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A point process model for generating biofilms with realistic microstructure and rheology. (English) Zbl 1407.92021

Summary: 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.

MSC:

92C10 Biomechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62P10 Applications of statistics to biology and medical sciences; meta analysis
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[1] [1]AbramsonI. S. (1982) On bandwidth variation in kernel estimates-a square root law. Ann. Stat.10(4), 1217-1223.10.1214/aos/1176345986 · Zbl 0507.62040 · doi:10.1214/aos/1176345986
[2] [2]AlpkvistE. & KlapperI. (2008) Description of mechanical response including detachment using a novel particle model of biofilm/flow interaction. Water Sci. Technol.55(8-9), 265-273.
[3] [3]BaddeleyA. & TurnerR. (2000) Practical maximum pseudolikelihood for spatial point patterns. Aust. N. Z. J. Stat.42(3), 283-322.10.1111/1467-842X.00128 · Zbl 0981.62078 · doi:10.1111/1467-842X.00128
[4] [4]BaddeleyA. J., MøllerJ. & WaagepetersenR. (2000) Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Stat. Neerlandica54(3), 329-350.10.1111/1467-9574.00144 · Zbl 1018.62027 · doi:10.1111/1467-9574.00144
[5] [5]BillingsleyP. (2008) Probability and Measure, John Wiley & Sons, New York.
[6] [6]ChristensenR. M. (1982) Theory of Viscoelasticity: An Introduction, 2nd ed, New York: Academic Press.
[7] [7]CoeurjollyJ. F., MøllerJ. & WaagepetersenR. (2017) A tutorial on Palm distribution for spatial point processes. Int. Stat. Rev.85(3), 404-420.10.1111/insr.12205 · doi:10.1111/insr.12205
[8] [8]ConradP. R., MarzoukY. M., PillaiN. S. & SmithA. (2016) Accelerating asymptotically exact MCMC for computationally intensive models via local approximations. J. Am. Stat. Assoc.111(516), 1591-1607.10.1080/01621459.2015.1096787 · doi:10.1080/01621459.2015.1096787
[9] [9]CourantR. & HilbertD. (1954) Methods of mathematical physics, Vol. I. Phys. Today7(5), 17-17.
[10] [10]CrockerJ. C. & GrierD. G. (1996) Methods of digital video microscopy for colloidal studies. J. Colloid Interface Sci.179(1), 298-310.10.1006/jcis.1996.0217 · doi:10.1006/jcis.1996.0217
[11] [11]CronieO. & van LieshoutM. N. M. (2018) A non-model-based approach to bandwidth selection for kernel estimators of spatial intensity functions. Biometrika.
[12] [12]DaleyD. J. & Vere-JonesD. (2007) An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure, Springer Science & Business Media, New York.
[13] [13]DzulS. P., ThorntonM. M., HohneD. N., StewartE. J., ShahA. A., BortzD. M., SolomonM. J. & YoungerJ. G. (2011) Contribution of the Klebsiella pneumoniae capsule to bacterial aggregate and biofilm microstructures. Appl. Environ. Microbiol.77(5), 1777-1782.10.1128/AEM.01752-10 · doi:10.1128/AEM.01752-10
[14] [14]EpanechnikovV. A. (1969) Non-parametric estimation of a multivariate probability density. Theory Probab. Appl.14(1), 153-158.10.1137/1114019 · doi:10.1137/1114019
[15] [15]FaiT. G., Leo-MaciasA., StokesD. L. & PeskinC. S. (2017) Image-based model of the spectrin cytoskeleton for red blood cell simulation. PLoS Comput. Biol.13(10), e1005790.10.1371/journal.pcbi.1005790 · doi:10.1371/journal.pcbi.1005790
[16] [16]FlemmingH. C. (2011) Microbial biofouling: Unsolved problems, insufficient approaches, and possible solutions. In: H.-C.Flemming, J.Wingender, U.Szewzyk (editors), Biofilm Highlights, Springer, pp. 81-109.10.1007/978-3-642-19940-0_5 · doi:10.1007/978-3-642-19940-0_5
[17] [17]GaboriaudF., GeeM. L., StrugnellR. & DuvalJ. F. L. (2008) Coupled electrostatic, hydrodynamic, and mechanical properties of bacterial interfaces in aqueous media. Langmuir24(19), 10988-10995.10.1021/la800258n · doi:10.1021/la800258n
