Qualitative studies of some biochemical models

Authors

  • Casian Pantea West Virginia University
  • Valery G. Romanovski University of Maribor; Center for Applied Mathematics and Theoretical Physics, University of Maribor

DOI:

https://doi.org/10.21638/11701/spbu01.2020.214

Abstract

A computational approach to detect Andronov — Hopf bifurcations in polynomial systems of ordinary differential equations depending on parameters is proposed. It relies on algorithms of computational commutative algebra based on the Groebner bases theory. The approach is applied to the investigation of two models related to the MAPK (mitogen-activated protein kinases) double phosphorylation, a biochemical network that occurs in many cellular pathways. For the models we perform the analysis of roots of the characteristic polynomials of the Jacobians at the steady states and prove the absence of Andronov — Hopf bifurcations for biochemically relevant values of parameters. We also performed a search for algebraic invariant subspaces in the systems (which represent “weak” conservations laws) and find all subfamilies admitting linear invariant subspaces. The search is done using the Darboux
method. That, is we look for Darboux polynomials and cofactors as polynomials with undetermined coefficients and then determine the coefficients using the algorithms of the elimination theory.

Keywords:

polynomial system of ODEs, Andronov — Hopf bifurcation, invariant subspace, biochemical reactions networks

Downloads

Download data is not yet available.
 

References

1. Feinberg M. Foundations of Chemical Reaction Network Theory. Springer, 2019.

2. Conradi C., Pantea C. Multistationarity in Biochemical Networks: Results, Analysis, and Examples. In: Algebraic and Combinatorial Computational Biology / Eds. R. Robeva, M. Macauley. Academic Press, 2019.

3. Errami H., Eiswirth M., Grigoriev D., Seiler W. M., Sturm T., Weber A. Detection of Hopf Bifurcations in Chemical Reaction Networks Using Convex Coordinates // J. Comput. Phys. 2015. Vol. 291. P. 279-302.

4. Niu W., Wang D. Algebraic Approaches to Stability Analysis of Biological Systems // Math. Comput. Sci. 2008. Vol. 1. P. 507-539.

5. Niu W., Wang D. Algebraic analysis of stability and bifurcation of a self-assembling micelle system // Appl. Math. Comput. 2012. Vol. 219. P. 108-121.

6. Sturm T., Weber A., Abdel-Rahman E. O., El Kahoui M. Investigating Algebraic and Logical Algorithms to Solve Hopf Bifurcation Problems in Algebraic Biology // Math. Comput. Sci. 2009. Vol. 2, no. 3. P. 493-515.

7. Kruff N., Walcher S. Coordinate-Independent Criteria for Hopf Bifurcations // Discrete and Continuous Dynamical Systems. 2020. Vol. 13, no. 4. P. 1319-1340. https://org.doi/ DOI: 10.3934/dcdss.2020075

8. Cox D., Little J., O'Shea D. Ideals, Varieties, and Algorithms. New York: Springer-Verlag, 1992.

9. Conradi C., Mincheva M., Shiu A. Emergence of Oscillations in a Mixed-Mechanism Phosphorylation System // Bull. Math. Biol. 2019. Vol. 81. P. 1829-1852.

10. Banaji M. I nheritance of Oscillation in Chemical Reaction Networks // Appl. Math. Comput. 2018. Vol. 325. P. 191-209.

11. Banaij M., Pantea C. The Inheritance of Nondegenerate Multistationarity in Chemical Reaction Networks // SIAM J. Appl. Math. 2018. Vol. 78. P. 1105-1130.

12. Decker W., Greuel G.-M., Pfister G., Shönemann H. Singular. 4-1-2-A Computer Algebra System for Polynomial Computations. 2019. URL: http://www.singular.uni-kl.de (accessed: December 12, 2019).

13. Decker W., Laplagne S., Pfister G., Schönemann H. Singular. 3-1 library for computing the prime decomposition and radical of ideals. 2010.

14. Gianni P., Trager B., Zacharias G. Gröbner Bases and Primary Decomposition of Polynomial Ideals // J. Symbolic Comput. 1988. Vol. 6. P. 146-167.

15. Arnold E. A. Modular Algorithms for Computing Gröbner Bases // J. Symbolic Comput. 2003. Vol. 35, no. 4. P. 403-419.

16. Romanovski V. G., Prešern M. An Approach to Solving Systems of Polynomials via Modular Arithmetics with Applications // J. Comput. Appl. Math. 2011. Vol. 236. P. 196-208.

Downloads

Published

2020-08-15

How to Cite

Pantea, C., & Romanovski, V. G. (2020). Qualitative studies of some biochemical models. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(2), 319–330. https://doi.org/10.21638/11701/spbu01.2020.214

Issue

Vol. 7 No. 2 (2020)

Section

In memoriam of V. A. Pliss

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.