# Pitfalls in Metaheuristics Solving Stoichiometric-Based Optimization Models for Metabolic Networks
## 5 Design of Experiments
### Experiments Definition  
Defining a methodology based on metaheuristics that improve the metabolic network flux information provided by the FBA method by redefining Flux Balance Analysis as a Multi-objective Optimization Problem is possible; four experiments were proposed, summarized in the following Table 4.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Experiment** & **Objective** & **Variables Involved** \\ \hline \multirow{2}{*}{Experiment\({}_{1}\)} & Using different optimization models produces & \begin{tabular}{l} _C. vulgaris_, NSGAII, MOFA\({}_{1}\), \\ MOFDA\({}_{2}\), MOFBA\({}_{3}\), MOFBA\({}_{4}\). \\ \end{tabular} \\ \hline \multirow{2}{*}{Experiment\({}_{2}\)} & Demonstrate that the use of metaheuristics & \begin{tabular}{l} _C. vulgaris_, FBA, NSGAII, \\ MOFBA\({}_{4}\) \\ \end{tabular} \\  & supports the understanding of & \begin{tabular}{l} _C. vulgaris_, FBA, NSGAII, \\ MOFBA\({}_{4}\) \\ \end{tabular} \\  & microalgae metabolism. & \\ \hline \multirow{2}{*}{Experiment\({}_{3}\)} & There are optimization algorithms more & \
\begin{tabular}{l} _C. vulgaris_, NSGAII, MOEA/D, \\ MOFBA\({}_{4}\). \\ \end{tabular} \\  &
\begin{tabular}{l} _C. vulgaris_, NSGAII, FBA, \\ random. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 4: Design of experiments to demonstrate metaheuristic support for understanding metabolism in microalgae.  
Figure 2: Flux distribution associated with glutamate metabolism in the chloroplast and cytoplasm [17].  
Experiment 1 demonstrated that different multi-objective optimization models offer different results. The algorithm was set to NSGAII, the microalgae was modified to demonstrate the approach's versatility, and finally, the four proposed optimization models were analyzed. For the cases studied, it was observed that the consistently best model was \(\text{MOFBA}_{4}\), which was used in the subsequent experiments.  
In Experiment 2, the algorithm, the microalgae, and the single-objective FBA model were compared against the multi-objective model defined by \(\text{MOFBA}_{4}\). It was observed that the metabolism of a microalgae can be described with different fluxes, not one, and these can be controlled to obtain information on different metabolites of interest. This confirms the ease of adaptation of the proposed methods to different types of metabolic networks, considering different configurations.  
Experiment 3 evaluated the performance of the NSGAII and MOEA/D algorithms on the same optimization problem. It was observed that, for the three objectives (i.e., three metabolites of interest) considered, NSGAII was the best. This result is consistent with the literature, given that for two or three objectives, NSGAII shows better performance than MOEA/D. This leaves open the question of whether MOEA/D will improve for a larger number of objectives, which is an open line of investigation for its application in the study of microalgal metabolic networks.  
Experiment 4 shows that simple random sampling is not sufficient to obtain a better distribution of solutions, which is possible through the use of metaheuristics.  
The metaheuristics NSGAII and MOEA/D were implemented with the aid of the jMetalPy framework [32]. The optimization models were developed in Python and used as part of the FBA implementation provided by the package COBRApy [33]. Graphics were recreated using the interface pyplot of matplotlib [34]. The computer used to run the experiments has a 64 bit 2.6 GHz processor with 32 RAM memory.