Fig 1.
TIObjFind utilizes MPA to identify weighting coefficients, termed Coefficients of Importance, on reaction (r) within a metabolic network (metabolites X).
(A) The framework consists of three main steps: Step 1: Identify a best-fit flux distribution using a single-stage (KKT) formulation of FBA, where candidate objectives c are tested against experimental fluxes (). The best-fit coefficient vector
results a feasible flux solution
for each reaction r. (B) Step 2: Generate the Mass Flow Graph,
, based on FBA solutions and apply MPA. The Mass Flow Graph is a directed, weighted graph where edge weights w represent reaction fluxes. minimum cut sets (MCs) are identified using min-cut analysis between nodes, representing the minimal reaction sets needed to sustain flow from source to target (e.g.,
). The hypothesis coefficients
are calculated as the normalized average weights of the
. (C) Step 3: Re-apply FBA using the hypothesis coefficients
as the weighted combination of the target reactions. The updated FBA solutions incorporate these coefficients, providing an optimized flux distribution that aligns with the essential pathways identified by the MPA.
Table 1.
Comparison of experimental and calculated metabolite yields in C. acetobutylicum glucose fermentation.
Fig 2.
Metabolic pathway of glucose fermentation by Cac (based on Papoutsakis [43]).
The key extracellular metabolites, highlighted in pink.
Fig 3.
Comparison of prediction errors in metabolite fluxes for C. acetobutylicum glucose fermentation in Experiment.
(A) Experiment 1, (B) Experiment 2. Three different weighting strategies were applied for flux prediction: (i) No weighting, (ii) Normalized experimental values as weights, (iii) TIObjFind-derived weights, and (iv) ObjFind method. Comparison of relative errors (%) in predicted fluxes for different weighting methods. Errors are calculated as .
Fig 4.
A graphical representation of the syntrophic consortium model.
(A) The core metabolic pathways for the syntrophic consortium pathway of an engineered Cac & Clj to produce isopropanol, butanol, and ethanol from glucose and carbon dioxide. (B) Microbial metabolic networks spy plot of the iCAC802, iJL680, and their hybrid types of models. The dots represent that the corresponding metabolites are associated with the reactions.
Table 2.
28 combinations of source (s) and a sink (t) for each species. Non-hybrid Cac was engineered to be an IBE producer. In contrast, non-hybrid Clj can only use fructose.
Fig 5.
Graphical representation of the implementation of the TIObjFind with experimental data.
Table 3.
The hypothesis coefficients as the weighted combination of the target reactions.
Fig 6.
Predicted vs. experimental titers using graph-based multi-objective FBA in DMMM.
Fig 7.
Predicted and experimental fermentation titers under single-objective function in the second stage of DMMM framework.
(A) The objective for the first 12 hr is butyrate and after that time is butanol. (B) The objective for the first 12 hr is butyrate and after that time is L-lactate.
Fig 8.
The optimization-based framework of TIObjFind.
(A) Step 1 reformulates the optimization problem as a single-level problem using the duality theorem of linear programming, subject to thermodynamic, mass balance, and uptake constraints. These dual variables, ui and g, reflect the sensitivity of the optimal objective value Zp to changes in their associated constraints. The reaction fluxes are the dual variables for the dual constraints, and
denotes weights for any potential cellular objective (e.g., biomass formation or energy production). The computed fluxes
are then mapped in the dual network. (B) Step 2 maps the FBA solutions,
, onto the Mass Flow Graph. In the dual formulation, primal reactions become metabolites in the dual network, while primal metabolites serve as constraints in the dual. Self-loops represent autocatalytic reactions, where products also act as reactants, capturing internal metabolic fluxes. (C) Step 3 shows the normalization of the pathway importance (represented as edge weights, w, in the Mass Flow Graph), leading to a new objective reaction flux distribution.