Interaction of two swimming Paramecia

It is important to understand and predict the flow of suspensions of micro-organisms that appear in massive plankton blooms in the ocean, harmful red tides in coastal regions and are used in bioreactors. The size of individual micro-organisms is often much smaller than that of the flow field of interest, so the suspension is modelled as a continuum in which the variables are volume-averaged quantities(Fasham et al., 1990; Pedley and Kessler, 1992; Metcalfe et al., 2004). Continuum models for suspensions of swimming micro-organisms have also been proposed for the analysis of phenomena such as bioconvection (e.g. Childress et al., 1975; Pedley and Kessler, 1990; Hillesdon et al., 1995; Bees and Hill, 1998; Metcalfe and Pedley, 2001). However, the continuum models proposed so far are restricted to dilute suspensions, in which cell-cell interactions are negligible. If one wishes to consider larger cell concentrations, it will be necessary to consider the interactions between micro-organisms. Then, the translational-rotational velocity of the micro-organisms, the particle stress tensor and the diffusion tensor in the continuum model will need to be refined.

To understand the interactions between micro-organisms, it is first necessary to clarify two-cell interactions. Thus, we investigated the interaction between two model micro-organisms analytically(Ishikawa et al., 2006),assuming that the cell-cell interaction is purely hydrodynamic; no biological reactions were considered. In practice, however, it is to be expected that in the presence of a nearby micro-organism, a given micro-organism would not behave as if it were alone. A micro-organism may consider reproducing sexually or attempting to consume (or avoid being consumed by) its neighbour. It may also move away from it, because of the increased competition for food. Although two-cell interactions are important when considering the interactions between many cells, it is not at all clear how cells behave when they are in close contact. Also, cell-cell interactions were not modelled precisely in any previous analytical studies dealing with a nondilute suspension of micro-organisms (Guell et al.,1988; Ramia et al.,1993; Nasseri and Phan-Thien,1997; Lega and Passot,2003; Jiang et al.,2002). In this study, we clarify how two micro-organisms interact when they come close to each other, both experimentally and numerically.

Paramecium caudatum was used in this study, because the behavior of individual cells of this organism is well understood. Naitoh and Sugino investigated two types of biological reactions of a solitary Paramecium cell to mechanical stimulations(Naitoh and Sugino, 1984). (i)Avoiding reactions occur when a cell bumps against a solid object with its anterior end. The cell swims backward first, gyrates about its posterior end,and then resumes normal forward locomotion. (ii) Escape reactions occur when the cell's posterior end is mechanically agitated. The cell increases its forward swimming velocity for a moment, then resumes normal forward locomotion. The change in the swimming motion is regulated by changes in membrane potential, because Paramecium cells, like other monads, have no nerves for transmitting stimulative information nor synapses to determine transmission direction. Machemer clarified the frequency and directional responses of cilia to changes in membrane potential in Parameciumcells (Machemer, 1974). There are many Ca²⁺ channels in the anterior end, whereas there are many K⁺ channels in the posterior end. The locality of the ion channels is the essential mechanism controlling Paramecium's biological reaction (Naitoh and Sugino,1984). In reality, the range of micro-organism lengths is large,and they alter their behaviour according to many environmental parameters. The variety of shapes both within and between species is also vast(Brennen and Winet, 1977). Many types of ciliate, however, show a similar biological reaction to that of Paramecium cells, because they also have Ca²⁺ channels in the anterior end and K⁺ channels in the posterior end. We focused on Paramecium cells in the present study, but we expect that our results will be applicable to other ciliates and micro-organisms.

The interaction between two P. caudatum cells was investigated both experimentally and numerically. Cell motion was restricted between flat plates with a gap of about 70 μm. We observed avoiding and escape reactions as well as hydrodynamic interactions. To conclude that the interaction is purely hydrodynamic, it is necessary to simulate the cells' motion numerically without any biological response. Thus, the P. caudatum cell was modelled as a rigid spheroid with prescribed surface tangential velocities measured by a particle image velocimetry (PIV) technique. The model is referred to as a `squirmer', and is explained briefly in Materials and methods(for details, see Ishikawa et al.,2006). The interaction between two squirmers was calculated by a boundary element method, and the results were compared with those of the experiments. The authors have used a squirmer model for other studies(Ishikawa et al., 2006; Pedley and Ishikawa, 2004) (J. Ishikawa and T. J. Pedley, manuscript submitted for publication), so a comparison was also made with the earlier data to check the reliability of the squirmer model.

