J. Exp. Biol. (1964), 41, I3S-IS4With 4 plates and 15 text-figuresPrinted in Great Britain

THE LOCOMOTION OF NEMATODES

BY J. GRAY AND H. W. LISSMANN

Department of Zoology, University of Cambridge

{Received 2 August 1963)

Apart from the fact that their locomotory activity determines the ability of parasiticforms to penetrate the tissues of their hosts, the movement of nematodes presentscertain features which are relevant to the general theory of undulatory propulsion.In the latter respect, two main types of problems arise: (i) To what extent are the formand frequency of the waves passing down the relatively long and cylindrical bodydetermined by the physical nature of the external medium? (ii) To what extent can thespeed of progression of an animal be related to the form and frequency of the waves?

That the form of the muscular waves depends on the physical properties of themedium was noted by Looss (1911, p. 420) who found that the larvae of Agchylostomawere only able to progress over the skin of the host and penetrate through a suitablecrevice if the skin was moist; when freely immersed in a bulk of water the larvae'threshed' actively from side to side without making much apparent headway andwithout being able to keep themselves in suspension by their own efforts. On the otherhand, when placed on or in a damp and suitably dense suspension of faecal particlesthe larvae exhibited ' quick graceful movements' leaving behind them well-markedsinusoidal tracks. Precisely similar phenomena have been observed in all the five typesof worms used in the present work {PanagreUus, Haemonchus, Rhabditis, Strongylus,Turbatrix); all these exhibited well-marked and graceful areeping or burrowingmovements in suitably dense suspensions of particles; all of them displayed very activemovements in water, but only one of them {Turbatrix) was able to support itself by itsown activity (see p. 142). More recently, Wallace (1958-60) has published an extensiveseries of observations on the factors which control the form of the waves and the speedof progression of Heterodera through suspensions of soil particles of varying sizes andin water films of varying depth. Wallace's observations and those described in thepresent paper suggest that there is no sharp line of demarcation between creepingand swimming but, in view of the well-marked contrast in the form of an animal whenburrowing through a dense suspension of particles and when swimming in water, itis convenient to retain these two well-established terms.

Table 1 shows the marked difference in the form and frequency of the waves(averaged over about twenty individuals in each case) exhibited by animals whencreeping—either over the surface of £— 1 % agar or through a suspension of starchparticles—and by individuals swimming in water. The following features are character-istic : (i) The wavelength (A)*, frequency (/), and speed of conduction over the body{Vw) are all very much less in creeping animals than in those swimming in water.

• The wavelength (A) being closely proportional to the length (0 of the body, it is usuallydesirable to express the wavelength in terms of body length thereby facilitating comparisons of the waveform in animals of different length.

136 J. GRAY AND H. W. LISSMANN

(ii) Although the absolute speed of progression of an animal is higher when swimmingthan when creeping, the relative distance moved forward during the period of a singlewave is very much less, (iii) The ratio of amplitude (b) to wavelength (A) is substantiallythe same during creeping and swimming.

Table 1

SpeciesPanagreUtu

silusiaeTurbatrixaceti.

Haemonckuicontortui

Patternof

movementCreepingSwimmingCreepingSwimmingCreepingSwimming

Ratioof wave-length

to bodylength(A//)0 4 70 8 0

o-330-850-400-78

Frequencyof wavesper sec.

(/)

0-853-0

0-635-20-271-7

Speedof con-ductionof waves

persecond

5122200

3123700

59770

Speedof pro-gression

persecond

412640

270718

52165

o-80 2 9

0 8 60-19o-88O-2I

Dis-tance

movedper

wave

to485213

429138218

97

Ratioof ampli-tude (b)to wavelength

(A)(6/A)0-21O-23

0'22(OI5)0-150 1 5

MECHANISM OF GLIDING

(a) Through a suspension of starch grains

When gliding through a relatively densely packed suspension of starch grains thehead of a nematode acts as a wedge operated by the forward thrust exerted by the restof the body. It displaces the grains normally to its own surface and drills a sinusoidalchannel through which the rest of the body glides tangentially (Plate i, figs. 1-4).Under optimum conditions the speed of the animal's forward progression is equal tothe speed at which the waves of muscular contraction pass backwards relative to thehead of the animal; the waves remain stationary relative to the ground and the animalleaves behind it a sinusoidal track whose wavelength and amplitude are the same asthose of the muscular waves. The mechanics of the movement are identical with thoseof a snake gliding through a sinusoidal tube with rigid walls (Gray, 1953).

If, after their displacement by the anterior end of the animal, the starch grains arenot sufficiently closely packed to resist further displacement by the forces which therest of the body exerts against them in a direction normal to its own surface eachelement of the body acquires a component of motion (Vn) normal to its own surfaceand no longer simply glides tangentially; this is equivalent to an increase in the rateof the element's tangential motion combined with a displacement of the element alongthe axis of progression; in other words, the waves are no longer stationary but movebackwards relative to the ground. If the waves move backward relative to the groundat speed Vs, the speed of the animal's forward progression (Vx) is Vw~ Vs, where Vw isthe speed of the waves relative to the head; the percentage 'slip' of the waves relativeto the ground is 100 [i-{VxIVw)]. As the consistency of a starch suspension isseldom uniform, it is not surprising that the amount of slip varies consider-ably in different parts of the suspension (Table 2). It may be noted that the wavelength(A,) of the track left behind the moving animal, relative to that of the wavespassing down the body, is determined by the amount of slip;

Al = V = V'

The locomotion of nematodes 137

Table 2. Rhabditis gliding between starch grains

Wave1 * -

Frequency persec. (/)

O-661-04

o-8o°'S50-690-57

Length (/*)(A)

650600700850

75°470

Speed ofwaves (FK)

Ox/sec) (V.=fXi

429624560467517270

Speed ofprogression(V*) 0*/sec.)

