Stability analysis of a bacterial growth model through computer algebra
Agustín G. Yabo1 ; Mohab Safey El Din2 ; Jean-Baptiste Caillau3 ; Jean-Luc Gouzé4
1 MISTEA, Université Montpellier, INRAE, Institut Agro, Montpellier, France
2 Sorbonne Université, LIP6 (UMR CNRS 7606), PolSys Team, France
3 Université Côte d’Azur, CNRS, Inria, LJAD, France
4 Université Côte d’Azur, Inria, INRAE, CNRS, Sorbonne Université, Biocore Team, Sophia Antipolis, France
MathematicS In Action, Maths Bio, Tome 12 (2023) no. 1, pp. 175-189.

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.

Publié le :
DOI : 10.5802/msia.37
Classification : 37N25, 92-08, 13P15, 14Q30, 68W30
Mots-clés : continuous bioreactors, self-replicator model, bacterial growth, local stability
Agustín G. Yabo 1 ; Mohab Safey El Din 2 ; Jean-Baptiste Caillau 3 ; Jean-Luc Gouzé 4

1 MISTEA, Université Montpellier, INRAE, Institut Agro, Montpellier, France
2 Sorbonne Université, LIP6 (UMR CNRS 7606), PolSys Team, France
3 Université Côte d’Azur, CNRS, Inria, LJAD, France
4 Université Côte d’Azur, Inria, INRAE, CNRS, Sorbonne Université, Biocore Team, Sophia Antipolis, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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, Tome 12 (2023) no. 1, pp. 175-189. doi : 10.5802/msia.37. https://msia.centre-mersenne.org/articles/10.5802/msia.37/

[1] David Angeli; Eduardo D Sontag Monotone control systems, IEEE Trans. Autom. Control, Volume 48 (2003) no. 10, pp. 1684-1698 | DOI | MR | Zbl

[2] Emil Artin Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Semin. Univ. Hamb., Volume 5 (1927) no. 1, pp. 100-115 | DOI | MR | Zbl

[3] Saugata Basu; Richard Pollack; Marie-Françoise Roy 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] Saugata Basu; Richard Pollack; Marie-Françoise Roy Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, 10, Springer, 2006 | Zbl

[5] George E. Collins 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] Charles L. Cooney; Konstantin B. Konstantinov White paper on continuous bioprocessing, International Symposium on Continuous Manufacturing of Pharmaceuticals (2014)

[7] Edmund X. DeJesus; Charles Kaufman 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] Attila Gábor; Julio R. Banga Robust and efficient parameter estimation in dynamic models of biological systems, BMC Syst. Biol., Volume 9 (2015) no. 1, pp. 1-25

[9] Nils Giordano; Francis Mairet; Jean-Luc Gouzé; Johannes Geiselmann; Hidde de Jong Dynamical Allocation of Cellular Resources as an Optimal Control Problem: Novel Insights into Microbial Growth Strategies, PLoS Comput. Biol. (2016) | DOI

[10] Antoine Giraud; Miroslav Radman; Ivan Matic; François Taddei The rise and fall of mutator bacteria, Curr. Opin. Microbiol., Volume 4 (2001) no. 5, pp. 582-585 | DOI

[11] Dimitri Grigoriev; Nikolai Vorobjov Solving systems of polynomial inequalities in subexponential time, J. Symb. Comput., Volume 5 (1988) no. 1-2, pp. 37-64 | DOI | MR

[12] Jérôme Izard; Cindy D. C. Gomez Balderas; Delphine Ropers; Stephan Lacour; Xiaohu Song; Yifan Yang; Ariel B. Lindner; Johannes Geiselmann; Hidde de Jong A synthetic growth switch based on controlled expression of RNA polymerase, Mol. Syst. Biol., Volume 11 (2015) no. 11, p. 840 | DOI

[13] Jean-Louis Krivine Anneaux préordonnés, J. Anal. Math., Volume 12 (1964), pp. 307-326 | DOI | MR | Zbl

[14] Jean-Bernard Lasserre Global Optimization with Polynomials and the Problem of Moments, SIAM J. Optim., Volume 11 (2001) no. 3, pp. 796-817 | DOI | MR | Zbl

