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

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.

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. AbbasR. 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.  Google Scholar

[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.  Google Scholar

[3]

D. AronsonM. 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.  Google Scholar

[4]

N. BarraudD. J. HassettS. H. HwangS. A. RiceS. 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.  Google Scholar

[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.   Google Scholar

[6]

A. Boyd and A. M. Chakrabarty, Role of alginate lyase in cell detachment of Pseudomonas Aeruginosa, Appl. Environ. Microbiol., 60 (1994), 2355-2359.   Google Scholar

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[8]

M. E. DaveyN. 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.  Google Scholar

[9]

D. A. D'ArgenioM. W. CalfeeP. 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.  Google Scholar

[10]

L. DemaretH. J. EberlM. 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.   Google Scholar

[11]

R. M. Donlan, Biofilms and device-associated infections, Emerging Infec. Dis., 7 (2001).   Google Scholar

[12]

R. DudduD. 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.  Google Scholar

[13]

H. J. EberlD. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

M. A. EfendievH. 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.   Google Scholar

[17]

M. A. EfendievH. 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.  Google Scholar

[18]

B. O. EmereniniB. A. HenseC. Kuttler and H. J. Eberl, A mathematical model of quorum sensing induced biofilm detachment, PLoS ONE., 10 (2015).  doi: 10.1371/journal.pone.0132385.  Google Scholar

[19]

A. FeketeC. KuttlerM. RothballerB. A. HenseD. FischerK. Buddrus-SchiemannM. LucioJ. MüllerP. 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.   Google Scholar

[20]

M. R. FrederickC. KuttlerB. 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.  Google Scholar

[21]

M. R. FrederickC. KuttlerB. A. HenseJ. 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.   Google Scholar

[22]

S. M. HuntM. A. HamiltonJ. T. SearsG. 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.  Google Scholar

[23]

S. M. HuntE. M. WernerB. HuangM. 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.  Google Scholar

[24]

H. KhassehkhanM. 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.  Google Scholar

[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.  Google Scholar

[26]

J. B. LangebrakeG. E. DilanjiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

G. A. O'Toole and P. S. Stewart, Biofilms strike back, Nature Biotechnology, 23 (2005), 1378-1379.  doi: 10.1038/nbt1105-1378.  Google Scholar

[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.  Google Scholar

[31]

C. PicioreanuM. 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.  Google Scholar

[32]

A. RaduJ. VrouwenvelderM. 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.  Google Scholar

[33]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edition, Springer Verlag, New York, 2004.  Google Scholar

[34]

S. A. RiceK. S. KohS. Y. QueckM. LabbateK. 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.  Google Scholar

[35]

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[36]

S. Sirca and M. Morvat, Computational Methods for Physicists, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-32478-9.  Google Scholar

[37]

SolanoEcheverz and LasaI, Biofilm dispersion and quorum sensing, Curr. Opin. Microbiol., 18 (2014), 96-104.  doi: 10.1016/j.mib.2014.02.008.  Google Scholar

[38]

S. SonnerM. 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.  Google Scholar

[39]

S. SonnerM. 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.  Google Scholar

[40]

P. S. Stewart, A model of biofilm detachment, Biotechnol. Bioeng., 41 (1993), 111-117.  doi: 10.1002/bit.260410115.  Google Scholar

[41]

M. G. Trulear and W. G. Characklis, Dynamics of biofilm processes, J. Water Pollut. Control Fed., 54 (1982), 1288-1301.   Google Scholar

[42]

B. L. Vaughan JrB. 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.   Google Scholar

[43]

O. Wanner and P. Reichert, Mathematical modelling of mixed-culture biofilm, Biotech. Bioeng., 49 (1996), 172-184.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[46]

J. B. XavierC. 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.  Google Scholar

[47]

J. B. XavierC. PicioreanuS. A. RaniM. 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.  Google Scholar

show all references

References:
[1]

F. AbbasR. 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.  Google Scholar

[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.  Google Scholar

[3]

D. AronsonM. 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.  Google Scholar

[4]

N. BarraudD. J. HassettS. H. HwangS. A. RiceS. 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.  Google Scholar

[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.   Google Scholar

[6]

