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
[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. |