Model for chemotactic bacterial bands

Suspensions of chemotactic bacteria develop spatial and temporal structures in response to an initial inhomogeneity in the medium. A theoretical model is presented for the analysis of spatial and temporal evolution of bacterial bands in response to several attractants. Applications of the model to various experimental cases give good agreement between theory and observation. The theoretical analysis provides further insight to the mechanisms governing band formation and band migration.     

Dynamics of bacterial population waves as dependent on the type of the chemotactic response function

A study was made of the time of formation and the rate of propagation of bacterial population waves as dependent on substrate concentration. Experimentally and theoretically it is shown that the time of bacterial wave formation grows with increasing concentration of the nutritive component. At the same time, the wave propagation rate drops at relatively high concentrations of the nutritive component. In numerical experiments with a mathematical model, comparative analysis has been made of the dynamics of the behavior of bacterial waves at different chemotactic responses.     

Model and analysis of chemotactic bacterial patterns in a liquid medium

 A variety of spatial patterns are formed chemotactically by the bacteria Escherichia coli and Salmonella typhimurium. We focus in this paper on patterns formed by E. coli and S. typhimurium in liquid medium experiments. The dynamics of the bacteria, nutrient and chemoattractant are modeled mathematically and give rise to a nonlinear partial differential equation system. We present a simple and intuitively revealing analysis of the patterns generated by our model. Patterns arise from disturbances to a spatially uniform solution state. A linear analysis gives rise to a second order ordinary differential equation for the amplitude of each mode present in the initial disturbance. An exact solution to this equation can be obtained, but a more intuitive understanding of the solutions can be obtained by considering the rate of growth of individual modes over small time intervals. Received: 10 March 1998 / Revised version: 7 June 1998     

Kinetic analysis of behavior of chemotactic bacterial population at the interface of two media

The behavior of a population of bacteria at the boundary with a chemotactically active water medium containing the sites of specific binding to cell receptors was analyzed. A kinetic model of chemotaxis was used for the analysis. Differences in the behavior of strains metabolizing and not metabolizing the substrate were revealed. Six phases of interface taxis were distinguished and characterized. The results of the analysis were confirmed by the densitometric data.     

Studies on bacterial chemotaxis. III. Effect of methyl esters on the chemotactic response of Escherichia coli.

The methylation-demethylation reaction of methyl-accepting chemotaxis protein (MCP) is tightly coupled to the appearance of the chemotactic response in Escherichia coli. The bacteria might therefore show a unique response upon the addition of a compound containing a methyl group. We selected methyl N-methyl anthranilate (NMMA) and its analogs for examination. When NMMA was added to a suspension of E. coli (wild type), the bacteria tumbled as it does in the presence of a repellent. NMMA caused tumbling of wild-type bacteria for at least 20 min, while a conventional repellent makes the bacteria tumble for at most one min. The effect of NMMA requires functional MCP, cheA gene product, cheB gene product, and possibly cheX gene product. A positive signal of NMMA (i.e. sudden dilution) was detected by cheZ mutants with much higher sensitivity than that of a conventional repellent, indole, while both signals were rather poorly but equally detected by cheB mutants. These results suggest that the drug is related to the function of cheB gene product, a possible demethylating enzyme of MCP.     

Model of the bacterial flagellar motor: response to varying viscous load.

A model of bacterial flagellar drive by cytomembrane streaming is described and applied to experiments of bacterial propulsion under varying viscous load. The theory predicts a linear dependence of the reciprocal propulsion velocity on the viscosity of the suspension medium, if the velocity of cytomembrane streaming far from the basal body of the flagella is assumed independent of the external viscosity. The theoretical predictions are in good agreement with experiments on free-swimming and on tethered bacteria. From comparison of the theory with experiments, the surface viscosity of the bacterial cytomembrane is evaluated for different bacterial species and turns out to be in the range observed experimentally on lipid monolayers.     

Motility response of Rhodobacter sphaeroides to chemotactic stimulation.

