We describe microbial growth and production of value-added chemical compounds in a continuous bioreactor through a dynamical system and we study the local stability of the equilibrium of interest by means of the classical Routh–Hurwitz criterion. The mathematical model considers various biological and structural parameters related to the bioprocess (concentration of substrate inflow, constants of the microchemical reactions, steady-state mass fractions of intracellular proteins, etc.) and thus, the stability condition is given in terms of these parameters. This boils down to deciding the consistency of a system of polynomial inequalities over the reals, which is challenging to solve from an analytical perspective, and out of reach even for traditional computational software designed to solve such problems. We show how to adapt classical techniques for solving polynomial systems to cope with this problem within a few minutes by leveraging its structural properties, thus completing the stability analysis of our model. The paper is accompanied by a Maple worksheet available online.
Keywords: continuous bioreactors, self-replicator model, bacterial growth, local stability
@article{MSIA_2023__12_1_175_0, author = {Agust{\'\i}n G. Yabo and Mohab Safey El Din and Jean-Baptiste Caillau and Jean-Luc Gouz\'e}, title = {Stability analysis of a bacterial growth model through computer algebra}, journal = {MathematicS In Action}, pages = {175--189}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {12}, number = {1}, year = {2023}, doi = {10.5802/msia.37}, language = {en}, url = {https://msia.centre-mersenne.org/articles/10.5802/msia.37/} }
TY - JOUR AU - Agustín G. Yabo AU - Mohab Safey El Din AU - Jean-Baptiste Caillau AU - Jean-Luc Gouzé TI - Stability analysis of a bacterial growth model through computer algebra JO - MathematicS In Action PY - 2023 SP - 175 EP - 189 VL - 12 IS - 1 PB - Société de Mathématiques Appliquées et Industrielles UR - https://msia.centre-mersenne.org/articles/10.5802/msia.37/ DO - 10.5802/msia.37 LA - en ID - MSIA_2023__12_1_175_0 ER -
%0 Journal Article %A Agustín G. Yabo %A Mohab Safey El Din %A Jean-Baptiste Caillau %A Jean-Luc Gouzé %T Stability analysis of a bacterial growth model through computer algebra %J MathematicS In Action %D 2023 %P 175-189 %V 12 %N 1 %I Société de Mathématiques Appliquées et Industrielles %U https://msia.centre-mersenne.org/articles/10.5802/msia.37/ %R 10.5802/msia.37 %G en %F MSIA_2023__12_1_175_0
Agustín G. Yabo; Mohab Safey El Din; Jean-Baptiste Caillau; Jean-Luc Gouzé. Stability analysis of a bacterial growth model through computer algebra. MathematicS In Action, Maths Bio, Volume 12 (2023) no. 1, pp. 175-189. doi : 10.5802/msia.37. https://msia.centre-mersenne.org/articles/10.5802/msia.37/
[1] Monotone control systems, IEEE Trans. Autom. Control, Volume 48 (2003) no. 10, pp. 1684-1698 | DOI | MR | Zbl
[2] Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Semin. Univ. Hamb., Volume 5 (1927) no. 1, pp. 100-115 | DOI | MR | Zbl
[3] A new algorithm to find a point in every cell defined by a family of polynomials, Quantifier elimination and cylindrical algebraic decomposition (Texts and Monographs in Symbolic Computation), Springer, 1998 | DOI | Zbl
[4] Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, 10, Springer, 2006 | Zbl
[5] Quantifier elimination for real closed fields by cylindrical algebraic decomposition, ATFL 2nd GI Conf. Kaiserslautern (Lecture Notes in Computer Science), Volume 33 (1975), pp. 134-183 | MR | Zbl
[6] White paper on continuous bioprocessing, International Symposium on Continuous Manufacturing of Pharmaceuticals (2014)
[7] Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Phys. Rev. A, Volume 35 (1987) no. 12, p. 5288 | DOI | MR
[8] Robust and efficient parameter estimation in dynamic models of biological systems, BMC Syst. Biol., Volume 9 (2015) no. 1, pp. 1-25
[9] Dynamical Allocation of Cellular Resources as an Optimal Control Problem: Novel Insights into Microbial Growth Strategies, PLoS Comput. Biol. (2016) | DOI
