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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Quantitative relationships between single-cell and cell-population model parameters for chemosensory migration responses of alveolar macrophages to C5a

Phenomenological parameters from a mathematical model of cell motility are used to quantitatively characterize chemosensory migration responses of rat alveolar macrophages migrating to C5a in the linear under-agarose assay, simultaneously at the levels of both single cells and cell populations. This model provides theoretical relationships between single-cell and cell-population motility parameters. Our experiments offer a critical test of these theoretical linking relationships, by comparison of results obtained at the cell population level to results obtained at the single-cell level. Random motility of a cell population is characterized by the random motility coefficient, mu (analogous to a particle diffusion coefficient), whereas single-cell random motility is described by cell speed, s, and persistence time, P (related to the period of time that a cell moves in one direction before changing direction). Population chemotaxis is quantified by the chemotactic sensitivity, chi 0, which provides a measure of the minimum attractant gradient necessary to elicit a specified chemotactic response. Single-cell chemotaxis is characterized by the chemotactic index, CI, which ranges from 0 for purely random motility to 1 for perfectly directed motility. Measurements of cell number versus migration distance were analyzed in conjunction with the phenomenological model to determine the population parameters while paths of individual cells in the same experiment were analyzed in order to determine the single-cell parameters. The parameter mu shows a biphasic dependence on C5a concentration with a maximum of 1.9 x 10(-8) cm2/sec at 10(-11) M C5a and relative minima of 0.86 x 10(-8) cm2/sec at 10(-7) M C5a and 1.1 x 10(-8) cm2/sec in the absence of Ca; s and P remain fairly constant with C5a concentration, with s ranging from 2.1 to 2.5 microns/min and P varying from 22 to 32 min. chi 0 is equal to 1.0 x 10(-6) cm/receptor for all C5a concentrations tested, corresponding to 60% correct orientation for a difference of 500 bound C5a receptors across a 20 microns cell length. The maximum CI measured was 0.2. Values for the population parameters mu and chi 0 were calculated from single-cell parameter values using the aforementioned theoretical linking relationships. The values of mu and chi 0 calculated from single-cell parameters agreed with values of mu and chi 0 determined independently from population migrations, over the full range of C5a concentrations, confirming the validity of the linking equations. Experimental confirmation of such relationships between single-cell and cell-population parameters has not previously been reported.     

Localized bacterial infection in a distributed model for tissue inflammation

Phagocyte motility and chemotaxis are included in a distributed mathematical model for the inflammatory response to bacterial invasion of tissue. Both uniform and non-uniform steady state solutions may occur for the model equations governing bacteria and phagocyte densities in a macroscopic tissue region. The non-uniform states appear to be more dangerous because they allow large bacteria densities concentrated in local foci, and in some cases greater total bacteria and phagocyte populations. Using a linear stability analysis, it is shown that a phagocyte chemotactic response smaller than a critical value can lead to a non-uniform state, while a chemotactic response greater than this critical value stabilizes the uniform state. This result is the opposite of that found for the role of chemotaxis in aggregation of slimemold amoebae because, in the inflammatory response, the chemotactic population serves as an inhibitor rather than an activator. We speculate that these non-uniform steady states could be related to the localized cell aggregation seen in chronic granulomatous inflammation. The formation of non-uniform states is not necessarily a consequence of defective phagocyte chemotaxis, however. Rather, certain values of the kinetic parameters can yield values for the critical chemotactic response which are greater than the normal response.Numerical computations of the transient inflammatory response to bacterial challenge are presented, using parameter values estimated from the experimental literature wherever possible.     

Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis.

The growth of new capillary blood vessels, or angiogenesis, is a prominent component of numerous physiological and pathological conditions. An understanding of the co-ordination of underlying cellular behaviors would be helpful for therapeutic manipulation of the process. A probabilistic mathematical model of angiogenesis is developed based upon specific microvessel endothelial cell (MEC) functions involved in vessel growth. The model focuses on the roles of MEC random motility and chemotaxis, to test the hypothesis that these MEC behaviors are of critical importance in determining capillary growth rate and network structure. Model predictions are computer simulations of microvessel networks, from which questions of interest are examined both qualitatively and quantitatively. Results indicate that a moderate MEC chemotactic response toward an angiogenic stimulus, similar to that measured in vitro in response to acidic fibroblast growth factor, is necessary to provide directed vascular network growth. Persistent random motility alone, with initial budding biased toward the stimulus, does not adequately provide directed network growth. A significant degree of randomness in cell migration direction, however, is required for vessel anastomosis and capillary loop formation, as simulations with an overly strong chemotactic response produce network structures largely absent of these features. The predicted vessel extension rate and network structure in the simulations are quantitatively consistent with experimental observations of angiogenesis in vivo. This suggests that the rate of vessel outgrowth is primarily determined by MEC migration rate, and consequently that quantitative in vitro migration assays might be useful tools for the prescreening of possible angiogenesis activators and inhibitors. Finally, reduction of MEC speed results in substantial inhibition of simulated angiogenesis. Together, these results predict that both random motility and chemotaxis are MEC functions critically involved in determining the rate and morphology of new microvessel network growth.     

