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Modeling bacterial attachment to surfaces as an early stage of biofilm development
1. | African Institute for Mathematical Sciences, 6 Melrose road, Muizenberg, 7945, South Africa, South Africa |
2. | The College of Saint Rose, Department of Mathematics, 432 Western Avenue, Albany, NY 12203, United States |
References:
[1] |
A. Bejan, "Convection Heat Transfer," John Wiley and Sons Inc., New York, NY Publisher, 1984. |
[2] | |
[3] |
A. P. Petroff, TD. Wu, B. Liang, J. Mui, JL. Guerquin-Kern, H. Vali, D. H. Rothman and T. Bosak, Reaction diffusion model of nutrient uptake in a biofilm: Theory and experiment, Journal of Theoretical Biology, 289 (2001), 90-95.
doi: 10.1016/j.jtbi.2011.08.004. |
[4] |
C. D. Nadell, J. B. Xavier and K. R. Foster, The sociobiologyof biofilms, FEMS Microbiol Review, (2009) 1-19. |
[5] |
C. Picioreanu, M. C. M. Van Loosdrecht and J. J. Heijnen, Mathematical modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach, Biotechnology and Bioengineering, 58 (1997).
doi: 10.1002/(SICI)1097-0290(19980405)58:1<101::AID-BIT11>3.0.CO;2-M. |
[6] |
C. S. Laspidou, A. Kungolos, P. Samaras C. S. Laspidou, A. Kungolos and P. Samaras, Cellular-automata and individual-based approaches for the modeling of biofilm structures: Pros and cons, Journal of Desalination, 250 (2010), 390-394.
doi: 10.1016/j.desal.2009.09.062. |
[7] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, 2003. |
[8] |
D. Jones, V. K. Hristo and D. Le and H. Smith, Bacterial wall attachment in a flow reactor, SIAM Journal, 62 (2002), 1728-1771.
doi: 10.1137/S0036139901390416. |
[9] |
D. Priscilla, Biofilms: The environmental playground of Legionella pneumophila, Journal of Environmental Microbiology, 12 (2010), 557-566. |
[10] |
D. V. Nicolau Jr., J. P. Armitage and P. K. Maini, Directional persistence and the optimality of run-and-tumble chemotaxis, Computational Biology and Chemistry, 33 (2009), 269-274.
doi: 10.1016/j.compbiolchem.2009.06.003. |
[11] |
E. A. J. F. Peters and Th. M. A. O. M. Barenbrug, Efficient brownian dynamics simulation of particles near walls. I. Reflecting and adsorbing walls, Physical Review E, 66 (2002).
doi: 10.1103/PhysRevE.66.056701. |
[12] |
E. A. J. F. Peters and Th. M. A. O. M. Barenbrug, Efficient brownian dynamics simulation of particles near walls. II. sticky walls, Physical Review E, 66 (2002).
doi: 10.1103/PhysRevE.66.056702. |
[13] |
E. Ben Jacob, O. Schochet, A. Tenenbaum, I. Cohen, A. Czirok and T. Vicsek, Generic modelling of cooperative growth patterns in bacterial colonies, Nature Journal, 368 (1994), 46-49. |
[14] |
E. F. Keller, Science as a medium for friendship: How the Keller-Segel models came about, Bull. Math. Biol., 68 (2009), 1033-1037.
doi: 10.1007/s11538-006-9097-5. |
[15] |
E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. |
[16] |
E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[17] |
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, Journal of Theoretical Biology, 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[18] |
G. D. Zacarias, C. P. Ferreira and J. X. Velasco-Hernandez, Porosity and tortuosity relations as revealed by a mathematical model of biofilm structure, Journal of Theoretical Biology, 233 (2005), 245-251.
doi: 10.1016/j.jtbi.2004.10.006. |
[19] |
H. Donnelly, Uniqueness of positive solutions of the heat equation, American Mathematical Society, 99 (1987).
doi: 10.1090/S0002-9939-1987-0870800-6. |
[20] |
H. F. Jenkinson and H. M. Lappin-Scott, Biofilms adhere to stay, Journal of Trends in Microbiology, 9 (2001), 9-10.
doi: 10.1016/S0966-842X(00)01891-6. |
[21] |
H. Stoodley, Luanne, W. Costerton and P. Stoodley, Bacterial biofilms: From the natural environment to infectious diseases, Review of Microbiology, 2 (2004), 1740-1526. |
[22] |
H. Tamboto, K. Vickery and A. K. Deva, Subclinical (Biofilm) infection causes capsular contracture in a porcine model following augmentation mammaplasty, Plastic & Reconstructive Surgery, 126 (2010), 835-842.
