Mathematical models for motile bacterial transport in cylindrical tubes

A computational framework for investigating bacteria transport in microvasculature

Blood-borne bacteria disseminate in tissue through microvasculature or capillaries. Capillary size, presence of red blood cells (RBCs), and bacteria motility affect bacteria intracapillary transport, an important yet largely unexplored phenomenon. Computational description of the system comprising interactions between plasma, RBCs, and motile bacteria in 5-10 μm diameter capillaries pose several challenges. The Immersed Boundary Method (IBM) was used to resolve the capillary, deformed RBCs, and bacteria. The challenge of disparate coupled time scales of flow and bacteria motion are reconciled by a temporal multiscale simulation method. Bacterium-wall and bacterium-RBC collisions were detected using a hierarchical contact- detection algorithm. Motile bacteria showed a net outward radial velocity of 2.8 µm/s compared to -0.5 µm/s inward for non-motile bacteria; thus, exhibiting a greater propensity to escape the bolus flow region between RBCs and marginate for potential extravasation, suggesting motility enhances extravasation of bacteria from capillaries.

Numerical Simulation of the Enrichment of Chemotactic Bacteria in Oil-Water Two-Phase Transfer Fields of Heterogeneous Porous Media

Oil pollution in soil-groundwater systems is difficult to remove, and a large amount of residual oil is trapped in the low permeable layer of the heterogeneous aquifer. Aromatic hydrocarbons in oil have high toxicity and low solubility in water, which are harmful to the ecological environment. Chemotactic degrading bacteria can perceive the concentration gradient of non-aqueous phase liquid (NAPL) pollutants in the groundwater environment, and enrich and proliferate around the pollutants, thus achieving a more efficient and thorough remediation effect. However, the existing theoretical models are relatively simple. The physical fields of oil–water two-phase flow and oil-phase solute convection and diffusion in water are not coupled, which further restricts the accuracy of studies on bacterial chemotaxis to NAPL. In this study, geometric models based on the actual microfluidic experimental study were constructed. Based on the phase field model, diffusion convection equation and chemotaxis velocity equation, the effects of heterogeneity of porous media, wall wettability and groundwater flow rate on the residual oil and the concentration distribution of chemotaxis bacteria were studied. Under all of the simulation conditions, the residual oil in the high permeable area was significantly lower than that in the low permeable area, and the wall hydrophilicity enhanced the water flooding effect. Chemotactic bacteria could react to the concentration gradient of pollutants dissolved into water in the oil phase, and enrich near the oil–water interface with high concentration of NAPL, and the density of chemotactic bacteria at the oil–water interface can be up to 1.8–2 times higher than that in the water phase at flow rates from 1.13 to 6.78 m/d.

Evaluation of bacterial run and tumble motility parameters through trajectory analysis

In this paper, a method for extraction of the behavior parameters of bacterial migration based on the run and tumble conceptual model is described. The methodology is applied to the microscopic images representing the motile movement of flagellated Azotobacter vinelandii. The bacterial cells are considered to change direction during both runs and tumbles as is evident from the movement trajectories. An unsupervised cluster analysis was performed to fractionate each bacterial trajectory into run and tumble segments, and then the distribution of parameters for each mode were extracted by fitting mathematical distributions best representing the data. A Gaussian copula was used to model the autocorrelation in swimming velocity. For both run and tumble modes, Gamma distribution was found to fit the marginal velocity best, and Logistic distribution was found to represent better the deviation angle than other distributions considered. For the transition rate distribution, log-logistic distribution and log-normal distribution, respectively, was found to do a better job than the traditionally agreed exponential distribution. A model was then developed to mimic the motility behavior of bacteria at the presence of flow. The model was applied to evaluate its ability to describe observed patterns of bacterial deposition on surfaces in a micro-model experiment with an approach velocity of 200 μm/s. It was found that the model can qualitatively reproduce the attachment results of the micro-model setting.

An alternative smooth particle hydrodynamics formulation to simulate chemotaxis in porous media

Chemotaxis, the microorganisms autonomous motility along or against the concentration gradients of a chemical species, is an important, yet often neglected factor controlling the transport of bacteria through saturated porous media. For example, chemotactic bacteria could enhance bioremediation by directing their own motion to residual contaminants trapped in low hydraulic conductive zones of contaminated aquifers. The aim of the present work is to develop an accurate numerical scheme to model chemotaxis in saturated porous media and other advective dominating flow systems. We propose to model chemotaxis by using a new class of meshless Lagrangian particle methods we recently developed for applications in fluid mechanics. The method is based on the Smooth Particle Hydrodynamics (SPH) formulation of (Ben Moussa et al., Int Ser Numer Math, 13(1):29-62, 2006), combined with a new Weighted Essentially Non-Oscillatory (WENO) reconstruction technique on moving point clouds in multiple space dimensions. The purpose of this new numerical scheme is to fully exploit the advantages of SPH among traditional mesh-based and mesh-free schemes and to overcome drawbacks related to the use of standard SPH for modeling chemotaxis in porous media. First, we test the new scheme against analytical reference solutions. Then, under the assumption of complete mixing at the Darcy scale, we perform two-dimensional conservative solute transport simulations under steady-state flow conditions, to show the capability of the proposed new scheme to model chemotaxis.

