2013, 10(3): 821-842. doi: 10.3934/mbe.2013.10.821

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

Received  June 2012 Revised  January 2013 Published  April 2013

Biofilms are present in all natural, medical and industrial surroundings where bacteria live. Biofilm formation is a key factor in the growth and transport of both beneficial and harmful bacteria. While much is known about the later stages of biofilm formation, less is known about its initiation which is an important first step in the biofilm formation. In this paper, we develop a non-linear system of partial differential equations of Keller-Segel type model in one-dimensional space, which couples the dynamics of bacterial movement to that of the sensing molecules. In this case, bacteria perform a biased random walk towards the sensing molecules. We derive the boundary conditions of the adhesion of bacteria to a surface using zero-Dirichlet boundary conditions, while the equation describing sensing molecules at the interface needed particular conditions to be set. The numerical results show the profile of bacteria within the space and the time evolution of the density within the free-space and on the surface. Testing different parameter values indicate that significant amount of sensing molecules present on the surface leads to a faster bacterial movement toward the surface which is the first step of biofilm initiation. Our work gives rise to results that agree with the biological description of the early stages of biofilm formation.
Citation: Fadoua El Moustaid, Amina Eladdadi, Lafras Uys. Modeling bacterial attachment to surfaces as an early stage of biofilm development. Mathematical Biosciences & Engineering, 2013, 10 (3) : 821-842. doi: 10.3934/mbe.2013.10.821
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E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis,, Journal of Theoretical Biology, 30 (1971), 235. doi: 10.1016/0022-5193(71)90051-8.

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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. doi: 10.1016/j.jtbi.2004.10.006.

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H. F. Jenkinson and H. M. Lappin-Scott, Biofilms adhere to stay,, Journal of Trends in Microbiology, 9 (2001), 9. doi: 10.1016/S0966-842X(00)01891-6.

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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. doi: 10.1097/PRS.0b013e3181e3b456.

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J. W. Costerton, Overview of microbial biofilms,, Journal of Industrial Microbiology and Biotechnology, 15 (1995), 137. doi: 10.1007/BF01569816.

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J. W. Costerton, Introduction to biofilm,, International Journal of Antimicrobial Agents, 11 (1999), 217. doi: 10.1016/S0924-8579(99)00018-7.

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J. W. Costerton, G. G. Geesey and G. K. Cheng, How bacteria stick,, Sci. Am., 238 (1978), 86. doi: 10.1038/scientificamerican0178-86.

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KJ. Engel, R. Nagel., "One Parameter Semigroups for Linear Evolution Equation,", S. Axler editor, (2000).

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K. K. Jefferson, What drives bacteria to produce a biofilm?,, FEMS Microbiology Letters, 236 (2004), 163.

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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.

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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.

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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. doi: 10.1128/jb.184.4.1140-1154.2002.

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L. R. Johnson, Microcolony and biofilm formation as a survival strategy for bacteria,, Journal of Theoretical Biology, 251 (2008), 24. 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. 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. 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. 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.

[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. 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. 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. 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. 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. doi: 10.1016/j.ijantimicag.2009.12.011.

[48]

O. Wanner and W. Gujer, A multispecies biofilm model,, Biotechnology and Bioengineering, 28 (1986), 314. doi: 10.1002/bit.260280304.

[49]

P. Carol, Microbiology: Biofilms invade microbiology,, Journal of Science, 273 (1996), 1795.

[50]

P. Watnick and R. Kolter, Biofilm, city of microbes,, Journal of Bacteriology, 182 (2000), 2675. 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. 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. 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.

[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. doi: 10.1128/JB.00107-07.

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Z. Lewandowski and H. Beyenal, Mechanisms of microbially influenced corrosion,, Springer Berlin Heidelberg, 4 (2009), 35.

[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., (1984).

[2]

A. Coghlan, Slime city,, New Scientist, 151 (1996), 32.

[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. 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.

[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. 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. doi: 10.1137/S0036139901390416.

[9]

D. Priscilla, Biofilms: The environmental playground of Legionella pneumophila,, Journal of Environmental Microbiology, 12 (2010), 557.

[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. 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.

[14]

E. F. Keller, Science as a medium for friendship: How the Keller-Segel models came about,, Bull. Math. Biol., 68 (2009), 1033. 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.

[16]

E. F. Keller and L. A. Segel, Model for chemotaxis,, Journal of Theoretical Biology, 30 (1971), 225. 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. 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. 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. 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.

[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. doi: 10.1097/PRS.0b013e3181e3b456.

[23]

I. Klapper and J. Dockery, Mathematical description of microbial biofilms,, SIAM Journal, 52 (2010), 221. doi: 10.1137/080739720.

[24]

J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications,", S. S. Antman editor, (2003).

[25]

J. D. Murray, "Mathematical Biology I: An Introduction,", S. S. Antman editor, (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. 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, (1986).

[29]

J. W. Costerton, Overview of microbial biofilms,, Journal of Industrial Microbiology and Biotechnology, 15 (1995), 137. doi: 10.1007/BF01569816.

[30]

J. W. Costerton, Introduction to biofilm,, International Journal of Antimicrobial Agents, 11 (1999), 217. 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. doi: 10.1038/scientificamerican0178-86.

[32]

KJ. Engel, R. Nagel., "One Parameter Semigroups for Linear Evolution Equation,", S. Axler editor, (2000).

[33]

K. K. Jefferson, What drives bacteria to produce a biofilm?,, FEMS Microbiology Letters, 236 (2004), 163.

[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.

[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. 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. 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. 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. 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. 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.

[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. 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. 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. 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. 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. doi: 10.1016/j.ijantimicag.2009.12.011.

[48]

O. Wanner and W. Gujer, A multispecies biofilm model,, Biotechnology and Bioengineering, 28 (1986), 314. doi: 10.1002/bit.260280304.

[49]

P. Carol, Microbiology: Biofilms invade microbiology,, Journal of Science, 273 (1996), 1795.

[50]

P. Watnick and R. Kolter, Biofilm, city of microbes,, Journal of Bacteriology, 182 (2000), 2675. 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. 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. 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.

[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. doi: 10.1128/JB.00107-07.

[58]

Z. Lewandowski and H. Beyenal, Mechanisms of microbially influenced corrosion,, Springer Berlin Heidelberg, 4 (2009), 35.

[59]

, http://grants.nih.gov/grants/guide/pa-files/PA-03-047.html, Last Accessed on June 11, (2012).

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