The movement of any short length of the tail of a spermatozoon of Psammechinus miliaris and the characteristic changes which it undergoes during each cycle of its displacement through the water can be described in terms of the form and speed of propagation of the bending waves which travel along the tail (Gray, 1953, 1955); the form of the wave depends on the maximum extent of bending reached by each element and on the difference in phase between adjacent elements. The object of this paper is to consider the forces exerted on the tail as it moves relative to the surrounding medium and to relate the propulsive speed of the whole spermatozoon to the form and speed of propagation of the bending waves generated by the tail. The mathematical theory of the propulsive properties of thin undulating filaments has recently been considered by Taylor (1951, 1952) and by Hancock (1953); the present study utilizes and extends their findings but approaches the problem from a somewhat different angle. resistance, and consequently the transverse displacement (Vy) elicits reactions tangential and normal to the surface of the element. The latter force (δNy) has a component(δNysinθ) acting forward along the axis (xx ′) of propulsion; it is this component which counteracts the retarding effect of all the forces acting tangentially to the surface.

As the dimensions of an element are extremely small and the speed of displacement very low, the reactions elicited from the water can (by analogy with those operating on a sphere) be regarded as directly proportional to the velocity of displacement and to the viscosity of the medium. If the velocity of displacement tangential to the body is Vy sinθ the tangential drag (δLy) can be expressed as CLVysin δ δs, and the force (δNy) acting normally to the surface as CNVy cos θ δs, where CL and CN are the coefficients of resistance to the surface of the element for a medium of known viscosity; the resultant forward thrust (δFy) along the axis (xx′) of propulsion is (CN —CL) Vu sin θ cos θ δs. In other words, the transverse displacement of the element will produce a forward thrust along the axis of propulsion provided the coefficient of resistance to displacement, normal to the surface of the element, is greater than that to displacement along the surface.

In an actively moving spermatozoon, each element is not only moving transversely across the axis of propulsion, it is also moving along the latter axis at a speed (Vx) which depends upon the speed at which the whole spermatozoon is progressing through the water (Fig. 1B). This forward displacement being equivalent to a tangential displacement Vxcosθ and a normal displacement Vxsinθ, the corresponding forces acting tangentially (δLX) and normally (δNX) to the surface are coséis and CNVxsinθδs respectively. The total forces (δN and δL), acting normally and tangentially to the surface owing to the element’s transverse and forward displacements, are
Fig. 1.

A. Diagram illustrating the forces impressed on an element δs when moving transversely across the axis of progression (xx1) at velocity Vy, the resultant propulsive thrust (δFy) being δNy sin θ — δL, cos θ, where δNy and δL, are the reactions from the water acting normally and tangentially to the surface of the element, and θ is the angle of inclination of the element to the axis xxi. For values of N, and L, see text. B. Diagram illustrating the forces impressed on an element when displaced along the axis at velocity Vx, the resultant drag (δFx) being δNx sin θ + δL, cos θ. The net propulsive thrust (δF) due to simultaneous transverse and forward movement is obtained by combining A and B.

Fig. 1.

A. Diagram illustrating the forces impressed on an element δs when moving transversely across the axis of progression (xx1) at velocity Vy, the resultant propulsive thrust (δFy) being δNy sin θ — δL, cos θ, where δNy and δL, are the reactions from the water acting normally and tangentially to the surface of the element, and θ is the angle of inclination of the element to the axis xxi. For values of N, and L, see text. B. Diagram illustrating the forces impressed on an element when displaced along the axis at velocity Vx, the resultant drag (δFx) being δNx sin θ + δL, cos θ. The net propulsive thrust (δF) due to simultaneous transverse and forward movement is obtained by combining A and B.

formula
The propulsive components of δN and δL along the axis of propulsion (xx′) being δN sin δ and δL cos θ respectively, the resultant forward thrust (δF) is
formula
Equation (i) shows the element can only exert a positive forward thrust if
formula
Equation (ii) shows that a positive forward thrust only develops if CN> CL.

