Mathematical Modeling of Chemotactic Bacterial Transport through a Two-Dimensional Heterogeneous Porous Medium

The impact of bacterial chemotaxis on in situ ground-water bioremediation remains an unanswered question. Although bacteria respond to chemical gradients in aqueous environments and under no-flow conditions, it is unclear whether they can also respond in porous media with advective flow to improve overall contaminant degradation. The effect of chemotaxis is most profound in regions with sharp chemical gradients, most notably around residual nonaqueous phase liquid (NAPL) ganglia and surrounding clay lenses or aquitards with trapped contamination. The purpose of this study is to simulate bacterial transport through a two-dimensional subsurface environment, containing one region of low permeability with trapped contaminant surrounded above and below by two regions of higher permeability. Using mathematical predictions of the effect of pore size on measured bacterial transport parameters, the authors observe a 50% decrease in both motility and chemotaxis in the finer-grained, low-permeability porous medium. The authors simulate how chemotaxis affects bacterial migration to the contaminated region under various flow and initial conditions. Results indicate that bacteria traveling through a high-permeability region with advective flow can successfully migrate toward and accumulate around a contaminant diffusing from a lower permeability region.     

Residence time calculation for chemotactic bacteria within porous media

Local chemical gradients can have a significant impact on bacterial population distributions within subsurface environments by evoking chemotactic responses. These local gradients may be created by consumption of a slowly diffusing nutrient, generation of a local food source from cell lysis, or dissolution of nonaqueous phase liquids trapped within the interstices of a soil matrix. We used a random walk simulation algorithm to study the effect of a local microscopic gradient on the swimming behavior of bacteria in a porous medium. The model porous medium was constructed using molecular dynamics simulations applied to a fluid of equal-sized spheres. The chemoattractant gradient was approximated with spherical symmetry, and the parameters for the swimming behavior of soil bacterium Pseudomonas putida were based on literature values. Two different mechanisms for bacterial chemotaxis, one in which the bacteria responded to both positive and negative gradients, and the other in which they responded only to positive gradients, were compared. The results of the computer simulations showed that chemotaxis can increase migration through a porous medium in response to microscopic-scale gradients. The simulation results also suggested that a more significant role of chemotaxis may be to increase the residence time of the bacteria in the vicinity of an attractant source.     

A Multiple-Relaxation-Time Lattice-Boltzmann Model for Bacterial Chemotaxis: Effects of Initial Concentration,Diffusion, and Hydrodynamic Dispersion on Traveling Bacterial Bands

Bacterial chemotaxis can enhance the bioremediation of contaminants in aqueous and subsurface environments if the contaminant is a chemoattractant that the bacteria degrade. The process can be promoted by traveling bands of chemotactic bacteria that form due to metabolism-generated gradients in chemoattractant concentration. We developed a multiple-relaxation-time (MRT) lattice-Boltzmann method (LBM) to model chemotaxis, because LBMs are well suited to model reactive transport in the complex geometries that are typical for subsurface porous media. This MRT-LBM can attain a better numerical stability than its corresponding single-relaxation-time LBM. We performed simulations to investigate the effects of substrate diffusion, initial bacterial concentration, and hydrodynamic dispersion on the formation, shape, and propagation of bacterial bands. Band formation requires a sufficiently high initial number of bacteria and a small substrate diffusion coefficient. Uniform flow does not affect the bands while shear flow does. Bacterial bands can move both upstream and downstream when the flow velocity is small. However, the bands disappear once the velocity becomes too large due to hydrodynamic dispersion. Generally bands can only be observed if the dimensionless ratio between the chemotactic sensitivity coefficient and the effective diffusion coefficient of the bacteria exceeds a critical value, that is, when the biased movement due to chemotaxis overcomes the diffusion-like movement due to the random motility and hydrodynamic dispersion.     

Lattice-Boltzmann model for bacterial chemotaxis

We present a new numerical approach for modeling bacterial chemotaxis and the fate and transport of a chemoattractant in bulk liquids. This Lattice-Boltzmann method represents the microorganisms and the chemoattractant by quasi-particles that move, collide, and react with each other on a two-dimensional numerical lattice. We use the model to simulate traveling bands of bacteria along self-generated gradients in substrate concentration in bulk liquids. Particularly, we simulate Pseudomonas putida that respond chemotactically to naphthalene dissolved in water. We find that only a fraction of a bacterial slug injected into a domain containing the chemoattractant at constant concentration forms a traveling band as the slug length exceeds a critical value. An expanding bacterial ring forms as one injects a droplet of bacteria into a two-dimensional domain.     

