Determination of Effective Transport Coefficients for Bacterial Migration in Sand Columns

A well-characterized experimental system was designed to evaluate the effect of porous media on macroscopic transport coefficients which are used to characterize the migration of bacterial populations. Bacterial density profiles of Pseudomonas putida PRS2000 were determined in the presence and absence of a chemical attractant (3-chlorobenzoate) gradient within sand columns having a narrow distribution of particle diameters. These experimental profiles were compared with theoretical predictions to evaluate the macroscopic transport coefficients. The effective random motility coefficient, used to quantify migration due to a random process in a porous medium, decreased nearly 20-fold as grain size in the columns decreased from 800 to 80 (mu)m. The effective random motility coefficient (mu)(infeff) was related to the random motility coefficient (mu), measured in a bulk aqueous system, according to (mu)(infeff) = ((epsilon)/(tau))(mu) with porosity (epsilon) and tortuosity (tau). Over the times and distances examined in these experiments, bacterial density profiles were unaffected by the presence of an attractant gradient. Theoretical profiles with the aqueous phase value of the chemotactic sensitivity coefficient (used to quantify migration due to a directed process) were consistent with this result and suggested that any chemotactic effect on bacterial migration was below the detection limits of our assay.     

Growth rate effects on fundamental transport properties of bacterial populations

In many natural environments, bacterial populations experience suboptimal growth due to the competition with other microorganisms for limited resources. The chemotactic response provides a mechanism by which bacterial populations can improve their situation by migrating toward more favorable growth conditions. For bacteria cultured under suboptimal growth conditions, evidence for an enhanced chemotactic response has been observed previously. In this article, for the first time, we have quantitatively characterized this behavior in terms of two macroscopic transport coefficients, the random motility and chemotactic sensitivity coefficients, measured in the stopped-flow diffusion chamber assay. Escherichia coli cultured over a range of growth rates in a chemostat exhibits a dramatic increase in the chemotactic sensitivity coefficient for D-fucose at low growth rates, while the random motility coefficient remains relatively constant by comparison. The change in the chemotactic sensitivity coefficient is accounted for by an independently measured increase in the number of galactose-binding proteins which mediate the chemotactic signal. This result is consistent with the relationship between macroscopic and microscopic parameters for chemotaxis, which was proposed in the mathematical model of Rivero and co-workers. (c) 1993 John Wiley & Sons, Inc.     

Quantification of random motility and chemotaxis bacterial transport coefficients using individual-cell and population-scale assays

A number of individual-cell and population-scale assays have been introduced to quantify bacterial motility and chemotaxis. The transport coefficients reported in the literature, however, span several orders of magnitude, making it difficult to ascertain to what degree variations in bacterial species/strain, growth medium, growth and experimental conditions, and experiment type contribute to the reported differences in coefficient values. We quantified the random motility of Escherichia coli AW405 using the capillary assay, stopped-flow diffusion chamber (SFDC), and tracking microscope. We obtained good agreement for the random motility coefficient between these assays when using the same bacterial strain and consistent growth and experimental conditions. Chemotaxis of E. coli toward the attractant alpha-methylaspartate was quantified using the SFDC and capillary assay. Good agreement for the chemotactic sensitivity coefficient between the SFDC and the capillary assay was obtained across a limited attractant concentration range. Three different mathematical models were considered for analyzing capillary assay data to obtain a chemotactic sensitivity coefficient. These models differed by their treatment of the bacterial concentration in the chamber and the attractant concentration at the mouth. Results from our study indicate that the capillary assay, the most commonly used bacterial random motility and chemotaxis assay, can be used to accurately quantify bacterial transport coefficients over a limited range of attractant concentrations, provided experiments are performed carefully and appropriate mathematical models are used to interpret the experimental data.     

