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.     

Characterization of two sets of subpolar flagella in Bradyrhizobium japonicum

Bradyrhizobium japonicum is one of the soil bacteria that form nodules on soybean roots. The cell has two sets of flagellar systems, one thick flagellum and a few thin flagella, uniquely growing at subpolar positions. The thick flagellum appears to be semicoiled in morphology, and the thin flagella were in a tight-curly form as observed by dark-field microscopy. Flagellin genes were identified from the amino acid sequence of each flagellin. Flagellar genes for the thick flagellum are scattered into several clusters on the genome, while those genes for the thin flagellum are compactly organized in one cluster. Both types of flagella are powered by proton-driven motors. The swimming propulsion is supplied mainly by the thick flagellum. B. japonicum flagellar systems resemble the polar-lateral flagellar systems of Vibrio species but differ in several aspects.     

Simultaneous measurement of bacterial flagellar rotation rate and swimming speed.

Swimming speeds and flagellar rotation rates of individual free-swimming Vibrio alginolyticus cells were measured simultaneously by laser dark-field microscopy at 25, 30, and 35 degrees C. A roughly linear relation between swimming speed and flagellar rotation rate was observed. The ratio of swimming speed to flagellar rotation rate was 0.113 microns, which indicated that a cell progressed by 7% of pitch of flagellar helix during one flagellar rotation. At each temperature, however, swimming speed had a tendency to saturate at high flagellar rotation rate. That is, the cell with a faster-rotating flagellum did not always swim faster. To analyze the bacterial motion, we proposed a model in which the torque characteristics of the flagellar motor were considered. The model could be analytically solved, and it qualitatively explained the experimental results. The discrepancy between the experimental and the calculated ratios of swimming speed to flagellar rotation rate was about 20%. The apparent saturation in swimming speed was considered to be caused by shorter flagella that rotated faster but produced less propelling force.     

Complex flagellar motions and swimming patterns of the flagellates Paraphysomonas vestita and Pteridomonas danica

Most flagellates with hispid flagella, that is, flagella with rigid filamentous hairs (mastigonemes), swim in the direction of the flagellar wave propagation with an anterior position of the flagellum. Previous analysis was based on planar wave propagation showing that the mastigonemes pull fluid along the flagellar axis. In the present study, we investigate the flagellar motions and swimming patterns for two flagellates with hispid flagella: Paraphysomonas vestita and Pteridomonas danica. Studies were carried out using normal and high-speed video recording, and particles were added to visualize flow around cells generating feeding currents. When swimming or generating flow, P. vestita was able to pull fluid normal to, and not just along, the flagellum, implying the use of the mastigonemes in an as yet un-described way. When the flagellum made contact with food particles, it changed the flagellar waveform so that the particle was fanned towards the ingestion area, suggesting mechano-sensitivity of the mastigonemes. Pteridomonas danica was capable of more complex swimming than previously described for flagellated protists. This was associated with control of the flagellar beat as well as an ability to bend the plane of the flagellar waveform.     

The Hydrodynamics of a Run-and-Tumble Bacterium Propelled by Polymorphic Helical Flagella

To study the swimming of a peritrichous bacterium such as Escherichia coli, which is able to change its swimming direction actively, we simulate the “run-and-tumble” motion by using a bead-spring model to account for: 1), the hydrodynamic and the mechanical interactions among the cell body and multiple flagella; 2), the reversal of the rotation of a flagellum in a tumble; and 3), the associated polymorphic transformations of the flagellum. Because a flexible hook connects the cell body and each flagellum, the flagella can take independent orientations with respect to the cell body. This simulation reproduces the experimentally observed behaviors of E. coli, namely, a three-dimensional random-walk trajectory in run-and-tumble motion and steady clockwise swimming near a wall. We show that the polymorphic transformation of a flagellum in a tumble facilitates the reorientation of the cell, and that the time-averaged flow-field near a cell in a run has double-layered helical streamlines, with a time-dependent flow magnitude large enough to affect the transport of surrounding chemoattractants.     

The shape and dynamics of the Leptospiraceae

Most swimming bacteria produce thrust by rotating helical filaments called flagella. Typically, the flagella stick out into the external fluid environment; however, in the spirochetes, a unique group that includes some highly pathogenic species of bacteria, the flagella are internalized, being incased in the periplasmic space; i.e., between the outer membrane and the cell wall. This coupling between the periplasmic flagella and the cell wall allows the flagella to serve a skeletal, as well as a motile, function. In this article, we propose a mathematical model for spirochete morphology based on the elastic interaction between the cell body and the periplasmic flagella. This model describes the mechanics of the composite structure of the cell cylinder and periplasmic flagella and accounts for the morphology of Leptospiraceae. This model predicts that the cell cylinder should be roughly seven times stiffer than the flagellum. In addition, we explore how rotation of the periplasmic flagellum deforms the cell cylinder during motility. We show that the transition between hook-shaped and spiral-shaped ends is purely a consequence of the change in direction of the flagellar motor and does not require flagellar polymorphism.     