Cultures of the unicellular freshwater ciliate Paramecium caudatumEhrenberg were used for the experiments. They were grown by inoculating 2 ml of a logarithmic-phase culture into 250 ml of culture fluid. The culture fluid was made by boiling demineralised water with 2 g of straw, then adding 0.2 g of yeast powder. It was kept at 20°C for 2 weeks before inoculation. Measurements were performed 2 weeks after inoculation of the culture at 19-21°C. Microscopic observation showed that the body length of an individual cell was in the range of approximately 200-250 μm and the width approximately 40-50 μm. The swimming speed of an individual cell was approximately 1 mm s^-1. The swimming motion of P. caudatumin a free space is not straight, but forms a left spiral. The pitch of the spiral was approximately 2 mm and the width was 0.4 mm.

The experimental setup was designed to measure the displacements of cells in a still fluid between flat plates (Fig. 1). The motion of cells was restricted to two-dimensions for the following two reasons: (i) the orientation vector of the cell and the distance between two surfaces of cells were easy to measure without a considerable error, and (ii) biological reactions were observed much more frequently in two-dimensional space, because cells experienced strong collisions when there was no height variation. The latter reason is important in order to analyse a large number of biological reactions. The experimental setup consisted of a digital video (DV) camera with a 24× macrolens, light sources, and inner and outer dishes. The test fluid was placed between the top of the outer dish and the bottom of the inner dish. The gap between the two dishes was about 70μm, so that cells could not overlap three-dimensionally. The test fluid was same as the culture fluid, but the volume fraction of cells was adjusted to within about 0.5-1% so that three-cell interactions rarely occurred. Cell movements were recorded by the DV camera over a 10-20 min period, and we did not observe any decrease in the cells' swimming velocity or aggregation due to the cell chemotaxis during the measurements.

Sample sequences of the interaction between two swimming P. caudatum observed in the experiment are shown in Fig. 2. The time interval for each sequence is Δt=1/3s, and the background is subtracted from the figure by image processing. We observed such interactions several times per 10-20 min experiment. Since the intention was to concentrate on two-cell interactions, data were deleted if there was a third cell within 750 μm of one of the two interacting cells. The velocity disturbance caused by a force-free cell near a wall boundary decays as r^-2, where r is the distance from the cell, so the effect of the third cell on the swimming velocities of the two interacting cells is less than about 10% if it is farther than 750 μm away. (The derivation of velocity disturbance due to a force-free particle near a wall boundary is shown in Appendix A.)Microscopic observation showed that some solitary cells occasionally stopped swimming, even though there seemed to be no mechanical stimulation. They then swam backward, gyrated about their posterior ends, and finally resumed normal forward locomotion. Such a reaction is rare if one uses a culture 2 weeks after the inoculation, but could not be eliminated completely. Since we intended to measure two-cell interactions, data were deleted if one of two interacting cells showed this type of solitary reaction before the two cells approached within a distance of L, where L is the body length of the shorter cell. The interaction data was recorded regardless of whether it was a biological reaction or a hydrodynamic interaction, and the total number of data recorded in this study was 301.

The velocity field around a swimming P. caudatum cell between flat plates was measured using a PIV technique. A small amount of milk was added to the culture fluid as tracer particles, and the flow field was recorded using a CMOS camera (Basler A602f; 659 493 pixels, 100 frames s^-1; Basler AG, Ahrensburg, Germany) through a microscope. A sample original image is shown in Fig. 3. The velocity vector was calculated from two successive images using a MatPIV(Grue et al., 1999), in which spurious vectors were removed by a signal-to-noise ratio filter (cf. Keane and Adrian, 1992), a global histogram operator and a median filter. The spurious vectors were replaced by vectors linearly interpolated from neighbouring points. The velocity vector was calculated for a time series of images, and approximately 80 000 velocity vectors around a swimming cell were obtained. These vectors were averaged by assuming that the velocity field is axisymmetric and time-independent. The velocity field relative to the cell's swimming velocity vector is shown in Fig. 4. We see that the cell swims smoothly without generating a recirculation region.

