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.     

Three Mutations in Escherichia coli That Generate Transformable Functional Flagella

Hydrodynamics predicts that swimming bacteria generate a propulsion force when a helical flagellum rotates because rotating helices necessarily translate at a low Reynolds number. It is generally believed that the flagella of motile bacteria are semirigid helices with a fixed pitch determined by hydrodynamic principles. Here, we report the characterization of three mutations in laboratory strains of Escherichia coli that produce different steady-state flagella without losing cell motility. E. coli flagella rotate counterclockwise during forward swimming, and the normal form of the flagella is a left-handed helix. A single amino acid exchange A45G and a double mutation of A48S and S110A change the resting flagella to right-handed helices. The stationary flagella of the triple mutant were often straight or slightly curved at neutral pH. Deprotonation facilitates the helix formation of it. The helical and curved flagella can be transformed to the normal form by torsion upon rotation and thus propel the cell. These mutations arose in the long-term laboratory cultivation. However, flagella are under strong selection pressure as extracellular appendages, and similar transformable flagella would be common in natural environments.     

A study of bacterial flagellar bundling

Certain bacteria, such as Escherichia coli (E. coli) and Salmonella typhimurium (S. typhimurium), use multiple flagella often concentrated at one end of their bodies to induce locomotion. Each flagellum is formed in a left-handed helix and has a motor at the base that rotates the flagellum in a corkscrew motion.We present a computational model of the flagellar motion and their hydrodynamic interaction. The model is based on the equations of Stokes flow to describe the fluid motion. The elasticity of the flagella is modeled with a network of elastic springs while the motor is represented by a torque at the base of each flagellum. The fluid velocity due to the forces is described by regularized Stokeslets and the velocity due to the torques by the associated regularized rotlets. Their expressions are derived. The model is used to analyze the swimming motion of a single flagellum and of a group of three flagella in close proximity to one another. When all flagellar motors rotate counterclockwise, the hydrodynamic interaction can lead to bundling. We present an analysis of the flow surrounding the flagella. When at least one of the motors changes its direction of rotation, the same initial conditions lead to a tumbling behavior characterized by the separation of the flagella, changes in their orientation, and no net swimming motion. The analysis of the flow provides some intuition for these processes.     

The Flat-Ribbon Configuration of the Periplasmic Flagella of Borrelia burgdorferi and Its Relationship to Motility and Morphology

Electron cryotomography was used to analyze the structure of the Lyme disease spirochete, Borrelia burgdorferi. This methodology offers a new means for studying the native architecture of bacteria by eliminating the chemical fixing, dehydration, and staining steps of conventional electron microscopy. Using electron cryotomography, we noted that membrane blebs formed at the ends of the cells. These blebs may be precursors to vesicles that are released from cells grown in vivo and in vitro. We found that the periplasmic space of B. burgdorferi was quite narrow (16.0 nm) compared to those of Escherichia coli and Pseudomonas aeruginosa. However, in the vicinity of the periplasmic flagella, this space was considerably wider (42.3 nm). In contrast to previous results, the periplasmic flagella did not form a bundle but rather formed a tight-fitting ribbon that wraps around the protoplasmic cell cylinder in a right-handed sense. We show how the ribbon configuration of the assembled periplasmic flagella is more advantageous than a bundle for both swimming and forming the flat-wave morphology. Previous results indicate that B. burgdorferi motility is dependent on the rotation of the periplasmic flagella in generating backward-moving waves along the length of the cell. This swimming requires that the rotation of the flagella exerts force on the cell cylinder. Accordingly, a ribbon is more beneficial than a bundle, as this configuration allows each periplasmic flagellum to have direct contact with the cell cylinder in order to exert that force, and it minimizes interference between the rotating filaments.     

Attachment of Vibrio alginolyticus to Glass Surfaces Is Dependent on Swimming Speed

The attachment of Vibrio alginolyticus to glass surfaces was investigated with special reference to the swimming speed due to the polar flagellum. This bacterium has two types of flagella, i.e., one polar flagellum and numerous lateral flagella. The mutant YM4, which possesses only the polar flagellum, showed much faster attachment than the mutant YM18, which does not possess flagella, indicating that the polar flagellum plays an important role. The attachment of YM4 was dependent on Na⁺ concentration and was specifically inhibited by amiloride, an inhibitor of polar flagellum rotation. These results are quite similar to those for swimming speed obtained under the same conditions. Observations with other mutants showed that chemotaxis is not critical and that the flagellum does not act as an appendage for attachment. From these results, it is concluded that the attachment of V. alginolyticus to glass surfaces is dependent on swimming speed.     

