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

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

doi:10.3934/mbe.2017036

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June 2017, 14(3): 625-653. doi: 10.3934/mbe.2017036

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

Blessing O. Emerenini ^1,, , Stefanie Sonner ^2, and Hermann J. Eberl ^3,

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

* Corresponding author: Blessing O. Emerenini

Received August 20, 2015 Accepted October 26, 2016 Published December 2016

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.

Keywords: Quorum sensing, biofilm, cell dispersal, density dependent diffusion, existence, uniqueness, numerical simulation.

Mathematics Subject Classification: Primary: 35Q92, 35K65; Secondary: 92D25.

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. Mathematical Biosciences & Engineering, 2017, 14 (3) : 625-653. doi: 10.3934/mbe.2017036

References:

[1]	F. Abbas, R. Sudarsan and H. J. Eberl, Longtime behaviour of one-dimensional biofilm moels with shear dependent detachment rates, Math. Biosc. Eng., 9 (2012), 215-239. doi: 10.3934/mbe.2012.9.215.
[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. doi: 10.1007/978-3-663-11336-2_1.
[3]	D. Aronson, M. G. Crandall and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal., 6 (1982), 1001-1022. doi: 10.1016/0362-546X(82)90072-4.
[4]	N. Barraud, D. J. Hassett, S. H. Hwang, S. A. Rice, S. Kjelleberg and J. S. Webb, Involvement of nitric oxide in biofilm dispersal of Pseudomonas Aeruginosa, J. Bacteriol, 188 (2006), 7344-7353. doi: 10.1128/JB.00779-06.
[5]	G. Boyadjiev and 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 and 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 and G. A. O'Toole, Rhamnolipid surfactant production affects biofilm architecture in Pseudomonas Aeruginosa PAO1, J. Bacteriol, 185 (2003), 1027-1036. doi: 10.1128/JB.185.3.1027-1036.2003.
[9]	D. A. D'Argenio, M. W. Calfee, P. B. Rainey and E. C. Pesci, Autolysis and autoaggregation in Pseudomonas Aeruginosa colony morphology mutants, J. Bacteriol., 184 (2002), 6481-6489. doi: 10.1128/JB.184.23.6481-6489.2002.
[10]	L. Demaret, H. J. Eberl, M. A. Efendiev and 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 and B. Moran, A two-dimensional continuum model of biofilm growth incorporating fluid flow and shear stress based detachment, Biotechnol. Bioeng., 103 (2009), 92-104. doi: 10.1002/bit.22233.
[13]	H. J. Eberl, D. F. Parker and M. C. M. van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med., 3 (2001), 161-175. doi: 10.1080/10273660108833072.
[14]	H. J. Eberl and 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 and R. Sudarsan, Exposure of biofilms to slow flow fields: The convective contribution to growth and disinfections, J. Theor. Biol., 253 (2008), 788-807. doi: 10.1016/j.jtbi.2008.04.013.
[16]	M. A. Efendiev, H. J. Eberl and 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 and S. V. Zelik, Existence and longtime behavior of a biofilm model, Comm. Pur. Appl. Math., 8 (2009), 509-531. doi: 10.3934/cpaa.2009.8.509.
[18]	B. O. Emerenini, B. A. Hense, C. Kuttler and H. J. Eberl, A mathematical model of quorum sensing induced biofilm detachment, PLoS ONE., 10 (2015). doi: 10.1371/journal.pone.0132385.
[19]	A. Fekete, C. Kuttler, M. Rothballer, B. A. Hense, D. Fischer, K. Buddrus-Schiemann, M. Lucio, J. Müller, P. Schmitt-Kopplin and 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 and H. J. Eberl, A mathematical model of quorum sensing regulated EPS production in biofilms, Theor. Biol. Med. Mod., 8 (2011). doi: 10.1186/1742-4682-8-8.