The surface velocity of P. caudatum has to be measured for use as a boundary condition for the numerical simulation, as explained in below. We intended to measure the surface velocity of the cell by the PIV technique;however, the method showed considerable instability at short wavelengths, so we could not accurately measure the velocity vector using this method. Thus,we interpolated surface velocity as follows. A cell was assumed to be an extended ellipsoid with a minor axis of 0.36, where length was nondimensionalised relative to the major axis. The surface velocity vectors were interpolated linearly from those at 0.18 and 0.36 from the surface with the same angle from the orientation vector of the cell. We assumed that surface velocity was mainly tangential, because cells did not deform and fluid did not penetrate its surface at high velocities. The tangential surface velocity, u_s, interpolated from the PIV data is shown in Fig. 5. We see that u_s is faster near the anterior end than near the posterior end.

where coefficients c_i were determined by the method of least squares (c₁=1.707, c₂=0.2400, c₃=0.2472, c₄=0.1506, c₅=0.1154). u_s (Eqn 1) is also shown in Fig. 5. The high modes decay faster than the low modes as the distance from the cell increases; thus, the far-field velocity is governed by the lowest mode (cf. Ishikawa et al., 2006). Moreover, the role of high modes in the near-field is to generate fluctuations in velocity. Hence, the overall properties, such as the trajectories of a pair of cells, may be captured by the first few modes. Thus, we used up to the fifth mode in Eqn 1, which was used as a boundary condition for the cell surface (cf. Numerical methods, below).

The P. caudatum cell was modelled as a rigid spheroid with the surface tangential velocity measured by the PIV technique, referred to as a`squirmer'. The squirmer model was first proposed by Lighthill(Lighthill, 1952), and his analysis was then extended (Blake,1971a). The numerical methods used in this study are similar to those reported elsewhere (Ishikawa et al.,2006), so only a brief explanation is given here. A squirmer is assumed to be neutrally buoyant and torque-free. The Reynolds number based on the swimming speed and the radius of individuals is small, so that the flow field can be assumed to be a Stokes flow. Brownian motion is not taken into account, because a cell is too large for Brownian effects to be important. The spheroid's surface is assumed to move purely tangentially and this tangential motion is assumed to be axisymmetric and time-independent. The tangential surface velocity of a squirmer is given by Eqn 1. We performed a trial simulation for a solitary squirmer swimming in a still fluid with the boundary condition of Eqn 1. The results showed that the swimming velocity of the squirmer was 1.04, although it should be 1.0 because the surface velocity is non-dimensionalized by the swimming velocity. The error may arise from the u_s approximation, although it is very small.

where U_m and Ω_m are the translational and rotational velocities of squirmer m, respectively. x_m is the centre of squirmer m, and u_s,m is the squirming velocity of squirmer m, given by Eqn 1. In simulating hydrodynamic interactions between two squirmers, we assume that the surface velocity is independent of the distance between the cells. Thus, no biological reaction is modelled, and the interaction is purely hydrodynamic. Similarly to Ishikawa et al. (Ishikawa et al., 2006),we employed another boundary condition of constant swimming power; however,the difference in trajectories was very small (see Appendix B).

In the experiment, two cells swim between flat plates, and the motion of cells was restricted to a two-dimensional plane. In the numerical simulation,on the other hand, squirmers swim in an infinite fluid, but the centres and orientation vectors of two squirmers were initially set in the same plane. The motion of squirmers remained in the plane, even though the flow field is three-dimensional, because the surface squirming velocity is axisymmetric. The two flat plates were omitted in the simulation, because adding them does not qualitatively affect the interaction provided that the two squirmers remain in the same plane. The quantitative effect of wall boundaries on the swimming speed was also small (see Appendix C).

where α₁ is a dimensional coefficient, andα ₂ is a dimensionless coefficient. is the minimum separation between two surfaces, which was calculated by the iterative methods proposed(Claeys and Brady, 1993). The separation vector h connects the minimum separation points on the two surfaces. The coefficients used in this study were α₁=0.1 andα ₂=10³. The minimum separation obtained using these parameters was in the range 10^-3-10^-4. The effect of the repulsive force on the trajectories of cells was very small, because it acts only in the very near field and changes the distance between particles by only approximately 10^-4. This was confirmed numerically by comparing three trajectories, which are shown in Figs 9 and 10, 11, with and without a repulsive force.