Cell orientation of swimming bacteria: From theoretical simulation to experimental evaluation

The motion of small bacteria consists of two phases: relatively long runs alternate with intermittent stops, back-ups, or tumbles, depending on the species. In polar monotrichous bacteria, the flagellum is anchored at the cell pole inherited from the parent generation (old pole) and is surrounded by a chemoreceptor cluster. During forward swimming, the leading pole is always the pole recently formed in cell division (new pole). The flagella of the peritrichous bacterium Escherichia coli often form a bundle behind the old pole. Its cell orientation and receptor positioning during runs generally mimic that of monotrichous bacteria. When encountering a solid surface, peritrichous bacteria exhibit a circular motion with the leading pole dipping downward. Some polar monotrichous bacteria also perform circular motion near solid boundaries, but during back-ups. In this case, the leading pole points upward. Very little is known about behavior near milieu-air interfaces. Biophysical simulations have revealed some of the mechanisms underlying these phenomena, but leave many questions unanswered. Combining biophysics with molecular techniques will certainly advance our understanding of bacterial locomotion.     

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.     

Direct Measurement of Helical Cell Motion of the Spirochete Leptospira

Leptospira are spirochete bacteria distinguished by a short-pitch coiled body and intracellular flagella. Leptospira cells swim in liquid with an asymmetric morphology of the cell body; the anterior end has a long-pitch spiral shape (S-end) and the posterior end is hook-shaped (H-end). Although the S-end and the coiled cell body called the protoplasmic cylinder are thought to be responsible for propulsion together, most observations on the motion mechanism have remained qualitative. In this study, we analyzed the swimming speed and rotation rate of the S-end, protoplasmic cylinder, and H-end of individual Leptospira cells by one-sided dark-field microscopy. At various viscosities of media containing different concentrations of Ficoll, the rotation rate of the S-end and protoplasmic cylinder showed a clear correlation with the swimming speed, suggesting that these two helical parts play a central role in the motion of Leptospira. In contrast, the H-end rotation rate was unstable and showed much less correlation with the swimming speed. Forces produced by the rotation of the S-end and protoplasmic cylinder showed that these two helical parts contribute to propulsion at nearly equal magnitude. Torque generated by each part, also obtained from experimental motion parameters, indicated that the flagellar motor can generate torque >4000 pN nm, twice as large as that of Escherichia coli. Furthermore, the S-end torque was found to show a markedly larger fluctuation than the protoplasmic cylinder torque, suggesting that the unstable H-end rotation might be mechanically related to changes in the S-end rotation rate for torque balance of the entire cell. Variations in torque at the anterior and posterior ends of the Leptospira cell body could be transmitted from one end to the other through the cell body to coordinate the morphological transformations of the two ends for a rapid change in the swimming direction.     

Direct Measurement of Helical Cell Motion of the Spirochete Leptospira

Leptospira are spirochete bacteria distinguished by a short-pitch coiled body and intracellular flagella. Leptospira cells swim in liquid with an asymmetric morphology of the cell body; the anterior end has a long-pitch spiral shape (S-end) and the posterior end is hook-shaped (H-end). Although the S-end and the coiled cell body called the protoplasmic cylinder are thought to be responsible for propulsion together, most observations on the motion mechanism have remained qualitative. In this study, we analyzed the swimming speed and rotation rate of the S-end, protoplasmic cylinder, and H-end of individual Leptospira cells by one-sided dark-field microscopy. At various viscosities of media containing different concentrations of Ficoll, the rotation rate of the S-end and protoplasmic cylinder showed a clear correlation with the swimming speed, suggesting that these two helical parts play a central role in the motion of Leptospira. In contrast, the H-end rotation rate was unstable and showed much less correlation with the swimming speed. Forces produced by the rotation of the S-end and protoplasmic cylinder showed that these two helical parts contribute to propulsion at nearly equal magnitude. Torque generated by each part, also obtained from experimental motion parameters, indicated that the flagellar motor can generate torque >4000 pN nm, twice as large as that of Escherichia coli. Furthermore, the S-end torque was found to show a markedly larger fluctuation than the protoplasmic cylinder torque, suggesting that the unstable H-end rotation might be mechanically related to changes in the S-end rotation rate for torque balance of the entire cell. Variations in torque at the anterior and posterior ends of the Leptospira cell body could be transmitted from one end to the other through the cell body to coordinate the morphological transformations of the two ends for a rapid change in the swimming direction.     

