Exploring the Dynamics of Simple Inhibition Systems in Continuous Stirred-Tank Reactor: Mathematical Modelling and Bifurcation Analysis

Afifi Md Desa; Mohd Hafiz Mohd; Mohamad Hekarl Uzir

doi:10.11113/mjfas.v19n5.3064

Authors

Afifi Md Desa ᵃSchool of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia; ᵇInstitute of Engineering Mathematics, Universiti Malaysia Perlis, 02600 Arau, Perlis, Malaysia
Mohd Hafiz Mohd School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia
Mohamad Hekarl Uzir School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia

DOI:

https://doi.org/10.11113/mjfas.v19n5.3064

Keywords:

Enzyme inhibitions, dynamical systems, bifurcation, phase plane analysis

Abstract

The simple enzyme inhibition systems consist of competitive inhibition, noncompetitive inhibition and uncompetitive inhibition. In this work, we incorporated these simple inhibition systems in the continuous stirred-tank reactor (CSTR) and analysed the models using some techniques from dynamical systems and bifurcation analysis. Our aim is to investigate the behaviours of such systems and compare their overall dynamics. The phase portrait is constructed to simulate possible behaviours such as stable steady states, stable limit cycle, bistability between the steady state and the stable limit cycle and bistability between two steady states. The systems undergo bifurcational changes in dynamics as enzyme concentration, dilution rate and proportional control constant are varied. Moreover, we conducted a codimension two bifurcation analysis to examine the joint effects of dilution rate and proportional control constant on the systems behaviours. Our results revealed distinct dynamics for each inhibition system. Increasing the dilution rate led to a transition from low to high substrate concentrations, with competitive inhibition showing the highest tipping (or bifurcation) point where dynamical regimes change due to intense substrate-inhibitor competition. Elevating enzyme concentration reduced substrate concentration, particularly in non-competitive inhibition systems due to higher conversion rates. Furthermore, the proportional control constant had varying effects depending on the specific inhibition system. These findings emphasize the on the combined influences of distinct chemical procoesses in controlling reactor heat and optimizing bioprocess efficiency, considering the unique characteristics of each inhibition system. Overall, the dynamical study on these simple inhibition systems enables us to improve our understanding on the chemical processes involving enzymes with multiple types of inhibitors and may give some insights in its controlling process.

References

... [1] T. Palmer and P. L. Bonner. (2007). Enzymes: Biochemistry, biotechnology, clinical chemistry. Second Edition. Woodhead Publishing Limited.

F. P. Guengerich. (2017). Mechanisms of enzyme catalysis and inhibition. Third Edit. Elsevier.

R. L. Woosley, Y. Chen, J. P. Freiman, and R. A. Gillis. (1993). Mechanism of the Cardiotoxic Actions of Terfenadine. JAMA J. Am. Med. Assoc., 269(12), 1532-1536.

C. H. Yun, R. A. Okerholm, and F. P. Guengerich. (1993). Oxidation of the antihistaminic drug terfenadine in human liver microsomes. Role of cytochrome P-450 3A(4) in N-dealkylation and C-hydroxylation, Drug Metab. Dispos., 21(3), 403 LP-409.

J. F. Brady et al. (1991). Modulation of rat hepatic microsomal monooxygenase enzymes and cytotoxicity by diallyl sulfide. Toxicol. Appl. Pharmacol., 108(2), 342-354.

U. Sharma, E. S. Roberts, and P. F. Hollenberg. (1996). Formation of a metabolic intermediate complex of cytochrome P4502B1 by clorgyline. Drug Metab. Dispos., 24(11), 1247 LP-1253.

J. P. Van Wauwe and P. A. J. Janssen. (1989). Is there a case for P-450 inhibitors in cancer treatment? J. Med. Chem., 32(10), 2231-2239.

H. Vanden Bossche. (1992). Inhibitors of P450-dependent steroid biosynthesis: From research to medical treatment. J. Steroid Biochem. Mol. Biol., 43(8), 1003-1021.

I. H. Segel. (1975). Enzyme kinetics: behavior and analysis of rapid equilibrium and steady state enzyme systems. New York: Wiley.

R. S. Ochs. (2018). Understanding enzyme inhibition. J. Chem. Educ., 77(11), 1453-1456.

N. S. Punekar. (2018). Enzymes: Catalysis, kinetics and mechanisms. Springer.

P. K. Robinson. (2015). Enzymes: Principles and biotechnological applications. Essays Biochem., 59,1-41.

C. G. Whiteley. (2000). Enzyme kinetics: Partial and complete uncompetitive inhibition. Biochem. Educ., 28(3), 144-147.

V. Q. Mai and T. A. Nhan. (2021). Numerical analysis of coupled systems of ODEs and applications to enzymatic competitive inhibition by product. Adv. Theory Nonlinear Anal. its Appl., 5(1), 58-71.

A. Akgül, S. H. A. Khoshnaw, and A. S. Abdalrahman. (2020). Mathematical modeling for enzyme inhibitors with slow and fast subsystems. Arab J. Basic Appl. Sci., 27(1), 442-449.

