American Institute of Mathematical Sciences

Advanced
May  2012, 17(3): 1027-1059. doi: 10.3934/dcdsb.2012.17.1027

Kinetic theories for biofilms

Qi Wang 1, and  Tianyu Zhang 2,

1. 

Department of Mathematics and NanoCenter at USC, University of South Carolina, Columbia, SC 29208
2. 

Department of Mathematical Sciences, Montana State University, P.O. Box 172400, Bozeman, MT 59717-2400

Received  January 2009 Revised  September 2011 Published  January 2012

We apply the kinetic theory formulation for binary complex fluids to develop a set of hydrodynamic models for the two-phase mixture of biofilms and solvent (water). It is aimed to model nonlinear growth and transport of the biomass in the mixture and the biomass-flow interaction. In the kinetic theory formulation of binary complex fluids, the biomass consisting of EPS (Extracellular Polymeric Substance) polymer networks and bacteria is coarse-grained into an effective fluid component, termed the effective polymer solution; while the other component, termed the effective solvent, is made up of the ensemble of nutrient substrates and the solvent. The mixture is modeled as an incompressible two-phase fluid in which the presence of the effective components are quantified by their respective volume fractions. The kinetic theory framework allows the incorporation of microscopic details of the biomass and its interaction with the coexisting effective solvent. The relative motion of the biomass and the solvent relative to an average velocity is described by binary mixing kinetics along with the intrinsic molecular elasticity of the EPS network strand modeled as an elastic dumbbell. This theory is valid in both the biofilm region which consists of the mixture of the biomass and solvent and the pure solvent region, making it convenient in numerical simulations of the biomass-flow interaction. Steady states and their stability are discussed under a growth condition. Nonlinear solutions of the three models developed in this study in simple shear are calculated and compared numerically in 1-D space.
Keywords: binary complex fluid., kinetic theory, Biofilms.
Mathematics Subject Classification: Primary: 35Q92, 92C45; Secondary: 92B0.
Citation: Qi Wang, Tianyu Zhang. Kinetic theories for biofilms. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1027-1059. doi: 10.3934/dcdsb.2012.17.1027
References:
[1]

E. Alpkvist and I. Klapper, A multidimensional multispecies continuum model for heterogeneous biofilm development, Bull. Math. Biol., 69 (2007), 765-789. Google Scholar

[2]

A. N. Beris and B. Edwards, "Thermodynamics of Flowing Systems with Internal Microstructure,'' Oxford Engineering Science Series, 36, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994.  Google Scholar

[3]

R. B. Bird, R. C. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids,'' Vol. 1 & 2, John Wiley & Sons, New York, 1987. Google Scholar

[4]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I: Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.  Google Scholar

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. III: Nucleation in a 2-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699. doi: 10.1063/1.1730447.  Google Scholar

[6]

P. M. Chaikin and T. C. Lubensky, "Principles of Condensed Matter Physics,'' Cambridge University Press, Cambridge, 1995. Google Scholar

[7]

Chen Chen, Mingming Ren, Ashok Srinivasan and Qi Wang, 3-D simulations of biofilm-solvent interaction, East Asian Journal on Applied Mathematics, 1 (2011), 197-214. Google Scholar

[8]

N. G. Cogan and J. Keener, Channel formation in gels, SIAM J. Applied Math., 65 (2005), 1839-1854. doi: 10.1137/040605515.  Google Scholar

[9]

N. Cogan and J. Keener, The role of biofilm matrix in structural development, Mathematical Medicine and Biology, 21 (2004), 147-166. doi: 10.1093/imammb/21.2.147.  Google Scholar

[10]

J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Korber and H. M. Lappin-Scott, Microbial biofilms, Annu. Rev. Microbiol., 49 (1995), 711-745. doi: 10.1146/annurev.mi.49.100195.003431.  Google Scholar

[11]

B. Costerton, "Medical Biofilm Microbiology: The Role of Microbial Biofilms in Disease, Chronic Infections, and Medical Device Failure," CD-ROM, Montana State University, 2003. Google Scholar

[12]

M. E. Davey and G. A. O'toole, Microbial biofilms: From ecology to molecular genetics, Microbiology and Molecular Biology Reviews, 64 (2000), 847-867. doi: 10.1128/MMBR.64.4.847-867.2000.  Google Scholar

[13]

E. De Lancey Pulcini, Bacterial biofilms: A review of current research, Nephrologie, 22 (2001), 439-441. Google Scholar

[14]

J. Dockery and I. Klapper, Finger formation in biofilm layers,, SIAM J. Appl. Math., 62 (): 853.   Google Scholar

[15]

