The chemotactic behavior of computer-based surrogate bacteria

BACKGROUND: Chemotaxis is the process by which organisms migrate toward nutrients and favorable environments and away from toxins and unfavorable environments. In many species of bacteria, this occurs when extracellular signals are detected by transmembrane receptors and relayed to flagellar motors, which control the cell's swimming behavior. RESULTS: We used a molecularly detailed reaction-kinetics model of the chemotaxis pathway in Escherichia coli coupled to a graphical display based on known swimming parameters to simulate the responses of bacteria to 2D gradients of attractants. The program gives the correct phenotype of over 60 mutants in which chemotaxis-pathway components are deleted or overexpressed and accurately reproduces the responses to pulses and step increases of attractant. In order to match the known sensitivity of bacteria to low concentrations of attractant, we had to introduce a set of "infectivity" reactions based on cooperative interactions between neighboring chemotaxis receptors in the membrane. In order to match the impulse response to a brief stimulus and to achieve an effective accumulation in a gradient, we also had to increase the activities of the adaptational enzymes CheR and CheB at least an order of magnitude greater than published values. Our simulations reveal that cells develop characteristic levels of receptor methylation and swimming behavior at different positions along a gradient. They also predict a distinctive "volcano" profile in some gradients, with peaks of cell density at intermediate concentrations of attractant. CONCLUSIONS: Our results display the potential use of computer-based bacteria as experimental objects for exploring subtleties of chemotactic behavior.     

Bacterial Thermotaxis by Speed Modulation

Naturally occurring gradients often extend over relatively long distances such that their steepness is too small for bacteria to detect. We studied the bacterial behavior in such thermal gradients. We find that bacteria migrate along shallow thermal gradients due to a change in their swimming speed resulting from the effect of temperature on the intracellular pH, which also depends on the chemical environment. When nutrients are scarce in the environment the bacteria''s intracellular pH decreases with temperature. As a result, the swimming speed of the bacteria decreases with temperature, which causes them to slowly drift toward the warm end of the thermal gradient. However, when serine is added to the medium at concentrations >300 μM, the intracellular pH increases causing the swimming speed to increase continuously with temperature, and the bacteria to drift toward the cold end of the temperature gradient. This directional migration is not a result of bacterial thermotaxis in the classical sense, because the steepness of the gradients applied is below the sensing threshold of bacteria. Nevertheless, our results show that the directional switch requires the presence of the bacterial sensing receptors. This seems to be due to the involvement of the receptors in regulating the intracellular pH.     

Dependence of Bacterial Chemotaxis on Gradient Shape and Adaptation Rate

Simulation of cellular behavior on multiple scales requires models that are sufficiently detailed to capture central intracellular processes but at the same time enable the simulation of entire cell populations in a computationally cheap way. In this paper we present RapidCell, a hybrid model of chemotactic Escherichia coli that combines the Monod-Wyman-Changeux signal processing by mixed chemoreceptor clusters, the adaptation dynamics described by ordinary differential equations, and a detailed model of cell tumbling. Our model dramatically reduces computational costs and allows the highly efficient simulation of E. coli chemotaxis. We use the model to investigate chemotaxis in different gradients, and suggest a new, constant-activity type of gradient to systematically study chemotactic behavior of virtual bacteria. Using the unique properties of this gradient, we show that optimal chemotaxis is observed in a narrow range of CheA kinase activity, where concentration of the response regulator CheY-P falls into the operating range of flagellar motors. Our simulations also confirm that the CheB phosphorylation feedback improves chemotactic efficiency by shifting the average CheY-P concentration to fit the motor operating range. Our results suggest that in liquid media the variability in adaptation times among cells may be evolutionary favorable to ensure coexistence of subpopulations that will be optimally tactic in different gradients. However, in a porous medium (agar) such variability appears to be less important, because agar structure poses mainly negative selection against subpopulations with low levels of adaptation enzymes. RapidCell is available from the authors upon request.     

