Measurement of leukocyte motility and chemotaxis parameters with a linear under-agarose migration assay

The interpretation of quantitative assays for leukocyte chemotactic migration is usually made in terms of measurements such as leading front distance, total migrating cells, and leukotactic index. These quantities allow comparison of cellular migration behavior under specified conditions. They are not useful; however, for comparisons between systems or for correlation with in vivo performance, because they depend upon specific physical aspects of the assay system, such as the geometry, chemoattractant concentration and diffusivity, and observation time. It would be more helpful to measure intrinsic properties of cell movement that could be used for comparison between systems, for correlation with in vivo studies, and to increase our understanding of the cell physiology. In this paper we demonstrate a means of quantitating leukocyte random motility, chemokinesis, and chemotaxis in terms of parameters that do characterize intrinsic cell properties. These parameters are the random motility coefficient and the chemotaxis coefficient, which appear in theoretical models of cell migration. We examine how well such a model describes the leukocyte density profile data observed in a modified under-agarose assay having a linear geometry. Furthermore, we obtain values for the random motility coefficient (and its dependence upon the concentration of the attractant peptide FNLLP) and for the chemotaxis coefficient for leukocytes responding to FNLLP.     

Transient behavior of a chemotaxis system modelling certain types of tissue inflammation

A spatially-distributed mathematical model for the inflammatory response to bacterial invasion of tissue is proposed which includes leukocyte motility and chemotaxis behavior and chemical mediator properties explicitly. This system involves three coupled nonlinear partial differential equations and so is not amenable to analysis. Using scaling arguments and singular perturbation techniques, an approximating system of two coupled nonlinear ordinary differential equations is developed. This system now permits analysis by phase plane methods. Using the approximating model, the dependence of the dynamic behavior of the inflammatory response upon key process parameters, including leukocyte chemotaxis, is studied.This work has been supported by the Deutsche Forschungsgemeinschaft     

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.     

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.     

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.     

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

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

Hydrodynamic effects on microcapillary motility and chemotaxis assays of Methylosinus trichosporium OB3b.

A study of the random motility and chemotaxis of Methylosinus trichosporium OB3b was conducted by using Palleroni-chamber microcapillary assay procedures. Under the growth conditions employed, this methanotroph was observed qualitatively with a microscope to be either slightly motile or essentially nonmotile. However, the cells did not not respond in the microcapillary assays in the manner expected for nonmotile Brownian particles. As a consequence, several hydrodynamic effects on these Palleroni microcapillary assays were uncovered. In the random-motility microcapillary assay, nondiffusive cell accumulations occurred that were strongly dependent upon cell concentration. An apparent minimal random-motility coefficient (mu) for this bacterial cell of 1.0 x 10(-7) cm2/s was estimated from microcapillary assays. A simple alternative spectrophotometric assay, based upon gravitational settling, was developed and shown to be an improvement over the Palleroni microcapillary motility assay for M. trichosporium OB3b in that it yielded a more-accurate threefold-lower random-motility coefficient. In addition, it provided a calculation of the gravitational-settling velocity. In the chemotaxis microcapillary assay, the apparent chemotactic responses were strongest for the highest test-chemical concentrations in the microcapillaries, were correlated with microcapillary fluid density, and were strongly dependent upon the microcapillary volume. A simple method to establish the maximal concentration of a chemical that can be tested and to quantify any contributions of abiotic convection is described. Investigators should be aware of the potential problems due to density-driven convection when using these commonly employed microcapillary assays for studying cells which have low motilities.     

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.     

Hydrodynamic effects on microcapillary motility and chemotaxis assays of Methylosinus trichosporium OB3b.

A study of the random motility and chemotaxis of Methylosinus trichosporium OB3b was conducted by using Palleroni-chamber microcapillary assay procedures. Under the growth conditions employed, this methanotroph was observed qualitatively with a microscope to be either slightly motile or essentially nonmotile. However, the cells did not not respond in the microcapillary assays in the manner expected for nonmotile Brownian particles. As a consequence, several hydrodynamic effects on these Palleroni microcapillary assays were uncovered. In the random-motility microcapillary assay, nondiffusive cell accumulations occurred that were strongly dependent upon cell concentration. An apparent minimal random-motility coefficient (mu) for this bacterial cell of 1.0 x 10(-7) cm2/s was estimated from microcapillary assays. A simple alternative spectrophotometric assay, based upon gravitational settling, was developed and shown to be an improvement over the Palleroni microcapillary motility assay for M. trichosporium OB3b in that it yielded a more-accurate threefold-lower random-motility coefficient. In addition, it provided a calculation of the gravitational-settling velocity. In the chemotaxis microcapillary assay, the apparent chemotactic responses were strongest for the highest test-chemical concentrations in the microcapillaries, were correlated with microcapillary fluid density, and were strongly dependent upon the microcapillary volume. A simple method to establish the maximal concentration of a chemical that can be tested and to quantify any contributions of abiotic convection is described. Investigators should be aware of the potential problems due to density-driven convection when using these commonly employed microcapillary assays for studying cells which have low motilities.     

