Unidirectional, intermittent rotation of the flagellum of Rhodobacter sphaeroides.

The single flagellum of the photosynthetic bacterium Rhodobacter sphaeroides was found to be medially located on the cell body. Observation of free-swimming bacteria, and bacteria tethered by their flagellar filaments, revealed that the flagellum could only rotate in the clockwise direction; switching of the direction of rotation was never observed. Flagellar rotation stopped periodically, typically several times a minute for up to several seconds each. Reorientation of swimming cells appeared to be the result of Brownian rotation during the stop periods. The flagellar filament displayed polymorphism; detached and nonrotating filaments were usually seen as large-amplitude helices of such short wavelength that they appeared as flat coils or circles, whereas the filaments on swimming cells showed a normal (small-amplitude, long-wavelength) helical form. With attached filaments, the transition from the normal to the coiled form occurred when the flagellar motor stopped rotating, proceeding from the distal end towards the cell body. It is possible that both the relaxation process and the smaller frictional resistance after relaxation may act to enhance the rate of reorientation of the cell. The transition from the coiled to the normal form occurred when the motor restarted, proceeding from the proximal end outwards, which might further contribute to the reorientation of the cell before it reaches a stable swimming geometry.     

Spirillum swimming: theory and observations of propulsion by the flagellar bundle.

The hydrodynamics and energetics of helical swimming by the bacterium Spirillum sp. is analysed using observations from medium speed cine photomicrography and theory. The photographic records show that the swimming organism's flagellar bundles beat in a helical fashion just as other bacterial flagella do. The data are analysed according to the rotational resistive theory of Chwang & Wu (1971) in a simple-to-use parametric form with the viscous coefficients Cs and Cn calculated according to the method of Lighthill (1975). Results of the analysis show that Spirillum dissipated biochemical energy in performing work against fluid resistance to motion at an average rate of about 6 X 10(-8) dyne cm s-1 with some 62-72% of the power dissipation due to the non-contractile body. These relationships yield a relatively low hydromechanical efficiency which is reflected in swimming speeds much smaller than a representative eukaryote. In addition the Cn/Cs ratio for the body is shown to lie in the range 0-86-1-51 and that for the flagellar bundle in the range 1-46-1-63. The implications of the power calculations for the Berg & Anderson (1973) rotating shaft model are discussed and it is shown that a rotational resistive theory analysis predicts a 5-cross bridge M ring for each flagellum of Spirillum.     

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.     

Simulating the Complex Cell Design of Trypanosoma brucei and Its Motility

The flagellate Trypanosoma brucei, which causes the sleeping sickness when infecting a mammalian host, goes through an intricate life cycle. It has a rather complex propulsion mechanism and swims in diverse microenvironments. These continuously exert selective pressure, to which the trypanosome adjusts with its architecture and behavior. As a result, the trypanosome assumes a diversity of complex morphotypes during its life cycle. However, although cell biology has detailed form and function of most of them, experimental data on the dynamic behavior and development of most morphotypes is lacking. Here we show that simulation science can predict intermediate cell designs by conducting specific and controlled modifications of an accurate, nature-inspired cell model, which we developed using information from live cell analyses. The cell models account for several important characteristics of the real trypanosomal morphotypes, such as the geometry and elastic properties of the cell body, and their swimming mechanism using an eukaryotic flagellum. We introduce an elastic network model for the cell body, including bending rigidity and simulate swimming in a fluid environment, using the mesoscale simulation technique called multi-particle collision dynamics. The in silico trypanosome of the bloodstream form displays the characteristic in vivo rotational and translational motility pattern that is crucial for survival and virulence in the vertebrate host. Moreover, our model accurately simulates the trypanosome''s tumbling and backward motion. We show that the distinctive course of the attached flagellum around the cell body is one important aspect to produce the observed swimming behavior in a viscous fluid, and also required to reach the maximal swimming velocity. Changing details of the flagellar attachment generates less efficient swimmers. We also simulate different morphotypes that occur during the parasite''s development in the tsetse fly, and predict a flagellar course we have not been able to measure in experiments so far.     

