Digitized precision measurements of the movements of sea urchin sperm flagella.

High speed cinemicrographs were made of sea urchin sperm at temperatures varying from 22 to 6 degrees C. Apparatus, combining a television camera and a video digitizer, was constructed to scan individual flagellar images and to digitize the flagellar waveforms. With appropriate smoothing and averaging procedures, the rough data were condensed by a microcomputer into the coordinates of 20 points along a flagellum, spaced 2 microns apart. The curvature of the flagellum at these points was also computed. The coordinates of the flagellar positions were obtained to an accuracy of approximately +/- 0.1 micron, flagellar curvature to an accuracy of approximately +/- 50 cm-1. At all temperatures the amplitude of the flagella was found to vary with time in a purely sinusoidal fashion to within +/- 2%. The local curvature of the flagella had basically a purely sinusoidal time course to within +/- 50 cm-1, but a varying amount of asymmetry was present in the distal and the proximal ends of the flagella. This asymmetry in the curvature was related to the radius of the circular path of the sperm. The flagellar waveforms can probably be summarized in simple algebraic functions.     

The equation of motion for sperm flagella.

The equation of motion for sperm flagella, in which the elastic bending moment and the active contractile moment are balanced by the moment from the viscous resistance of the surrounding fluid, is solved for a wave solution that superimposes partial solutions. Substitution of the expression for the wave solution into the equation leads to an expression for the active contractile moment. This active moment can be decomposed into two parts. The first part describes an active moment that travels over the flagellum with the mechanical flagellar wave, the second part represents a moment in phase over the entire length of the flagellum, which decreases linearly towards the distal tip. The linear synchronous moment, to which an amount of traveling moment has been added as a perturbation, leads to wave solutions that closely resemble flagellar waves. Properties such as wavelength and wave amplitudes and also the shape of the waves in sea urchin sperm flagella at different frequencies are accurately described by the theory. The change in wave shape in sea urchin sperm flagella at raised viscosity is predicted well by the theory. The different wave properties caused in bull sperm flagella by different boundary conditions at the proximal junction are explained. When only a traveling active moment is present in a flagellum, the wave solutions describe waves of a small wave length in a long flagellum. Some properties of the wave motion of sperm flagella are derived from the theory and verified experimentally.     

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.     

Motion characteristics of flagellar fragments of long insect sperm.

The flagellar length of cricket spermatozoa was reduced in steps from congruent to 1,000 micrometer (intact length) to 50 micrometer. In intact sperm the flagellar wave properties were largely independent of the viscosity of the external medium. When the flagellar length had been reduced to less than 100 micrometer the flagellar frequency was reduced at a raised external viscosity. Independent motion of different sections of a flagellum was not observed when its length is less than 100 micrometer. It is concluded that in long thin flagella, transverse viscous forces cannot exert a moment beyond a lever length of approximately 100 micrometer. It is shown that the existence of a maximum lever length, beyond which no moment can be transmitted, leads to the absence of a standing active contractile moment in the long insect sperm.     

Intraflagellar transport particle size scales inversely with flagellar length: revisiting the balance-point length control model

The assembly and maintenance of eukaryotic flagella are regulated by intraflagellar transport (IFT), the bidirectional traffic of IFT particles (recently renamed IFT trains) within the flagellum. We previously proposed the balance-point length control model, which predicted that the frequency of train transport should decrease as a function of flagellar length, thus modulating the length-dependent flagellar assembly rate. However, this model was challenged by the differential interference contrast microscopy observation that IFT frequency is length independent. Using total internal reflection fluorescence microscopy to quantify protein traffic during the regeneration of Chlamydomonas reinhardtii flagella, we determined that anterograde IFT trains in short flagella are composed of more kinesin-associated protein and IFT27 proteins than trains in long flagella. This length-dependent remodeling of train size is consistent with the kinetics of flagellar regeneration and supports a revised balance-point model of flagellar length control in which the size of anterograde IFT trains tunes the rate of flagellar assembly.     

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.     

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 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.     

Bend propagation in flagella. I. Derivation of equations of motion and their simulation.

A set of nonlinear differential equations describing flagellar motion in an external viscous medium is derived. Because of the local nature of these equations and the use of a Crank-Nicolson-type forward time step, which is stable for large deltat, numerical solution of these equations on a digital computer is relatively fast. Stable bend initiation and propagation, without internal viscous resistance, is demonstrated for a flagellum containing a linear elastic bending resistance and an elastic shear resistance that depends on sliding. The elastic shear resistance is derived from a plausible structural model of the radial link system. The active shear force for the dynein system is specified by a history-dependent functional of curvature characterized by the parameters m0, a proportionality constant between the maximum active shear moment and curvature, and tau, a relaxation time which essentially determines the delay between curvature and active moment.     

