Run and Tumble Chemotaxis in a Shear Flow: The Effect of Temporal Comparisons,Persistence, Rotational Diffusion,and Cell Shape

Escherichia coli is a motile bacterium that moves up a chemoattractant gradient by performing a biased random walk composed of alternating runs and tumbles. This paper presents calculations of the chemotactic drift velocity v _d (the mean velocity up the chemoattractant gradient) of an E. coli cell performing chemotaxis in a uniform, steady shear flow, with a weak chemoattractant gradient at right angles to the flow. Extending earlier models, a combined analytic and numerical approach is used to assess the effect of several complications, namely (i) a cell cannot detect a chemoattractant gradient directly but rather makes temporal comparisons of chemoattractant concentration, (ii) the tumbles exhibit persistence of direction, meaning that the swimming directions before and after a tumble are correlated, (iii) the cell suffers random re-orientations due to rotational Brownian motion, and (iv) the non-spherical shape of the cell affects the way that it is rotated by the shear flow. These complications influence the dependence of v _d on the shear rate γ. When they are all included, it is found that (a) shear disrupts chemotaxis and shear rates beyond γ≈2 s⁻¹ render chemotaxis ineffective, (b) in terms of maximizing drift velocity, persistence of direction is advantageous in a quiescent fluid but disadvantageous in a shear flow, and (c) a more elongated body shape is advantageous in performing chemotaxis in a shear flow. J.T. Locsei is supported by an Oliver Gatty Studentship from the University of Cambridge.     

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.     

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.     

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.     

Tumble Kinematics of Escherichia coli near a Solid Surface

Bacteria tumble periodically, following environmental cues. Whether they can tumble near a solid surface is a basic issue for the inception of infection or mineral biofouling. Observing freely swimming Escherichia coli near and parallel to a glass surface imaged at high magnification (×100) and high temporal resolution (500 Hz), we identified tumbles as events starting (or finishing, respectively) in abrupt deceleration (or reacceleration, respectively) of the body motion. Selected events show an equiprobable clockwise (CW) or counterclockwise change in direction that is superimposed on a surface CW path because of persistent propulsion. These tumbles follow a common long (about 300 ± 100 ms, N = 52) deceleration-reorientation-acceleration pattern. A wavelet transform multiscale analysis shows these tumbles cause in-plane diffusive reorientations with 1.5 rad²/s rotational diffusivity, a value that compares with that measured in bulk tumbles. In half of the cases, additional few-millisecond bursts of an almost equiprobable CW or counterclockwise change of direction (12 ± 90°, N = 89) occur within the reorientation stage. The highly dispersed absolute values of change of direction (70 ± 66°, N = 89) of only a few bursts destabilize the cell-swimming direction. These first observations of surface tumbles set a foundation for statistical models of run-and-tumble surface motion different from that in bulk and lend support for chemotaxis near solid surface.     

Enhancement of Swimming Speed Leads to a More-Efficient Chemotactic Response to Repellent

Negative chemotaxis refers to the motion of microorganisms away from regions with high concentrations of chemorepellents. In this study, we set controlled gradients of NiCl₂, a chemorepellent, in microchannels to quantify the motion of Escherichia coli over a broad range of concentrations. The experimental technique measured the motion of the bacteria in space and time and further related the motion to the local concentration profile of the repellent. Results show that the swimming speed of bacteria increases with an increasing concentration of repellent, which in turn enhances the drift velocity. The contribution of the increased swimming speed to the total drift velocity was in the range of 20 to 40%, with the remaining contribution coming from the modulation of the tumble frequency. A simple model that incorporates receptor dynamics, including adaptation, intracellular signaling, and swimming speed variation, was able to qualitatively capture the observed trend in drift velocity.     

Variation of swimming speed enhances the chemotactic migration of Escherichia coli

Studies on chemotaxis of Escherichia coli have shown that modulation of tumble frequency causes a net drift up the gradient of attractants. Recently, it has been demonstrated that the bacteria is also capable of varying its runs speed in uniform concentration of attractant. In this study, we investigate the role of swimming speed on the chemotactic migration of bacteria. To this end, cells are exposed to gradients of a non-metabolizable analogue of glucose which are sensed via the Trg sensor. When exposed to a gradient, the cells modulate their tumble duration, which is accompanied with variation in swimming speed leading to drift velocities that are much higher than those achieved through the modulation of the tumble duration alone. We use an existing intra-cellular model developed for the Tar receptor and incorporate the variation of the swimming speed along with modulation of tumble frequency to predict drift velocities close to the measured values. The main implication of our study is that E. coli not only modulates the tumble frequency, but may also vary the swimming speed to affect chemotaxis and thereby efficiently sample its nutritionally rich environment.

Electronic supplementary material

The online version of this article (doi:10.1007/s11693-015-9174-x) contains supplementary material, which is available to authorized users.     