Comparative analysis of movement characteristics of lancelet and fish spermatozoa having different morphologies

The movement characteristics of the sperm and their flagella obtained from a lancelet and 35 species from almost all orders of fishes were examined using high-speed video microscopy. The aim was to clarify the relationship between the motility parameters of the spermatozoa having different morphologies and how these motility parameters affect the swimming speed of the spermatozoa. The motility parameters representing the flagellar waveform, the wavelength, and the amplitude were neither very different between the spermatozoa of the different species nor related to the swimming speed. In contrast, the beat frequency was remarkably changed in the different spermatozoa and was proportional to the swimming speed. The maximum shear angle of the flagellar wave, which is directly related to the maximum sliding displacement between the doublet microtubules, remained nearly constant while the beat frequency varied widely; therefore, the spermatozoa beat in the constant sliding displacement mode. An analysis of the relationship between swimming speed and flagellar length revealed that short flagella were at a disadvantage in developing swimming speed; however, so were extra-long flagella. The ratio of the swimming speed to the wave velocity calculated from the wavelength and the beat frequency depended on the distance from the glass surface. The swimming speeds calculated using the original resistive-force theory were greater than the measured values. To rationalize the measured values, the ratio between the normal and tangential drag coefficient in the resistive-force theory was corrected; namely, 1.99 at 1 μm and 1.63 at 3 μm from the glass surface.     

Coupling biochemistry and hydrodynamics captures hyperactivated sperm motility in a simple flagellar model

Hyperactivation in mammalian sperm is characterized by highly asymmetrical waveforms and an increase in the amplitude of flagellar bends. It is important for the sperm to be able to achieve hyperactivated motility in order to reach and fertilize the egg. Calcium (Ca²⁺) dynamics are known to play a large role in the initiation and maintenance of hyperactivated motility. Here we present an integrative model that couples the CatSper channel mediated Ca²⁺ dynamics of hyperactivation to a mechanical model of an idealized sperm flagellum in a 3-d viscous, incompressible fluid. The mechanical forces are due to passive stiffness properties and active bending moments that are a function of the local Ca²⁺ concentration along the length of the flagellum. By including an asymmetry in bending moments to reflect an asymmetry in the axoneme's response to Ca²⁺, we capture the transition from activated motility to hyperactivated motility. We examine the effects of elastic properties of the flagellum and the Ca²⁺ dynamics on the overall swimming patterns. The swimming velocities of the model flagellum compare well with data for hyperactivated mouse sperm.     

A theory of piezoelectric activity and ion-movements in the relation of flagellar structures and their movements to the phototaxis of Euglena

A review of the literature on the flagellar undulations and phototactic movements of Euglena indicates that the flagellum functions as an ATP-using motor, triggered and mediated by cations, especially H₃O⁺, K^+, Mg²⁺ and Ca²⁺, and driven by energy from ATP. The undulatory waves are assumed to be started by means of repetitive pulses due to a redox reaction at the base of the flagellum. It is also assumed that the axoneme and paraflagellar rod are composed of asymmetrically-crystalline proteinaceous fibrils which are piezoelectric, i.e. they bend when energy passes through or along them, thus acting as a motor, and when bending they deliver a current, thus acting as a generator. This piezoelectric activity displaces cations and drives them ahead of it, triggering sequential bending and straightening of segments of the flagellum from base to tip. The paraflagellar swelling (“photoreceptor”) is also assumed to be piezoelectric, reactive to light, acting as a capacitor. It discharges as the intensity of light striking it is changed by the alternative shading effect of the stigma (“eyespot”) and exposure to light as the Euglena gyrates in swimming. The charge delivered by the photoreceptor augments the effects of ion-movements along the flagellum, also augmenting the amplitude and force of the flagellar undulations and altering the position of the flagellum relative to the body and the direction of swimming. The body is tipped away from the original path and swims either toward or away from the light, depending on the ultimate alteration of the path of swimming.     

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 role of hydrodynamic interaction in the locomotion of microorganisms.