260520420

33°2 2 0

240

VJVWo-6o0-850-750-710-420 8 9

% »

40

IS25295811

Gliding on or within a jelly

Panagrellus, Turbatrix, Haemonchus and Rhabditis all glide readily when placed onthe surface of £-1 % agar or gelatine. With suitably oblique illumination it can be seenthat the animal incises a sinusoidal track on the surface of the jelly; in some cases thewavelength of the track is the same as that of the waves on the body, indicating a totalabsence of slip (Text-fig. 1). In other cases (see Table 3) appreciable slip is observed.Not infrequently the animals penetrate into the jelly, moving forwards or backwardswith equal facility.

0-1 mm.

Text-fig. 1. Haemonchus amtortus creeping with zero slip over the surface of an agar gel.Tracings from successive photographs taken at approx. J sec. intervals. Note that thewaves were stationary relative to the ground.

(c) Movement on a damp rigid surface

When a culture of Panagrellus silusiae is kept in a covered glass container, the animalscreep up the walls as long as these are damp, and they display the same type of wavesas on the surface of an agar jelly; the wavelength, amplitude and frequency of themovements appear to be the same in both cases. On the other hand, the amount ofslip on the surface of the glass is always very much greater than on the surface of thejelly and no trace of tracks is observed. A comparable amount of slip occurs on a dampsheet of solid anhydrous gelatine and on the surface of mercury. In all such cases there

J. GRAY AND H. W. LISSMANN

Table 3Length Wave No. of Frequency Speed of Speed ofof body length Ampl. (A) waves of waves waves progress

SpeciesPanagrellus*

silusiae (8)

P. silusiae (13)

P. silusiae (7)

P. silusiae (10)

Turbatrixaceti (8)

T. aceti (6)

Haemonchuscontortus (23)

MediumOni%

agarIn i%agar

Oni%gelatine

In 1 %gelatine

OnJ%agar

W%agar

On 3%agar

toIIOO

1333

1270

1340

1570

1500

575

(A) to620

641

540

579

484

508

218

to1 2 0

192

94 '

116

1 2 0

90

33

6/AO-2

0 3

O l 8

O'2

O-25

O-2

0-15

visible'"4

i-6

i-8

i-8

2 - 0

2 +

2-O

per sec.1-05

0 8 8

0 7 0

0 7 9

0-63

0-63

0 2 7

Oi/sec.)650

563

378

457

315

320

58-8

(ft/sec.)

502

466

291

387

256

287

52

VJVW

0 7 7

0 8 3

0 7 7

0-85

0 8 1

0-90

0 8 9

% slip23

17

23

15

19

10

11

•The figures show average values for the number of individuals shown in parentheses.

is a marked tendency for a number of animals to adhere together by a film of waterand to exhibit a striking degree of co-ordinated movement (Text-fig 2).

These observations suggest that the ability of Panagrellus or Turbatrix to creep overa damp but smooth surface without appreciable slip only occurs when the animal isable to incise a groove on the surface of the medium. How far this is effected by activedisplacement of the medium by the head of the animal or how far the body is passivelypressed into the medium by the surface tension of the air/water interface is uncertain.That this interface exerts a force against the animal is shown by the fact that animalsin a drop of water are unable to move out of it although they can deform the surface ofthe drop when their heads come into contact with it. It should be noted that otherspecies of nematodes, notably Heterodera, are able to creep over a damp rigid surfacewith very little slip (Wallace, 1958, 1959). It seems just possible that the failure ofPanagrellus and Turbatrix is due to the presence of small traces of surface-activesubstances—derived from the culture medium—which change the physical propertiesof the water film surrounding the body.

Forces which determine the amount of slip

As explained elsewhere (Gray & Hancock, 1955) the propulsive force of an undulatingorganism moving in an aqueous medium is derived from the resistance encountered byeach element of the body to displacement in a direction normal to its own surface.Provided the coefficient of resistance to such displacement (CN) is greater than thatto displacement tangentially to its surface (CL), a propulsive force is generated. Thiscondition is satisfied when a nematode glides through a suspension of particles sinceparticles in the vicinity of the element have to be displaced relative to each otherwhen the element moves normally to its own surface, but not when it moves tangen-tially. On theoretical grounds Hancock (1953) showed that for very small organismsoperating in a homogeneous aqueous medium the expected value of CN would be twicethat of CL; consequently, for waves of the form in such animals as nematodes theexpected amount of slip would be about 80 % A corollary of Hancock's argument is,

The locomotion of nematodes 139

however, that if CN is not twice CL, the amount of slip depends on the ratio betweenthem. If CN = kCL the expected percentage slip is [100 (B + i)]/[kB + 1], where B is27r2i2/A2; b being the amplitude and A the wavelength of the propagated waves. Forcreeping nematodes such as were used in the present study B = 0-5 (approx.) and con-sequently the percentage slip may be expected to be 100 [3 /(k + 2)]. Table 4 gives thecalculated percentage slip for various values of k.

