## 3 Results
### Combining DFs and Fourier Surrogate Modeling  
As a second group of tests, we applied the new surF algorithm to all the previous fitness landscapes to optimize a filtered and smoothed version of the optimization problem. Figure 8 shows the effect of surF using \(\gamma=2\): in the case of the original landscape, the low number of Fourier coefficients prevents the reconstruction of the high frequency optimum, a circumstance already discussed in [5; 12]; in the case of the landscape dilated with the anti-log DF, the smoothed landscape is denoted by a global optimum corresponding to the actual target parameterization; and finally, in the case of surF applied to the fitness landscape dilated using the evolved DF, the problem of the missing high frequency components again prevents a correct reconstruction of the landscape, as in the case of the original problem.  
To further investigate the effect of the parameter \(\gamma\) on the fitness landscape, we created different surrogate models starting from the fitness landscape obtained with the evolved DF (Figures 9).  
Figure 8: Effect of the surrogate modeling on the fitness landscape of the MM model. (**a**) Surrogate model of the original fitness landscape. (**b**) Surrogate model of the fitness landscape dilated by means of the anti-log DF. (**c**) Surrogate model of the fitness landscape dilated by means of the evolved DF.  
Figure 7: (**Left**): convergence plot of the PE of the MM model obtained with FST-PSO (no \(v_{min}\)) (coral solid line), FST-PSO using the analytic DF (green dashed line), and FST-PSO using the evolved DF (blue dotted line). The lines correspond to the ABF calculated over 30 runs. (**Right**): boxplots representing the distributions of the best solutions found at the end of each run, for each methodology; red dashes denote the mean of the distributions, while diamonds denote the outliers. The asterisks denote the \(p\)-values obtained by comparing the distributions by means of the Mann–Whitney U tests ( \(\star\)\(p\)-value \(\leq\) 0.0001).  
The plots highlight that, even in the case of \(\gamma=100\), the global optimum cannot be properly reconstructed, as the excessively strong dilation caused by the evolved DF is detrimental to the optimization results when coupled to Fourier surrogate modeling. Figure 10 confirms this insight by showing that FST-PSO cannot converge in the case of the surrogate model created with surf, starting from the fitness landscape dilated with the evolved DF (blue dotted line); on the contrary, FST-PSO using the surrogate model created with surf starting from the fitness landscape dilated with the analytical anti-log DF, is capable of reaching the global optimum. In any case, it is worth mentioning that, thanks to the strategy of co-evolving the \(\gamma\) value with the rest of the candidate solution, the methodology no longer requires domain knowledge to achieve accurate solutions for the PE problem of biochemical systems.  
Figure 10: (**Left**): convergence plot of the PE of the MM model obtained with FST-PSO (no \(v_{min}\)) (coral solid line), FST-PSO using the analytic DF + surf (green dashed line), FST-PSO using the evolved  
DF + surf (blue dotted line). The lines correspond to the ABF calculated over 30 runs. (**Right**): boxplots representing the distribution of the best solutions found at the end of each run; red dashes denote the mean of the distributions, while diamonds denote outliers. The asterisks denote the \(p\)-values obtained by comparing the distributions by means of the Mann-Whitney U tests (* \(p\)-value \(\leq\) 0.5, ** \(p\)-value \(\leq\) 0.0001).