Before analysing the results of two-cell interactions, we show some typical observation results in order to help readers understand the phenomena. Fig. 7A shows an avoiding reaction, in which arrows are drawn to schematically show cell motion. The time interval for each sequence is Δt=1/6s and the colour is reversed from the original figure (as shown in Fig. 2). We see that two cells collide with their anterior ends. They first swim backward, then gyrate about their posterior ends, then resume normal forward locomotion. We defined an avoiding reaction as one involving backward swimming during the interaction.

Fig. 7B shows an escape reaction, which occurs when the cell's posterior end is strongly agitated. We see that cell 2 increases its forward swimming velocity after the collision. The original movie was taken 30 frames s^-1, and the velocity vector of a cell was calculated by tracking the centre of the major axis of the cell in each frame. If a cell's swimming velocity after the collision became 1.3 times faster than the initial value, we classified the interaction as an escape reaction, where the initial value was defined when two cells' surfaces were at a distance of L before the collision.

When a cell-cell interaction did not satisfy the definition of an avoiding reaction (AR) nor that of an escape reaction (ER), it was classified as a hydrodynamic interaction (HI). Four kinds of typical HI results are shown in Fig. 8. When two cells are initially facing, they come close to each other at first, then they change their directions slightly in the near field, and finally move away from each other (see Fig. 8A). The change in direction is small in this case. When the initial angle between the orientation vectors of the two cells is less than about 3π/4, there are two main kinds of swimming motions. If one cell collides with the posterior end of the other cell, the two cells do not significantly change their swimming direction, as shown in Fig. 8B. On the other hand, if one cell collides with the anterior end of the other cell, the two cells tend to swim side by side at first, then move away from each other with an acute angle, as shown in Fig. 8C,D. The final angle in this case seems to be independent of the initial angle, which will be discussed in detail below.

All the interacting cells were classified as HI, AR or ER, and the results are shown in Table 1. The total number of interacting cells recorded in this study was 602 (the total number of cases was 301). It was found that 84.7% of cells interact hydrodynamically. The ratio of ER is slightly higher than that of AR.

Although we have assumed that the experimental data not satisfying the definition of AR nor ER are HI, there is no evidence that two cells do not actively change their swimming motions during the interaction. Thus, we also performed numerical simulations. The authors used a squirmer model for some previous studies (Ishikawa et al.,2006; Pedley and Ishikawa,2004) (T. Ishikawa and T. J. Pedley, manuscript submitted for publication), so a comparison with the earlier data was also made to check the reliability of the squirmer model.

We first show the interactions between two squirmers withθ _in≈0 with a small distance between the squirmers in the perpendicular direction to the orientation vectors, as shown in Fig. 9. The orientation vectors of the squirmers are shown as large arrows on the ellipsoids, and a thin solid line is added so that one can easily compare the angle between the two squirmers. It is found from the figure that the two squirmers come very close to each other, then change their orientation in the near field, and finally move away from each other. This tendency is similar to the experimental results shown in Fig. 8A.

Fig. 10 shows the interactions between two squirmers with θ_in≈2π/3, in which one squirmer collides with the posterior end of the other. We see that the two cells do not significantly change their swimming direction. This tendency is the same as the experimental results shown in Fig. 8B. If one squirmer collides with the anterior end of the other, on the other hand, the two cells tend to swim side by side at first, then move away from each other with an acute angle, as shown in Fig. 11. This tendency is again the same as the experimental results shown in Fig. 8C,D.

We can conclude from Table 1that the cell-cell interaction is mainly hydrodynamic and that the biological reaction accounts for a minority of incidents. In the present experiment, the motion of cells was restricted between flat plates. In a real cell suspension,however, cells often swim freely in threedimensional space. Since the effect of the walls on the cell behaviour is unclear, we performed a trial experiment in which cells swim three-dimensionally in a large container. In the trial experiment, we observed fewer biological reactions, because cells could avoid each other using three dimensions, and strong collisions occurred infrequently. We expect, therefore, that the conclusion obtained here is also valid for cell-cell interactions in a cell suspension.