Text-fig. 2. Co-ordinated movement of an aggregate of Panagrellus when on a dampsurface of glass. Scale i mm.

Table 4

I

1*52-O

4-01 0 0

2 0 0

50-000

00 1 4O'2S0 5 1

o-75o-860 9 41 0

1 0 086755°25' 460

In order to account for the low amount of slip of an animal moving through a densesuspension of particles it would be necessary to assume that the ratio CNICL was notless than 20.

So far as is known, there is no observational evidence concerning the resistance ofrelatively dense suspensions to shearing forces of the order of magnitude likely to begenerated by very small and slowly moving cylinders comparable with nematodes;

140 J. GRAY AND H. W. LISSMANN

but, using thin wires, and assuming that the relative value of the resistance coef-ficients can be determined by observing the rate at which the wires fall steadily undertheir own weight when orientated normally and tangentially to their own direction offall, the following observations are perhaps of some interest.

With the equipment available, it was necessary to use media (glycerine, GoldenSyrup) with relatively high viscosity in order to reduce the rates of fall to readilyobservable values. As shown in Table 5, the calculated ratio CNjCL for wires varyingfrom 0-2 to i-6 mm. in diameter was found to be remarkably constant at 1-4 to i*6instead of an expected value of 2-0. No appreciable change in the rate of vertical fallcould be effected by sharpening the leading end of the wire to a sharp point. When thewires were allowed to fall with their long axes inclined to the vertical there was amarked but constant difference between the axis of the wire and its path of motionduring the course of its movements (Text-fig. 3).

It may be concluded that changes in the viscosity of the medium are unlikelyto induce marked changes in the ratio of propulsive speed to wave speed in the

-

cm.

N1

1

\

\

• \

t1

Text-fig. 3. Successive positions at i min. intervals of a wire (20 mm. long; wt. 0-075 8-) fallingthrough Golden Syrup when the axis of the wire was inclined to the vertical. Note the markedbut constant angle between the axis of the wire and its path of motion.

Table 5

Medium

Glycerine

Syrup

Lengthof wire(mm.)

ao2020

20202020

Diameterof wire(mm.)

0320-25O-2O

1 61180-780-50

Weightof wire

(g-)0-015o-oi0-005

0-320-1700750-03

Velocity of fall, (mm./see.)

Wirevertical

2-6

0-460-300-150-07

Wirehorizontal

i-o07

032

O-2IO-IOO-O5

1-4

1'4

The locomotion of nematodes 141case of such animals as nematodes. This conclusion is confirmed by direct obser-vation of the movements of Panagrellus in a homogeneous medium (' Golden Syrup')having a viscosity approximately 1000 times that of water. As shown in Table 6, theaverage percentage slip was effectively the same as in water although the length,frequency and speed of conduction of the waves were markedly reduced.

Table 6

Medium

WaterGolden Syrup

Ratio:wavelength Frequency

per sec.body lengtho-8o-43

Averaged for twenty individuals.

3-o0 1 3

Speed ofconduction

(/i/sec)

2250650

slip

80

Text-fig. 4. Movement of pointed wires through i% agar. The figure shows the rate anddirection of thick wire (25 mm. diameter) and a thin wire (1-4 mm. diameter) when falling withtheir axes vertical and when these are orientated at an angle to the vertical. Note that in thelatter case there was only a very slight angle between the axia of the wire and its path of motion.

Just as there is a marked contrast in the behaviour of nematodes in syrup and in agar,so there is a corresponding difference in the movement of iron wires when submergedin these two media. When an iron rod 22 mm. long, 1*4 mm. in diameter and weighing0-23 g. was gently submerged either vertically or horizontally into a relatively longcolumn of £ % agar, it remained stationary for an indefinite period. On the otherhand, a similar rod whose leading end had been sharpened to a point fell verticallythrough the agar at a steady speed of about 0-33 mm./sec. Further, when such a rodwas submerged obliquely in the medium its rate of displacement in a direction normalto its own axis was extremely small, in other words, it glided through the agar withvery little slip (Text-fig. 4). If a pointed rod was allowed to fall vertically along thetrack which had previously been traversed by a similar rod, its rate of fall was greatlyincreased. A reasonable explanation of these results is available on the assumptionthat \ to 1 % agar gel. can be regarded as a network of fibres which yield when sub-jected to critical shearing forces. These forces are only reached when the effectiveweight of the rod is concentrated at the sharpened end, which, like the head of a nema-tode, acts as a wedge; once the structure of the agar has been ruptured some time is

142 J. GRAY AND H. W. LISSMANN

required before its original structure is regained. The same principle could be appliedto nematodes if the forward thrust exerted by the animal is sufficient to disrupt thestructure of the agar.

The general conclusion to be reached from the study of creeping nematodes is thatthe amount of slip depends on the relative resistance which the medium offers to theshearing forces applied to it by the normal and tangential movements of each part ofthe animal's body. Further analysis probably depends on a more precise knowledgeof the physical properties of agar gels, and of thin films of water. In the case of nema-todes moving between starch grains the work of Oldroyd (1955) and of Kynch (1956)on the effective viscosity of suspension is of considerable biological interest.

