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.     

Modeling microbial chemotaxis in a diffusion gradient chamber

The diffusion gradient chamber (DGC) has proven to be a useful experimental tool for studying population-level microbial growth and chemotaxis. A mathematical model capable of reproducing the population-level patterns formed as a result of cellular growth and chemotaxis in the DGC has been developed. The model consists of coupled partial differential balance equations for cells, chemoattractants, and a nutrient, which are solved simultaneously by the alternating direction implicit method. Modeling simulation results were compared with population-level migration patterns of Escherichia coli growing on glycerol and responding to a gradient of the chemoattractant aspartate for two different initial conditions. To accurately reproduce the experimental results, a second chemoattractant equation was necessary. The second chemoattractant has been identified as oxygen by directly measuring oxygen gradients in the DGC. Important trends observed experimentally and reproduced by the model include the formation of a chemotactic wave, a reduction in the wave velocity as it encounters higher chemoattractant concentrations, and chemotaxis in response to two different chemoattractants simultaneously. The model was also used to study the relative magnitude of cell fluxes due to random motility and chemotaxis, and the suppression of chemotaxis due to receptor saturation. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 191-205, 1997.     

Analysis of a Vector-Bias Model on Malaria Transmission

We incorporate a vector-bias term into a malaria-transmission model to account for the greater attractiveness of infectious humans to mosquitoes in terms of differing probabilities that a mosquito arriving at a human at random picks that human depending on whether he is infectious or susceptible. We prove that transcritical bifurcation occurs at the basic reproductive ratio equalling 1 by projecting the flow onto the extended centre manifold. We next study the dynamics of the system when incubation time of malaria parasites in mosquitoes is included, and find that the longer incubation time reduces the prevalence of malaria. Also, we incorporate a random movement of mosquitoes as a diffusion term and a chemically directed movement of mosquitoes to humans expressed in terms of sweat and body odour as a chemotaxis term to study the propagation of infected population to uninfected population. We find that a travelling wave occurs; its speed is calculated numerically and estimated for the lower bound analytically.     

A comparison of random vs. chemotaxis-driven contacts of T cells with dendritic cells during repertoire scanning

Generating adaptive immunity after infection or immunization requires physical interactions within a lymph node (LN) T-zone between antigen-bearing dendritic cells (DCs) that arrive from peripheral tissues and rare cognate T cells entering via high endothelial venules (HEVs). This interaction results in activation of cognate T cells, expansion of that T cell lineage and their exit from the LN T-zone via efferent lymphatics (ELs). How antigen-specific T cells locate DCs within this complex environment is controversial, and both random T cell migration and chemotaxis have been proposed. We developed an agent-based computational model of a LN that captures many features of T cell and DC dynamics observed by two-photon microscopy. Our simulations matched in vivo two-photon microscopy data regarding T cell speed, short-term directional persistence of motion and cell motility. We also obtained in vivo data regarding density of T cells and DCs within a LN and matched our model environment to measurements of the distance from HEVs to ELs. We used our model to compare chemotaxis with random motion and showed that chemotaxis increased total number of T cell DC contacts, but decreased unique contacts, producing fewer activated T cells. Our results suggest that, within a LN T-zone, a random search strategy is optimal for a rare cognate T cell to find its DC match and maximize production of activated T cells.     

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.     

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.     

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.     

Integrodifference equations in patchy landscapes

We analyze integrodifference equations (IDEs) in patchy landscapes. Movement is described by a dispersal kernel that arises from a random walk model with patch dependent diffusion, settling, and mortality rates, and it incorporates individual behavior at an interface between two patch types. Growth follows a simple Beverton–Holt growth or linear decay. We obtain explicit formulae for the critical domain-size problem, and we illustrate how different individual behavior at the boundary between two patch types affects this quantity. We also study persistence conditions on an infinite, periodic, patchy landscape. We observe that if the population can persist on the landscape, the spatial profile of the invasion evolves into a discontinuous traveling periodic wave that moves with constant speed. Assuming linear determinacy, we calculate the dispersion relation and illustrate how movement behavior affects invasion speed. Numerical simulations justify our approach by showing a close correspondence between the spread rate obtained from the dispersion relation and from numerical simulations.     