Predicted Auxiliary Navigation Mechanism of Peritrichously Flagellated Chemotactic Bacteria

Chemotactic movement of Escherichia coli is one of the most thoroughly studied paradigms of simple behavior. Due to significant competitive advantage conferred by chemotaxis and to high evolution rates in bacteria, the chemotaxis system is expected to be strongly optimized. Bacteria follow gradients by performing temporal comparisons of chemoeffector concentrations along their runs, a strategy which is most efficient given their size and swimming speed. Concentration differences are detected by a sensory system and transmitted to modulate rotation of flagellar motors, decreasing the probability of a tumble and reorientation if the perceived concentration change during a run is positive. Such regulation of tumble probability is of itself sufficient to explain chemotactic drift of a population up the gradient, and is commonly assumed to be the only navigation mechanism of chemotactic E. coli. Here we use computer simulations to predict existence of an additional mechanism of gradient navigation in E. coli. Based on the experimentally observed dependence of cell tumbling angle on the number of switching motors, we suggest that not only the tumbling probability but also the degree of reorientation during a tumble depend on the swimming direction along the gradient. Although the difference in mean tumbling angles up and down the gradient predicted by our model is small, it results in a dramatic enhancement of the cellular drift velocity along the gradient. We thus demonstrate a new level of optimization in E. coli chemotaxis, which arises from the switching of several flagellar motors and a resulting fine tuning of tumbling angle. Similar strategy is likely to be used by other peritrichously flagellated bacteria, and indicates yet another level of evolutionary development of bacterial chemotaxis.     

Steady-state running rate sets the speed and accuracy of accumulation of swimming bacteria

We study the chemotaxis of a population of genetically identical swimming bacteria undergoing run and tumble dynamics driven by stochastic switching between clockwise and counterclockwise rotation of the flagellar rotary system, where the steady-state rate of the switching changes in different environments. Understanding chemotaxis quantitatively requires that one links the measured steady-state switching rates of the rotary system, as well as the directional changes of individual swimming bacteria in a gradient of chemoattractant/repellant, to the efficiency of a population of bacteria in moving up/down the gradient. Here we achieve this by using a probabilistic model, parametrized with our experimental data, and show that the response of a population to the gradient is complex. We find the changes to the steady-state switching rate in the absence of gradients affect the average speed of the swimming bacterial population response as well as the width of the distribution. Both must be taken into account when optimizing the overall response of the population in complex environments.     

Mathematical Modeling of Chemotactic Bacterial Transport through a Two-Dimensional Heterogeneous Porous Medium

The impact of bacterial chemotaxis on in situ ground-water bioremediation remains an unanswered question. Although bacteria respond to chemical gradients in aqueous environments and under no-flow conditions, it is unclear whether they can also respond in porous media with advective flow to improve overall contaminant degradation. The effect of chemotaxis is most profound in regions with sharp chemical gradients, most notably around residual nonaqueous phase liquid (NAPL) ganglia and surrounding clay lenses or aquitards with trapped contamination. The purpose of this study is to simulate bacterial transport through a two-dimensional subsurface environment, containing one region of low permeability with trapped contaminant surrounded above and below by two regions of higher permeability. Using mathematical predictions of the effect of pore size on measured bacterial transport parameters, the authors observe a 50% decrease in both motility and chemotaxis in the finer-grained, low-permeability porous medium. The authors simulate how chemotaxis affects bacterial migration to the contaminated region under various flow and initial conditions. Results indicate that bacteria traveling through a high-permeability region with advective flow can successfully migrate toward and accumulate around a contaminant diffusing from a lower permeability region.     

Mathematical Modeling of Chemotactic Bacterial Transport through a Two-Dimensional Heterogeneous Porous Medium

The impact of bacterial chemotaxis on in situ ground-water bioremediation remains an unanswered question. Although bacteria respond to chemical gradients in aqueous environments and under no-flow conditions, it is unclear whether they can also respond in porous media with advective flow to improve overall contaminant degradation. The effect of chemotaxis is most profound in regions with sharp chemical gradients, most notably around residual nonaqueous phase liquid (NAPL) ganglia and surrounding clay lenses or aquitards with trapped contamination. The purpose of this study is to simulate bacterial transport through a two-dimensional subsurface environment, containing one region of low permeability with trapped contaminant surrounded above and below by two regions of higher permeability. Using mathematical predictions of the effect of pore size on measured bacterial transport parameters, the authors observe a 50% decrease in both motility and chemotaxis in the finer-grained, low-permeability porous medium. The authors simulate how chemotaxis affects bacterial migration to the contaminated region under various flow and initial conditions. Results indicate that bacteria traveling through a high-permeability region with advective flow can successfully migrate toward and accumulate around a contaminant diffusing from a lower permeability region.     

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.     