FLAGELLAR DEVELOPMENT AND REGENERATION IN VOLVOX CARTERI (CHLOROPHYTA)1

The biflagellate somatic cells of Volvox carteri f. nagariensis lyengar exhibit an asymmetric pattern of flagellar development. Initiallt each somatic cell has two short (4 μm) flagella but after several hours one flagellum on each cell elongates unitl it reaches a length of 12 μm. Due to the regular arrangement of somatic cells in the Volvox spheroid it is apparent that the same flagellum on each somatic is the first to elongale. The asymmetric flagellar length is maintained for about 8 h after which the second flagellum on each somatic cell elongates. When the second flagellum attains the same length (12 μm) as the first flagellum, both flagella elongale at the same rate until reaching a final length of 22 μm. Experimental removal of somatic cell flagella results in their regeneration. Somatis cells regenerate both flagella simultaneously and full length flagella are produced in about 2 h. The intial rate of flagellar regeneration is about ten times faster than the intial rate of flagllar growth in development. Cycloheximide, an inhibitor of protein synthesis, has no effect on the initial rate of flagellar regeneration but the flagella produced in the presence of the drug are half the length of flagella produced in its absence. Somatic cells are able to regenerate flagella up to the time of α and β tubulin, the major structural proteins of the flagellar axoneme, and other cellular proteins.     

Fibrous Flagellar Hairs of Chlamydomonas reinhardtii Do Not Enhance Swimming

The flagella of Chlamydomonas reinhardtii possess fibrous ultrastructures of a nanometer-scale thickness known as mastigonemes. These structures have been widely hypothesized to enhance flagellar thrust; however, detailed hydrodynamic analysis supporting this claim is lacking. In this study, we present a comprehensive investigation into the hydrodynamic effects of mastigonemes using a genetically modified mutant lacking the fibrous structures. Through high-speed observations of freely swimming cells, we found the average and maximum swimming speeds to be unaffected by the presence of mastigonemes. In addition to swimming speeds, no significant difference was found for flagellar gait kinematics. After our observations of swimming kinematics, we present direct measurements of the hydrodynamic forces generated by flagella with and without mastigonemes. These measurements were conducted using optical tweezers, which enabled high temporal and spatial resolution of hydrodynamic forces. Through our measurements, we found no significant difference in propulsive flows due to the presence of mastigonemes. Direct comparison between measurements and fluid mechanical modeling revealed that swimming hydrodynamics were accurately captured without including mastigonemes on the modeled swimmer’s flagella. Therefore, mastigonemes do not appear to increase the flagella’s effective area while swimming, as previously thought. Our results refute the longstanding claim that mastigonemes enhance flagellar thrust in C. reinhardtii, and so, their function still remains enigmatic.     

Mesoscopic modeling of bacterial flagellar microhydrodynamics

A particle-based hybrid method of elastic network model and smooth-particle hydrodynamics has been employed to describe the propulsion of bacterial flagella in a viscous hydrodynamic environment. The method explicitly models the two aspects of bacterial propulsion that involve flagellar flexibility and long-range hydrodynamic interaction of low-Reynolds-number flow. The model further incorporates the molecular organization of the flagellar filament at a coarse-grained level in terms of the 11 protofilaments. Each of these protofilaments is represented by a collection of material points that represent the flagellin proteins. A computational model of a single flexible helical segment representing the filament of a bacterial flagellum is presented. The propulsive dynamics and the flow fields generated by the motion of the model filament are examined. The nature of flagellar deformation and the influence of hydrodynamics in determining the shape of deformations are examined based on the helical filament.     

A 3D Motile Rod-Shaped Monotrichous Bacterial Model

We introduce a 3D model for a motile rod-shaped bacterial cell with a single polar flagellum which is based on the configuration of a monotrichous type of bacteria such as Pseudomonas aeruginosa. The structure of the model bacterial cell consists of a cylindrical body together with the flagellar forces produced by the rotation of a helical flagellum. The rod-shaped cell body is composed of a set of immersed boundary points and elastic links. The helical flagellum is assumed to be rigid and modeled as a set of discrete points along the helical flagellum and flagellar hook. A set of flagellar forces are applied along this helical curve as the flagellum rotates. An additional set of torque balance forces are applied on the cell body to induce counter-rotation of the body and provide torque balance. The three-dimensional Navier–Stokes equations for incompressible fluid are used to describe the fluid dynamics of the coupled fluid–microorganism system using Peskin’s immersed boundary method. A study of numerical convergence is presented along with simulations of a single swimming cell, the hydrodynamic interaction of two cells, and the interaction of a small cluster of cells.     

