Mathematical analysis of a quorum sensing induced biofilm dispersal model and numerical simulation of hollowing effects

Blessing O. Emerenini; Stefanie Sonner; Hermann J. Eberl; Blessing O. Emerenini; Stefanie Sonner; Hermann J. Eberl

doi:10.3934/mbe.2017036

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 3: 625-653. doi: 10.3934/mbe.2017036

Previous Article Next Article

Mathematical analysis of a quorum sensing induced biofilm dispersal model and numerical simulation of hollowing effects

1.
Biomedical Physics, Dept. Physics, Ryerson University, 350 Victoria Street Toronto, ON, M5B 2K3, Canada
2.
Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36,8010 Graz, Austria
3.
Dept. Mathematics and Statistics, University of Guelph, 50 Stone Road East, ON, N1G 2W1, Canada

Received: 20 August 2015 Accepted: 26 October 2016 Published: 01 June 2017
MSC : Primary: 35Q92, 35K65; Secondary: 92D25

We analyze a mathematical model of quorum sensing induced biofilm dispersal. It is formulated as a system of non-linear, density-dependent, diffusion-reaction equations. The governing equation for the sessile biomass comprises two non-linear diffusion effects, a degeneracy as in the porous medium equation and fast diffusion. This equation is coupled with three semi-linear diffusion-reaction equations for the concentrations of growth limiting nutrients, autoinducers, and dispersed cells. We prove the existence and uniqueness of bounded non-negative solutions of this system and study the behavior of the model in numerical simulations, where we focus on hollowing effects in established biofilms.
- Quorum sensing,
- biofilm,
- cell dispersal,
- density dependent diffusion,
- existence,
- uniqueness,
- numerical simulation
Citation: Blessing O. Emerenini, Stefanie Sonner, Hermann J. Eberl. Mathematical analysis of a quorum sensing induced biofilm dispersal model and numerical simulation of hollowing effects[J]. Mathematical Biosciences and Engineering, 2017, 14(3): 625-653. doi: 10.3934/mbe.2017036

Related Papers:

Abstract

We analyze a mathematical model of quorum sensing induced biofilm dispersal. It is formulated as a system of non-linear, density-dependent, diffusion-reaction equations. The governing equation for the sessile biomass comprises two non-linear diffusion effects, a degeneracy as in the porous medium equation and fast diffusion. This equation is coupled with three semi-linear diffusion-reaction equations for the concentrations of growth limiting nutrients, autoinducers, and dispersed cells. We prove the existence and uniqueness of bounded non-negative solutions of this system and study the behavior of the model in numerical simulations, where we focus on hollowing effects in established biofilms.

