Modeling of Penicillin Fermentation Process Based on a Multi-stage LHS-EPRCC Method

doi:10.16182/j.issn1004731x.joss.25-0948E

Abstract

Abstract:

This paper focused on the modeling of microbial fermentation processes under varying production environments and proposed a novel approach. Considering that the dynamic characteristics of microorganisms differ across growth stages, we introduced the concept of multi-stage sensitivity analysis, in which each stage was investigated separately. The fuzzy C-means (FCM) algorithm was employed to cluster process data under nominal conditions, thereby dividing the penicillin fermentation process into distinct growth stages. Based on this division, the Latin hypercube sampling with partial rank correlation coefficient (LHS-EPRCC) method was applied to conduct sensitivity analysis for each stage, identifying an importance parameter set (IPS) that corresponds to the stage-specific growth characteristics. Re-estimation and correction of the IPS were then performed to enhance the predictive accuracy of the model. In a penicillin fermentation process deviating from nominal conditions, the proposed method was applied for model correction. Simulation results demonstrate that the corrected model aligns well with the actual process, thereby verifying the effectiveness of the proposed multi-stage sensitivity analysis approach in addressing complex fermentation processes and environmental uncertainties.

Key words: penicillin fermentation process, FCM algorithm, LHS-EPRCC method, importance parameter set, model correction

CLC Number:

TP391.9

Li Quan, Su Peng, Wan Haiying, Zhang Chengxi, He Zhijian, Ni Yiyang, Zhao Zhonggai, Liu Fei. Modeling of Penicillin Fermentation Process Based on a Multi-stage LHS-EPRCC Method[J]. Journal of System Simulation, 2026, 38(5): 1255-1276.

Figures/Tables 16

Table 1

Initial operating conditions

Variable	Value
Biomass concentration $X 0$ /(g/L)	0.1
Penicillin concentration $P 0$ /(g/L)	0
Glucose concentration $S 0$ /(g/L)	15
Initial culture volume $V 0$ /L	100
Feed flow rate $F$ /(L/h)	0.04

Table 1

Table 2

Nominal model parameter values

Parameter	Value	Parameter	Value
$u X$	0.092	$K H$	0.04
$K X$	0.15	$Y X / S$	0.45
$u P$	0.005	$Y P / S$	0.9
$K P$	0.000 2	$m X$	0.14
$K I$	0.1	$S f$	600

Table 2

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 3

LHS-EPRCC sensitivity index ranking for all dynamic model parameters during biomass growth stage

Rank	Glucose				Biomass					Penicillin
Rank	$k i$	$γ ¯ k 1, y 1 1$	$k i$	$G ¯ k 1, y 1 1$	$k i$	$γ ¯ k 1, y 2 1$	$k i$	$γ ¯ k 1, y 2 1$	$k i$	$γ ¯ k 1, y 3 1$	$k i$	$γ ¯ k 1, y 3 1$
1	$u X *$	-0.980 5	$u X * *$	0.446 2	$u X *$	0.999 9	$u X * *$	0.999 3	$K I *$	0.960 4	$K I * *$	0.4232
2	$S f *$	0.973 5	$S f * *$	0.510 4	$K X *$	-0.657 1	$Y X / S$	0.000 1	$u X *$	0.910 0	$u X * *$	0.416 6
3	$Y X / S *$	0.909 0	$Y X / S * *$	0.045 1	$S f *$	0.295 6	$K X$	6.34×10^-5	$u P *$	0.908 4	$u P * *$	0.152 0
4	$m X$	-0.169 5	$m X$	0.000 2	$Y X / S *$	0.202 1	$S f$	3.99×10^-5	$K H *$	-0.486 3	$K H * *$	0.006 4
5	$K X$	0.049 4	$K X$	5.24×10^-5	$K I$	-0.028 2	$m X$	4.53×10^-7	$S f *$	-0.451 1	$S f * *$	0.006 2
6	$u P$	0.038 9	$K I$	1.07×10^-8	$Y P / S$	0.023 2	$K I$	5.75×10^-9	$Y X / S$	-0.224 2	$Y X / S * *$	0.004 9
7	$k d u m m y$	-0.028 0	$u P$	1.93×10^-9	$u P$	-0.017 9	$u P$	1.39×10^-9	$Y P / S$	0.038 4	$K X$	0.000 2
8	$Y P / S$	-0.023 1	$Y P / S$	1.21×10^-9	$m X$	-0.012 9	$K H$	2.25×10^-10	$k d u m m y$	-0.029 6	$m X$	2.23×10^-5
9	$K H$	-0.017 7	$K H$	1.86×10^-10	$k d u m m y$	-0.011 9	$Y P / S$	1.88×10^-11	$m X$	0.024 8	$Y P / S$	1.02×10^-8
10	$K I$	-0.012 4	$K P$	2.33×10^-18	$K H$	-0.010 6	$K P$	5.09×10^-20	$K X$	-0.024 9	$K P$	6.20×10^-13
11	$K P$	-0.006 9			$K P$	-0.006 3			$K P$	-0.023 0

