# Simulation-Based Exploration of Quorum Sensing Triggered Resistance of Biofilms to Antibiotics

## Abstract

We present a mathematical model of biofilm response to antibiotics, controlled by a quorum sensing system that confers increased resistance. The model is a highly nonlinear system of partial differential equations that we investigate in computer simulations. Our results suggest that an adaptive, quorum sensing-controlled, mechanism to switch between modes of fast growth with little protection and protective modes of slow growth may confer benefits to biofilm populations. It enhances the formation of micro-niches in the inner regions of the biofilm in which bacteria are not easily reached by antibiotics. Whereas quorum sensing inhibitors can delay the onset of increased resistance, their advantage is lost after up-regulation. This emphasizes the importance of timing for treatment of biofilms with antibiotics.

## Keywords

Antibiotics Biofilm Mathematical model Nonlinear diffusion Quorum sensing Simulation## Mathematics Subject Classification

92D25 35K65 65N08 34C60## Notes

### Acknowledgements

This study was supported in parts by the Natural Science and Engineering Research Council of Canada (NSERC) with a Discovery Grant and a Research Tools and Infrastructure Grant awarded to HJE, and a postgraduate scholarship awarded to MG. A visit of MG to the TU Munich for collaborative work was supported in parts by the Technical University of Munich with an Entrepreneurial Award for Global Challenges in The Mathematical Sciences, awarded to CK.