Tethered rotating cells of Rhodobacter sphaeroides varied widely in their stopping frequency; 45% of cells showed no stops of longer than 1 s, whereas others showed stops of up to several seconds. Individual cells alternated between stops and rotation at a fairly constant rate, without continuous variation. Addition of the chemoattractant propionate to free-swimming cells of R. sphaeroides increased the mean population swimming speed from 15 to 23 microns s-1. After correction for nonmotile cells, the percentage swimming at less than 5 microns s-1 dropped from approximately 22 to 8, whereas the percentage swimming at greater than 50 microns s-1 increased from 6 to 15. However, cells already swimming did not swim faster after propionate addition; the increase in the mean population speed after propionate addition was caused by an increase in the mean run length between stops from 25 to 101 microns. The increased run length was the result of a drop in both the stopping frequency and the length of a stop. Addition of propionate over the range of 10 microM to 1 mM decreased the stopping frequency; this decrease was almost entirely blocked by benzoate, a competitive inhibitor of propionate transport. The chemoattractants acetate and potassium had the same effect as propionate on the distribution of stopping frequency, which demonstrated that this is a general behavioral response to chemotactic stimulation. Adaptation to propionate stimulation was slow and very variable, cultures frequently showing little adaptation over 30 min. This characteristic may be the result of the lack of a highly specific chemosensory system in R. sphaeroides.     

Neutrophil hyperpolarization in response to a chemotactic peptide

The chemotactic peptide formylmethionyl-leucyl-phenylalanine (fMLP), at concentrations below 10(-9) M, elicits a sustained increase in the human neutrophil's membrane potential within 10 s of its addition. This hyperpolarization, detected with the fluorescent cationic potentiometric probes, 3,3'-dipentyloxacarbocyanine (diO-C5-(3)), and 1,1'-dipropyl-3,3,3',3'-tetramethylindocarbocyanine iodide (diI-C3-(3)), and with the anionic probe bis-(1,3-diethylthiobarbituric)trimethine oxonol (bis-oxonol), is immediately followed by a large depolarization when [fMLP] greater than 10(-9) M. By extracellular substitution of sodium ions with potassium ions or choline or by pretreatment of the cells with ionophores, we report here that the hyperpolarization is primarily dependent on an intact potassium ion gradient and is accompanied by a concurrent acidification of the cytoplasm (approximately 0.05 pH unit) Although the latter occurs simultaneously with a large, transient increase in cytosolic Ca2+ at [fMLP] greater than 10(-10) M, it occurs without a detectable increase in cytosolic Ca2+ at [fMLP] less than 10(-10) M. The hyperpolarization is neither affected nor initiated by the chemotactic peptide antagonist tert-butyloxycarbonyl-methionyl-leucyl-phenylalanine, whereas the depolarization is completely inhibited. Neutrophils isolated from patients with X-linked chronic granulomatous disease exhibit normal hyperpolarizations and cytosolic Ca2+ increases in response to chemotactic peptides but exhibit no depolarization or oxidative burst. The hyperpolarization appears earlier in the ontogeny of differentiating myeloid precursor cells than either the rise in cytosolic Ca2+ or the depolarization response. Together, these findings indicate that an increase in transmembrane potential is one of the earliest events in the neutrophil response to chemotactic peptides, coinciding temporally with increases in cytoplasmic Ca2+ and H+ concentrations but preceding detectable oxidative burst activity.     

The chemotactic response to PDGF-BB: evidence of a role for Ras

The PDGF receptor-beta mediates both mitogenic and chemotactic responses to PDGF-BB. Although the role of Ras in tyrosine kinase- mediated mitogenesis has been characterized extensively, its role in PDGF-stimulated chemotaxis has not been defined. Using cells expressing a dominant-negative ras, we find that Ras inhibition suppresses migration toward PDGF-BB. Overexpression of either Ras-GTPase activating protein (Ras-GAP) or a Ras guanine releasing factor (GRF) also inhibited PDGF-stimulated chemotaxis. In addition, cells producing excess constitutively active Ras failed to migrate toward PDGF-BB, consistent with the observation that either excess ligand or excess signaling intermediate can suppress the chemotactic response. These results suggest that Ras can function in normal cells to support chemotaxis toward PDGF-BB and that either too little or too much Ras activity can abrogate the chemotactic response. In contrast to Ras overexpression, cells producing excess constitutively active Raf, a downstream effector of Ras, did migrate toward PDGF-BB. Cells expressing dominant-negative Ras were able to migrate toward soluble fibronectin demonstrating that these cells retained the ability to migrate. These results suggest that Ras is an intermediate in PDGF- stimulated chemotaxis but may not be required for fibronectin- stimulated cell motility.     

Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients

The mathematical model developed by Riveroet al. (1989,Chem. Engng Sci. 44, 2881–2897) is applied to literature data measuring chemotactic bacterial population distributions in response to steep as well as shallow attractant gradients. This model is based on a fundamental picture of the sensing and response mechanisms of individual bacterial cells, and thus relates individual cell properties such as swimming speed and tumbling frequency to population parameters such as the random motility coefficient and the chemotactic sensitivity coefficient. Numerical solution of the model equations generates predicted bacterial density and attractant concentration profiles for any given experimental assay. We have previously validated the mathematical model from experimental work involving a step-change in the attractant gradient (Fordet al., 1991Biotechnol. Bioengng.37, 647–660; For and Lauffenburger, 1991,Biotechnol. Bioengng,37, 661–672). Within the context of this experimental assay, effects of attractant diffusion and consumption, random motility, and chemotactic sensitivity on the shape of the profiles are explored to enhance our understanding of this complex phenomenon. We have applied this model to various other types of gradients with successful intepretation of data reported by Dalquistet al. (1972,Nature New Biol. 236, 120–123) forSalmonella typhimurum validating the mathematical model and supportin the involvement of high and low affinity receptors for serine chemotaxis by these cells.     

Utilization of triaryl phosphates by a mixed bacterial population.

A mixed bacterial population that has been isolated by enrichment culture is capable of growth on Fyrquel 220, a commercial triaryl phosphate lubricant, as sole carbon source. The mixture was dominated by a yellow, Gram-negative rod which made up greater than 60% of the mixture. However, all attempts to grow this organism in pure culture on triaryl phosphate were unsuccessful. The mixed population was also capable of growth on tri-o-cresyl phosphate, trixylenyl phosphate, and triphenyl phosphate as sole carbon sources. Viable cell numbers increased 20- to 30-fold, reaching a maximum after 72-96 h growth. Only a small portion of the triaryl phosphate was used for growth; the major part was emulsified and remained in the culture medium. No evidence of extracellular enzymes capable of triaryl phosphate degradation could be found in concentrates of the culture supernatant after growth, though traces of what may have been triaryl phosphate breakdown products were observed. Cell-free extracts of the mixed culture catalyzed the release of inorganic phosphate when incubated with Fyrquel 220, tri-o-cresyl phosphate, trixylenyl phosphate, or triphenyl phosphate, indicating the presence of a phosphotriesterase or of a phosphodiesterase of wide specificity.     

Development and applications of a model for cellular response to multiple chemotactic cues

The chemotactic response of a cell population to a single chemical species has been characterized experimentally for many cell types and has been extensively studied from a theoretical standpoint. However, cells frequently have multiple receptor types and can detect and respond chemotactically to more than one chemical. How these signals are integrated within the cell is not known, and we therefore adopt a macroscopic phenomenological approach to this problem. In this paper we derive and analyze chemotactic models based on partial differential (chemotaxis) equations for cell movement in response to multiple chemotactic cues. Our derivation generalizes the approach of Othmer and Stevens [29], who have recently developed a modeling framework for studying different chemotactic responses to a single chemical species. The importance of such a generalization is illustrated by the effect of multiple chemical cues on the chemotactic sensitivity and the spatial pattern of cell densities in several examples. We demonstrate that the model can generate the complex patterns observed on the skin of certain animal species and we indicate how the chemotactic response can be viewed as a form of positional indicator. Received: 15 February 1999 / Revised version: 1 February 2000 / Published online: 14 September 2000     

Bias in the gradient-sensing response of chemotactic cells

We apply linear stability theory and perform perturbation studies to better characterize, and to generate new experimental predictions from, a model of chemotactic gradient sensing in eukaryotic cells. The model uses reaction-diffusion equations to describe 3(') phosphoinositide signaling and its regulation at the plasma membrane. It demonstrates a range of possible gradient-sensing mechanisms and captures such characteristic behaviors as strong polarization in response to static gradients, adaptation to differing mean levels of stimulus, and plasticity in response to changing gradients. An analysis of the stability of polarized steady-state solutions indicates that the model is most sensitive to off-axis perturbations. This biased sensitivity is also reflected in responses to localized external stimuli, and leads to a clear experimental prediction, namely, that a cell which is polarized in a background gradient will be most sensitive to transient point-source stimuli lying within a range of angles that are oblique with respect to the polarization axis. Stimuli at angles below this range will elicit responses whose directions overshoot the stimulus angle, while responses to stimuli applied at larger angles will undershoot the stimulus angle. We argue that such a bias is likely to be a general feature of gradient sensing in highly motile cells, particularly if they are optimized to respond to small gradients. Finally, an angular bias in gradient sensing might lead to preferred turn angles and zigzag movements of cells moving up chemotactic gradients, as has been noted under certain experimental conditions.     