A general Boundary Element Method is presented and benchmarked with existing Slender Body Theory results and reflection solutions for the motion of spheres and slender bodies near plane boundaries. This method is used to model the swimming of a microorganism with a spherical cell body, propelled by a single rotating flagellum. The swimming of such an organism near a plane boundary, midway between two plane boundaries or in the vicinity of another similar organism, is investigated. It is found that only a small increase (less than 10%) results in the mean swimming speed of an organism swimming near and parallel to another identical organism. Similarly, only a minor propulsive advantage (again, less than 10% increase in mean swimming speed) is predicted when an organism swims very close and parallel to plane boundaries (such as a microscopic plate and (or) a coverslip, for example). This is explained in terms of the flagellar propulsive advantage derived from an increase in the ratio of the normal to tangential resistance coefficients of a slender body being offset by the apparently equally significant increase in the cell body drag. For an organism swimming normal to and toward a plane boundary, however, it is predicted that (assuming it is rotating its flagellum, relative to its cell body, with a constant angular frequency) the resulting swimming speed decreases asymptotically as the organism approaches the boundary.     

Computer Simulation of Flagellar Movement: I. Demonstration of Stable Bend Propagation and Bend Initiation by the Sliding Filament Model

A program has been developed for digital computer simulation of the movement of a flagellar model consisting of straight segments connected by joints at which bending occurs. The program finds values for the rate of bending at each joint by solving equations which balance active, viscous, and elastic bending moments at each joint. These bending rates are then used to compute the next position of the model. Stable swimming movements, similar to real flagellar movements, can be generated routinely with a 25-segment model using 16 time steps/beat cycle. These results depend on four assumptions about internal flagellar mechanisms: (a) Bending is generated by a sliding filament process. (b) The active process is controlled locally by the curvature of the flagellum. (c) Nonlinear elastic resistances stabilize the amplitude of the movement. (d) Internal viscous resistances stabilize the wavelength of the movement and explain the relatively low sensitivity of flagellar movement to changes in external viscosity.     

Hydrodynamic theory of swimming of flagellated microorganisms.

A theory of the type commonly used in polymer hydrodynamics is developed to calculate swimming properties of flagellated microorganisms. The overall shape of the particle is modeled as an array of spherical beads which act, at the same time, as frictional elements. The fluid velocity field is obtained as a function of the forces acting at each bead through Oseen-type, hydrodynamic interaction tensors. From the force and torque equilibrium conditions, such quantities as swimming velocity, angular velocity, and efficiency can be calculated. Application is made to a spherical body propelled by a helical flagellum. A recent theory by Lighthill, and earlier formulations based on tangential and normal frictional coefficients of a curved cylinder, CT and CN, are analyzed along with our theory. Although all the theories predict similar qualitative characteristics, such as optimal efficiency and the effect of fluid viscosity, they lead to rather different numerical values. In agreement with Lighthill, we found the formalisms based on CN and CT coefficients to be somewhat inaccurate, and head-flagellum interactions are shown to play an important role.     

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.     

Transformation of the flagella and associated flagellar components during cell division in the coccolithophoridPleurochrysis carterae

Summary Immunofluorescence microscopy, conventional and high voltage transmission electron microscopy were used to describe changes in the flagellar apparatus during cell division in the motile, coccolithbearing cells ofPleurochrysis carterae (Braarud and Fagerlund) Christensen. New basal bodies appear alongside the parental basal bodies before mitosis and at prophase the large microtubular (crystalline) roots disassemble as their component microtubules migrate to the future spindle poles. By prometaphase the crystalline roots have disappeared; the flagellar axonemes shorten and the two pairs of basal bodies (each consisting of one parental and one daughter basal body) separate so that each pair is distal to a spindle pole. By late prometaphase the pairs of basal bodies bear diminutive flagellar roots for the future daughter cells. The long flagellum of each daughter cell is derived from the parental basal bodies; thus, the basal body that produces a short flagellum in the parent produces a long flagellum in the daughter cell. We conclude that each basal body in these cells is inherently identical but that a first generation basal body generates a short flagellum and in succeeding generations it produces a long flagellum. At metaphase a fibrous band connecting the basal bodies appears and the roots and basal bodies reorient to their interphase configuration. By telophase the crystalline roots have begun to reform and the rootlet microtubules have assumed their interphase appearance by early cytokinesis.Abbreviations CR1, CR2 crystalline roots 1 and 2 - CT cytoplasmic tongue microtubules - DIC differential interference contrast light microscopy - H haptonema - HVEM high voltage transmission electron microscopy - IMF immunofluorescence microscopy - L left flagellum/basal body - M metaphase plate - MT microtubule - N nucleus - R right flagellum/basal body - R1, R2, R3 roots 1, 2, and 3 - TEM transmission electron microscopy     

Flagellar apparatus duplication and partition, flagellar transformation during division in Entosiphon sulcatum.