Ciliary beat and cell motility of Dunaliella: computer analysis of high speed microcinematography

A high-speed microcinematographic study was performed on the biflagellate unicellular alga Dunaliella. A frame-by-frame analysis has shown that the two flagella never beat at the same frequency. For a better characterization of the bending pattern of the two flagella, a new automated method of image analysis has been developed. The method allowed an automatic acquisition of a line characterizing the Dunaliella flagellum and its mathematical modelling. From this model, some binding parameters could be automatically measured, which have permitted determination of the velocities of formation and propagation of flagellar waves and the variation of the curvature radius of the bends between the two flagella. Both flagella showed a similar pattern of ciliary beat. The most important difference was the lengthening of the initiation phase for the slower flagella.     

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.     

Calcium regulation of flagellar curvature and swimming pattern in triton X-100--extracted rat sperm

Free Ca2+ changes the curvature of epididymal rat sperm flagella in demembranated sperm models. The radius of curvature of the flagellar midpiece region was measured and found to be a continuous function of the free Ca2+ concentration. Below 10(-7) M free Ca2+, the sperm flagella assumed a pronounced curvature in the same direction as the sperm head. The curvature reversed direction at 2.5 x 10(-6) M Ca2+ to assume a tight, hook-like bend at concentrations of 10(-5) to 10(-4) M free Ca2+. Sodium vanadate at 2 x 10(-6) M blocked flagellar motility, but did not inhibit the Ca2+-mediated change in curvature. Nickel ion at 0.2 mM and cadmium ion at 1 microM interfered with the transition and induced the low Ca2+ configuration of the flagellum. The forces that maintain the Ca2+-dependent curvature are locally produced, as dissection of the flagella into segments did not significantly alter the curvature of the excised portions. Irrespective of the induced pattern of curvature, the sperm exhibited coordinated, repetitive flagellar beating in the presence of ATP and cAMP. At 0.3 mM ATP the flagellar waves propagated along the principal piece while the level of free Ca2+ controlled the overall curvature. When Ca2+-treated sperm models with hooked midpieces were subjected to higher concentrations of ATP (1-5 mM), some cells exhibited a pattern of movement similar to hyperactivated motility in capacitated live sperm. This type of motility involved repetitive reversals of the Ca2+-induced bend in the midpiece, as well as waves propagated along the principal piece. The free Ca2+ available to the flagellum therefore appeared to modify both the pattern of motility and the flagellar curvature.     

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.     

Light dependence of sexual agglutinability in Chlamydomonas eugametos

The mating activity of mating-type plus gametes of Chlamydomonas eugametos depends on light. Cells lost their ability to agglutinate with mating-type minus gametes after a dark period of 30 min. They regained their agglutinability after 10 min exposure to light. Other mating reactions, such as tipping and flagellar tip activation, were not dependent upon light. Since cycloheximide and tunicamycin did not affect the light-induced activation of flagellar agglutinability, no protein synthesis or glycosylation is involved in this process. Equal amounts of biologically active agglutination factor could be extracted from cells placed either in light or in darkness. A minor portion of the active material was found to be located on the flagellar surface of illuminated cells. No active material was found on the flagellar surface of dark-exposed cells, whereas their cell bodies contained the same amount of active material as the cell bodies of illuminated cells. Since a light-induced flow of agglutination factors from the cell body to the flagella could not be detected and dark-exposed cells could be slightly activated by amputation or fixation by glutaraldehyde, we propose that light affects flagellar agglutinability by an in-situ modification of the agglutination factor on the flagella. When mt ⁺ and mt ^- strains were crossed and the progeny examined for light-sensitivity, it was apparent that this phenomenon is not mating type-linked.Abbreviations and symbols FTA flagellar tip activation - mt ^+/- mating type plus or minus - WGA wheat-germ agglutinin     

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.     

Three-dimensional mechanics of eukaryotic flagella.

Equations are derived that account for the contribution of internal structure of cilia and flagella to motion in three dimensions according to a sliding filament model of the motile system. It is shown that for reasonable amounts of bending and twisting, the bending properties of an axoneme can be described by a linear elastic bending resistance, and approximate values for the bending and twisting resistances are computed. Expressions for the shear moments contributed by purely elastic or pinned links between filaments are also derived. It is shown that within the confines of a strict sliding filament model such internal structures cannot by themselves produce twist. Thus planar bending will occur if the internal shear force lies in a plane. Application of an external force, however, will in general produce twisting. Computer simulations of flagellar shape in response to a constant external force applied to the distal end of the axoneme are presented. It is shown that a small amount of twist may arise because of acylindrical bend resistance. Large twists, however, result when the external force is applied to an axoneme with internal shear resistant links.     

Control of phobic behavioral responses by rhodopsin-induced photocurrents in Chlamydomonas.

