Keywords
Sel’kov model
Substrate inhibition
Period-doubling bifurcation
Chaos control
Comparative dynamical analysis
How to Cite
Abstract
An investigation of the dynamic effects of substrate inhibition in a modified Sel’kov model of glycolytic oscillations is presented in this paper. With a saturated nonlinear term in place of classical polynomial feedback, the discrete-time formulation captures enzymatic regulation more realistically. It exhibits complicated dynamics, including period-doubling bifurcations, compared to the classical continuous Sel’kov model, which undergoes Hopf bifurcation. In this study, the model’s behavior is investigated in multiple ways, including fixed point determination, stability assessment based on the Schur-Cohn criterion, and comprehensive numerical bifurcation analysis. Dynamic transitions from stability to periodic cycles, and then to chaos are revealed. Based on a comparative analysis with the classical model, we demonstrate how substrate inhibition induces complex nonlinear behavior through successive bifurcations. A deeper understanding of feedback regulation in biochemical systems can be gained from this study.
References
Abbas, A. and A. Khaliq, 2023. Analyzing predator–prey interaction in chaotic and bifurcating environments. Chaos Theory and Applications, 5: 207–218.
Alligood, K. T., T. D. Sauer, and J. A. Yorke, 1996. Chaos: An Introduction to Dynamical Systems. Springer, New York.
Bastin, G. and D. Dochain, 1990. On-line Estimation and Adaptive Control of Bioreactors. Elsevier, Amsterdam.
Bertram, R., L. S. Satin, M. Zhang, P. Smolen, and A. Sherman, 2007. Metabolic and electrical oscillations: partners in controlling pulsatile insulin secretion. American Journal of Physiology-Endocrinology and Metabolism, 293: E890–E900.
Chen, L., R. Wang, and K. Aihara, 2021. Nonlinear dynamical systems in biology: Analysis and control. Annual Review of Biomedical Engineering, 23: 75–106.
De Vries, G., T. Hillen, M. Lewis, J. Muller, and B. Schönfisch, 2006. A Course in Mathematical Biology: Quantitative Modeling with Mathematical and Computational Methods. SIAM, Philadelphia.
Elaydi, S., 2005. An Introduction to Difference Equations. Springer, New York, third edition.
Fall, C. P., E. S. Marland, J. M. Wagner, and J. J. Tyson, 2002. Computational Cell Biology. Springer, New York.
Gambino, G., M. C. Lombardo, and M. Sammartino, 2022. Discrete dynamical modeling of ecological and biological systems. Mathematics, 10: 1154.
Gaspers, L. D. and A. P. Thomas, 2000. Mitochondrial metabolism of pyruvate is required for glucose-stimulated insulin secretion and is regulated by calcium in β-cells. Molecular and Cellular Endocrinology, 167: 1–10.
Glass, L. and M. C. Mackey, 1988. From Clocks to Chaos: The Rhythms of Life. Princeton University Press, Princeton.
Goldbeter, A., 1996. Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour. Cambridge University Press, Cambridge.
Guckenheimer, J. and P. Holmes, 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York.
Hassard, B. D., N. D. Kazarinoff, and Y.-H. Wan, 1981. Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge.
Heinrich, R. and S. Schuster, 1996. The Regulation of Cellular Systems. Springer, New York.
Hofmeyr, J.-H. S. and A. Cornish-Bowden, 1986. The thermodynamic basis of metabolic control analysis. European Journal of Biochemistry, 155: 265–270.
Kuznetsov, Y. A., 2004. Elements of Applied Bifurcation Theory. Springer, New York, third edition.
Merdan, H. and O. Duman, 2022. Stability and period-doubling bifurcation in a modified commensal symbiosis model with Allee effect. Mathematical Modelling of Natural Phenomena, 17: 1–16.
Murray, J. D., 2002. Mathematical Biology I: An Introduction. Springer, New York, third edition.
Nicolis, G. and I. Prigogine, 1977. Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations. Wiley, New York.
Richard, P., 2003. Metabolic regulation: a precision-control mechanism of the cell. Biochemical Journal, 375: 1–16.
Sataric, M. V., T. Nemes, and J. A. Tuszynski, 2024. Re-examination of the Sel’kov model of glycolysis and its symmetry-breaking instability due to diffusion. Biophysica, 4: 545–560.
Schnakenberg, J., 1979. Simple chemical reaction systems with limit cycle behaviour. Journal of Theoretical Biology, 81: 389–400.
Segel, L. A., 1988. Modeling Dynamic Phenomena in Molecular and Cellular Biology. Cambridge University Press, Cambridge.
Sel’kov, E. E., 1968. Self-oscillations in glycolysis: I. A simple kinetic model. European Journal of Biochemistry, 4: 79–86.
Strogatz, S. H., 2015. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press, Boca Raton, second edition.
Teusink, B. et al., 2000. Can yeast glycolysis be understood in terms of in vitro kinetics of the constituent enzymes? Testing biochemistry. European Journal of Biochemistry, 267: 5313–5329.
Tornheim, K. and J. M. Lowenstein, 1979. Oscillations of glycolysis in yeast: evidence for an ATP-initiated feedback system. Proceedings of the National Academy of Sciences of the USA, 76: 6602–6606.
Wiggins, S., 2003. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York.
Yoshino, M., K. Murakami, and A. Tsuji, 2015. Analysis of the substrate inhibition of complete and partial types. Biochemical and Biophysical Reports, 3: 1–6.
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