Tuning sperm chemotaxis by calcium burst timing

Marine invertebrate oocytes establish chemoattractant gradients that guide spermatozoa towards their source. In sea urchin spermatozoa, this relocation requires coordinated motility changes initiated by Ca²⁺-driven alterations in sperm flagellar curvature. We discovered that Lytechinus pictus spermatozoa undergo chemotaxis in response to speract, an egg-derived decapeptide previously noted to stimulate non-chemotactic motility alterations in Strongylocentrotus purpuratus spermatozoa. Sperm of both species responded to speract gradients with a sequence of turning episodes that correlate with transient flagellar Ca²⁺ increases, yet only L. pictus spermatozoa accumulated at the gradient source. Detailed analysis of sperm behavior revealed that L. pictus spermatozoa selectively undergo Ca²⁺ fluctuations while swimming along negative speract gradients while S. purpuratus sperm generate Ca²⁺ fluctuations in a spatially non-selective manner. This difference is attributed to the selective suppression of Ca²⁺ fluctuations of L. pictus spermatozoa as they swim towards the source of the chemoattractant gradient. This is the first study to compare and characterize the motility components that differ in chemotactic and non-chemotactic spermatozoa. Tuning of Ca²⁺ fluctuations and associated turning episodes to the chemoattractant gradient polarity is a central feature of sea urchin sperm chemotaxis and may be a feature of sperm chemotaxis in general.     

Motility, chemokinesis, and methylation-independent chemotaxis in Azospirillum brasilense.

Observations of free-swimming and antibody-tethered Azospirillum brasilense cells showed that their polar flagella could rotate in both clockwise and counterclockwise directions. Rotation in a counterclockwise direction caused forward movement of free-swimming cells, whereas the occasional change in the direction of rotation to clockwise caused a brief reversal in swimming direction. The addition of a metabolizable chemoattractant, e.g., malate or proline, had two distinct effects on the swimming behavior of the bacteria: (i) a short-term decrease in reversal frequency from 0.33 to 0.17 s-1 and (ii) a long-term increase in the mean population swimming speed from 13 to 23 microns s-1. A. brasilense therefore shows both chemotaxis and chemokinesis in response to temporal gradients of some chemoeffectors. Chemokinesis was dependent on the growth state of the cells and may depend on an increase in the electrochemical proton gradient above a saturation threshold. Analysis of behavior of a methionine auxotroph, assays of in vivo methylation, and the use of specific antibodies raised against the sensory transducer protein Tar of Escherichia coli all failed to demonstrate the methylation-dependent pathway for chemotaxis in A. brasilense. The range of chemicals to which A. brasilense shows chemotaxis and the lack of true repellents indicate an alternative chemosensory pathway probably based on metabolism of chemoeffectors.     

Biased reorientation in the chemotaxis of peritrichous bacteria Salmonella enterica serovar Typhimurium

Many kinds of peritrichous bacteria that repeat runs and tumbles by using multiple flagella exhibit chemotaxis by sensing a difference in the concentration of the attractant or repellent between two adjacent time points. If a cell senses that the concentration of an attractant has increased, their flagellar motors decrease the switching frequency from counterclockwise to clockwise direction of rotation, which causes a longer run in swimming up the concentration gradient than swimming down. We investigated the turn angle in tumbles of peritrichous bacteria swimming across the concentration gradient of a chemoattractant because the change in the switching frequency in the rotational direction may affect the way tumbles. We tracked several hundreds of runs and tumbles of single cells of Salmonella enterica serovar Typhimurium in the concentration gradient of L-serine and found that the turn angle depends on the concentration gradient that the cell senses just before the tumble. The turn angle is biased toward a smaller value when the cells swim up the concentration gradient, whereas the distribution of the angle is almost uniform (random direction) when the cells swim down the gradient. The effect of the observed bias in the turn angle on the degree of chemotaxis was investigated by random walk simulation. In the concentration field where attractants diffuse concentrically from the point source, we found that this angular distribution clearly affects the reduction of the mean-square displacement of the cell that has started at the attractant source, that is, the bias in the turn angle distribution contributes to chemotaxis in peritrichous bacteria.     

Bacterial swimming strategies and turbulence

Most bacteria in the ocean can be motile. Chemotaxis allows bacteria to detect nutrient gradients, and hence motility is believed to serve as a method of approaching sources of food. This picture is well established in a stagnant environment. In the ocean a shear microenvironment is associated with turbulence. This shear flow prevents clustering of bacteria around local nutrient sources if they swim in the commonly assumed "run-and-tumble" strategy. Recent observations, however, indicate a "back-and-forth" swimming behavior for marine bacteria. In a theoretical study we compare the two bacterial swimming strategies in a realistic ocean environment. The "back-and-forth" strategy is found to enable the bacteria to stay close to a nutrient source even under high shear. Furthermore, rotational diffusion driven by thermal noise can significantly enhance the efficiency of this strategy. The superiority of the "back-and-forth" strategy suggests that bacterial motility has a control function rather than an approach function under turbulent conditions.     

