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.     

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.     

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.     

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.     

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.     

Fucose-binding Lotus tetragonolobus lectin binds to human polymorphonuclear leukocytes and induces a chemotactic response.

Fucose-binding L. tetragonolobus lectin to the surface of human polymorphonuclear leukocytes (PMN) and induces a chemotactic response. Both surface binding and chemotaxis are inhibited by free fucose but not by fructose, mannose, or galactose. The lectin-binding sites on PMN are unrelated to the A, B, or O blood group antigen. Utilization of this lectin should be a useful tool in isolating PMN membrane components and in analyzing the mechanism of neutrophil chemotaxis.     

The effect of protein modification reagents on the chemotactic response in Salmonella typhimurium.

Several classes of reagents that covalently modify proteins have been shown to inhibit the ability of Salmonella typhimurium to reverse the direction of its flagellar rotation. Such reversal normally allows the bacterium to tumble and reorient its movement in a more favorable direction. These reagents include those that react with amino, guanidino, sulfhydryl, and disulfide groups on proteins. At high concentrations, most of these compounds also cause the paralysis of flagellar rotation. Tumbling in bacterial chemotaxis has been shown to be dependent on the methylation of a class of membrane proteins. The effects of these reagents in an in vitro methylation system have been studied. The results obtained suggest that most of these compounds are probably not acting on intact cells by inhibiting the activities of the cheR methyltransferase or the methyl-accepting proteins.     

Optimal noise filtering in the chemotactic response of Escherichia coli

Information-carrying signals in the real world are often obscured by noise. A challenge for any system is to filter the signal from the corrupting noise. This task is particularly acute for the signal transduction network that mediates bacterial chemotaxis, because the signals are subtle, the noise arising from stochastic fluctuations is substantial, and the system is effectively acting as a differentiator which amplifies noise. Here, we investigated the filtering properties of this biological system. Through simulation, we first show that the cutoff frequency has a dramatic effect on the chemotactic efficiency of the cell. Then, using a mathematical model to describe the signal, noise, and system, we formulated and solved an optimal filtering problem to determine the cutoff frequency that bests separates the low-frequency signal from the high-frequency noise. There was good agreement between the theory, simulations, and published experimental data. Finally, we propose that an elegant implementation of the optimal filter in combination with a differentiator can be achieved via an integral control system. This paper furnishes a simple quantitative framework for interpreting many of the key notions about bacterial chemotaxis, and, more generally, it highlights the constraints on biological systems imposed by noise.     

Estimation of the thermal coefficient in the decline of a bacterial population under heat stress.

The "most probable number" (MPN) technique for estimating numbers of bacteria in suspensions is well known and has been used for decades by microbiologists, food researchers, and other laboratory scientists. A related procedure, involving an infinite number of serial dilutions at each of times ti, i = 0,1,...,n, is considered, and the joint probability law of the index numbers of the last tubes showing growth at these times is derived through use of probability-generating functions. Maximum likelihood estimates of the initial density lambda and the thermal death rate mu are computed. Another estimate, mu, of mu is given, using a simple weighting scheme. Finally, the thermal death time (TDT) is estimated by D = (1n 10)/mu.