J. Mirón, M. P. González, J. A. Vázquez, L. Pastrana, and M. A. Murado. (2004). A mathematical model for glucose oxidase kinetics, including inhibitory, deactivant and diffusional effects, and their interactions. Enzyme Microb. Technol., 34(5), 513-522.

G. L. Waldrop. (2009). A qualitative approach to enzyme inhibition. Biochem. Mol. Biol. Educ., 37(1), 11-15.

M. Yoshino and K. Murakami. (2015). Analysis of the substrate inhibition of complete and partial types. Springerplus, 4(1).

M. Danish, M. Khaloofah, A. Mesfer, M. Rashid, and M. K. Al Mesfer. (2015). Effect of operating conditions on CSTR performance: An experimental study. Int. J. Eng. Res. Appl., 5(2), 74-78.

C. Kravaris and I. K. Kookos. (2021). Understanding process dynamics and control. Underst. Process Dyn. Control.

A. Molnár, M. Krajčiová, J. Markoš, and Ľ. Jelemenský. (2004). Use of bifurcation analysis for identification of a safe CSTR operability. J. Loss Prev. Process Ind., 17(6), 489-498.

C. Van Heerden. (1953). Autothermic processes. Ind. Eng. Chem., 45(6), 1242-1247.

A. S. Bommarius and B. R. Riebel. (2004). Biocatalysis, Fundamentals and Applications. John Wiley & Sons.

P. F. Hollenberg. (2022). Enzyme inhibition. Drug Metabolism Handbook, 407-425.

R. Aris and N. R. Amundson. (1958). An analysis of chemical reactor stability and control—II: The evolution of proportional control. Chem. Eng. Sci., 7(3), 132-147.

M. R. M. Radzi and M. H. Uzir. (2009). Stability study of an exothermic biocatalytic reaction and its application in bioprocess systems, 17(June 2007), 95-115.

A. Uppal, W. H. Ray, and A. B. Poore. (1974). On the dynamic behavior of continuous stirred tank reactors. Chem. Eng. Sci., 29(4), 967-985.

M. K. Purkait and D. Haldar. (2021). Strategies to improve enzymatic production of sugars. Lignocellul. Biomass to Value-Added Prod., 95-109.

R. Aris and N. R. Amundson. (1958). An analysis of chemical reactor stability and control—I: The possibility of local control, with perfect or imperfect control mechanisms. Chem. Eng. Sci., 7(3), 121-131.

Seborg. (2019). Process dynamics and control. Fourth Edition. Wiley.

M. Danish, M. K. Al Mesfer, and M. Rashid. (2015). Effect of operating conditions on CSTR performance: An experimental study. J. Eng. Res. Appl., 5(2), 74-78.

N. Miložič, M. Lubej, M. Lakner, P. Žnidaršič-Plazl, and I. Plazl. (2017). Theoretical and experimental study of enzyme kinetics in a microreactor system with surface-immobilized biocatalyst. Chem. Eng. J., 313, 374-381.

A. Matin, A. Grootjans, and H. Hogenhuis. (1976). Influence of dilution rate on enzymes of intermediary metabolism in two freshwater bacteria grown in continuous culture. J. Gen. Microbiol., 94(2), 323-332.

C. Zhang and X. H. Xing. (2011). Enzyme Bioreactors. Second Edi. Elsevier.

R. Konnur and S. Pushpavanam. (1994). The dynamics of a fed-batch reactor: The transition from the batch to the CSTR. Chem. Eng. Sci., 49(3), 383-392.

J. W. Swarts, R. C. Kolfschoten, M. C. A. A. Jansen, A. E. M. Janssen, and R. M. Boom. (2010). Effect of diffusion on enzyme activity in a microreactor. Chem. Eng. J., 162(1), 301-306.

A. Uppal, W. H. Ray, and A. B. Poore. (1976). The classification of the dynamic behavior of continuous stirred tank reactors-influence of reactor residence time. Chem. Eng. Sci., 31(3), 205-214.

D. Paolucci-Jeanjean, M. P. Belleville, G. M. Rios, and N. Zakhia. (2000). The effect of enzyme concentration and space time on the performance of a continuous recycle membrane reactor for one-step starch hydrolysis. Biochem. Eng. J., 5(1), 17-22.

R. Ball and B. F. Gray. (2013). Thermal instability and runaway criteria: The dangers of disregarding dynamics. Process Saf. Environ. Prot., 91(3), 221-226.

R. Ball and B. F. Gray. (1999). Thermal stabilization of chemical reactors II. Bifurcation analysis of the Endex CSTR. Proc. R. Soc. A Math. Phys. Eng. Sci., 455(1992), 4223-4243.

R. M. Daniel, M. J. Danson, R. Eisenthal, C. K. Lee, and M. E. Peterson. (2008). The effect of temperature on enzyme activity: New insights and their implications. Extremophiles, 12(1), 51-59.

K. T. Lu, K. M. Luo, T. F. Yeh, and P. C. Lin. (2008). The kinetic parameters and safe operating conditions of nitroglycerine manufacture in the CSTR of Biazzi process. Process Saf. Environ. Prot., 86(1B), 37-4