## References

- Abel zur Wiesch P, Clarelli F, Cohen T (2017) Using chemical reaction kinetics to predict optimal antibiotic treatment strategies. PLoS Comput Biol 13(1):e1005321. https://doi.org/10.1371/journal.pcbi.1005321 CrossRefGoogle Scholar
- Anderl JN, Franklin MJ, Stewart PS (2000) Role of antibiotic penetration limitation in
*Klebsiella pneumoniae*biofilm resistance to ampicillin and ciprofloxacin. Antimicrob Agents Chemother 4(7):1818–1824CrossRefGoogle Scholar - Balaban NQ, Merrin J, Chait R, Kowalik L, Leibler S (2004) Bacterial persistence as a phenotypic switch. Science 305:1622–1625CrossRefGoogle Scholar
- Brackman G, Cos P, Maes L, Nelis HJ, Coenye T (2011) Quorum sensing inhibitors increase the susceptibility of bacterial biofilms to antibiotics in vitro and in vivo. Antimicrob Agents Chemother 55:2655–2661CrossRefGoogle Scholar
- Brackman G, Coenye T (2015) Quorum sensing inhibitors as anti-biofilm agents. Curr Pharm Des 21(1):5–11CrossRefGoogle Scholar
- Brown MR et al (1988) Resistance of bacterial biofilms to antibiotics: a growth-rate related effect? J Antimicrob Chemother 22:777–780CrossRefGoogle Scholar
- Bruchmann J, Kirchen S, Schwartz T (2013) Sub-inhibitory concentrations of antibiotics and wastewater influencing biofilm formation and gene expression of multi-resistant
*Pseudomonas aeruginosa*wastewater isolates. Environ Sci Pollut Res 20:3539–3549CrossRefGoogle Scholar - Chambless JD, Hunt SM, Stewart PS (2006) A three-dimensional computer model of four hypothetical mechanisms protecting biofilms from antimicrobials. Appl Environ Microbiol 72(3):2005–2013CrossRefGoogle Scholar
- Chopp DL, Kirisits MJ, Parsek MR, Moran B (2002) A mathematical model of quorum sensing in a growing
*P. aeruginosa*biofilm. J Ind Microbiol Biotechnol 29(6):339–346CrossRefzbMATHGoogle Scholar - Chopp DL, Kirisits MJ, Parsek MR, Moran B (2003) The dependence of quorum sensing on the depth of a growing biofilm. Bull Math Biol 65(6):1053–1079CrossRefzbMATHGoogle Scholar
- Cogan NG, Cortez R, Fauci L (2005) Modeling physiological resistance in bacterial biofilms. Bull Math Biol 67(4):831–853MathSciNetCrossRefzbMATHGoogle Scholar
- Cogan NG (2008) Two-fluid model of biofilm disinfection. Bull Math Biol 70:800–819MathSciNetCrossRefzbMATHGoogle Scholar
- Cogan NG, Szomolay B, Dindos M (2013) Effect of periodic disinfection on persisters in a one-dimensional biofilm model. Bull Math Biol 75:94–123MathSciNetCrossRefzbMATHGoogle Scholar
- Cogan NG, Rath H, Kommerein N, Stumpp SN, Stiesch M (2016) Theoretical and experimental evidence for eliminating persister bacteria by manipulating killing timing. FEMS Microbiol Lett 363(23):fnw264CrossRefGoogle Scholar
- Czaran T, Hoekstra RF (2009) Microbial communication, cooperation and cheating: Quorum sensing dries the evolution of cooperation in bacteria. PLoS One 4:e6655CrossRefGoogle Scholar
- Demaret L, Eberl HJ, Efendiev M, Lasser R (2008) Analysis and simulation of a meso-scale model of diffusive resistance of bacterial biofilms to penetration of antibiotics. Adv Math Sci Appls 18(1):269–304MathSciNetzbMATHGoogle Scholar
- Dillon R, Fauci L, Fogelson A, Gaver D (1996) Modelling biofilm processes using the immersed boundary method. J Comput Phys 129(1):57–73CrossRefzbMATHGoogle Scholar
- Eberl HJ, Parker DF, Van Loosdrecht CM (2001) A new deterministic spatio-temporal continuum model for biofilm development. J Theor Med 3:161–175CrossRefzbMATHGoogle Scholar
- Eberl HJ, Demaret L (2007) A finite difference scheme for a degenerated diffusion equation arising in microbial ecology. El J Differ Equ CS 15:77–95MathSciNetzbMATHGoogle Scholar
- Eberl HJ, Sudarsan R (2008) Exposure of biofilms to slow flow fields: the convective contribution to growth and disinfection. J Theor Biol 253:788–807MathSciNetCrossRefGoogle Scholar
- Eberl HJ, Collinson S (2009) A modelling and simulation study of siderophore mediated antagonsim in dual-species biofilms. Theor Biol Med Mod 6:30CrossRefGoogle Scholar
- Efendiev MA, Zelik SV, Eberl HJ (2009) Existence and longtime behavior of a biofilm model. Commun Pure Appl Anal 8(2):509–531MathSciNetCrossRefzbMATHGoogle Scholar
- Emerenini B, Hense BA, Kuttler C, Eberl HJ (2015) A mathematical model of quorum sensing induced biofilm detachment. Plos ONE 10(7):e0132385CrossRefGoogle Scholar
- Englmann M, Fekete A, Kuttler C, Frommberger M, Li X, Gebefügi I, Fekete J, Schmitt-Kopplin P (2007) The hydrolysis of unsubstituted N-acylhomoserine lactones to their homoserine metabolites. Analytical approaches using ultra performance liquid chromatography. J Chromatogr A 1160(1–2):184–93CrossRefGoogle Scholar
- Frederick M, Kuttler C, Hense BA, Müller J, Eberl HJ (2010) A mathematical model of quorum sensing in patchy biofilm communities with slow background flow. Can Appl Math Q 18(3):267–298MathSciNetzbMATHGoogle Scholar
