Quantitative studies of bacterial chemotaxis and microbial population dynamics

Although there is a long history of conjecture regarding the role and significance of bacterial chemotaxis in microbial ecology, only recently has a significant body of work appeared attempting to address this issue. The purpose of this paper is to provide a concise overview of this work, which combined mathematical modeling of bacterial population migration and experimental measurement of the model parameters with modeling of competitive microbial population dynamics in a nonmixed environment. Predictions from the population dynamics models, based on experimental estimates of the various motility and growth parameter values, are related to the small number of experimental observations available to date dealing with the effects of bacterial motility on competition in a nonmixed environment. Current results indicate that cell motility and chemotaxis properties can be as important to population dynamics as cell growth kinetic properties, so that greater attention to this aspect of microbial behavior is warranted in future studies of microbial ecology.     

The Role of Adaptation in Bacterial Speed Races

Evolution of biological sensory systems is driven by the need for efficient responses to environmental stimuli. A paradigm among prokaryotes is the chemotaxis system, which allows bacteria to navigate gradients of chemoattractants by biasing their run-and-tumble motion. A notable feature of chemotaxis is adaptation: after the application of a step stimulus, the bacterial running time relaxes to its pre-stimulus level. The response to the amino acid aspartate is precisely adapted whilst the response to serine is not, in spite of the same pathway processing the signals preferentially sensed by the two receptors Tar and Tsr, respectively. While the chemotaxis pathway in E. coli is well characterized, the role of adaptation, its functional significance and the ecological conditions where chemotaxis is selected, are largely unknown. Here, we investigate the role of adaptation in the climbing of gradients by E. coli. We first present theoretical arguments that highlight the mechanisms that control the efficiency of the chemotactic up-gradient motion. We discuss then the limitations of linear response theory, which motivate our subsequent experimental investigation of E. coli speed races in gradients of aspartate, serine and combinations thereof. By using microfluidic techniques, we engineer controlled gradients and demonstrate that bacterial fronts progress faster in equal-magnitude gradients of serine than aspartate. The effect is observed over an extended range of concentrations and is not due to differences in swimming velocities. We then show that adding a constant background of serine to gradients of aspartate breaks the adaptation to aspartate, which results in a sped-up progression of the fronts and directly illustrate the role of adaptation in chemotactic gradient-climbing.     

Imprecision of Adaptation in Escherichia coli Chemotaxis

Adaptability is an essential property of many sensory systems, enabling maintenance of a sensitive response over a range of background stimulus levels. In bacterial chemotaxis, adaptation to the preset level of pathway activity is achieved through an integral feedback mechanism based on activity-dependent methylation of chemoreceptors. It has been argued that this architecture ensures precise and robust adaptation regardless of the ambient ligand concentration, making perfect adaptation a celebrated property of the chemotaxis system. However, possible deviations from such ideal adaptive behavior and its consequences for chemotaxis have not been explored in detail. Here we show that the chemotaxis pathway in Escherichia coli shows increasingly imprecise adaptation to higher concentrations of attractants, with a clear correlation between the time of adaptation to a step-like stimulus and the extent of imprecision. Our analysis suggests that this imprecision results from a gradual saturation of receptor methylation sites at high levels of stimulation, which prevents full recovery of the pathway activity by violating the conditions required for precise adaptation. We further use computer simulations to show that limited imprecision of adaptation has little effect on the rate of chemotactic drift of a bacterial population in gradients, but hinders precise accumulation at the peak of the gradient. Finally, we show that for two major chemoeffectors, serine and cysteine, failure of adaptation at concentrations above 1 mM might prevent bacteria from accumulating at toxic concentrations of these amino acids.     

