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.     

Chemotaxis and random motility in unsteady chemoattractant fields: a computational study

We discuss a generic computational model which captures the effects of transient chemoattractant concentration on the chemotactic motility of individual cells. The model solves the appropriate unsteady chemoattractant transport equation using finite differences, while simultaneously executing biased random walks representing individual cells. The simulations were implemented for a 2D homogeneous domain, and two case studies were considered. In the first case study, we consider a single-point source at the origin of the domain which produces chemoattractant, while other cells execute biased random walks toward this point source. We observe that for continuous chemoattractant production, chemoattractant diffusivity has no effect on cell motility, as measured by the mean of time to reach the source. However, in the case of pulsed random production with a specific average duty cycle, the mean time-to-contact is generally minimal with respect to chemoattractant diffusivity over a moderate range of diffusivities. In the second case study, two mobile cells which simultaneously secrete chemoattractant are initially placed a certain distance apart and are then allowed to execute biased random walks. Our model shows that a pulsed random protocol for chemoattractant production facilitates the two cells "finding" one another compared to continuous production. From this case study we also learn that there exists a range of moderate chemoattractant diffusivities for which the mean time-to-contact is minimal when cells both produce/detect chemoattractant and chemotactically migrate. Using these case studies, we discuss how transience in chemoattractant concentration becomes important in characterizing the effectiveness of chemotaxis.     

Excitable Signal Transduction Induces Both Spontaneous and Directional Cell Asymmetries in the Phosphatidylinositol Lipid Signaling System for Eukaryotic Chemotaxis

Intracellular asymmetry in the signaling network works as a compass to navigate eukaryotic chemotaxis in response to guidance cues. Although the compass variable can be derived from a self-organization dynamics, such as excitability, the responsible mechanism remains to be clarified. Here, we analyzed the spatiotemporal dynamics of the phosphatidylinositol 3,4,5-trisphosphate (PtdInsP3) pathway, which is crucial for chemotaxis. We show that spontaneous activation of PtdInsP3-enriched domains is generated by an intrinsic excitable system. Formation of the same signal domain could be triggered by various perturbations, such as short impulse perturbations that triggered the activation of intrinsic dynamics to form signal domains. We also observed the refractory behavior exhibited in typical excitable systems. We show that the chemotactic response of PtdInsP3 involves biasing the spontaneous excitation to orient the activation site toward the chemoattractant. Thus, this biased excitability embodies the compass variable that is responsible for both random cell migration and biased random walk. Our finding may explain how cells achieve high sensitivity to and robust coordination of the downstream activation that allows chemotactic behavior in the noisy environment outside and inside the cells.     

Excitable Signal Transduction Induces Both Spontaneous and Directional Cell Asymmetries in the Phosphatidylinositol Lipid Signaling System for Eukaryotic Chemotaxis

Intracellular asymmetry in the signaling network works as a compass to navigate eukaryotic chemotaxis in response to guidance cues. Although the compass variable can be derived from a self-organization dynamics, such as excitability, the responsible mechanism remains to be clarified. Here, we analyzed the spatiotemporal dynamics of the phosphatidylinositol 3,4,5-trisphosphate (PtdInsP3) pathway, which is crucial for chemotaxis. We show that spontaneous activation of PtdInsP3-enriched domains is generated by an intrinsic excitable system. Formation of the same signal domain could be triggered by various perturbations, such as short impulse perturbations that triggered the activation of intrinsic dynamics to form signal domains. We also observed the refractory behavior exhibited in typical excitable systems. We show that the chemotactic response of PtdInsP3 involves biasing the spontaneous excitation to orient the activation site toward the chemoattractant. Thus, this biased excitability embodies the compass variable that is responsible for both random cell migration and biased random walk. Our finding may explain how cells achieve high sensitivity to and robust coordination of the downstream activation that allows chemotactic behavior in the noisy environment outside and inside the cells.     

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.     

Three-dimensional simulation of obstacle-mediated chemotaxis

Amoeboid cells exhibit a highly dynamic motion that can be directed by external chemical signals, through the process of chemotaxis. Here, we propose a three-dimensional model for chemotactic motion of amoeboid cells. We account for the interactions between the extracellular substances, the membrane-bound proteins, and the cytosolic components involved in the signaling pathway that originates cell motility. We show two- and three-dimensional simulations of cell migration on planar substrates, flat surfaces with obstacles, and fibrous networks. The results show that our model reproduces the main features of chemotactic amoeboid motion. Our simulations unveil a complicated interplay between the geometry of the cell’s environment and the chemoattractant dynamics that tightly regulates cell motion. The model opens new opportunities to simulate the interactions between extra- and intra-cellular compounds mediated by the matrix geometry.     

