E. coli fate and transport in the Happel sphere-in-cell model

Abstract Rates of mass and gene transfer reactions involving biotic phases are often expressed as proportional to local number densities of bacteria. When the reactions involve attached bacteria, reaction rates depend on local densities of bacteria attached to surfaces. Such may be the case with microbially-facilitated redox reactions involving mineral electron donors and mineral electron receivers (e.g., Sani RK, Peyton BM, Amonette JE, Dohnalkova A. Reoxidation of uranium in the presence of iron(III)-(hydr)oxides under sulfate reducing conditions. Environ Sci Technol 2005;39:2059–66), biofilm formation induced by quorum sensing (Purevdorj B, Costerton JW, Stoodley P. Influence of hydrodynamics and cell signaling on the structure and behavior of Pseudomonas aeruginosa Biofilms. Appl Environ Microbiol 2002;68(9):4457–64) and horizontal gene transfer among attached phase bacteria (Beaudoin DL, Bryers JD, Cunningham AB, Peretti SW. Mobilization of broad host range plasmid from Pseudomonas putida to established biofilm of Bacillus azotoformans. I. Experiments. Biotech Bioeng 1998a;57(3):272–79; Beaudoin DL, Bryers JD, Cunningham AB, Peretti SW. Mobilization of broad host range plasmid from Pseudomonas putida to established biofilm of Bacillus azotoformans. II. Modeling Biotech Bioeng 1998b;57(3):280–86). Here we use the conceptual Happel sphere-in-cell model to determine the microscopic distribution of attached bacteria on idealized spherical grains of porous media, assuming azimuthal symmetry. We extend a Lagrangian model of colloid filtration to investigate the effects of motility of Escherichia coli on attachment rate and on the attachment distribution as a function of location on grain surface. The hydrodynamics of the Happel model is implicitly 3D and represented in 2D polar coordinates under the assumption of axisymmetric flow, while the motility of the E. coli cells is explicitly 3D. The model incorporates the fate and transport processes of colloid filtration theory in the particle tracking approach of Nelson and Ginn (Nelson KE, Ginn TR. Theoretical investigation of bacterial chemotaxis in porous media. Langmuir 2001;17:5636–45; Nelson KE, Ginn TR. Colloid filtration theory and the Happel sphere-in-cell model revisited with direct numerical simulation. Langmuir 2005;21:2173–84) and includes statistics reported in Biondi et al. (Biondi SA, Quinn JA, Goldfine H. Random motility of swimming bacteria in restricted geometries. AIChE J 1998;44(8):1923–29) to describe the “run & tumble” behavior of individual E. coli cells. Incorporation of the motility allowed for comparative modeling of diffusion-controlled vs. motility-controlled transport and attachment in the Happel sphere. It was found that, whereas the Levy distribution is often associated with the distribution of run lengths for this species, the data of Biondi is better fit by a Gamma distribution due to the low frequency of short runs, and that the turn distribution is non-uniform. Further, the motility mean free path is much larger than that corresponding to Brownian diffusion and the attachment frequency is higher in the case of motile cells as a result. However the distribution of attached cells is similar in both motile and diffusive cases.