Behavioral responses of Schistosoma mansoni miracidia in concentration gradients of snail-conditioned water.

Movement patterns of Schistosoma mansoni miracidia were examined in several concentrations and gradients of snail-conditioned water (SCW). Miracidia surrounded by uniform concentrations of SCW swam at the same speed and exhibited the same rate of turning (angular velocity) as did control miracidia swimming in spring water. However, miracidia in gradients of SCW exhibited a 3-fold increase in their angular velocity without altering their swimming speed. Miracidia ascending gradients of SCW did not increase their angular velocity and failed to orient to the gradient of the stimulant. In contrast, miracidia which encountered sufficiently abrupt decreases in SCW concentration, while descending the gradient, sharply increased their angular velocity. This behavior caused miracidia to remain in regions of high concentration of stimulant. The magnitude of decrease in SCW concentration needed to evoke this response depended on the absolute concentration of SCW. Thus, the miracidial response is a "boundary reaction", a form of chemoklinokinesis, and not a chemotaxis.     

Chemotaxis of Ralstonia sp. SJ98 towards p-nitrophenol in soil

Bioremediation of contaminated sites has been accepted as an efficient and cheaper alternative to physicochemical means of remediation in several cases. Although chemotactic behaviour of many bacteria has been studied earlier and assays have been developed to study bacterial chemotaxis in semi-solid media, this phenomenon has never been demonstrated in soil. For bioremediation application it is important to know whether bacteria actually migrate through the heterogenous soil medium towards a gradient of a particular chemoattractant. In the present study we have successfully demonstrated bacterial chemotaxis of a Ralstonia sp. SJ98 in soil microcosm using qualitative and quantitative plate and tray assays. The migration of bacteria has been established using several methods such as plate counting, vital staining and flow cytometry and slot blot hybridization. A non-chemotactic p-nitrophenol utilizing strain Burkholderia cepacia RKJ200 has been used as negative control. Our work clearly substantiates the hypothesis that chemotactic bacteria may enhance in situ bioremediation of toxic pollutants from soils and sediments.     

A microscale model of bacterial and biofilm dynamics in porous media

A microscale model for the transport and coupled reaction of microbes and chemicals in an idealized two-dimensional porous media has been developed. This model includes the flow, transport, and bioreaction of nutrients, electron acceptors, and microbial cells in a saturated granular porous media. The fluid and chemicals are represented as a continuum, but the bacterial cells and solid granular particles are represented discretely. Bacterial cells can attach to the particle surfaces or be advected in the bulk fluid. The bacterial cells can also be motile and move preferentially via a run and tumble mechanism toward a chemoattractant. The bacteria consume oxygen and nutrients and alter the profiles of these chemicals. Attachment of bacterial cells to the soil matrix and growth of bacteria can change the local permeability. The coupling of mass transport and bioreaction can produce spatial gradients of nutrients and electron acceptor concentrations. We describe a numerical method for the microscale model, show results of a convergence study, and present example simulations of the model system.     

Logarithmic Sensing in Escherichia coli Bacterial Chemotaxis

We studied the response of swimming Escherichia coli (E. coli) bacteria in a comprehensive set of well-controlled chemical concentration gradients using a newly developed microfluidic device and cell tracking imaging technique. In parallel, we carried out a multi-scale theoretical modeling of bacterial chemotaxis taking into account the relevant internal signaling pathway dynamics, and predicted bacterial chemotactic responses at the cellular level. By measuring the E. coli cell density profiles across the microfluidic channel at various spatial gradients of ligand concentration grad[L] and the average ligand concentration near the peak chemotactic response region, we demonstrated unambiguously in both experiments and model simulation that the mean chemotactic drift velocity of E. coli cells increased monotonically with grad [L]/ or ∼grad(log[L])—that is E. coli cells sense the spatial gradient of the logarithmic ligand concentration. The exact range of the log-sensing regime was determined. The agreements between the experiments and the multi-scale model simulation verify the validity of the theoretical model, and revealed that the key microscopic mechanism for logarithmic sensing in bacterial chemotaxis is the adaptation kinetics, in contrast to explanations based directly on ligand occupancy.     

