Wall Drag On Free-Moving Ciliated Micro-Organisms

Velocity profiles of a self-propelled (P_w) and a passively moving (P_e) body at low R_e. A body of minor semi-axis a moves its centroid from the base to the tip of the horizontal arrow at a constant velocity U_∞. If passive, the body entrains fluid by an outwardly decreasing amount indicated by velocity profile P_e. If oscillating, it entrains very little fluid as indicated by profile P_w. Theoretically, P_e becomes asymptotic to the vertical axis while P_w merges with it at a distance S_a from the translational axis. An experimentally practical cutoff or pseudomerging point is determined at from the translational axis. S_e and S_w are, then, radii of the spheres of viscous influence of passive and propulsive movers, respectively. Note, however, that the S_e and S_w referred to in the text are the radii of the spheres of viscous influence minus one equivalent cell radius (R) or minor semi-axis (a) ; i.e. they are clearances. The solid boundary at R₀ from the translational axis has no significant effect on U_∞ until R₀< S_e or R_o< S_w for a sinking or swimming cell, respectively. The equivalent sphere (of radius R) can represent the passive body with no effect on the data until R_o ⩽ S_e.

Fig. 1 shows the essential differences between S (of radius S_e) for sinking cells (of equivalent radius R) and S (of radius S_w) for swimming cells (of minor semi-axis a). As is apparent from the diagram, S_w< S_e, because as indicated in the velocity profiles, a self-propelling body disturbs less of its surrounding fluid than a passive body does (see Blake, 1973 and Jahn & Votta, 1972, for more detailed velocity profiles of swimming bodies).

It is curious that even though wall effects are obviously significant at some clearance (R₀ − (R or a), or the form used here which is the non-dimensionalized ‘reduced clearance’ R₀/(R or a) − 1); and even though there must be a difference between and S_w, there are apparently no measurements of these quantities. Accordingly, it is the purpose of this work to obtain a measure of S_e and S_w by comparing the velocities (U) of swimming and sinking ciliates down tubes of decreasing bore with their velocities in an unbounded fluid (U_∞) and with each other. After determining S_e and S_w, their values will be compared with corresponding theoretical models which estimate spheres of viscous influence from the point of view of the validity of Stokes equations for motion of a body at low R_e.

MATERIALS AND METHODS

Paramecium caudatum and P. multimicronucleatum grown in wheat-fortified Paramecium medium (Ward ‘s) and Tetrahymena sp. (from Ward ‘s) grown in 2 % proteose-peptone were observed in glass tubes at 20 °C. In all cases the portion of the tube being viewed was immersed in immersion oil to reduce refraction and to enable measurement to be made of the tube bore.

(A) Sedimentation

Ciliates to be sedimented were fixed in formaldehyde and rinsed in Paramecium medium at least 3 times by hand centrifugation, decantation and re-suspension. Sedimentation in vertical (determined by eye with a plumb line) constricted tubes was followed with a horizontal binocular dissecting microscope fitted with a calibrated Whipple disc and timed with a hand-held stopwatch. Only cells at the point of narrowest bore or below and those tilted ∼45° or less from the vertical axis were timed. In the swimming experiments cells were timed over a course of constant bore ; in the sinking experiments the bore width of the vertical tubes increased to a maximum below the constriction. Accordingly, only the average width for each course length was utilized. An example of the tube structure is presented in Fig. 2. Sample sizes ranged from 13 upwards with one exception: only 2 measurements could be obtained for P. multimicronucleatum at the narrowest tube bore.

Fig. 2.

A sedimentation tube. The tube is immersed in immersion oil to allow resolution of the inner bore which varies continuously from the point of constriction to the original tube width. Sinking paths were less than 1 mm, so the fluid volume containing the cells could be approximated by a truncated cone. The average diameter of the cone is, then, the bore width (2R₀ in figure).

The physical sedimentation model consisted of 2 mm plastic spheres sinking in a series of vertical glass tubes of different bores which were filled with Dow Corning DC 200 fluid (1000 centistokes). Sinking velocity was determined from photographs of stroboscopic flashes taken during the descent of the sphere.

