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.     

The Bacillus subtilis catabolite control protein CcpA exerts all its regulatory functions by DNA-binding

Swimming speed (v) and flagellar-bundle rotation rate (f) of Salmonella typhimurium, which has peritrichous flagella, were simultaneously measured by laser dark-field microscopy (LDM). Clear periodic changes in the LDM signals from a rotating bundle indicated in-phase rotation of the flagella in the bundle. A roughly linear relation between v and f was observed, though the data points were widely distributed. The ratio of v to f (v-f ratio), which indicates the propulsive distance during one flagellar rotation, was 0.27 microm (11% of the flagellar pitch) on average. The experimental v-f ratio was twice as large as the calculated one on the assumption that a cell had a single flagellum. A flagellar bundle was considered to propel a cell more efficiently than a single flagellum.     

Reversible intracellular ATP changes in intact rat spermatozoa and effects on flagellar sperm movement.

The initiation of motility and modification of energy metabolism of rat caudal epididymal spermatozoa can be induced by dilution in a saline medium. We have investigated in these cells the relationships between the energy reserve (sperm ATP content measured by bioluminescence) and flagellar movement (high speed videomicrography, 200 frames/sec). A steady state was observed in sperm ATP content, progressive velocity (Vp) and flagellar beat frequency (F) with sperm dilution in a medium with glucose, lactate, pyruvate and acetate substrates after 30 minutes of incubation. Without these substrates, changes in metabolic pathways occurred immediately and initially disturbed the relationship between ATP levels and F, suggesting differences in motility initiation when energy is from an endogenous origin via mitochondrial oxidative phosphorylation. This "energy crisis" was reversed by the addition of substrates to the medium. The three-dimensional flagellar movement observed in the presence of substrates quickly became two-dimensional in their absence. The flagellar beat envelope became more splayed, the mean amplitude of lateral head displacement increased and F decreased. The resulting high flagellar beat efficiency can be compared to that observed during hyperactivation which is a physiological event related to a fall in intracellular ATP level. In both media, the displacement of the flagellum in relation to the wave axis varied sinusoidally. The sine period increased with time when the spermatozoa were incubated in the medium without substrates. These results suggest a gradual slowing-down of the velocity of wave formation in the proximal part of the flagellum.     

High-speed video observation of swimming behavior and flagellar motility ofProrocentrum minimum (Dinophyceae)

Summary To understand the functions of the longitudinal and transverse flagella of dinoflagellates, the flagellar waveform and frequency of each flagellum were observed by high-speed video-recording. The longitudinal flagellum emerged from the anterior end of the cell and beat with a planar undulating wave whose plane was perpendicular to the valval sutural plane. The transverse flagellum curved around the anterior end of the cell and beat with a helical wave, with different alternating half pitches. The half pitch corresponding to the parts farther from the cellular antero-posterior axis was shorter than that of the parts closer to the axis. This pattern is described by the ratio of the outer-parts half pitch to the pitch of the whole period of the helix and seems to be characteristic of the dinoflagellates' transverse flagellum.Abbreviations p _in half pitch corresponding to the inner parts of the transverse flagellum - p _out half pitch corresponding to the outer parts of the transverse flagellum - P _p pitch of helical swimming trajectory - R _p radius of helical swimming trajectory - _c rotational frequency of the cell     

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 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.     

Trypanosome Motion Represents an Adaptation to the Crowded Environment of the Vertebrate Bloodstream

Blood is a remarkable habitat: it is highly viscous, contains a dense packaging of cells and perpetually flows at velocities varying over three orders of magnitude. Only few pathogens endure the harsh physical conditions within the vertebrate bloodstream and prosper despite being constantly attacked by host antibodies. African trypanosomes are strictly extracellular blood parasites, which evade the immune response through a system of antigenic variation and incessant motility. How the flagellates actually swim in blood remains to be elucidated. Here, we show that the mode and dynamics of trypanosome locomotion are a trait of life within a crowded environment. Using high-speed fluorescence microscopy and ordered micro-pillar arrays we show that the parasites mode of motility is adapted to the density of cells in blood. Trypanosomes are pulled forward by the planar beat of the single flagellum. Hydrodynamic flow across the asymmetrically shaped cell body translates into its rotational movement. Importantly, the presence of particles with the shape, size and spacing of blood cells is required and sufficient for trypanosomes to reach maximum forward velocity. If the density of obstacles, however, is further increased to resemble collagen networks or tissue spaces, the parasites reverse their flagellar beat and consequently swim backwards, in this way avoiding getting trapped. In the absence of obstacles, this flagellar beat reversal occurs randomly resulting in irregular waveforms and apparent cell tumbling. Thus, the swimming behavior of trypanosomes is a surprising example of micro-adaptation to life at low Reynolds numbers. For a precise physical interpretation, we compare our high-resolution microscopic data to results from a simulation technique that combines the method of multi-particle collision dynamics with a triangulated surface model. The simulation produces a rotating cell body and a helical swimming path, providing a functioning simulation method for a microorganism with a complex swimming strategy.     

