*3.4. Settings*

For all optimization methods, we set the population size to *N* = 100. All optimizers run maximally 1000 generations. In the case of ES, we use the initial value of *σ* equal 0.1. For DE, RevDE, and RevDE+, we use *F* = 0.5, and *p* = 0.9. For EDA we take *M* = 100. In the case of EDA+ and RevDE+, we use the *K*-NN as the surrogate model with *K* = 3, and we do not store more than 10,000 evaluated individuals.

#### **4. Results & Discussion**

**Fitness value**: In Figure 4 we present convergence of the methods in Figure 4. We notice that all methods were able to converge and achieve very similar fitness values. However, the (1 + 1)-ES method was slowest due to the slow exploration capabilities. EDA also required more evaluations to obtain better results. Interestingly, DE, RevDE, RevDE+, and EDA+ achieved almost identical values of the fitness function (the differences were beyond the three-digit precision). An important observation is that application of the surrogate model (the *K*-NN regressor) allowed to significantly speed up the convergence of RevDE+ and EDA+ compared to RevDE and EDA, respectively. We conclude that all population-based methods were able to converge and achieved almost identical scores, and our proposition of applying the surrogate model led to improving both RevDE and EDA.

**Figure 4.** The convergence of the population-based optimization methods over 3 runs. In the legends, we indicate the value of the fitness function after the methods converged.

**Timecourses**: The final value of the fitness function tells us how well the simulator models the observed timecourses for given parameters provided by an optimizer. Additionally, we can also qualitatively inspect the timecourses both the observed and unobserved metabolites. In Figure 5 we present timecourses for the unobserved metabolites, for parameter values found the five methods.

For all unobserved metabolites, the average over 3 repetitions of the experiments overlapped with the real value or laid within the confidence interval (3× standard deviation). This is a result that we hoped for since being able to generate unobserved metabolite is extremely important for analyzing biological systems. However, we notice that DE and

RevDE+ led to almost identical timecourses, thus, they were able to properly identify parameters.

**Figure 5.** A comparison of the timecourses of the unobserved metabolites. Real timecourses are depicted in red, and the average value and a confidence interval (3× standard deviation) over 3 runs of the simulator is depicted in blue. The titles of the plots indicate optimization methods.

**Differences in parameters**: In this paper, we know precisely the values of the parameters since they were measured in [16]. Hence, we can compare the parameter values found by the optimization methods with the real parameter values. We use the absolute

value of the difference of two values. We calculate the mean and the standard deviations of the difference from three runs, and use the cumulative distribution function of the folded normal distribution to visualize the distribution of differences (the ideal case is 0). The difference between two real-valued random variable is normally distributed. However, taking the absolute value of a normally distributed random variable results in the folded normal distribution.

In Figure 6 we present difference of all parameters. In general, the differences are marginal and we can conclude that all parameter values were rather properly identified. The biggest problems though appear for parameters that have very large values, e.g., *k*8 or *k*33. This result is very promising because it seems to confirm the promise of the paper that it is possible to identify parameters of a complex biological network for only partially observable metabolites.

**Figure 6.** The cumulative distribution functions (cdfs) of the differences for all parameters. Ideally, a cdf of an optimization method should resemble a step-function centered at 0. The averages and the scales are calculated over 3 repetitions of the experiment.
