**3. Experimental Setup**

The experiments have been carried out by means of computer simulation in which the performance of the selected algorithms has been assessed.

#### *Experiment Setup (Computational Approach)*

Three di fferent algorithms; PSOHS, SA, and the downhill simplex method, have been used to estimate the parameter values. The algorithms were executed in MATLAB R2010a on a 1024 MB (1 GB) RAM and Intel Pentium 4 processor laptop. The result of PSOHS was compared with SA and the downhill simplex method. The total run for estimating all the kinetic values was 50 individual runs. The accuracy, consistency, nonlinear least squared error, and standard deviation were calculated and compared with both algorithms to evaluate the performance of PSOHS. The formula for calculating the nonlinear least squared error and standard deviation is given below:

$$\varepsilon = \sum\_{i=1}^{N} (y - y\_i)^2 \tag{3}$$

In Equation (3),*e* means the squares of the errors, *y* is the measurement result, and *yi* is the simulated result. Equation (4) is used to calculate the average squared error, where *A* represents the average squared error and *N* represents the number of samples.

$$A = \frac{\varepsilon}{N} \tag{4}$$

Equation (5) is used to calculate the standard deviation.

$$STD = \sqrt{\frac{\varepsilon}{N}}\tag{5}$$

Aspartate metabolism from *Arabidopsis thaliana* is the dataset that has been used in this paper. *Arabidopsis thaliana* has been chosen as a model organism due to its advantages for genetic experiments, such as (i) a short generation time; (ii) its small size; and (iii) its prolific seed production. In microorganisms, aspartate acts as the precursor to several amino acids, including methionine, threonine, isoleucine, and lysine, which are essential for humans. Threonine, isoleucine, and lysine have been selected in this paper because all of them cannot be produced by the body, but they are important in almost all body functions. Dataset details are shown in Table 1. In this paper, the values of a total of 31 kinetic parameters are estimated using PSOHS.

**Table 1.** Information on the dataset.


#### **4. Result and Discussion**

The performance of PSOHS was compared with SA and the downhill simplex method. Tables 2–4 show the kinetic parameter values that are estimated using PSOHS, SA, and the downhill simplex method on the basis of the experimental value [19]. The parameter values might range wildly in scale as they originate from the previous work [19]. It should be noted that Equation (3) is used to calculate the distance between the experimental data and model simulation for each reaction in each amino acid (isoleucine, lysine, and threonine). There are many reactions involved in each amino acid as well as ODEs in *Arabidopsis thaliana*. Tables 2–4 summarize the experimental results. The average squared error in Tables 2–4 shows the average reaction in the ODE system for each amino acid that involves a number of kinetic parameters. Meanwhile, Tables 5–7 report the parameter values obtained from the experimental data, as well as those generated from the proposed PSOHS, the downhill simplex method, and SA.

**Table 2.** Comparison between PSOHS, the downhill simplex method, and Simulated Annealing (SA) in estimating six parameters for isoleucine in terms of computational time, average squared error, and standard deviation.


Note: The bold numbers represent the best result.



Note: The best results are in bold.

**Table 4.** Comparison between the performance of PSOHS, the downhill simplex method, and SA in estimating sixteen parameters for threonine in terms of computational time, average squared error, and standard deviation.


Note: The best results are in bold.

**Table 5.** List of kinetic parameter values for isoleucine with the experimental values.


Note: The best results are in bold.

**Table 6.** List of kinetic parameter values for lysine with the experimental values.


Note: The bold numbers represent the best result.


**Table 7.** List of kinetic parameter values for threonine with the experimental values.

Note: The bold numbers represent the best result.

This paper focuses on parameter estimation using the proposed PSOHS. The performance of PSOHS was measured in terms of computational time, model accuracy, and precision of the algorithms. Model accuracy is measured by the distance value between the experimental data and model simulation using the nonlinear least squared method. In addition, the average squared error was calculated to ge<sup>t</sup> the average of all ODEs in each amino acid. To test the algorithm's precision, standard deviation was used for 50 individual runs. High standard deviations demonstrate low precision, and low standard deviations demonstrate high precision. The experiments were carried out in 50 individual runs to test the algorithms, and the result shown is the best multivariate solution among the runs. The average squared error and standard deviation were calculated from the runs. Tables 2–4 show the comparisons of computational time in seconds, average squared error, and standard deviation among PSOHS, SA, and the downhill simplex method. Table 2 shows the execution time for PSOHS to estimate the six kinetic parameters of isoleucine is 100.23 s, which is the lowest compared to 130.56 s for the downhill simplex method and 778.00 s for SA. The standard deviation of PSOHS is 0.0002, which is the closest to zero compared to the downhill simplex method and SA, which are 0.0004 and 0.002. Based on these comparisons, PSOHS shows the lowest average squared error and a low standard deviation, and this proves that PSOHS is more consistent, precise, and reliable in parameter values estimation compared to SA and the downhill simplex method.

Table 3 shows that the average nonlinear least squared error for the three algorithms is 0.0211, 0.084, and 0.0406. The standard deviations of the three algorithms are 0.0133, 0.0998, and 0.0347. Besides, the computational time for PSOHS is 184.03; for the downhill simplex method it is 376.59 and for SA it is 1518.05. From among the three algorithms for estimating the kinetic parameter of lysine, PSOHS shows the best result.

Threonine has the greatest number of kinetic parameters to be estimated among the selected amino acids. Table 4 presents the comparison among PSOHS, the downhill simplex method, and SA. It seems that the average nonlinear least squared error for PSOHS is smaller than the downhill simplex method and SA. The average nonlinear least squared error for PSOHS is 0.0024, while the average nonlinear least squared errors for the downhill simplex method and SA are 0.012 and 0.0066, respectively. Furthermore, to estimate all kinetic parameters, PSOHS takes 255.37 s, which is a considerably lower computational time than the downhill simplex method, which takes 362.32 s, and SA, which takes 1794.91 s. In terms of standard deviation, the downhill simplex method and SA show a high standard deviation compared to PSOHS. The standard deviation for PSOHS, downhill simplex method, and SA are 0.0037, 0.017, and 0.0079, respectively. The results show that PSOHS outperforms the downhill simplex method and SA in estimating the sixteen kinetic parameters of threonine.