Propulsive forces on the flagellum during locomotion of Chlamydomonas reinhardtii

The distributed propulsive forces exerted on the flagellum of the swimming alga Chlamydomonas reinhardtii by surrounding fluid were estimated from experimental image data. Images of uniflagellate mutant Chlamydomonas cells were obtained at 350 frames/s with 125-nm spatial resolution, and the motion of the cell body and the flagellum were analyzed in the context of low-Reynolds-number fluid mechanics. Wild-type uniflagellate cells, as well as uniflagellate cells lacking inner dynein arms (ida3) or outer dynein arms (oda2) were studied. Ida3 cells exhibit stunted flagellar waveforms, whereas oda2 cells beat with lower frequency. Image registration and sorting algorithms provided high-resolution estimates of the motion of the cell body, as well as detailed kinematics of the flagellum. The swimming cell was modeled as an ellipsoid in Stokes flow, propelled by viscous forces on the flagellum. The normal and tangential components of force on the flagellum (f_N and f_T) were related by resistive coefficients (C_N and C_T) to the corresponding components of velocity (V_N and V_T).The values of these coefficients were estimated by satisfying equilibrium requirements for force and torque on the cell. The estimated values of the resistive coefficients are consistent among all three genotypes and similar to theoretical predictions.     

The bidirectional polar and unidirectional lateral flagellar motors of Vibrio alginolyticus are controlled by a single CheY species

The bacterial flagellar motor is an elaborate molecular machine that converts ion-motive force into mechanical force (rotation). One of its remarkable features is its swift switching of the rotational direction or speed upon binding of the response regulator phospho-CheY, which causes the changes in swimming that achieve chemotaxis. Vibrio alginolyticus has dual flagellar systems: the Na(+)-driven polar flagellum (Pof) and the H(+)-driven lateral flagella (Laf), which are used for swimming in liquid and swarming over surfaces respectively. Here we show that both swimming and surface-swarming of V. alginolyticus involve chemotaxis and are regulated by a single CheY species. Some of the substitutions of CheY residues conserved in various bacteria have different effects on the Pof and Laf motors, implying that CheY interacts with the two motors differently. Furthermore, analyses of tethered cells revealed that their switching modes are different: the Laf motor rotates exclusively counterclockwise and is slowed down by CheY, whereas the Pof motor turns both counterclockwise and clockwise, and CheY controls its rotational direction.     

Effects of Geometric Parameters on Swimming of Micro Organisms with Single Helical Flagellum in Circular Channels

We present a computational fluid dynamics (CFD) model for the swimming of micro organisms with a single helical flagellum in circular channels. The CFD model is developed to obtain numerical solutions of Stokes equations in three dimensions, validated with experiments reported in literature, and used to analyze the effects of geometric parameters, such as the helical radius, wavelength, radii of the channel and the tail and the tail length on forward and lateral swimming velocities, rotation rates, and the efficiency of the swimmer. Optimal shapes for the speed and the power efficiency are reported. Effects of Brownian motion and electrostatic interactions are excluded to emphasize the role of hydrodynamic forces on lateral velocities and rotations on the trajectory of swimmers. For thin flagella, as the channel radius decreases, forward velocity and the power efficiency of the swimmer decreases as well; however, for thick flagella, there is an optimal radius of the channel that maximizes the velocity and the efficiency depending on other geometric parameters. Lateral motion of the swimmer is suppressed as the channel is constricted below a critical radius, for which the magnitude of the lateral velocity reaches a maximum. Results contribute significantly to the understanding of the swimming of bacteria in micro channels and capillary tubes.     

Effects of Geometric Parameters on Swimming of Micro Organisms with Single Helical Flagellum in Circular Channels

We present a computational fluid dynamics (CFD) model for the swimming of micro organisms with a single helical flagellum in circular channels. The CFD model is developed to obtain numerical solutions of Stokes equations in three dimensions, validated with experiments reported in literature, and used to analyze the effects of geometric parameters, such as the helical radius, wavelength, radii of the channel and the tail and the tail length on forward and lateral swimming velocities, rotation rates, and the efficiency of the swimmer. Optimal shapes for the speed and the power efficiency are reported. Effects of Brownian motion and electrostatic interactions are excluded to emphasize the role of hydrodynamic forces on lateral velocities and rotations on the trajectory of swimmers. For thin flagella, as the channel radius decreases, forward velocity and the power efficiency of the swimmer decreases as well; however, for thick flagella, there is an optimal radius of the channel that maximizes the velocity and the efficiency depending on other geometric parameters. Lateral motion of the swimmer is suppressed as the channel is constricted below a critical radius, for which the magnitude of the lateral velocity reaches a maximum. Results contribute significantly to the understanding of the swimming of bacteria in micro channels and capillary tubes.     