DYNAMICS OF THE HISPID FLAGELLUM OF OCHROMONAS DANICA : The Role of Mastigonemes

High speed cinephotographic techniques were used to determine the pattern of fluid flow about the hispid flagellum of Ochromonas danica and to investigate the behavior of this flagellum in media of increased viscosity. The fluid currents are consistent with the hypothesis that the mastigonemes are passive, rigid, remain normal to the flagellar surface, and lie in the plane of flagellar undulation during motility.     

Flagellar assembly in Salmonella typhimurium

The bacterial flagellum is a motility apparatus in which a long helical filament - the propeller - is driven by a rotary motor embedded in the cell surface. Out of more than 40 genes required for construction of a fully functional flagellum in Salmonella typhimurium, only 18 gene products have been identified in the mature structure. Some other flagellar proteins play logistical roles during construction, which involves the selective export of flagellar components through a central hole in the flagellum. The whole structure is constructed from base to tip by linear assembly; that is, by adding new components on the growing end, resulting in the distal growth of each substructure. Components of the substructures do not necessarily self-assemble, but often demand the help of other proteins. Recent progress in the understanding of flagellar assembly, which has been most extensively studied in S. typhimurium, is reviewed.     

Bacterial flagellar microhydrodynamics: Laminar flow over complex flagellar filaments,analog archimedean screws and cylinders,and its perturbations

The flagellar filament, the bacterial organelle of motility, is the smallest rotary propeller known. It consists of 1), a basal body (part of which is the proton driven rotary motor), 2), a hook (universal joint-allowing for off-axial transmission of rotary motion), and 3), a filament (propeller-a long, rigid, supercoiled helical assembly allowing for the conversion of rotary motion into linear thrust). Helically perturbed (so-called "complex") filaments have a coarse surface composed of deep grooves and ridges following the three-start helical lines. These surface structures, reminiscent of a turbine or Archimedean screw, originate from symmetry reduction along the six-start helical lines due to dimerization of the flagellin monomers from which the filament self assembles. Using high-resolution electron microscopy and helical image reconstruction methods, we calculated three-dimensional density maps of the complex filament of Rhizobium lupini H13-3 and determined its surface pattern and boundaries. The helical symmetry of the filament allows viewing it as a stack of identical slices spaced axially and rotated by constant increments. Here we use the closed outlines of these slices to explore, in two dimensions, the hydrodynamic effect of the turbine-like boundaries of the flagellar filament. In particular, we try to determine if, and under what conditions, transitions from laminar to turbulent flow (or perturbations of the laminar flow) may occur on or near the surface of the bacterial propeller. To address these questions, we apply the boundary element method in a manner allowing the handling of convoluted boundaries. We tested the method on several simple, well-characterized cylindrical structures before applying it to real, highly convoluted biological surfaces and to simplified mechanical analogs. Our results indicate that under extreme structural and functional conditions, and at low Reynolds numbers, a deviation from laminar flow might occur on the flagellar surface. These transitions, and the conditions enabling them, may affect flagellar polymorphism and the formation and dispersion of flagellar bundles-factors important in the chemotactic response.     

Gliding movement in Peranema trichophorum is powered by flagellar surface motility

A colorless euglenoid flagellate Peranema trichophorum shows unique unidirectional gliding cell locomotion on the substratum at velocities up to 30 micro m/s by an as yet unexplained mechanism. In this study, we found that (1) treatment with NiCl(2) inhibited flagellar beating without any effect on gliding movement; (2) water currents applied to a gliding cell from opposite sides caused detachment of the cell body from the substratum. With only the anterior flagellum adhering to the substratum, gliding movement continued along the direction of the anterior flagellum; (3) gentle pipetting induced flagellar severance into various lengths. In these cells, gliding velocity was proportional to the flagellar length; and (4) Polystyrene beads were translocated along the surface of the anterior flagellum. All of these results indicate that a cell surface motility system is present on the anterior flagellum, which is responsible for cell gliding in P. trichophorum.     