Directional Migration of Recirculating Lymphocytes through Lymph Nodes via Random Walks

Naive T lymphocytes exhibit extensive antigen-independent recirculation between blood and lymph nodes, where they may encounter dendritic cells carrying cognate antigen. We examine how long different T cells may spend in an individual lymph node by examining data from long term cannulation of blood and efferent lymphatics of a single lymph node in the sheep. We determine empirically the distribution of transit times of migrating T cells by applying the Least Absolute Shrinkage & Selection Operator () or regularised to fit experimental data describing the proportion of labelled infused cells in blood and efferent lymphatics over time. The optimal inferred solution reveals a distribution with high variance and strong skew. The mode transit time is typically between 10 and 20 hours, but a significant number of cells spend more than 70 hours before exiting. We complement the empirical machine learning based approach by modelling lymphocyte passage through the lymph node . On the basis of previous two photon analysis of lymphocyte movement, we optimised distributions which describe the transit times (first passage times) of discrete one dimensional and continuous (Brownian) three dimensional random walks with drift. The optimal fit is obtained when drift is small, i.e. the ratio of probabilities of migrating forward and backward within the node is close to one. These distributions are qualitatively similar to the inferred empirical distribution, with high variance and strong skew. In contrast, an optimised normal distribution of transit times (symmetrical around mean) fitted the data poorly. The results demonstrate that the rapid recirculation of lymphocytes observed at a macro level is compatible with predominantly randomised movement within lymph nodes, and significant probabilities of long transit times. We discuss how this pattern of migration may contribute to facilitating interactions between low frequency T cells and antigen presenting cells carrying cognate antigen.     

“Conjugate Channeling” Effect in Dislocation Core Diffusion: Carbon Transport in Dislocated BCC Iron

Dislocation pipe diffusion seems to be a well-established phenomenon. Here we demonstrate an unexpected effect, that the migration of interstitials such as carbon in iron may be accelerated not in the dislocation line direction , but in a conjugate diffusion direction. This accelerated random walk arises from a simple crystallographic channeling effect. is a function of the Burgers vector b, but not , thus a dislocation loop possesses the same everywhere. Using molecular dynamics and accelerated dynamics simulations, we further show that such dislocation-core-coupled carbon diffusion in iron has temperature-dependent activation enthalpy like a fragile glass. The 71° mixed dislocation is the only case in which we see straightforward pipe diffusion that does not depend on dislocation mobility.     

Cell transit analysis of ligand-induced stiffening of polymorphonuclear leukocytes.

A mathematical treatment of the mechanical behavior of transiently bonded polymer networks is used to interpret measurements of the pressure-induced passage of plant cells through microporous membranes. Cell transit times are inferred to be proportional to the instantaneous shear modulus of the cell cortex, a parameters that we then relate to properties of the cortical F-actin matrix. These theoretical results are used to analyze published data on chemoattractant-induced changes of rigidity of polymorphonuclear leukocytes. We thereby rationalize previously noted, peculiar, power-law logarithmic dependences of transit time on ligand concentration. As a consequence, we are able to deduce a linear relationship between the extent of F-actin polymerization and the logarithm of the chemoattractant concentration. The latter is examined with regard to the G-protein activation that is known to occur when chemoattractants bind to receptors on the surfaces of polymorphonuclear cells.     

Degranulation and superoxide production depend on cholesterol in PLB-985 cells

The effect of agents disrupting cholesterol-rich microdomains of the cell membrane was studied on the chemoattractant receptor (FPR and FRPL1) coupled effector responses of promyelocytic PLB-985 cells. Both methyl-beta-cyclodextrin (MbetaCD) and filipin III inhibited exocytosis of primary granules and O(2)(.-) production induced by stimulation of either chemotactic receptor. Alteration of calcium homeostasis of MbetaCD-treated cells does not account for the impairment of the effector responses. Disruption of microfilaments by cytochalasin B (CB) partially reverses the inhibitory effect of cholesterol depletion. Our results provide functional support for the involvement of cholesterol-rich membrane domains in the signaling of chemotactic receptors and call the attention to the possible role of microfilaments in the organization of lipid microdomains.     

Random Wiring,Ganglion Cell Mosaics,and the Functional Architecture of the Visual Cortex