[18] [18]GangopadhyayA. & CheungK. (2002) Bayesian approach to the choice of smoothing parameter in kernel density estimation. J. Nonparametric Stat.14(6), 655-664.10.1080/10485250215320 · Zbl 1013.62038 · doi:10.1080/10485250215320
[19] [19]GeyerC. J. & MøllerJ. (1994) Simulation procedures and likelihood inference for spatial point processes. Scand. J. Stat.21(4), 359-373. · Zbl 0809.62089
[20] [20]GuanY. (2007) A least-squares cross-validation bandwidth selection approach in pair correlation function estimations. Stat. Probab. Lett.77(18), 1722-1729.10.1016/j.spl.2007.04.016 · Zbl 1129.62027 · doi:10.1016/j.spl.2007.04.016
[21] [21]GuanY. (2008) On consistent nonparametric intensity estimation for inhomogeneous spatial point processes. J. Am. Stat. Assoc.103(483), 1238-1247.10.1198/016214508000000526 · Zbl 1205.62139 · doi:10.1198/016214508000000526
[22] [22]GuélonT., MathiasJ. D. & StoodleyP. (2011) Advances in biofilm mechanics. In: H.-C.Flemming, J.Wingender, U.Szewzyk (editors), Biofilm Highlights, Springer, pp. 111-139.10.1007/978-3-642-19940-0 · doi:10.1007/978-3-642-19940-0
[23] [23]Guizar-SicairosM. & Gutiérrez-VegaJ. C. (2004) Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields. J. Opt. Soc. Am. A21(1), 53-58.10.1364/JOSAA.21.000053 · doi:10.1364/JOSAA.21.000053
[24] [24]HallP. & MarronJ. S. (1991) Local minima in cross-validation functions. J. R. Stat. Soc. Ser. B (Methodological)53(1), 245-252. · Zbl 0800.62216
[25] [25]HammondJ. F., StewartE. J., YoungerJ. G., SolomonM. J. & BortzD. 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. · Zbl 1356.76052
[26] [26]HansenJ. P. & McDonaldI. R. (1990) Theory of Simple Liquids, Elsevier, London, UK. · Zbl 0756.00004
[27] [27]HardleW., MarronJ. S. & WandM. P. (1990) Bandwidth choice for density derivatives. J. R. Stat. Ser. B (Methodological)52(1), 223-232. · Zbl 0699.62036
[28] [28]JonesM. C. (1993) Simple boundary correction for kernel density estimation. Stat. Comput.3(3), 135-146.10.1007/BF00147776 · doi:10.1007/BF00147776
[29] [29]KerscherM., SzapudiI. & SzalayA. S. (2000) A comparison of estimators for the two-point correlation function. Astrophys. J. Lett.535(1), L13.10.1086/312702 · doi:10.1086/312702
[30] [30]LandyS. D. & SzalayA. S. (1993) Bias and variance of angular correlation functions.Astrophys. J.412, 64-71.10.1086/172900 · doi:10.1086/172900
[31] [31]LaspidouC. S. & RittmannB. E. (2004) Modeling the development of biofilm density including active bacteria, inert biomass, and extracellular polymeric substances. Water Res.38(14), 3349-3361.10.1016/j.watres.2004.04.037 · doi:10.1016/j.watres.2004.04.037
[32] [32]LovettR., MouC. Y. & BuffF. P. (1976) The structure of the liquid-vapor interface. J. Chem. Phys. 65, 2377.10.1063/1.433352 · doi:10.1063/1.433352
[33] [33]MollerJ. & WaagepetersenR. P. (2003) Statistical Inference and Simulation for Spatial Point Processes, CRC Press, Boca Raton, FL. · Zbl 1039.62089
[34] [34]OrnsteinL. S. & ZernikeF. (1914) The influence of accidental deviations of density on the equation of state. Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings19(2), 1312-1315.
[35] [35]ParzenE. (1962) On estimation of a probability density function and mode. Ann. Math. Stat.33(3), 1065-1076.10.1214/aoms/1177704472 · Zbl 0116.11302 · doi:10.1214/aoms/1177704472
[36] [36]PavlovskyL., YoungerJ. G. & SolomonM. J. (2013) In situ rheology of Staphylococcus epidermidis bacterial biofilms. Soft Matter9(1), 122-131.10.1039/C2SM27005F · doi:10.1039/C2SM27005F
[37] [37]RipleyB. D. (1991) Statistical Inference for Spatial Processes, Cambridge University Press. · Zbl 0782.62091
[38] [38]RosenblattM.et al. (1956) Remarks on some nonparametric estimates of a density function. Ann. Math. Stat.27(3), 832-837.10.1214/aoms/1177728190 · Zbl 0073.14602 · doi:10.1214/aoms/1177728190
[39] [39]SilvermanB. W. (1981) Using Kernel density estimates to investigate multimodality. J. R. Stat. Soc.43(1), 97-99.