Table 3

Table 5

LHS-EPRCC sensitivity index ranking for all dynamic model parameters during growth stabilization stage

Rank	Glucose				Biomass				Penicillin
Rank	$k i$	$γ ¯ k 3, y 1 1$	$k i$	$G ¯ k 3, y 1 1$	$k i$	$γ ¯ k 3, y 2 1$	$k i$	$γ ¯ k 3, y 2 1$	$k i$	$γ ¯ k 3, y 3 1$	$k i$	$γ ¯ k 3, y 3 1$
1	$Y X / S *$	-0.973 2	$Y X / S * *$	0.510 4	$S f *$	0.967 8	$S f * *$	0.627 0	$S f *$	0.919 1	$S f * *$	0.350 2
2	$m X *$	-0.947 8	$m X * *$	0.243 7	$m X *$	-0.921 7	$m X * *$	0.260 8	$K H *$	-0.886 4	$m X * *$	0.261 0
3	$S f *$	0.907 8	$S f * *$	0.133 5	$Y X / S *$	0.725 0	$Y X / S * *$	0.077 0	$m X *$	-0.879 9	$K H * *$	0.230 2
4	$u X *$	0.755 4	$u X * *$	0.042 0	$u P *$	-0.638 3	$Y P / S * *$	0.033 1	$u P *$	0.774 9	$u P * *$	0.102 2
5	$u P *$	-0.738 8	$Y P / S * *$	0.037 3	$Y P / S *$	0.624 2	$u P * *$	0.031 0	$Y P / S *$	0.579 5	$Y X / S * *$	0.074 4
6	$Y P / S *$	0.732 2	$u P * *$	0.035 9	$u X *$	-0.360 3	$S f *$	0.014 5	$Y X / S *$	-0.449 2	$Y P / S * *$	0.032 6
7	$K P$	0.257 8	$K I * *$	0.003 4	$K I$	-0.116 4	$K I$	0.000 6	$u X *$	0.177 6	$u X * *$	0.018 22
8	$K X$	0.239 7	$K X * *$	0.002 2	$k d u m m y$	0.014 2	$K P$	8.82×10^-5	$K I$	0.144 3	$K I * *$	0.001
9	$K I$	0.234 7	$K P * *$	0.002 1	$K P$	0.013 7	$K X$	3.42×10^-5	$K P$	-0.078 1	$K P$	0.000 5
10	$K H$	-0.033 9	$K H$	5.13×10^-6	$K H$	0.013 7	$K H$	1.04×10^-7	$K X$	0.055 4	$K X$	0.000 1
11	$k d u m m y$	0.027 5			$K X$	-0.008 1			$k d u m m y$	0.017 5

Table 5

Table 4

LHS-EPRCC sensitivity index ranking for all dynamic model parameters during penicillin synthesis stage