[21]	M. R. Frederick, C. Kuttler, B. A. Hense, J. Müller and 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 and J. Reno, A computer investigation of chemically mediated detachment in bacterial biofilms, J. Microbiol., 149 (2003), 1155-1163. doi: 10.1099/mic.0.26134-0.
[23]	S. M. Hunt, E. M. Werner, B. Huang, M. A. Hamilton and P. S. Stewart, Hypothesis for the role of nutrient starvation in biofilm detachment, J. Appl. Environ. Microb., 70 (2004), 7418-7425. doi: 10.1128/AEM.70.12.7418-7425.2004.
[24]	H. Khassehkhan, M. A. Efendiev and 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. doi: 10.3934/dcdsb.2009.12.371.
[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 and 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. doi: 10.1016/j.jtbi.2014.07.033.
[27]	P. D. Marsh, Dental plaque as a biofilm and a microbial community implications for health and disease, BMC Oral Health, 6 (2006), S14. doi: 10.1186/1472-6831-6-S1-S14.
[28]	N. Muhammad and 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. doi: 10.1007/978-3-642-12659-8_14.
[29]	G. A. O'Toole and P. S. Stewart, Biofilms strike back, Nature Biotechnology, 23 (2005), 1378-1379. doi: 10.1038/nbt1105-1378.
[30]	M. R. Parsek and P. K. Singh, Bacterial biofilms: An emerging link to disease pathogenesis, Annu. Rev. Microbiol., 57 (2003), 677-701. doi: 10.1146/annurev.micro.57.030502.090720.
[31]	C. Picioreanu, M. C. M. van Loosdrecht and J. J. Heijnen, Two-dimensional model of biofilm detachment caused by internal stress from liquid flow, Biotechnol. Bioeng., 72 (2001), 205-218. doi: 10.1002/1097-0290(20000120)72:2<205::AID-BIT9>3.0.CO;2-L.
[32]	A. Radu, J. Vrouwenvelder, M. C. M. van Loosdrecht and 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. doi: 10.1016/j.cej.2012.01.133.
[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 and S. Kjelleberg, Biofilm formation and sloughing in Serratia marcescens are controlled by quorum sensing and nutrient cues, J. Bacteriol, 187 (2005), 3477-3485. doi: 10.1128/JB.187.10.3477-3485.2005.
[35]	Y. Saad, Iterative Methods for Sparse Linear Systems, 2^nd edition, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718003.
[36]	S. Sirca and M. Morvat, Computational Methods for Physicists, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-32478-9.
[37]	Solano, Echeverz and LasaI, Biofilm dispersion and quorum sensing, Curr. Opin. Microbiol., 18 (2014), 96-104. doi: 10.1016/j.mib.2014.02.008.
[38]	S. Sonner, M. A. Efendiev and 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. doi: 10.1002/mma.1475.
[39]	S. Sonner, M. A. Efendiev and H. J. Eberl, On the well-posedness of mathematical models for multicomponent biofilms, Math. Methods Appl. Sci., 38 (2015), 3753-3775. doi: 10.1002/mma.3315.
[40]	P. S. Stewart, A model of biofilm detachment, Biotechnol. Bioeng., 41 (1993), 111-117. doi: 10.1002/bit.260410115.
[41]	M. G. Trulear and W. G. Characklis, Dynamics of biofilm processes, J. Water Pollut. Control Fed., 54 (1982), 1288-1301.
[42]	B. L. Vaughan Jr, B. G. Smith and 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 and 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 and M. C. M. van Loosdrecht, A general description of detachment for multidimensional modelling of biofilms, Biotechnol. Bioeng., 91 (2005), 651-669. doi: 10.1002/bit.20544.
[47]	J. B. Xavier, C. Picioreanu, S. A. Rani, M. C. M. van Loosdrecht and P. S. Stewart, Biofilm-control strategies based on enzymic disruption of the extracellular polymeric substance matrix a modelling study, Microbiol., 151 (2005), 3817-3832. doi: 10.1099/mic.0.28165-0.

show all references

References:

[1]	F. Abbas, R. Sudarsan and H. J. Eberl, Longtime behaviour of one-dimensional biofilm moels with shear dependent detachment rates, Math. Biosc. Eng., 9 (2012), 215-239. doi: 10.3934/mbe.2012.9.215.
[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. doi: 10.1007/978-3-663-11336-2_1.
[3]	D. Aronson, M. G. Crandall and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal., 6 (1982), 1001-1022. doi: 10.1016/0362-546X(82)90072-4.
[4]	N. Barraud, D. J. Hassett, S. H. Hwang, S. A. Rice, S. Kjelleberg and J. S. Webb, Involvement of nitric oxide in biofilm dispersal of Pseudomonas Aeruginosa, J. Bacteriol, 188 (2006), 7344-7353. doi: 10.1128/JB.00779-06.
[5]	G. Boyadjiev and 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 and 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 and G. A. O'Toole, Rhamnolipid surfactant production affects biofilm architecture in Pseudomonas Aeruginosa PAO1, J. Bacteriol, 185 (2003), 1027-1036. doi: 10.1128/JB.185.3.1027-1036.2003.
[9]	D. A. D'Argenio, M. W. Calfee, P. B. Rainey and E. C. Pesci, Autolysis and autoaggregation in Pseudomonas Aeruginosa colony morphology mutants, J. Bacteriol., 184 (2002), 6481-6489. doi: 10.1128/JB.184.23.6481-6489.2002.
[10]	L. Demaret, H. J. Eberl, M. A. Efendiev and 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 and B. Moran, A two-dimensional continuum model of biofilm growth incorporating fluid flow and shear stress based detachment, Biotechnol. Bioeng., 103 (2009), 92-104. doi: 10.1002/bit.22233.
[13]	H. J. Eberl, D. F. Parker and M. C. M. van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med., 3 (2001), 161-175. doi: 10.1080/10273660108833072.
[14]	H. J. Eberl and 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 and R. Sudarsan, Exposure of biofilms to slow flow fields: The convective contribution to growth and disinfections, J. Theor. Biol., 253 (2008), 788-807. doi: 10.1016/j.jtbi.2008.04.013.
[16]	M. A. Efendiev, H. J. Eberl and 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 and S. V. Zelik, Existence and longtime behavior of a biofilm model, Comm. Pur. Appl. Math., 8 (2009), 509-531. doi: 10.3934/cpaa.2009.8.509.
[18]	B. O. Emerenini, B. A. Hense, C. Kuttler and H. J. Eberl, A mathematical model of quorum sensing induced biofilm detachment, PLoS ONE., 10 (2015). doi: 10.1371/journal.pone.0132385.
[19]	A. Fekete, C. Kuttler, M. Rothballer, B. A. Hense, D. Fischer, K. Buddrus-Schiemann, M. Lucio, J. Müller, P. Schmitt-Kopplin and 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 and H. J. Eberl, A mathematical model of quorum sensing regulated EPS production in biofilms, Theor. Biol. Med. Mod., 8 (2011). doi: 10.1186/1742-4682-8-8.
[21]	M. R. Frederick, C. Kuttler, B. A. Hense, J. Müller and 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 and J. Reno, A computer investigation of chemically mediated detachment in bacterial biofilms, J. Microbiol., 149 (2003), 1155-1163. doi: 10.1099/mic.0.26134-0.
[23]	S. M. Hunt, E. M. Werner, B. Huang, M. A. Hamilton and P. S. Stewart, Hypothesis for the role of nutrient starvation in biofilm detachment, J. Appl. Environ. Microb., 70 (2004), 7418-7425. doi: 10.1128/AEM.70.12.7418-7425.2004.
[24]	H. Khassehkhan, M. A. Efendiev and 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. doi: 10.3934/dcdsb.2009.12.371.
[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 and 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. doi: 10.1016/j.jtbi.2014.07.033.
[27]	P. D. Marsh, Dental plaque as a biofilm and a microbial community implications for health and disease, BMC Oral Health, 6 (2006), S14. doi: 10.1186/1472-6831-6-S1-S14.
[28]	N. Muhammad and 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. doi: 10.1007/978-3-642-12659-8_14.
[29]	G. A. O'Toole and P. S. Stewart, Biofilms strike back, Nature Biotechnology, 23 (2005), 1378-1379. doi: 10.1038/nbt1105-1378.
[30]	M. R. Parsek and P. K. Singh, Bacterial biofilms: An emerging link to disease pathogenesis, Annu. Rev. Microbiol., 57 (2003), 677-701. doi: 10.1146/annurev.micro.57.030502.090720.
[31]	C. Picioreanu, M. C. M. van Loosdrecht and J. J. Heijnen, Two-dimensional model of biofilm detachment caused by internal stress from liquid flow, Biotechnol. Bioeng., 72 (2001), 205-218. doi: 10.1002/1097-0290(20000120)72:2<205::AID-BIT9>3.0.CO;2-L.
[32]	A. Radu, J. Vrouwenvelder, M. C. M. van Loosdrecht and 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. doi: 10.1016/j.cej.2012.01.133.
[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 and S. Kjelleberg, Biofilm formation and sloughing in Serratia marcescens are controlled by quorum sensing and nutrient cues, J. Bacteriol, 187 (2005), 3477-3485. doi: 10.1128/JB.187.10.3477-3485.2005.
[35]	Y. Saad, Iterative Methods for Sparse Linear Systems, 2^nd edition, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718003.
[36]	S. Sirca and M. Morvat, Computational Methods for Physicists, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-32478-9.
[37]	Solano, Echeverz and LasaI, Biofilm dispersion and quorum sensing, Curr. Opin. Microbiol., 18 (2014), 96-104. doi: 10.1016/j.mib.2014.02.008.
[38]	S. Sonner, M. A. Efendiev and 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. doi: 10.1002/mma.1475.
[39]	S. Sonner, M. A. Efendiev and H. J. Eberl, On the well-posedness of mathematical models for multicomponent biofilms, Math. Methods Appl. Sci., 38 (2015), 3753-3775. doi: 10.1002/mma.3315.
[40]	P. S. Stewart, A model of biofilm detachment, Biotechnol. Bioeng., 41 (1993), 111-117. doi: 10.1002/bit.260410115.
[41]	M. G. Trulear and W. G. Characklis, Dynamics of biofilm processes, J. Water Pollut. Control Fed., 54 (1982), 1288-1301.
[42]	B. L. Vaughan Jr, B. G. Smith and 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 and 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 and M. C. M. van Loosdrecht, A general description of detachment for multidimensional modelling of biofilms, Biotechnol. Bioeng., 91 (2005), 651-669. doi: 10.1002/bit.20544.
[47]	J. B. Xavier, C. Picioreanu, S. A. Rani, M. C. M. van Loosdrecht and P. S. Stewart, Biofilm-control strategies based on enzymic disruption of the extracellular polymeric substance matrix a modelling study, Microbiol., 151 (2005), 3817-3832. doi: 10.1099/mic.0.28165-0.