- Frederick MR, Kuttler C, Hense BA, Eberl HJ (2011) A mathematical model of quorum sensing regulated EPS production in biofilms. Theor Biol Med Mod 8:8CrossRefGoogle Scholar
- Fuqua W, Winans S, Greenberg E (1994) Quorum sensing in bacteria: the LuxR-LuxI family of cell density-responsive transcriptional regulators. J Bacteriol 176:269–275CrossRefGoogle Scholar
- Ghasemi M, Eberl HJ (2017) Extension of a regularization based time-adaptive numerical method for a degenerate diffusion–reaction biofilm growth model to systems involving quorum sensing. Proc Comput Sci 108:1893–1902CrossRefGoogle Scholar
- Ghasemi M, Eberl HJ (2018) Time adaptive numerical solution of a highly degenerate diffusion–reaction biofilm model based on regularisation. J Sci Comput 74:1060–1090MathSciNetCrossRefzbMATHGoogle Scholar
- Hall-Stoodley L, Costerton JW, Stoodley P (2004) Bacterial biofilms: from the natural environment to infectious diseases. Nat Rev Microbiol 2(2):95–108CrossRefGoogle Scholar
- Hense BA, Kuttler C, Müller J, Rothballer M, Hartman A, Kreft JU (2007) Does efficiency sensing unify diffusion and quorum sensing? Nat Rev Microbiol 5:230–239CrossRefGoogle Scholar
- Hense BA, Schuster M (2015) Core principles of bacterial autoinducer systems. Microbiol Mol Biol Rev 79:153–169CrossRefGoogle Scholar
- Hentzer M, Givskov M (2003) Pharmacological inhibition of quorum sensing for the treatment of chronic bacterial infections. J Clin Invest 112(9):1300–1307CrossRefGoogle Scholar
- Imran M, Smith H (2014) A model of optimal dosing of antibiotic treatment in biofilm. Math Biosci Eng 11(3):547–571MathSciNetCrossRefzbMATHGoogle Scholar
- Janakiraman V, Englert D, Jayaraman A, Baskaran H (2009) Modeling growth and quorum sensing in biofilms grown in microfluidic chamber. Ann Biomed Eng 37(6):1206–1216CrossRefGoogle Scholar
- Kalia VC (2013) Quorum sensing inhibitors: an overview. Biotechnol Adv 31(2):224–245CrossRefGoogle Scholar
- Khassehkhan H, Hillen T, Eberl HJ (2009a) A non-linear master equation for a degenerate diffusion model of biofilm growth. LNCS 5544:735–744Google Scholar
- Khassehkhan H, Efendiev MA, Eberl HJ (2009b) Existence and simulations of solutions to a degenerate diffusion–reaction model of an amensalistic biofilm control system. Discrete Contin Dyn Syst B 12(2):371–388MathSciNetCrossRefzbMATHGoogle Scholar
- Ladyženskaja OA, Solonnikov A, Ural’ceva NN (1968) Linear and quasi-linear equations of parabolic type. AMS, ProvidenceCrossRefGoogle Scholar
- Lear G, Lewis GD (2012) Microbial biofilms: current research and applications. Caister Academic Press, Poole. ISBN 978-1-904455-96-7Google Scholar
- Lin LH, Wang JH, Yo JL, Li YY, Liu GX (2013) Effects of Allicin on the formation of
*Pseudomonas aeruginosa*biofilm and the production of Quorum-sensing controlled virulence factors. Pol J Microbiol 62:243–251Google Scholar - Machineni L, Rajapantul A, Nandamuri V, Pawar PD (2017) Influence of nutrient availability and Quorum sensing on the formation of metabolically inactive microcolonies within structurally heterogeneous bacterial biofilms: an individual-based 3D cellular automata model. Bull Math Biol 79(3):594–618MathSciNetCrossRefzbMATHGoogle Scholar
- Mah TC, O’Toole GA (2001) Mechanisms of biofilm resistance to antimicrobial agents. Trends Microbiol 9(1):34–39CrossRefGoogle Scholar
- Martins dos Santos VA, Yakimov MM, Timmis KN, Golyshin PN (2008) Genomic insights into oil biodegradation in marine systems. In Diaz E (ed) Microbial biodegradation: genomics and molecular biology. Horizon Scientific Press, Poole, p 1971. ISBN: 978-1-904455-17-2Google Scholar
- Matur MG, Müller J, Kuttler C, Hense BA (2015) An approximative approach for single cell spatial modeling of quorum sensing. J Comput Biol 22:227–235CrossRefGoogle Scholar
- Muhammad N, Eberl HJ (2011) Model parameter uncertainties in a dual-species biofilm competition model affect ecological output parameters much stronger than morphological ones. Math Biosci 233(1):1–18MathSciNetCrossRefzbMATHGoogle Scholar
- Müller J, Hense BA, Fuchs TM, Utz M, Pötzsche C (2013) Bet-hedging in stochastically switching environments. J Theor Biol 336:144–157MathSciNetCrossRefGoogle Scholar
- Mund A, Kuttler C, Perez-Velazquez J, Hense BA (2016) An age-dependent model to analyse the evolutionary stability of bacterial quorum sensing. J Theor Biol 405:104–115MathSciNetCrossRefzbMATHGoogle Scholar
- Nealson K, Platt T, Hastings J (1970) Cellular control of the synthesis and activity of the bacterial luminescence system. J Bacteriol 104:313–322Google Scholar
- Ngamsaad W, Sunatai S (2016) Mechanically-driven spreading of bacterial populations. Commun Nonlinear Sci Num Simul 35:88–96MathSciNetCrossRefGoogle Scholar
