Journal of Scientific Computing

, Volume 74, Issue 2, pp 1060–1090 | Cite as

Time Adaptive Numerical Solution of a Highly Degenerate Diffusion–Reaction Biofilm Model Based on Regularisation

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Abstract

We consider a quasilinear degenerate diffusion–reaction system that describes biofilm formation. The model exhibits two non-linear effects: a power law degeneracy as one of the dependent variables vanishes and a super diffusion singularity as it approaches unity. Biologically relevant solutions are characterized by a moving interface and gradient blow-up there. Discretisation of the PDE in space by a standard finite volume scheme leads to a singular system of ordinary differential equations. We show that regularisation of this system allows the application of error controlled adaptive integration techniques to solve the underlying PDE. This overcomes the major limitation of existing methods for this type of problem which work with fixed time-steps. We apply the resulting numerical method to study the effect of signal diffusion in the aqueous phase on quorum sensing induction in a biofilm colony.

Keywords

Biofilm Degenerate diffusion–reaction equation Quorum sensing Regularization Semi-discretization Time adaptivity 

Mathematics Subject Classification

Primary 35K65 65M08 Secondary 68U20 92D25 

Notes

Acknowledgements

This study has been financially supported by the Natural Science and Engineering Research Council of Canada (NSERC): MG holds a PGS-D graduate scholarship, HJE a Discovery Grant, the equipment was purchased with a Research Tools and Infrastructure Grant (HJE).

