Quasi-elastic light scattering from migrating chemotactic bands of Escherichia coli. III. Studies of band formation propagation and motility in oxygen and serine substrates.

A series of light scattering experiments have been performed to study both macroscopic aspects of band formation and propagation and microscopic motility parameters of Escherichia coli in the combined substrate gradients of oxygen and serine. From the band formation experiment the conclusion is drawn that a minimum threshold gradient of the substrate is required for bacteria to form a band. From the band propagation experiment in the serine substrate the motility coefficient mu and chemotactic coefficient delta are determined. A separate quasi-elastic scattering experiment has been made with a propagating band to obtain three microscopic motility parameters: mean twiddle time tau 1, mean run time tau 2, and mean run speed V2. Finally, a scaling argument is made to connect the macroscopic parameters mu and delta with the microscopic parameters tau 1, tau 2, and V2, thus achieving a unified understanding of macroscopic and microscopic aspect of chemotaxis.     

Quasi-elastic light scattering from migrating chemotactic bands of Escherichia coli.

We report the observation of migrating chemotactic bands of Escherichia coli in a buffer solution. The temporal development of the bacterial density profile is observed by the scattered light intensity as the band migrates through a stationary laser beam. We have made a preliminary analysis of the observed band profile with help of the Keller-Segel theory. The model accounts for only some aspects of the observed time evolution of the density profile. The microscopic motility characteristics of the E. coli in the band are simultaneously studied by photon correlation. The measured correlation functions are analyzed to obtain the spatial dependence of the half-width within the band. A simple analytical model is proposed to account for the contribution of the twiddle motion to the correlation function. By analyzing the correlation function as a superposition of straight-line and twiddle motions, we obtain a satisfactory agreement between the theory and the measured angular dependence of the line shape. As a consequence we are able to extract a parameter beta, which measures the average fraction of twiddling bacteria in the center of the band at a given time.     

Chemotactic transport theory for neutrophil leukocytes

The simplest admissible phenomenological transport theory for the chemotactic migration of a population of neutrophil leukocytes is formulated along the lines of the original Keller-Segel model for bacterial chemotaxis, but with appropriate specialization of the motility and chemotactic flux coefficient to reflect their dependence on the local cytotaxin (chemoattractant) concentration, as observed experimentally by Wilkinson and other workers. By supplementing deductions from the governing transport equation with inferences from measurements and then reasoning both forwards and backwards, the functional forms of the motility and chemotactic flux coefficient can be established for any prescribed cytotaxin. This analysis is performed here with numerical details for casein, a cytotaxin which gives rise to a motility function with an increasing-then-decreasing form of dependence on the concentration and a chemotactic flux coefficient that is essentially constant with variations in the concentration. Three dimensionless numbers are associated with the chemotactic response of neutrophil leukocytes to casein.     

Measurement of bacterial random motility and chemotaxis coefficients: I. Stopped-flow diffusion chamber assay

Bacterial chemotaxis, the directed movement of a cell population in response to a chemical gradient, plays a critical role in the distribution and dynamic interaction of bacterial populations in nonmixed systems. Therefore, in order to make reliable predictions about the migratory behavior of bacteria within the environment, a quantitative characterization of the chemotactic response in terms of intrinsic cell properties is needed.The design of the stopped-flow diffusion chamber (SFDC) provides a well-characterized chemical gradient and reliable method for measuring bacterial migration behavior. During flow through the chamber, a step change in chemical concentration is imposed on a uniform suspension of bacteria. Once flow is stopped, diffusion causes a transient chemical gradient to develop, and bacteria respond by forming a band of high cell density which travels toward higher concentrations of the attractant. Changes in bacterial spatial distributions observed through light scattering are recorded on photomicrographs during a 10-min period. Computer-aided image analysis converts absorbance of the photographic negatives to a digital representation of bacterial density profiles. A mathematical model (part II) is used to quantitatively characterize these observations in terms of intrinsic cell parameters: a chemotactic sensitivity coefficient, mu(0), from the aggregate cell density accumulated in the band and a random motility coefficient, mu, from population dispersion in the absence of a chemical gradient.Using the SFDC assay and an individual-cell-based mathematical model, we successfully determined values for both of these population parameters for Escherichia coli K12 responding to fucose. The values obtained were mu = 1.1 +/- 0. 4 x 10(-5) cm(2)/s and chi(o) = 8 +/- 3 +/- 10(-5) cm(2)/s. We have demonstrated a method capable of determining these parameter values from the now validated mathematical model which will be useful for predicting bacterial migration in application systems.     

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.     

A simple expression for quantifying bacterial chemotaxis using capillary assay data: application to the analysis of enhanced chemotactic responses from growth-limited cultures.