S W I M M I N G IN AQUEOUS MEDIA

As already mentioned, marked changes occur in the form and frequency of the waveswhen nematodes are transferred from a dense suspension of particles to a homo-geneous aqueous medium; wavelength, amplitude and frequency are greatly increased,whereas the ratio of propulsive speed to wave speed is greatly reduced (see Table 7).Typical swimming movements are illustrated in PL 2.

Table 7Waves

Speed ofBody progression

Length Ampl. (6)Length Diam. (A) (JJ.) Freq. Speed , * \ Av. Av. Max.

Species (p.) (ji) Qi) b 6/A alt. sec. to Obs. Calc. % slip ob». obs. Calc.f

Haemonckus 547 20 453 68 0-15 1*7 770 165 115 79 0-21 0-36 0-15contortus (29)

Turbatrix 840 28 712 107 0-15 5-2 3700 718 592 81 0-19 0-40 0-16aceti (20)

PanagreJlus 962 40 735 172 023 3'O 2200 640 440 71 0-29 0-40 0-20silusiae (19)

f Calculated values from Hancock (1935, fig. 5.)

(i) Turbatrix aceti

Turbatrix swims actively to the surface of vinegar when confined in a long verticaltube (Peters, 1928). During the present study, visual observations in a tube 60 cm. longand o-8 cm. wide yielded an average speed of vertical progression of 15 individuals of738/x/sec. (max. 1070; min. 220); as determined photographically for horizontal move-ment over a glass slide the average speed of progression of twenty individuals was718/i/sec. (max. 1055; min. 220). These figures suggest that the speed of progressionis not greatly affected by proximity to a glass surface; but as neither the form nor thefrequency of the waves could be determined when animals were swimming verticallyin a bulk of fluid, it is impossible to be sure that these were the same as for similaranimals moving horizontally over a slide. As shown by Table 7 the observed ratio ofpropulsive speed (PQ to wave-speed (PJ,,) was sometimes surprisingly high.

A frequent, but not invariable, characteristic of Turbatrix, when swimming invinegar, is a change in the form of the wave as it passes along the body; the amplitude

The locomotion of nematodes 143of the waves increasing as they pass posteriorly along the body. In the individualshown in Text-fig. 5 the amplitude of lateral displacement of the head was about 3 5/a,whereas that of the tip of the tail was 125/x. As explained elsewhere (Gray, 1958) thischaracteristic is associated with an ability of the animal to propel itself without yawingfrom side to side.

1-0 -

0-5

1 1 1 1

Text-fig. 5. Successive positions at intervals of TV sec. of a specimen of Turbatrix aceti in whichthe amplitude of transverse movements of the tail was about four times that of the head.

1615

Text-fig. 6. Motion relative to fixed axes of an individual Haemonchus contortui derived fromphotographs taken at intervals of A sec. The tracks of the head and tail are shown by dottedlines. Note the two characteristic nodes.

(ii) Haemonchus, Strongylus and Panagrellus

In all these forms the body of a swimming animal displays less than one completewave during certain phases of each complete cycle, and consequently it is not surprisingthat its motion relative to the ground should differ from that of a system in which allthe transverse force operating against the body are assumed to be in equilibrium(see p. 149).

Relative to the ground, the envelope of a complete cycle of a swimming Haemonchusexhibits two distinct 'nodes' (Text-fig. 6) round which the anterior and posterior

144 J- GRAY AND H. W. LISSMANN

regions of the body glide tangentially. The existence of these nodes is associated withtwo characteristic features of the animal's movement, (i) The whole body is yawingfrom side to side in the plane of its transverse movement (Text-fig. 76). (ii) Relative tothe head of the animal, the waves are accelerating over the anterior end of the bodyand slowing down over the posterior end. As shown by Text-fig. 6, the motion of anelement relative to the surrounding water depends on its position relative to the headof the animal and and is by no means the same for all elements; the motion of thewhole body differs very substantially from that which forms the basis of calculationof propulsive speed. On general grounds one might have expected the observed speedof progression of a yawing organism to be less, and not greater, than the calculatedvalue.

W (b)

Text-fig. 7. The periodic yawing movement of the body of Haemonchut. (a) shows thesinusoidal track along which the animal appeared to glide when a series of successive photo-graphs were superimposed so that the outline of individual waves remained stationary tofixed axes passing through their respective crests. The axis of the sinusoidal track ( in a)defines the axia of the waves, (b) shows the periodic yawing of this axis about the axis of theanimal's progression; the line ab is drawn normal to the axis of progression.

MOVEMENTS OF WATER IN THE VICINITY OF A SWIMMING NEMATODE

The changes of external bodily form which a nematode produces by its own internalefforts inevitably induces displacement of the surrounding water and a precise know-ledge of the nature and extent of this displacement is of very great significance in anytheoretical analysis of undulatory propulsion.