References

  1. 1.
    Dillon, R., Fauci, L., Fogelson, A., Gaver, D.: Modelling biofilm processes using the immersed boundary method. J. Comput. Phys. 129(1), 57–73 (1996)CrossRefMATHGoogle Scholar
  2. 2.
    Hall-Stoodley, L., Costerton, J.W., Stoodley, P.: Bacterial biofilms: from the natural environment to infectious diseases. Nat. Rev. Microbiol. 2(2), 95–108 (2004)CrossRefGoogle Scholar
  3. 3.
    Imran, M., Smith, H.: A model of optimal dosing of antibiotic treatment in biofilm. Math. Biosci. Eng. 11(3), 547–571 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Lear, G., Lewis, G.D.: Microbial Biofilms: Current Research and Applications. Caister Academic, Berlin (2012). ISBN 978-1-904455-96-7Google Scholar
  5. 5.
    Martins dos Santos, V.A.P., Yakimov, M.M., Timmis, K.N., Golyshin, P.N.: Genomic insights into oil biodegradation in marine systems. In: Diaz, E. (ed), Microbial Biodegradation: Genomics and Molecular Biology, p. 1971. Horizon Scientific Press. ISBN 978-1-904455-17-2 (2008)Google Scholar
  6. 6.
    Watnick, P., Kolter, R.: Biofilm-city of microbes (minireview). J. Bacteriol. 182(10), 2675–2679 (2000)CrossRefGoogle Scholar
  7. 7.
    Stewart, P.S., Costerton, J.W.: Antibiotic resistance of bacteria in biofilms. Lancet 358(9276), 135–8 (2001)CrossRefGoogle Scholar
  8. 8.
    Anderl, J.N., Franklin, M.J., Stewart, P.S.: Role of antibiotic penetration limitation in Klebsiella pneumoniae biofilm resistance to ampicillin and ciprofloxacin. Antimicrob. Agents Chemother. 44, 1818–1824 (2000)Google Scholar
  9. 9.
    Schwermer, C.U., Lavik, G., Abed, R.M., et al.: Impact of nitrate on the structure and function of bacterial biofilm communities in pipelines used for injection of seawater into oil fields. Appl. Environ. Microbiol. 74(9), 2841–51 (2008)CrossRefGoogle Scholar
  10. 10.
    Donlan, R.M., Costerton, J.W.: Biofilms: survival mechanisms of clinically relevant microorganisms. Clin. Microbiol. Rev. 15(2), 167–193 (2002)CrossRefGoogle Scholar
  11. 11.
    Andersen, P.C., Brodbeck, B.V., Oden, S., Shriner, A., Leite, B.: Influence of xylem fluid chemistry on planktonic growth, biofilm formation and aggregation of Xylella fastidiosa. FEMS Microbiol. Lett. 274(2), 210–217 (2007)Google Scholar
  12. 12.
    Wanner, O., Eberl, H.J., Van Loosdrecht, M.C.M., Morgenroth, E., Noguera, D.R., Picioreanu, C., Rittmann, B.E.: Mathematical Modelling of Biofilms. IWA Publishing, London (2006)Google Scholar
  13. 13.
    van Loosdrecht, M.C.M., Heijnen, J.J., Eberl, H., Kreft, J., Picioreanu, C.: Mathematical Modelling of Biofilm Structures. Antonie Van Leeuwenhoek 81(1), 245–256 (2002)CrossRefGoogle Scholar
  14. 14.
    Eberl, H.J., Parker, D.F., Van Loosdrecht, C.M.: A new deterministic spatio-temporal continuum model for biofilm development. J. Theor. Med. 3, 161–175 (2001)CrossRefMATHGoogle Scholar
  15. 15.
    Klapper, I., Dockery, J.: Mathematical description of microbial biofilms. SIAM Rev. 52(2), 221–265 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Wang, Q., Zhang, T.: Review of mathematical models for biofilms. Solid State Commun. 150(21/22), 1009–1022 (2010)CrossRefGoogle Scholar
  17. 17.
    Eberl, H.J., Collinson, S.: A modelling and simulation study of siderophore mediated antagonsim in dual-species biofilms. Theor. Biol. Med. Mod. 6, 30 (2009)CrossRefGoogle Scholar
  18. 18.
    Eberl, H.J., Sudarsan, R.: Exposure of biofilms to slow flow fields: the convective contribution to growth and disinfection. J. Theor. Biol. 253(4), 788–807 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Emerenini, B., Hense, B.A., Kuttler, C., Eberl, H.J.: A mathematical model of quorum sensing induced biofilm detachment. PLoS ONE 10(7), e0132385 (2015)CrossRefGoogle Scholar
  20. 20.
    Frederick, M., Kuttler, C., Hense, B.A., Müller, J., Eberl, H.J.: A mathematical model of quorum sensing in patchy biofilm communities with slow background flow. Can. Appl. Math. Quart. 18(3), 267–298 (2010)MathSciNetMATHGoogle Scholar
  21. 21.
    Frederick, M.R., Kuttler, C., Hense, B.A., Eberl, H.J.: A mathematical model of quorum sensing regulated EPS production in biofilms. Theor. Biol. Med. Mod. 8, 8 (2011)CrossRefGoogle Scholar
  22. 22.
    Khassehkhan, H., Efendiev, M.A., Eberl, H.J.: A degenerate diffusion–reaction model of an amensalistic biofilm control system: existence and simulation of solution. Discrete Cont. Dy. Syst. B 12(2), 371–388 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Macias-Diaz, J.E.: A positive finite-difference model in the computational simulation of complex biological film models. J. Differ. Equ. Appl. 20(4), 548–569 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rahman, K.A., Sudarsan, R., Eberl, H.J.: A mixed culture biofilm model with cross-diffusion. Bull. Math. Biol. 77(11), 2086–2124 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Khassehkhan, H., Hillen, T., Eberl, H.J.: A non-linear master equation for a degenerate diffusion model of biofilm growth. LNCS 5544, 5–744 (2009)Google Scholar
  26. 26.
    Ngamsaad, W., Sunatai, S.: Mechanically-driven spreading of bacterial populations. Commun. Nonlinear Sci. Numer. Simul. 35, 88–96 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Efendiev, M.A., Zelik, S.V., Eberl, H.J.: Existence and longtime behaviour of a biofilm model. Commun. Pure Appl. Anal. 8(2), 509–531 (2009)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Duvnjak, A., Eberl, H.J.: Time-discretisation of a degenerate reaction–diffusion equation arising in biofilm modelling. El. Trans Num. Anal. 23, 15–38 (2006)MATHGoogle Scholar