Systems biology of bacterial chemotaxis

Motile bacteria regulate chemotaxis through a highly conserved chemosensory signal-transduction system. System-wide analyses and mathematical modeling are facilitated by extensive experimental observations regarding bacterial chemotaxis proteins, including biochemical parameters, protein structures and protein-protein interaction maps. Thousands of signaling and regulatory chemotaxis proteins within a bacteria cell form a highly interconnected network through distinct protein-protein interactions. A bacterial cell is able to respond to multiple stimuli through a collection of chemoreceptors with different sensory modalities, which interact to affect the cooperativity and sensitivity of the chemotaxis response. The robustness or insensitivity of the chemotaxis system to perturbations in biochemical parameters is a product of the system's hierarchical network architecture.     

E. coli Superdiffusion and Chemotaxis—Search Strategy, Precision, and Motility

Escherichia coli motion is characterized by a sequence of consecutive tumble-and-swim events. In the absence of chemical gradients, the length of individual swims is commonly believed to be distributed exponentially. However, recently there has been experimental indication that the swim-length distribution has the form of a power-law, suggesting that bacteria might perform superdiffusive Lévy-walk motion. In E. coli, the power-law behavior can be induced through stochastic fluctuations in the level of CheR, one of the key enzymes in the chemotaxis signal transmission pathway. We use a mathematical model of the chemotaxis signaling pathway to study the influence of these fluctuations on the E. coli behavior in the absence and presence of chemical gradients. We find that the population with fluctuating CheR performs Lévy-walks in the absence of chemoattractants, and therefore might have an advantage in environments where nutrients are sparse. The more efficient search strategy in sparse environments is accompanied by a generally larger motility, also in the presence of chemoattractants. The tradeoff of this strategy is a reduced precision in sensing and following gradients, as well as a slower adaptation to absolute chemoattractant levels.     

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.     

Behaviors and Strategies of Bacterial Navigation in Chemical and Nonchemical Gradients

Navigation of cells to the optimal environmental condition is critical for their survival and growth. Escherichia coli cells, for example, can detect various chemicals and move up or down those chemical gradients (i.e., chemotaxis). Using the same signaling machinery, they can also sense other external factors such as pH and temperature and navigate from both sides toward some intermediate levels of those stimuli. This mode of precision sensing is more sophisticated than the (unidirectional) chemotaxis strategy and requires distinctive molecular mechanisms to encode and track the preferred external conditions. To systematically study these different bacterial taxis behaviors, we develop a continuum model that incorporates microscopic signaling events in single cells into macroscopic population dynamics. A simple theoretical result is obtained for the steady state cell distribution in general. In particular, we find the cell distribution is controlled by the intracellular sensory dynamics as well as the dependence of the cells'' speed on external factors. The model is verified by available experimental data in various taxis behaviors (including bacterial chemotaxis, pH taxis, and thermotaxis), and it also leads to predictions that can be tested by future experiments. Our analysis help reveal the key conditions/mechanisms for bacterial precision-sensing behaviors and directly connects the cellular taxis performances with the underlying molecular parameters. It provides a unified framework to study bacterial navigation in complex environments with chemical and non-chemical stimuli.     

Universal Response-Adaptation Relation in Bacterial Chemotaxis

The bacterial strategy of chemotaxis relies on temporal comparisons of chemical concentrations, where the probability of maintaining the current direction of swimming is modulated by changes in stimulation experienced during the recent past. A short-term memory required for such comparisons is provided by the adaptation system, which operates through the activity-dependent methylation of chemotaxis receptors. Previous theoretical studies have suggested that efficient navigation in gradients requires a well-defined adaptation rate, because the memory time scale needs to match the duration of straight runs made by bacteria. Here we demonstrate that the chemotaxis pathway of Escherichia coli does indeed exhibit a universal relation between the response magnitude and adaptation time which does not depend on the type of chemical ligand. Our results suggest that this alignment of adaptation rates for different ligands is achieved through cooperative interactions among chemoreceptors rather than through fine-tuning of methylation rates for individual receptors. This observation illustrates a yet-unrecognized function of receptor clustering in bacterial chemotaxis.     