Coordinated Flagellar and Ciliary Beating in the Protozoon Tritrichomonas foetus1

ABSTRACT. Tritrichomonas foetus is a flagellated protozoon found in urogenital tract of cattle. Its free movement in liquid medium is powered by the coordinated movement of three flagella projecting towards the anterior region of the cell, and one recurrent flagellum that forms a junction with the cell body and ends as a free projection in the posterior region of the cell. We have used video microscopy and digital image processing to analyze the relationships between the movements of these flagella. The anterior flagella beat in a ciliary type pattern displaying effective and recovery strokes, while the recurrent flagellum beats in a typical flagellar wave form. One of the three anterior flagella has a distinctive pattern of beating. It beats straight in its forward direction as opposed to the ample beats performed by the others. Frequency measurements obtained from cells swimming in a viscous medium shows that the beating frequency of the recurrent flagelium is approximate twice the frequency for the three anterior flagella. We also observed that the costa and the axostyle do not show any active motion. On the contrary, they form a cytoskeletal base for the anchoring and orientation of the flagella.     

Swimming behaviour of the unicellular biflagellate Oxyrrhis marina: in vivo and in vitro movement of the two flagella

The movement of the 2 flagella of Oxyrrhis marina was examined with respect to their individual waveforms and the swimming behavior of the organism. The longitudinal flagella propagated helicoidal waves whose amplitude decreased toward the tip of th flagellum. Their beat frequencies were 50-60 Hz. The transverse flagella beat helicoidally within a furrow. Sudden changes in the direction of the cell trajectories were generated by transient arrests of the longitudinal flagellum beat, which were accompanied by a switch from the backward orientation to a forward one. This sweeping motion generated the rotation of the cell body. Ca2+ ions highly stimulated the frequencies of this arrest response, which compared to the "walking-stick" behavior of sea urchin spermatozoa. Isolated flagella were ATA-reactivated after detergent treatment. They exhibited 2 types of motion within the same experimental conditions. A progressive helicoidal motion was generated upon longitudinal flagellum reactivation, whereas a rolling motion with little progression characterized transverse flagellum reactivation. The differences in motile behavior reflect regulations of flagellar movement which were not destroyed by the isolation procedure and may be indicative of regulation by accessory structures.     

Difference in bacterial motion between forward and backward swimming caused by the wall effect

A bacterial cell that has a single polar flagellum alternately repeats forward swimming, in which the flagellum pushes the cell body, and backward swimming, in which the flagellum pulls the cell body. We have reported that the backward swimming speeds of Vibrio alginolyticus are on average greater than the forward swimming speeds. In this study, we quantitatively measured the shape of the trajectory as well as the swimming speed. The trajectory shape in the forward mode was almost straight, whereas that in the backward mode was curved. The same parameters were measured at different distances from a surface. The difference in the motion characteristics between swimming modes was significant when a cell swam near a surface. In contrast, the difference was indistinguishable when a cell swam >60 microm away from any surfaces. In addition, a cell in backward mode tended to stay near the surface longer than a cell in forward mode. This wall effect on the bacterial motion was independent of chemical modification of the glass surface. The macroscopic behavior is numerically simulated on the basis of experimental results and the significance of the phenomenon reported here is discussed.     