Mechanisms of flagellar propulsion

Summary Flagellar propulsion takes place in the viscosity-dominated realm of low Reynolds number fluid dynamics. Volumes of fluid are carried in a capture zone around the moving regions of the flagellum, and the flagellar motion achieves propulsion because some of that water is shed from the capture zone, either from the flagellar tip in typical flagellar motion or to the side reached at the end of the effective stroke in the case of ciliary motion. Helical flagellar motion is in principle more efficient than planar beating, and the rotation caused by the former introduces complications in propulsion that may be advantageous, e.g., inEuglena, or disadvantageous, e.g., in a fixed cell. The presence of a surface near to the moving organelle restricts the fluid motion, but this effect enhances ciliary propulsion. There is a great variety of beat patterns, functionally adapted hydrodynamically or in other ways for locomotion, feeding, and other more restricted roles.Abbreviations Re Reynolds number - C_N coefficient of resistance to normal motion - C_T coefficient of resistance to tangential motion - l length - v velocity - fluid density - fluid viscosity - L an element of flagellar length moving at velocity V_L - V_W velocity of a wave - V_N velocity of element L in perpendicular (normal) direction - V_T velocity of element L in tangential direction - F_N force in normal direction - F_T force in tangential direction - F_P propulsive force - F_D drag force - E effective stroke - R recovery stroke - angular velocity of flagellum - angular velocity of body     

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-dimensional motion of avian spermatozoa

Observations have been made on spermatozoa from the domestic fowl, quail and pigeon (non-passerine birds) and also from the starling and zebra finch (passerine birds). In free motion, all these spermatozoa roll (spin) continuously about the progression axis, whether or not they are close to a plane surface. Furthermore, the direction of roll is consistently clockwise (as seen from ahead). The flagellar wave has been shown to be helical and dextral (as predicted) for domestic fowl sperm when they swim rapidly in low viscosity salines. Calculations have shown that their forward velocity is consistent with their induced angular velocity but that the size of the sperm head is suboptimal for progression speed under these conditions. Dextrally helical waves also occur on the distal flagellum of fowl, quail and pigeon sperm in high viscosity solutions. But in other cases, the mechanism of torque-generation is more problematical. The problem is most profound for passerine sperm, in that typically these cells spin rapidly while seeming to remain virtually straight. Because there is no evidence for a helical wave on these flagella, we have considered other possible means whereby rotation about the local flagellar axis (self-spin) might be achieved. Sometimes, passerine sperm, while maintaining their spinning motion, adopt a fixed curvature; this must be an instance of bend-transfer circumferentially around the axonemal cylinder-though the mechanism is obscure. It is suggested that the self-spin phenomenon may be occurring in non-passerine sperm that in some circumstances spin persistently, yet without expressing regular helical waves. More complex waves are apparent in non-passerine sperm swimming in high viscosity solutions: added to the small scale bends is a large scale, sinistrally helical curvature of the flagellum. It is argued that the flagellum follows this sinistrally helical path (i.e. "screws" though the fluid) because of the shape of the sperm head and the angle at which the flagellum is inserted into it. These conclusions concerning avian sperm motility are thought to have relevance to other animal groups. Also reported are relevant aspects of flagellar ultrastructure for pigeon and starling sperm.     

Flagellar waveform and rotational orientation in a Chlamydomonas mutant lacking normal striated fibers

The Chlamydomonas mutant vfl-3 lacks normal striated fibers and microtubular rootlets. Although the flagella beat vigorously, the cells rarely display effective forward swimming. High speed cinephotomicrography reveals that flagellar waveform, frequency, and beat synchrony are similar to those of wild-type cells, indicating that neither striated fibers nor microtubular rootlets are required for initiation or synchronization of flagellar motion. However, in contrast to wild type, the effective strokes of the flagella of vfl-3 may occur in virtually any direction. Although the direction of beat varies between cells, it was not observed to vary for a given flagellum during periods of filming lasting up to several thousand beat cycles, indicating that the flagella are not free to rotate in the mature cell. Structural polarity markers in the proximal portion of each flagellum show that the flagella of the mutant have an altered rotational orientation consistent with their altered direction of beat. This implies that the variable direction of beat is not due to a defect in the intrinsic polarity of the axoneme, and that in wild-type cells the striated fibers and/or associated structures are important in establishing or maintaining the correct rotational orientation of the basal bodies to ensure that the inherent functional polarity of the flagellum results in effective cellular movement. As in wild type, the flagella of vfl-3 coordinately switch to a symmetrical, flagellar-type waveform during the shock response (induced by a sudden increase in illumination), indicating that the striated fibers are not directly involved in this process.     