Diffusion of Bacterial Cells in Porous Media

The chemotaxis signal transduction network regulates the biased random walk of many bacteria in favorable directions and away from harmful ones through modulating the frequency of directional reorientations. In mutants of diverse bacteria lacking the chemotaxis response, migration in classic motility agar, which constitutes a fluid-filled porous medium, is compromised; straight-swimming cells unable to tumble become trapped within the agar matrix. Spontaneous mutations that restore spreading have been previously observed in the enteric bacterium Escherichia coli, and recent work in other bacterial species has isolated and quantified different classes of nonchemotacting mutants exhibiting the same spreading phenotype. We present a theoretical description of bacterial diffusion in a porous medium-the natural habitat for many cell types-which elucidates how diverse modifications of the motility apparatus resulting in a nonzero tumbling frequency allows for unjamming of otherwise straight-swimming cells at internal boundaries and leads to net migration. A unique result of our analysis is increasing diffusive spread with increasing tumbling frequency in the small pore limit, consistent with earlier experimental observations but not captured by previous models. Our theoretical results, combined with a simple model of bacterial diffusion and growth in agar, are compared with our experimental measurements of swim ring expansion as a function of time, demonstrating good quantitative agreement. Our results suggest that the details of the cellular tumbling process may be adapted to enable bacteria to propagate efficiently through complex environments. For engineered, self-propelled microswimmers that navigate via alternating straight runs and changes in direction, these results suggest an optimal reorientation strategy for efficient migration in a porous environment with a given microarchitecture.

Quantitative analysis of chemotaxis towards toluene by Pseudomonas putida in a convection-free microfluidic device

Chemotaxis has been shown to be beneficial for the migration of soil-inhabiting bacteria towards industrial chemical pollutants, which they degrade. Many studies have demonstrated the importance of this microbial property under various circumstances; however, few quantitative analyses have been undertaken to measure the two essential parameters that characterize the chemotaxis of bioremediation bacteria: the chemotactic sensitivity coefficient χ(0) and the chemotactic receptor constant K(c). The main challenge to determine these parameters is that χ(0) and K(c) are coupled together in non-linear mathematical models used to evaluate them. In this study we developed a method to accurately measure these parameters for Pseudomonas putida in the presence of toluene, an important pollutant in groundwater contamination. Our approach uses a multilayer microfluidic device to expose bacteria to a convection-free linear chemical gradient of toluene that is stable over time. The bacterial distribution within the gradient is measured in terms of fluorescence intensity, and is then used to fit the parameters Kc and χ(0) with mathematical models. Critically, bacterial distributions under chemical gradients at two different concentrations were used to solve for both parameters independently. To validate the approach, the chemotaxis parameters of Escherichia coli strains towards α-methylaspartate were experimentally derived and were found to be consistent with published results from related work.

Theoretical insights into bacterial chemotaxis

Research into understanding bacterial chemotactic systems has become a paradigm for Systems Biology. Experimental and theoretical researchers have worked hand-in-hand for over 40 years to understand the intricate behavior driving bacterial species, in particular how such small creatures, usually not more than 5 µm in length, detect and respond to small changes in their extracellular environment. In this review we highlight the importance that theoretical modeling has played in providing new insight and understanding into bacterial chemotaxis. We begin with an overview of the bacterial chemotaxis sensory response, before reviewing the role of theoretical modeling in understanding elements of the system on the single cell scale and features underpinning multiscale extensions to population models.

Motility of Pseudomonas aeruginosa in saturated granular media as affected by chemoattractant

To examine and quantify the effects of glass beads and chemoattractant on bacterial motility in granular media, we examined the motile behavior of P. aeruginosa in a saturated granular medium and quantified the effects of glass beads and the presence of a chemoattractant. By recording individual cell trajectories in microfluidic channels under a high-speed confocal microscope, we directly measured the cell's run direction and corresponding run-length, speed and turn angle. Bacterial run speed increased in the presence of chemoattractant in both aqueous and granular media. But it decreased in glass-beads compared to in aqueous media due to the restricted pore geometry and interactions between bacteria and grain surfaces. Notably, the relatively higher frequency distribution at turn angles of 170° decreased dramatically, while the smaller peak at 70° increased and became dominant on a bimodal distribution, showing more bacteria changed directions at smaller turn angles rather than reverse their swimming directions. Additionally, the presence of glass beads also decreased the chemotactic velocity and random motility by similar proportions due to the restrictive geometry and the interactions between bacteria and glass beads surface. Our study indicates that the swimming parameters measured from aqueous media cannot be directly adopted in models for predicting bacteria travel in granular media.