Effects of cell motility and chemotaxis on microbial population growth.

A mathematical model is developed to elucidate the effects of biophysical transport processes (nutrient diffusion, cell motility, and chemotaxis) along with biochemical reaction processes (cell growth and death, nutrient uptake) upon steady-state bacterial population growth in a finite one-dimensional region. The particular situation considered is that of growth limitation by a nutrient diffusing from an adjacent phase not accessible to the bacteria. It is demonstrated that the cell motility and chemotaxis properties can have great influence on steady-state population size. In fact, motility effects can be as significant as growth kinetic effects, in a manner analogous to diffusion- and reaction-limited regimes in chemically reacting systems. In particular, the following conclusions can be drawn from our analysis for bacterial populations growing at steady-state in a confined, unmixed region: (a) Random motility may lead to decreased population density; (b) chemotaxis can allow increased population density if the chemotactic response is large enough; (c) a species with superior motility properties can outgrow a species with superior growth kinetic properties; (d) motility effects become greater as the size of the confined growth region increases; and (e) motility effects are diminished by significant mass-transfer limitation of the nutrient from the adjacent source phase. The relationships of these results for populations to previous conclusions for individual cells is discussed, and implications for microbial competition are suggested.     

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 model for traveling bands of chemotactic bacteria.

A theoretical model is used to study band formation by chemotactic populations of Escherichia coli. The model includes the bacterial response to attractant gradients, the chemotactic sensitivity of the bacteria to the concentration of the attractant, and population growth. For certain values of the parameters in the model, traveling bands of bacteria form and propagate with or without growth. Under specific growth conditions the band profile is maintained and the band propagates at constant speed. These predictions are in general agreement with the experiment results of J. Adler and earlier theoretical work by L. Segel and his collaborators. However, our theory differs in several important respects from the latter efforts. Suggestions are made for further experiments to test the proposed model and to clarify the nature of the processes which lead to band formation.     

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.     

Quantification of bacterial chemotaxis by measurement of model parameters using the capillary assay

The capillary assay for quantitative characterization of bacterial motility and chemotaxis is analyzed in terms of a mathematical model for cell population migration, in order to determine values for the cell random motility coefficient, mu and the cell chemotaxis coefficient, chi. The analysis involves both analytical perturbation methods and numerical finite-difference techniques. Transient cell density profiles within the capillary tube are determined as they depend upon mu and chi, providing a means for estimating mu and chi from the common protocol measurements of cell accumulation in the tube at specified observation times. The effects of extraneous factors such as assay geometry, stimulus diffusivity, bacterial density, and observation time are thus separated from the intrinsic cell-stimulus interaction and response. This allows independent population measurements of cell chemosensory movement properties to be extrapolated to situations involving growth and competition of populations, for purposes of better understanding microbial population dynamics in systems of biotechnological and microbial ecological importance.     

How the Motility Pattern of Bacteria Affects Their Dispersal and Chemotaxis

Most bacteria at certain stages of their life cycle are able to move actively; they can swim in a liquid or crawl on various surfaces. A typical path of the moving cell often resembles the trajectory of a random walk. However, bacteria are capable of modifying their apparently random motion in response to changing environmental conditions. As a result, bacteria can migrate towards the source of nutrients or away from harmful chemicals. Surprisingly, many bacterial species that were studied have several distinct motility patterns, which can be theoretically modeled by a unifying random walk approach. We use this approach to quantify the process of cell dispersal in a homogeneous environment and show how the bacterial drift velocity towards the source of attracting chemicals is affected by the motility pattern of the bacteria. Our results open up the possibility of accessing additional information about the intrinsic response of the cells using macroscopic observations of bacteria moving in inhomogeneous environments.     

The effect of cell sedimentation on measuring chondrocyte population migration using a Boyden chamber

The Boyden chamber assay provides a convenient method of assessing cell migration and measuring cell motility coefficients at the population level. Previous models of this assay completely ignore cell sedimentation in the suspension, assuming that all cells have already settled on the filter surface before commencing migration within the filter. However, ignoring cell sedimentation could lead to poor data interpretation because the time required for cells to settle through the suspension is close to the incubation period of only a few hours. This study models the Boyden chamber assay by incorporating the cell settling process to account for the cells remaining in the upper well when other cells migrate in the filter. The simulations in this study elucidate the experiments in the literature that test the haptotactic and chemotactic responses of rabbit chondrocytes to type II collagen. This study determines the cell population random motility, as well as the haptotaxis and chemotaxis coefficients, by fitting the experimental data. Results show that the chemotactic motility coefficient is 100 times greater than the haptotactic coefficient, and the equilibrium collagen-receptor dissociation constant is about 10-fold the haptotactic counterpart. Diffusion causes the soluble collagen gradients in the chemotactic case to decline over time, while the coated collagen gradients in the haptotactic assay are likely to remain fixed. As a result, the chemotactic case exhibits a lower number of migrated cells than the haptotactic assay. This study also demonstrates the influences of the dimensionless parameters that control cell behavior in the Boyden assay, providing a reference for future experiment designs.