[15] Bingtuan Li 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] John Mallet-Paret; Hal L. Smith The Poincaré–Bendixson theorem for monotone cyclic feedback systems, J. Dyn. Differ. Equations, Volume 2 (1990) no. 4, pp. 367-421 | DOI | Zbl

[17] Andreas Milias-Argeitis; Marc Rullan; Stephanie K. Aoki; Peter Buchmann; Mustafa Khammash 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] Aaron Novick; Leo Szilard Description of the chemostat, Science, Volume 112 (1950) no. 2920, pp. 715-716 | DOI

[19] Bernhard O. Palsson; Edwin N. Lightfoot 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] Pablo A. Parrilo Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, Ph. D. Thesis, California Institute of Technology, Pasadena, California (2000)

[21] Helfried Peyrl; Pablo A. Parrilo Computing sum of squares decompositions with rational coefficients, Theor. Comput. Sci., Volume 409 (2008) no. 2, pp. 269-281 | DOI | MR | Zbl

[22] Mohab Safey El Din RAGlib – A library for real solving polynomial systems of equations and inequalities (https://www-polsys.lip6.fr/~safey/RAGLib/)

[23] Mohab Safey El Din; Éric Schost 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] Matthew Scott; Stefan Klumpp; Eduard M. Mateescu; Terence Hwa Emergence of robust growth laws from optimal regulation of ribosome synthesis, Mol. Syst. Biol., Volume 10 (2014) no. 8, p. 747 | DOI

[25] Hal L. Smith Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, 2008

[26] Hal L. Smith; Paul Waltman The theory of the chemostat: dynamics of microbial competition, Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, 1995 | DOI

[27] Gilbert Stengle A nullstellensatz and a positivstellensatz in semialgebraic geometry, Math. Ann., Volume 207 (1974), pp. 87-97 | DOI | MR | Zbl

[28] Adam Strzeboński Solving systems of strict polynomial inequalities, J. Symb. Comput., Volume 29 (2000) no. 3, pp. 471-480 | DOI | MR | Zbl

[29] Horst R. Thieme 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] Darren J. Wilkinson Bayesian methods in bioinformatics and computational systems biology, Brief. Bioinfor., Volume 8 (2007) no. 2, pp. 109-116 | DOI

[31] Agustín Gabriel Yabo Optimal resource allocation in bacterial growth: theoretical study and applications to metabolite production, Ph. D. Thesis, Université Côte d’Azur, Nice, France (2021)

[32] Agustín Gabriel Yabo; Jean-Baptiste Caillau; Jean-Luc Gouzé Optimal bacterial resource allocation: metabolite production in continuous bioreactors, Math. Biosci. Eng., Volume 17 (2020) no. 6, pp. 7074-7100 | DOI | MR | Zbl

[33] Agustín Gabriel Yabo; Jean-Baptiste Caillau; Jean-Luc Gouzé Hierarchical MPC applied to bacterial resource allocation and metabolite synthesis, 2021 IEEE 60th Conference on Decision and Control (CDC), IEEE (2021) | DOI

[34] Agustín Gabriel Yabo; Jean-Baptiste Caillau; Jean-Luc Gouzé Optimal allocation of bacterial resources in fed-batch reactors, 2022 European Control Conference (ECC), IEEE (2022) | DOI

[35] Agustín Gabriel Yabo; Jean-Baptiste Caillau; Jean-Luc Gouzé Optimal bacterial resource allocation strategies in batch processing (2022) (preprint, https://hal.inria.fr/hal-03710681)

[36] Agustín Gabriel Yabo; Jean-Baptiste Caillau; Jean-Luc Gouzé; Hidde de Jong; Francis Mairet 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] Agustín Gabriel Yabo; Jean-Luc Gouzé Optimizing bacterial resource allocation: metabolite production in continuous bioreactors, IFAC-PapersOnLine, Volume 53 (2020) no. 2, pp. 16753-16758 | DOI

[38] Ivan Yegorov; Francis Mairet; Hidde de Jong; Jean-Luc Gouzé Optimal control of bacterial growth for the maximization of metabolite production, J. Math. Biol. (2018) | DOI | Zbl

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