Rank	Glucose				Biomass				Penicillin
Rank	$k i$	$γ ¯ k 2, y 1 1$	$k i$	$G ¯ k 2, y 1 1$	$k i$	$γ ¯ k 2, y 2 1$	$k i$	$γ ¯ k 2, y 2 1$	$k i$	$γ ¯ k 2, y 3 1$	$k i$	$γ ¯ k 2, y 3 1$
1	$m X *$	-0.899 5	$Y X / S * *$	0.464 77	$u X *$	-0.743 0	$u X * *$	0.085 5	$u P *$	0.726 2	$Y X / S * *$	0.266 4
2	$S f *$	0.898 5	$m X * *$	0.209 0	$S f *$	0.731 2	$S f * *$	0.469 8	$K H *$	-0.615	$u P * *$	0.193 3
3	$Y X / S *$	-0.855 6	$S f * *$	0.149 44	$Y X / S *$	0.677 3	$Y X / S * *$	0.325 9	$Y X / S *$	0.413	$u X * *$	0.192 2
4	$u P *$	-0.696 1	$u X * *$	0.114 55	$m X *$	-0.562 5	$m X * *$	0.130 2	$S f *$	0.364 3	$S f * *$	0.185 2
5	$Y P / S *$	0.690 2	$Y P / S * *$	3.80×10^-2	$u P *$	-0.318 1	$Y P / S * *$	0.014 1	$K I *$	0.292 7	$m X * *$	0.161 5
6	$u X *$	0.648 4	$u P * *$	3.55×10^-2	$Y P / S *$	0.314 1	$u P * *$	0.013 2	$u X *$	0.183	$K H * *$	0.150 0
7	$K X$	0.198 7	$K X * *$	4.86×10^-3	$K X *$	-0.228 7	$K I$	0.000 9	$m X *$	-0.155 3	$Y P / S * *$	0.022 7
8	$K P$	0.063 2	$K I$	8.99×10^-4	$K I$	-0.112 3	$K X$	0.000 3	$K X *$	-0.119 3	$K I * *$	0.014 0
9	$K H$	-0.027 2	$K P$	1.19×10^-4	$K P$	-0.019 4	$K P$	1.26×10^-5	$Y P / S$	0.074 3	$K X$	0.000 3
10	$K I$	-0.027 1	$K H$	5.72×10^-7	$k d u m m y$	0.017 4	$K H$	6.03×10^-9	$K P$	-0.042 7	$K P$	0.000 2
11	$k d u m m y$	0.021 2			$K H$	-0.009 5			$k d u m m y$	-0.010 9

Table 4

Table 6

Parameter values describing actual process

Parameter	Value	Parameter	Value
$u X$	0.094 8	$K H$	0.036 7
$K X$	0.137 4	$Y X / S$	0.409 3
$u P$	0.005 5	$Y P / S$	0.881 3
$K P$	1.988 0×10^-4	$m X$	0.014 0
$K I$	0.098 1	$S f$	647.081 1

Table 6

Fig. 7

Fig. 8

Table 7

Mean squared error and mean absolute error

Metabolite	Nominal model		Modified model
	Nominal model		IPS $(K 1, K 2, K 3)$		IPS $(K)$ ^[24]		Total parameter
	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Biomass	0.158 6	0.681 7	2.455 3×10^-6	0.000 9	1.859 7×10^-5	0.411 0	7.170 5×10^-7	0.000 3
Penicillin	0.072 5	0.240 9	1.164 8×10^-8	0.000 8	8.193 8×10^-6	0.022 1	2.978 2×10^-8	0.000 2
Glucose	0.243 1	0.226 0	1.238 8×10^-6	0.000 3	5.420 0×10^-5	0.113 9	8.161 7×10^-7	0.000 3

Table 7

Table 8

Wilcoxon signed-rank test results

Modified model	P-value	Signed rank
IPS $(K 1, K 2, K 3)$	1.800 8×10^-216	1.220 9×10⁶
IPS $(K)$	1.852 8×10^-194	905 46
Total parameter	2.261 7×10^-216	1.220 7×10⁶