The nature of the movement induced in the water by a swimming animal can bedetermined by observation of the movements of small suspended particles. A seriesof photographs (at intervals of about ^ j sec. and j ^ sec. exposure) of a swimmingPanagrellus showed that a starch grain originally situated close to the head on theright-hand side of the body travelled tangentially backwards along the surface of thebody until it reached the vicinity of the tail and remained on the right side of the body

The locomotion of nematodes 145

during the whole of its displacement (Text-fig. 8); relative to the head, the particlemoved tangentially over the surface of the body for a distance approximately equalto one wavelength. The photographs showed that during the period required for theparticle to travel this distance, four complete waves had passed along the body, andconsequently the rate of backward movement of the particle relative to the head wasone quarter of the rate of propagation of the waves over the surface of the body.

Text-fig. 8. The movement of a particle ( *) along the right side of the body. Between position iand position 41 the particle moved along the body for a distance of approx. one wavelength.The displacement of the particle relative to the ground is shown by the dotted line; the particleexecuted three rather irregular loops traversing the loops in an anticlockwise direction, andduring this time four waves passed over the body.

When the displacements of the head, the tail and the particle shown in Text-fig. 8 areplotted relative to the ground their respective tracks are relatively complex (Text-fig. 9); this is partly due to the periodic yawing movements of the whole body in atransverse horizontal plane and partly to the fact that the movements of this animal wereless regular than those shown in figs. 5 and 6. As shown in Text-fig. 9 the head and tail of

10 Exp. BioL 41, 1

146 J. GRAY AND H. W. LISSMANN

the animal move forward along somewhat irregular sinusoidal tracks whose amplitudesare considerably greater than their wavelengths, the frequency of transverse displacementbeing the same as that at which the waves are propagated over the body. At first sight thetrack of the particle appears to have little relationship to that of the head; it exhibits a seriesof rather irregular loops with the particle travelling forwards relative to the ground(i.e. in the direction in which the animal is moving forwards) when the particle is inthe vicinity of a wave crest, and backwards (i.e. in the direction in which the wavestravel relative to the head) when in the trough of a wave. As will be seen in Text-figs. 8 and 10, the average rate at which the particle moves backwards (relative to theground) is very much less than that at which the waves travel relative to the ground.The whole track of the particle is the resultant of three types of displacement (i) alongitudinal harmonic motion, (ii) a transverse harmonic motion, (iii) a longitudinaldisplacement at a speed much less than that at which the waves travel relative to theground.

Text-fig. 9. Track of the head and tail of the animal shown in Text-fig. 8 andof a suspended particle.

Some of the characteristic features of the displacement of a particle can be illus-trated by plotting its displacement and that of the head of the animal relative to astationary wave. For this purpose a series of photographs were mounted so that theoutline of individual waves remained stationary in respect to fixed axes passing throughtheir respective crests; in order to do this it is necessary to rotate each photographin a transverse plane to compensate for the transverse harmonic yawing movementswhich occur during the passage of each wave (see p. 144). Most of the outline of thewhole body then falls on a relatively smooth sinusoidal curve (Text-fig, yd) alongwhich the head and all other elements of the body move forward, at the same speed(and with the same frequency and form of transverse displacement) as the waves arepropagated along the body. When successive positions of the particle (referred to inthe previous paragraph) were plotted along the sinusoidal track followed by the body,the particle moved forward relative to the fixed axes at an average speed about three

The locomotion of nematodes 147

quarters of that of the head and executed three cycles of transverse movement. Therate of a particle's forward displacement relative to a stationary wave is, however, notconstant; it is greatest when the particle is in the vicinity of a wave crest and leastwhen in the vicinity of a trough—the ratio of the two speeds being of the order ofthree to one. The movements of the head and particle relative to a stationary wave areshown diagrammatically in Text-fig. 10.

48 421

361

301

241

181

121

61

010

m aText-fig. 10. The movement of a particle relative to astationary wave. If such a filament as thatshown in Text-fig. 13 is moved anteriorly along the axis ofthe wave at the same speed (6cm./sec.)as the wave travels posteriorly relative to the head, the waves become stationary relative to theground and the body glides forward without slip. Both head(-«-) and particle (•) follow the sametrack; in the diagram the positions of the head and particle are shown at successive intervals of 1sec. In 12 sec. the head travels 1 \ wavelengths (72 cm.), whilst the particles travel 1 wavelength(54 cm.). In 36 sec. the head completes 4 cycles of transverse displacement, while the particlecompletes 3 cycles.

The physical basis of longitudinal harmonic movements of the particle will be con-sidered later, but the fact that fluid surrounding the body exhibits transverse harmonicdisplacements is illustrated by PI. 1, figs 5-8, which is a photograph of an animalswimming into a drop of water from one of 50% Golden Syrup; the body is surroundedby a relatively wide layer of fluid which undergoes transverse movements during thepassage of the wave.

For analytical purposes, it is important to know how far the disturbances producedin the water by the movements of the body extend beyond its surface. This is illustratedby PI. 3, figs. 1-4, which shows four consecutive photographs of Turbatrix swimmingin a suspension of starch grains; the duration of each exposure was approx. xS^h sec-and the interval between them £th sec.; the frequency of the waves was 3 per second.Starch grains at rest appear sharp, whilst those set in motion appear as streaks.Regions of the body situated near the end of their transverse displacement appearsharp, whilst those in active transverse movement are, as might be expected, blurred.The most striking feature of the photographs is the circulation of particles centredround regions of the body which have reached their maximum transverse displacement,i.e. round the crests and troughs of the waves. The direction of the circulation cannotbe determined from the photographs but when the latter are interpreted in the light ofthe observations of single particles (as in Text-fig 8) there can be very little doubt thatthe directions of circulation in PI. 3 is as shown in Text-fig. 11; the particles movingforward (i.e. in the direction of the animal's progression) when in the vicinity of a wavecrest and backwards when in a trough. Thus members of any pair of adjacent 'vortices'rotate in opposite directions. It will be noted that the circulation extends beyond thesurface of the body for a distance equal to ten or fifteen times the diameter of the body.Not the least surprising feature of PI. 3 is the regularity of the rotational flow patterns

148 J. GRAY AND H. W. LISSMANN

despite the relatively complex and somewhat irregular track of individual particlesrelative to the ground.