In order to determine S_e in a bounded fluid the measured velocities U had to compared with the corresponding velocity U_∞ in an unbounded fluid. The latter exists, of course, only hypothetically. Accordingly, U_∞ had to be determined theoretically, and three methods were utilized because each could claim validity, given suitable data.

The fastest measured U may be taken as the maximum U value. Thus, if this U is given the symbol U_∞meas we have
If cell density is known the sedimentation equation may be utilized.

where the sphere radius r may be replaced for a non-spherical body by the equivalent radius R under low R_e conditions (see Happel & Brenner, 1965, p. 223) and p for the ciliates is assumed to be that of Tetrahymena pyriformis (calculated as 1·076 from Dunham & Child, 1961).

3. An interpolated U_∞int= U_∞ may be calculated from the ratio of Stokes drag on a spheroid in an unbounded fluid (F_s∞) to the numerical solution for Stokes drag on a spheroid settling in a cylinder with the major axis of the spheroid collinear with the axis of the cylinder (F_sb) (Happel & Brenner, 1965, p. 340).

Here R_o is the internal radius of the tube, a is the minor semi-axis of the spheroid and c is its major semi-axis. A more exact formula is available for the sphere model

(Happel & Brenner, 1965, pp. 318-20) where U_∞= U_∞esi which is readily calculated from the measured density of the plastic sphere and the known kinematic viscosity of DC 200.

To avoid geometric effects due to the asymmetry of a ciliate the values of U chosen for calculating U_∞int were limited to those measured at maximum R_o.

Finally, the determination of the radius of S_e as the distance for significant wall drag followed a study of the plots of U/ U_∞ as a function of R_o, the details of which are reserved for the conclusion section.

(B) Swimming

Swimming cells were timed with a hand-held stopwatch as they swam over a premeasured distance in a horizontal capillary tube. The tubes of narrower bore were obtained by drawing out the tubing over a flame. For trials with large-bore tubes, only those cells having swimming paths of small amplitude were measured (ciliates swim along helical paths, i.e. they gyrate). The dimensions of the cell and of the tube bore were determined from photomicrographs or measured directly through the microscope with the aid of a calibrated Whipple disk. Sample sizes ranged from 20 upwards.

RESULTS

(A) Sedimentation

Data from sinking cells were collected until R_o reached the practical limits of focal length and optical resolution. Measurements are summarized in Table 1. As expected from the drag-ratio models (3, 4), U increases with R₀. Similarly, R_e increases with R₀ by an order of magnitude in the case of Paramecium.

Table 1.

Sinking velocity ratios

While U_∞est and U_∞int are almost identical for the physical model, as expected, there is significant discrepancy between the two measures for all the ciliates. In contrast, the discrepancies between U_∞meas and U_∞int fall well within the standard deviation of the former for all three ciliates.

These relationships are more striking when sinking velocity ratio is plotted as a function of reduced clearance, as shown in Figs. 3–5. In all three graphs the solid lines represent the mathematical models utilizing the measured values of R. The agreement of the physical model with (4) is quite good. The spheroid mathematical model (3) does, however, tend to overestimate sinking velocity for all three ciliates regardless of the method for calculating U_∞. Except in the case of P. multimicronucleatum, this tendency does remain within the standard deviations of U/U_∞meas for clearance values beyond 10(R₀/R − 1) (namely, 10 cell equivalent radii) and deviates less than 3% from the upper limit of U/U_∞int for the same clearance values. In contrast, U/U_∞est approaches that for the model only in the case of Tetrahymena, being the lowest curve in all three cases.

Fig. 3.

Velocity ratio of sinking Tetrahymena sp. and plastic spheres as a function of reduced clearance. The two solid lines represent plots of equation (3) and (4) which represent the spheroid approximating the ciliate body and the plastic sphere respectively. Variances are standard deviations of the sample. In the figure the open circles represent plots of U/U_∞where U_∞ is defined in (1), the closed circles represent plots of (3) or (4) and the squares represent plots of (2). All symbols except R_s are defined in Materials and Methods. The Stokes radius R_s is defined in the Discussion. Of the three forms of U/U_∞ it will be noted that UIU_œest underestimates F_s∞/F_sb for this ciliate while showing good agreement for the physical model. The other two forms of U/U_∞ which were used for the ciliates only show good agreement with the mathematical model (3).