doi: 10.1097/PRS.0b013e3181e3b456. |
[23] |
I. Klapper and J. Dockery, Mathematical description of microbial biofilms, SIAM Journal, 52 (2010), 221-265.
doi: 10.1137/080739720. |
[24] |
J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," S. S. Antman editor, Springer Publisher, 2003. |
[25] |
J. D. Murray, "Mathematical Biology I: An Introduction," S. S. Antman editor, Springer publisher, 2002. |
[26] |
J. E. Guyer, D. Wheeler and J. A. Warren, FiPy: Partial differential equations with python, Journal of Computer Science and Engineering, 11 (2009), 6-15.
doi: 10.1109/MCSE.2009.52. |
[27] |
J. L. Goldberg and A. J. Schwartz, "Systems of Ordinary Differential Equations: An Introduction," I. N. Herstein and Gian-Carlo Rota editor Harper and Row publisher. |
[28] |
J. S. Poindexter and E. R. Leadbetter, " Bacteria in Nature 2: Methods and Special Applications in Bacterial Ecology," Spring Street Editor, Plenum Press, New York, 1986. |
[29] |
J. W. Costerton, Overview of microbial biofilms, Journal of Industrial Microbiology and Biotechnology, 15 (1995), 137-140.
doi: 10.1007/BF01569816. |
[30] |
J. W. Costerton, Introduction to biofilm, International Journal of Antimicrobial Agents, 11 (1999), 217-221.
doi: 10.1016/S0924-8579(99)00018-7. |
[31] |
J. W. Costerton, G. G. Geesey and G. K. Cheng, How bacteria stick, Sci. Am., 238 (1978), 86-95.
doi: 10.1038/scientificamerican0178-86. |
[32] |
KJ. Engel, R. Nagel., "One Parameter Semigroups for Linear Evolution Equation," S. Axler editor, Springer publisher, 2000. |
[33] |
K. K. Jefferson, What drives bacteria to produce a biofilm?, FEMS Microbiology Letters, 236 (2004), 163-173. |
[34] |
K. Kang, T. Kolokolnikov and J. Ward, The stability and dynamics of a spike in the 1D Keller Segel model, IMA Journal of Applied Mathematics, (2007).
doi: 10.1093/imamat/hxl028. |
[35] |
K. Kawasaki, A. Mochizuki, M. Matsushita, T. Umeda and N. Shigesada, Modeling spatio-temporal patterns generated by bacillus subtilis, Journal of Theoretical Biology, 188 (1997), 177-185. |
[36] |
K. Sauer, A. K. Camper, G. D. Ehrlich, J. W. Costerton and D. G. Davies, Pseudomonas aeruginosa displays multiple phenotypes during development as a biofilm, Journal of Bacteriology, 184 (2002), 1140-1154.
doi: 10.1128/jb.184.4.1140-1154.2002. |
[37] |
L. R. Johnson, Microcolony and biofilm formation as a survival strategy for bacteria, Journal of Theoretical Biology, 251 (2008), 24-34.
doi: 10.1016/j.jtbi.2007.10.039. |
[38] |
M. Ballyk and H. Smith, A model of microbial growth in a plug flow reactor with wall attachment, Mathematical Biosciences, 158 (1999), 95-126.
doi: 10.1016/S0025-5564(99)00006-1. |
[39] |
M. Burmolle, T. Rolighed Thomsen, M. Fazli, I. Dige, L. Christensen, P. Homoe, M. Tvede, B. Nyvad, T. Tolker-Nielsen, M. Givskov, C. Moser, K. Kirketerp-Moller, H. Krogh Johansen, N. Hoiby, P. Ostrup Jensen, S. J. Sorensen and T. Bjarnsholt, Biofilms in chronic infections a matter of opportunity monospecies biofilms in multispecies infections, FEMS Immunol. Med. Microbiol, 59 (2010), 324-336.