The Orientation of Swimming Biflagellates in Shear Flows

Biflagellated algae swim in mean directions that are governed by their environments. For example, many algae can swim upward on average (gravitaxis) and toward downwelling fluid (gyrotaxis) via a variety of mechanisms. Accumulations of cells within the fluid can induce hydrodynamic instabilities leading to patterns and flow, termed bioconvection, which may be of particular relevance to algal bioreactors and plankton dynamics. Furthermore, knowledge of the behavior of an individual swimming cell subject to imposed flow is prerequisite to a full understanding of the scaled-up bulk behavior and population dynamics of cells in oceans and lakes; swimming behavior and patchiness will impact opportunities for interactions, which are at the heart of population models. Hence, better estimates of population level parameters necessitate a detailed understanding of cell swimming bias. Using the method of regularized Stokeslets, numerical computations are developed to investigate the swimming behavior of and fluid flow around gyrotactic prolate spheroidal biflagellates with five distinct flagellar beats. In particular, we explore cell reorientation mechanisms associated with bottom-heaviness and sedimentation and find that they are commensurate and complementary. Furthermore, using an experimentally measured flagellar beat for Chlamydomonas reinhardtii, we reveal that the effective cell eccentricity of the swimming cell is much smaller than for the inanimate body alone, suggesting that the cells may be modeled satisfactorily as self-propelled spheres. Finally, we propose a method to estimate the effective cell eccentricity of any biflagellate when flagellar beat images are obtained haphazardly.     

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.     

Dynamic Model and Motion Mechanism of Magnetotactic Bacteria with Two Lateral Flagellar Bundles

Magnetotactic Bacteria (MTB) propel themselves by rotating their flagella and swim along the magnetic field lines.To analyze the motion of MTB,MTB magneto-ovoid strain MO-1 cells,each with two bundles of flagella,were taken as research object.The six-degrees-of-freedom (6-DoF) dynamic model of MO-1 was established based on the Newton-Euler dynamic equations.In particular,the interaction between the flagellum and fluid was considered by the resistive force theory.The simulated motion trajectory of MTB was found to consist of two kinds of helices:small helices resulting from the imbalance of force due to flagellar rotation,and large helices arising from the different directions of the rotation axis of the cell body and the propulsion axis of the flagellum.The motion behaviours of MTB in various magnetic fields were studied,and the simulation results agree well with the experiment results.In addition,the rotation frequency of the flagella was estimated at 1100 Hz,which is consistent with the average rotation rate for Na+-driven flagellar motors.The included angle of the magnetosome chain was predicted at 40° that is located within 20° to 60° range of the observed results.The results indicate the correctness of the dynamic model,which may aid research on the operation and control of MTB-propelled micro-actuators.Meanwhile,the motion behaviours of MTB may inspire the development of micro-robots with new driving mechanisms.     

Anomalies in the motion dynamics of long-flagella mutants of Chlamydomonas reinhardtii

Chlamydomonas reinhardtii has long been used as a model organism in studies of cell motility and flagellar dynamics. The motility of the well-conserved ‘9+2’ axoneme in its flagella remains a subject of immense curiosity. Using high-speed videography and morphological analyses, we have characterized long-flagella mutants (lf1, lf2-1, lf2-5, lf3-2, and lf4) of C. reinhardtii for biophysical parameters such as swimming velocities, waveforms, beat frequencies, and swimming trajectories. These mutants are aberrant in proteins involved in the regulation of flagellar length and bring about a phenotypic increase in this length. Our results reveal that the flagellar beat frequency and swimming velocity are negatively correlated with the length of the flagella. When compared to the wild-type, any increase in the flagellar length reduces both the swimming velocities (by 26–57%) and beat frequencies (by 8–16%). We demonstrate that with no apparent aberrations/ultrastructural deformities in the mutant axonemes, it is this increased length that has a critical role to play in the motion dynamics of C. reinhardtii cells, and, provided there are no significant changes in their flagellar proteome, any increase in this length compromises the swimming velocity either by reduction of the beat frequency or by an alteration in the waveform of the flagella.     

Bacterial cell-body rotation driven by a single flagellar motor and by a bundle

Using self-trapped Escherichia coli bacteria that have intact flagellar bundles on glass surfaces, we study statistical fluctuations of cell-body rotation in a steady (unstimulated) state. These fluctuations underline direction randomization of bacterial swimming trajectories and plays a fundamental role in bacterial chemotaxis. A parallel study is also conducted using a classical rotation assay in which cell-body rotation is driven by a single flagellar motor. These investigations allow us to draw the important conclusion that during periods of counterclockwise motor rotation, which contributes to a run, all flagellar motors are strongly correlated, but during the clockwise period, which contributes to a tumble, individual motors are uncorrelated in long times. Our observation is consistent with the physical picture that formation and maintenance of a coherent flagellar bundle is provided by a single dominant flagellum in the bundle.