References

[1]	[ F. Abbas,R. Sudarsan,H. J. Eberl, Longtime behaviour of one-dimensional biofilm moels with shear dependent detachment rates, Math. Biosc. Eng., 9 (2012): 215-239.
[2]	[ H. Amman, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, Teubner-Texte Math., 133 (1993): 9-126.
[3]	[ D. Aronson,M. G. Crandall,L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal., 6 (1982): 1001-1022.
[4]	[ N. Barraud,D. J. Hassett,S. H. Hwang,S. A. Rice,S. Kjelleberg,J. S. Webb, Involvement of nitric oxide in biofilm dispersal of Pseudomonas Aeruginosa, J. Bacteriol, 188 (2006): 7344-7353.
[5]	[ G. Boyadjiev,N. Kutev, Comparison principle for quasilinear elliptic and parabolic systems, Comptes rendus de l'Académie bulgare des Sciences, 55 (2002): 9-12.
[6]	[ A. Boyd,A. M. Chakrabarty, Role of alginate lyase in cell detachment of Pseudomonas Aeruginosa, Appl. Environ. Microbiol., 60 (1994): 2355-2359.
[7]	[ V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.
[8]	[ M. E. Davey,N. C. Caiazza,G. A. O'Toole, Rhamnolipid surfactant production affects biofilm architecture in Pseudomonas Aeruginosa PAO1, J. Bacteriol, 185 (2003): 1027-1036.
[9]	[ D. A. D'Argenio,M. W. Calfee,P. B. Rainey,E. C. Pesci, Autolysis and autoaggregation in Pseudomonas Aeruginosa colony morphology mutants, J. Bacteriol., 184 (2002): 6481-6489.
[10]	[ L. Demaret,H. J. Eberl,M. A. Efendiev,R. Lasser, Analysis and simulation of a meso-scale model of diffusive resistance of bacterial biofilms to penetration of antibiotics, Adv. Math. Sci. Appl., 18 (2008): 269-304.
[11]	[ R. M. Donlan, Biofilms and device-associated infections, Emerging Infec. Dis., 7 (2001).
[12]	[ R. Duddu,D. L. Chopp,B. Moran, A two-dimensional continuum model of biofilm growth incorporating fluid flow and shear stress based detachment, Biotechnol. Bioeng., 103 (2009): 92-104.
[13]	[ H. J. Eberl,D. F. Parker,M. C. M. van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med., 3 (2001): 161-175.
[14]	[ H. J. Eberl,L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, Electron. J. Differential Equations, 15 (2007): 77-96.
[15]	[ H. J. Eberl,R. Sudarsan, Exposure of biofilms to slow flow fields: The convective contribution to growth and disinfections, J. Theor. Biol., 253 (2008): 788-807.
[16]	[ M. A. Efendiev,H. J. Eberl,S. V. Zelik, Existence and longtime behaviour of solutions of a nonlinear reaction-diffusion system arising in the modeling of biofilms, Nonlin. Diff. Sys. Rel. Topics, RIMS Kyoto, 1258 (2002): 49-71.
[17]	[ M. A. Efendiev,H. J. Eberl,S. V. Zelik, Existence and longtime behavior of a biofilm model, Comm. Pur. Appl. Math., 8 (2009): 509-531.
[18]	[ B. O. Emerenini,B. A. Hense,C. Kuttler,H. J. Eberl, A mathematical model of quorum sensing induced biofilm detachment, PLoS ONE., 10 (2015).
[19]	[ A. Fekete,C. Kuttler,M. Rothballer,B. A. Hense,D. Fischer,K. Buddrus-Schiemann,M. Lucio,J. Müller,P. Schmitt-Kopplin,A. Hartmann, Dynamic regulation of N-acyl-homoserine lactone production and degradation in Pseudomonas putida IsoF., FEMS Microbiology Ecology, 72 (2010): 22-34.
[20]	[ M. R. Frederick,C. Kuttler,B. A. Hense,H. J. Eberl, A mathematical model of quorum sensing regulated EPS production in biofilms, Theor. Biol. Med. Mod., 8 (2011).
[21]	[ M. R. Frederick,C. Kuttler,B. A. Hense,J. Müller,H. J. Eberl, A mathematical model of quorum sensing in patchy biofilm communities with slow background flow, Can. Appl. Math. Quarterly, 18 (2011): 267-298.
[22]	[ S. M. Hunt,M. A. Hamilton,J. T. Sears,G. Harkin,J. Reno, A computer investigation of chemically mediated detachment in bacterial biofilms, J. Microbiol., 149 (2003): 1155-1163.
[23]	[ S. M. Hunt,E. M. Werner,B. Huang,M. A. Hamilton,P. S. Stewart, Hypothesis for the role of nutrient starvation in biofilm detachment, J. Appl. Environ. Microb., 70 (2004): 7418-7425.