The architecture of iso-orientation domains in the primary visual cortex (V1) of placental carnivores and primates apparently follows species invariant quantitative laws. Dynamical optimization models assuming that neurons coordinate their stimulus preferences throughout cortical circuits linking millions of cells specifically predict these invariants. This might indicate that V1’s intrinsic connectome and its functional architecture adhere to a single optimization principle with high precision and robustness. To validate this hypothesis, it is critical to closely examine the quantitative predictions of alternative candidate theories. Random feedforward wiring within the retino-cortical pathway represents a conceptually appealing alternative to dynamical circuit optimization because random dimension-expanding projections are believed to generically exhibit computationally favorable properties for stimulus representations. Here, we ask whether the quantitative invariants of V1 architecture can be explained as a generic emergent property of random wiring. We generalize and examine the stochastic wiring model proposed by Ringach and coworkers, in which iso-orientation domains in the visual cortex arise through random feedforward connections between semi-regular mosaics of retinal ganglion cells (RGCs) and visual cortical neurons. We derive closed-form expressions for cortical receptive fields and domain layouts predicted by the model for perfectly hexagonal RGC mosaics. Including spatial disorder in the RGC positions considerably changes the domain layout properties as a function of disorder parameters such as position scatter and its correlations across the retina. However, independent of parameter choice, we find that the model predictions substantially deviate from the layout laws of iso-orientation domains observed experimentally. Considering random wiring with the currently most realistic model of RGC mosaic layouts, a pairwise interacting point process, the predicted layouts remain distinct from experimental observations and resemble Gaussian random fields. We conclude that V1 layout invariants are specific quantitative signatures of visual cortical optimization, which cannot be explained by generic random feedforward-wiring models.     

Residence time calculation for chemotactic bacteria within porous media

Local chemical gradients can have a significant impact on bacterial population distributions within subsurface environments by evoking chemotactic responses. These local gradients may be created by consumption of a slowly diffusing nutrient, generation of a local food source from cell lysis, or dissolution of nonaqueous phase liquids trapped within the interstices of a soil matrix. We used a random walk simulation algorithm to study the effect of a local microscopic gradient on the swimming behavior of bacteria in a porous medium. The model porous medium was constructed using molecular dynamics simulations applied to a fluid of equal-sized spheres. The chemoattractant gradient was approximated with spherical symmetry, and the parameters for the swimming behavior of soil bacterium Pseudomonas putida were based on literature values. Two different mechanisms for bacterial chemotaxis, one in which the bacteria responded to both positive and negative gradients, and the other in which they responded only to positive gradients, were compared. The results of the computer simulations showed that chemotaxis can increase migration through a porous medium in response to microscopic-scale gradients. The simulation results also suggested that a more significant role of chemotaxis may be to increase the residence time of the bacteria in the vicinity of an attractant source.     

Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms

We study kinetic models for chemotaxis, incorporating the ability of cells to assess temporal changes of the chemoattractant concentration as well as its spatial variations. For prescribed smooth chemoattractant density, the macroscopic limit is carried out rigorously. It leads to a drift equation with a chemotactic sensitivity depending on the time derivative of the chemoattractant density. As an application it is shown by numerical experiments that the new model can resolve the chemotactic wave paradox. For this purpose, the macroscopic equation is coupled to a simple activation-inhibition model for the chemoattractant which produces the chemoattractant waves typical for the slime mold Dictyostelium discoideum.     

Compartmentalized structure of the plasma membrane for receptor movements as revealed by a nanometer-level motion analysis

Movements of transferrin and alpha 2-macroglobulin receptor molecules in the plasma membrane of cultured normal rat kidney (NRK) fibroblastic cells were investigated by video-enhanced contrast optical microscopy with 1.8 nm spatial precision and 33 ms temporal resolution by labeling the receptors with the ligand-coated nanometer-sized colloidal gold particles. For both receptor species, most of the movement trajectories are of the confined diffusion type, within domains of approximately 0.25 microns2 (500-700 nm in diagonal length). Movement within the domains is random with a diffusion coefficient approximately 10(-9) cm2/s, which is consistent with that expected for free Brownian diffusion of proteins in the plasma membrane. The receptor molecules move from one domain to one of the adjacent domains at an average frequency of 0.034 s-1 (the residence time within a domain approximately 29 s), indicating that the plasma membrane is compartmentalized for diffusion of membrane receptors and that long- range diffusion is the result of successive intercompartmental jumps. The macroscopic diffusion coefficients for these two receptor molecules calculated on the basis of the compartment size and the intercompartmental jump rate are approximately 2.4 x 10(-11) cm2/s, which is consistent with those determined by averaging the long-term movements of many particles. Partial destruction of the cytoskeleton decreased the confined diffusion mode, increased the simple diffusion mode, and induced the directed diffusion (transport) mode. These results suggest that the boundaries between compartments are made of dynamically fluctuating membrane skeletons (membrane-skeleton fence model).     

Cells responding to chemoattractant on a structured substrate

Cell migration on an adhesive substrate surface comprises actin-based protrusion at the front and retraction of the tail in combination with coordinated adhesion to, and detachment from, the substrate. To study the effect of cell-to-substrate adhesion on the chemotactic response of Dictyostelium discoideum cells, we exposed the cells to patterned substrate surfaces consisting of adhesive and inert areas, and forced them by a gradient of chemoattractant to enter the border between the two areas. Wild-type as well as myosin II-deficient cells stop at the border of an adhesive area. They do not detach with their rear part, while on the nonadhesive area they protrude pseudopods at their front toward the source of chemoattractant. Avoidance of the nonadhesive area may cause a cell to move in tangential direction relative to the attractant gradient, keeping its tail at the border of the adhesive surface.     

