The Propulsion of Sea-Urchin Spermatozoa

Abstract

1. The general theory of flagellar propulsion is discussed and an expression obtained whereby the propulsive speed of a spermatozoon can be expressed in terms of the amplitude, wave-length and frequency of the waves passing down the tail of a spermatozoon of Psammechinus miliaris. 2. The expression obtained is applicable to waves of relatively large amplitude, and allowance is made for the presence of an inert head. 3. The calculated propulsive speed is almost identical with that derived from observational data. Unless the head of a spermatozoon is very much larger than that of Psammechinus, its presence makes relatively little difference to the propulsive speed. Most of the energy of the cell is used up in overcoming the tangential drag of the tail. 4. Although the amplitude may change as a wave passes along the tail, the propulsive properties of the latter may be expected to be closely similar to those of a tail generating waves of the same average amplitude.

Modelling Motility: The Mathematics of Spermatozoa

“…It is also linear, which in this context means that, ceteris paribus, doubling the frequency of flagellar beating will also double the swimming speed. The existence of such simple relations has enabled numerous rapid developments of Gray and Hancock (1955)'s seminal method, known as resistive force theory (Figure 1C), establishing quantitative links between the flagellar beat pattern and both the sperm's swimming speed and behaviour (for example, Rikmenspoel, 1965;Brokaw, 1972b;Suarez et al, 1991;Elgeti et al, 2010;Ishijima, 2011;Curtis et al, 2012). A key component of this theory is the notion of anisotropic drag, with approximately twice as much force being required to push the flagellum in its normal direction compared to its tangential direction (Gray and Hancock, 1955).…”

“…The existence of such simple relations has enabled numerous rapid developments of Gray and Hancock (1955)'s seminal method, known as resistive force theory (Figure 1C), establishing quantitative links between the flagellar beat pattern and both the sperm's swimming speed and behaviour (for example, Rikmenspoel, 1965;Brokaw, 1972b;Suarez et al, 1991;Elgeti et al, 2010;Ishijima, 2011;Curtis et al, 2012). A key component of this theory is the notion of anisotropic drag, with approximately twice as much force being required to push the flagellum in its normal direction compared to its tangential direction (Gray and Hancock, 1955). This ultimately gives rise to the propulsion of a swimming spermatozoon (Figure 1) and can be observed by performing the simple experiment of moving a thin stick through syrup parallel and perpendicular to its length.…”

“…More generally, the need for a mechanical perspective on the swimming of spermatozoa was recognised in the 1950s, with initial application to sea urchin sperm based on microscopic imaging (Gray and Hancock, 1955). These pioneer studies have been extended and generalised in numerous directions over the past six to seven decades, with recent refinement in particular driven by improvements in the digital microscopy of the flagellum beat and increased computational power, overcoming many of the technological limitations of previous studies.…”

Regularized Stokeslets Lines Suitable for Slender Bodies in Viscous Flow

“…In the small a limit, asymptotic theories have been developed for the flow around slender bodies [1][2][3][4][5][6][7][8][9][10]. These theories are called slender-body theories (SBTs), and they expand the system in powers of a or 1/ ln a. Algebraically accurate SBTs have proven to be very accurate and provide exact results for prolate spheroids [3,5,7,[11][12][13][14][15][16], while logarithmically accurate SBTs are useful for analytical estimations [17][18][19][20][21][22][23][24]. Many of these theories place the singularity solutions of Stokes equations along the centreline of the body.…”

Hydrodynamics and direction change of tumbling bacteria

“…An early study of bacterial hydrodynamics appeared in the seminal work of Chwang and Wu [21]. They used the resistive-force theory of Gray and Hancock [22], valid for slender filaments, and enforced the balance of force and moments on the cell, which allows one to compute simultaneously the linear and angular velocity of the organism. Since then, there has been a lot of research in bacterial hydrodynamics aimed at understanding the interactions of bacteria with surfaces [23][24][25][26][27][28][29][30][31], with other cells [32] and their collective motion [33][34][35][36].…”