Swimming of spermatozoa in a linear viscoelastic fluid.

A modified resistive force theory is developed for a spermatozoon swimming in a general linear viscoelastic fluid. The theory is based on a Fourier decomposition of the flagellar velocity, which leads to solving the Stokes flow equations with a complex viscosity. We use a model spermatozoon with a spherical head which propagates small amplitude sinusoidal waves along its flagellum. Results are obtained for the velocity of propulsion and the rate of working for a free swimming spermatozoon and the thrust on a fixed spermatozoon. There is no change in propulsive velocity for a viscoelastic fluid compared to a Newtonian fluid. The rate of working does change however, decreasing with increasing elasticity of the fluid, for a Maxwell fluid. Thus the theory predicts that a spermatozoon can swim faster in a Maxwell fluid with the same expenditure of energy for a Newtonian fluid.     

FLAGELLAR REGENERATION IN SPERMATOZOPSIS SIMILIS (CHLOROPHYTA)

The unicellular green alga Spermatozopsis similis Preisig et Melkonian bears two flagella of unequal length. After deflagellation, cells first regenerated the longer flagellum to about one third of its original length, before the shorter flagellum started to develop. Growth rates were similar for both flagella. Thus, the length difference between both flagella was restored by a lag-phase during regeneration of the shorter flagellum. To explain the lag-phase, we have considered a gating mechanism near the flagellar base that controls the entry of precursors into the flagellum. This would allow cells to restrict the time of effective flagellar growth and thereby control flagellar length. Our data indicated that cells are capable of individually regulating flagellar assembly onto basal bodies. We discuss a recent model of flagellar length regulation based on a balance of assembly and disassembly and conclude that flagellar length is controlled by additional factors, including the availability of flagellar proteins and the developmental status of basal bodies.     

Analysis of motility parameters from paddlefish and shovelnose sturgeon spermatozoa

Ninety to 100% of paddlefish Polyodon spathula were motile just after transfer into distilled water, with a velocity of 175 μm s^-1, a flagellar beat frequency of 50 Hz and motility lasting 4–6 min. Similarly, 80–95% of shovelnose sturgeon Scaphirhynchus platorynchus spermatozoa were motile immediately when diluted in distilled water, with a velocity of 200 μm s^-1, a flagellar beat frequency of 48 Hz and a period of motility of 2–3 min. In both species, after sperm dilution in a swimming solution composed of 20 mM Tris–HCl (pH 8·2) and 20 mM NaCl, a majority of the samples showed 100% motility of spermatozoa with flagella beat frequency of 50 Hz within the 5 s following activation and a higher velocity than in distilled water. In such a swimming medium, the time of motility was prolonged up to 9 min for paddlefish and 5 min for sturgeon and a lower proportion of sperm cells had damage such as blebs of the flagellar membrane or curling of the flagellar tip, compared with those in distilled water. The shape of the flagellar waves changed during the motility phase, mostly through a restriction at the part of the flagellum most proximal to the head. A rotational movement of whole cells was observed for spermatozoa of both species. There were significant differences in velocity of spermatozoa between swimming media and distilled water and between paddlefish and shovelnose sturgeon.     

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.     

Kinematics of human spermatozoa

A microcinematographic (50 f/s) study was performed on motile human spermatozoa. Eighty percent were found to have a linear trajectory and a pseudo-sinusoidal head displacement pattern. Throughout their progression, the spermatozoa periodically rotated on their longitudinal axis at a frequency equal to that of flagellar wave formation. These waves were found always to begin on the same side of the cell and to propagate in the flattened plane of the head until the moment of rotation. At this time the wave had reached a point near the middle of the flagellum. Beyond this point, the flagellum moves out of the plane of the head. Different variables used to characterize the movement of spermatozoa included the velocity of progression, amplitude and velocity of head displacement, frequency of rotation, wave amplitude, and propagation velocity of the flagellar wave. Among these variables, it was the propagation velocity of the wave that was found to be best correlated with the velocity of spermatozoan progression. This flagellar wave exhibited two stages, one of initiation and one of propagation.