Flux balanced analysis was based on the reconstruction based on Oberhardt et al. [7]. The model incorporates 1056 genes related with 883 reactions including anabolic pathways necessary for the synthesis of most cellular biomass from basic metabolic precursors. Moreover the model takes into account the metabolic processes includeing phenazine synthesis, which is the main focus for our work.

Biomass as reported by Oberhardt and phenazine flux was selected as the main objective for the biological system studied [8,9]. Moreover, special restrictions were imposed over oxygen flux as the electron shuttles are mainly produced upon anaerobic conditions, to achieve this conditions the flux that represented the oxygen intake by the cell was reduced to almost zero. The optimization problem posed “Eqs.1 and 2” was solved using Xpress MP Professional^®. The inputs for this model were the stoichiometric matrix, the number of reactions and metabolites.

Multilayered optimization frameworks are often applied to optimize systems where multi-objective hierarchy is present. In this case biomass growth is prioritized by the cell regardless of the internal and external conditions, then a bi-level optimization approach seems to be ideal. Nevertheless, this approach demands an intensive exploration in a intangible search space. Here in this work, we switched our bi-level approach to a single level optimization problem by imposing an additional restriction to the biomass flux based on the steady state value from the in silico wild type strain. The optimization problem could be posed as:

Maximize

Subject to:

Subject to:

where is the maximum value of biomass flux and the restrictions refer to substratum flux and micro-aerobic conditions to promote phenanize flux.

First, the value of biomass restriction is determined to the solution of a single level optimization problemwhere the biomass flux is the objective function to maximize, with the same characteristics of the bi-level optimization model. As a second step, this value of biomass flux is used as a restriction in a new optimization problem. This optimization model has the objective function to maximize the phenanize flux with the biomass flux fixed. Then, using Xpress MP Professional^® coupled with a MFC COMSOL^® model we finally determined candidates susceptible of mutating for improving the voltage of the cell.

Substrate variation in time was modeled by fixing the substrate rate consumption and the phenazine-glucose yield coefficient. The mathematical model is described by the following “Eqs.5 and 6”

where is the substrate consumption rate in is formance coefficient, is the cellular growth rate in,

is the specific growth rate and is the cell concentration. Standard values were obtained from literature [10].

In order to determine the power generated inside the MFC we built a two-dimensional model that could be further coupled with the FBA analysis [11]. This model consists of four compartments: an anode, a proton exchange membrane, a cathode and culture medium. Each of the porous electrodes is in contact with a current collector [12,13].

The mathematical model used to describe the microbial fuel cell was adapted from the model proposed by Shi et al. [14] and Comsol Chemical Engineering Module [13]. A conductive medium model describes the potential distributions in the four compartments using Laplace equations for electric potential.

where is the effective electronic conductivity of the media (S/m) (They are specified for each media in Table 1) The potential (V) at the electrodes and the membrane is denoted by for the electric potential. Ion current density at both anode and cathode is described with the following equation:

where the subscript for cathode or anode is e, , represents the active layer thickness, the macroscopic porosity. represents the current densities that are given by the agglomerate model. Local current density is based on Butler-Volmer Eq.9:

For the culture medium domain the phenazine concentration reference level was given by the linear programming problem solution. Over-voltages at the anode and cathode are given by the following equations:

Table 1. Variable and constants for the kinetics and microbial fuel cell model with P. aeruginosa.

Boundary conditions for the electric potential of the electrodes conserve with differences in the sign. The potential difference between the current collectors on the cathode and the anode matches the total voltage of the fuel cell. Here we selected the current collector in the anode as the reference level (zero value). Hence the total voltage of the cell serves as a boundary condition in the current collector at the cathode.

In order to model the gas flow at the cathode we used Darcy’s law. The gas velocity is given by the continuity equation:

where represents gas mixture density (Kg/m³) and represents the gas velocity (m/s). Darcy’s law for porous media sates [15].

Assuming ideal gas at the cathode the density can be calculated using the following equation:

where R is the ideal gas constant, P is the pressure, T is the temperature and M is the molecular weight of each gas. Also and constitute the boundary conditions for the Darcy equation (inlet and outlet pressure respectively)

On the border of the cathode, the gas velocity is calculated from the total mass flow given by the electrochemical reactions.

