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Bulletin of Mathematical Biology

, Volume 70, Issue 3, pp 745–768 | Cite as

Klebsiella pneumoniae Flocculation Dynamics

  • D. M. Bortz
  • T. L. Jackson
  • K. A. Taylor
  • A. P. Thompson
  • J. G. Younger
Original Article

Abstract

The bacterial pathogen Klebsiella pneumoniae is a cause of community- and hospital-acquired lung, urinary tract and blood stream infections. It is a common contaminant of indwelling catheters and it is theorized in that context that systemic infection follows shedding of aggregates off of surface-adherent biofilm colonies. In an effort to better understand bacterial proliferation in the host bloodstream, we develop a PDE model for the flocculation dynamics of Klebsiella pneumoniae in suspension. Existence and uniqueness results are provided, as well as a brief description of the numerical approximation scheme. We generate artificial data and illustrate the requirements to accurately identify proliferation, aggregation, and fragmentation of flocs in the experimental domain of interest.

Keywords

Klebsiella pneumoniae Aggregation Flocculation Parameter identification 

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References

  1. Ackleh, A.S., 1997. Estimation of parameters in a structured algal coagulation-fragmentation model. Nonlinear Anal. Theory Methods Appl. 28(5), 836–854. CrossRefMathSciNetGoogle Scholar
  2. Ackleh, A.S., 1999. Parameter identification in size-structured population models with nonlinear individual rates. Math. Comput. Model. 30, 81–92. zbMATHCrossRefMathSciNetGoogle Scholar
  3. Ackleh, A.S., Fitzpatrick, B.G., 1997. Modeling aggregation and growth processes in an algal population model: analysis and computations. J. Math. Biol. 35, 480–502. zbMATHCrossRefMathSciNetGoogle Scholar
  4. Ackleh, A.S., Fitzpatrick, B.G., Hallam, T.G., 1994. Approximation and parameter estimation problems for algal aggregation models. Math. Models Methods Appl. Sci. 4(3), 291–311. zbMATHCrossRefMathSciNetGoogle Scholar
  5. Ackleh, A.S., Banks, H.T., Deng, K., Hu, S., 2005. Parameter estimation in a coupled system of nonlinear size-structured populations. Math. Biosci. Eng. 2(2), 289–315. zbMATHMathSciNetGoogle Scholar
  6. Aldous, D.J., 1999. Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48. zbMATHCrossRefMathSciNetGoogle Scholar
  7. Anderl, J.N., Zahller, J., Roe, F., Stewart, P.S., 2003. Role of nutrient limitation and stationary-phase existence in Klebsiella pneumoniae biofilm resistance to ampicillin and ciproflaxacin. Antimicrob. Agents Chemother. 47(4), 1251–1256. CrossRefGoogle Scholar
  8. Balestrino, D., Haagensen, J.A.J., Rich, C., Forestier, C., 2005. Characterization of type 2 quorum sensing in Klebsiella pneumoniae and relationship to biofilm formation. J. Bacteriol. 187(8), 2870–2880. CrossRefGoogle Scholar
  9. Banks, H.T., Kappel, F., 1989. Transformation semigroups and L 1-approximation for size structured population models. Semigroup Forum 38, 141–155. zbMATHCrossRefMathSciNetGoogle Scholar
  10. Banks, H.T., Kunisch, K., 1989. Estimation Techniques for Distributed Parameter Systems. Systems & Control: Foundations & Applications, vol. 1. Birkhäuser, Boston. zbMATHGoogle Scholar
  11. Carroll, R.J., Ruppert, D., 1988. Transformation and Weighting in Regression. Chapman & Hall, London. zbMATHGoogle Scholar
  12. Coleman, T.F., Li, Y., 1994. On the convergence of reflective newton methods for large-scale nonlinear minimization subject to bounds. Math. Program. 67(2), 189–224. CrossRefMathSciNetGoogle Scholar
  13. Coleman, T.F., Li, Y., 1996. An interior, trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6, 418–445. zbMATHCrossRefMathSciNetGoogle Scholar
  14. Drake, R.L., 1972. A general mathematical survey of the coagulation equation. In: G.M. Hidy, J.R. Brock (Eds.), Topics in Current Aerosol Research (Part 2). International Reviews in Aerosol Physics and Chemistry, vol. 3, pp. 201–376. Pergamon, New York. Google Scholar
  15. Dubovskii, P.B., 1994. Mathematical Theory of Coagulation. Lecture Notes Series, vol. 23, pp. 151–742. Research Institute of Mathematics: Global Analysis Center, Seoul National University, Seoul. zbMATHGoogle Scholar
  16. Filbert, F., Laurençot, P., 2004. Numerical simulation of the Smoluchowski coagulation equation. SIAM J. Control Optim. 25(6), 2004–2028. Google Scholar
  17. Han, B., Akeprathumchai, S., Wickramasinghe, S.R., Qian, X., 2003. Flocculation of biological cells: experiment vs. theory. AIChE J. 49(7), 1687–1701. CrossRefGoogle Scholar