Bacterial chemotaxis transverse to axial flow in a microfluidic channel

Swimming bacteria sense and respond to chemical signals in their environment. Chemotaxis is the directed migration of a bacterial population toward increasing concentrations of a chemical that they perceive to be beneficial to their survival. Bacteria that are indigenous to groundwater environments exhibit chemotaxis toward chemical contaminants such as hydrocarbons, which they are also able to degrade. This phenomenon may facilitate bioremediation processes by bringing bacteria into closer proximity to these contaminants. A microfluidic device was assembled to study chemotaxis transverse to advective flow. Using a T-shaped channel design (T-sensor), two fluid streams were brought into contact by impinging flow. They then flowed adjacent to each other along a transparent channel. An advantage to this design is that it allows real-time visualization of bacterial distributions within the channel. Under laminar flow conditions a chemotactic driving force was created perpendicular to the direction of flow by diffusion of the chemical attractant from one input stream to the other. A comparison of the chemotactic band behavior in the absence and presence of flow showed that fluid velocity did not significantly impede chemotactic migration in the transverse direction.     

Transport of Algal Cells in Hyporheic Sediments of the River Elbe (Germany)

The advective transport of algal cells into the interstices of the hyporheic zone of the River Elbe was spatially and temporally heterogenous. Even deep sediment layers were reached by large phytoplankton species. Therefore, it is suggested that (i) the advective interstitial transport patterns vary between different algal sizes and morphotypes and (ii) sediment characteristics, expressed by the permeability coefficient k_f of porous media, affect retention and retardation of surface water algae during subsurface transport. The transport behaviour of different green algae (Chlorella sp., Scenedesmus acuminatus, Desmodesmus communis, and Pediastrum duplex) and algal sized microspheres was tested in flow‐through column experiments with hyporheic sediments. The algal cell transport was directly related to the permeability of the column sediments. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A microscale model of bacterial and biofilm dynamics in porous media

A microscale model for the transport and coupled reaction of microbes and chemicals in an idealized two-dimensional porous media has been developed. This model includes the flow, transport, and bioreaction of nutrients, electron acceptors, and microbial cells in a saturated granular porous media. The fluid and chemicals are represented as a continuum, but the bacterial cells and solid granular particles are represented discretely. Bacterial cells can attach to the particle surfaces or be advected in the bulk fluid. The bacterial cells can also be motile and move preferentially via a run and tumble mechanism toward a chemoattractant. The bacteria consume oxygen and nutrients and alter the profiles of these chemicals. Attachment of bacterial cells to the soil matrix and growth of bacteria can change the local permeability. The coupling of mass transport and bioreaction can produce spatial gradients of nutrients and electron acceptor concentrations. We describe a numerical method for the microscale model, show results of a convergence study, and present example simulations of the model system.     

Mathematical model for characterization of bacterial migration through sand cores

The migration of chemotactic bacteria in liquid media has previously been characterized in terms of two fundamental transport coefficients-the random motility coefficient and the chemotactic sensitivity coefficient. For modeling migration in porous media, we have shown that these coefficients which appear in macroscopic balance equations can be replaced by effective values that reflect the impact of the porous media on the swimming behavior of individual bacteria. Explicit relationships between values of the coefficients in porous and liquid media were derived. This type of quantitative analysis of bacterial migration is necessary for predicting bacterial population distributions in subsurface environments for applications such as in situ bioremediation in which bacteria respond chemotactically to the pollutants that they degrade.We analyzed bacterial penetration times through sand columns from two different experimental studies reported in the literature within the context of our mathematical model to evaluate the effective transport coefficients. Our results indicated that the presence of the porous medium reduced the random motility of the bacterial population by a factor comparable to the theoretical prediction. We were unable to determine the effect of the porous medium on the chemotactic sensitivity coefficient because no chemotactic response was observed in the experimental studies. However, the mathematical model was instrumental in developing a plausible explanation for why no chemotactic response was observed. The chemical gradients may have been too shallow over most of the sand core to elicit a measurable response. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 53: 487-496, 1997.     