- Nielsen EI, Cars O, Friberg LE (2011) Pharmacokinetic/pharmacodynamic (PK/PD) indices of antibiotics predicted by a semimechanistic PKPD model: a step toward model-based dose optimization. Antimicrob Agents Chemother 55(10):4619–4630CrossRefGoogle Scholar
- Poulsen LV (1999) Microbial biofilm in food processing. LWT Food Sci Technol 32(6):321–326CrossRefGoogle Scholar
- Prakash B, Veeregowda B, Krishnappa G (2003) Biofilms: a survival strategy of bacteria. Curr Sci India 85:1299–1307Google Scholar
- Rang J (2013) Improved traditional Rosenbrock–Wanner methods for stiff ODEs and DAEs. ‘. Institute of Scientific Computing, GermanyzbMATHGoogle Scholar
- Redfield RJ (2002) Is quorum sensing a side effect of diffusion sensing? Trends Microbial 10:365–370CrossRefGoogle Scholar
- Ross-Gillespie A, Kümmerli R (2014) Collective decision-making in microbes. Front Microbiol 5:54CrossRefGoogle Scholar
- Saad Y (1994) SPARSKIT: a basic tool for sparse matrix computations. http://www.users.cs.umn.edu/saad/software/SPARSKIT/sparskit.html
- Schwermer CU, Lavik G, Abed RM et al (2008) Impact of nitrate on the structure and function of bacterial biofilm communities in pipelines used for injection of seawater into oil fields. Appl Environ Microbiol 74(9):2841–51CrossRefGoogle Scholar
- Sengupta S, Chattopadhyay MK, Grossart HP (2013) The multifaceted roles of antibiotics and antibiotic resistance in nature. Front Micriobiol 4:47Google Scholar
- Sonner S, Efendiev MA, Eberl HJ (2011) On the well-posedness of a mathematical model of Quorum-sensing in patchy biofilm communities. Math Methods Appl Sci 34(13):1667–1684MathSciNetCrossRefzbMATHGoogle Scholar
- Stewart PS, Raquepas JB (1995) Implications of reaction–diffusion theory for disinfection of microbial biofilms by reactive antimicrobial agents. Chem Eng Sci 50(19):3099–3104CrossRefGoogle Scholar
- Stewart PS, Hamilton MA, Goldstein BR, Schneider BT (1996) Modelling biocide action against biofilms. Biotech Bioeng 49:445–455CrossRefGoogle Scholar
- Stewart PS (1996) Theoretical aspects of antimicrobial diffusion into microbial biofilms. Antimicrob Agents Chemother 40(11):2517–2522Google Scholar
- Stewart PS, Costerton JE (2001) Antibiotic resistance of bacteria in biofilms. Lancet 358:135–38CrossRefGoogle Scholar
- Stewart PS (2003) Diffusion in biofilms. J Bacteriol 185:1485–1491CrossRefGoogle Scholar
- Stewart PS, Davison WM, Steenbergen JN (2009) Daptomycin rapidly penetrates a staphylococcus epidermidis biofilm. Antimicrob Agents Chemother 53(8):3505–3507CrossRefGoogle Scholar
- Szomolay B, Klapper I, Dindos M (2010) Analysis of adaptive response to dosing protocols for biofilm control. SIAM J Appl Math 70:3175–3202MathSciNetCrossRefzbMATHGoogle Scholar
- Ulitzur S (1989) The regulatory control of the bacterial luminescence system—a new view. J Biolumin Chemilum 4:317–325CrossRefGoogle Scholar
- van Loosdrecht MCM, Heijnen JJ, Eberl H, Kreft J, Picioreanu C (2002) Mathematical modelling of biofilm structures. Antonie van Leeuwenhoek 81(1):245–256CrossRefGoogle Scholar
- Vaughan BL, Smith BG, Chopp DL (2010) The Influence of fluid flow on modeling Quorum sensing in bacterial biofilms. Bull Math Biol 72:1143–1165CrossRefzbMATHGoogle Scholar
- Van der Vorst HA (1992) Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J Sci Stat Comput 13(2):631–644CrossRefzbMATHGoogle Scholar
- Wang LL, Zhang CL, Gong FY, Li HT, Xie XH, Xia C, Chen J, Song Y, Shen AX, Song JX (2013) Influence of
*Pseudomonas aeruginosa*pvdQ gene on altering antibiotic susceptibility under swarming conditions. Curr Microbiol 66:152–161CrossRefGoogle Scholar - Wanner O, Eberl HJ, Van Loosdrecht MCM, Morgenroth E, Noguera DR, Picioreanu C, Rittmann BE (2006) Mathematical modelling of biofilms. IWA Publishing, LondonGoogle Scholar
- Ward JP, King JR, Koerber AJ, Croft JM, Sockett RE, Williams P (2003) Early development and quorum sensing in bacterial biofilms. J Math Biol 47:23–55MathSciNetCrossRefzbMATHGoogle Scholar
- Watnick P, Kolter R (2000) Biofilm—city of microbes (minireview). J Bacteriol 182(10):2675–2679CrossRefGoogle Scholar
- Williams P, Winzer K, Chan W, Cámara M (2007) Look who’s talking: communication and quorum sensing in the bacterial world. Philos Trans R Soc B 362:1119–1134CrossRefGoogle Scholar
- Yates EA et al (2002) N-acylhomoserine lactones undergo lactonolysis in a pH-, temperature-, and acyl chain length-dependent manner during growth of
*Yersinia pseudotuberculosis*and*Pseudomonas aeruginosa*. Infect Immun 70:5635–5646CrossRefGoogle Scholar - Yurtsev EA, Chao HX, Datta MS, Artemova T, Gore J (2013) Bacterial cheating drives the population dynamics of cooperative antibiotic resistance plasmids. Mol Sys Biol 9:683Google Scholar
- Zhao J, Wang Q (2017) Three-dimensional numerical simulations of biofilm dynamics with quorum sensing in a flow cell. Bull Math Biol 79(4):884–919MathSciNetCrossRefzbMATHGoogle Scholar