Electron microscopic examination of serial sections of developmental stages of the flagellar apparatus during the cell cycle indicates that the basal bodies replicate in a semi-conservative manner and that there is a flagellar transformation over two cell cycles in euglenoids as in other algal flagellate groups. Two new pairs of basal bodies are formed, each pair comprising one parental and one newly developed basal body. There is a transformation of the parental dorsal flagellum containing a thin paraxonemal rod into a ventral flagellum bearing a large paraxonemal rod. Observation of the roots associated with the basal bodies shows that the dorsal root transforms into an intermediate root over two cell cycles following the transformation of the dorsal basal body/flagellum to a ventral one. Also the two ventral roots are newly formed in relation to the formation of two new phagotrophic apparatuses during the division. After the breakage of the connection between the parental basal bodies the two new pairs move apart and are guided/drawn by transverse microfibrillar bundles which connect them to opposite sides of the pellicle. The axis of the separation/migration of the pairs of basal bodies is parallel to the axis of elongation of the dividing nucleus.     

THE FLAGELLAR DEVELOPMENT CYCLE OF THE UNIFLAGELLATE PELAGOMONAS CALCEOLATA (PELAGOPHYCEAE)1

Vegetative cells of Pelagomonas calceolata Andersen & Saunders were confirmed to possess a reduced flagellar apparatus, consisting of a single basal body/flagellum that is not accompanied by either flagellar roots or a barren basal body. Just prior to division, the parental flagellum retracts (or is abscised) as two new basal bodies/flagella arise de novo. During cytokinesis the parental basal body segregates with a new basal body/flagellum, briefly producing a progeny cell typical of other known uniflagellates (i.e. containing a basal body/flagellum and accompanying barren basal body). The parental basal body then disintegrates or "transforms" out of existence, leaving both progeny cells with a single basal body/flagellum (i.e. neither progeny cell possesses any vestige of a mature flagellum/basal body ). Pelagomonas calceolata belongs to a lineage of chromophyte algae characterized by having a reduced flagellar apparatus, but it is the only known species, not only in this lineage but among all eukaryotes, to have undergone the complete elimination of the mature flagellum /basal body .     

Bending-wave propagation by microtubules and flagella

The movement of an elastic filament in a viscous medium can be computed from the fourth-order nonlinear partial differential equation obtained by balancing bending moments at all points along the length of the filament. These bending moments result from active forces, elastic resistance to bending, and viscous resistance to movement through the medium. I have studied numerical solutions obtained for two situations of biological interest: For the movement of individual microtubules, the active force is generated by interaction between the microtubule and the substratum over which it is moving, and is directed along the axis of the microtubule. The computations can reproduce the gliding movement of unrestrained microtubules, and also the periodic bending and bend propagation seen when the leading end of the microtubule is restrained. No modulation of active force is required to generate bending waves. For the movement of flagella, the active forces are generated internally as sliding forces between adjacent members of a cylinder of nine microtubular doublets. Without some additional control assumptions, the forces will be balanced and no bending moments will be generated. The problem faced by investigators of flagellar motility is to determine the control mechanisms that operate to make the system asymmetric, so that active bending moments are generated. Computations with models in which the curvature of the flagellum modulates the active-force generators have indicated that this control specification is sufficient to generate oscillation and bend propagation, but is insufficient to completely determine the movement.     

