Quorum Sensing pp 253-271 | Cite as

# Differential Equations Models to Study Quorum Sensing

## Abstract

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

### Key words

Quorum sensing Differential equations Derivatives Ordinary differential equations Partial differential equations Mathematical models### References

- 1.James S, Nilsson P, James G, Kjelleberg S, Fagerström T (2000) Luminescence control in the marine bacterium
*Vibrio fischeri*: an analysis of the dynamics of*lux*regulation. J Mol Biol 296:1127–1137CrossRefPubMedGoogle Scholar - 2.Nilsson P, Olofsson A, Fagerlind M, Fagerström T, Rice S, Kjelleberg S et al (2001) Kinetics of the
*ahl*regulatory system in a model biofilm system: how many bacteria constitute a “quorum”? J Mol Biol 309:631–640CrossRefPubMedGoogle Scholar - 3.Dockery J, Keener J (2001) A mathematical model for quorum sensing in
*Pseudomonas aeruginosa*. Bull Math Biol 63:95–116CrossRefPubMedGoogle Scholar - 4.Ward JP, King JR, Koerber AJ, Williams P, Croft JM, Sockett RE (2001) Mathematical modelling of quorum sensing in bacteria. Math Med Biol 18:263–292CrossRefGoogle Scholar
- 5.Kuttler C, Hense BA (2008) Interplay of two quorum sensing regulation systems of
*Vibrio fischeri*. J Theor Biol 251:167–180CrossRefPubMedGoogle Scholar - 6.Fekete A, Kuttler C, Rothballer M, Hense BA, Fischer D, Buddrus-Schiemann K et al (2010) Dynamic regulation of
*N*-acyl-homoserine lactone production and degradation in*Pseudomonas putida*IsoF. FEMS Microbiol Ecol 72:22–34CrossRefPubMedGoogle Scholar - 7.Pérez-Velázquez J, Gölgeli M, García-Contreras R (2016) Mathematical modelling of bacterial quorum sensing: a review. Bull Math Biol 78:1585–1639CrossRefPubMedGoogle Scholar
- 8.Ward JP, King JR, Koerber AJ, Croft JM, Sockett RE, Williams P (2004) Cell-signalling repression in bacterial quorum sensing. Math Med Biol 21:169–204CrossRefPubMedGoogle Scholar
- 9.Müller J, Kuttler C, Hense BA, Rothballer M, Hartmann A (2006) Cell-cell communication by quorum sensing and dimension-reduction. J Math Biol 53:672–702CrossRefPubMedGoogle Scholar
- 10.Lupp C, Urbanowski M, Greenberg EP, Ruby EG (2003) The
*Vibrio fischeri*quorum-sensing systems*ain*and*lux*sequentially induce luminescence gene expression and are important for persistence in the squid host. Mol Microbiol 50:319–331CrossRefPubMedGoogle Scholar - 11.Kuo A, Callahan SM, Dunlap PV (1996) Modulation of luminescence operon expression by
*N*-octanoyl-L-homoserine lactone in*ainS*mutants of*Vibrio fischeri*. J Bacteriol 178:971–976CrossRefPubMedPubMedCentralGoogle Scholar - 12.Buddrus-Schiemann K, Rieger M, Mühlbauer M, Barbarossa MV, Kuttler C, Hense BA et al (2014) Analysis of
*N*-acylhomoserine lactone dynamics in continuous cultures of*Pseudomonas putida*IsoF by use of ELISA and UHPLC/qTOF-MS-derived measurements and mathematical models. Anal Bioanal Chem 25:6373–6383CrossRefGoogle Scholar - 13.Koerber AJ, King JR, Ward JP, Williams P, Croft JM, Sockett RE (2002) A mathematical model of partial-thickness burn-wound infection by
*Pseudomonas aeruginosa*: quorum sensing and the build-up to invasion. Bull Math Biol 64:239–259CrossRefPubMedGoogle Scholar - 14.Hense BA, Müller J, Kuttler C, Hartmann A (2012) Spatial heterogeneity of autoinducer regulation systems. Sensors 12:4156–4171CrossRefPubMedPubMedCentralGoogle Scholar
- 15.Fujimoto K, Sawai S (2013) A design principle of group-level decision making in cell populations. PLoS Comput Biol 9:1–13CrossRefGoogle Scholar
- 16.Brown (2013) Linking molecular and population processes in mathematical models of quorum sensing. Bull Math Biol 75:1813–1839CrossRefPubMedGoogle Scholar
- 17.Hense BA, Schuster M (2015) Core principles of bacterial autoinducer systems. Microbiol Mol Biol Rev 79:153–169CrossRefPubMedPubMedCentralGoogle Scholar