Before discussing the general nature of the flow of water in the vicinity of an undu-lating body it may be useful to summarize the main features of the movement ofparticles in the neighbourhood of a swimming nematode as revealed by directobservation: (i) The particles move tangentially along the surface of the body and mayremain on the same side of the body during the whole of their displacement, (ii) Therate of tangential displacement is not constant; it is greatest when the particle issituated in the trough of a wave; it is reversed in direction when situated near a wavecrest, (iii) The average speed of posterior displacement relative to the head is aboutone quarter of that at which the waves are propagated along the body; in order that aparticle should be displaced posteriorly for a distance of one wavelength, four completewaves must pass along the body and during this period the particle executes threecycles of transverse movement, (iv) Relative to the ground, particles traverse loopedtracks; particles on the right side of the body move along these loops in an anticlock-wise direction, those on the left side move in a clockwise direction, (v) The flow extendsfor a considerable distance away from the surface of the body.

(<*)

Text-fig. 11. The pattern of the stream lines in the neighbourhood of the body of Turbatrix (seePI. 3). The centres of rotation move posteriorly relative to the ground at the same speed as thewaves.

DISCUSSION

As applied to small organisms such as those considered in this paper, the generaltheory of undulatory propulsion (Taylor, 1951, 1952; Gray & Hancock, 1955) makesa number of assumptions, four of which are of immediate significance: (i) That thedisturbance which the moving body induces in the surrounding water is restricted toa very short distance away from the surface of the body (Taylor, 1951). (ii) That themotion of an element of the body relative to the main mass of the undisturbed wateris the same for all elements, (iii) That the forces which an element exerts against

The locomotion of nematodes 149

the surrounding water are the same as those exerted by a corresponding element ofa long straight cylinder moving at the same speed and with its surface inclined at thesame angle to its direction of motion, (iv) That, at all phases of movement, the trans-verse forces acting on the body summate to zero; the body does not yaw from sideto side and the axis of progression always coincides with the axis of the waves. Noneof these assumptions are justified in the case of nematodes and it may be open todoubt how far they can be safely applied to other organisms.

12i

-*- Direction of waves18 24

Anterior Posterior

Text-fig. 12. Displacement of particles by a filament undulating within a close fitting rect-angular channel whose width is twice the amplitude of the waves and whose depth is equalto the diameter of the filament. All particles lying between the filament and two successivewave-crests must travel posteriorly at the same speed as the waves and each particle mnintgiimits position relative to the crests of the waves. Wavelength 54 cm.; amplitude 6 cm.; velocityrelative to head and ground 6 cm./sec. Period 9 sec. Forward displacement of filamentrelative to ground nil.

The pattern of flow in the neighbourhood of a swimming nematode is essentially thesame as that described some years ago for the flow past the surface of an undulatingcylinder of rubber (Gray, 1935). The main factor responsible for the reversed flow inthe neighbourhood of the wave crests was revealed when the cylinder was replaced byan undulating rubber sheet \ in. thick, 4 in. wide and 24 in. long. By attachmentto the operating rods of a wave machine the sheet generated waves 30 cm. in wave-length and 5-5 cm. amplitude, thus giving a ratio of amplitude to wavelength approxi-mately the same as that typical of nematodes. The whole apparatus was placed in alarge rectangular tank containing sufficient water to submerge the lower half of thesheet—leaving the upper half unsubmerged. A rigid plate was then placed tangentiallyto two successive wave crests, and a small number of ' particles' (parsnip seeds) werescattered on the surface of the water. The movement of these individual seeds wasrecorded photographically using exposures of 16/sec. and of jfa sec. duration. As thewhole of the water between two successive wave crests was completely containedwithin the boundaries of the model and the rigid plate, all particles moved posteriorlyin straight lines parallel to the axis of the waves and at a speed equal to that of thewaves (see Text-fig. 12.) When photographed with appropriately long exposures, theparticles appear as streaks all aligned parallel to the axis of the waves (PI. 3, fig. 5). Assoon as the plate was removed, the pattern of flow of all the particles changed to thatillustrated in PI. 3, fig. 6. The track of individual particles relative to the ground showedcharacteristic loops. When viewed from above and looking towards the anterior endof the model, all particles passing along the left side of the model traverse their loopsin a clockwise direction whilst particles on the right side move anti-clockwise.