Persistence of direction increases the drift velocity of run and tumble chemotaxis

Escherichia coli is a motile bacterium that moves up a chemoattractant gradient by performing a biased random walk composed of alternating runs and tumbles. Previous models of run and tumble chemotaxis neglect one or more features of the motion, namely (a) a cell cannot directly detect a chemoattractant gradient but rather makes temporal comparisons of chemoattractant concentration, (b) rather than being entirely random, tumbles exhibit persistence of direction, meaning that the new direction after a tumble is more likely to be in the forward hemisphere, and (c) rotational Brownian motion makes it impossible for an E. coli cell to swim in a straight line during a run. This paper presents an analytic calculation of the chemotactic drift velocity taking account of (a), (b) and (c), for weak chemotaxis. The analytic results are verified by Monte Carlo simulation. The results reveal a synergy between temporal comparisons and persistence that enhances the drift velocity, while rotational Brownian motion reduces the drift velocity. This work was supported by an Oliver Gatty Studentship from the University of Cambridge.     

Wave front migration of endothelial cells in a bone-implant interface

The neo-vascularization of the host site is crucial for the primary fixation and the long-term stability of the bone-implant interface. Our aim was to investigate the progression of endothelial cell population in the first weeks of healing. We proposed a theoretical reactive model to study the role of initial conditions, random motility, haptotaxis and chemotaxis in interactions with fibronectin factors and transforming angiogenic factors. The application of governing equations concerned a canine experimental implant and numerical experiments based upon statistical designs of experiments supported the discussion. We found that chemotaxis due to transforming angiogenic factors was attracting endothelial cells present into the host bone. Haptotaxis conditioned by fibronectin factors favored cells adhesion to the host bone. The combination of diffusive and reactive effects nourished the wave front migration of endothelial cells from the host bone towards the implant. Angiogenesis goes together with new-formed bone formation in clinics, so the similarity of distribution patterns of mineralized tissue observed in-vivo and the spatio-temporal concentration of endothelial cells predicted by the model, tended to support the reliability of our theoretical approach.     

“Pseudochemotaxis” by micro-organisms in an attractant gradient

The random behavior of microorganisms in defined attractant gradients may exhibit apparent chemotaxis, or pseudochemotaxis, if the random motility of the population is a function of the attractant concentration. Several examples are studied using computer techniques to solve the differential equations of motion for populations of micro-organisms responding chemokinetically to variable attractant concentrations. The chemokinetic coefficient is a convenient parameter to characterize the time course of experiments in which micro-organisms move in attractant gradients. Measurements of the chemokinetic coefficient as a function of time may be used to directly determine the random mobility as a function of attractant concentration.     

Localization of submembranous cations to the leading end of human neutrophils during chemotaxis

Potassium pyroantimonate was used to localize sites of bound cations in human neutrophils under conditions of random migration, stimulated random migration (chemokinesis), and directed migration (chemotaxis). The cells were placed in a standard chamber in which 0.45-micron micropore filters separated the cells from the stimulus (buffer, Escherichia coli endotoxin-activated serum or the synthetic chemotactic peptide N-formyl-Met-Leu-Phe). The small pore filters permitted pseudopod formation but impeded cell imgration through the filter. Cells examined under all conditions had electron-dense precipitates of antimonate salts in some granules. However, antimonate deposits were localized in the condensed chromatin of the nucleus during random migration and associated to a large extent with the uncondensed nuclear chromatin during chemokinesis and chemotaxis. Under conditions of chemokinesis deposition of antimonate procipitates appeared on the cytoplasmic side of the plasma membrane of neutrophils whereas under conditions of chemotaxis cation deposits beneath the cell membrane were localized to the pseudopods which were directed toward the chemoattractant. In addition to endotoxin-activated serum, concentrations of N-formyl-Met-Leu-Phe which caused neutrophil chemotaxis (10(-8) M) also caused cation deposition beneath the cell membrane at the leading end of the cell regardless of whether albumin was present in the incubation media. However, with higher concentrations of the synthetic peptide (10(-5) M) which caused granule release and were not chemotactic, submembranous cation deposition was not seen. EDTA (10 mM) and EGTA (10 mM) removed nuclear, granular, and submembranous cation deposits from neutrophils examined under conditions of chemotaxis. X-ray microprobe analysis of antimonate deposits revealed the possible presence of calcium but did not detect sodium or magnesium. The data indicate that chemotactic factors induce submembranous deposition of cations, most likely Ca++, which localize to the leading edge of cells exposed to a gradient of chemoattractant.     