Oscillatory Behavior of Neutrophils under Opposing Chemoattractant Gradients Supports a Winner-Take-All Mechanism

Neutrophils constitute the largest class of white blood cells and are the first responders in the innate immune response. They are able to sense and migrate up concentration gradients of chemoattractants in search of primary sites of infection and inflammation through a process known as chemotaxis. These chemoattractants include formylated peptides and various chemokines. While much is known about chemotaxis to individual chemoattractants, far less is known about chemotaxis towards many. Previous studies have shown that in opposing gradients of intermediate chemoattractants (interleukin-8 and leukotriene B₄), neutrophils preferentially migrate toward the more distant source. In this work, we investigated neutrophil chemotaxis in opposing gradients of chemoattractants using a microfluidic platform. We found that primary neutrophils exhibit oscillatory motion in opposing gradients of intermediate chemoattractants. To understand this behavior, we constructed a mathematical model of neutrophil chemotaxis. Our results suggest that sensory adaptation alone cannot explain the observed oscillatory motion. Rather, our model suggests that neutrophils employ a winner-take-all mechanism that enables them to transiently lock onto sensed targets and continuously switch between the intermediate attractant sources as they are encountered. These findings uncover a previously unseen behavior of neutrophils in opposing gradients of chemoattractants that will further aid in our understanding of neutrophil chemotaxis and the innate immune response. In addition, we propose a winner-take-all mechanism allows the cells to avoid stagnation near local chemical maxima when migrating through a network of chemoattractant sources.     

AgentCell: a digital single-cell assay for bacterial chemotaxis

MOTIVATION: In recent years, single-cell biology has focused on the relationship between the stochastic nature of molecular interactions and variability of cellular behavior. To describe this relationship, it is necessary to develop new computational approaches at the single-cell level. RESULTS: We have developed AgentCell, a model using agent-based technology to study the relationship between stochastic intracellular processes and behavior of individual cells. As a test-bed for our approach we use bacterial chemotaxis, one of the best characterized biological systems. In this model, each bacterium is an agent equipped with its own chemotaxis network, motors and flagella. Swimming cells are free to move in a 3D environment. Digital chemotaxis assays reproduce experimental data obtained from both single cells and bacterial populations.     

The three adaptation systems of Bacillus subtilis chemotaxis

Adaptation has a crucial role in the gradient-sensing mechanism that underlies bacterial chemotaxis. The Escherichia coli chemotaxis pathway uses a single adaptation system involving reversible receptor methylation. In Bacillus subtilis, the chemotaxis pathway seems to use three adaptation systems. One involves reversible receptor methylation, although quite differently than in E. coli. The other two involve CheC, CheD and CheV, which are chemotaxis proteins not found in E. coli. Remarkably, no one system is absolutely required for adaptation or is independently capable of generating adaptation. In this review, we discuss these three novel adaptation systems in B. subtilis and propose a model for their integration.     

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.     

Perfect and near-perfect adaptation in a model of bacterial chemotaxis

The signaling apparatus mediating bacterial chemotaxis can adapt to a wide range of persistent external stimuli. In many cases, the bacterial activity returns to its prestimulus level exactly, and this perfect adaptability is robust against variations in various chemotaxis protein concentrations. We model the bacterial chemotaxis signaling pathway, from ligand binding to CheY phosphorylation. By solving the steady-state equations of the model analytically, we derive a full set of conditions for the system to achieve perfect adaptation. The conditions related to the phosphorylation part of the pathway are discovered for the first time, while other conditions are generalizations of the ones found in previous works. Sensitivity of the perfect adaptation is evaluated by perturbing these conditions. We find that, even in the absence of some of the perfect adaptation conditions, adaptation can be achieved with near-perfect precision as a result of the separation of scales in both chemotaxis protein concentrations and reaction rates, or specific properties of the receptor distribution in different methylation states. Since near-perfect adaptation can be found in much larger regions of the parameter space than that defined by the perfect adaptation conditions, their existence is essential to understand robustness in bacterial chemotaxis.     