doi: 10.1111/j.1574-695X.2010.00714.x. |
[40] |
M. G. Fagerlind, J. S. Webb, N. Barraud, D. McDougald, A. Jansson, P. Nilsson, M. Harln, S. Kjelleberg and S. A. Rice, Dynamic modelling of cell death during biofilm development, Journal of Theoretical Biology, 259 (2012), 23-36.
doi: 10.1016/j.jtbi.2011.10.007. |
[41] |
M. M. Ballyk, D. A. Jones and H. L. Smith, Microbial competition in reactors with wall attachment, Microbial Ecology, 41 (2001), 210-221. |
[42] |
M. Mimura, H. Sakaguchi and M. Matsushita, Reaction diffusion modeling of bacterial colony patterns, Physica A: Statistical Mechanics and its Applications, 282 (2000), 283-303.
doi: 10.1016/S0378-4371(00)00085-6. |
[43] |
M. R. Rahbar, I. Rasooli, S. Latif, M. Gargari, J. Amani and Y. Fattahian, In silico analysis of antibody triggering biofilm associated protein in Acinetobacter baumannii, Journal of Theoretical Biology, 266 (2010), 275-290.
doi: 10.1016/j.jtbi.2010.06.014. |
[44] |
M. Tindall, P. Maini, S. Porter and J. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations, Bulletin of Mathematical Biology, 70 (2008), 1570-1607.
doi: 10.1007/s11538-008-9322-5. |
[45] |
N. Balaban, "Control of Biofilom Infections by Signal Manipulation," J. William Costerton Editor Springer Publisher, 2008.
doi: 10.1007/978-3-540-73853-4. |
[46] |
N. Hoiby, T. Bjarnsholt, M. Givskov, S. Molin and O. Ciofu, Antibiotic resistance of bacterial biofilms, International Journal of Antimicrobial Agents, 35 (2010), 322-332.
doi: 10.1016/j.ijantimicag.2009.12.011. |
[47] |
N. Hoiby, T. Bjarnsholt, M. L. Givskov, S. Molin and O. Ciofu, Antibiotic resistance of bacterial biofilms, International Journal of Antimicrobial Agents, 35 (2010), 322-332.
doi: 10.1016/j.ijantimicag.2009.12.011. |
[48] |
O. Wanner and W. Gujer, A multispecies biofilm model, Biotechnology and Bioengineering, 28 (1986), 314-328.
doi: 10.1002/bit.260280304. |
[49] |
P. Carol, Microbiology: Biofilms invade microbiology, Journal of Science, 273 (1996), 1795-1797. |
[50] |
P. Watnick and R. Kolter, Biofilm, city of microbes, Journal of Bacteriology, 182 (2000), 2675-2679.
doi: 10.1128/JB.182.10.2675-2679.2000. |
[51] |
Q. Wang and T. Zhang, Review of mathematical models for biofilms, Solid State Communications, 150 (2010), 21-22.
doi: 10.1016/j.ssc.2010.01.021. |
[52] |
R. Erban and G. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, multiscale model, Journal of Simul., 3 (2005), 362-394.
doi: 10.1137/040603565. |
[53] |
R. J. Leveque, "Finite Volume Methods for Hyperbolic Problems," Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253. |
[54] |
R. M. Donlan and J. W. Costerton, Survival mechanisms of clinically relevant microorganisms, Clinical Microbiology Reviews, 15 (2002).
doi: 10.1128/CMR.15.2.167-193.2002. |
[55] |
T. R. de Kievit, Quorum sensing in Pseudomonas aeruginosa biofilms, Environmental Microbiology, 11 (2009), 279-288. |
[56] |
T. Tolker-Nielsen, U. C. Brinch, P. C. Ragas, J. B. Andersen, C. S. Jacobsen and S. Molin, Development and dynamics of Pseudomonas sp. biofilms, Journal of Bacteriology, 182 (2000). |
[57] |
S. Abdul Rani, B. Pitts, H. Beyenal, R. A. Veluchamy, Z. Lewandowski, W. M. Davison, K. Buckingham-Meyer and P. S. Stewart, Spatial patterns of DNA replication, protein synthesis, and oxygen concentration within bacterial biofilms reveal diverse physiological states, Journal of Bacteriology, 189 (2007), 4223-4233.
doi: 10.1128/JB.00107-07. |
[58] |
Z. Lewandowski and H. Beyenal, Mechanisms of microbially influenced corrosion, Springer Berlin Heidelberg, 4 (2009), 35-64. |
[59] |
, http://grants.nih.gov/grants/guide/pa-files/PA-03-047.html, Last Accessed on June 11, (2012).