ULTRASTRUCTURAL STUDIES ON BIGELOWIELLA NATANS, GEN. ET SP. NOV., A CHLORARACHNIOPHYTE FLAGELLATE

Three isolates from the Provasoli-Guillard National Center for Culture of Marine Phytoplankton at Bigelow Laboratory, previously labeled Pedinomonas sp. and Pedinomonas minutissima from the green algal class Pedinophyceae, have been examined by light microscopy and TEM and shown to belong to the Chlorarachniophyceae, a class of nucleomorph-containing amebae. The three isolates represent the first chlorarachniophycean flagellates to be discovered. The ultrastructure of the cells has been examined in detail, with particular emphasis on the flagellar apparatus, a feature not examined in detail in chlorarachniophytes before. Cells are basically biflagellate, but the second flagellum is represented by a very short basal body only. Flagellar replication has shown this flagellum to be the mature stage, that is, the no. 1 flagellum, whereas the long emergent flagellum is the no. 2 flagellum that shortens into a short basal body during cell division. Mitosis is open with a pair of centrioles at each pole. Emergent flagella are absent during mitosis. Cells may form cysts, and the flagellar basal bodies and part of the flagellar roots are maintained in the cysts. Four microtubular roots emanate from the basal bodies, and the path of one of them is very unusual and very unlike any other known flagellate. No striated roots were observed. Other fine-structural features of the cell include a very unusual type of pyrenoid and a special type of extrusome. Cells are mixotrophic. The three isolates are very similar and are described as Bigelowiella natans , gen. et sp. nov. Ultrastructurally, chlorarachniophytes do not show close relationship to any known group of algae or other protists.     

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.     

Computer simulation of flagellar movement. VI. Simple curvature-controlled models are incompletely specified.

Computer simulation is used to examine a simple flagellar model that will initiate and propagate bending waves in the absence of viscous resistances. The model contains only an elastic bending resistance and an active sliding mechanism that generates reduced active shear moment with increasing sliding velocity. Oscillation results from a distributed control mechanism that reverses the direction of operation of the active sliding mechanism when the curvature reaches critical magnitudes in either direction. Bend propagation by curvature-controlled flagellar models therefore does not require interaction with the viscous resistance of an external fluid. An analytical examination of moment balance during bend propagation by this model yields a solution curve giving values of frequency and wavelength that satisfy the moment balance equation and give uniform bend propagation, suggesting that the model is underdetermined. At 0 viscosity, the boundary condition of 0 shear rate at the basal end of the flagellum during the development of new bends selects the particular solution that is obtained by computer simulations. Therefore, the details of the pattern of bend initiation at the basal end of a flagellum can be of major significance in determining the properties of propagated bending waves in the distal portion of a flagellum. At high values of external viscosity, the model oscillates at frequencies and wavelengths that give approximately integral numbers of waves on the flagellum. These operating points are selected because they facilitate the balance of bending moments at the ends of the model, where the external viscous moment approaches 0. These mode preferences can be overridden by forcing the model to operate at a predetermined frequency. The strong mode preferences shown by curvature-controlled flagellar models, in contrast to the weak or absent mode preferences shown by real flagella, therefore do not demonstrate the inapplicability of the moment-balance approach to real flagella. Instead, they indicate a need to specify additional properties of real flagella that are responsible for selecting particular operating points.     

A complete set of flagellar genes acquired by horizontal transfer coexists with the endogenous flagellar system in Rhodobacter sphaeroides

Bacteria swim in liquid environments by means of a complex rotating structure known as the flagellum. Approximately 40 proteins are required for the assembly and functionality of this structure. Rhodobacter sphaeroides has two flagellar systems. One of these systems has been shown to be functional and is required for the synthesis of the well-characterized single subpolar flagellum, while the other was found only after the genome sequence of this bacterium was completed. In this work we found that the second flagellar system of R. sphaeroides can be expressed and produces a functional flagellum. In many bacteria with two flagellar systems, one is required for swimming, while the other allows movement in denser environments by producing a large number of flagella over the entire cell surface. In contrast, the second flagellar system of R. sphaeroides produces polar flagella that are required for swimming. Expression of the second set of flagellar genes seems to be positively regulated under anaerobic growth conditions. Phylogenic analysis suggests that the flagellar system that was initially characterized was in fact acquired by horizontal transfer from a gamma-proteobacterium, while the second flagellar system contains the native genes. Interestingly, other alpha-proteobacteria closely related to R. sphaeroides have also acquired a set of flagellar genes similar to the set found in R. sphaeroides, suggesting that a common ancestor received this gene cluster.