Directional sensing during chemotaxis

Cells have the innate ability to sense and move towards a variety of chemoattractants. We investigate the pathways by which cells sense and respond to chemoattractant gradients. We focus on the model system Dictyostelium and compare our understanding of chemotaxis in this system with recent advances made using neutrophils and other mammalian cell types, which share many molecular components and signaling pathways with Dictyostelium. This review also examines models that have been proposed to explain how cells are able to respond to small differences in ligand concentrations between the anterior leading edge and posterior of the cell. In addition, we highlight the overlapping functions of many signaling components in diverse processes beyond chemotaxis, including random cell motility and cell division.     

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.     

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.     

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.     

A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations

Two central features of polymorphonuclear leukocyte chemosensory movement behavior demand fundamental theoretical understanding. In uniform concentrations of chemoattractant, these cells exhibit a persistent random walk, with a characteristic "persistence time" between significant changes in direction. In chemoattractant concentration gradients, they demonstrate a biased random walk, with an "orientation bias" characterizing the fraction of cells moving up the gradient. A coherent picture of cell movement responses to chemoattractant requires that both the persistence time and the orientation bias be explained within a unifying framework. In this paper, we offer the possibility that "noise" in the cellular signal perception/response mechanism can simultaneously account for these two key phenomena. In particular, we develop a stochastic mathematical model for cell locomotion based on kinetic fluctuations in chemoattractant/receptor binding. This model can simulate cell paths similar to those observed experimentally, under conditions of uniform chemoattractant concentrations as well as chemoattractant concentration gradients. Furthermore, this model can quantitatively predict both cell persistence time and dependence of orientation bias on gradient size. Thus, the concept of signal "noise" can quantitatively unify the major characteristics of leukocyte random motility and chemotaxis. The same level of noise large enough to account for the observed frequency of turning in uniform environments is simultaneously small enough to allow for the observed degree of directional bias in gradients.     

Contact-inhibited chemotaxis in de novo and sprouting blood-vessel growth

Blood vessels form either when dispersed endothelial cells (the cells lining the inner walls of fully formed blood vessels) organize into a vessel network (vasculogenesis), or by sprouting or splitting of existing blood vessels (angiogenesis). Although they are closely related biologically, no current model explains both phenomena with a single biophysical mechanism. Most computational models describe sprouting at the level of the blood vessel, ignoring how cell behavior drives branch splitting during sprouting. We present a cell-based, Glazier-Graner-Hogeweg model (also called Cellular Potts Model) simulation of the initial patterning before the vascular cords form lumens, based on plausible behaviors of endothelial cells. The endothelial cells secrete a chemoattractant, which attracts other endothelial cells. As in the classic Keller-Segel model, chemotaxis by itself causes cells to aggregate into isolated clusters. However, including experimentally observed VE-cadherin-mediated contact inhibition of chemotaxis in the simulation causes randomly distributed cells to organize into networks and cell aggregates to sprout, reproducing aspects of both de novo and sprouting blood-vessel growth. We discuss two branching instabilities responsible for our results. Cells at the surfaces of cell clusters attempting to migrate to the centers of the clusters produce a buckling instability. In a model variant that eliminates the surface-normal force, a dissipative mechanism drives sprouting, with the secreted chemical acting both as a chemoattractant and as an inhibitor of pseudopod extension. Both mechanisms would also apply if force transmission through the extracellular matrix rather than chemical signaling mediated cell-cell interactions. The branching instabilities responsible for our results, which result from contact inhibition of chemotaxis, are both generic developmental mechanisms and interesting examples of unusual patterning instabilities.     

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.     

Positive and negative signaling components involved in TNFalpha-induced NF-kappaB activation

Cell migration is involved in diverse physiological processes including embryogenesis, immunity, and diseases such as cancer and chronic inflammatory disease. The movement of many cell types is directed by extracellular gradients of diffusible chemicals. This phenomenon, referred to as "chemotaxis", was first described in 1888 by Leber who observed the movement of leukocytes toward sites of inflammation. We now know that a large family of small proteins, chemokines, serves as the extracellular signals and a family of G-protein-coupled receptors (GPCRs), chemokine receptors, detects gradients of chemokines and guides cell movement in vivo. Currently, we still know little about the molecular machineries that control chemokine gradient sensing and migration of immune cells. Fortunately, the molecular mechanisms that control these fundamental aspects of chemotaxis appear to be evolutionarily conserved, and studies in lower eukaryotic model systems have allowed us to form concepts, uncover molecular components, develop new techniques, and test models of chemotaxis. These studies have helped our current understanding of this complicated cell behavior. In this review, we wish to mention landmark discoveries in the chemotaxis research field that shaped our current understanding of this fundamental cell behavior and lay out key questions that remain to be addressed in the future.     

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.     

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 thermodynamical model of cell distributions in the slug of cellular slime mold

A theoretical model is proposed for the formation of cell distribution patterns in the slug stage of the cellular slime moldDictyostelium discoideum. The equilibrium distribution of two types of cells, prestalk and prespore, is obtained by minimizing the free energy, which is defined in terms of differential chemotaxis, differential cell adhesion and randomness of cell movement. Resulting distributions show various segregation patterns of cell types. The condition for cell sorting is obtained from stability analysis of the set of diffusion equations governing the evolution of cell type distribution and the concentration of chemoattractant. The intensities of differential chemotaxis and random cell movement are quantitatively evaluated from experimental data to show that two cell types can sort themselves completely by these forces.