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A point process model for generating biofilms with realistic microstructure and rheology

Published online by Cambridge University Press:  16 May 2018

JAY ALEXANDER STOTSKY ,
VANJA DUKIC  and
DAVID M. BORTZ
JAY ALEXANDER STOTSKY
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA emails: Jay.Stotsky@colorado.edu, Vanja.Dukic@colorado.edu, dmbortz@colorado.edu
VANJA DUKIC
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA emails: Jay.Stotsky@colorado.edu, Vanja.Dukic@colorado.edu, dmbortz@colorado.edu
DAVID M. BORTZ
Affiliation:
Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA emails: Jay.Stotsky@colorado.edu, Vanja.Dukic@colorado.edu, dmbortz@colorado.edu
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Abstract

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Biofilms are communities of bacteria that exhibit a multitude of multiscale biomechanical behaviours. Recent experimental advances have led to characterisations of these behaviours in terms of measurements of the viscoelastic moduli of biofilms grown in bioreactors and the fracture and fragmentation properties of biofilms. These properties are macroscale features of biofilms; however, a previous work by our group has shown that heterogeneous microscale features are critical in predicting biofilm rheology. In this paper, we use tools from statistical physics to develop a generative statistical model of the positions of bacteria in biofilms. Specifically, the model is a type of pairwise interaction model (PIM). We show through simulation that the macroscopic mechanical properties of biofilms depend on the choice of microscale spatial model. A key finding is that uniform and non-uniform sets of points lead to differing mechanical properties. This distinction appears not to have been previously considered in mathematical biofilm literature. We also found that realisations of a biologically informed PIM have realistic in silico mechanical properties, and have statistical properties that closely match experimental data. We also note that a Poisson spatial point process of suitable number density also yields realistic mechanical properties, but that the spatial distribution of points does not reflect those occurring in our experimentally observed biofilm.

Keywords

BiofilmsBiomechanicsNonparametric Density EstimationSpatial Stochastic Processes
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 6: Biofilms , December 2018 , pp. 1141 - 1177
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This work was supported in part by the National Science Foundation grants PHY-0940991 and DMS-1225878 to DMB, and by the Department of Energy through the Computational Science Graduate Fellowship program, DE-FG02-97ER25308, to JAS.