A tactile response in Staphylococcus aureus

It is well established that bacteria are able to respond to temporal gradients (e.g., by chemotaxis). However, it is widely held that prokaryotes are too small to sense spatial gradients. This contradicts the common observation that the vast majority of bacteria live on the surface of a solid substrate (e.g., as a biofilm). Herein we report direct experimental evidence that the nonmotile bacterium Staphylococcus aureus possesses a tactile response, or primitive sense of touch, that allows it to respond to spatial gradients. Attached cells recognize their substrate interface and localize adhesins toward that region. Braille-like avidity maps reflect a cell's biochemical sensory response and reveal ultrastructural regions defined by the actual binding activity of specific proteins.     

Ecology and Physics of Bacterial Chemotaxis in the Ocean

Summary: Intuitively, it may seem that from the perspective of an individual bacterium the ocean is a vast, dilute, and largely homogeneous environment. Microbial oceanographers have typically considered the ocean from this point of view. In reality, marine bacteria inhabit a chemical seascape that is highly heterogeneous down to the microscale, owing to ubiquitous nutrient patches, plumes, and gradients. Exudation and excretion of dissolved matter by larger organisms, lysis events, particles, animal surfaces, and fluxes from the sediment-water interface all contribute to create strong and pervasive heterogeneity, where chemotaxis may provide a significant fitness advantage to bacteria. The dynamic nature of the ocean imposes strong selective pressures on bacterial foraging strategies, and many marine bacteria indeed display adaptations that characterize their chemotactic motility as “high performance” compared to that of enteric model organisms. Fast swimming speeds, strongly directional responses, and effective turning and steering strategies ensure that marine bacteria can successfully use chemotaxis to very rapidly respond to chemical gradients in the ocean. These fast responses are advantageous in a broad range of ecological processes, including attaching to particles, exploiting particle plumes, retaining position close to phytoplankton cells, colonizing host animals, and hovering at a preferred height above the sediment-water interface. At larger scales, these responses can impact ocean biogeochemistry by increasing the rates of chemical transformation, influencing the flux of sinking material, and potentially altering the balance of biomass incorporation versus respiration. This review highlights the physical and ecological processes underpinning bacterial motility and chemotaxis in the ocean, describes the current state of knowledge of chemotaxis in marine bacteria, and summarizes our understanding of how these microscale dynamics scale up to affect ecosystem-scale processes in the sea.     

Modeling of microorganisms transport in a cylindrical pore

A mathematical model accounting for key parameters as microbial propagation, metabolite formation, dispersion, microbial chemotaxis and water flooding has been proposed to simulate the transport of microorganisms and their metabolites in a cylindrical pore with oil adhered to its inside surface. The model focuses on the transport and the concentration distributions of microorganisms and their metabolites in the cylindrical pore, especially the concentrations that on the oil-water interface. Results from the present model indicate that microorganisms and their metabolites assembled on the oil-water interface during the water flooding process, and the concentration gradients of microorganisms and their metabolites from the pore center region up to the oil-water interface in radial direction of the cylindrical pore were consequently formed. Equilibrium concentrations of microorganisms and their metabolites in the cylindrical pore were obtained when water flow rate within a certain scope, and there existed a critical water flow rate at which the maximum equilibrium concentration of microorganisms on the oil-water interface was developed. Investigations carried out in this study may provide better understanding on the transport mechanism of microorganisms in porous media.     

Effect of bacterial heterogeneity on adhesion to uniform collectors by monoclonal populations

Abstract Transport of bacteria over significant distances through aquifer sediments occurs primarily among bacteria with low affinity for sediment materials. Bacterial affinity for a uniform collector surface has been represented quantitatively by a collision efficiency (α), defined as the fraction of colliding cells that adhere to the collector surface. Using a new method for estimating α during advective transport of monoclonal bacterial populations through a uniform bed of 40-μm borosilicate glass spheres, we found that α decreased 10-fold over a bed depth of only 1 cm. Depth-dependent differences in α were not related to variation in bacterial size or intra-strain genetic variation. Intra-population heterogeneity in biocolloid-collector affinity may be important determinant of subsurface bacterial transport characteristics, with critical implications for pathogen transport and dispersal of bacteria for the remediation of hazardous waste.     