Fig. 4.

Velocity ratio of sinking P. caudatum and plastic spheres as a function of reduced clearance. All plots and symbols have the same meaning as in Fig. 2. Here the underestimation of Fs _∞/F_sbby ciliate U/U_∞ is more marked than the case for Tetrahymena. In addition a tendency for both U/U_∞ and U/U_∞ to underestimate the drag ratio is detectable between U/U_∞= 0·54 and 0·92. Deviation of the model (3) from data at reduced clearance values below 10 are due to the breakdown of the model (which is, after all, a polynomial numerical solution) close to the moving surface.

The plot of (3) is virtually identical in Figs. 3–5. Accordingly, one set of velocity-ratio values and corresponding clearances obtained using the model may be used with all three ciliates. Such a set is presented in Table 2.

Table 2.

Sample spheroid velocity ratios for given reduced clearances

(B) Swimming

Data from swimming ciliates are summarized in Table 3. A comparison with Table 1 shows a maximum velocity ratio between swimming and sinking cells (U_{∞ swim}/ U_{∞ int}) ranging from 7·5 to 50 with decrease in R or a. Unlike sinking cells, the swimmers achieve a maximum U at R_o values well within the focal length and optical resolution of the observation system. Accordingly, U_∞ was measured directly, enabling U/U_∞ to be plotted independently of any model and simply as a function of clearance. The resultant graphs are presented in Figs. 6–8. Here it is apparent that U/U_∞= 1 is maintained to within the following distances of the ciliate body surface : Tetrahymena, 4’ia; P. caudatum, 0·64a; P. multimicronucleatum, 4·2a. A plot of the sedimentation data is included in the figures for comparison.

Table 3.

Swimming velocity ratios

Fig. 5.

Velocity ratio of sinking P. multimicronucleatum and plastic spheres as a function of reduced clearance. All plots here are similar to those for P. caudatum only more deviant. Only one value of U/U_∞int and one of U/U_∞meas fall on the F_s∞/F_sb plot. Overestimation by the model between velocity-ratio values of 0-35 and 0-92 reaches 22 % at one point. Nevertheless, the agreement of U/U_{∞ meas} with the model at the maximum measured value does establish an upper limit for the clearance at which there is ‘safety’ from wall drag.

Fig. 6.

Velocity ratio of swimming Tetrahymena sp. as a function of reduced clearance is traced by the heavy dashed line. Velocity in an ‘unbounded’ fluid U_{∞ swim} is reached at a clearance of S_w. The light dashed line represents corresponding data from sinking Tetrahymena which reaches an effective maximum velocity at a clearance of Se. The symbol 8 represents the radius of the region of validity of the quasi-steady Stokes equation or the depth of penetration of the vorticity which is dealt with in the Discussion. All variances are standard deviations of the sample.

Fig. 7.

Velocity ratio of swimming P. caudatum as a function of reduced clearance. The pattern of this figure is the same as that for Fig. 5.

CONCLUSIONS

Equation (3) may be considered adequate for determining the value of 5 safe from wall drag on the body of a sinking ciliate. If there is error in the estimate it is in the desirable direction of overrating, with a maximum error of about 22 % (see Fig. 5). The difference between the plots for the sphere model and for the prolate spheroid model, together with the dependence of both (3) and (4) on sphericity and axisymmetry, leads to the conclusion that this error is a function of geometry. That is to say, given the shape of Tetrahymena, essentially an intersection of a paraboloid and ellipsoid of revolution (Winet, 1969), and the shape of Paramecium, being non-axisymmetric because of the large oral groove, one should not expect to obtain velocity measurements in agreement with those obtained using a model based upon viscous stresses over a spheroid. However, the measure of S_e as the distance at which wall drag is significant is the concern here, and the errors described above are important only to the extent that they determine S_e by indicating the point at which the wall can ‘discriminate’ departures of the sinking bodies from the spheroid shape. The clearest indication of such a point appears in Fig. 5 where the measured values for P. multimicronucleatum deviate markedly from those for the model and indicate that shape ‘discrimination’ can occur up to about U/U_∞ = 0·92. This velocity ratio corresponds to a reduced clearance of 100 R.