Table 8

References 36

[1]	Di Massimo C, Montague G A, Willis M J, et al. Towards Improved Penicillin Fermentation via Artificial Neural Networks[J]. Computers & Chemical Engineering, 1992, 16(4): 283-291.
[2]	Larsen Jesper, Raisen C L, Ba Xiaoliang, et al. Emergence of Methicillin Resistance Predates the Clinical Use of Antibiotics[J]. Nature, 2022, 602(7895): 135-141.
[3]	Rodman A D, Gerogiorgis D I. Multi-objective Process Optimisation of Beer Fermentation via Dynamic Simulation[J]. Food and Bioproducts Processing, 2016, 100, Part A: 255-274.
[4]	叶凌箭, 程江华. 基于Matlab/Simulink的青霉素发酵过程仿真平台[J]. 系统仿真学报, 2015, 27(3): 515-520.
	Ye Lingjian, Cheng Jianghua. Simulator of Penicillin Fermentation Process in Matlab/Simulink Environment[J]. Journal of System Simulation, 2015, 27(3): 515-520.
[5]	李文亮, 纪成, 梁晨, 等. 基于TDMN的青霉素浓度在线软测量[J]. 化工学报, 2025, 76(6): 2848-2858.
	Li Wenliang, Ji Cheng, Liang Chen, et al. On-line Soft Measurement of Penicillin Concentration Based on TDMN[J]. CIESC Journal, 2025, 76(6): 2848-2858.
[6]	Bajpai R K, Reuβ M. A Mechanistic Model for Penicillin Production[J]. Journal of Chemical Technology and Biotechnology, 1980, 30(1): 332-344.
[7]	Birol Gülnur, Ündey Cenk, Çinar Ali. A Modular Simulation Package for Fed-batch Fermentation: Penicillin Production[J]. Computers & Chemical Engineering, 2002, 26(11): 1553-1565.
[8]	Goldrick S, Ştefan Andrei, Lovett D, et al. The Development of an Industrial-scale Fed-batch Fermentation Simulation[J]. Journal of Biotechnology, 2015, 193: 70-82.
[9]	Goldrick S, Duran-Villalobos C A, Jankauskas K, et al. Modern Day Monitoring and Control Challenges Outlined on an Industrial-scale Benchmark Fermentation Process[J]. Computers & Chemical Engineering, 2019, 130: 106471.
[10]	Marino S, Hogue I B, Ray C J, et al. A Methodology for Performing Global Uncertainty and Sensitivity Analysis in Systems Biology[J]. Journal of Theoretical Biology, 2008, 254(1): 178-196.
[11]	Hille Rubin, Mandur J, Budman Hector M. Robust Batch-to-batch Optimization in the Presence of Model-plant Mismatch and Input Uncertainty[J]. AIChE Journal, 2017, 63(7): 2660-2670.
[12]	Mohammad Reza Naghavi, Jalali Alireza, Raisee Mehrdad. Uncertainty Quantification and Sensitivity Analysis of an Ultraviolet Water Disinfection Photo-reactor[J]. Chemical Engineering Science, 2021, 245: 116830.
[13]	温亮, 周平. 基于多参数灵敏度分析与遗传优化的铁水质量无模型自适应控制[J]. 自动化学报, 2021, 47(11): 2600-2613.
	Wen Liang, Zhou Ping. Model Free Adaptive Control of Molten Iron Quality Based on Multi-parameter Sensitivity Analysis and GA Optimization[J]. Acta Automatica Sinica, 2021, 47(11): 2600-2613.
[14]	Zhang Y, Rundell A. Comparative Study of Parameter Sensitivity Analyses of the TCR-activated Erk-MAPK Signalling Pathway[J]. Systems Biology, 2006, 153(4): 201-211.
[15]	刘博, 尚晓兵, 晁涛, 等. 基于PCE的飞行器不确定性量化方法[J]. 系统仿真学报, 2018, 30(12): 4555-4562.
	Liu Bo, Shang Xiaobing, Chao Tao, et al. Uncertainty Quantification Method for Aircraft Using PCE Model[J]. Journal of System Simulation, 2018, 30(12): 4555-4562.
[16]	Shu Zhen, Jirutitijaroen Panida. Latin Hypercube Sampling Techniques for Power Systems Reliability Analysis with Renewable Energy Sources[J]. IEEE Transactions on Power Systems, 2011, 26(4): 2066-2073.
[17]	李伟达, 刘琳琳, 张磊, 等. 基于灵敏度分析的页岩气净化流程的模拟与优化[J]. 化工学报, 2018, 69(3): 1008-1013.
	Li Weida, Liu Linlin, Zhang Lei, et al. Simulation and Optimization of Shale Gas Purification Process by Sensitivity Analysis[J]. CIESC Journal, 2018, 69(3): 1008-1013.
[18]	Lei Zhipeng, Wang Feiyu, Li Chuanyang. A Denoising Method of Partial Discharge Signal Based on Improved SVD-VMD[J]. IEEE Transactions on Dielectrics and Electrical Insulation, 2023, 30(5): 2107-2116.