Quantitative analysis of transverse bacterial migration induced by chemotaxis in a packed column with structured physical heterogeneity

A two-dimensional mathematical model was developed to simulate transport phenomena of chemotactic bacteria in a sand-packed column designed with structured physical heterogeneity in the presence of a localized chemical source. In contrast to mathematical models in previous research work, in which bacteria were typically treated as immobile colloids, this model incorporated a convective-like chemotaxis term to represent chemotactic migration. Consistency between experimental observation and model prediction supported the assertions that (1) dispersion-induced microbial transfer between adjacent conductive zones occurred at the interface and had little influence on bacterial transport in the bulk flow of the permeable layers and (2) the enhanced transverse bacterial migration in chemotactic experiments relative to nonchemotactic controls was mainly due to directed migration toward the chemical source zone. On the basis of parameter sensitivity analysis, chemotactic parameters determined in bulk aqueous fluid were adequate to predict the microbial transport in our intermediate-scale porous media system. Additionally, the analysis of adsorption coefficient values supported the observation of a previous study that microbial deposition to the surface of porous media might be decreased under the effect of chemoattractant gradients. By quantitatively describing bacterial transport and distribution in a heterogeneous system, this mathematical model serves to advance our understanding of chemotaxis and motility effects in granular media systems and provides insights for modeling microbial transport in in situ microbial processes.

Dynamics of microbial growth and coexistence on variably saturated rough surfaces

The high degree of microbial diversity found in soils is attributed to the highly heterogeneous pore space and the dynamic aqueous microenvironments. Previous studies have shown that spatial and temporal variations in aqueous diffusion pathways play an important role in shaping microbial habitats and biological activity in unsaturated porous media. A new modeling framework was developed for the quantitative description of diffusion-dominated microbial interactions focusing on competitive growth of two microbial species inhabiting partially saturated rough surfaces. Surface heterogeneity was represented by patches with different porosities and water retention properties, yielding heterogeneous distribution of water contents that varies with changes in relative humidity or soil matric potential. Nutrient diffusion and microbial growth on the variably hydrated and heterogeneous surface was modeled using a hybrid method that combines a reaction diffusion method for nutrient field with individual based model for microbial growth and expansion. The model elucidated the effects of hydration dynamics and heterogeneity on nutrient fluxes and mobility affecting microbial population growth, expansion, and coexistence at the microscale. In contrast with single species dominance under wet conditions, results demonstrated prolonged coexistence of two competing species under drier conditions where nutrient diffusion and microbial movement were both limited. The uneven distribution of resources and diffusion pathways in heterogeneous surfaces highlighted the importance of position in the landscape for survival that may compensate for competitive disadvantages conferred by physiological traits. Increased motility was beneficial for expansion and survival. Temporal variations in hydration conditions resulted in fluctuations in microbial growth rate and population size. Population growth dynamics of the dominant species under wet-dry cycles were similar to growth under average value of diffusion coefficients for dry and wet conditions, respectively, suggesting that the time-averaged diffusion coefficient could serve as a useful indicator for estimation of microbial activities in a highly dynamic system such as that found in soils.

Classical solutions and pattern formation for a volume filling chemotaxis model

We establish the global existence of classical solutions to a generalized chemotaxis model, which includes the volume filling effect expressed through a nonlinear squeezing probability. This novel choice of squeezing probability reflects the elastic properties of cells. Necessary and sufficient conditions for spatial pattern formation are given and the underlying bifurcations are analyzed. In numerical simulations, the complex dynamics of merging and emerging patterns are shown for zero cell kinetics and nonzero cell kinetics, respectively. We conclude that the emerging process of pattern formation is due to cell growth.

Bacterial motility: links to the environment and a driving force for microbial physics

Bacterial motility was recognized 300 years ago. Throughout this history, research into motility has led to advances in microbiology and physics. Thirty years ago, this union helped to make run and tumble chemotaxis the paradigm for bacterial movement. This review highlights how this paradigm has expanded and changed, and emphasizes the following points. The absolute magnitude of swimming speed is ecologically important because it helps determine vulnerability to Brownian motion, sensitivity to gradients, the type of receptors used and the cost of moving, with some bacteria moving at 1 mm s(-1). High costs for high speeds are offset by the benefit of resource translocation across submillimetre redox and other environmental gradients. Much of environmental chemotaxis appears adapted to respond to gradients of micrometres, rather than migrations of centimetres. In such gradients, control of ion pumps is particularly important. Motility, at least in the ocean, is highly intermittent and the speed is variable within a run. Subtleties in flagellar physics provide a variety of reorientation mechanisms. Finally, while careful physical analysis has contributed to our current understanding of bacterial movement, tactic bacteria are increasingly widely used as experimental and theoretical model systems in physics.

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.