Quantitative analysis of experiments on bacterial chemotaxis to naphthalene

A mathematical model was developed to quantify chemotaxis to naphthalene by Pseudomonas putida G7 (PpG7) and its influence on naphthalene degradation. The model was first used to estimate the three transport parameters (coefficients for naphthalene diffusion, random motility, and chemotactic sensitivity) by fitting it to experimental data on naphthalene removal from a discrete source in an aqueous system. The best-fit value of naphthalene diffusivity was close to the value estimated from molecular properties with the Wilke-Chang equation. Simulations applied to a non-chemotactic mutant strain only fit the experimental data well if random motility was negligible, suggesting that motility may be lost rapidly in the absence of substrate or that gravity may influence net random motion in a vertically oriented experimental system. For the chemotactic wild-type strain, random motility and gravity were predicted to have a negligible impact on naphthalene removal relative to the impact of chemotaxis. Based on simulations using the best-fit value of the chemotactic sensitivity coefficient, initial cell concentrations for a non-chemotactic strain would have to be several orders of magnitude higher than for a chemotactic strain to achieve similar rates of naphthalene removal under the experimental conditions we evaluated. The model was also applied to an experimental system representing an adaptation of the conventional capillary assay to evaluate chemotaxis in porous media. Our analysis suggests that it may be possible to quantify chemotaxis in porous media systems by simply adjusting the model's transport parameters to account for tortuosity, as has been suggested by others.     

Measurement of bacterial random motility and chemotaxis coefficients: I. Stopped-flow diffusion chamber assay

Bacterial chemotaxis, the directed movement of a cell population in response to a chemical gradient, plays a critical role in the distribution and dynamic interaction of bacterial populations in nonmixed systems. Therefore, in order to make reliable predictions about the migratory behavior of bacteria within the environment, a quantitative characterization of the chemotactic response in terms of intrinsic cell properties is needed.The design of the stopped-flow diffusion chamber (SFDC) provides a well-characterized chemical gradient and reliable method for measuring bacterial migration behavior. During flow through the chamber, a step change in chemical concentration is imposed on a uniform suspension of bacteria. Once flow is stopped, diffusion causes a transient chemical gradient to develop, and bacteria respond by forming a band of high cell density which travels toward higher concentrations of the attractant. Changes in bacterial spatial distributions observed through light scattering are recorded on photomicrographs during a 10-min period. Computer-aided image analysis converts absorbance of the photographic negatives to a digital representation of bacterial density profiles. A mathematical model (part II) is used to quantitatively characterize these observations in terms of intrinsic cell parameters: a chemotactic sensitivity coefficient, mu(0), from the aggregate cell density accumulated in the band and a random motility coefficient, mu, from population dispersion in the absence of a chemical gradient.Using the SFDC assay and an individual-cell-based mathematical model, we successfully determined values for both of these population parameters for Escherichia coli K12 responding to fucose. The values obtained were mu = 1.1 +/- 0. 4 x 10(-5) cm(2)/s and chi(o) = 8 +/- 3 +/- 10(-5) cm(2)/s. We have demonstrated a method capable of determining these parameter values from the now validated mathematical model which will be useful for predicting bacterial migration in application systems.     

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.     

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.     

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.     

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.     

Chemotactic transport theory for neutrophil leukocytes

The simplest admissible phenomenological transport theory for the chemotactic migration of a population of neutrophil leukocytes is formulated along the lines of the original Keller-Segel model for bacterial chemotaxis, but with appropriate specialization of the motility and chemotactic flux coefficient to reflect their dependence on the local cytotaxin (chemoattractant) concentration, as observed experimentally by Wilkinson and other workers. By supplementing deductions from the governing transport equation with inferences from measurements and then reasoning both forwards and backwards, the functional forms of the motility and chemotactic flux coefficient can be established for any prescribed cytotaxin. This analysis is performed here with numerical details for casein, a cytotaxin which gives rise to a motility function with an increasing-then-decreasing form of dependence on the concentration and a chemotactic flux coefficient that is essentially constant with variations in the concentration. Three dimensionless numbers are associated with the chemotactic response of neutrophil leukocytes to casein.     

A simple expression for quantifying bacterial chemotaxis using capillary assay data: application to the analysis of enhanced chemotactic responses from growth-limited cultures.