During steady motion the resultant thrust (δF) from an element, when integrated over a complete cycle, is zero ; it is now possible to express the propulsive speed in terms of the wave speed, if the values of CN, CL and θ are known and if the propulsive speed remains constant during the whole cycle of the element’s motion.

The possibility of applying the above principles to an undulating flagellum was first pointed out to one of us (J. G.) in 1933 by the late Dr R. C. Howland. If the transverse velocity of an element be dy/dt and the tangent of its angle of inclination be dy/dx, equation (ii) becomes
formula
the total thrust (F) exerted by the flagellum over one wave-length o ⩽ x ⩽ λ is
formula
To proceed further Howland restricted attention to cases in which the angle of inclination of an element was sufficiently small to eliminate terms containing (dy/dx)2 and to regard the length of the flagellum constituting one wave-length as approximately equal to the wave-length. Equation (iii) then reduces to
formula
If the form of the waves generated by the tail conforms to that of a sine curve
formula
where b = amplitude, λ =wave-length, Vw=velocity of wave relatively to the head.
The transverse velocity (dy/dt) of the element is and the tangent of the angle of inclination (dy/dx) of the element is
formula
Equation (vi) gives the forward thrust exerted by each complete wave.
When an undulating organism (without a head) is propelling itself forward at a steady speed (Vx), F=o
formula
or since Vw = . (where f is the frequency of the waves)
formula
The active tail of a spermatozoon exerts its effort against the drag of an inert head, and consequently
formula
where n is the number of waves exhibited simultaneously by the tail and D is the drag of the head. The drag of the head can be denoted by hCLaVx, where hCL is the drag coefficient and a is a linear dimension of the head. From equations (vi) and (ix)
formula
or
formula
Howland thus reached the conclusion that an organism comparable with a spermatozoon should propel itself through the water at a speed which depended on six factors : (1) the frequency of the waves ; (2) the square of the amplitude of the waves and (3) their wave-length; (4) the difference between the coefficients of normal and tangential resistance; (5) the drag coefficient and size of the head; (6) the length of the tail.

Until recently, Howland’s results appeared to have only limited significance from an experimental point of view, for, apart from the limitation to waves of small amplitude, there seemed no means of determining the values of CN,CL and h and no observational data on the amplitude, wave-length and frequency of the waves. It is now known, however (Hancock, 1953), that for very thin filaments (such as the tail of a spermatozoon) CN is effectively twice CL. It is also known that the waves passing down the tail of a spermatozoon of Psammechinus sometimes conform closely to sine waves of amplitude 4/2 and wave-length 24μ, and that their frequency is about 30‒40 per sec. (Gray, 1955).

It will be noted that if CN = 2CLequation (vii) becomes
formula
Equation (vii) is thus identical with that derived by Taylor (1952) and by Hancock (1953) for waves of small amplitude. Howland’s argument has been set out in some detail in view of the fact that its underlying physical principles can readily be visualized from a biological point of view.

If a filament is deformed into a single complete sine curve of relatively large amplitude, the length (s) of the filament is considerably greater than the wave-length (λ), and the length (δs) of the filament intercepted by a short distance (δx) along the axis of propulsion varies with the phase of transverse motion. Consequently each point on a non-extensible filament executes a figure of eight movement relative to the head of the filament. The motion of each element relative to fixed axes is therefore a figure of eight superimposed upon a forward propulsive velocity (V̄x) defined by the velocity of forward propulsion when averaged over a complete cycle of activity.