[40] [40]SobczykK. & KirknerD. J. (2012) Stochastic Modeling of Microstructures, Springer Science & Business Media, Boston, MA. · Zbl 1013.60073
[41] [41]StewartE. J., GanesanM., YoungerJ. G. & SolomonM. J. (2015) Artificial biofilms establish the role of matrix interactions in Staphylococcal biofilm assembly and disassembly. Sci. Rep.5, 13081; doi: 10.1038/srep13081. · doi:10.1038/srep13081
[42] [42]StewartE. J., SatoriusA. E., YoungerJ. G. & SolomonM. J. (2013) Role of environmental and antibiotic stress on Staphylococcus epidermidis biofilm microstructure. Langmuir29(23), 7017-7024.10.1021/la401322k · doi:10.1021/la401322k
[43] [43]StotskyJ. A., HammondJ. F., PavlovskyL., StewartE. J., YoungerJ. G., SolomonM. J. & BortzD. 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, 204-222.10.1016/j.jcp.2016.04.027 · Zbl 1349.76536 · doi:10.1016/j.jcp.2016.04.027
[44] [44]StoyanD., BertramU. & WendrockH. (1993) Estimation variances for estimators of product densities and pair correlation functions of planar point processes. Ann. Inst. Stat. Math.45(2), 211-221.10.1007/BF00775808 · Zbl 0777.62084 · doi:10.1007/BF00775808
[45] [45]StoyanD., KendallW. S. & MeckeJ. (1995) Stochastic Geometry and its Applications, Akademie-Verlag, Berlin. · Zbl 0838.60002
[46] [46]SudarsanR., GhoshS., StockieJ. M. & EberlH. J. (2016) Simulating biofilm deformation and detachment with the immersed boundary method. Commun. Comput. Phys.19(3), 682-732.10.4208/cicp.161214.021015a · Zbl 1373.76014 · doi:10.4208/cicp.161214.021015a
[47] [47]SzapudiI. & SzalayA. S. (1998) A new class of estimators for the n-point correlations. Astrophys. J. Lett.494(1), L41.10.1086/311146 · doi:10.1086/311146
[48] [48]TorquatoS. (2013) Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Vol. 16, Springer Science & Business Media, New York. · Zbl 0988.74001
[49] [49]TruskettT. M., TorquatoS. & DebenedettiP. G. (1998) Density fluctuations in many-body systems. Phys. Rev. E58(6), 7369.10.1103/PhysRevE.58.7369 · doi:10.1103/PhysRevE.58.7369
[50] [50]VoG. D., BrindleE. & HeysJ. (2010) An experimentally validated immersed boundary model of fluid – biofilm interaction. Water Sci. Technol.61(12), 3033-3040.10.2166/wst.2010.213 · doi:10.2166/wst.2010.213
[51] [51]WandM. P. & JonesM. C. (1993) Comparison of smoothing parameterizations in bivariate kernel density estimation. J. Am. Stat. Assoc.88(422), 520-528.10.1080/01621459.1993.10476303 · Zbl 0775.62105 · doi:10.1080/01621459.1993.10476303
[52] [52]WróbelJ. K., CortezR. & FauciL. (2014) Modeling viscoelastic networks in stokes flow. Phys. Fluids (1994-present)26(11), 113102.10.1063/1.4900941 · Zbl 1323.76132 · doi:10.1063/1.4900941
[53] [53]YeongC. L. Y. & TorquatoS. (1998) Reconstructing random media. Phys. Rev. E57(1), 495.10.1103/PhysRevE.57.495 · doi:10.1103/PhysRevE.57.495
[54] [54]ZhangT., CoganN. G. & WangQ. (2008) Phase field models for biofilms. I. Theory and one-dimensional simulations. SIAM J. Appl. Math.69(3), 641-669.10.1137/070691966 · Zbl 1186.92015 · doi:10.1137/070691966
[55] [55]ZhangT., CoganN. G. & WangQ. (2008) Phase field models for biofilms. ii. 2-d numerical simulations of biofilm-flow interaction. Commun. Comput. Phys4(1), 72-101. · Zbl 1365.92024
[56] [56]ZhaoJ., ShenY., HaapasaloM., WangZ. & WangQ. (2016) A 3d numerical study of antimicrobial persistence in heterogeneous multi-species biofilms.J. Theor. Biol.392, 83-98.10.1016/j.jtbi.2015.11.010 · Zbl 1347.92048 · doi:10.1016/j.jtbi.2015.11.010
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