An individual cell-based mathematical model of Rivero et al. provides a framework for determining values of the chemotactic sensitivity coefficient chi 0, an intrinsic cell population parameter that characterizes the chemotactic response of bacterial populations. This coefficient can theoretically relate the swimming behavior of individual cells to the resulting migration of a bacterial population. When this model is applied to the commonly used capillary assay, an approximate solution can be obtained for a particular range of chemotactic strengths yielding a very simple analytical expression for estimating the value of chi 0, [formula: see text] from measurements of cell accumulation in the capillary, N, when attractant uptake is negligible. A0 and A infinity are the dimensionless attractant concentrations initially present at the mouth of the capillary and far into the capillary, respectively, which are scaled by Kd, the effective dissociation constant for receptor-attractant binding. D is the attractant diffusivity, and mu is the cell random motility coefficient. NRM is the cell accumulation in the capillary in the absence of an attractant gradient, from which mu can be determined independently as mu = (pi/4t)(NRM/pi r2bc)2, with r the capillary tube radius and bc the bacterial density initially in the chamber. When attractant uptake is significant, a slightly more involved procedure requiring a simple numerical integration becomes necessary. As an example, we apply this approach to quantitatively characterize, in terms of the chemotactic sensitivity coefficient chi 0, data from Terracciano indicating enhanced chemotactic responses of Escherichia coli to galactose when cultured under growth-limiting galactose levels in a chemostat.     

Chemotactic migration of bacteria in porous media

Chemotactic migration of bacteria—their ability to direct multicellular motion along chemical gradients—is central to processes in agriculture, the environment, and medicine. However, current understanding of migration is based on studies performed in bulk liquid, despite the fact that many bacteria inhabit tight porous media such as soils, sediments, and biological gels. Here, we directly visualize the chemotactic migration of Escherichia coli populations in well-defined 3D porous media in the absence of any other imposed external forcing (e.g., flow). We find that pore-scale confinement is a strong regulator of migration. Strikingly, cells use a different primary mechanism to direct their motion in confinement than in bulk liquid. Furthermore, confinement markedly alters the dynamics and morphology of the migrating population—features that can be described by a continuum model, but only when standard motility parameters are substantially altered from their bulk liquid values to reflect the influence of pore-scale confinement. Our work thus provides a framework to predict and control the migration of bacteria, and active matter in general, in complex environments.     

Stopped-flow chamber and image analysis system for quantitative characterization of bacterial population migration: Motility and chemotaxis ofEscherichia coli K12 to fucose

The directed movement of a bacterial population in response to a chemical gradient is known as bacterial chemotaxis and plays a critical role in the distribution and dynamic interaction of bacterial populations. A quantitative characterization of the chemotactic response in terms of intrinsic cell properties is necessary for making reliable predictions about the migratory behavior of bacterial populations within the environment. The design of the stopped-flow diffusion chamber (SFDC) provides a well-characterized chemical gradient and reliable method for measuring bacterial migration behavior. During flow through the chamber a step change in the chemical concentration is imposed on a uniform suspension of bacteria. Once flow is stopped a transient chemical gradient forms due to diffusion; bacteria respond by forming a band of high cell density that travels toward higher concentrations of the attractant. Sequential observations of bacterial spatial distributions over a period of about ten minutes are recorded on photomicrographs. Computer-aided image analysis of the photographic negatives converts light-scattering information to a digital representation of the bacterial density profiles. A mathematical model is used to quantitatively characterize these observations in terms of intrinsic cell parameters: a chemotactic sensitivity coefficient, χ₀, from the aggregate cell density accumulated in the band and a random motility coefficient, μ₀, from population dispersion in the absence of a chemical gradient. Using the SFDC assay and an individual cell-based mathematical model we successfully determined values for both of these population parameters forEscherichia coli K12 responding to fucose. The values we obtained were μ₀=1.1 ± 0.4 x 10^-5 cm²/sec and χ₀=8 ± 3 x 10^-5 cm²/sec. These parameters will be useful for predicting population behavior in application systems such as biofilm development, population dynamics of genetically-engineered bacteria released into the environment, and in situ bioremediation technologies.     

