hierarchically clustered by minimal Euclidean distance. For the compositions average values are given and values below 1% are omitted for clarity. Gray bars scale with the frequency of the clusters. The length of the edges (distance) represents the dissimilarities of the mixtures. Enlarged versions of the chord diagrams are provided in S4 Text.
Fig 8.
Pareto fronts obtained for LC-MS/MS filtered for solutions with full model dimensionality (p = 21).
Colored scatter plots represent 2D projections of the Pareto front approximation for the 5D-MO-ED scenario, the 3D-MO-ED Pareto front is shown in black. Design costs are color-coded. For comparability, all criterion values are normalized to [0,1]. Complete Pareto fronts are provided in S1 Data.