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.     

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.     

Traveling waves in a simple population model involving growth and death

The structure of solutions to a simple spatially dependent population model involving growth and death is investigated. Two forms of motility of the population are considered: (1) random motion only modeled by a Fickian law, and (2) a directed component of motion (chemotaxis), included in addition to the random motion. Under certain growth conditions a traveling wave of constant speed is approached. This speed can be increased by the addition of the chemotaxis with a corresponding increase in the asymptotic population. Development of initial conditions into a wave is illustrated numerically.     

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.     

Cell balance equation for chemotactic bacteria with a biphasic tumbling frequency

Alts three-dimensional cell balance equation characterizing the chemotactic bacteria was analyzed under the presence of one-dimensional spatial chemoattractant gradients. Our work differs from that of others who have developed rather general models for chemotaxis in the use of a non-smooth anisotropic tumbling frequency function that responds biphasically to the combined temporal and spatial chemoattractant gradients. General three-dimensional expressions for the bacterial transport parameters were derived for chemotactic bacteria, followed by a perturbation analysis under the planar geometry. The bacterial random motility and chemotaxis were summarized by a motility tensor and a chemotactic velocity vector, respectively. The consequence of invoking the diffusion-approximation assumption and using intrinsic one-dimensional models with modified cellular swimming speeds was investigated by numerical simulations. Characterizing the bacterial random orientation after tumbles by a turn angle probability distribution function, we found that only the first-order angular moment of this turn angle probability distribution is important in influencing the bacterial long-term transport. Mathematics Subject Classification (2000):60G05, 60J60, 82A70     

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.     

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.     

Motility and chemotaxis of filamentous cells of Escherichia coli

Filamentous cells of Escherichia coli can be produced by treatment with the antibiotic cephalexin, which blocks cell division but allows cell growth. To explore the effect of cell size on chemotactic activity, we studied the motility and chemotaxis of filamentous cells. The filaments, up to 50 times the length of normal E. coli organisms, were motile and had flagella along their entire lengths. Despite their increased size, the motility and chemotaxis of filaments were very similar to those properties of normal-sized cells. Unstimulated filaments of chemotactically normal bacteria ran and stopped repeatedly (while normal-sized bacteria run and tumble repeatedly). Filaments responded to attractants by prolonged running (like normal-sized bacteria) and to repellents by prolonged stopping (unlike normal-sized bacteria, which tumble), until adaptation restored unstimulated behavior (as occurs with normal-sized cells). Chemotaxis mutants that always ran when they were normal sized always ran when they were filament sized, and those mutants that always tumbled when they were normal sized always stopped when they were filament sized. Chemoreceptors in filaments were localized to regions both at the poles and at intervals along the filament. We suggest that the location of the chemoreceptors enables the chemotactic responses observed in filaments. The implications of this work with regard to the cytoplasmic diffusion of chemotaxis components in normal-sized and filamentous E. coli are discussed.     

Effect of bacterial chemotaxis on dynamics of microbial competition

Although the dynamic behavior of microbial populations in nonmixed systems is a central aspect of many problems in biochemical engineering and microbiology, the factors that govern this behavior are not well understood. In particular, the effects of bacterial chemotaxis (biased migration of cells in the direction of chemical concentration gradients) have been the subject of much speculation but very little quantitative investigation. In this paper, we provide the first theoretical analysis of the effects of bacterial chemotaxis on the dynamics of competition between two microbial populations for a single rate-limiting nutrient in a confined nonmixed system. We use a simple unstructured model for cell growth and death, and the most soundly based current model for cell population migration. Using numerical finite element techniques, we examine both transient and steady-state behavior of the competing populations, focusing primarily on the influence of the cell random motility coefficient,, and the cell chemotaxis coefficient, . We find that, in general, there are four possible steady-state outcomes: both populations die out, population 1 exists alone, population 2 exists alone, and the two populations coexist. We find that, in contrast to well-mixed systems, the slower-growing population can coexist and even exist alone if it possesses sufficiently superior motility and chemotaxis properties. Our results allow estimation of the value of necessary to allow coexistence and predominance for reasonable values of growth and random motility parameters in common systems. An especially intriguing finding is that there is a minimum value of necessary for a chemotactic population to have a competitive advantage over an immotile population in a confined nonmixed system. Further, for typical system parameter values, this minimum value of is the range of values that can be estimated from independent experimental assays for chemotaxis.Thus, in typical nonmixed systems, cell motility and chemotaxis properties can be the determining factors in governing population dynamics.     

Traveling waves in a chemotactic model

A model for chemotaxis in a bacteria-substrate mixture introduced by Keller and Segel, which is described by nonlinear partial differential equations, is studied analytically. The existence of traveling waves is shown for the system in which the substrate diffusion is taken into account and the chemotactic coefficient is greater than the motility one, and the instability of traveling waves is discussed.     

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.     