Cell balance equation for chemotactic bacteria with a biphasic tumbling frequency

Alts three-dimensional cell balance equation characterizing the chemotactic bacteria was analyzed under the presence of one-dimensional spatial chemoattractant gradients. Our work differs from that of others who have developed rather general models for chemotaxis in the use of a non-smooth anisotropic tumbling frequency function that responds biphasically to the combined temporal and spatial chemoattractant gradients. General three-dimensional expressions for the bacterial transport parameters were derived for chemotactic bacteria, followed by a perturbation analysis under the planar geometry. The bacterial random motility and chemotaxis were summarized by a motility tensor and a chemotactic velocity vector, respectively. The consequence of invoking the diffusion-approximation assumption and using intrinsic one-dimensional models with modified cellular swimming speeds was investigated by numerical simulations. Characterizing the bacterial random orientation after tumbles by a turn angle probability distribution function, we found that only the first-order angular moment of this turn angle probability distribution is important in influencing the bacterial long-term transport. Mathematics Subject Classification (2000):60G05, 60J60, 82A70     

A diffusion gradient chamber for studying microbial behavior and separating microorganisms

The natural habitats of most microbes are dynamic and include spatial gradients of growth substrates, electron acceptors, pH, salts, and inhibitory compounds. To mimic this diffusive aspect of nature, we developed an analytical diffusion gradient chamber (DGC) that can be used to separate, enrich for, isolate, and study the behavior of microorganisms. The chamber is a polycarbonate box containing an arena (5 by 5 by 2 cm) into which is cast a semisolid growth medium. Continuously replenished solute reservoirs positioned on each side of the arena but separated from it by a porous membrane enable the formation throughout the gel of multiple, intersecting gradients of solutes in two dimensions. With glucose as the solute, a gradient which spanned a 100-fold range in concentration was established across the arena in about 4 days. The shape of the glucose gradient was accurately predicted by a mathematical model based on Fickian diffusion. The growth and migratory behavior of Escherichia coli in response to imposed gradients of attractants (aspartate, alpha-methyl aspartate, and serine) and a repellent (valine) were examined. Cells responded in predictable ways to such gradients by forming distinctive growth and migration patterns in the DGC. This was true for wild-type E. coli as well as specific chemotaxis and motility mutants. The patterns yielded information about the threshold concentration of chemoeffectors needed to elicit a response as well as their saturating concentration. It was also evident that the metabolism of attractants significantly affected the gradients and, hence, the movement of cells. Finally, it was possible to separate E. coli and Pseudomonas fluorescens in the DGC on the basis of their differential responses to gradients of various chemoeffectors.     

A Multiple-Relaxation-Time Lattice-Boltzmann Model for Bacterial Chemotaxis: Effects of Initial Concentration,Diffusion, and Hydrodynamic Dispersion on Traveling Bacterial Bands

Bacterial chemotaxis can enhance the bioremediation of contaminants in aqueous and subsurface environments if the contaminant is a chemoattractant that the bacteria degrade. The process can be promoted by traveling bands of chemotactic bacteria that form due to metabolism-generated gradients in chemoattractant concentration. We developed a multiple-relaxation-time (MRT) lattice-Boltzmann method (LBM) to model chemotaxis, because LBMs are well suited to model reactive transport in the complex geometries that are typical for subsurface porous media. This MRT-LBM can attain a better numerical stability than its corresponding single-relaxation-time LBM. We performed simulations to investigate the effects of substrate diffusion, initial bacterial concentration, and hydrodynamic dispersion on the formation, shape, and propagation of bacterial bands. Band formation requires a sufficiently high initial number of bacteria and a small substrate diffusion coefficient. Uniform flow does not affect the bands while shear flow does. Bacterial bands can move both upstream and downstream when the flow velocity is small. However, the bands disappear once the velocity becomes too large due to hydrodynamic dispersion. Generally bands can only be observed if the dimensionless ratio between the chemotactic sensitivity coefficient and the effective diffusion coefficient of the bacteria exceeds a critical value, that is, when the biased movement due to chemotaxis overcomes the diffusion-like movement due to the random motility and hydrodynamic dispersion.     

The energetics and scaling of search strategies in bacteria

The influence of body size on the energetic cost of movement is well studied in animals but has been rarely investigated in bacteria. Here, I calculate the cost of four chemotactic strategies for different-sized bacteria by adding the costs of their locomotion and reorientation. Size differences of 0.1 microm result in 100,000-fold changes in the energetic cost of chemotaxis. The exact cost for any given size is a nonlinear function of flagella length, the minimum speed necessary to detect and respond to a signal, and the gradient of the signal. These parameters are interlinked in such a way that body size and strategy are tightly coupled to particular environmental gradients, offering avenues for explaining and exploring diversity and competition. The analysis here has implications beyond bacteria. Power-law regression through the minimum costs of transport for different kinds of chemotaxis has the same slope as that for swimming animals, suggesting a universal allometric equation for all swimming organisms.