[19]	Kourosh Nasr Esfahani, Pérez-Moya Montserrat, Graells Moisès. Modelling of the Photo-fenton Process with Flexible Hydrogen Peroxide Dosage: Sensitivity Analysis and Experimental Validation[J]. Science of the Total Environment, 2022, 839: 155941.
[20]	Jones J W, Naab J, Fatondji Dougbedji, et al. Uncertainties in Simulating Crop Performance in Degraded Soils and Low Input Production Systems[M]//Job Kihara, Dougbedji Fatondji, Jones J W, et al. Improving Soil Fertility Recommendations in Africa Using the Decision Support System for Agrotechnology Transfer (DSSAT). Dordrecht: Springer Netherlands, 2012: 43-59.
[21]	Saltelli Andrea, Bolado Ricardo. An Alternative Way to Compute Fourier Amplitude Sensitivity Test (FAST)[J]. Computational Statistics & Data Analysis, 1998, 26(4): 445-460.
[22]	Tarantola S, Mara Thierry A. Variance-based Sensitivity Indices of Computer Models with Dependent Inputs: The Fourier Amplitude Sensitivity Test[J]. International Journal for Uncertainty Quantification, 2017, 7(6): 511-523.
[23]	Song Min, Finley S D. ERK and Akt Exhibit Distinct Signaling Responses Following Stimulation by Pro-angiogenic Factors[J]. Cell Communication and Signaling, 2020, 18(1): 114.
[24]	Li Quan, Wan Haiying, Zhao Zhonggai, et al. Sensitivity Analysis of the Penicillin Fermentation Process Model Based on LHS-EPRCC[J]. Computers & Chemical Engineering, 2023, 179: 108405.
[25]	Al-Sultana Khaled S, Maroof Khan M. Computational Experience on Four Algorithms for the Hard Clustering Problem[J]. Pattern Recognition Letters, 1996, 17(3): 295-308.
[26]	Pakgohar Naghmeh, Javad Eshaghi Rad, Gholami Gholamhossein, et al. A Comparative Study of Hard Clustering Algorithms for Vegetation Data[J]. Journal of Vegetation Science, 2021, 32(3): e13042.
[27]	张瑞垚, 周平. 基于鲁棒加权模糊聚类的污水处理过程监测方法[J]. 自动化学报, 2022, 48(9): 2198-2211.
	Zhang Ruiyao, Zhou Ping. Robust Weighted Fuzzy Clustering for Sewage Treatment Process Monitoring[J]. Acta Automatica Sinica, 2022, 48(9): 2198-2211.
[28]	Ruspini E H, Bezdek J C, Keller J M. Fuzzy Clustering: A Historical Perspective[J]. IEEE Computational Intelligence Magazine, 2019, 14(1): 45-55.
[29]	Bezdek J C, Ehrlich R, Full W. FCM: The Fuzzy C-means Clustering Algorithm[J]. Computers & Geosciences, 1984, 10(2/3): 191-203.
[30]	刘阁, 陈彬, 张贤明. 基于T_S模糊辨识的油液真空脱水效率研究[J]. 系统仿真学报, 2017, 29(1): 27-33, 42.
	Liu Ge, Chen Bin, Zhang Xianming. Study on Vacuum Dehydration Rate from Oil Based on T_S Fuzzy Identifying Model[J]. Journal of System Simulation, 2017, 29(1): 27-33, 42.
[31]	Doan Xuan-Tien, Srinivasan Rajagopalan. Online Monitoring of Multi-phase Batch Processes Using Phase-based Multivariate Statistical Process Control[J]. Computers & Chemical Engineering, 2008, 32(1/2): 230-243.
[32]	Wang Bo, He Mengyi, Wang Xingyu, et al. A Multi-model Predictive Control Method for the Pichia pastoris Fermentation Process Based on Relative Error Weighting Algorithm[J]. Alexandria Engineering Journal, 2022, 61(12): 9649-9660.
[33]	Mclachlan G J, Do K, Ambroise C. Wiley Series in Probability and Statistics[M]. [S.l.] : John Wiley & Sons, 2005: 301-308.
[34]	Paleari Livia, Movedi Ermes, Zoli Michele, et al. Sensitivity Analysis Using Morris: Just Screening or an Effective Ranking Method?[J]. Ecological Modelling, 2021, 455: 109648.
[35]	Blower S M, Dowlatabadi H. Sensitivity and Uncertainty Analysis of Complex Models of Disease Transmission: an HIV Model, as an Example[J]. International Statistical Review, 1994, 62(2): 229-243.
[36]	Saltelli Andrea, Annoni Paola, Azzini Ivano, et al. Variance Based Sensitivity Analysis of Model Output. Design and Estimator for the Total Sensitivity Index[J]. Computer Physics Communications, 2010, 181(2): 259-270.