An improved chamber for direct visualisation of chemotaxis

There has been a growing appreciation over the last decade that chemotaxis plays an important role in cancer migration, invasion and metastasis. Research into the field of cancer cell chemotaxis is still in its infancy and traditional investigative tools have been developed with other cell types and purposes in mind. Direct visualisation chambers are considered the gold standard for investigating the behaviour of cells migrating in a chemotactic gradient. We therefore drew up a list of key attributes that a chemotaxis chamber should have for investigating cancer cell chemotaxis. These include (1) compatibility with thin cover slips for optimal optical properties and to allow use of high numerical aperture (NA) oil immersion objectives; (2) gradients that are relatively stable for at least 24 hours due to the slow migration of cancer cells; (3) gradients of different steepnesses in a single experiment, with defined, consistent directions to avoid the need for complicated analysis; and (4) simple handling and disposability for use with medical samples. Here we describe and characterise the Insall chamber, a novel direct visualisation chamber. We use it to show GFP-lifeact transfected MV3 melanoma cells chemotaxing using a 60x high NA oil immersion objective, which cannot usually be done with other chemotaxis chambers. Linear gradients gave very efficient chemotaxis, contradicting earlier results suggesting that only polynomial gradients were effective. In conclusion, the chamber satisfies our design criteria, most importantly allowing high NA oil immersion microscopy to track chemotaxing cancer cells in detail over 24 hours.     

Spatial organization in bacterial chemotaxis

Spatial organization of signalling is not an exclusive property of eukaryotic cells. Despite the fact that bacterial signalling pathways are generally simpler than those in eukaryotes, there are several well‐documented examples of higher‐order intracellular signalling structures in bacteria. One of the most prominent and best‐characterized structures is formed by proteins that control bacterial chemotaxis. Signals in chemotaxis are processed by ordered arrays, or clusters, of receptors and associated proteins, which amplify and integrate chemotactic stimuli in a highly cooperative manner. Receptor clusters further serve to scaffold protein interactions, enhancing the efficiency and specificity of the pathway reactions and preventing the formation of signalling gradients through the cell body. Moreover, clustering can also ensure spatial separation of multiple chemotaxis systems in one bacterium. Assembly of receptor clusters appears to be a stochastic process, but bacteria evolved mechanisms to ensure optimal cluster distribution along the cell body for partitioning to daughter cells at division.     

Modeling microbial chemotaxis in a diffusion gradient chamber

The diffusion gradient chamber (DGC) has proven to be a useful experimental tool for studying population-level microbial growth and chemotaxis. A mathematical model capable of reproducing the population-level patterns formed as a result of cellular growth and chemotaxis in the DGC has been developed. The model consists of coupled partial differential balance equations for cells, chemoattractants, and a nutrient, which are solved simultaneously by the alternating direction implicit method. Modeling simulation results were compared with population-level migration patterns of Escherichia coli growing on glycerol and responding to a gradient of the chemoattractant aspartate for two different initial conditions. To accurately reproduce the experimental results, a second chemoattractant equation was necessary. The second chemoattractant has been identified as oxygen by directly measuring oxygen gradients in the DGC. Important trends observed experimentally and reproduced by the model include the formation of a chemotactic wave, a reduction in the wave velocity as it encounters higher chemoattractant concentrations, and chemotaxis in response to two different chemoattractants simultaneously. The model was also used to study the relative magnitude of cell fluxes due to random motility and chemotaxis, and the suppression of chemotaxis due to receptor saturation. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 191-205, 1997.     

