Propulsion of micro-organisms by three-dimensional flagellar waves

Hydrodynamic evolution of sperm swimming: Optimal flagella by a genetic algorithm

Swimming performance of spermatozoa is an important index for the success of fertilization. For many years, numerous studies have reported the optimal swimming of flagellar organisms. Nevertheless, there is still a question as to which is optimal among planar, circular helical and ellipsoidal helical beating. In this paper, we use a genetic algorithm to investigate the beat pattern with the best swimming efficiency based on hydrodynamic dissipation and internal torque exertion. For the parameters considered, our results show that the planar beat is optimal for small heads and the helical flagellum is optimum for a larger heads, while the ellipsoidal beat is never optimal. Also, the genetic optimization reveals that the wavenumber and shape of wave envelope are relevant parameters, whereas the wave shape and head geometry have relatively minor effects on efficiency. The optimal beat with respect to the efficiency based on the internal torque exertion of an active elastic flagellum is characterized by a small-wavenumber and large-amplitude wave in a lower-viscosity medium. The obtained results on the optimal waveform are consistent with observations for planar waveforms, but in many respects, the results suggest the necessity of a detailed flagellar structure-fluid interaction to address whether real spermatozoa exhibit hydrodynamically efficient swimming. The evolutional optimization approach used in this study has distinguished biologically important parameters, and the methodology can potentially be applicable to various swimmers.

Functional roles of the transverse and longitudinal flagella in the swimming motility ofProrocentrum minimum(Dinophyceae)

Equations describing the motion of the dinoflagellate Prorocentrum minimum, which has both a longitudinal and a transverse flagellum, were formulated and examined using numerical calculations based on hydrodynamic resistive force theory. The calculations revealed that each flagellum has its own function in cell locomotion. The transverse flagellum works as a propelling device that provides the main driving force or thrust to move the cell along the longitudinal axis of its helical swimming path. The longitudinal flagellum works as a rudder, giving a lateral force to the cell in a direction perpendicular to the longitudinal axis of the helix. Combining these functions results a helical swimming motion similar to the observed motion. Flagellar hairs present on the transverse flagellum are necessary to make the calculated cell motion agree with the observed cell motion.

Motility modes of Spiroplasma melliferum BC3: a helical, wall-less bacterium driven by a linear motor

Spiroplasma are members of the Mollicutes (Mycoplasma, Acholeplasma and Spiroplasma) - the simplest, minimal, free-living and self-replicating forms of life. The mollicutes are unique among bacteria in completely lacking cell walls and flagella and in having an internal, contractile cytoskeleton, which also functions as a linear motor. Spiroplasma are helical, chemotactic and viscotactic active swimmers. The Spiroplasmal cytoskeleton is a flat ribbon composed of seven pairs of fibrils. The ribbon is attached to the inner side of the cell membrane along its innermost (shortest) helical line. The cell's geometry and dynamic helical parameters, and consequently motility, can be controlled by changing differentially and in a co-ordinated manner, the length of the fibrils. We identified several consistent modes of cell movements and motility originating, most likely, as a result of co-operative or local molecular switching of fibrils: (i). regular extension and contraction within the limits of helical symmetry (this mode also includes straightening, beyond what is allowed by helical symmetry, and reversible change of helical sense); (ii). spontaneous and random change of helical sense originating at random sites along the cell (these changes propagate along the cell in either direction and hand switching is completed within approximately 0.08 second); (iii). forming a deformation on one of the helical turns and propagating it along the cell (these helical deformations may travel along the cell at a speed of up to approximately 40 microm s-1); (iv). random bending, flexing and twitching (equivalent to tumbling). In standard medium (viscosity = 1.147 centipoise) the cells run at approximately 1.5 microm s-1, have a Reynolds number of approximately 3.5 x 10-6 and consume approximately 30 ATP molecules s-1. Running velocity, duration, persistence and efficiency increase with viscosity upon adding ficoll, dextran and methylcellulose to standard media. Relative force measurements using optical tweezers confirm these findings.

Unidirectional motility of Escherichia coli in restrictive capillaries

In a 6-microns capillary filled with buffer and in the absence of any chemotactic stimuli, Escherichia coli K-12 cells swim persistently in only one direction. This behavior of E. coli can be simply explained by means of the length and relative rigidity of their flagella. Single-cell motility parameters--swimming speed, turn angle, and run length time--were measured. Compared with the motility parameters measured in bulk phase, turn angle was influenced because of the effect of the geometrical restriction.

Biophysical aspects and modelling of ciliary motility

The dominance of viscous forces in the generation of propulsive thrust by cilia is emphasised. Fourier analysis indicates that ciliary bends consist of circular arcs joined by linear segments; this arc-line shape appears to be a property associated with the molecular mechanism responsible for bending the cilium and is unchanged by variations in the external viscous loading on the organelle. The flexibility of a computer-generated model of axonemal structure is demonstrated by the incorporation of recent data concerning the surface lattice of the microtubules. Computer simulations using the model show that predictions based on stochastic, rather than co-ordinated, dynein arm activity provide a qualitative match to experimental observations of microtubules gliding over fields of dynein molecules.

Three-dimensional bend propagation in hamster sperm models and the direction of roll in free-swimming cells

Iontophoretic application of ATP to the flagellum of the demembranated hamster spermatozoon produced a planar pair of bends at the two ends of the stimulated site. During bend propagation, torsion appeared in the vicinity of the interbend in some responses such that the distal bend was twisted clockwise when viewed from the base of the flagellum. This pattern of propagation is consistent with the instantaneous configurations of free-swimming cells previously described. The technique used here establishes that the three dimensionality arises from propagation per se, and does not depend on forces developed during swimming. The rolling of both free-swimming intact and demembranated spermatozoa was examined by two-color darkground videomicroscopy and the direction of rotation was, as predicted, always anticlockwise. A hypothetical mechanism, involving differential speeds of propagation of active sliding within the active microtubule subset, is proposed to account for the observed waveforms.

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.

Bacterial behaviour

Bacteria swim by rotating their flagella. They back up or choose new directions at random by changing the direction of the rotation. The probability of such changes is biased by sensory reception. The bias depends on the way in which the intensity ofthe stimulus changes with time, so that the bacteria tend to swim up a gradient of attractant and down a gradient of repellent chemicals.

Reversal of flagellar rotation in monotrichous and peritrichous bacteria: generation of changes in direction

Reversal of flagellar rotation can explain both the "backing up" of monoflagellated Pseudomonas citronellolis and the tumbling of multiflagellated Salmonella typhimurium. Reversals occur spontaneously and can be induced by negative gradients of attractant and by high-intensity light.

Effect of viscosity on bacterial motility

The behavior of a number of motile flagellated bacteria toward viscosity characteristics of their fluid environments was observed. All showed an increase in velocity (micrometers per second) in more viscous solutions. Velocity reached a maximum at a characteristic value, however, and thereafter decreased with higher viscosities. Peritrichously flagellated bacteria had maximum velocities at higher viscosities than polarly flagellated bacteria. Effects of temperature, and possible utilization of chemical constituents in the viscous solutions, were studied and found to be negligible factors under the experimental conditions used. Different agents produced the same phenomenon, thus indicating that there probably were no chemically induced metabolic effects. Loss of available water and the possibility of a variable energy supply to the flagellar propulsive system were considered but are believed minimal. Theoretically derived thermodynamic equations were utilized and suggest that the conformation of the flagellar helix affects efficiency of propulsion. Such a relationship between helix waveform and velocity was experimentally observed with Thiospirillum jenese.