[10] The rise and fall of mutator bacteria, Curr. Opin. Microbiol., Volume 4 (2001) no. 5, pp. 582-585 | DOI
[11] Solving systems of polynomial inequalities in subexponential time, J. Symb. Comput., Volume 5 (1988) no. 1-2, pp. 37-64 | DOI | MR
[12] A synthetic growth switch based on controlled expression of RNA polymerase, Mol. Syst. Biol., Volume 11 (2015) no. 11, p. 840 | DOI
[13] Anneaux préordonnés, J. Anal. Math., Volume 12 (1964), pp. 307-326 | DOI | MR | Zbl
[14] Global Optimization with Polynomials and the Problem of Moments, SIAM J. Optim., Volume 11 (2001) no. 3, pp. 796-817 | DOI | MR | Zbl
[15] Global asymptotic behavior of the chemostat: general response functions and different removal rates, SIAM J. Appl. Math., Volume 59 (1998) no. 2, pp. 411-422 | MR | Zbl
[16] The Poincaré–Bendixson theorem for monotone cyclic feedback systems, J. Dyn. Differ. Equations, Volume 2 (1990) no. 4, pp. 367-421 | DOI | Zbl
[17] Automated optogenetic feedback control for precise and robust regulation of gene expression and cell growth, Nat. Commun., Volume 7 (2016) no. 1, pp. 1-11
[18] Description of the chemostat, Science, Volume 112 (1950) no. 2920, pp. 715-716 | DOI
[19] Mathematical modelling of dynamics and control in metabolic networks. I. On Michaelis-Menten kinetics, J. Theor. Biol., Volume 111 (1984) no. 2, pp. 273-302 | DOI | MR
[20] Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, Ph. D. Thesis, California Institute of Technology, Pasadena, California (2000)
[21] Computing sum of squares decompositions with rational coefficients, Theor. Comput. Sci., Volume 409 (2008) no. 2, pp. 269-281 | DOI | MR | Zbl
[22] RAGlib – A library for real solving polynomial systems of equations and inequalities (https://www-polsys.lip6.fr/~safey/RAGLib/)
[23] Polar varieties and computation of one point in each connected component of a smooth real algebraic set, ISSAC’03, ACM Press (2003), pp. 224-231 | Zbl
[24] Emergence of robust growth laws from optimal regulation of ribosome synthesis, Mol. Syst. Biol., Volume 10 (2014) no. 8, p. 747 | DOI
[25] Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, 2008
[26] The theory of the chemostat: dynamics of microbial competition, Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, 1995 | DOI
[27] A nullstellensatz and a positivstellensatz in semialgebraic geometry, Math. Ann., Volume 207 (1974), pp. 87-97 | DOI | MR | Zbl
[28] Solving systems of strict polynomial inequalities, J. Symb. Comput., Volume 29 (2000) no. 3, pp. 471-480 | DOI | MR | Zbl
[29] Convergence results and a Poincaré–Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., Volume 30 (1992) no. 7, pp. 755-763 | Zbl
[30] Bayesian methods in bioinformatics and computational systems biology, Brief. Bioinfor., Volume 8 (2007) no. 2, pp. 109-116 | DOI
[31] Optimal resource allocation in bacterial growth: theoretical study and applications to metabolite production, Ph. D. Thesis, Université Côte d’Azur, Nice, France (2021)
[32] Optimal bacterial resource allocation: metabolite production in continuous bioreactors, Math. Biosci. Eng., Volume 17 (2020) no. 6, pp. 7074-7100 | DOI | MR | Zbl
[33] Hierarchical MPC applied to bacterial resource allocation and metabolite synthesis, 2021 IEEE 60th Conference on Decision and Control (CDC), IEEE (2021) | DOI
[34] Optimal allocation of bacterial resources in fed-batch reactors, 2022 European Control Conference (ECC), IEEE (2022) | DOI
[35] Optimal bacterial resource allocation strategies in batch processing (2022) (preprint, https://hal.inria.fr/hal-03710681)
[36] Dynamical analysis and optimization of a generalized resource allocation model of microbial growth, SIAM J. Appl. Dyn. Syst., Volume 21 (2022) no. 1, pp. 137-165 | DOI | MR | Zbl
[37] Optimizing bacterial resource allocation: metabolite production in continuous bioreactors, IFAC-PapersOnLine, Volume 53 (2020) no. 2, pp. 16753-16758 | DOI
[38] Optimal control of bacterial growth for the maximization of metabolite production, J. Math. Biol. (2018) | DOI | Zbl
Cited by Sources: