Figure 1.
(a) A schematic diagram of KFP applied to a toy metabolic network. At , the system is switched from a
C-labeled environment to
C-labeled one, and
is measured at a few time points thereafter. (b) For a given trajectory of
(the black solid curve), the three time regimes (linear, mixed and constant) are marked and three measurements are made (two in the mixed regime and one in the constant). Normalizing it gives
between 0 and 1 (the black dashed curve), parameterized by a single parameter
, which can be estimated by comparing the normalized measurements to
's of different
's (the red and blue dashed curves). (c) A schematic diagram of rKFP applied to the same network in (a). Relative quantitation is performed on
in two conditions (with subscripts
and
respectively) with the goal of estimating
. (d) The ratio in
between
and
is
(Eq. 6), and since
's and
are identifiable from relative quantitation, so is
.
Table 1.
Definitions of variables and their identifiabilities in rKFP.
Figure 2.
(a) A schematic diagram of getting the reduced model through metabolite removal in KFP. Dashed squares represent metabolites removed in the reduced model; thick dark arrow represents reduction; represents the estimated
(potentially biased). (b) The estimation results for the three options. The solid curves represent
, the dashed curves represent the cost of fitting (normalized by the number of data points to be comparable across options), and three colors represent the three options. Parameter values used for generating the simulated data:
(overall patterns independent of the choice here). (c) The estimation results in (b) in terms of the three summary statistics.
Figure 3.
(a) A schematic diagram of getting the reduced model through metabolite removal in rKFP. The same figure scheme as in Figure 2a applies here. Parameter values used in generating the simulated data: ,
, and
. (b) Dependence of bias on the pool size of the missing metabolite in two conditions. (c) Dependence of error ratio on the pool size of the missing metabolite in two conditions.
Figure 4.
Modeling reversible reactions in KFP and rKFP.
(a) A schematic diagram of the toy system considered here; for rKFP two copies of each network are similarly made as in Figure 3a. Parameter values used in generating the simulated data: ,
, and
. (b) Dependence of the summary statistics on
in KFP. (c) Dependence of the bias of
on
of the two conditions in rKFP. The red solid diagonal line corresponds to
where there is no bias. The red dashed curves correspond to a five-fold difference in the relative pool size changes between the substrate and product, a range we observe in our data.
Figure 5.
Analysis of experimental data.
(a) The diagram of glycolysis and its two branching pathways used in the analysis, where dashed squares (e.g., BPG) represent missing metabolites, dashed arrows represent missing pathways, and dashed rectangles containing solid squares represent undistinguished metabolites. Abbreviations for metabolites: GLU: glucose; G6P: glucose-6-phosphate; F6P: fructose-6-phosphate; FBP: fructose-1,6-bisphosphate; DHAP: dihydroxyacetone phosphate; GAP: glyceraldehyde-3-phosphate; BPG: 1,3-biphosphglycerate; 3PG: 3-phosphoglycerate; 2PG: 2-phosphoglycerate; PEP: phosphoenolpyruvate; PYR: pyruvate; PGL: 6-phosphogluconolactone; 6PG: phosphogluconate; R5P: the pool of ribose 5-phosphate, ribulose 5-phosphate and xylulose 5-phosphate; PHP: phosphohydroxypyruvate; 3PS: 3-phosphoserine; SER: serine; GLY: glycine. (b) An exemplary plot of the data of a metabolite and its fit. Plots of all metabolites and their fits can be found in Figure S3. (c) Histograms of 's as generated by sampling the corresponding posterior distributions in a way detailed in the Methods. Glycolysis flux refers to
in the diagram, PPP flux
, and serine synthesis flux
. The three histograms for each flux correspond to three different modeling choices described in the text.