# American Institute of Mathematical Sciences

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

## Kinetic theories for biofilms

 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.
Citation: Qi Wang, Tianyu Zhang. Kinetic theories for biofilms. Discrete and 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. [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. [3] R. B. Bird, R. C. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids,'' Vol. 1 & 2, John Wiley & Sons, New York, 1987. [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. [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. [6] P. M. Chaikin and T. C. Lubensky, "Principles of Condensed Matter Physics,'' Cambridge University Press, Cambridge, 1995. [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. [8] N. G. Cogan and J. Keener, Channel formation in gels, SIAM J. Applied Math., 65 (2005), 1839-1854. doi: 10.1137/040605515. [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. [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. [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. [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. [13] E. De Lancey Pulcini, Bacterial biofilms: A review of current research, Nephrologie, 22 (2001), 439-441. [14] J. Dockery and I. Klapper, Finger formation in biofilm layers, SIAM J. Appl. Math., 62 (2001/02), 853-869. [15] M. Doi and S. F. Edwards, "The Theory of Polymer Dynamics,'' Oxford Science Publications, Oxford, 1986. [16] M. Doi, "Introduction to Polymer Physics,'' Oxford Science Publications, Oxford, 1995. [17] P. J. Flory, "Principles of Polymer Chemistry,'' Cornell University Press, Ithaca, NY, 1953. [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. [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. [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. [21] I. Klapper and J. Dockery, Role of cohesion in the material description of biofilms, Phys. Rev. E (3), 74 (2006), 031902, 8 pp. [22] R. G. Larson, "The Rheology of Complex Fluids,'' Oxford University Press, New York, 1998. [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. [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. [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. [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. [27] G. O'Toole, H. B. Kaplan and R. Kolter, Biofilm formation as microbial development, Annual Review of Microbiology, 54 (2000), 49-79. [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. [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. [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. [31] C. Picioreanu, J. B. Xavier and M. van Loosdrecht, Advances in mathematical modeling of biofilm structure, Biofilm, 1 (2004), 337-349. [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. [33] H. Tanaka, Viscoelastic model of phase separation, Phys. Rev. E, 56 (1997), 4451-4462. doi: 10.1103/PhysRevE.56.4451. [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. [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. [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. [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. [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.

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. [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. [3] R. B. Bird, R. C. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids,'' Vol. 1 & 2, John Wiley & Sons, New York, 1987. [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. [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. [6] P. M. Chaikin and T. C. Lubensky, "Principles of Condensed Matter Physics,'' Cambridge University Press, Cambridge, 1995. [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. [8] N. G. Cogan and J. Keener, Channel formation in gels, SIAM J. Applied Math., 65 (2005), 1839-1854. doi: 10.1137/040605515. [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. [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. [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. [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. [13] E. De Lancey Pulcini, Bacterial biofilms: A review of current research, Nephrologie, 22 (2001), 439-441. [14] J. Dockery and I. Klapper, Finger formation in biofilm layers, SIAM J. Appl. Math., 62 (2001/02), 853-869. [15] M. Doi and S. F. Edwards, "The Theory of Polymer Dynamics,'' Oxford Science Publications, Oxford, 1986. [16] M. Doi, "Introduction to Polymer Physics,'' Oxford Science Publications, Oxford, 1995. [17] P. J. Flory, "Principles of Polymer Chemistry,'' Cornell University Press, Ithaca, NY, 1953. [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. [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. [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. [21] I. Klapper and J. Dockery, Role of cohesion in the material description of biofilms, Phys. Rev. E (3), 74 (2006), 031902, 8 pp. [22] R. G. Larson, "The Rheology of Complex Fluids,'' Oxford University Press, New York, 1998. [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. [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. [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. [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. [27] G. O'Toole, H. B. Kaplan and R. Kolter, Biofilm formation as microbial development, Annual Review of Microbiology, 54 (2000), 49-79. [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. [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. [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. [31] C. Picioreanu, J. B. Xavier and M. van Loosdrecht, Advances in mathematical modeling of biofilm structure, Biofilm, 1 (2004), 337-349. [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. [33] H. Tanaka, Viscoelastic model of phase separation, Phys. Rev. E, 56 (1997), 4451-4462. doi: 10.1103/PhysRevE.56.4451. [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. [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. [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. [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. [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.
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