An individual cell-based mathematical model of Rivero et al. provides a framework for determining values of the chemotactic sensitivity coefficient chi 0, an intrinsic cell population parameter that characterizes the chemotactic response of bacterial populations. This coefficient can theoretically relate the swimming behavior of individual cells to the resulting migration of a bacterial population. When this model is applied to the commonly used capillary assay, an approximate solution can be obtained for a particular range of chemotactic strengths yielding a very simple analytical expression for estimating the value of chi 0, [formula: see text] from measurements of cell accumulation in the capillary, N, when attractant uptake is negligible. A0 and A infinity are the dimensionless attractant concentrations initially present at the mouth of the capillary and far into the capillary, respectively, which are scaled by Kd, the effective dissociation constant for receptor-attractant binding. D is the attractant diffusivity, and mu is the cell random motility coefficient. NRM is the cell accumulation in the capillary in the absence of an attractant gradient, from which mu can be determined independently as mu = (pi/4t)(NRM/pi r2bc)2, with r the capillary tube radius and bc the bacterial density initially in the chamber. When attractant uptake is significant, a slightly more involved procedure requiring a simple numerical integration becomes necessary. As an example, we apply this approach to quantitatively characterize, in terms of the chemotactic sensitivity coefficient chi 0, data from Terracciano indicating enhanced chemotactic responses of Escherichia coli to galactose when cultured under growth-limiting galactose levels in a chemostat.     

Kinetic analysis of the chemotaxis of bacteria

On the basis of a kinetic model of bacterial chemotactic movement the system of differential equations was reduced to describe the phenomenon of bacterial bonds migration. It follows that Keller-Segel equation is a private case of a more general "diffusion approximation" of the kinetic model. The functional parameters of the reduced equation for E. coli K-12 are estimated.     

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.     

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.     

The sodium cycle--a new type of bacterial energetics

The literature data and experimental results of the author's laboratory on the role of Na+ in bacterial energetics are reviewed. It was shown that certain bacterial species utilize the transmembrane difference of Na+ electrochemical potentials (delta mu Na+) as a convertible membrane-linked form of energy. The membranes of such bacteria were found to contain delta mu Na+ generators (e. g., decarboxylases of some carboxylic acids of NADH-menaquinone reductase). It was shown that delta mu Na+ formed by these generators may support all the three main types of work of the bacterial cell, i. e., chemical (ATP synthesis), osmotic (substrate accumulation) and mechanical (motility).     

Critical band in auditory lateralization.

A new and powerful procedure for determining frequency analysis in the auditory system, as evidence by the critical band, is described. The onset time difference, delta T, needed to lateralize 30-msec tone bursts toward the leading ear was measured as a function of the frequency difference, delta F, between the brust in one ear and the burst in the other ear. When delta F was less than the critical band, threshold delta T was constant at 100 mu sec or less, depending on center frequency; beyond the critical band, delta T increased with delta F. These dichotically measured critical bandwidths increased from 110 Hz at a center frequency of 500 Hz to 1100 Hz at a center frequency of 6000 Hz. They were unaffected by varying signal level from 25 to 80 dB or signal duration from 10 to 300 msec. The sam e critical-band values have been measured with monaural stimuli in loudness summation, maskin, detection, phase perception, consonance, and so forth.     

Model of bacterial band formation in aerotaxis

Aerotaxis is a particular form of "energy taxis". It is based on a largely elusive signal transduction machinery. In aerotaxis, oxygen dissolved in water plays the role of both attractant (at moderate concentrations) and repellent (at high and low concentrations). Cells swimming from favorable oxygen concentrations into regions with unfavorable concentrations increase the frequency of reversals, turn back into the favorable domain, and become effectively trapped there. At the same time, bacteria consume oxygen, creating an oxygen gradient. This behavior leads to a pattern formation phenomenon: bacteria self-organize into a dense band at a certain distance from the air-water interface. We incorporate experimental observations of the aerotactic bacterium, Azospirillum brasilense, into a mathematical model. The model consists of a system of differential equations describing swimming bacterial cells and diffusing oxygen. The cells' frequency of reversals depends on the concentration of oxygen and its time derivative while oxygen is depleted by the bacteria. We suggest a hypothetical model of energy sensing mediated by aerotactic receptors Aer and Tsr. Computer simulations and analysis of the model equations allow comparisons of theoretical and experimental results and provide insight into the mechanisms of bacterial pattern formation and underlying signal transduction machinery. We make testable predictions about position and density of the bacterial band.     

Quasielastic light scattering from migrating chemotactic bands of Escherichia coli. II. Analysis of anisotropic bacterial motions.

Chemotactic effects of dissolved oxygen on motions of Escherichia coli in a motility buffer solution have been studied by measurements of quasielastic light scattering. Under conditions where the bacteria form a sharp band in an oxygen concentrations gradient created by their metabolism, components of motions along the direction of the gradient and perpendicular to it were studied separately at each point within the band profile. A theoretical model for bacterial self correlation function based on two-state motions has been developed to extract the mean square speed of run motion and the relative probability of twiddle vs. run at each point of the band profile. A combined novel experimental set-up and new data analysis method allowed us to extract also the mean square displacements at short times along and perpendicular to the direction of the gradient. Parameters extracted from the measured correlation functions have been discussed in the framework of the established picture of bacterial motions under chemotaxis.     

