Effect of α-stable sorptive waiting times on microbial transport in microflow cells

Abstract

The interaction of bacteria in the fluid phase with pore walls of a porous material involves a wide range of effective reaction times which obey a diversity of substrate-bacteria adhesion conditions, and adhesive mechanisms. For a transported species, this heterogeneity in sorption conditions occurs both in time and space. Modern experimental methods allow one to measure adhesive reaction times of individual bacteria. This detailed information may be incorporated into nonequilibrium transport-sorption models that capture the heterogeneity in reaction times caused by varying chemical conditions. We have carried out particle (Brownian dynamic) simulations of adhesive, self-motile bacteria convected between two infinite plates as a model for a microflow cell. The adhesive heterogeneity is included by introducing adhesive reaction time (understood as time spent at a solid boundary once the particle collides against it) as a random variable that can be infinite (irreversible sorption) or vary over a wide range of values. This is made possible by treating this reaction time random variable as having an alpha-stable probability distribution whose properties (e.g., infinite moments and long tails) are distinctive from the standard exponential distribution commonly used to model reversible sorption. In addition, the alpha-stable distribution is renormalizable and hence upscalable to complex porous media. Simulations are performed in a pressure-driven microflow cell. Bacteria motility (driven by an effective Brownian force) acts as a dispersive component in the convective field. Upon collision with the pore wall, bacteria attachment or detachment occurs. The time bacteria spend at the wall varies over a wide range of time scales. This model has the advantage of being parsimonious, that is, involving very few parameters to model complex irreversible or reversible adhesion in heterogeneous environments. It is shown that, as in Taylor dispersion, the ratio of the channel half width b to the Brownian bacteria motility coefficient (D0 or dispersion coefficient) t(b)=b(2)/D(0) controls the different adhesion regimes along with the value of alpha. Universal scalings (with respect to dimensionless time t(*)=t/t(b)) for the mean position, <X>=V(*)(eff)t(theta)(*), and mean-square displacement, <DeltaX2>=D(*)(eff)t(gamma)(*) exist for long-time dispersion and the coefficients were obtained. The model can account for a great many sorptive processes including reversible and irreversible sorption, and sub- and superdispersive regimes with just a few parameters.

Microfluidic aspects of adhesive microbial dynamics: A numerical exploration of flow-cell geometry, Brownian dynamics, and sticky boundaries

“…We study several different, but realistic, flow-cell geometries and allow reversible adhesion. This differs from previous work by Bonilla and Cushman [4] which involved flow between two infinite planes, but allowed more general adhesion including irreversible adhesion and a different model. The microbes may be non-spherical.…”

“…Most hydrodynamic aspects thought to be relevant to microbe transport at the pore scale have been discussed in Section 2. Although the simulations in this contribution have been limited to reversible adhesion, a more general sticking boundary has been modeled elsewhere [4]. The simulations allow one to incorporate biochemical processes acting at submicron and nanoscales through a hypothetical stochastic model supported by both theory and experiments.…”

“…However, the qualitative comparable behavior suggests bimodal behavior can average out to one of these fractional probability models (such as the Weibull) with long tails. Bonilla and Cushman [4] have used this idea and constructed a model with a-stable sticking behavior. This long tail behavior has been observed in column experiments designed to study desorption of tracer solutes [25].…”

“…However, dealing with various boundary conditions is not a trivial exercise when studying Brownian particles. While sticky Brownian motion has been studied in the mathematical literature [34,35], others have not applied the results to reversible sorption until recently [4].…”

Scaling the fractional advective–dispersive equation for numerical evaluation of microbial dynamics in confined geometries with sticky boundaries

“…Hence for dilute concentrations of microbes swimming through a given fluid at standard conditions in an experimental flow cell, a probability distribution can be applied to describe the motility pattern. It has been observed that Levy motion defined by a-stable probability distributions of increments best describes the runs and tumbles of microbial motion [6,18,19]. The mixing measure within the definition of a Levy process may be employed to account for microbe food (energy) sources by skewing the movement along a gradient.…”