Accordingly, S_e= 100R which in terms of a yields the following: Tetrahymena, 108a; P. caudatum, 127a; P. multimicronucleatum, 132a.

In contrast, when these ciliates are swimming the wall cannot ‘discriminate’ them until it is within ∼ 4a of their body surfaces. Such values of S_w when compared with S_e are consistent with the qualitative relationship between the velocity profiles described in Fig. 1. They are also summarized in Table 4 in the form of a passive/propulsive ratio from which it is evident that the height of S_e for these ciliates is 27–198 times as high as that of S_w. The specific profiles for each ciliate are approximated by their velocity-ratio graphs as presented in Figs. 6-8. The comparison may best be made by rotating the figures 90 ° counter-clockwise.

Table 4.

Comparison of velocity profiles or radii spheres of viscous influence (S) for sinking and for swimming ciliates

The most obvious conclusion to be drawn from the U/U_∞ data is that formalin-fixed specimens of the tested ciliates do not have the same value of ρ as T. pyriformis. Either these organisms are inherently less dense than their model or they were rendered so by the formalin. Although both mechanisms may be valid, the similarity of the Paramecium plots and their divergence from the Tetrahymena plot point to an inherent density difference.

DISCUSSION

The values of S_e and S_w obtained above are useful as ‘boundary conditions’ for experimental work with moving micro-organisms. But the question arises as to their utility for the hydromechanical models being applied to these experiments. This question is most pertinent because the present model, the Stokes equations, is valid only within a certain distance of the moving body, and data on fluid flow outside this distance will not conform to the model. Hence, there is just as much need to establish a ‘safe’ sphere of ‘influence’ for the theoretical model as there is for the experimental objects. The analogy between validity limits on theoretical and experimental results is worth stressing as it makes the former seem more ‘real’ and, hence, easier to accept as a model for observations. Actually, solutions to Stokes equations in their quasi-steady form (for oscillating bodies translating at steady velocities) for non-rotating waving bodies (Gray & Hancock, 1955) and for rotating waving bodies (Chwang & Wu, 1971) have already generated formulas which agree satisfactorily with observations (Gray & Hancock, 1955; Gray, 1962; Chwang, Wu & Winet, 1972; Chwang, Winet & Wu, 1973).

Hence, just as it is necessary for the purpose of gathering data to know when the assumption of ‘no wall effect’ is invalidated, it is similarly important for the purpose of matching data with an approximate model to know where the two are expected to diverge ; e.g. it will be noted that (3) diverges from the data at reduced clearance values below 10 (see Figs. 4, 5). This result is expected because (3) is a polynomial approximation which begins to oscillate at low values of the independent variable. For a body in steady motion (a sinking cell at terminal velocity) the critical distance is termed the Stokes radius (R_s) and the corresponding distance for a body in quasi-steady motion is the ‘depth of penetration of the vorticity’ (δ). Both R_s and δ are mathematical quantities not intended for expression in physical terms. Nevertheless, some physical concept of these quantities is necessary for the comparison to be made below; and, accordingly, they may be represented by the models shown in Figs. 9, 10. In addition, an estimate of these critical distances may be calculated from

Fig. 8.

Velocity ratio of swimming P. multimicronucleatum as a function of reduced clearance. The pattern of this figure is the same as that for Fig. 5.

Fig. 9.

A physical conception of the Stokes radius (R_s). Stokes equations for steady motion of a body are valid out to the point where r = R_s. Beyond this point inertial and convective stresses are so large in comparison with diffusion and viscous stresses that the Oseen correction must be introduced to the Stokes equations making them much more difficult to solve. The question taken up in the Discussion is:’Is S_e of the same order as R_s?’.