The length of filament (δs) intercepted by a small fraction (δx) of a wave-length is given by equation (xiii)
formula
Equation (iii) now becomes equation (xiv)
formula
If CN=2CL
formula
For a sine wave
formula
Substituting in equation (xv) and denoting dy/dx as A
formula
The forward thrust (F) from one complete wave is
formula
Putting
formula
If F = 0
formula
were
formula
and
formula
If is replaced by the equivalent expression , the contribution to the integrals I and J of the term will be smaller than the remaining terms ; if this term is neglected I/J approximates (over all the range of values of 2πb/λ)to
formula
hence
formula
or
formula
It will be noted that equation (xx) is identical with that already obtained from a different line of approach (Hancock, 1952, p. 106) and yields the same result as equation (xii) for waves of small amplitude.
In the case of a filament propelling an inert head
formula
where n = number of waves exhibited by the whole tail and CH = drag coefficient of the head. Assuming the head to be spherical CH = 6πaμ, where a is the radius of the head and μ, the viscosity of the medium.
From equations (xvii) and (xxii)
formula
If CN = 2CL
formula
where
formula
To utilize equation (xxiii) it is necessary to know the absolute value of CL. This can be derived from equation (7) of a previous paper (Hancock, 1953, p. 102), which gives the fluid velocity V in the vicinity of an element of length θs moving with a velocity Vt tangentially to itself in a fluid which is at rest :
formula
where V is the velocity of the fluid parallel to the element surface, d = the radius of the filament, r = the distance from the filament surface, 2π/k = λ = the wave-length. The tangential drag force dF on the element is therefore
formula
Substituting this value for CL in (xxiii) and using the integrals already defined
formula
reducing approximately, by a similar argument as before, to
formula
or
formula

The calculated propulsive speed of the spermatozoa of P. miliaris can be derived from equation (xxvi) by utilizing the data provided in a previous paper (Gray, 1955), bearing in mind the uncertainty of some of the values involved.

As a general test of agreement between the calculated and observed speeds, the following values will be used :

From equation(xxvi)
formula
The average observed speed (as derived from 89 cells) was 191·4μ per sec. for an average frequency of 34·5 per sec. For thirty-three of these cells the speed of propulsion was determined photographically, whilst the frequency was measured stroboscopically ; for 56 cells both speed and frequency were measured photographically. The significance of the very remarkable agreement between the calculated and observed results must not be over-emphasized. Quite apart from the theoretical considerations referred to below, it must be borne in mind that the values used for the effective radius of the head and tail are necessarily approximate, and that most of the other values represent the arithmetic mean of populations showing considerable variation between individuals ; thus in a population of 29 cells the frequency of the waves varies between 25 and 46 per sec., and in the population of 89 cells referred to above, the propulsive speed varied from 100 to 290μ per sec. Probably the safest estimate of agreement between observed and calculated values is provided by the ratio , since this eliminates any error in an estimate of wave frequency; the calculated value of 0·23 indicates that the passage of each wave of 24μ wave-length over the tail should propel the spermatozoon through a distance of 5·52μ; a very large number of records show that the observed distance lies between 5 and 6 μ.
Table 1.
graphic
graphic

The theoretical analysis involves at least three assumptions which differ from the observed facts:

  1. The form of the wave—both in amplitude and wave-length—only conforms to a sine curve during part of its propagation along the tail; the movements executed by elements of the tail at the two ends are different from those situated more centrally. The values given for amplitude and wave-length are, therefore, average values not only for a number of cells but also averages over one complete cycle. As shown in the Appendix, however, this limitation is not likely to be serious as far as variations in amplitude are concerned.

  2. In order that the form of the tail should conform to that of a sine wave the changes in curvature and rate of transverse displacement about the axis of progression must be symmetrical. In many instances this is not the case (Gray, 1955).

  3. The formula assumes that the head travels along the axis of progression without oscillating from side to side. In fact, the head always oscillates during each cycle for a distance comparable with its forward displacement. It may be noted, however, that on theoretical grounds the effect of the head on the rate of forward propulsion is likely to be small. Equation (xxvi) indicates that the presence of the head of the sperm of P. miliaris only reduces the speed of propulsion by about 15 %. In the absence of the head, the last term in the denominator becomes zero, and the calculated speed of progression at 35 waves per sec. rises to about 220 /i per sec. In other words, the propulsive component of the forces acting normally to the surface of the tail largely operates against the tangential drag of the tail—and only to a minor extent against the drag of the head.