|
show all references
References:
[1] |
A. Bejan, "Convection Heat Transfer," John Wiley and Sons Inc., New York, NY Publisher, 1984. |
[2] | |
[3] |
A. P. Petroff, TD. Wu, B. Liang, J. Mui, JL. Guerquin-Kern, H. Vali, D. H. Rothman and T. Bosak, Reaction diffusion model of nutrient uptake in a biofilm: Theory and experiment, Journal of Theoretical Biology, 289 (2001), 90-95.
doi: 10.1016/j.jtbi.2011.08.004. |
[4] |
C. D. Nadell, J. B. Xavier and K. R. Foster, The sociobiologyof biofilms, FEMS Microbiol Review, (2009) 1-19. |
[5] |
C. Picioreanu, M. C. M. Van Loosdrecht and J. J. Heijnen, Mathematical modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach, Biotechnology and Bioengineering, 58 (1997).
doi: 10.1002/(SICI)1097-0290(19980405)58:1<101::AID-BIT11>3.0.CO;2-M. |
[6] |
C. S. Laspidou, A. Kungolos, P. Samaras C. S. Laspidou, A. Kungolos and P. Samaras, Cellular-automata and individual-based approaches for the modeling of biofilm structures: Pros and cons, Journal of Desalination, 250 (2010), 390-394.
doi: 10.1016/j.desal.2009.09.062. |
[7] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, 2003. |
[8] |
D. Jones, V. K. Hristo and D. Le and H. Smith, Bacterial wall attachment in a flow reactor, SIAM Journal, 62 (2002), 1728-1771.
doi: 10.1137/S0036139901390416. |
[9] |
D. Priscilla, Biofilms: The environmental playground of Legionella pneumophila, Journal of Environmental Microbiology, 12 (2010), 557-566. |
[10] |
D. V. Nicolau Jr., J. P. Armitage and P. K. Maini, Directional persistence and the optimality of run-and-tumble chemotaxis, Computational Biology and Chemistry, 33 (2009), 269-274.
doi: 10.1016/j.compbiolchem.2009.06.003. |
[11] |
E. A. J. F. Peters and Th. M. A. O. M. Barenbrug, Efficient brownian dynamics simulation of particles near walls. I. Reflecting and adsorbing walls, Physical Review E, 66 (2002).
doi: 10.1103/PhysRevE.66.056701. |
[12] |
E. A. J. F. Peters and Th. M. A. O. M. Barenbrug, Efficient brownian dynamics simulation of particles near walls. II. sticky walls, Physical Review E, 66 (2002).
doi: 10.1103/PhysRevE.66.056702. |
[13] |
E. Ben Jacob, O. Schochet, A. Tenenbaum, I. Cohen, A. Czirok and T. Vicsek, Generic modelling of cooperative growth patterns in bacterial colonies, Nature Journal, 368 (1994), 46-49. |
[14] |
E. F. Keller, Science as a medium for friendship: How the Keller-Segel models came about, Bull. Math. Biol., 68 (2009), 1033-1037.
doi: 10.1007/s11538-006-9097-5. |
[15] |
E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. |
[16] |
E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[17] |
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, Journal of Theoretical Biology, 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[18] |
G. D. Zacarias, C. P. Ferreira and J. X. Velasco-Hernandez, Porosity and tortuosity relations as revealed by a mathematical model of biofilm structure, Journal of Theoretical Biology, 233 (2005), 245-251.
doi: 10.1016/j.jtbi.2004.10.006. |
[19] |
H. Donnelly, Uniqueness of positive solutions of the heat equation, American Mathematical Society, 99 (1987).
doi: 10.1090/S0002-9939-1987-0870800-6. |
[20] |
H. F. Jenkinson and H. M. Lappin-Scott, Biofilms adhere to stay, Journal of Trends in Microbiology, 9 (2001), 9-10.
doi: 10.1016/S0966-842X(00)01891-6. |
[21] |
H. Stoodley, Luanne, W. Costerton and P. Stoodley, Bacterial biofilms: From the natural environment to infectious diseases, Review of Microbiology, 2 (2004), 1740-1526. |
[22] |
H. Tamboto, K. Vickery and A. K. Deva, Subclinical (Biofilm) infection causes capsular contracture in a porcine model following augmentation mammaplasty, Plastic & Reconstructive Surgery, 126 (2010), 835-842.