Modeling cell gradient sensing and migration in competing chemoattractant fields

Directed cell migration mediates physiological and pathological processes. In particular, immune cell trafficking in tissues is crucial for inducing immune responses and is coordinated by multiple environmental cues such as chemoattractant gradients. Although the chemotaxis mechanism has been extensively studied, how cells integrate multiple chemotactic signals for effective trafficking and positioning in tissues is not clearly defined. Results from previous neutrophil chemotaxis experiments and modeling studies suggested that ligand-induced homologous receptor desensitization may provide an important mechanism for cell migration in competing chemoattractant gradients. However, the previous mathematical model is oversimplified to cell gradient sensing in one-dimensional (1-D) environment. To better understand the receptor desensitization mechanism for chemotactic navigation, we further developed the model to test the role of homologous receptor desensitization in regulating both cell gradient sensing and migration in different configurations of chemoattractant fields in two-dimension (2-D). Our results show that cells expressing normal desensitizable receptors preferentially orient and migrate toward the distant gradient in the presence of a second local competing gradient, which are consistent with the experimentally observed preferential migration of cells toward the distant attractant source and confirm the requirement of receptor desensitization for such migratory behaviors. Furthermore, our results are in qualitative agreement with the experimentally observed cell migration patterns in different configurations of competing chemoattractant fields.     

Chemoattraction and chemotaxis in Dictyostelium discoideum: myxamoeba cannot read spatial gradients of cyclic adenosine monophosphate

Myxamoebae of the morphogenetic cellular slime mold Dictyostelium discoideum are thought to be able to accurately read and respond to directional information in spatial gradients of cyclic AMP. We examined the spatial and temporal mechanisms proposed for chemotaxis by comparing the behavior of spreading or evenly distributed cell populations after exposure to well-defined spatial gradients. The effects of gradient generation on cells were avoided by using predeveloped gradients. Qualitatively different responses were obtained using (a) isotropic, (b) static spatial, or (c) temporal (impulse) gradients in a simple chamber of penetrable micropore filters. We simulated models of chemotaxis and chemokinesis to aid our interpretations. The attractive and locomotory responses of populations were maximally stimulated by 0.05 microM cyclic AMP, provided that cellular phosphodiesterase was inhibited. But a single impulse of cyclic AMP during gradient development caused a greater and qualitatively different attraction. Attraction in spatial gradients was only transient, in that populations eventually developed a random distribution when confined to a narrow territory. Populations never accumulated nor lost their random distribution even in extremely steep spatial gradients. Attraction in spatial gradients was inducible only in spreading populations, not randomly distributed ones. Thus, spatial gradients effect biased-random locomotion: i.e., chemokinesis without adaptation. Cells cannot read gradients; the reaction of the cells is stochastic. Spatial gradients do not cause chemotaxis, which probably requires a sharp stimulant concentration increase (a temporal gradient) as a pulse or impulse. The results also bear on concepts of how embryonic cells might be able to decipher the positional information in a morphogen spatial gradient during development.     

The role of methylation in chemotaxis. An explanation of outstanding anomalies

The role of methylation in chemotaxis is understood generally, but several anomalies exist which bring into question the timing of methylation relative to sensing. A double mutant bacterium, deficient in both methyltransferase and methylesterase (Tr-Es-) is capable of chemotaxis even though the respective single mutants (Tr- and Es-) are not. This Tr-Es- mutant will accumulate in capillaries containing aspartic acid but not in capillaries containing serine despite the fact that both the aspartate and serine receptors are part of the methylation-dependent pathway. To understand these anomalies, a combination of theoretical analyses and experimental studies was performed. A mathematical analysis of the gradients of aspartate and serine in the capillary assay shows that outside the capillary the gradients are shallow, but just inside the mouth of the capillary they are very steep. Also, when the number of bacteria accumulated in the capillary is at a maximum, the range of attractant concentrations in the steep gradient just inside the mouth of the capillary is optimal for response and partial adaptation by the Tr-Es- mutant. We postulate that random motion brings the Tr-Es- mutant into the capillary, where it is able to move up the steep gradient. The difference in timing of the responses to serine and aspartate explains why the Tr-Es- mutant accumulates in aspartate- but not in serine-containing capillaries. A simple diffusion-capture model incorporating these concepts can account for experimental values of the number of Tr-Es- bacteria accumulating in the capillary. These studies provide a rational explanation for all of the apparent anomalies and lead to the conclusion that methylation/demethylation plays a crucial role in sensing as well as setting the zero point of the receptor.