A model for aggregation-dispersion dynamics of a population

An important feature that distinguishes the movement of living systems from the random motion of inorganic material is a delicate balance between spreading and concentrating. This movement is based on the kind of interactions which a bacterial colony may establish during migration. Namely, the antagonistic effects of dispersal which take place preferentially down the population gradient and the tendency in grouping together. In this work a model is proposed which considers these effects. The phase plane analysis and the numerical calculations reveal the existence of stable sharp wave front solutions. The speed of the wave front is modulated by the compromise between the tendencies of spreading and aggregating. The results obtained were compared with experimental observations in cultures of Escherichia coli and Streptococcus faecalis. The agreement between both types of results supports the hypothesis on which the model was based.     

The effect of oxygen on chemotaxis to naphthalene by Pseudomonas putida G7

Chemotactic bacteria can be attracted to electron donors they consume. In systems where donor is heterogeneously distributed, chemotaxis can lead to enhanced removal of donor relative to that achieved in the absence of chemotaxis. However, simultaneous consumption of an electron acceptor may result in the formation of an acceptor gradient to which the bacteria also respond, thus diminishing the positive effect of chemotaxis. Depletion of an electron acceptor can also reduce the rate of electron donor consumption in addition to its effect on chemotaxis. In this study, we examined the effect of oxygen on chemotaxis to naphthalene and on naphthalene consumption by Pseudomonas putida G7. The organism was able to move up an oxygen gradient when there was a naphthalene gradient in the opposite direction. In the absence of an oxygen gradient, low levels of oxygen attenuated chemotaxis to naphthalene but did not affect random motility. The rate of naphthalene consumption decreased at dissolved oxygen concentrations similar to those at which chemotaxis was attenuated. These results suggest that low dissolved oxygen concentrations can reduce naphthalene removal by P. putida G7 in systems where naphthalene is heterogeneously distributed by simultaneously attenuating chemotactic motion toward naphthalene and decreasing the rate of naphthalene degradation.     

Impact of fluorochrome stains used to study bacterial transport in shallow aquifers on motility and chemotaxis of Pseudomonas species

One of the most common methods of tracking movement of bacteria in groundwater environments involves a priori fluorescent staining. A major concern in using these stains to label bacteria in subsurface injection-and-recovery studies is the effect they may have on the bacterium's transport properties. Previous studies investigated the impact of fluorophores on bacterial surface properties (e.g. zeta potential). However, no previous study has looked at the impact of fluorescent staining on swimming speed and chemotaxis. It was found that DAPI lowered the mean population swimming speed of Pseudomonas putida F1 by 46% and Pseudomonas stutzeri by 55%. DAPI also inhibited the chemotaxis in both strains. The swimming speeds of P. putida F1 and P. stutzeri were diminished slightly by CFDA/SE, but not to a statistically significant extent. CFDA/SE had no effect on chemotaxis of either strain to acetate. SYBR(?) Gold had no effect on swimming speed or the chemotactic response to acetate for either strain. This research indicates that although DAPI may not affect sorption to grain surfaces, it adversely affects other potentially important transport properties such as swimming and chemotaxis. Consequently, bacterial transport studies conducted using DAPI are biased to nonchemotactic conditions and do not appear to be suitable for monitoring the effect of chemotaxis on bacterial transport in shallow aquifers.     