M. Doi and S. F. Edwards, "The Theory of Polymer Dynamics,'' Oxford Science Publications, Oxford, 1986. Google Scholar

[16]

M. Doi, "Introduction to Polymer Physics,'' Oxford Science Publications, Oxford, 1995. Google Scholar

[17]

P. J. Flory, "Principles of Polymer Chemistry,'' Cornell University Press, Ithaca, NY, 1953. Google Scholar

[18]

D. J. Hassett, P. A. Limbach, R. F. Hennigan, K. E. Klose, R. E. Hancock, M. D. Platt and D. F. Hunt, Bacterial biofilms of importance to medicine and bioterrorism: Proteomic techniques to identify novel vaccine components and drug targets, Expert Opin. Biol. Ther., 3 (2003), 1201-1207. doi: 10.1517/14712598.3.8.1201.  Google Scholar

[19]

I. Klapper, Effect of heterogeneous structure in mechanically unstressed biofilms on overall growth, Bulletin of Mathemstical Biology, 66 (2004), 809-824. doi: 10.1016/j.bulm.2003.11.008.  Google Scholar

[20]

I. Klapper, C. J. Rupp, R. Cargo, B. Purvedorj and P. Stoodley, Viscoelastic fluid description of bacterial biofilm material properties, Biotechnology and Bioengineering, 80 (2002), 289-296. doi: 10.1002/bit.10376.  Google Scholar

[21]

I. Klapper and J. Dockery, Role of cohesion in the material description of biofilms, Phys. Rev. E (3), 74 (2006), 031902, 8 pp.  Google Scholar

[22]

R. G. Larson, "The Rheology of Complex Fluids,'' Oxford University Press, New York, 1998. Google Scholar

[23]

C. S. Laspidou and B. E. Rittmann, Modeling biofilm complexity by including active and inert biomass and extracelluar polymeric substances, Biofilm, 1 (2004), 285-291. Google Scholar

[24]

C. A. A. Lima, R. Ribeiro, E. Foresti and M. Zaiat, Morphological study of biomass during the start-up period of a fixed-bed anaerobic reactor treating domestic sewage,, Brazilian Archives of Biology and Technology., ().   Google Scholar

[25]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654.  Google Scholar

[26]

S. T. Milner, Dynamical theory of concentration fluctuations in polymer solutions under shear, Phys. Rev. E, 48 (1993), 3674-3691. doi: 10.1103/PhysRevE.48.3674.  Google Scholar

[27]

G. O'Toole, H. B. Kaplan and R. Kolter, Biofilm formation as microbial development, Annual Review of Microbiology, 54 (2000), 49-79. Google Scholar

[28]

C. Picioreanu, M. van Loosdrecht and J. Heijnen, Mathematical modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach, Biotech. Bioeng., 58 (1998), 101-116. doi: 10.1002/(SICI)1097-0290(19980405)58:1<101::AID-BIT11>3.0.CO;2-M.  Google Scholar

[29]

C. Picioreanu, M. van Loosdrecht and J. Heijnen, Multidimensional modelling of biofilm structure, in "Biotech. Microbial Biosystems: New Frontiers" (eds. C. R. Bell, M. Brylinsky and P. Johnson-Green), Proceedings of the 8th International Symposium on Microbial Ecology, Atlantic Canada Society for Microbial Ecology, Halifax, Canada, 1999. Google Scholar

[30]

C. Picioreanu, M. J.-U. Kreft and M. van Loosdrecht, Particle-based multidimensional multispecies biofilm models, Applied and Envrionmental Microbiology, 70 (2004), 3024-3040. doi: 10.1128/AEM.70.5.3024-3040.2004.  Google Scholar

[31]

C. Picioreanu, J. B. Xavier and M. van Loosdrecht, Advances in mathematical modeling of biofilm structure, Biofilm, 1 (2004), 337-349. Google Scholar

[32]

P. Stoodley, Z. Lewandowski, J. D. Boyle and H. M. Lappin-Scott, The formation of migratory ripples in a mixed species bacterial biofilm growing in turbulent flows, Environ. Microbiol., 1 (1999), 447-457. doi: 10.1046/j.1462-2920.1999.00055.x.  Google Scholar

[33]

H. Tanaka, Viscoelastic model of phase separation, Phys. Rev. E, 56 (1997), 4451-4462. doi: 10.1103/PhysRevE.56.4451.  Google Scholar

[34]