Figure 1. Schematic of the biofilm system cf [18]: The aqueous phase is the domain $\Omega_1(t) = \{x\in \Omega: M(t;x) =0 \}$, the biofilm phase $\Omega_2(t) = \{x\in \Omega: M(t;x) >0 \}$. These regions change over time as the biofilm grows. Biofilm colonies form attached to the substratum, which is a part of the boundary of the domain

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Figure 2. 2-D structural representation of the microbial floc growth for autoinducer production rate $\alpha=30.7$ and maximum dispersal rate $\eta_1=0.6$ for selected time instances t. Color coded is the biomass density $M$, iso-lines of the autoinducer concentration $A$ are plotted in grayscale

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Figure 3. 1-D Spatial representation of the development and dispersal of bacterial cells from the microbial floc: The snapshots are taken at different computational time $t$, with an autoinducer production rate $\alpha=30.7$ and a dispersal rate of $\eta_1=0.6$

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Figure 4. Temporal plots of simulations computed for a non-quorum sensing producing microfloc (Non-QS) and a quorum sensing producing microfloc using seven different constitutive autoinducer production rate $\alpha = \{92.0,46.0,30.7,23.0,18.4,15.3,13.1\}$ and fixed maximum dispersal rate $\eta_1=0.6$. Shown are (a) the total sessile biomass fraction $M_{tot}$ in the floc, (b) the floc size $\omega$ (c) dispersed cells $N_{tot}$, (d) relative variation $R$, and (e) signal concentration $A_{ave}$