Continuous-Flow Capillary Assay for Measuring Bacterial Chemotaxis

Bacterial chemotaxis may have a significant impact on the structure and function of bacterial communities. Quantification of chemotactic motion is necessary to identify chemoeffectors and to determine the bacterial transport parameters used in predictive models of chemotaxis. When the chemotactic bacteria consume the chemoeffector, the chemoeffector gradient to which the bacteria respond may be significantly perturbed by the consumption. Therefore, consumption of the chemoeffector can confound chemotaxis measurements if it is not accounted for. Current methods of quantifying chemotaxis use bacterial concentrations that are too high to preclude chemoeffector consumption or involve ill-defined conditions that make quantifying chemotaxis difficult. We developed a method of quantifying bacterial chemotaxis at low cell concentrations (~10⁵ CFU/ml), so metabolism of the chemoeffector is minimized. The method facilitates quantification of bacterial-transport parameters by providing well-defined boundary conditions and can be used with volatile and semivolatile chemoeffectors.     

Chemotaxis increases vertical migration and apparent transverse dispersion of bacteria in a bench-scale microcosm

The success of in situ bioremediation is often limited by the inability to bring bacteria in contact with the pollutant, which they will degrade. A bench-scale model aquifer was used to evaluate the impact of chemotaxis on the migration of bacteria toward the source of a chemical pollutant. The model was packed with sand and aqueous media was pumped across horizontally, simulating groundwater flow in a homogenous aquifer. A vertical gradient in chemoattractant was created by either a continuous injection of sodium benzoate or a pulse injection of sodium acetate. A pulse of chemotactic Pseudomonas putida F1 or a non-chemotactic mutant of the same species was injected below the attractant. The eluent was sampled at the microcosm outlet to generate vertical concentration profiles of the bacteria and chemoattractant. Moment analysis was used to determine the center and variance of the bacterial profiles. The center of the chemotactic bacterial population was located at an average of 0.74 ± 0.07 cm closer to the level at which the chemoattractant was injected than its non-chemotactic mutant in benzoate experiments (P < 0.015) and 0.4 ± 0.2 cm closer in acetate experiments (P < 0.05). The transverse dispersivity of the chemotactic bacteria was 4 ± 1 × 10(-3) cm higher in benzoate experiments than the transverse dispersivity of the non-chemotactic mutant and 1 ± 2 × 10(-3) cm higher in acetate experiments. These results underscore the contribution of chemotaxis to improve transport of bacteria to contaminant sources, potentially enhancing the effectiveness of in situ bioremediation.     

Continuous-flow capillary assay for measuring bacterial chemotaxis

Bacterial chemotaxis may have a significant impact on the structure and function of bacterial communities. Quantification of chemotactic motion is necessary to identify chemoeffectors and to determine the bacterial transport parameters used in predictive models of chemotaxis. When the chemotactic bacteria consume the chemoeffector, the chemoeffector gradient to which the bacteria respond may be significantly perturbed by the consumption. Therefore, consumption of the chemoeffector can confound chemotaxis measurements if it is not accounted for. Current methods of quantifying chemotaxis use bacterial concentrations that are too high to preclude chemoeffector consumption or involve ill-defined conditions that make quantifying chemotaxis difficult. We developed a method of quantifying bacterial chemotaxis at low cell concentrations ( approximately 10(5) CFU/ml), so metabolism of the chemoeffector is minimized. The method facilitates quantification of bacterial-transport parameters by providing well-defined boundary conditions and can be used with volatile and semivolatile chemoeffectors.     