Fig. 10.

https://doi.org/10.1016/0021-9290(73)90082-1

A physical conception of the depth of penetration of the vorticity (δ). Stokes equations for quasi-steady motion of a body are valid out to the distance δ_r which is greater than δ by one minor semi-axis of the cell. It has been shown (see Taylor, 1952) that micro-organism propulsion is generated by a viscous-stress-induced rotation of fluid elements surrounding an oscillating body. In effect, a torus of fluid develops around each infinitesimal body segment which rotates about its material axis such that the torus surface in contact with the body thrusts it forward in the direction opposite $\vec{d F}$ ⁠. This thrust can be shown mathematically to be due entirely to the cross product of the vorticity $\vec{ω}$ or rotation tendency (actually equal to twice the angular velocity of the fluid elements on the axis just described) and the gradient $\vec{V}$ along which such vorticity diffuses outward from the body. Beyond δ, as in the case for R_s, the uncorrected Stokes equations are not valid. The question taken up in the Discussion is :’Is S_w of the same order as δ?’.

(Rosenhead, 1963) where R_e= U_∞R/v, v is the kinematic viscosity,

R_eω= ω/vk² is the oscillatory Reynolds Number, ω is the frequency of ciliary beat (obtained from Machemer, 1972, and Preston, 1972), k = 2π/λ, and λ is the metachronal wavelength (obtained from Machemer, 1972, and Parducz, 1966).

It will be noted that both R_s and δ are functions of R_e while (3) and (4) within the confines of a ‘low R_e’, are not. Yet, when wall drag slows a cell it must be altering body R_e (see Table 1). Consequently, R_s and δ may be considered as functions of wall effect (i.e. clearance) through resultant changes in body R_e. Furthermore, since it is easier to perform experiments like those reported here than to obtain some of the parameters in (5) and (6), it is a clear advantage to be able to estimate R_s with S_e and δ with S_w. In effect, such a technique would match the models in Figs. 9, 10 with those in Fig. 1.

Toward this end, calculated values for R_s are plotted in Figs. 3–5, those for δ are plotted in Figs. 6–8, and a comparison of all four quantities is listed in Table 5.

Table 5.

Comparison of S radii with region of validity of steady Stokes equations region (radius is R_s) and region of validity of quasi-steady Stokes equations (region radius is δ)

Examination of corresponding values of R_s and S_e reveals a poor match for Tetrahymena but a not unreasonable fit for Paramecium, given the asymptotic nature of the steady-motion Stokes equations. The results for swimming cells are much more encouraging, with relatively small difference between S_w and δ for each ciliate. Accordingly, S_w is proposed as a useful measure of δ.

Meanings of Symbols

a
minor semi-axis of organism (body radius)

c
half length of organism (major semi-axis)

F_sb
Stokes drag on a body in a bounded fluid

F_s∞
Stokes drag on a body in an unbounded fluid

R
equivalent sphere radius

R_e
translatory Reynolds Number

R_eω
oscillatory Reynolds Number

R_s
Stokes radius

R_o
cylinder bore radius

r
sphere radius

S
radius of the sphere of viscous influence (non-specific)

S_e
measured radius of sphere of viscous influence for a self-propelled body

S_w
measured radius of sphere of viscous influence for a passively moving body

U
velocity of the moving body

U_∞
velocity of a body in an unbounded fluid

U_{∞ eat}
velocity of a body in an unbounded fluid calculated from sedimentation equation (2)

U_{∞ int}
velocity of a body in an unbounded fluid calculated from (3)

U_{∞ meas}
velocity of a body in an unbounded fluid calculated from (1)

δ
depth of penetration of diffusing vorticity

δ_r
depth of penetration of the vorticity plus one minor semi-axis of the cell

λ
metachronal wavelength

v
kinematic viscosity

The author is indebted to Mr Long Wa Lee and Mr Stuart R. Keller for their assistance in performing the experiments. This investigation was supported by Grant GK31161X from the National Science Foundation and Contract N00014-67-A-0094-0012 from the Office of Naval Research.

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