A. Boyd and A. M. Chakrabarty, Role of alginate lyase in cell detachment of Pseudomonas Aeruginosa, Appl. Environ. Microbiol., 60 (1994), 2355-2359.   Google Scholar

[7]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[8]

M. E. DaveyN. 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.  Google Scholar

[9]

D. A. D'ArgenioM. W. CalfeeP. 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.  Google Scholar

[10]

L. DemaretH. J. EberlM. 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.   Google Scholar

[11]

R. M. Donlan, Biofilms and device-associated infections, Emerging Infec. Dis., 7 (2001).   Google Scholar

[12]

R. DudduD. 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.  Google Scholar

[13]

H. J. EberlD. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

M. A. EfendievH. 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.   Google Scholar

[17]

M. A. EfendievH. 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.  Google Scholar

[18]

B. O. EmereniniB. A. HenseC. Kuttler and H. J. Eberl, A mathematical model of quorum sensing induced biofilm detachment, PLoS ONE., 10 (2015).  doi: 10.1371/journal.pone.0132385.  Google Scholar

[19]

A. FeketeC. KuttlerM. RothballerB. A. HenseD. FischerK. Buddrus-SchiemannM. LucioJ. MüllerP. 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.   Google Scholar

[20]

M. R. FrederickC. KuttlerB. 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.  Google Scholar

[21]

M. R. FrederickC. KuttlerB. A. HenseJ. 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.   Google Scholar

[22]

S. M. HuntM. A. HamiltonJ. T. SearsG. 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.  Google Scholar

[23]

S. M. HuntE. M. WernerB. HuangM. 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.  Google Scholar

[24]

H. KhassehkhanM. 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.  Google Scholar

[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.  Google Scholar

[26]

J. B. LangebrakeG. E. DilanjiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

G. A. O'Toole and P. S. Stewart, Biofilms strike back, Nature Biotechnology, 23 (2005), 1378-1379.  doi: 10.1038/nbt1105-1378.  Google Scholar

[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.  Google Scholar

[31]

C. PicioreanuM. 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.  Google Scholar

[32]

A. RaduJ. VrouwenvelderM. 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.  Google Scholar

[33]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edition, Springer Verlag, New York, 2004.  Google Scholar

[34]

S. A. RiceK. S. KohS. Y. QueckM. LabbateK. 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.  Google Scholar

[35]

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[36]

S. Sirca and M. Morvat, Computational Methods for Physicists, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-32478-9.  Google Scholar

[37]

SolanoEcheverz and LasaI, Biofilm dispersion and quorum sensing, Curr. Opin. Microbiol., 18 (2014), 96-104.  doi: 10.1016/j.mib.2014.02.008.  Google Scholar

[38]

S. SonnerM. 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.  Google Scholar

[39]

S. SonnerM. 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.  Google Scholar

[40]

P. S. Stewart, A model of biofilm detachment, Biotechnol. Bioeng., 41 (1993), 111-117.  doi: 10.1002/bit.260410115.  Google Scholar

[41]

M. G. Trulear and W. G. Characklis, Dynamics of biofilm processes, J. Water Pollut. Control Fed., 54 (1982), 1288-1301.   Google Scholar

[42]

B. L. Vaughan JrB. 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.   Google Scholar

[43]

O. Wanner and P. Reichert, Mathematical modelling of mixed-culture biofilm, Biotech. Bioeng., 49 (1996), 172-184.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[46]

J. B. XavierC. 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.  Google Scholar

[47]

J. B. XavierC. PicioreanuS. A. RaniM. 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.  Google Scholar

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
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
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 $\eta_1=0.6$
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 $A_{ave}$
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
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
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 $A_{ave}$
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
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 ratevaried[18]
$\lambda$quorum sensing abiotic decay rate $0.02218$[39]
$\alpha$constitutive autoinducer production ratevaried-
$\beta$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
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 ratevaried[18]
$\lambda$quorum sensing abiotic decay rate $0.02218$[39]
$\alpha$constitutive autoinducer production ratevaried-
$\beta$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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