Transport of bacteria in porous media: I. An experimental investigation

The convective transport of concentrated suspension of bacteria in porous media is of interest for several processes such as microbial enhanced oil recovery and in situ bioremediation. The parameters which affect the transport of the bacterium Bacillus licheniformis JF-2, a candidate microorganism for microbial enhanced oil recovery, were investigated experimentally in sandpacks. Bacteria retention and permeability reduction occurred primarily in the first few centimeters upon entering the porous medium. In downstream sections of the sandpack, the permeability reduction was low, even in cases in which high cell concentrations (10(8) cfu/mL) were detected in the effluent. The effect of (i) addition of a dispersant, (ii) linear velocity of injection, (iii) cell concentration, (iv) salinity (v) temperature, and (vi) the presence of a residual oleic phase were determined experimentally. A lower reduction in permeability and a higher effluent bacterial concentration were obtained in the presence of dispersant, high injection velocities, low salinities, and at a higher temperature. Macroscopic measurements at different linear velocities and in the presence or absence of dispersants suggest that the formation of reversible microaggregates and multiparticle hydrodynamic exclusion may be the primary mechanisms for bacterial retention and permeability reduction. (c) 1994 John Wiley & Sons, Inc.     

Effects of leukocyte random motility and chemotaxis in tissue inflammatory response.

A rapidly growing body of experimental evidence indicates that defects in leukocyte motility and chemotactic response correlate with increased susceptibility to and severity of bacterial infection in tissue. While this is understandable in qualitative terms, the sensitivity of the correlation is remarkable.In the present study, a theoretical analysis has been developed to relate the dynamics of bacterial growth to the growth and transport parameters of bacteria and leukocytes in tissue. The model considers a local tissue region in the vicinity of a venule and applies continuum unsteady state species conservation equations to the bacterial population, the phagocytic leukocytes, and a chemotactically active chemical mediator assumed to be produced by the bacteria. The analysis quantifies the effects of key parameters, such as leukocyte random motility and chemotactic coefficients, phagocytic and growth rate constants, and leukocyte vessel wall permeability, upon host ability to eliminate the bacteria.As an example, the model's predictions are compared to experimental results correlating inhibition of leukocyte chemotaxis by hemoglobin with its adjuvant action in experimental peritoneal infection by E. coli.     

Computational analysis of confined jet flow and mass transport in a blind tube.

A computational analysis of confined nonimpinging jet flow in a blind tube is performed as an initial investigation of the underlying fluid and mass transport mechanics of tracheal gas insufflation. A two-dimensional axisymmetric model of a laminar steady jet flow into a concentric blind-end tube is put forth and the governing continuity, momentum, and convection-diffusion equations are solved with a finite element code. The effects of the jet diameter based Reynolds number (Re(j)), the ratio of the jet-to-outer tube diameters (epsilon), and the Schmidt number (Sc) are evaluated with the determined velocity and contaminant concentration fields. The normalized penetration depth of the jet is found to increase linearly with increasing Re(j) for epsilon = O(0.1). For a given epsilon, a ring vortex that develops is observed to be displaced downstream and radially outward from the jet tip for increasing Re(j). The axial shear stress profile along the inside wall of the outer tube possesses regions of fixed shear stress in addition to a local minimum and maximum in the vicinity of the jet tip. Corresponding regions of axial shear stress gradients exist between the fixed shear stress regions and the local extrema. Contaminant concentration gradients develop across the ring vortex indicating the inward diffusion of contaminant into the jet flow. For fixed epsilon and Sc and Re(j) approximately 900, normalized contaminant flow rate is observed to be approximately twice that of simple diffusion. This model predicts modest net axial contaminant transport enhancement due to convection-diffusion interaction in the region of the ring vortex.     

Transport of bacteria in porous media: II. A model for convective Transport and growth

A model is presented for the coupled processes of bacterial growth and convective transport of bacteria has been modeled using a fractional flow approach. The various mechanisms of bacteria retention can be incorporated into the model through selection of an appropriate shape of the fractional flow curve. Permeability reduction due to pore plugging by bacteria was simulated using the effective medium theory. In porous media, the rates of transport and growth of bacteria, the generation of metabolic products, and the consumption of nutrients are strongly coupled processes. Consequently, the set of governing conservation equations form a set of coupled, nonlinear partial differential equations that were solved numerically. Reasonably good agreement between the model and experimental data has been obtained indicating that the physical processes incorporated in the model are adequate. The model has been used to predict the in situ transport and growth of bacteria, nutrient consumption, and metabolite production. It can be particularly useful in simulating laboratory experiments and in scaling microbial-enhanced oil recovery or bioremediation processes to the field. (c) 1994 John Wiley & Sons, Inc.