Gyrase and Topo IV modulate chromosome domain size in vivo

In bacteria, DNA supercoil movement is restricted to subchromosomal regions or 'domains.' To elucidate the nature of domain boundaries, we analysed reaction kinetics for γδ site-specific resolution in six chromosomal intervals ranging in size from 14 to 90 kb. In stationary cultures of Salmonella typhimurium , resolution kinetics were rapid for both short and long intervals, suggesting that random stationary barriers occur with a 30% probability at approximately 80 kb intervals along DNA. To test the biochemical nature of domain barriers, a genetic screen was used to look for mutants with small domains. Rare temperature-sensitive alleles of DNA gyrase and Topo IV (the two essential type II topoisomerases) had more supercoil barriers than wild-type strains in all growth states. The most severe gyrase mutants were found to have twice as many barriers in growing cells as wild type throughout a 90 kb interval of the chromosome. We propose that knots and tangles in duplex DNA restrain supercoil diffusion in living bacteria.     

Human galectin-3 is a novel chemoattractant for monocytes and macrophages

Galectin-3 is a beta-galactoside-binding protein implicated in diverse biological processes. We found that galectin-3 induced human monocyte migration in vitro in a dose-dependent manner, and it was chemotactic at high concentrations (1.0 microM) but chemokinetic at low concentrations (10-100 nM). Galectin-3-induced monocyte migration was inhibited by its specific mAb and was blocked by lactose and a C-terminal domain fragment of the protein, indicating that both the N-terminal and C-terminal domains of galectin-3 are involved in this activity. Pertussis toxin (PTX) almost completely blocked monocyte migration induced by high concentrations of galectin-3. Galectin-3 caused a Ca2+ influx in monocytes at high, but not low, concentrations, and both lactose and PTX inhibited this response. There was no cross-desensitization between galectin-3 and any of the monocyte-reactive chemokines examined, including monocyte chemotactic protein-1, macrophage inflammatory protein-1alpha, and stromal cell-derived factor-1alpha. Cultured human macrophages and alveolar macrophages also migrated toward galectin-3, but not monocyte chemotactic protein-1. Finally, galectin-3 was found to cause monocyte accumulation in vivo in mouse air pouches. These results indicate that galectin-3 is a novel chemoattractant for monocytes and macrophages and suggest that the effect is mediated at least in part through a PTX-sensitive (G protein-coupled) pathway.     

Study of the Chemotactic Response of Multicellular Spheroids in a Microfluidic Device

We report the first application of a microfluidic device to observe chemotactic migration in multicellular spheroids. A microfluidic device was designed comprising a central microchamber and two lateral channels through which reagents can be introduced. Multicellular spheroids were embedded in collagen and introduced to the microchamber. A gradient of fetal bovine serum (FBS) was established across the central chamber by addition of growth media containing serum into one of the lateral channels. We observe that spheroids of oral squamous carcinoma cells OSC–19 invade collectively in the direction of the gradient of FBS. This invasion is more directional and aggressive than that observed for individual cells in the same experimental setup. In contrast to spheroids of OSC–19, U87-MG multicellular spheroids migrate as individual cells. A study of the exposure of spheroids to the chemoattractant shows that the rate of diffusion into the spheroid is slow and thus, the chemoattractant wave engulfs the spheroid before diffusing through it.     

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.     

Stochastic signal inputs for chemotactic response in Dictyostelium cells revealed by single molecule imaging techniques

Chemotactic cells can exhibit extreme sensitivity to chemical gradients. Theoretical estimations of the signal inputs required for chemotaxis suggest that the response can be achieved under the strong influence of stochastic input noise generated by the receptors during the transmembrane signaling. This arises a fundamental question regarding the mechanisms for directional sensing: how do cells obtain reliable information regarding gradient direction by using stochastically operating receptors and the downstream molecules? To address this question, we have developed single molecule imaging techniques to visualize signaling molecules responsible for chemotaxis in living Dictyostelium cells, allowing us to monitor the stochastic signaling processes directly. Single molecule imaging of a chemoattractant bound to a receptor demonstrates that signal inputs fluctuate with time and space. Downstream signaling molecules, such as PTEN and a PH domain-containing protein that are constituent parts of chemotactic signaling system, can also be followed at single molecule level in living cells, illuminating the stochastic nature of chemotactic signaling processes. In this report, we start with a brief introduction of chemotactic response of the eukaryotic cells, followed by an explanation for single molecule imaging techniques, and finally discuss these applications to chemotactic signaling system of Dictyostelium cells.