Comparison across growth kinetic models of alkaline protease production in batch and fed-batch fermentation using hybrid genetic algorithm and particle swarm optimization

Figures & data

Table 1. Growth models considered for the data evaluation.[30–32]

Growth kinetic model	µ = f(S,X)	Equation number
Aiba	$\mu =\frac{{\mu }_{\max }S}{k_S+S}\left(\exp \left(\frac{-S}{k_{I,S}}\right)\right)$	(5)
Andrews	$\mu =\frac{{\mu }_{\max }}{\left(1+\frac{k_S}{S}\right)\left(1+\frac{S}{k_{I,S}}\right)}$	(6)
Contois	$\mu =\frac{{\mu }_{\max }S}{k_{X}X+S}$	(7)
Haldane	$\mu =\frac{{\mu }_{\max }S}{k_S+S+\frac{S^2}{k_{I,S}}}$	(8)
Monod	$\mu =\frac{{\mu }_{\max }S}{k_S+S}$	(9)
Moser	$\mu =\frac{{\mu }_{\max }{S}^n}{k_S+S^n}$	(10)
Tessier	$\mu ={\mu }_{\max }\left(1-\exp \left(\frac{-S}{k_S}\right)\right)$	(11)

Figure 1. Chromosome structure in GA.

Figure 2. Depiction of the velocity and position updates in PSO.

Figure 3. The flowchart of hybrid GA/PSO.

Table 2. Estimated kinetic parameters by hybrid GA/PSO method and value of objective function for various kinetic models.

		Parameters
Kinetic model	Batch/Fed-batch	µmax	kd	YX/S	mS	α	kP	kS	kX	k	kI,S	n	Value of objective function$(\phi )$
Aiba	Batch	0.6330	0.0017	1.4661	0.8809	1.0151	0.0445	1.1000	–	3.0868	29.1400	–	1.0685
	Fed-batch	0.6872	0.0022	1.1855	0.9590	1.6421	0.1833	1.5607	–	1.6190	28.9070	–	1.5056
Andrews	Batch	0.6901	0.0010	2.2957	1.0000	0.8085	0.0271	3.1285	–	4.9996	29.0110	–	0.5141
	Fed-batch	0.6543	0.0010	2.9797	1.0000	0.9267	0.0647	2.7398	–	4.9984	27.8234	–	0.5749
Contois	Batch	0.7000	0.0146	0.5825	0.0134	0.7774	0.0372	–	7.4482	1.1238	–	–	0.0582
	Fed-batch	0.6384	0.0010	0.7387	0.0139	0.5848	0.0010	–	4.3388	1.1006	–	–	0.0518
Haldane	Batch	0.6210	0.0020	1.5893	0.9700	0.7942	0.0250	2.2483	–	4.8792	30.0000	–	0.4363
	Fed-batch	0.6645	0.0020	9.1142	1.0000	0.8172	0.0443	4.0648	–	5.0000	30.0000	–	0.5490
Monod	Batch	0.5645	0.0083	0.4612	0.0100	0.7619	0.0287	2.0000	–	5.0000	–	–	0.2869
	Fed-batch	0.3945	0.0010	9.9918	0.8042	0.6539	0.0137	3.2875	–	5.0000	–	–	0.3252
Moser	Batch	0.3566	0.0036	10.0000	0.8948	0.7561	0.0229	2.0542	–	4.3100	–	2.0000	0.3413
	Fed-batch	0.3716	0.0010	0.5805	0.0100	0.5903	0.0010	2.0598	–	5.0000	–	2.0000	0.3294
Tessier	Batch	0.4398	0.0091	0.4680	0.0100	0.7634	0.0298	2.0340	–	4.0430	–	–	0.2716
	Fed-batch	0.3528	0.0010	10.0000	0.8987	0.6740	0.0177	2.6554	–	4.9990	–	–	0.3501

Figure 4. Comparison of the experimental and simulated data of batch fermentation by various growth kinetic models: Aiba (a), Andrews (b), Contois (c), Haldane (d), Monod (e), Moser (f), and Tessier (g). Experimental concentrations of X (٭), S (○), and P (◊) vs. hybrid GA/PSO simulated data (solid lines). Note: X, biomass (B. licheniformis); S, substrate (glucose); P, product (protease). Initial concentration of substrate is 40 g/L.

Figure 5. Comparison of the experimental and simulated data of fed-batch fermentation by various growth kinetic models: Aiba (a), Andrews (b), Contois (c), Haldane (d), Monod (e), Moser (f), and Tessier (g). Experimental concentrations of X (٭), S (○), and P (◊) vs. hybrid GA/PSO simulated data (solid lines). Note: X, biomass (B. licheniformis); S, substrate (glucose); P, product (protease). Initial concentration of substrate is 40 g/L; feed rate = 0.01 L/h.

Figure 6. Comparison of the experimental and simulated data of batch fermentation by the Contois model by using GA (a) and PSO (b). Experimental concentrations of X (٭), S (○), and P (◊) vs. simulated data (solid lines). Note: X, biomass (B. licheniformis); S, substrate (glucose); P, product (protease). Initial concentration of substrate is 40 g/L.

Figure 7. Comparison of the experimental and simulated data of fed-batch fermentation by the Contois model by using GA (a) and PSO (b). Experimental concentrations of X (٭), S (○), and P (◊) vs. simulated data (solid lines). Note: X, biomass (B. licheniformis); S, substrate (glucose); P, product (protease). Initial concentration of substrate is 40 g/L; feed rate = 0.01 L/h.

Table 3. Comparison of performance across GA, PSO, and hybrid GA/PSO.

		Performance
Algorithm	Batch/Fed-batch	Computation time (s)	Objective function$(\phi )$
GA	Batch	117.59	0.0989
	Fed-batch	126.34	0.0932
PSO	Batch	102.63	0.0778
	Fed-batch	115.84	0.0789
Hybrid GA/PSO	Batch	53.21	0.0582
	Fed-batch	57.02	0.0518