References

[1] Abramson, I. S. (1982) On bandwidth variation in kernel estimates-a square root law. Ann. Stat. 10 (4), 1217–1223.Google Scholar
[2] Alpkvist, E. & Klapper, I. (2008) Description of mechanical response including detachment using a novel particle model of biofilm/flow interaction. Water Sci. Technol. 55 (8–9), 265–273.Google Scholar
[3] Baddeley, A. & Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Aust. N. Z. J. Stat. 42 (3), 283–322.Google Scholar
[4] Baddeley, A. J., Møller, J. & Waagepetersen, R. (2000) Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Stat. Neerlandica 54 (3), 329–350.Google Scholar
[5] Billingsley, P. (2008) Probability and Measure, John Wiley & Sons, New York.Google Scholar
[6] Christensen, R. M. (1982) Theory of Viscoelasticity: An Introduction, 2nd ed, New York: Academic Press.Google Scholar
[7] Coeurjolly, J. F., Møller, J. & Waagepetersen, R. (2017) A tutorial on Palm distribution for spatial point processes. Int. Stat. Rev. 85 (3), 404–420.Google Scholar
[8] Conrad, P. R., Marzouk, Y. M., Pillai, N. S. & Smith, A. (2016) Accelerating asymptotically exact MCMC for computationally intensive models via local approximations. J. Am. Stat. Assoc. 111 (516), 1591–1607.Google Scholar
[9] Courant, R. & Hilbert, D. (1954) Methods of mathematical physics, Vol. I. Phys. Today 7 (5), 17–17.Google Scholar
[10] Crocker, J. C. & Grier, D. G. (1996) Methods of digital video microscopy for colloidal studies. J. Colloid Interface Sci. 179 (1), 298–310.Google Scholar
[11] Cronie, O. & van Lieshout, M. N. M. (2018) A non-model-based approach to bandwidth selection for kernel estimators of spatial intensity functions. Biometrika.Google Scholar
[12] Daley, D. J. & Vere-Jones, D. (2007) An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure, Springer Science & Business Media, New York.Google Scholar
[13] Dzul, S. P., Thornton, M. M., Hohne, D. N., Stewart, E. J., Shah, A. A., Bortz, D. M., Solomon, M. J. & Younger, J. G. (2011) Contribution of the Klebsiella pneumoniae capsule to bacterial aggregate and biofilm microstructures. Appl. Environ. Microbiol. 77 (5), 1777–1782.Google Scholar
[14] Epanechnikov, V. A. (1969) Non-parametric estimation of a multivariate probability density. Theory Probab. Appl. 14 (1), 153–158.Google Scholar
[15] Fai, T. G., Leo-Macias, A., Stokes, D. L. & Peskin, C. S. (2017) Image-based model of the spectrin cytoskeleton for red blood cell simulation. PLoS Comput. Biol. 13 (10), e1005790.Google Scholar
[16] Flemming, H. C. (2011) Microbial biofouling: Unsolved problems, insufficient approaches, and possible solutions. In: Flemming, H.-C., Wingender, J., Szewzyk, U. (editors), Biofilm Highlights, Springer, pp. 81–109.Google Scholar
[17] Gaboriaud, F., Gee, M. L., Strugnell, R. & Duval, J. F. L. (2008) Coupled electrostatic, hydrodynamic, and mechanical properties of bacterial interfaces in aqueous media. Langmuir 24 (19), 10988–10995.Google Scholar
[18] Gangopadhyay, A. & Cheung, K. (2002) Bayesian approach to the choice of smoothing parameter in kernel density estimation. J. Nonparametric Stat. 14 (6), 655–664.Google Scholar
[19] Geyer, C. J. & Møller, J. (1994) Simulation procedures and likelihood inference for spatial point processes. Scand. J. Stat. 21 (4), 359–373.Google Scholar
[20] Guan, Y. (2007) A least-squares cross-validation bandwidth selection approach in pair correlation function estimations. Stat. Probab. Lett. 77 (18), 1722–1729.Google Scholar
[21] Guan, Y. (2008) On consistent nonparametric intensity estimation for inhomogeneous spatial point processes. J. Am. Stat. Assoc. 103 (483), 1238–1247.Google Scholar
[22] Guélon, T., Mathias, J. D. & Stoodley, P. (2011) Advances in biofilm mechanics. In: Flemming, H.-C., Wingender, J., Szewzyk, U. (editors), Biofilm Highlights, Springer, pp. 111–139.Google Scholar
[23] Guizar-Sicairos, M. & Gutiérrez-Vega, J. C. (2004) Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields. J. Opt. Soc. Am. A 21 (1), 53–58.Google Scholar
[24] Hall, P. & Marron, J. S. (1991) Local minima in cross-validation functions. J. R. Stat. Soc. Ser. B (Methodological) 53 (1), 245–252.Google Scholar
[25] Hammond, J. F., Stewart, E. J., Younger, J. G., Solomon, M. J. & Bortz, D. M. (2014) Variable viscosity and density biofilm simulations using an immersed boundary method, Part I: Numerical scheme and convergence results. Comput. Model. Eng. Sci. 98 (3), 295–340.Google Scholar
[26] Hansen, J. P. & McDonald, I. R. (1990) Theory of Simple Liquids, Elsevier, London, UK.Google Scholar
[27] Hardle, W., Marron, J. S. & Wand, M. P. (1990) Bandwidth choice for density derivatives. J. R. Stat. Ser. B (Methodological) 52 (1), 223–232.Google Scholar
[28] Jones, M. C. (1993) Simple boundary correction for kernel density estimation. Stat. Comput. 3 (3), 135–146.Google Scholar