In order to analyse the experimental results quantitatively, we defined three angles during cell-cell interactions (see Fig. 12). θ_inis the angle between e_1,in and e_2,in,where e_i,in is the orientation vector of cell iwhen the two cell surfaces are at a distance of L before the collision. θ_out is the angle between e_1,outand e_2,out, where e_i,out is the orientation vector of cell i when the two cell surfaces are at a distance of L after the collision. In order to describe the change in orientation of cell 2 relative to cell 1, a frame is fixed to cell 1 so that the frame rotates when cell 1 rotates, and dθ is defined as the change in the angle of cell 2 relative to this rotating frame. Data for AR and ER are classified into seven θ_in ranges as well as three contact positions, where a cell is divided into three equal-length sections; head,body and tail, respectively from the anterior end. The reaction rates for eachθ _in range and each contact position are shown in Fig. 13, where reaction rates are calculated by dividing the number of AR or ER for a certainθ _in range and a certain contact position by the total number of cells under the same conditions. In Fig. 13, head-tail, for instance, indicates that the collision occurs between the head of one cell and the tail of the other (see Fig. 14). We see that AR occurs mostly when θ_in2π/3 and under head-head or head-body conditions. AR rarely occurs under a head-tail condition. This is apparently because there are many Ca²⁺channels in the anterior end, as explained in the Introduction.

ER occurs only when one cell collides with the tail of the other, and the reaction rate is not sensitive to θ_in, as shown in Fig. 13. The dependence of contact position can be explained by the locality of K⁺ channels,as explained in the Introduction. The reaction rate of ER is higher than that of AR, so we can say that ER occurs more frequently in the cell-cell interaction. The temporal change of the dimension-free swimming velocity under ER was measured, and the results are shown in Fig. 15. Since the escaping velocity depends on how strongly the posterior end of the cell is agitated,the error bar becomes very long. We see from the figure that the escaping velocity is maximum at t=0.2-0.3 s, and its value is about 1.8 times the initial velocity.

When a cell-cell interaction does not satisfy the definition of AR or ER,it is classified as HI. The correlations between d and θ_infor HI for the three contact positions are shown in Fig. 16. We see that dθfor head-tail (white squares) is distributed near dθ=0, which means that the two cells do not change their orientations significantly after collision(cf. Fig. 8B). The cases for head-head and head-body, however, show a different tendency. Whenθ _in is smaller than approximately 3π/4, dθincreases almost linearly with θ_in. A broken line with slope one is drawn in Fig. 16 for comparison. We see that dθ is slightly larger than θ_in,which means that the two cells tend to swim side by side at first, then move away from each other with a small angle, as shown in Fig. 8C,D. Whenθ _in is larger than approximately 3π/4, dθ is distributed near dθ=0, which again means that the two cells do not change their orientations significantly (cf. Fig. 8A). Consequently a tendency of the hydrodynamic interaction between two cells clearly appears,and is dependent on θ_in and contact position.

In order to compare numerical and experimental results quantitatively, we performed simulations for various θ_in conditions and for two contact positions (head-head and head-tail). The correlations between dθ and θ_in obtained in the simulations are shown in Fig. 17. We see that dθ for head-tail (large white squares) is distributed near dθ=0,which is the same tendency as in the experiments. In the case of head-head(large white circles), dθ increases almost linearly withθ _in when θ_in is smaller than approximately 3π/4. This tendency is also the same as in the experiments. The final angleθ _out for these squirmers is about 0.4 rad; thus, the numerical results fit well with y=x+0.4, as shown in the figure. When θ_in is larger than approximately 3π/4,dθ is distributed near dθ=0, which is again the same as in the experiments. We can conclude, therefore, that the HI data in the experiments agree well with the numerical results and that the interaction is purely hydrodynamic. Moreover, we can say that a squirmer model is appropriate for expressing the motion of Paramecia, and hopefully for some other ciliates as well.

In most of the previous analytical studies on cell-cell interactions(Guell et al., 1988; Ramia et al., 1993; Nasseri and Phan-Thien, 1997; Jiang et al., 2002), two cells in close contact were not discussed. [Lega and Passot(Lega and Passot, 2003)included an ad hoc interactive force acting between cells, which in practice is unlikely to exist.] The present results show, however, that the near-field interaction dramatically changes the orientation of cells. Since orientation change affects the macroscopic properties of the suspension, such as the diffusivity, the near-field interaction has to be solved accurately. The present results also show that the near-field interaction can be accurately solved hydrodynamically by the boundary element method and lubrication theory.