Response of Bdellovibrio and Like Organisms (BALOs) to the Migration of Naturally Occurring Bacteria to Chemoattractants

A dual culture-based and non–culture-based approach was applied to characterize predator bacterial groups in surface water samples collected from Apalachicola Bay, Florida. Chemotaxis drop assays were performed on concentrated samples in an effort to isolate predator bacteria by their chemotactic ability. Yeast extract (YE) and casamino acids (CA) proved to be strong chemoattractants and resulted in three visibly distinct bands; however, dextrose, succinate, pyruvate, and concentrated cells of Vibrio parahaemolyticus P5 as prey did not elicit any response. The three distinct bands from YE and CA were separately collected to identify the chemotactic microbial assemblages. Plaque-forming unit assays from different chemotaxis bands with P5 as prey indicated 5- (CA) to 10-fold (YE) higher numbers of predator bacteria in the outermost chemotactic bands. Polymerase chain reaction–restriction fragment length polymorphism and 16S rDNA sequencing of clones from different chemotaxis bands resulted in identification of Pseudoalteromonas spp., Marinomonas spp., and Vibrio spp., with their numbers inversely proportional to the numbers of predators—i.e., Bdellovibrio spp. and Bacteriovorax spp—in the chemotaxis bands. This study indicates that predatorial bacteria potentially respond to high densities of microbial biomass in aquatic ecosystems and that chemotaxis drop assay may be an alternate culture-independent method to characterize predatorial bacterial guilds from the environment.     

Chemotaxis is required for virulence and competitive fitness of the bacterial wilt pathogen Ralstonia solanacearum

Ralstonia solanacearum, a soilborne plant pathogen of considerable economic importance, invades host plant roots from the soil. Qualitative and quantitative chemotaxis assays revealed that this bacterium is specifically attracted to diverse amino acids and organic acids, and especially to root exudates from the host plant tomato. Exudates from rice, a nonhost plant, were less attractive. Eight different strains from this heterogeneous species complex varied significantly in their attraction to a panel of carbohydrate stimuli, raising the possibility that chemotactic responses may be differentially selected traits that confer adaptation to various hosts or ecological conditions. Previous studies found that an aflagellate mutant lacking swimming motility is significantly reduced in virulence, but the role of directed motility mediated by the chemotaxis system was not known. Two site-directed R. solanacearum mutants lacking either CheA or CheW, which are core chemotaxis signal transduction proteins, were completely nonchemotactic but retained normal swimming motility. In biologically realistic soil soak virulence assays on tomato plants, both nonchemotactic mutants had significantly reduced virulence indistinguishable from that of a nonmotile mutant, demonstrating that directed motility, not simply random motion, is required for full virulence. In contrast, nontactic strains were as virulent as the wild-type strain was when bacteria were introduced directly into the plant stem through a cut petiole, indicating that taxis makes its contribution to virulence in the early stages of host invasion and colonization. When inoculated individually by soaking the soil, both nontactic mutants reached the same population sizes as the wild type did in the stems of tomato plants just beginning to wilt. However, when tomato plants were coinoculated with a 1:1 mixture of a nontactic mutant and its wild-type parent, the wild-type strain outcompeted both nontactic mutants by 100-fold. Together, these results indicate that chemotaxis is an important trait for virulence and pathogenic fitness in this plant pathogen.     

A differential role of extracellular signal-regulated kinase in stimulated PC12 pheochromocytoma cell movement

Rat pheochromocytoma PC12 cells have been widely used as a cell system for study of growth factor-stimulated cell functions. We report here that nerve growth factor (NGF) stimulated both chemotaxis (directional migration) and chemokinesis (random migration) of PC12 cells. Treatment with a MEK1/2-specific inhibitor (PD98059) or expression of a dominant negative variant of Ras differentially inhibited NGF-stimulated chemotaxis but not chemokinesis of PC12 cells. Priming of PC12 cells with NGF resulted in reduced extracellular signal-regulated kinase (ERK) activation and loss of chemotactic, but not chemokinetic, response. In addition, NGF stimulation of ERK is known to involve an early transient phase of activation followed by a late sustained phase of activation; in contrast, epidermal growth factor (EGF) elicits only early transient ERK activation. We observed that like NGF, EGF also stimulated both chemotaxis and chemokinesis, and treatment with PD98059 abolished the EGF-stimulated chemotaxis. Therefore, the early transient phase of ERK activation functioned in signaling chemotaxis; the late sustained phase of ERK activation did not seem to have an essential role. In addition, our results suggested that chemotactic signaling required a threshold level of ERK activation; at below threshold level of ERK activation, chemotaxis would not occur.     

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.     

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.     

Modelling run-and-tumble chemotaxis in a shear flow

The biased random walk undergone by chemotactic bacteria such as Escherichia coli will be influenced at the microscopic level by flow in the ambient medium. In this paper, we model swimming bacteria being advected and rotated by a simple shear flow. Under certain scaling assumptions, we obtain an advection—diffusion equation for cell density, when the chemotactic response is small, which shows a coupling between the rotation and chemotaxis. We also present an alternative method for calculating the chemotactic flux in an unbounded region which is valid for more general chemotactic responses.