Influence of confinement on the spreading of bacterial populations

The spreading of bacterial populations is central to processes in agriculture, the environment, and medicine. However, existing models of spreading typically focus on cells in unconfined settings—despite the fact that many bacteria inhabit complex and crowded environments, such as soils, sediments, and biological tissues/gels, in which solid obstacles confine the cells and thereby strongly regulate population spreading. Here, we develop an extended version of the classic Keller-Segel model of bacterial spreading via motility that also incorporates cellular growth and division, and explicitly considers the influence of confinement in promoting both cell-solid and cell-cell collisions. Numerical simulations of this extended model demonstrate how confinement fundamentally alters the dynamics and morphology of spreading bacterial populations, in good agreement with recent experimental results. In particular, with increasing confinement, we find that cell-cell collisions increasingly hinder the initial formation and the long-time propagation speed of chemotactic pulses. Moreover, also with increasing confinement, we find that cellular growth and division plays an increasingly dominant role in driving population spreading—eventually leading to a transition from chemotactic spreading to growth-driven spreading via a slower, jammed front. This work thus provides a theoretical foundation for further investigations of the influence of confinement on bacterial spreading. More broadly, these results help to provide a framework to predict and control the dynamics of bacterial populations in complex and crowded environments.     

Finite time blow-up in some models of chemotaxis

We consider a class of models of chemotactic bacterial populations, introduced by Keller-Segel. For those models, we investigate the possibility of chemotactic collapse, in other words, the possibility that in finite time the population of predators aggregates to form a delta-function. To study this phenomenon, we construct self-similar solutions, which may or may not blow-up (in finite time), depending on the relative strength of three mechanisms in competition: (i) the chemotactic attraction of bacteria towards regions of high concentration in substrate (ii) the rate of consumption of the substrate by the bacteria and (iii) (possibly) the diffusion of bacteria. The solutions we construct are radially symmetric, and therefore have no relation with the classical traveling wave solutions. Our scaling can be justified by a dimensional analysis. We give some evidence of numerical stability.     

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.     

A new interpretation of the Keller-Segel model based on multiphase modelling

In this paper an alternative derivation and interpretation are presented of the classical Keller-Segel model of cell migration due to random motion and chemotaxis. A multiphase modelling approach is used to describe how a population of cells moves through a fluid containing a diffusible chemical to which the cells are attracted. The cells and fluid are viewed as distinct components of a two-phase mixture. The principles of mass and momentum balance are applied to each phase, and appropriate constitutive laws imposed to close the resulting equations. A key assumption here is that the stress in the cell phase is influenced by the concentration of the diffusible chemical. By restricting attention to one-dimensional cartesian geometry we show how the model reduces to a pair of nonlinear coupled partial differential equations for the cell density and the chemical concentration. These equations may be written in the form of the Patlak-Keller-Segel model, naturally including density-dependent nonlinearities in the cell motility coefficients. There is a direct relationship between the random motility and chemotaxis coefficients, both depending in an inter-related manner on the chemical concentration. We suggest that this may explain why many chemicals appear to stimulate both chemotactic and chemokinetic responses in cell populations. After specialising our model to describe slime mold we then show how the functional form of the chemical potential that drives cell locomotion influences the ability of the system to generate spatial patterns. The paper concludes with a summary of the key results and a discussion of avenues for future research.     

Uncoupling of oxidative phosphorylation. 1. Protonophoric effects account only partially for uncoupling

The mechanism of uncoupling of oxidative phosphorylation by carbonyl cyanide p-trifluoromethoxy)phenylhydrazone (FCCP), a typical weak acid protonophore, oleic acid, a fatty acid, and chloroform, a general anesthetic, has been investigated by measuring in mitochondria their effect on (i) the transmembrane proton electrochemical potential gradient (delta mu H) and the rates of electron transfer and adenosine 5'-triphosphate (ATP) hydrolysis in static head, (ii) delta mu H and the rates of electron transfer and ATP synthesis in state 3, and (iii) the membrane proton conductance. Both FCCP and oleic acid increase the membrane proton conductance, and accordingly, they cause a depression of delta mu H [generated by either the redox proton pumps or the adenosinetriphosphatase (ATPase) proton pumps]. Although their effects on ATP synthesis/hydrolysis, respiration, and delta mu H are qualitatively consistent with a pure protonophoric uncoupling mechanism and an additional inhibitory action of oleic acid on both the ATPases and the electron-transfer enzymes, a quantitative comparison between the dissipative proton influx and the rate of either electron transfer or ATP hydrolysis (multiplied by either the H+/e- or the H+/ATP stoichiometry, respectively) at the same delta mu H shows that the increase in membrane conductance induced by FCCP and oleic acid accounts for the stimulation of the rate of ATP hydrolysis but not for that of the rate of electron transfer. Chloroform (at concentrations that fully inhibit ATP synthesis) only very slightly increases the proton conductance of the mitochondrial membrane and causes only a little depression of delta mu H.(ABSTRACT TRUNCATED AT 250 WORDS)