Orientation in two dimensions: chemosensory motile behaviour of Euplotes vannus

Most studies on chemosensory motile behaviour of protists apply to free-swimming species moving in 3-dimensional space. But many protists are associated with surfaces and this excludes the use of helical klinotaxis for orientation in chemical gradients. It is here shown that the predominantly surface-dwelling ciliated protozoon Euplotes vannus orients itself in chemical gradients by simple temporal gradient sensing (equivalent to the run and tumble mechanism described for bacteria) and they react only to a temporal decrease in attractant concentration. The motility of Euplotes can be described as a classical 2-dimensional random walk with Poisson distribution of run lengths punctuated by random changes in walking direction. The chemosensory motile behaviour allows cells that are distributed within about 1 cm² to accumulate at point sources of an attractant within 3–4 min.     

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     

Swimming patterns of the quadriflagellate Tetraflagellochloris mauritanica (Chlamydomonadales,Chlorophyceae)

Chlamydomonadales are elective subjects for the investigation of the problems related to locomotion and transport in biological fluid dynamics, whose resolution could enhance searching efficiency and assist in the avoidance of dangerous environments. In this paper, we elucidate the swimming behavior of Tetraflagellochloris mauritanica, a unicellular–multicellular alga belonging to the order Chlamydomonadales. This quadriflagellate alga has a complex swimming motion consisting of alternating swimming phases connected by in‐place random reorientations and resting phases. It is capable of both forward and backward swimming, both being normal modes of swimming. The complex swimming behavior resembles the run‐and‐tumble motion of peritrichous bacteria, with in‐place reorientation taking the place of tumbles. In the forward swimming, T. mauritanica shows a very efficient flagellar beat, with undulatory retrograde waves that run along the flagella to their tip. In the backward swimming, the flagella show a nonstereotypical synchronization mode, with a pattern that does not fit any of the modes present in the other Chlamydomonadales so far investigated.     

Collective motion enhances chemotaxis in a two-dimensional bacterial swarm

In a dilute liquid environment in which cell-cell interaction is negligible, flagellated bacteria, such as Escherichia coli, perform chemotaxis by biased random walks alternating between run-and-tumble. In a two-dimensional crowded environment, such as a bacterial swarm, the typical behavior of run-and-tumble is absent, and this raises the question whether and how bacteria can perform chemotaxis in a swarm. Here, by examining the chemotactic behavior as a function of the cell density, we showed that chemotaxis is surprisingly enhanced because of cell crowding in a bacterial swarm, and this enhancement is correlated with increase in the degree of cell body alignment. Cells tend to form clusters that move collectively in a swarm with increased effective run length, and we showed analytically that this resulted in increased drift velocity toward attractants. We also explained the enhancement by stochastically simulating bacterial chemotaxis in a swarm. We found that cell crowding in a swarm enhances chemotaxis if the cell-cell interactions used in the simulation induce cell-cell alignment, but it impedes chemotaxis if the interactions are collisions that randomize cell moving direction. Therefore, collective motion in a bacterial swarm enhances chemotaxis.     

The physics of flagellar motion of E. coli during chemotaxis

Flagellar motion has been an active area of study right from the discovery of bacterial chemotaxis in 1882. During chemotaxis, E. coli moves with the help of helical flagella in an aquatic environment. Helical flagella are rotated in clockwise or counterclockwise direction using reversible flagellar motors situated at the base of each flagellum. The swimming of E. coli is characterized by a low Reynolds number that is unique and time reversible. The random motion of E. coli is influenced by the viscosity of the fluid and the Brownian motion of molecules of fluid, chemoattractants, and chemorepellants. This paper reviews the literature about the physics involved in the propulsion mechanism of E. coli. Starting from the resistive-force theory, various theories on flagellar hydrodynamics are critically reviewed. Expressions for drag force, elastic force and velocity of flagellar elements are derived. By taking the elastic nature of flagella into account, linear and nonlinear equations of motions are derived and their solutions are presented.     

The effect of amino acids on the motile behavior of Bacillus subtilis

Constant levels of amino acids enhanced the velocity of Bacillus subtilis 60015 cells about 2-fold and stimulated the response in motility assays. The stimulation of velocity did not occur via the receptors for chemotaxis. Cysteine and methionine, general inhibitors of chemotaxis, both completely inhibited the smooth response in a temporal gradient of attractant. After methionine starvation B. subtilis 60015 showed no measurable response in a temporal gradient of attractant, this in contrast to the effect observed with some other bacteria. Addition of methionine to starved cells restored the response toward attractant. Revertants of B. subtilis 60015 for methionine requirement could not be starved and showed a normal behavior toward temporal gradients of attractant.Abbreviation O.D.₆₀₀ optical density measured at 600 nm     

A stochastic model for chemosensory cell movement: application to neutrophil and macrophage persistence and orientation