A material-balance approach for modeling bacterial chemotaxis to a consumable substrate in the capillary assay

A mathematical model was developed to simulate the movement of chemotactic bacteria into and within a capillary tube containing a metabolizable chemoattractant. The model was based on a material balance that accounts for the transport of bacteria and chemoattractant as well as consumption of the substrate throughout the capillary assay system. By solving the model with a numerical method, it was possible to predict the concentration of substrate and bacteria at points within the capillary and throughout the chamber. The model was tested for its ability to simulate the results of capillary assay experiments performed with Pseudomonas putida G7 and one of its chemoattractants, naphthalene, under conditions wherein naphthalene consumption was expected to affect the flux of bacteria into the capillary. While variations in the chemotactic responses were evident among different experiments, the model could simulate the accumulation of cells in the capillary using previously determined parameters, including the chemotactic sensitivity and random motility coefficients, chi(0) and mu. In particular, model predictions were consistent with the experimental observation that the chemotactic response in the capillary is diminished under conditions wherein consumption would be expected to be significant.     

Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients

The mathematical model developed by Riveroet al. (1989,Chem. Engng Sci. 44, 2881–2897) is applied to literature data measuring chemotactic bacterial population distributions in response to steep as well as shallow attractant gradients. This model is based on a fundamental picture of the sensing and response mechanisms of individual bacterial cells, and thus relates individual cell properties such as swimming speed and tumbling frequency to population parameters such as the random motility coefficient and the chemotactic sensitivity coefficient. Numerical solution of the model equations generates predicted bacterial density and attractant concentration profiles for any given experimental assay. We have previously validated the mathematical model from experimental work involving a step-change in the attractant gradient (Fordet al., 1991Biotechnol. Bioengng.37, 647–660; For and Lauffenburger, 1991,Biotechnol. Bioengng,37, 661–672). Within the context of this experimental assay, effects of attractant diffusion and consumption, random motility, and chemotactic sensitivity on the shape of the profiles are explored to enhance our understanding of this complex phenomenon. We have applied this model to various other types of gradients with successful intepretation of data reported by Dalquistet al. (1972,Nature New Biol. 236, 120–123) forSalmonella typhimurum validating the mathematical model and supportin the involvement of high and low affinity receptors for serine chemotaxis by these cells.     

Transport of bacteria in porous media and its enhancement by surfactants for bioaugmentation: A review

The success of bioaugmentation processes for groundwater bioremediation requires efficient transport of bacteria in the subsurface environment. In this paper, the factors that influence transport of bacterial cells in porous media are reviewed and the effects of surfactants on the transport are discussed. Movement of bacterial cells in porous media is a process driven by advection and hydrodynamic dispersion forces of fluids. Immobilization of bacterial cells takes place due to processes such as adsorption and straining. Blocking and ripening along with bacterial migration process decrease and increase the retention of cells in porous media, respectively. Physicochemical properties of the porous media, groundwater chemistry, and properties of the bacterial cells affect the transport behavior. Surfactants have the potential to modify bacterial surface properties for both bacterial cells and medium solids, and thus enhance bacterial transport.     

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.     

Traveling bands of chemotactic bacteria in the context of population growth

A mathematical model for traveling bands of motile and chemotactic bacteria in the presence of cell growth and death is examined. It is found that asymptotic traveling wave solutions exist in the absence of chemotaxis, due to the balance of growth, death and random motility. Thus random motility confers the ecological advantage of population propagation through migration into nutrient-rich regions. The presence of chemotaxis amplifies this advantage by moving more cells into higher nutrient concentration regions, resulting in larger and faster bands. Therefore there seem to be two types of traveling bands that can be attained by chemotactic bacteria in the presence of growth and death: (1) these growth/death/motility bands; and (2) pure chemotactic ‘Keller-Segel'-type bands. Comparison to experimental observations by Chapman in 1973 indicate that the latter seem to be formed. The relationship between these two types of solution is at present uncertain. The growth/death/motility bands may have relevance on longer time or distance scales characteristic of microbial ecological systems.