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Discrete attachment to a cellulolytic biofilm modeled by an Itô stochastic differential equation

1 Department of Mathematics and Statistics, University of Guelph, 50 Stone Rd E, Guelph ON, N1G 2W1, Canada
2 Radboud University, Department of Mathematics, Postbus 9010, 6500 GL Nijmegen, The Netherlands

We propose a mathematical framework for introducing random attachment of bacterial cells in a deterministic continuum model of cellulosic biofilms. The underlying growth model is a highly nonlinear coupled PDE-ODE system. It is regularised and discretised in space. Attachment is described then via an auxiliary stochastic process that induces impulses in the biomass equation. The resulting system is an Itô stochastic differential equation. Unlike the more direct approach of modeling attachment by additive noise, the proposed model preserves non-negativity of solutions. Our numerical simulations are able to reproduce characteristic features of cellulolytic biofilms with cell attachment from the aqueous phase. Grid refinement studies show convergence for the expected values of spatially integrated biomass density and carbon concentration. We also examine the sensitivity of the model with respect to the parameters that control random attachment.
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Keywords attachment; cellulolytic biofilm; cellulosic biofilm; mathematical model; nonlinear diffusion; numerical simulation; stochastic differential equation

Citation: Yousef Rohanizadegan, Stefanie Sonner, Hermann J. Eberl. Discrete attachment to a cellulolytic biofilm modeled by an Itô stochastic differential equation. Mathematical Biosciences and Engineering, 2020, 17(3): 2236-2271. doi: 10.3934/mbe.2020119