Two central features of leukocyte chemosensory movement behavior demand fundamental theoretical understanding. In uniform concentrations of chemoattractant, these cells exhibit a persistent random walk, with a characteristic “persistence time” between significant changes in direction. In chemoattractant concentration gradients, they demonstrate a biased random walk, with an “orientation bias” characterizing the fraction of cells moving up the gradient. A coherent picture of cell-movement responses to chemoattractant requires that both the persistence time and the orientation bias be explained within a unifying framework. In this paper we offer the possibility that “noise” in the cellular signal perception/response mechanism can simultaneously account for these two key phenomena. In particular, we report on a stochastic mathematical model for cell locomotion based on kinetic fluctuations in chemoattractant receptor binding. This model proves to be capable of stimulating cell paths similar to those observed experimentally for two cell types examined to date: neutrophils and alveolar macrophages, under conditions of uniform chemoattractant concentrations as well as chemoattractant concentration gradients. Further, this model can quantitatively predict both cell persistence time and dependence of orientation bias on gradient size. The model also successfully predicts that an increase in persistence time is associated with a decrease in orientation for typical system parameter values, as is observed for alveolar macrophages in comparison to neutrophils. Thus, the concept of signal “noise” can quantitatively unify the major characteristics of leukocyte random motility and chemotaxis. The same level of noise large enough to account for the observed frequency of turning in uniform environments is simultaneously small enough to allow for the observed degree of directional bias in gradients. This suggests that chemosensory cell movement behavior may be based on a “usefully” imperfect integrated signal response system, which allows both random and directed searches under appropriate conditions.     

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.     

Effect of Bacterial Chemotaxis on Biodegradation in a Porous Medium

This research investigates the effect of bacterial chemotaxis on biodegradation rate in an experimental model for in situ bioremediation. The novel experimental methodology of this work has provided for the systematic evaluation of the effect of the chemotaxis phenomenon in a saturated porous medium. The methodology has been developed to measure enhancement of degradation rate of serine, a simulated contaminant and chemoattractant. Escherichia coli RP437 was used as a representative, chemotactic, in situ bacterium, whereas E. coli RP5700, a tsr? mutant strain of RP437 that lacks the serine chemoreceptor, was used as the specifically nonchemotactic control strain. These two strains were highly characterized for this work. Swimming speeds, run lengths, and turn angles were compared using a tracking microscope and were statistically similar, as were serine uptake rates, making this pair of strains an excellent choice for chemotaxis studies. For these experiments, a model aquifer introduces bacteria to serine in saturated sand via a sharp gradient. The aquifer successfully achieved biodegradation at an 18% level; however, the degradation rate of serine was similar for both strains over 21 h, indicating that chemotaxis enhancement did not occur. This result is in agreement with certain prior works which did not detect enhanced chemotactic migration.     

Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms

We study kinetic models for chemotaxis, incorporating the ability of cells to assess temporal changes of the chemoattractant concentration as well as its spatial variations. For prescribed smooth chemoattractant density, the macroscopic limit is carried out rigorously. It leads to a drift equation with a chemotactic sensitivity depending on the time derivative of the chemoattractant density. As an application it is shown by numerical experiments that the new model can resolve the chemotactic wave paradox. For this purpose, the macroscopic equation is coupled to a simple activation-inhibition model for the chemoattractant which produces the chemoattractant waves typical for the slime mold Dictyostelium discoideum.     

Understanding Streaming in <Emphasis Type="Italic">Dictyostelium discoideum</Emphasis>: Theory Versus Experiments

Recent experimental work involving Dictyostelium discoideum seems to contradict several theoretical models. Experiments suggest that localization of the release of the chemoattractant cyclic adenosine monophosphate to the uropod of the cell is important for stream formation during aggregation. Yet several mathematical models are able to reproduce streaming as the cells aggregate without taking into account localization of the chemoattractant. A careful analysis of the experiments and the theory suggests the two major features of the system which are important to stream formation are random cell motion and chemotaxis to regions of higher cell density. Random cell motion acts to reduce streaming, whereas chemotaxis to regions of higher cell density reinforces streaming. With this understanding, the experimental results can be explained in a manner consistent with the theoretical results. In all the experiments, alterations in the two main factors of random motion and chemotaxis to regions of higher cell density, not the localization of the release of the chemoattractant, can explain the results as they relate to streaming. Additionally, a comparison of results from a mathematical model that simulates cells which localize the chemoattractant and cells which do not shows little difference in the streaming patterns.     

A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations

Two central features of polymorphonuclear leukocyte chemosensory movement behavior demand fundamental theoretical understanding. In uniform concentrations of chemoattractant, these cells exhibit a persistent random walk, with a characteristic "persistence time" between significant changes in direction. In chemoattractant concentration gradients, they demonstrate a biased random walk, with an "orientation bias" characterizing the fraction of cells moving up the gradient. A coherent picture of cell movement responses to chemoattractant requires that both the persistence time and the orientation bias be explained within a unifying framework. In this paper, we offer the possibility that "noise" in the cellular signal perception/response mechanism can simultaneously account for these two key phenomena. In particular, we develop a stochastic mathematical model for cell locomotion based on kinetic fluctuations in chemoattractant/receptor binding. This model can simulate cell paths similar to those observed experimentally, under conditions of uniform chemoattractant concentrations as well as chemoattractant concentration gradients. Furthermore, this model can quantitatively predict both cell persistence time and dependence of orientation bias on gradient size. Thus, the concept of signal "noise" can quantitatively unify the major characteristics of leukocyte random motility and chemotaxis. The same level of noise large enough to account for the observed frequency of turning in uniform environments is simultaneously small enough to allow for the observed degree of directional bias in gradients.