In the case of the model it seems clear that the looped track of a particle relative tothe ground is an expression of the fact that water situated near the leading surface of

The locomotion of nematodes 151

speed one quarter of that at which the waves are propagated, (ii) That the frequency ofa particle's transverse displacement is three-quarters of that at which the waves aregenerated. The arbitrary values selected for the wave characteristics were adopted on

Direction of wmve

Pen it ton of particle and wmve after successive intervals of 1 uc .Direction of wmve

Position of particle and wive after successive intervals of 1 seci _ 1? 1,8 24 30 36 42 48 54'on

23

24

2S

26

Text-fig. 13. For legend see opposite.

the basis of simplicity of geometrical construction; wavelength 54 cm., amplitude6 cm., frequency one wave in 9 sec. and speed of propagation 6 cm./sec. In using thediagrams it is important to remember that all the particles shown invariably lie closeto the surface of the body.

Text-fig. 13 shows the movement of the waves and of a particle at thirty-six succes-sive intervals of one second. A particle originally situated on the right anterior side ofthe body opposite a wave crest travels tangentially over the body's surface through adistance of 54 cm. (one wavelength) in 36 sec. during which time a wave travelled216 cm. (four wavelengths). In 36 sec. the particle executes three complete cycles oftransverse movements whilst the head completes four complete cycles. The diagramalso shows that the particle moves anteriorly, relative to both head and ground, whenin the vicinity of a wave crest (phases 0/1,11/13, 23/25) and posteriorly at its maximumspeed when in a wave trough (phases 5/7,17/19, 29/31) and that it traversed its looped

152 J. GRAY AND H. W. LISSMANN

track in an anticlockwise direction. A particle on the left side of the body traversedits track (see phase 36ft) in a clockwise direction.

Text-fig. 14 shows the displacement of particles in 1 sec. during which the wavemoves 6 cm. In Fig. 14 a all the particles lie on the right side of the body and movealong their tracks in an anticlockwise direction; in Fig. 146 the particles lie on the leftside of the body and are moving clockwise. The arrows show the displacement of theparticles relative to the ground and exhibit a marked contrast with Text-fig. 12.

Direction of waves. •Velocity of waves 6 cm./sec. Wavelength 54 cm. Period of wave, 9 sec

0 6 12 18 24 30 36 42 48 54

Anterior

36 42 48 54 60 cm

y v i. j )Text-fig. 14. The movement of particles relative to the ground and to the surface of the body.All particles on the right side of the body move anticlockwise (a); all particles on left side (b)travel clockwise. The length and direction of the arrows show the displacement of theparticles relative to the ground and to the surface of the body when the crest of a wave travels16 cm. (from position , to position ) in 1 sec. Compare this diagram with Text-fig. 12.

Text-fig. 15 shows that the track of the particle not only depends on the rate atwhich the waves move relative to the head but also in the rate at which the whole bodymoves forward relative to the ground; in other words, the form of the track dependson the rate at which the waves travel relative to the ground.

From a theoretical point of view the most significant conclusion to be derived fromthe present study is that the water in the vicinity of a swimming nematode not onlymoves backward relative to the ground at a much lower speed than the waves them-selves travel relative to the ground, but also exhibits two harmonic motions: (i) Trans-verse harmonic motion—due to the transverse drag exerted by adjacent elements ofthe body; the frequency of these movements being less than that of the propagation ofthe waves along the body, (ii) A longitudinal harmonic motion due to the reversal ofthe flow over the crests of the waves. It may well be that these characteristics of theflow may be typical of all undulating surfaces.*

It may be noted that whereas Text-fig. 11 and PI. 3 depict the displacement (duringa short period of time of a large number of particles at varying distances from thesurface of the body, the dotted lines in Text-fig. 14 show diagrammatically the dis-placement of individual particles all close to the surface of the body during arelatively long period of time. At first sight it appears strange that the observed pathsof displacement (Text-fig. 8) should show great irregularity, whereas Text-fig. 11

• See Ewald, POschl & Prantl (1930), p. 264, fig. 65, and Pohl (1932), p. 219, figs. 45, 46.

The locomotion of nematodes 153

shows a very regular pattern. This anomaly would, presumably, disappear if it werelegitimate to assume that the flow of water relative to the surface of the body is notaffected by yawing movements of the body relative to the ground. A morecomplete analysis of the flow of water in the neighbourhood of an undulatingorganism would involve the observation of the movement of individual particlessituated at known distances away from the surface of the body. The incorporation ofall the data into a generalized theory of undulatory propulsion would appear to presentformidable problems in hydrodynamics.

Displacement of head Displacement of waves relative to groundrelative to ground at successive Intervals of 1 sec

10 5 0 5 10 15 20 25 30 35 « -45 50 55 60 cm• • l.t.' . . I . . t i I I I | | J ^ 1 1 1_

Text-fig. 15. Displacement of a particle relative to the ground when the waves are travelling6 cm./sec relative to the head of the animal and the head travelling 1 cm./sec. relative to theground; the speed of the waves relative to the ground is 5 cm./sec. In 13 sec. all particles moveanteriorly 54 cm. relative to the waves (as in Text-fig. 13), whilst the wave travels 60 cm.posteriorly relative to the ground; hence, the particles move posteriorly 6 cm. relative to theground. The positions of the particle after successive intervals of 1 sec. are marked by thenumerals along the looped tracks; the relative positions of the head are marked along thedotted line. Note that, as in Text-fig. 13, the particle executes one complete transverse cyclewhilst the head executes i j cycles; also that the track of the particle relative to the grounddepends on the speed at which the waves themselves travel relative to the ground.