  18. Kaku, V.J., Boufadel, M.C., Venosa, A.D., 2006. Evaluation of mixing energy in laboratory flasks used for dispersant effectiveness testing. J. Environ. Eng. 132(1), 93–101. CrossRefGoogle Scholar
  19. Makino, J., Fukushige, T., Funato, Y., Kokubo, E., 1998. On the mass distribution of planetesimals in the early runaway stage. New Astron. 3, 411–417. CrossRefGoogle Scholar
  20. Menon, G., Pego, R.L., 2006. Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence. SIAM Rev. 48(4), 745–768. zbMATHCrossRefMathSciNetGoogle Scholar
  21. Moré, J.J., Wright, S.J., 1993. Optimization Software Guide. Frontiers in Applied Mathematics, vol. 14. SIAM, Philadelphia. zbMATHGoogle Scholar
  22. Müller, H., 1928. Zur allgemeinen theorie der raschen koagulation. Kolloidchem. Beih. 27, 257–311. CrossRefGoogle Scholar
  23. Pawar, P., Shin, P.K., Mousa, S.A., Ross, J.M., Konstantopoulos, K., 2004. Fluid shear regulates the kinetics and receptor specificity of Staphylococcus aureus binding to activated platelets. J. Immunol. 173, 1258–1265. Google Scholar
  24. Pazy, A., 1992. Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York. Google Scholar
  25. Podschun, R., Ullmann, U., 1998. Klebsiella spp. as nosocomial pathogens: epidemiology, taxonomy, typing methods, and pathogenicity factors. Clin. Microbiol. Rev. 11(4), 589–603. Google Scholar
  26. Poppele, E.H., Hozalski, R.M., 2003. Micro-cantilever method for measuring the tensile strength of biofilms and microbial flocs. J. Microbiol. Methods 55(3), 607–615. CrossRefGoogle Scholar
  27. Pruppacher, H.R., Klett, J.D., 1980. Microphysics of Clouds and Precipitation. Riedel, Boston. Google Scholar
  28. Riebesell, U., Wolf-Gladrow, D.A., 1992. The relationship between physical aggregation of phytoplankton and particle flux: a numerical model. Deep-Sea Res. 39(7/8), 1085–1102. CrossRefGoogle Scholar
  29. Saffman, P.G., Turner, J.S., 1956. On the collision of drops in turbulent clouds. J. Fluid Mech. 1, 16–30. zbMATHCrossRefGoogle Scholar
  30. Seber, G.A.F., Wild, C.J., 1989. Nonlinear Regression. Wiley, New York. zbMATHGoogle Scholar
  31. Smit, D.J., Hounslow, M.J., Paterson, W.R., 1994a. Aggregation and gelation—I. Analytical solutions for CST and batch operation. Chem. Eng. Sci. 49(7), 1025–1035. CrossRefGoogle Scholar
  32. Smit, D.J., Hounslow, M.J., Paterson, W.R., 1994b. Aggregation and gelation—II. Mixing effects in continuous flow vessels. Chem. Eng. Sci. 49(18), 3147–3167. CrossRefGoogle Scholar
  33. Smit, D.J., Hounslow, M.J., Paterson, W.R., 1995. Aggregation and gelation—III. Numerical classification of kernels and case studies of aggregation and growth. Chem. Eng. Sci. 50(5), 849–862. CrossRefGoogle Scholar
  34. Somasundaran, P., Runkanan, V., Kapur, P.C., Flocculation and dispersion of colloidal suspensions by polymers and surfactants: experimental and modeling studies. In: Coagulation and Flocculation (Stechemesser and Dobiáš, 2005), pp. 767–803. Google Scholar
  35. Stechemesser, H., Dobiáš, B., 2005. Coagulation and Flocculation. Surfactant Science Series, 2nd edn., vol. 126. Taylor and Francis, Boca Raton. Google Scholar
  36. Thomaseth, K., Cobelli, C., 1999. Generalized sensitivity functions in physiological system identification. Ann. Biomed. Eng. 27, 607–616. CrossRefGoogle Scholar
  37. van Smoluchowski, M., 1916. Drei vorträge über diffusion, brownsche bewegung und koagulation von kolloidteilchen. Zeit. Phys. 17, 557–571, 585–599. Google Scholar
  38. van Smoluchowski, M., 1917. Versuch einer mathematischen theorie der koagulation kinetic kolloider losungen. Zeit. Phys. Chem. 92, 129–168. Google Scholar
  39. Wentland, E.J., Stewart, P.S., Huang, C.-T., McFeters, G.A., 1996. Spatial variations in growth rate within Klebsiella pneumoniae colonies and biofilm. Biotechnol. Prog. 12, 316–321. CrossRefGoogle Scholar
  40. Zahller, J., Stewart, P.S., 2002. Transmission electron microscope study of antibiotic action on Klebsiella pneumoniae biofilm. Antimicrob. Agents Chemother. 46(8), 2679–2683. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2007

Authors and Affiliations

  • D. M. Bortz
    • 1
  • T. L. Jackson
    • 2
  • K. A. Taylor
    • 3
  • A. P. Thompson
    • 4
  • J. G. Younger
    • 4
  1. 1.Department of Applied MathematicsUniversity of ColoradoBoulderUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Columbia UniversityNew YorkUSA
  4. 4.Department of Emergency MedicineUniversity of MichiganAnn ArborUSA

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