Here, we take into account two species at the anode and three at the cathode (air). For the mass transport we use of the MaxwellStefan convective and diffusion model for each electrode. The mass transport can be described from the following three mass transport equations of Maxwell-Stefan who assume that diffusion over temperature gradients is negligible (oxygen = 1, water = 2, nitrogen = 3). The following equations model the mass transport at the cathode, at the anode the equations are similar but without convective terms.

Metabolic fluxes were calculated for P. aeruginosa in silico model using MFA. Maximum growth rate obtained was 1.04793 and 0.0846 h-1 for aerobic and anaerobic conditions respectively with a glucose uptake of 10, falling within the values that previously reported for similar conditions [19]. Phenazine and biomass production were estimated for different glucose consumption rates in order to assess the sensitivity of the model.

FBA based model was then coupled to the 2D MFC model previously described utilizing the kinetic equations described in Section 2.3. Phenazine flux was calculated at different intervals of glucose using the FBA model and voltage was evaluated from the flux predicted in the FBA model with the 2D MFC model. All parameters were obtained from the literature (Table 1) [20-29]. Our model was also validated through comparing the power density profile obtained assuming the kinetics explained in Eqs.5 and 6 with an experimental setup at 2.5 gr·l⁻¹ of glucose. The results obtained during the first cycle of glucose agreed with the profile provided by our model (Figure 1). Therefore, our model was validated for the next goal: to determine candidate that enable to improve MFC performance.

The multi-objective optimization approach was utilized to identify reactions susceptible of deletion so the synthesis of electron shuttles could be augmented. For this work we fixed the initial concentration of glucose at 2.5 gr/l and estimated the glucose flux assuming passive transport. Then, FBA was utilized to calculate the phenazine flux that was consider the input for the 2D MFC model. In order to decrease the search space, we restricted each run down to two possible mutations and a genetic algorithm developed in Matlab^® was used to select candidates that both maintain the biomass growth and increase the shuttle flux. Table 2 displays the results obtained for each run. The fact of the appearance of different genes for each run exhibits the presence of local optima which are often elucidated when using combinatorial approaches such as genetic algorithms also, our algorithm always provided one single mutations as best candidates. In this case, we found an interesting trend towards sulfur transport and metabolism as a target to increase the synthesis of this electron shuttle as the model predicts around 1.8 fold increase compared to the value when no deletions occur (1.41 mmol phenazine·h- 1·g cell) when deleting cysT, cysW, sbp, and cysP.

P. aeruginosa cysP-C10::ISlacZ/hah, P. aeruginosa cysA-A03::ISlacZ/hah, and P. aeruginosa rpoS-B03:: ISlacZ/hah with transposons inserted in the sulfate transporter cysP, sulfate binding precursor cysA, and stress response factor rpoS respectively, were utilized in order to corroborate the outcome of the proposed algorithm. Even though RpoS is not an outcome of the model as it is not directly metabolic related, it is reported that regulates sulfate related genes [30]. The performance of the cell, evaluated thorough determining the power density average and coulombic efficiency was then compared to the performance of the wild type P. aeruginosa PA01 reported by Garcia et al. [31]. Moreover, we obtained voltametric cyclic curves in order to detect qualitative changes in the production of the electron shuttles. Interestingly, for all cases, the modification of the strains leads to an increase in both peak and average power density, demonstrating the effect of these genes on the phenazine production (Table 3). Nevertheless, the relation between sulfate transport and phenazine is not trivial, as this molecule does not content sulfur in its structure. So it seems that the connectivity of the network causes a modification on the flux of this metabolite. In order to corroborate our results, we ran the

Table 2. Outcome of the Multiobjective optimization platform for the improvement of the performance of a P. aeruginosa based air cathode microbial fuel cell.

Table 3. Power density peak and average and Columbic efficiency for air cathode microbial fuel cell with P. aeruginosa PAO1 and mutants.

algorithm for several times finding the same trend. Regarding the voltametric cyclic curves we did not detect a significant difference regarding the place and the intensity of the oxidation and reduction peaks for cysP and cysA mutants (see Figure 2(a) for cysP mutant). On the other hand rpoS mutant displayed an interesting difference in this diagram as the peaks (Figure 2(b)), which represent the presence of the shuttles are higher for the mutant, suggesting an increasing in its concentration. This dramatical increment was also reflected on the power density average as the mutation of these genes leads to a 13-fold increment. Also rpoS deletion enabled the cell to use the electrons wisely and this was revealed in the coulumbic efficiency (2.5-fold increment)