SUMMARY

1. The form and frequency of the waves passing down the bodies of small free-living nematodes (Panagrellus, Rhabditis and Turbatrix) depend on the nature of theexternal medium.

2. Observations of animals moving in such media as syrup, agar gels, and densesuspensions of particles suggest that the relationship between the speed of progressionof the animal to the speed of propagation of the waves along the body, depends on therelative resistance exerted by the medium to displacement of the body in directionsnormal to and tangential to its own surface.

3. When swimming in water the body of a nematode yaws periodically in a trans-verse plane and the axis of progression does not coincide with that of the waves; thedisplacement of an element of the body relative to the medium depends on its positionon the body. The envelope of one complete wave exhibits two characteristic nodes.

4. A suspended particle originally situated near the anterior end of a swimminganimal moves tangentially backwards along the surface of the body, but the speed atwhich it does so is not constant and is always much less than that at which the wavestravel relative to the ground. The average speed of a particle close to the surface ofthe body is about one quarter of that of the waves.

5. When plotted relative to the ground the path of displacement of a particleexhibits characteristic loops. When viewed from above and in the direction of theanimal's progression, all particles on the left side of the body traverse their loops in aclockwise direction; all those on the right side move anticlockwise.

154 J- GRAY AND H. W. LISSMANN

6. There is a highly characteristic pattern of circulation round the body of theanimal. Water in the vicinity of a wave crest moves in the opposite direction to that ofthe propagation of the waves, but in the same direction as the waves when situated ina wave trough. The circulation extends for a considerable distance from the surface ofthe body.

7. The flow round the body of a swimming nematode is essentially the same asthat in the neighbourhood of an undulating sheet of rubber. Its analysis presents aninteresting but difficult hydrodynamical problem.

REFERENCES

EWALD, P. P., POSCHL, T. & PRANTL, L. (1930). The Pkyria of Solids and Fluids. London: Blackie & Sons.GRAY, J. (1935). The propulsive powers of the dolphin J. Exp. BioL 13, 192-9.GRAY, J. (1953). Undulatory propulsion Quart. J. Micr. Set. 94, 551-78.GRAY, J. (1958). The movement of the spermatozoa of the bull. J. Exp. Biol. 35, 96-108.GRAY, J. & HANCOCK, G. J. (1955). The propulsion of sea-urchin spermatozoa. J. Exp. Biol. 3a, 802-14.HANCOCK, G. J. (1953). The self-propulsion of microscopic organisms through liquids. Proc. Roy. Soe.

A. 317, 96-121.KYNCH, G. J. (1956). The effective viscosity of suspensions of spherical particles. Proc. Roy. Soc. A,

337, 00-116.Looss, A. (1911). The anatomy and life history of Agckylostoma duodenale Dub. Pt. II. Cairo Records

Eg. Govt. Sch. Med. 4, 167H507.OLROYD, J. G. (1955). The effect of interracial stabilizing fjlma on the elastic and viscous properties of

emulsions. Proc. Roy. Soc. A, 333, 567-77.PETERS, B. G. (1928). On the bionomics of the vinegar eelworm. J. Helminth. 6, 1-38.POHL, R. W. (1932). Physical Principles of Mechanics and Acoustics. London: Blackie & Sons.TAYLOR, G. I. (1951) Analysis of the swimming of microscopic organisms. Proc. Roy. Soc. A, 309,

447-61.TAYLOR, G. I. (1952). Analysis of the swimming of long and narrow animals. Proc. Roy. Soc. A, 314,

158-83.WALLACE, H. R. (1958). Movement of eelworms. I. The influence of pore size and moisture content of

the soil on the migration of larvae of the beet eelworm, Heterodera schachtii Schmidt. Ann. Appl.Biol. 46, 74-85.

WALLACE, H. R. (1959). Movement of eelwonns in water films. Arm. Appl. Biol. 47, 366-370.WALLACE, H. R. (i960). Movement of eelworms. VI. The influence of soil type, moisture gradients and

host plant roots on the migration of the potato root eelworm Heterodera rostochiensis Wollemoeber.Arm. Appl. Biol. 48, 107-20.

EXPLANATION OF PLATESPLATE I

Figs. 1—4. Panagrellus silusiae moving through a thin layer of a dense suspension of starch grains; theanimal is moving from right to left. Note the track left behind the body. Scale 550^1. Interval betweenphotographs approx. TV sec.Figs. 5-8. Panagrellus swimming into a drop of water from one of Golden Syrup. Interval betweenphotographs • sea

PLATE 3

Figs. 1-12. Haemonchus contortus swimming in water. Interval between photographs A sec.

PLATE 3

Figs. 1-14. Turbatrix aceti swimming in a dilute suspension of starch grains. Interval between photo-graphs i sec.; exposure time TV sec.Fig. 5. Movement of particles when the flow induced by an undulating rubber lamina is restricted by arigid plate in contact with two wave crests. Exposure time Ar sec. Note that the particles move alongstraight lines.Fig. 6. Flow of particles after removal of plate. Note the circulation round the wave crest on the rightof the figure. Exposure time A sec.

PLATE 4

Figs. 1-9. Flow of particles induced by an undulating rubber lamina. In fig. 1 the model is at rest.The waves are travelling from right to left. Interval between photographs J sec.; exposure time i sec.

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