  29. 29.
    Khassehkhan, H., Eberl, H.J.: Interface tracking for a non-linear degenerated diffusion–reaction equation describing biofilm formation. Dyn. Cont. Disc. Imp. Sys. A 13SA, 131–144 (2006)Google Scholar
  30. 30.
    Khassehkhan, H., Eberl, H.J.: Modelling and simulation of a bacterial biofilm that is controlled by pH and protonated lactic acids. Comput. Math. Methods Med. 9(1), 47–67 (2008)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Eberl, H.J., Demaret, L.: A finite difference scheme for a degenerated diffusion equation arising in microbial ecology. El. J. Diff. Equs. CS 15, 77–95 (2007)MathSciNetMATHGoogle Scholar
  32. 32.
    Sirca, S., Horvat, M.: Computational Methods for Physicists. Springer, Berlin (2012)CrossRefMATHGoogle Scholar
  33. 33.
    Muhammad, N., Eberl, H.J.: OpenMP parallelization of a mickens time-integration scheme for a mixed-culture biofilm model and its performance on multi-core and multi-processor computers. LNCS 5976, 180–195 (2010)Google Scholar
  34. 34.
    Rahman, K.A., Eberl, H.J.: Numerical treatment of a cross-diffusion model of biofilm exposure to antimicrobials. LNCS 8384, 134–144 (2014)MathSciNetGoogle Scholar
  35. 35.
    Morales-Hernandez, M.D., Medina-Ramirez, I.E., Avelar-Gonzalez, F.J., Macias-Dias, J.E.: An efficient recursive algorithm in the computational simulation of the bounded growth of biological films. Int. J. Comp. Meth. 9(4), 1250050 (2012)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Jalbert, E.M.: Comparison of a semi-implicit and a fully-implicit time integration method for a highly degenerate diffusion–reaction equation coupled with an ordinary differential equation. M.Sc. Thesis, University of Guelph (2016)Google Scholar
  37. 37.
    Balsa-Canto, E., Lopez-Nunez, A., Vazquez, C.: Numerical methods for a nonlinear reaction–diffusion system modelling a batch culture of biofilm. Appl. Math. Model. 41, 164–179 (2017)Google Scholar
  38. 38.
    Medina-Ramirez, I.E., Macias-Diaz, J.E.: On a fully discrete finite-difference approximation of a non-linear diffusionreaction model in microbial ecology. Int. J. Comput. Math. 90(9), 1915–1937 (2013)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Sun, G.F., Liu, G.R., Li, M.: An Efficient Explicit Finite-Difference Scheme for Simulating Coupled Biomass Growth on Nutritive Substrates, Mathematical Problems in Engineering, p. 708497 (2015)Google Scholar
  40. 40.
    Hense, B.A., Kuttler, C., Müller, J., Rothballer, M., Hartmann, A., Kreft, J.: Does efficiency sensing unify diffusion and quorum sensing? Nat. Rev. Microbiol. 5, 230–239 (2007)CrossRefGoogle Scholar
  41. 41.
    Redfield, R.J.: Is quorum sensing a side effect of diffusion sensing? Trends Microbial. 10, 365–370 (2002)CrossRefGoogle Scholar
  42. 42.
    Trovato, A., Seno, F., Zanardo, M., Alberghini, S., Tondello, A., Squartini, A.: Quorum vs. diffusion sensing: a quantitative analysis of the relevance of absorbing or reflecting boundaries. FEMS Microbiol. Lett. 352(2), 198–203 (2014)CrossRefGoogle Scholar
  43. 43.
    Chopp, D.L., Kirisits, M.J., Moran, B., Parsek, M.R.: A mathematical model of quorum sensing in a growing bacterial biofilm. J. Ind. Microbiol. Biotech. 29, 339–346 (2002)CrossRefMATHGoogle Scholar
  44. 44.
    Chopp, D.L., Kirisits, M.J., Moran, B., Parsek, M.R.: The dependence of quorum sensing on the depth of a growing biofilm. Bull. Math. Biol. 65(6), 1053–1079 (2003)CrossRefMATHGoogle Scholar
  45. 45.
    Vaughan, B.L., Smith, B.G., Chopp, D.L.: The influence of fluid flow on modelling quorum sensing in bacterial biofilms. Bull. Math. Biol. 72(5), 1143–1165 (2010)CrossRefMATHGoogle Scholar
  46. 46.
    Walter, W.: Ordinary Differential Equations. Springer, Berlin (1997)Google Scholar
  47. 47.
    Hackbusch, W.: Theorie und Numerik Elliptischer Differentialgleichungen. Teubner, Stuttgart (1986)CrossRefMATHGoogle Scholar
  48. 48.
    Rang, J.: Improved Traditional Rosenbrock–Wanner Methods for Stiff ODEs and DAEs. Institute of Scientific Computing, Heidelberg (2013)MATHGoogle Scholar
  49. 49.
    Van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992)CrossRefMATHGoogle Scholar
  50. 50.
    Saad, Y.: SPARSKIT: a basic tool for sparse matrix computations (1994). http://www.users.cs.umn.edu/saad/software/SPARSKIT/sparskit.html
  51. 51.
    Polyanin, A.D., Zaitsev, V.F.: Handbook of Non-linear Partial Differential Equations, 2nd edn. CRC Press, Boca Raton (2011)MATHGoogle Scholar
  52. 52.
    Picioreanu, C., Van Loosdrecht, C.M., Heijnen, J.J.: Mathematical modelling of biofilm structure with a hybrid differential-discrete cellular automaton approach. Biotechnol. Bioeng. 58(1), 101–116 (1998)CrossRefGoogle Scholar
  53. 53.
    Ward, J.P., King, J.R., Koerber, A.J., Williams, P., Croft, J.M., Sockett, R.E.: Mathematical modeling of quorum sensing bacteria. IMA J. Math. Appl. Med. Biol. 18, 263–292 (2001)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.University of GuelphGuelphCanada

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