Role of metabolism in the chemotactic response of Rhodobacter sphaeroides to ammonia.

Rhodobacter sphaeroides only showed chemotaxis towards ammonia if grown under nitrogen-limited conditions. This chemotactic response was completely inhibited by the addition of methionine sulfoximine. There was no effect of methionine sulfoximine treatment on motility or taxis towards propionate, demonstrating that the effect is specific to ammonia taxis. It is known that methionine sulfoximine inhibits glutamine synthetase and hence blocks ammonia assimilation. Methionine sulfoximine does not inhibit ammonia transport in R. sphaeroides; therefore, these results suggest that limited metabolism via a specific pathway is required subsequent to transport to elicit a chemotactic response to ammonia. Bacteria grown on high ammonia show transport but no chemotactic response to ammonia, suggesting that the pathway of assimilation is important in eliciting a chemotactic response.     

Theoretical results for chemotactic response and drift of E. coli in a weak attractant gradient

The bacterium Escherichia coli (E. coli) moves in its natural environment in a series of straight runs, interrupted by tumbles which cause change of direction. It performs chemotaxis towards chemo-attractants by extending the duration of runs in the direction of the source. When there is a spatial gradient in the attractant concentration, this bias produces a drift velocity directed towards its source, whereas in a uniform concentration, E. coli adapts, almost perfectly in case of methyl aspartate. Recently, microfluidic experiments have measured the drift velocity of E. coli in precisely controlled attractant gradients, but no general theoretical expression for the same exists. With this motivation, we study an analytically soluble model here, based on the Barkai-Leibler model, originally introduced to explain the perfect adaptation. Rigorous mathematical expressions are obtained for the chemotactic response function and the drift velocity in the limit of weak gradients and under the assumption of completely random tumbles. The theoretical predictions compare favorably with experimental results, especially at high concentrations. We further show that the signal transduction network weakens the dependence of the drift on concentration, thus enhancing the range of sensitivity.     

Calcium modulation and chemotactic response: divergent stimulation of neutrophil chemotaxis and cytosolic calcium response by the chemotactic peptide receptor

Stimulation of neutrophils by chemoattractants is followed by a rapid, transient rise in cytosolic calcium concentration. The role of calcium in activation of cell movement and related responses was examined by selectively chelating extracellular or both extra- and intracellular calcium. Removal of calcium from the extracellular medium did not alter the cytosolic calcium concentration (Quin 2 fluorescence, 110 to 120 nM) of unstimulated neutrophils and did not dramatically affect the rise induced by formyl peptide. Despite the intact Quin 2 response, depletion of extracellular calcium partially inhibited chemotaxis, adherence to substrate, and polarization (increased forward light scatter) in response to formyl peptide. Loading neutrophils with Quin 2 in the absence of calcium depressed cytosolic Ca2+ to 10 to 20 nM and abrogated a detectable rise with formyl peptide stimulation. Depletion of intracellular calcium further inhibited chemotaxis and polarization, although neutrophils still demonstrated significant directed migration and shape change to formyl peptide (30 to 40% of control) without an increase in Quin 2 fluorescence. Other neutrophil responses related to chemotaxis (decreased right-angle light scatter, actin polymerization) were minimally affected by depletion of calcium from either site. The data indicate that neutrophil chemotaxis and related responses to formyl peptide may be activated by intracellular signals not detectable with Quin 2.     

The bacterial population of a blanket peat

The number of aerobic bacteria in a blanket peat decreased with depth from 26 times 10⁶/g dry peat in the surface layers to 0.5 times 10⁶/g dry peat at 30–40 cm down the profile, thereafter remaining roughly constant. Obligate psychrophiles comprised <2.5% of this population. Anaerobes were most numerous, 9 times 10⁶/g dry peat at 6–10 cm depth, decreasing to 0.5 times 10⁶/g at 20–30 cm. Calculations indicated that these counts, 10³–10⁴-fold lower than the direct counts, substantially underestimated the active microbial population. Gram negative rods, the predominant aerobes in the surface layers, were replaced by unidentified Bacillus strains at 10–20 cm depth but became increasingly more numerous further down the profile. The Gram negative rods were the most numerous organisms/m² but the Bacillus strains, one third of which were present as spores, made the largest contribution to the biomass/m². Gram positive cocci, Arthrobacter and, infrequently, Nocardia were also isolated. Actinomyces -like forms were the predominant obligate anaerobes and were approximately three times more numerous than clostridia and a curved Gram negative rod.