The movement of spermatozoa with helical head: Theoretical analysis and experimental results

Dipartimento di Biologia, Università di Milano, Italy.
Biophysical Journal (Impact Factor: 3.97). 11/1994; 67(4):1767-74. DOI: 10.1016/S0006-3495(94)80651-4
Source: PubMed


The present work is concerned with the study of the swimming of flagellated microscopic organisms with a helical head and a helical pattern of flagellar beating, such as Xenopus sperms. The theoretical approach is similar to that taken by Chang and Wu (1971) in the study of helical flagellar movement. The model used in the present study allows us to determine the velocity of propulsion (U) and the frequency of rotation of the sperm head (fh) as a function of the frequency of the wave of motion (ft) traveling along the tail. The results relative to the case of helical and planar flagellar waves are compared. Our main finding is that the helical shape of the head seems to increase the efficiency of propulsion of the spermatozoon when compared with the more commonly shaped spherical head. Experimentally measured values of fh versus U may be fitted by a linear plot whose slope is much higher than that corresponding to the case of planar flagellar beating. This fact is consistent with an effectively three-dimensional (nonplanar) movement of the flagellar tail. However, the results do not fit those predicted from a circular helix, suggesting that a different shape of the flagellar beating should be considered.

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    • "To characterize 3D motion, researchers usually extend two-dimensional (2D) measurement schemes which are commonly based on standard optical systems, such as video cameras and microscopes [11]. Other researchers employ numerical analysis [12] [13], theoretical models [14] [15] or digital holography with extensive numerical computations [16] [17] [18]. Also, standard techniques based on the classical non-relativistic Doppler effect have been widely applied to determine velocity components perpendicular to the direction of propagation of the illuminating light beam, relying upon measurements of the longitudinal component for a large set of directions [19]. "
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    ABSTRACT: We measure the rotational and translational velocity components of particles moving in helical motion using the frequency shift they induced to the structured light beam illuminating them. Under Laguerre-Gaussian mode illumination, a particle with a helical motion reflects light that acquires an additional frequency shift proportional to the angular velocity of rotation in the transverse plane, on top of the usual frequency shift due to the longitudinal motion. We determined both the translational and rotational velocities of the particles by switching between two modes: by illuminating with a Gaussian beam, we can isolate the longitudinal frequency shift; and by using a Laguerre-Gaussian mode, the frequency shift due to the rotation can be determined. Our technique can be used to characterize the motility of microorganisms with a full three-dimensional movement.
    Optics Express 03/2014; 22(13). DOI:10.1364/OE.22.016504 · 3.49 Impact Factor
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    • "Similar nonlinear effects are noted in polymers (Sekimoto et al. 1995);(Goldstein and Langer 1995), hair bundles (Boal 2002), or in vitro symmetry breaking mechanisms (Bourdieu et al. 1995) using similar types of nonlinear models. Not surprisingly, three-dimensional flagellar movements also appear to have nonlinearities, as indicated by the observations that a circular helix with linear relationships does not accurately describe the 3D motion of Xenopus sperm (Andrietti and Bernardini 1994) and that not all cycles of avian sperm bending lead to production of the same type of 3-D waveform shape (Woolley and Osborn 1984). Nonlinear waveforms can have a variety of shapes and dynamic states that necessitates characterizing the observed parametric limits of the nonlinear forms within an experimental system in order to build equations and expressions to describe that specific system. "
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    ABSTRACT: A quantitative description of the flagellar dynamics in the procyclic T. brucei is presented in terms of stationary oscillations and traveling waves. By using digital video microscopy to quantify the kinematics of trypanosome flagellar waveforms. A theoretical model is build starting from a Bernoulli-Euler flexural-torsional model of an elastic string with internal distribution of force and torque. The dynamics is internally driven by the action of the molecular motors along the string, which is proportional to the local shift and consequently to the local curvature. The model equation is a nonlinear partial differential wave equation of order four, containing nonlinear terms specific to the Korteweg-de Vries (KdV) equation and the modified-KdV equation. For different ranges of parameters we obtained kink-like solitons, breather solitons, and a new class of solutions constructed by smoothly piece-wise connected conic functions arcs (e.g. ellipse). The predicted amplitude and wavelengths are in good match with experiments. We also present the hypotheses for a step-wise kinematical model of swimming of procyclic African trypanosome.
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    ABSTRACT: A method is proposed for analyzing the swimming motions of bacteria. It is assumed that a bacterium consists of a spherical cell body and several rotating flagella. The forces and torques exerted on each flagellum are evaluated in a coordinate system fixed to the flagellum, using the resistive force theory. They are transformed into a coordinate system fixed to the cell body. Then, the velocity and angular velocity of the cell body can be determined because the net force and torque on the bacterium are zero. This method is applicable to the motion of bacteria with rotating flagella in any form of curve attached at any point on the cell body. A numerical example shows that the trajectory of bacteria with a flagellum slanting from the radial direction of the cell body follows the pattern of a double helix. Another example shows that a doubly flagellated bacterium swims faster than a singly flagellated bacterium.
    Nihon Kikai Gakkai Ronbunshu, B Hen/Transactions of the Japan Society of Mechanical Engineers, Part B 01/1997; 63(606):410-416. DOI:10.1299/kikaib.63.410
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