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This study presents the simulation of a MFC with Pseudomonas aeruginosa based on a metabolic flux analysis (MFA) which arises as a linear programming model that served as input for the fuel cell model. The linear model was implemented in Xpress MP^{?} and the coupling model in Comsol Multiphysics^{?}. The in silico model predicted maximum potentials of 0.135 V which were consistent with those obtained in the experimental cell. Afterwards, an optimization platform based on multiobjective optimization approach was implemented aimed to determine genes that increase the average cell power. cysA, cysP and rpoS mutants which were predicted to increase the power of the cell, were experimentally tested with an air cathode fuel cell finding an augment up to 35-fold in the average power density for the rpoS mutant. Power densities were obtained through experimentation in the range of [0.13 - 5] mW/m^{2}. Electron shuttle rise was qualitative corroborated trough cyclic voltammetry tests, which allowed to visualize the augment of the peaks for rpoS mutation.

The increasing demand of global energy has set a dramatical searching for alternatives which could be based on renewable sources yet cheap in terms of materials needed. Global fuels currently satisfy 80% of the energy demand and this keeps increasing causing an increment in greenhouse gases. It is also a fact that this demand is scaled-based regarding its application, as this energy must be supplied according to the device or the system that would be using it.

Fuel cells have been identified as one of the most promising technologies to supplement energy to low scale devices. Generally, they generate electricity through oxidizing substrates coupled with an adequate electrons sinks. The based is electrochemistry as they are composed of a cathode and an anode. Its classification is derived from the substrate, chemical reaction and materials which are based on. Hydrogen fuel cells on one hand have been popularized thoroughly. On the other hand bacteria can be used for analogous purposes as they are capable of catalyze the conversion of organic matter into electricity using microbial fuel cells (MFC). In this case, the anode promotes the formation of biofilms while allowing electron transfer. However, electron transfer demands the participation of electron shuttles and not all bacterial species have the ability to synthesize; hence the addition of artificial shuttle becomes unavoidable. Fortunately, there exist facultative anaerobic species such as Pseudomonas aeruginosa (P. aeruginosa) known to produce phenazine derivates partially regulated by quorum sensing [

The important role of phenazine was established utilizing a P. aeruginosa phenazine mutant MFC as the performance of the cell decreased 95% [

Maximize

Subject to:

where and are upper and lower bounds and the stoichiometric matrix.

The objective function constitutes a critical step when formulating the problem as it establishes the direction where the biological system evolves. Generally, biomass synthesis is utilized as objective function, but this has to be reviewed carefully as biological systems could behave differently when the electron receptor changes and this is the case for P. aeruginosa in MFC where the presence of oxygen must be reduced. Different objective functions have also been tested in biological systems such as ATP synthesis, redox potential and starch synthesis [

FBA models should be framed in the environment where the cells evolve hence if one want to determine which genes should be deleted or over-expressed in the system in order to enhance the performance of the fuel cell it is necessary to develop a model that connect the kinetics of the cell, mass transport, and the electrochemistry behind the current generation process. Picioreanu et al. [

Flux balanced analysis was based on the reconstruction based on Oberhardt et al. [

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 [

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 [

The mathematical model used to describe the microbial fuel cell was adapted from the model proposed by Shi et al. [

where is the effective electronic conductivity of the media (S/m) (They are specified for each media in

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:

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/m^{3}) and represents the gas velocity (m/s). Darcy’s law for porous media sates [

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.

Wild type strain P. aeruginosa PA01, P. aeruginosa cysP-C10::ISlacZ/hah, P. aeruginosa cysA-A03::ISlacZ/ hah, and P. aeruginosa rpoS-B03::ISlacZ/hah were kindly provided by University of Washington Genome Center and contains an inserted tranposon in the stress response genes cysP, cysA, and rpoS respectively. The strain of P. aeruginosa was maintained in a petri dish with nutrient agar. To ensure that the experimental results were not erroneous due to contamination, all manipulations were performed inside a vertical flow chamber in completely sterile conditions.

The MFC utilized in experiments consists of an air cathode single chamber fuel cell made in acrylic glass. The cell has an anode compartment of 3 cm of length and a tube length of 4 cm giving a total cell volume of 28.27 cm3 and total anode area of 7 cm^{2}. Proton exchange membrane consists of J-Cloth fabric type [

MFC was sterilized with UV light exposition during 24 h. Cultures were grown in nutrient broth (Oxoid, Basingstoke, UK) for 24 h previous to inoculation (3 mL for 20 hours at 150 rpm and 30˚C). Once inoculated, both, cathode and anode are hooked to a circuit with an external resistor of 500 W and incubated at 30˚C for periods of approximately 15 days. From the third day of incubation 5 mL of the internal environment of the cell was removed and 5 mL of 2.5 g/L sterile glucose solution of was added. The cells were monitored every 30 seconds during the whole incubation period with a data acquisition card (PCI-6221, National Instruments) and a connection module (SCB-68) with the Labview^{®} acquisition software (National Instruments^{®}).

Current, power density and columbic efficiency were calculated based on equations reported by Logan [

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 [

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 (^{−1} of glucose. The results obtained during the first cycle of glucose agreed with the profile provided by our model (

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.

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 [

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

Here in this work we demonstrate that by coupling FBA and spatial models which include mass transport and other physics such as electrochemistry is possible to improve the performance of different systems where kinetics and mass transport are limiting such as MFC. We also demonstrate that multi-objective optimization strategies where the second level is switched to a restriction for the first level seems to work satisfactorily with heuristics approaches such as genetic algorithms because it reduces computation infrastructure and time. On the other hand we also found that the sulfate transport a stress play a big role in P. aeruginosa MFC performance as we reached a 13-fold increment in the performance through deleting rpoS.

The authors wish to thank FICO for providing the Xpress-MP Professional licenses under the institutional arrangement with the Universidad de los Andes.