[29] Kerscher, M., Szapudi, I. & Szalay, A. S. (2000) A comparison of estimators for the two-point correlation function. Astrophys. J. Lett. 535 (1), L13.Google Scholar
[30] Landy, S. D. & Szalay, A. S. (1993) Bias and variance of angular correlation functions. Astrophys. J. 412, 64–71.Google Scholar
[31] Laspidou, C. S. & Rittmann, B. E. (2004) Modeling the development of biofilm density including active bacteria, inert biomass, and extracellular polymeric substances. Water Res. 38 (14), 3349–3361.Google Scholar
[32] Lovett, R., Mou, C. Y. & Buff, F. P. (1976) The structure of the liquid-vapor interface. J. Chem. Phys. 65, 2377.Google Scholar
[33] Moller, J. & Waagepetersen, R. P. (2003) Statistical Inference and Simulation for Spatial Point Processes, CRC Press, Boca Raton, FL.Google Scholar
[34] Ornstein, L. S. & Zernike, F. (1914) The influence of accidental deviations of density on the equation of state. Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings 19 (2), 1312–1315.Google Scholar
[35] Parzen, E. (1962) On estimation of a probability density function and mode. Ann. Math. Stat. 33 (3), 1065–1076.Google Scholar
[36] Pavlovsky, L., Younger, J. G. & Solomon, M. J. (2013) In situ rheology of Staphylococcus epidermidis bacterial biofilms. Soft Matter 9 (1), 122–131.Google Scholar
[37] Ripley, B. D. (1991) Statistical Inference for Spatial Processes, Cambridge University Press.Google Scholar
[38] Rosenblatt, M. et al. (1956) Remarks on some nonparametric estimates of a density function. Ann. Math. Stat. 27 (3), 832–837.Google Scholar
[39] Silverman, B. W. (1981) Using Kernel density estimates to investigate multimodality. J. R. Stat. Soc. 43 (1), 97–99.Google Scholar
[40] Sobczyk, K. & Kirkner, D. J. (2012) Stochastic Modeling of Microstructures, Springer Science & Business Media, Boston, MA.Google Scholar
[41] Stewart, E. J., Ganesan, M., Younger, J. G. & Solomon, M. J. (2015) Artificial biofilms establish the role of matrix interactions in Staphylococcal biofilm assembly and disassembly. Sci. Rep. 5, 13081; doi: 10.1038/srep13081.Google Scholar
[42] Stewart, E. J., Satorius, A. E., Younger, J. G. & Solomon, M. J. (2013) Role of environmental and antibiotic stress on Staphylococcus epidermidis biofilm microstructure. Langmuir 29 (23), 7017–7024.Google Scholar
[43] Stotsky, J. A., Hammond, J. F., Pavlovsky, L., Stewart, E. J., Younger, J. G., Solomon, M. J. & Bortz, D. M. (2016) Variable viscosity and density biofilm simulations using an immersed boundary method, Part II: Experimental validation and the heterogeneous rheology-IBM. J. Comput. Phys. 317, 204–222.Google Scholar
[44] Stoyan, D., Bertram, U. & Wendrock, H. (1993) Estimation variances for estimators of product densities and pair correlation functions of planar point processes. Ann. Inst. Stat. Math. 45 (2), 211–221.Google Scholar
[45] Stoyan, D., Kendall, W. S. & Mecke, J. (1995) Stochastic Geometry and its Applications, Akademie-Verlag, Berlin.Google Scholar
[46] Sudarsan, R., Ghosh, S., Stockie, J. M. & Eberl, H. J. (2016) Simulating biofilm deformation and detachment with the immersed boundary method. Commun. Comput. Phys. 19 (3), 682–732.Google Scholar
[47] Szapudi, I. & Szalay, A. S. (1998) A new class of estimators for the n-point correlations. Astrophys. J. Lett. 494 (1), L41.Google Scholar
[48] Torquato, S. (2013) Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Vol. 16, Springer Science & Business Media, New York.Google Scholar
[49] Truskett, T. M., Torquato, S. & Debenedetti, P. G. (1998) Density fluctuations in many-body systems. Phys. Rev. E 58 (6), 7369.Google Scholar
[50] Vo, G. D., Brindle, E. & Heys, J. (2010) An experimentally validated immersed boundary model of fluid–biofilm interaction. Water Sci. Technol. 61 (12), 3033–3040.Google Scholar
[51] Wand, M. P. & Jones, M. C. (1993) Comparison of smoothing parameterizations in bivariate kernel density estimation. J. Am. Stat. Assoc. 88 (422), 520–528.Google Scholar
[52] Wróbel, J. K., Cortez, R. & Fauci, L. (2014) Modeling viscoelastic networks in stokes flow. Phys. Fluids (1994-present) 26 (11), 113102.Google Scholar
[53] Yeong, C. L. Y. & Torquato, S. (1998) Reconstructing random media. Phys. Rev. E 57 (1), 495.Google Scholar
[54] Zhang, T., Cogan, N. G. & Wang, Q. (2008) Phase field models for biofilms. I. Theory and one-dimensional simulations. SIAM J. Appl. Math. 69 (3), 641–669.Google Scholar
[55] Zhang, T., Cogan, N. G. & Wang, Q. (2008) Phase field models for biofilms. ii. 2-d numerical simulations of biofilm-flow interaction. Commun. Comput. Phys 4 (1), 72–101.Google Scholar
[56] Zhao, J., Shen, Y., Haapasalo, M., Wang, Z. & Wang, Q. (2016) A 3d numerical study of antimicrobial persistence in heterogeneous multi-species biofilms. J. Theor. Biol. 392, 83–98.Google Scholar