doi: 10.1097/PRS.0b013e3181e3b456. |
[23] |
I. Klapper and J. Dockery, Mathematical description of microbial biofilms, SIAM Journal, 52 (2010), 221-265.
doi: 10.1137/080739720. |
[24] |
J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," S. S. Antman editor, Springer Publisher, 2003. |
[25] |
J. D. Murray, "Mathematical Biology I: An Introduction," S. S. Antman editor, Springer publisher, 2002. |
[26] |
J. E. Guyer, D. Wheeler and J. A. Warren, FiPy: Partial differential equations with python, Journal of Computer Science and Engineering, 11 (2009), 6-15.
doi: 10.1109/MCSE.2009.52. |
[27] |
J. L. Goldberg and A. J. Schwartz, "Systems of Ordinary Differential Equations: An Introduction," I. N. Herstein and Gian-Carlo Rota editor Harper and Row publisher. |
[28] |
J. S. Poindexter and E. R. Leadbetter, " Bacteria in Nature 2: Methods and Special Applications in Bacterial Ecology," Spring Street Editor, Plenum Press, New York, 1986. |
[29] |
J. W. Costerton, Overview of microbial biofilms, Journal of Industrial Microbiology and Biotechnology, 15 (1995), 137-140.
doi: 10.1007/BF01569816. |
[30] |
J. W. Costerton, Introduction to biofilm, International Journal of Antimicrobial Agents, 11 (1999), 217-221.
doi: 10.1016/S0924-8579(99)00018-7. |
[31] |
J. W. Costerton, G. G. Geesey and G. K. Cheng, How bacteria stick, Sci. Am., 238 (1978), 86-95.
doi: 10.1038/scientificamerican0178-86. |
[32] |
KJ. Engel, R. Nagel., "One Parameter Semigroups for Linear Evolution Equation," S. Axler editor, Springer publisher, 2000. |
[33] |
K. K. Jefferson, What drives bacteria to produce a biofilm?, FEMS Microbiology Letters, 236 (2004), 163-173. |
[34] |
K. Kang, T. Kolokolnikov and J. Ward, The stability and dynamics of a spike in the 1D Keller Segel model, IMA Journal of Applied Mathematics, (2007).
doi: 10.1093/imamat/hxl028. |
[35] |
K. Kawasaki, A. Mochizuki, M. Matsushita, T. Umeda and N. Shigesada, Modeling spatio-temporal patterns generated by bacillus subtilis, Journal of Theoretical Biology, 188 (1997), 177-185. |
[36] |
K. Sauer, A. K. Camper, G. D. Ehrlich, J. W. Costerton and D. G. Davies, Pseudomonas aeruginosa displays multiple phenotypes during development as a biofilm, Journal of Bacteriology, 184 (2002), 1140-1154.
doi: 10.1128/jb.184.4.1140-1154.2002. |
[37] |
L. R. Johnson, Microcolony and biofilm formation as a survival strategy for bacteria, Journal of Theoretical Biology, 251 (2008), 24-34.
doi: 10.1016/j.jtbi.2007.10.039. |
[38] |
M. Ballyk and H. Smith, A model of microbial growth in a plug flow reactor with wall attachment, Mathematical Biosciences, 158 (1999), 95-126.
doi: 10.1016/S0025-5564(99)00006-1. |
[39] |
M. Burmolle, T. Rolighed Thomsen, M. Fazli, I. Dige, L. Christensen, P. Homoe, M. Tvede, B. Nyvad, T. Tolker-Nielsen, M. Givskov, C. Moser, K. Kirketerp-Moller, H. Krogh Johansen, N. Hoiby, P. Ostrup Jensen, S. J. Sorensen and T. Bjarnsholt, Biofilms in chronic infections a matter of opportunity monospecies biofilms in multispecies infections, FEMS Immunol. Med. Microbiol, 59 (2010), 324-336.
doi: 10.1111/j.1574-695X.2010.00714.x. |
[40] |
M. G. Fagerlind, J. S. Webb, N. Barraud, D. McDougald, A. Jansson, P. Nilsson, M. Harln, S. Kjelleberg and S. A. Rice, Dynamic modelling of cell death during biofilm development, Journal of Theoretical Biology, 259 (2012), 23-36.