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Figure 5. 2-D structural representation of the microbial biofilm growth for autoinducer production rate $\alpha=30.7$ and maximum dispersal rate $\eta_1=0.6$ for selected time instances $t$. Color coded is the biomass density $M$, iso-lines of the autoinducer concentration $A$ are plotted in grayscale

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Figure 6. 2-D structural representation of the microbial biofilm growth for autoinducer production rate $\alpha=92.0$ and maximum dispersal rate $\eta_1=0.6$ for selected time instances $t$. Color coded is the biomass density $M$, iso-lines of the autoinducer concentration $A$ are plotted in grayscale

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Figure 7. Temporal plots of simulations computed for a non-quorum sensing producing biofilm (Non-QS) and a quorum sensing producing biofilm using seven different constitutive autoinducer production rate $\alpha = \{92.0,46.0,30.7,23.0,18.4,15.3,13.1\}$ and fixed maximum dispersal rate $\eta_1=0.6$. Shown are (a) the total sessile biomass fraction $M_{tot}$ in the floc, (b) the floc size $\omega$ (c) dispersed cells $N_{tot}$, (d) relative variation $R$, and (e) signal concentration $A_{ave}$

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Figure 8. Comparison of the sessile biomass $M_{tot}$ and the dispersed cells $N_{tot}$ under different boundary conditions for the signal molecule $A$: Homogenous Dirichlet conditions and Neumann conditions. The left panel is for a microbial floc while the right panel is for a biofilm

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Table 1. Parameters used in the numerical simulations

Parameter	Description	Value	Source
$k_1$	half saturation concentration (growth)	$0.4$	[44]
$k_2$	lysis rate	$0.067$	assumed
$\sigma$	nutrient consumption rate	$793.65$	[19]
$\eta_1$	maximum dispersal rate	varied	[18]
$\lambda$	quorum sensing abiotic decay rate	$0.02218$	[39]
$\alpha$	constitutive autoinducer production rate	varied	-
$\beta$	induced autoinducer production rate	$10 \times \alpha$	[19]
$m$	degree of polymerization	$2.5$	[19]
$d_1$	constant diffusion coefficients for $N$	$4.1667$	assumed
$d_2$	constant diffusion coefficients for $C$	$4.1667$	[15]
$d_3$	constant diffusion coefficients for $A$	$3.234$	[15]
$d$	biomass motility coefficient	$4.2 \times 10^{-8}$	[13]
$a$	biofilm diffusion exponent	$4.0$	[13]
$b$	biofilm diffusion exponent	$4.0$	[13]
$L$	system length	$1.0$	[15]
$H$	system height	$1.0$	assumed

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Parameter	Description	Value	Source
$k_1$	half saturation concentration (growth)	$0.4$	[44]
$k_2$	lysis rate	$0.067$	assumed
$\sigma$	nutrient consumption rate	$793.65$	[19]
$\eta_1$	maximum dispersal rate	varied	[18]
$\lambda$	quorum sensing abiotic decay rate	$0.02218$	[39]
$\alpha$	constitutive autoinducer production rate	varied	-
$\beta$	induced autoinducer production rate	$10 \times \alpha$	[19]
$m$	degree of polymerization	$2.5$	[19]
$d_1$	constant diffusion coefficients for $N$	$4.1667$	assumed
$d_2$	constant diffusion coefficients for $C$	$4.1667$	[15]
$d_3$	constant diffusion coefficients for $A$	$3.234$	[15]
$d$	biomass motility coefficient	$4.2 \times 10^{-8}$	[13]
$a$	biofilm diffusion exponent	$4.0$	[13]
$b$	biofilm diffusion exponent	$4.0$	[13]
$L$	system length	$1.0$	[15]
$H$	system height	$1.0$	assumed

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2018 Impact Factor: 1.313

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2018 Impact Factor: 1.313

American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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