Finally, it must be remembered that all the data on speeds of translation are derived from spermatozoa which are moving in close proximity to a glass or air surface, and the assumption is made that these figures also apply to spermatozoa moving freely in a bulk of fluid.

Bearing in mind the above limitations it seems, nevertheless, remarkable that the forward speed of propulsion of a spermatozoon can be calculated (well within the limits of experimental error) in terms of the form and frequency of the bending waves generated by the tail, on the basis that the coefficient of resistance of displacement of the tail normal to its surface is twice that of its displacement tangential to this surface and that all regions of the tail contribute equally towards the necessary propulsive thrust.

APPENDIX

Effect of variation in amplitude of waves

In many undulatory organisms the amplitude of the waves tends to increase as the waves pass posteriorly along the body (Gray, 1953). This feature can be incorporated into the theoretical analysis by assuming that the wave form is described by
formula
where the amplitude function b(x) depends on x, and k = 2π/π
The increment of force dF acting in the x direction may be defined by a similar argument as before :
formula
where the dash denotes differentiation with respect to x. The velocity of propulsion can be obtained from equation (ia) by taking the integral ∫dF = 0 as before. A number of applications are now developed :
(i) Consider a semi-infinite filament (o ⩽ x ⩽ ∞) with steadily decreasing amplitude as x increases, assuming that then
formula
then
formula
where a is the parameter defining the rate at which the amplitude changes and in particular the ratio of any two amplitudes distance one wave-length apart is e-2π/aa = ∞ gives the case of constant wave-length. This wave form is shown below in Fig. 2. Hence from equation (i a), using the fact that for extremely thin filaments CN = 2CL
formula
where
formula
and
formula
Hence
formula
since I is finite and J is infinite. This result is to be expected in this hypothetical case since the propulsive elements at the rear of the filament (in the neighbourhood of finite x) cannot propel the non-propulsive elements of filament at x = infinity.

(ii) Consider a finite filament of n wave-lengths (o ∞ x ∞ 2πn/k), the upper limit of the integrals I and J is 2πn/k instead of infinity. This formula is complicated for all values of bk, but if the assumption is made that bk is small (so that b3k3 can be neglected), then a formula is given which can be compared with that given by the theory of constant amplitude which is valid over the same set of values of bk.

Equation (ii a), on the basis of this assumption, reduces to
formula
This equation becomes
formula
for no variation of amplitude (this follows by expanding the function e‒4πn/a and letting a→ ∞) which is the standard formula for this case. Equation (iv a) also tends to zero as n, the number of wave-lengths, tend to infinity, which agrees with equation (iii a).
(iii) Taking the special case a = 4 π then a table of the mean values of depending on n the number of wave-lengths can be determined. Note that in this case the ratio of successive amplitudes between each wave-length is 0·6. For a = 4π
formula
It is interesting to note how the above values compare with the formula
formula
where is the mean amplitude of the motion, defined by
formula
where ba are the successive amplitudes after each wave-length, then
formula
in the special case considered here. This approximation is equivalent to regarding the motion with varying amplitude as equivalent to that with the constant amplitude equal to the mean amplitude of the original motion. For a = 4π
formula
It is seen that the correspondence between these two sets of results is very good, and gives an easy approximate method for obtaining the velocity of propulsion. This agreement is only valid for small values of bk, but it will be assumed that the same approximation of taking the mean amplitude will give reasonable results for all the larger values of bk.

SUMMARY

  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 mtUutas.

  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.

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* If V1V2 tan d the element has no component of motion normal to its surface ; motion of this type could only ensue if the element were exerting its effort against a rigid medium—and were equivalent to a snake gliding through a rigid close-fitting tube.