C. Wolgemuth, E. Hoiczyk, D. Kaiser and G. Oster, How myxobacteria glide, Current Biology, 12 (2002), 369-377. doi: 10.1016/S0960-9822(02)00716-9.  Google Scholar

[35]

Pengtao Yue, James J. Feng, Chun Liu and Jie Shen, A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515 (2004), 293-317.  Google Scholar

[36]

Pengtao Yue, James J. Feng, Chun Liu and Jie Shen, Viscoelastic effects on drop deformation in steady shear, J. Fluid Mech., 540 (2005), 427-437. doi: 10.1017/S0022112005006166.  Google Scholar

[37]

T. Zhang, N. Cogan and Q. Wang, Phase-field models for biofilms. I. Theory and one-dimensional simulations, SIAM J. Appl. Math., 69 (2008), 641-669. doi: 10.1137/070691966.  Google Scholar

[38]

T. Zhang, N. Cogan and Q. Wang, Phase-field models for biofilms. II. 2-D numerical simulations of biofilm-blow interaction, Communications in Computational Physics, 4 (2008), 72-101. Google Scholar

show all references

References:
[1]

E. Alpkvist and I. Klapper, A multidimensional multispecies continuum model for heterogeneous biofilm development, Bull. Math. Biol., 69 (2007), 765-789. Google Scholar

[2]

A. N. Beris and B. Edwards, "Thermodynamics of Flowing Systems with Internal Microstructure,'' Oxford Engineering Science Series, 36, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994.  Google Scholar

[3]

R. B. Bird, R. C. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids,'' Vol. 1 & 2, John Wiley & Sons, New York, 1987. Google Scholar

[4]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I: Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.  Google Scholar

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. III: Nucleation in a 2-component incompressible fluid, J. Chem. Phys., 31 (1959), 688-699. doi: 10.1063/1.1730447.  Google Scholar

[6]

P. M. Chaikin and T. C. Lubensky, "Principles of Condensed Matter Physics,'' Cambridge University Press, Cambridge, 1995. Google Scholar

[7]

Chen Chen, Mingming Ren, Ashok Srinivasan and Qi Wang, 3-D simulations of biofilm-solvent interaction, East Asian Journal on Applied Mathematics, 1 (2011), 197-214. Google Scholar

[8]

N. G. Cogan and J. Keener, Channel formation in gels, SIAM J. Applied Math., 65 (2005), 1839-1854. doi: 10.1137/040605515.  Google Scholar

[9]

N. Cogan and J. Keener, The role of biofilm matrix in structural development, Mathematical Medicine and Biology, 21 (2004), 147-166. doi: 10.1093/imammb/21.2.147.  Google Scholar

[10]

J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Korber and H. M. Lappin-Scott, Microbial biofilms, Annu. Rev. Microbiol., 49 (1995), 711-745. doi: 10.1146/annurev.mi.49.100195.003431.  Google Scholar

[11]

B. Costerton, "Medical Biofilm Microbiology: The Role of Microbial Biofilms in Disease, Chronic Infections, and Medical Device Failure," CD-ROM, Montana State University, 2003. Google Scholar

[12]

M. E. Davey and G. A. O'toole, Microbial biofilms: From ecology to molecular genetics, Microbiology and Molecular Biology Reviews, 64 (2000), 847-867. doi: 10.1128/MMBR.64.4.847-867.2000.  Google Scholar

[13]

E. De Lancey Pulcini, Bacterial biofilms: A review of current research, Nephrologie, 22 (2001), 439-441. Google Scholar

[14]

J. Dockery and I. Klapper, Finger formation in biofilm layers,, SIAM J. Appl. Math., 62 (): 853.   Google Scholar

[15]

M. Doi and S. F. Edwards, "The Theory of Polymer Dynamics,'' Oxford Science Publications, Oxford, 1986. Google Scholar

[16]

M. Doi, "Introduction to Polymer Physics,'' Oxford Science Publications, Oxford, 1995. Google Scholar

[17]

P. J. Flory, "Principles of Polymer Chemistry,'' Cornell University Press, Ithaca, NY, 1953. Google Scholar

[18]

D. J. Hassett, P. A. Limbach, R. F. Hennigan, K. E. Klose, R. E. Hancock, M. D. Platt and D. F. Hunt, Bacterial biofilms of importance to medicine and bioterrorism: Proteomic techniques to identify novel vaccine components and drug targets, Expert Opin. Biol. Ther., 3 (2003), 1201-1207. doi: 10.1517/14712598.3.8.1201.  Google Scholar

[19]