Dynamics of Bacterial Swarming

When vegetative bacteria that can swim are grown in a rich medium on an agar surface, they become multinucleate, elongate, synthesize large numbers of flagella, produce wetting agents, and move across the surface in coordinated packs: they swarm. We examined the motion of swarming Escherichia coli, comparing the motion of individual cells to their motion during swimming. Swarming cells' speeds are comparable to bulk swimming speeds, but very broadly distributed. Their speeds and orientations are correlated over a short distance (several cell lengths), but this correlation is not isotropic. We observe the swirling that is conspicuous in many swarming systems, probably due to increasingly long-lived correlations among cells that associate into groups. The normal run-tumble behavior seen in swimming chemotaxis is largely suppressed, instead, cells are continually reoriented by random jostling by their neighbors, randomizing their directions in a few tenths of a second. At the edge of the swarm, cells often pause, then swim back toward the center of the swarm or along its edge. Local alignment among cells, a necessary condition of many flocking theories, is accomplished by cell body collisions and/or short-range hydrodynamic interactions.     

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.     

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.     

Transition of bacterial flagella from helical to straight forms with different subunit arrangements.

We have found that several kinds of helical flagella from Salmonella and Escherichia become straight in the presence of 0·5 m-citric acid at pH values below 4·0, while the straight flagella from a mutant Salmonella (SJ814) are transformed into a helical shape under the same conditions. These transformations are reversible and transitional.Current models of bacterial flagella (Calladine, 1976,1978; Kamiya, 1976) predict that the family of distinct wave-forms should include two types of straight flagella, which have either an extreme right-handed twist (about 7 ° at the surface of the flagellum) or an extreme left-handed twist (2 ° to 3 °). As the inclination of the near-longitudinal rows of subunits in the Salmonella SJ814 flagellum (O'Brien &; Bennett, 1972) agrees closely with the degree of twisting predicted for the right-handed type, this flagellum has been considered to be the right-handed type. We have determined that the basic (1-start) helix in flagella is right-handed, using the method of Finch (1972). This fact, together with the selection rule (O'Brien &; Bennett, 1972), strongly suggests that the near-longitudinal rows in an SJ814 flagellum are right-handed, in agreement with the prediction. However, our optical diffraction and X-ray diffraction studies have revealed that the near-longitudinal rows of subunits in the citric acid-induced straight flagella and in the straight flagella from a mutant E. coli (Kondoh &; Yanagida, 1975) tilt at an angle of 2 ° to 3 ° with respect to the flagellar axis. This inclination is probably left-handed. Thus the predicted presence of the two types of straight flagella seems to be proved.     

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.     

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.     

Dinoflagellate flagella adopt various conformations in response to different needs

The two flagella of Dinoflagellates have, up to now, been poorly described.They display different structures and different patterns of behaviour compared with other organisms. In addition, the two flagella are different from each other: the transverse flagellum is ribbon-shaped and beats with a spiral undulation inside a furrow located around the cell body while the longitudinal flagellum has a larger diameter than simple flagella because it contains structures in addition to the axoneme and propagates essentially sinusoidal waves to push the cell. Ceratium flagella are particularly interesting to study because they both show different types of movements and have complex structures in addition to the axoneme. We propose that the additional structures are responsible for the particular movements of Dinoflagellate flagella. The presence of food particles in vacuoles in the vicinity of the flagella pocket suggests that their flagellar apparatus may not only be a propulsive organelle but could also be involved in prey capture.     

Flagellin phase‐dependent swimming on epithelial cell surfaces contributes to productive Salmonella gut colonisation

The flagellum is a sophisticated nanomachine and an important virulence factor of many pathogenic bacteria. Flagellar motility enables directed movements towards host cells in a chemotactic process, and near‐surface swimming on cell surfaces is crucial for selection of permissive entry sites. The long external flagellar filament is made of tens of thousands subunits of a single protein, flagellin, and many Salmonella serovars alternate expression of antigenically distinct flagellin proteins, FliC and FljB. However, the role of the different flagellin variants during gut colonisation and host cell invasion remains elusive. Here, we demonstrate that flagella made of different flagellin variants display structural differences and affect Salmonella's swimming behaviour on host cell surfaces. We observed a distinct advantage of bacteria expressing FliC‐flagella to identify target sites on host cell surfaces and to invade epithelial cells. FliC‐expressing bacteria outcompeted FljB‐expressing bacteria for intestinal tissue colonisation in the gastroenteritis and typhoid murine infection models. Intracellular survival and responses of the host immune system were not altered. We conclude that structural properties of flagella modulate the swimming behaviour on host cell surfaces, which facilitates the search for invasion sites and might constitute a general mechanism for productive host cell invasion of flagellated bacteria.