Behaviors and Strategies of Bacterial Navigation in Chemical and Nonchemical Gradients

Navigation of cells to the optimal environmental condition is critical for their survival and growth. Escherichia coli cells, for example, can detect various chemicals and move up or down those chemical gradients (i.e., chemotaxis). Using the same signaling machinery, they can also sense other external factors such as pH and temperature and navigate from both sides toward some intermediate levels of those stimuli. This mode of precision sensing is more sophisticated than the (unidirectional) chemotaxis strategy and requires distinctive molecular mechanisms to encode and track the preferred external conditions. To systematically study these different bacterial taxis behaviors, we develop a continuum model that incorporates microscopic signaling events in single cells into macroscopic population dynamics. A simple theoretical result is obtained for the steady state cell distribution in general. In particular, we find the cell distribution is controlled by the intracellular sensory dynamics as well as the dependence of the cells'' speed on external factors. The model is verified by available experimental data in various taxis behaviors (including bacterial chemotaxis, pH taxis, and thermotaxis), and it also leads to predictions that can be tested by future experiments. Our analysis help reveal the key conditions/mechanisms for bacterial precision-sensing behaviors and directly connects the cellular taxis performances with the underlying molecular parameters. It provides a unified framework to study bacterial navigation in complex environments with chemical and non-chemical stimuli.     

Travelling Waves in Hybrid Chemotaxis Models

Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant), which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations.     

In chemotaxing fibroblasts, both high-fidelity and weakly biased cell movements track the localization of PI3K signaling

Cell movement biased by a chemical gradient, or chemotaxis, coordinates the recruitment of cells and collective migration of cell populations. During wound healing, chemotaxis of fibroblasts is stimulated by platelet-derived growth factor (PDGF) and certain other chemoattractants. Whereas the immediate PDGF gradient sensing response has been characterized previously at the level of phosphoinositide 3-kinase (PI3K) signaling, the sensitivity of the response at the level of cell migration bias has not yet been studied quantitatively. In this work, we used live-cell total internal reflection fluorescence microscopy to monitor PI3K signaling dynamics and cell movements for extended periods. We show that persistent and properly aligned (i.e., high-fidelity) fibroblast migration does indeed correlate with polarized PI3K signaling; accordingly, this behavior is seen only under conditions of high gradient steepness (>10% across a typical cell length of 50 μm) and a certain range of PDGF concentrations. Under suboptimal conditions, cells execute a random or biased random walk, but nonetheless move in a predictable fashion according to the changing pattern of PI3K signaling. Inhibition of PI3K during chemotaxis is accompanied by loss of both cell-substratum contact and morphological polarity, but after a recovery period, PI3K-inhibited fibroblasts often regain the ability to orient toward the PDGF gradient.     

Continuous-flow capillary assay for measuring bacterial chemotaxis

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

The chemokine receptor CCR7 activates in dendritic cells two signaling modules that independently regulate chemotaxis and migratory speed

CCR7 is necessary to direct dendritic cells (DCs) to secondary lymphoid nodes and to elicit an adaptative immune response. Despite its importance, little is known about the molecular mechanisms used by CCR7 to direct DCs to lymph nodes. In addition to chemotaxis, CCR7 regulates the migratory speed of DCs. We investigated the intracellular pathways that regulate CCR7-dependent chemotaxis and migratory speed. We found that CCR7 induced a G(i)-dependent activation of MAPK members ERK1/2, JNK, and p38, with ERK1/2 and p38 controlling JNK. MAPK members regulated chemotaxis, but not the migratory speed, of DCs. CCR7 induced activation of PI3K/Akt; however, these enzymes did not regulate either chemotaxis or the speed of DCs. CCR7 also induced activation of the GTPase Rho, the tyrosine kinase Pyk2, and inactivation of cofilin. Pyk2 activation was independent of G(i) and Src and was dependent on Rho. Interference with Rho or Pyk2 inhibited cofilin inactivation and the migratory speed of DCs, but did not affect chemotaxis. Interference with Rho/Pyk2/cofilin inhibited DC migratory speed even in the absence of chemokines, suggesting that this module controls the speed of DCs and that CCR7, by activating its components, induces an increase in migratory speed. Therefore, CCR7 activates two independent signaling modules, one involving G(i) and a hierarchy of MAPK family members and another involving Rho/Pyk2/cofilin, which control, respectively, chemotaxis and the migratory speed of DCs. The use of independent signaling modules to control chemotaxis and speed can contribute to regulate the chemotactic effects of CCR7.