doi: 10.1016/j.jtbi.2011.10.007. |
[41] |
M. M. Ballyk, D. A. Jones and H. L. Smith, Microbial competition in reactors with wall attachment, Microbial Ecology, 41 (2001), 210-221. |
[42] |
M. Mimura, H. Sakaguchi and M. Matsushita, Reaction diffusion modeling of bacterial colony patterns, Physica A: Statistical Mechanics and its Applications, 282 (2000), 283-303.
doi: 10.1016/S0378-4371(00)00085-6. |
[43] |
M. R. Rahbar, I. Rasooli, S. Latif, M. Gargari, J. Amani and Y. Fattahian, In silico analysis of antibody triggering biofilm associated protein in Acinetobacter baumannii, Journal of Theoretical Biology, 266 (2010), 275-290.
doi: 10.1016/j.jtbi.2010.06.014. |
[44] |
M. Tindall, P. Maini, S. Porter and J. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations, Bulletin of Mathematical Biology, 70 (2008), 1570-1607.
doi: 10.1007/s11538-008-9322-5. |
[45] |
N. Balaban, "Control of Biofilom Infections by Signal Manipulation," J. William Costerton Editor Springer Publisher, 2008.
doi: 10.1007/978-3-540-73853-4. |
[46] |
N. Hoiby, T. Bjarnsholt, M. Givskov, S. Molin and O. Ciofu, Antibiotic resistance of bacterial biofilms, International Journal of Antimicrobial Agents, 35 (2010), 322-332.
doi: 10.1016/j.ijantimicag.2009.12.011. |
[47] |
N. Hoiby, T. Bjarnsholt, M. L. Givskov, S. Molin and O. Ciofu, Antibiotic resistance of bacterial biofilms, International Journal of Antimicrobial Agents, 35 (2010), 322-332.
doi: 10.1016/j.ijantimicag.2009.12.011. |
[48] |
O. Wanner and W. Gujer, A multispecies biofilm model, Biotechnology and Bioengineering, 28 (1986), 314-328.
doi: 10.1002/bit.260280304. |
[49] |
P. Carol, Microbiology: Biofilms invade microbiology, Journal of Science, 273 (1996), 1795-1797. |
[50] |
P. Watnick and R. Kolter, Biofilm, city of microbes, Journal of Bacteriology, 182 (2000), 2675-2679.
doi: 10.1128/JB.182.10.2675-2679.2000. |
[51] |
Q. Wang and T. Zhang, Review of mathematical models for biofilms, Solid State Communications, 150 (2010), 21-22.
doi: 10.1016/j.ssc.2010.01.021. |
[52] |
R. Erban and G. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, multiscale model, Journal of Simul., 3 (2005), 362-394.
doi: 10.1137/040603565. |
[53] |
R. J. Leveque, "Finite Volume Methods for Hyperbolic Problems," Cambridge University Press, 2002.
doi: 10.1017/CBO9780511791253. |
[54] |
R. M. Donlan and J. W. Costerton, Survival mechanisms of clinically relevant microorganisms, Clinical Microbiology Reviews, 15 (2002).
doi: 10.1128/CMR.15.2.167-193.2002. |
[55] |
T. R. de Kievit, Quorum sensing in Pseudomonas aeruginosa biofilms, Environmental Microbiology, 11 (2009), 279-288. |
[56] |
T. Tolker-Nielsen, U. C. Brinch, P. C. Ragas, J. B. Andersen, C. S. Jacobsen and S. Molin, Development and dynamics of Pseudomonas sp. biofilms, Journal of Bacteriology, 182 (2000). |
[57] |
S. Abdul Rani, B. Pitts, H. Beyenal, R. A. Veluchamy, Z. Lewandowski, W. M. Davison, K. Buckingham-Meyer and P. S. Stewart, Spatial patterns of DNA replication, protein synthesis, and oxygen concentration within bacterial biofilms reveal diverse physiological states, Journal of Bacteriology, 189 (2007), 4223-4233.
doi: 10.1128/JB.00107-07. |
[58] |
Z. Lewandowski and H. Beyenal, Mechanisms of microbially influenced corrosion, Springer Berlin Heidelberg, 4 (2009), 35-64. |
[59] |
, http://grants.nih.gov/grants/guide/pa-files/PA-03-047.html, Last Accessed on June 11, (2012).
|
[1] |
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