I. Klapper, Effect of heterogeneous structure in mechanically unstressed biofilms on overall growth, Bulletin of Mathemstical Biology, 66 (2004), 809-824. doi: 10.1016/j.bulm.2003.11.008.  Google Scholar

[20]

I. Klapper, C. J. Rupp, R. Cargo, B. Purvedorj and P. Stoodley, Viscoelastic fluid description of bacterial biofilm material properties, Biotechnology and Bioengineering, 80 (2002), 289-296. doi: 10.1002/bit.10376.  Google Scholar

[21]

I. Klapper and J. Dockery, Role of cohesion in the material description of biofilms, Phys. Rev. E (3), 74 (2006), 031902, 8 pp.  Google Scholar

[22]

R. G. Larson, "The Rheology of Complex Fluids,'' Oxford University Press, New York, 1998. Google Scholar

[23]

C. S. Laspidou and B. E. Rittmann, Modeling biofilm complexity by including active and inert biomass and extracelluar polymeric substances, Biofilm, 1 (2004), 285-291. Google Scholar

[24]

C. A. A. Lima, R. Ribeiro, E. Foresti and M. Zaiat, Morphological study of biomass during the start-up period of a fixed-bed anaerobic reactor treating domestic sewage,, Brazilian Archives of Biology and Technology., ().   Google Scholar

[25]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654.  Google Scholar

[26]

S. T. Milner, Dynamical theory of concentration fluctuations in polymer solutions under shear, Phys. Rev. E, 48 (1993), 3674-3691. doi: 10.1103/PhysRevE.48.3674.  Google Scholar

[27]

G. O'Toole, H. B. Kaplan and R. Kolter, Biofilm formation as microbial development, Annual Review of Microbiology, 54 (2000), 49-79. Google Scholar

[28]

C. Picioreanu, M. van Loosdrecht and J. Heijnen, Mathematical modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach, Biotech. Bioeng., 58 (1998), 101-116. doi: 10.1002/(SICI)1097-0290(19980405)58:1<101::AID-BIT11>3.0.CO;2-M.  Google Scholar

[29]

C. Picioreanu, M. van Loosdrecht and J. Heijnen, Multidimensional modelling of biofilm structure, in "Biotech. Microbial Biosystems: New Frontiers" (eds. C. R. Bell, M. Brylinsky and P. Johnson-Green), Proceedings of the 8th International Symposium on Microbial Ecology, Atlantic Canada Society for Microbial Ecology, Halifax, Canada, 1999. Google Scholar

[30]

C. Picioreanu, M. J.-U. Kreft and M. van Loosdrecht, Particle-based multidimensional multispecies biofilm models, Applied and Envrionmental Microbiology, 70 (2004), 3024-3040. doi: 10.1128/AEM.70.5.3024-3040.2004.  Google Scholar

[31]

C. Picioreanu, J. B. Xavier and M. van Loosdrecht, Advances in mathematical modeling of biofilm structure, Biofilm, 1 (2004), 337-349. Google Scholar

[32]

P. Stoodley, Z. Lewandowski, J. D. Boyle and H. M. Lappin-Scott, The formation of migratory ripples in a mixed species bacterial biofilm growing in turbulent flows, Environ. Microbiol., 1 (1999), 447-457. doi: 10.1046/j.1462-2920.1999.00055.x.  Google Scholar

[33]

H. Tanaka, Viscoelastic model of phase separation, Phys. Rev. E, 56 (1997), 4451-4462. doi: 10.1103/PhysRevE.56.4451.  Google Scholar

[34]

C. Wolgemuth, E. Hoiczyk, D. Kaiser and G. Oster, How myxobacteria glide, Current Biology, 12 (2002), 369-377. doi: 10.1016/S0960-9822(02)00716-9.  Google Scholar

[35]

Pengtao Yue, James J. Feng, Chun Liu and Jie Shen, A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515 (2004), 293-317.  Google Scholar

[36]

Pengtao Yue, James J. Feng, Chun Liu and Jie Shen, Viscoelastic effects on drop deformation in steady shear, J. Fluid Mech., 540 (2005), 427-437. doi: 10.1017/S0022112005006166.  Google Scholar

[37]

T. Zhang, N. Cogan and Q. Wang, Phase-field models for biofilms. I. Theory and one-dimensional simulations, SIAM J. Appl. Math., 69 (2008), 641-669. doi: 10.1137/070691966.  Google Scholar

[38]

T. Zhang, N. Cogan and Q. Wang, Phase-field models for biofilms. II. 2-D numerical simulations of biofilm-blow interaction, Communications in Computational Physics, 4 (2008), 72-101. Google Scholar

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