[24]	[ H. Khassehkhan,M. A. Efendiev,H. J. Eberl, A degenerate diffusion-reaction model of an amensalistic biofilm control system: existence and simulation of solutions, Disc. Cont. Dyn. Sys. Series B, 12 (2009): 371-388.
[25]	[ O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of parabolic Type, American Mathematical Society, Providence RI, 1968.
[26]	[ J. B. Langebrake,G. E. Dilanji,S. J. Hagen,P. de Leenheer, Traveling waves in response to a diffusing quorum sensing signal in spatially-extended bacterial colonies, J. Theor. Biol., 363 (2014): 53-61.
[27]	[ P. D. Marsh, Dental plaque as a biofilm and a microbial community implications for health and disease, BMC Oral Health, 6 (2006): S14.
[28]	[ N. Muhammad,H. J. Eberl, OpenMP parallelization of a Mickens time-integration scheme for a mixed-culture biofilm model and its performance on multi-core and multi-processor computers, LNCS, 5976 (2010): 180-195.
[29]	[ G. A. O'Toole,P. S. Stewart, Biofilms strike back, Nature Biotechnology, 23 (2005): 1378-1379.
[30]	[ M. R. Parsek,P. K. Singh, Bacterial biofilms: An emerging link to disease pathogenesis, Annu. Rev. Microbiol., 57 (2003): 677-701.
[31]	[ C. Picioreanu,M. C. M. van Loosdrecht,J. J. Heijnen, Two-dimensional model of biofilm detachment caused by internal stress from liquid flow, Biotechnol. Bioeng., 72 (2001): 205-218.
[32]	[ A. Radu,J. Vrouwenvelder,M. C. M. van Loosdrecht,C. Picioreanu, Effect of flow velocity, substrate concentration and hydraulic cleaning on biofouling of reverse osmosis feed channels, Chem. Eng. J., 188 (2012): 30-39.
[33]	[ M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2^nd edition, Springer Verlag, New York, 2004.
[34]	[ S. A. Rice,K. S. Koh,S. Y. Queck,M. Labbate,K. W. Lam,S. Kjelleberg, Biofilm formation and sloughing in Serratia marcescens are controlled by quorum sensing and nutrient cues, J. Bacteriol, 187 (2005): 3477-3485.
[35]	[ Y. Saad, Iterative Methods for Sparse Linear Systems, 2^nd edition, SIAM, Philadelphia, 2003.
[36]	[ S. Sirca and M. Morvat, Computational Methods for Physicists, Springer, Heidelberg, 2012.
[37]	[ Solano,Echeverz,LasaI, Biofilm dispersion and quorum sensing, Curr. Opin. Microbiol., 18 (2014): 96-104.
[38]	[ S. Sonner,M. A. Efendiev,H. J. Eberl, On the well-posedness of a mathematical model of quorum-sensing in patchy biofilm communities, Math. Methods Appl. Sci., 34 (2011): 1667-1684.
[39]	[ S. Sonner,M. A. Efendiev,H. J. Eberl, On the well-posedness of mathematical models for multicomponent biofilms, Math. Methods Appl. Sci., 38 (2015): 3753-3775.
[40]	[ P. S. Stewart, A model of biofilm detachment, Biotechnol. Bioeng., 41 (1993): 111-117.
[41]	[ M. G. Trulear,W. G. Characklis, Dynamics of biofilm processes, J. Water Pollut. Control Fed., 54 (1982): 1288-1301.
[42]	[ B. L. Vaughan Jr,B. G. Smith,D. L. Chopp, The Influence of Fluid Flow on Modeling Quorum Sensing in Bacterial Biofilms, Bull. Math. Biol., 72 (2010): 1143-1165.
[43]	[ O. Wanner,P. Reichert, Mathematical modelling of mixed-culture biofilm, Biotech. Bioeng., 49 (1996): 172-184.
[44]	[ O. Wanner, H. J. Eberl, E. Morgenroth, D. R. Noguera, C. Picioreanu, B. E. Rittmann and M. C. M. van Loosdrecht, Mathematical Modelling of Biofilms, IWA Publishing, London, 2006.
[45]	[ J. S. Webb, Differentiation and dispersal in biofilms, Book chapter in The Biofilm Mode of Life: Mechanisms and Adaptations, Horizon Biosci., Oxford (2007), 167–178.
[46]	[ J. B. Xavier,C. Piciroeanu,M. C. M. van Loosdrecht, A general description of detachment for multidimensional modelling of biofilms, Biotechnol. Bioeng., 91 (2005): 651-669.
[47]	[ J. B. Xavier,C. Picioreanu,S. A. Rani,M. C. M. van Loosdrecht,P. S. Stewart, Biofilm-control strategies based on enzymic disruption of the extracellular polymeric substance matrix a modelling study, Microbiol., 151 (2005): 3817-3832.

Reader Comments

Your name:*

Email:*
© 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)