  • 1. Z. Lewandowski, H. Beyenal, Fundamentals of Biofilm research, CRC Press, Boca Raton, 2007.
  • 2. B. D'Acunto, L. Frunzo, V. Luongo, M. R. Mattei, Free boundary approach for the attachment in the initial phase of multispecies biofilm growth, Zeitschrift für angewandte Mathematik und Physik, 70 (2019), 91.
  • 3. B. D'Acunto, L. Frunzo, M. R. Mattei, Continuum approach to mathematical modellingof multispecies biofilms, Ric. di Mat., 66 (2017), 153-169.
  • 4. A. Masic, H. J. Eberl, Persistence in a single species cstr model with suspended flocs and wall attached biofilms, Bull. Math. Biol., 74 (2012), 1001-1026.
  • 5. A. Masic, H. J. Eberl, A modeling and simulation study of the role of suspended microbial populations in nitrification in a biofilm reactor, Bull. Math. Biol., 76 (2014), 27-58.
  • 6. H. J. Gaebler, H. J. Eberl, A simple model of biofilm growth in a porous medium that accounts for detachment and attachment of suspended biomass and their contribution to substrate degradation, Eur. J. Appl. Math., 29 (2018), 1110-1140.
  • 7. E. Alkvist, I. Klapper, A multidimensional multispecies continuum modelfor heterogeneous biofilm development, Bull. Math. Biol., 69 (2007), 765-789.
  • 8. B. V. Merkey, B. E. Rittmann, D. L. Chopp, Modeling how soluble microbial products (smp) support heterotrophic bacteria in autotroph-based biofilms, J. Theor. Biol., 259 (2009), 670-683.
  • 9. N. G. Cogan, Two-fluid model of biofilm disinfection, Bull. Math. Biol., 70 (2008), 800-819.
  • 10. J. Dockery, I. Klapper, Finger formation in biofilms, SIAM J. Appl. Math., 62 (2002), 853-869.
  • 11. J. P. Ward, J. R. King, Thin-film modelling of biofilm growth and quorum sensing, J. Eng. Math., 73 (2012), 71-92.
  • 12. H. J. Eberl, D. F. Parker, M. Van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, Comput. Math. Methods Med., 3 (2001), 161-175.
  • 13. K. A. Rahman, R. Sudarsan, H. J. Eberl, A mixed-culture biofilm model with cross-diffusion, Bull. Math. Biol., 77 (2015), 2086-2124.
  • 14. M. Whidden, N. Cogan, M. Donahue, F. Navarrete, L. D. L. Fuente, A two-dimensional multiphase model of biofilmformation in microfluidic chambers, Bull. Math. Biol., 77 (2015), 2161-2179.
  • 15. F. El-Moustaid, A. Eladdadi, L. Uys, Modeling bacterial attachment to surfaces at and early stage of biofilm development, Math. Biosci. Eng., 10 (2013), 821-841.
  • 16. H. J. Eberl, H. Khassehkhan, L. Demaret, A mixed-culture model of a probiotic biofilm control system, Comput. Math. Methods Med., 11 (2010), 99-118.
  • 17. A. Carroll, C. Somerville, Cellulosic biofuels, Annu. Rev. Plant Biol., 60 (2009), 165-182.
  • 18. M. H. Langholtz, B. J. Stokes, L. M. Eaton, 2016 billion-ton report: Advancing domestic resources for a thriving bioeconomy, available from: https://bit.ly/2LLjdBS, 2016, URL https://doi.org/10.2172/1271651, (accessed 12-June-2018).
  • 19. J. G. Linger, A. Darzins, Consolidated Bioprocessing, 267-280, Springer New York, New York, NY, 2013.
  • 20. B. G. Schuster, M. S. Chinn, Consolidated bioprocessing of lignocellulosic feedstocks for ethanol fuel production, BioEnergy Res., 6 (2013), 416-435.
  • 21. A. Dumitrache, G. Wolfaardt, G. Allen, S. N. Liss, L. R. Lynd, Form and function of Clostridium thermocellum biofilms, Appl. Environ. Microbiol., 79 (2013), 231-239.
  • 22. V. Mbaneme-Smith, M. S. Chinn, Consolidated bioprocessing for biofuel production: recent advances, Energy Emission Control Technol., 3 (2015), 23-44.
  • 23. Z.-W. Wang, S.-H. Lee, J. G. Elkins, J. L. Morrell-Falvey, Spatial and temporal dynamics of cellulose degradation and biofilm formation by caldicellulosiruptor obsidiansis and clostridium thermocellum, AMB Express, 1 (2011), 30.
  • 24. M. van Loosdrecht, J. Heijnen, H. Eberl, J. Kreft, C. Picioreanu, Mathematical modelling of biofilm structures, Antonie van Leeuwenhoek, 81 (2002), 245-256.
  • 25. D. G. Davies, M. R. Parsek, J. P. Pearson, B. H. Iglewski, J. W. Costerton, E. P. Greenberg, The involvement of cell-to-cell signals in the development of a bacterial biofilm, Science, 280 (1998), 295-298.
  • 26. S. M. Hunt, E. M. Werner, B. Huang, M. A. Hamilton, P. S. Stewart, Hypothesis for the role of nutrient starvation in biofilm detachment, Appl. Environ. Microbiol., 70 (2004), 7418-7425.
  • 27. J. Zhu, J. J. Mekalanos, Quorum sensing-dependent biofilms enhance colonization in vibrio cholerae, Dev. Cell, 5 (2003), 647-656.
  • 28. Y. H. An, R. J. Friedman, Concise review of mechanisms of bacterial adhesion to biomaterial surfaces, J. Biomed. Mater. Res., 43 (1998), 338-348.
  • 29. H. Eberl, E. Jalbert, A. Dumitrache, G. Wolfaardt, A spatially explicit model of inverse colony formation of cellulolytic biofilms, Biochem. Eng. J., 122 (2017), 141-151.
  • 30. P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin Heidelberg, 1992.
  • 31. B. Øksendal, Stochastic differential equations: An introduction with applications, Springer, Berlin Heidelberg, 2003.
  • 32. A. Dumitrache, G. Wolfaardt, G. Allen, S. N. Liss, L. R. Lynd, Tracking the cellulolytic activity of Clostridium thermocellum biofilms, Biotech. Biofuels, 6 (2013), 175.
  • 33. G. J. Lord, C. E. Powell, T. Shardlow, An introduction to computational stochastic PDEs, 50, Cambridge University Press, New York, NY, 2014.
  • 34. J. Cresson, M. Efendiev, S. Sonner, On the positivity of solutions of systems of stochastic pdes, ZAMM - J. Appl. Math. Mech. / Zeitschrift für Angewandte Mathematik und Mechanik, 93 (2013), 414-422.
  • 35. J. Cresson, B. Puig, S. Sonner, Stochastic models in biology and the invariance problem, Discrete Cont. Dyn- B, 21 (2016), 2145-2168.
  • 36. R. Fox, Stochastic versions of the Hodgkin-Huxley equations, Biophys. J., 72 (1997), 2068-2074.
  • 37. A. Kamina, R. W. Makuch, H. Zhao, A stochastic modeling of early HIV-1 population dynamics, Math. Biosci., 170 (2001), 187-198.
  • 38. H. C. Tuckwell, E. L. Corfec, A stochastic model for early hiv-1 population dynamics, J. Theor. Biol., 195 (1998), 451-463.
  • 39. H. J. Eberl, L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, Electron. J. Differ. Equ, 15 (2007), 77-95.
  • 40. A. Dumitrache, H. J. Eberl, D. G. Allen, G. M. Wolfaardt, Mathematical modeling to validate on-line co2 measurements as a metric for cellulolytic biofilm activity in continuous-flow bioreactors, Biochem. Eng. J., 101 (2015), 55-67.
  • 41. M. A. Efendiev, S. V. Zelik, H. J. Eberl, Existence and longtime behavior of a biofilm model, Commun. Pure Appl. Anal., 8 (2009), 509-531.
  • 42. M. Ghasemi, H. J. Eberl, Time adaptive numerical solution of a highly degenerate diffusion-reaction biofilm model based on regularisation, J. Sci. Comput., 74 (2018), 1060-1090.
  • 43. J. Cresson, B. Puig, S. Sonner, Validating stochastic models: invariance criteria for systems of stochastic differential equations and the selection of a stochastic Hodgkin-Huxley type model, Internat. J. Biomath. Biostat., 2 (2013), 111-222.
  • 44. X. Han, P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Springer, Singapore, 2017.
  • 45. P. E. Kloeden, E. Platen, H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer-Verlag, Berlin Heidelberg, 1994.
  • 46. J. Monod, The growth of bacterial cultures, Ann. Rev. Microbiol., 3 (1949), 371-394.
  • 47. Y. Rohanizadegan, A Stochastic Formulation of Bacterial Attachment in a Spatially Explicit Model of Cellulolytic Biofilm Formation, Master's thesis, University of Guelph, Canada, 2018, available from: URL https://atrium.lib.uoguelph.ca/xmlui/handle/10214/14673.
  • 48. W. Walter, Ordinary Differential Equations, Springer New York, 1998.


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