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Flow optimization in vascular networks
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 |
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 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, 2nd 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, 2nd 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, 2nd 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, 2nd 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. |








Parameter | Description | Value | Source |
half saturation concentration (growth) | | [44] | |
| lysis rate | | assumed |
nutrient consumption rate | | [19] | |
| maximum dispersal rate | varied | [18] |
| quorum sensing abiotic decay rate | | [39] |
constitutive autoinducer production rate | varied | - | |
| induced autoinducer production rate | | [19] |
| degree of polymerization | | [19] |
| constant diffusion coefficients for | | assumed |
| constant diffusion coefficients for | | [15] |
| constant diffusion coefficients for | | [15] |
| biomass motility coefficient | | [13] |
| biofilm diffusion exponent | | [13] |
| biofilm diffusion exponent | | [13] |
| system length | | [15] |
| system height | | assumed |
Parameter | Description | Value | Source |
half saturation concentration (growth) | | [44] | |
| lysis rate | | assumed |
nutrient consumption rate | | [19] | |
| maximum dispersal rate | varied | [18] |
| quorum sensing abiotic decay rate | | [39] |
constitutive autoinducer production rate | varied | - | |
| induced autoinducer production rate | | [19] |
| degree of polymerization | | [19] |
| constant diffusion coefficients for | | assumed |
| constant diffusion coefficients for | | [15] |
| constant diffusion coefficients for | | [15] |
| biomass motility coefficient | | [13] |
| biofilm diffusion exponent | | [13] |
| biofilm diffusion exponent | | [13] |
| system length | | [15] |
| system height | | assumed |
[1] |
Richard L Buckalew. Cell cycle clustering and quorum sensing in a response / signaling mediated feedback model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 867-881. doi: 10.3934/dcdsb.2014.19.867 |
[2] |
Hassan Khassehkhan, Messoud A. Efendiev, Hermann J. Eberl. A degenerate diffusion-reaction model of an amensalistic biofilm control system: Existence and simulation of solutions. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 371-388. doi: 10.3934/dcdsb.2009.12.371 |
[3] |
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