2.4. FBA and Shadow Prices as Tools for Sensitivity Analysis of Metabolic Networks

FBA has been used very often to study the metabolic flux distribution of an entire metabolic network [55,56,57,58]. Under the assumption of a pseudo-steady state, the net flux of production and consumption of a metabolite is assumed to be zero. In order to simulate the present experimental data of this paper, a two-step optimization approach was used. First, the maximization of CA production was defined as an objective function under the assumption that during phosphate limitation, CA production is achieved; thus, configuring the FBA primal problem. Second, the minimization of the Manhattan norm of the absolute value of all the intracellular fluxes was done by using as a constraint the maximum CA value obtained in the first step optimization. This FBA optimization was implemented in the cobra toolbox function “optimizeCbModel” [54]. Additional to the vector of fluxes, the vector of shadow prices per metabolite was calculated with the solution of the optimization problem. Shadow prices were interpreted as the sensitivity of FBA to flux imbalances obtained during the solution of the dual problem of linear optimization. The shadow prices vector relates the change in the objective function of the primal problem (i.e., CA maximization) when the flux of one of the intracellular metabolites increase or decrease, resulting in a deviation from the steady state; that is the sensitivity of FBA to flux imbalances obtained during the solution of the dual problem of the linear optimization. Reznik et al., 2013 showed the importance of shadow prices in the analysis of metabolic networks. For instance, a negative value of the shadow price for a metabolite implies that additional outflow of this metabolite would increase the value of the objective function showing that the metabolite is actually a limiting compound for the objective [42]. The biological interpretation of shadow prices in metabolic modeling has been summarized as follows: (i) negative values of shadow prices are obtained for those metabolites whose flux is limiting the objective; (ii) zero value implies that the objective function is not sensitive to that metabolite; and (iii) positive values of shadow prices are obtained for those metabolites with sufficient intracellular flux for reaching the objective.

Further details on shadow prices utilization in metabolic modeling are available (see references [25,54]). The experimental growth rate was set as a hard constraint in the optimization problem with the aim to get a better representation of the metabolic scenarios, while the measured uptake and secretion rates (when available) were set as lower (O₂, glycerol, succinate, oxaloacetate, and malate) and upper (CO₂, pyruvate and acetate) bounds, respectively. The IBM CPLEX Studio Optimizer v12.10.0 was used for the solution of the optimization problems. It is worth mentioning that the sign of shadow prices calculated by CPLEX solver was negative (−λ), and it was considered for the further biological interpretation of shadow prices results.

Furthermore, the solution space for the fluxes’ distributions of two different metabolic states of S. clavuligerus was explored by using FVA and flux sampling using the sclav_red metabolic model. The model was constrained with uptake and secretion rates corresponding to 36 h (batch stage) and 48 h (12 h after starting the fed-batch stage) of cultivation. The model predictions were contrasted with experimental exchange fluxes for the assayed metabolites and nutrients using the squared error (SE) as indicated in Equation (1) for a given metabolite i. MSE was calculated as indicated in Equation (2) to assess the deviation of model predictions and experimental fluxes for a given metabolic scenario.

where p is the number of experimental fluxes considered in the scenario, X is the experimental flux value, and X^ is the model-predicted flux value.

In addition, the feasible flux range was determined for each reaction via FVA. Alternatively to the unique solution provided by the FBA problem, the coordinate hit-and-run with rounding algorithm (CHRR) [59] was used for sampling the solution space for the explored cultivation conditions (batch and fed-batch stages). The following CHRR parameters were set: the sampling density, nStepsPerPoint = 1848, and the number of samples, nPointsReturned = 5000. A Kruskal—Wallis test was used to assess whether flux samples generated using either the batch (36 h) or fed-batch (48 h) constraints stemmed from the same statistical distribution [60]. A principal component analysis (PCA) was applied to metabolite shadow prices for the identification of those metabolites that contributes to the changes in flux distributions or phenotypes between the culture conditions under study.

All simulations were carried out in RAVEN 2.0 [61] and COBRA Toolbox v3.0 [54] under MATLAB 2020b (see Supplementary File S3).