Both phototactic and photophobic responses of Chlamydomonas are mediated by a visual system comprising a rhodopsin photoreceptor. Suction pipette recordings have revealed that flash stimulation causes calcium currents into the eyespot and the flagella. These photocurrents have been suggested to be the trigger for all behavioral light responses of the cell. But this has never been shown experimentally. Here we describe a detection technique that combines electrical and optical measurements from individual algae held in a suction pipette. Thus it is possible to record photocurrents and flagellar beating simultaneously and establish a direct link between the two. We demonstrate that in Chlamydomonas only the photoreceptor current in conjuction with a fast flagellar current constitutes the trigger for photophobic responses. Within the time of the action-potential-like flagellar current, the flagella switch from forward to backward swimming, which constitutes the beginning of the photoshock reaction. The switch is accompanied by a complex frequency change and beating pattern modulation. The results are interpreted in terms of a general model for phototransduction in green algae (Chlorophyceae).     

Flagellar waveforms of gametes in the brown alga Ectocarpus siliculosus

Brown algae are members of the Stramenopiles and their gametes generally have two heterogeneous flagella: a long anterior flagellum (AF) with mastigonemes and a short posterior flagellum (PF). In this study, swimming paths and flagellar waveforms in free-swimming and thigmotactic-swimming male and female gametes and in male gametes during chemotaxis, were quantitatively analysed in the model brown alga Ectocarpus siliculosus. This analysis was performed using a high-speed video camera. It was revealed that the AF plays a role in changing the locomotion of male and female gametes from free-swimming to thigmotactic-swimming and also in changing the swimming path of male gametes from linear to circular during chemotaxis. In the presence of a sex pheromone, male gametes changed their swimming path from linear (swimming path curvature, 0–0.02 µm^–1) to middle and small circular path (swimming path curvature, 0.04–0.20 µm^–1). The flagellar asymmetry and the deflection angle of the AF became larger, whereas the oscillation pattern of the AF was stable. However, there was no correlation between the flagellar asymmetry and the deflection angle of the AF and the path curvature when the male gametes showed middle to small circular paths. The PF irregularly changed the deflection angle and the oscillation pattern was unstable depending on the gradient of the sex pheromone concentration. AF waveforms were independent of PF locomotion during chemotaxis. This means that the AF has the ability to change the swimming path of male gametes – for example, from a highly linear path to a circular path – while changes in locomotion from a middle circle path to a small circle path is the result of beating of the PF.     

Elastic Properties of the Sea Urchin Sperm Flagellum

The theory of flexural vibrations in thin rods, applied to the movement of flagella, has been extended to include an investigation of the influence of the boundary conditions on the theoretical waveforms. It was found that for flagella which are flexible enough, the flexibility can be estimated solely from the wavelength of the wave traveling in it. This can be expected to hold for those flagella which do not possess a fibrous sheath. The bending moment in flagella in which the ampitude of the wave is maintained as the wave travels distally is almost completely produced by active contractile elements. This means that the active bending moment can be estimated from the radius of curvature of the flagellum and the stiffness. The above findings were applied to the case of the sea urchin sperm flagellum. One finds that the stiffness of the flagellum is caused mainly by the nine longitudinal fibers which must have a Young's modulus of slightly less than 10⁸dyne/cm². The longitudinal fibers need to develop a tension of 1.6 × 10⁸dyne/cm² to account for the bending moment in the flagellum. These two figures are in line with those found for muscle fibers.     

Plain and Complex Flagella of Pseudomonas rhodos: Analysis of Fine Structure and Composition

Cells of Pseudomonas rhodos 9-6 produce two morphologically distinct flagella termed plain and complex, respectively. Fine structure analyses by electron microscopy and optical diffraction showed that plain flagellar filaments are cylinders of 13-nm diameter composed of globular subunits like normal bacterial flagella. The structure comprises nine large-scale helical rows of subunits intersecting four small-scale helices of pitch angle 25 degrees . Complex filaments have a conspicuous helical sheath, 18-nm wide, of three close-fitting helical bands, each about 4.7-nm wide, separated by axial intervals, 4.7 nm wide, running at an angle of 27 degrees . The internal core has similar but not identical substructure to plain filaments. Unlike plain flagella, the complex species is fragile and does not aggregate in bundles. Mutants bearing only one of two types of flagellum were isolated. Cells with plain flagella showed normal translational motion, and cells with complex flagella showed rapid spinning. Isolated plain flagella consist of a 37,000-dalton subunit separable into two isoproteins. Complex filaments consist of a 55,000-dalton protein; a second 43,000-dalton protein was assigned to complex flagellar hooks. The results indicate that plain and complex flagella are entirely different in structure and composition and that the complex type represents a novel flagellar species. Its possible mode of action is discussed.