List of symbols
 
• A_m

  surface area of squirmer m

 
• AR

  avoiding reaction

 
• e

  orientation vector of cell

 
• ER

  escape reaction

 
• F_rep

  non-hydrodynamic interparticle repulsive force

 
• h

  separation vector

 
• HI

  hydrodynamic interaction

 
• K

  Oseen tensor

 
• L

  length

 
• q

  single-layer potential (traction force on the inner surface minus that on the outer surface)

 
• U_m

  translational velocity of squirmer m

 
• u_s

  tangential surface velocity

 
• u_s,m

  squirming velocity of squirmer m

 
• x_m

  centre of squirmer m

 
• Ωm

  rotational velocity of squirmer m

 
• α1, α2

  dimensionless coefficients

 
• ϵ

  minimum separation between two surfaces

 
• θ

  angle from the orientation vector of the cell

In Appendix A, we show that the velocity disturbance due to a force-free particle decays as r^-2, even though a no-slip wall boundary exists near the particle.

where μ is the viscosity, r=x-x′, xis the position vector of the field point, and x′ is the point where the traction force is generated.

where R=x-X, and X is the position vector of the image point that is the mirror image of x′ about the wall boundary. The direction of subscript 3 is taken normal to the wall, and b is the minimum distance of x′ or X from the wall (for details, see fig. 1in Blake, 1971b). Eqn A2 indicates that the effect of the wall boundary is equivalent to inducing a point force, a dipole and a source doublet at the image point.

where K̂=K′-K, and F, L and S are, respectively, the monopole, the anti-symmetric dipole, and the symmetric dipole. Subscript α indicates real particles, and β indicates image particles. In the case of a force-free particle, F^α=F^β=0. Hence, the leading-order term in the right-hand side of Eqn A4 is r^-2 or R^-2. Therefore, the velocity disturbance due to a force-free particle decays as r^-2,even though a no-slip wall boundary exists near the particle.

In Appendix B, we show that the difference between two boundary conditions,constant surface velocity and constant swimming power, in the trajectories of two cells is small. A similar discussion has been published(Ishikawa et al., 2006).

where τ is the traction force on the surface.

Because log(ϵ^-1) is a very weak singularity,λchanges very slowly. Thus, the difference between the two boundary conditions in the trajectories of two cells is small.

There may be some other boundary conditions for a swimming cell. Short et al., for instance, assumed constant surface stress instead of constant surface velocity (Short et al., 2006). However, the effect of this boundary condition should also be small, because the lubrication force between two cells is again not sufficiently strong, even if ϵ becomes the length scale of molecules, as mentioned above.

In Appendix C, we show that the effect of a wall boundary on the translational velocity of a squirmer parallel to the wall is small if the distance between the two surfaces ϵ is larger than 10^-4 or 10^-5.

We derived the first-order solution for the lubrication flow between a spherical squirmer and a sphere with arbitrary radius, in which a flat plate corresponds to infinitely large radius(Ishikawa et al., 2006). Although the squirmer model used in this study is spheroidal, we exploit the lubrication theory for a spherical squirmer in the following discussion for simplicity. The lubrication force due to the translational motion parallel to the wall is proportional to δu log(ϵ^-1) to the leading order, where δu is the velocity difference between the two surfaces in the lubrication region (cf. Ishikawa et al., 2006).δ u can be rewritten asδ u=U+u_s|_lub,where U is the translational velocity of the squirmer parallel to the wall and u_s|_lub is the squirming velocity in the lubrication region. When the squirmer touches the wall, i.e.ϵ→0, δu log(ϵ^-1) diverges to infinity unless δu=0; thus, the squirmer swims with a velocity of-u_s|_lub. When the surface squirming velocity is given by Eqn 1 and the orientation vector of the squirmer is parallel to the wall, the swimming velocity of the squirmer attached to the wall is about 1.5. Therefore, the wall boundary increases the swimming velocity of the squirmer by up to 50% when ϵ=0.

Next, we show how δu decays withϵ. The lubrication force is generated by the relative motion between two surfaces in the lubrication region. This force has to be canceled by the viscous drag force acting outside the lubrication region, because the particle is force-free. Since the viscous drag force is not sensitive toϵ, we can assume that the lubrication force is also unaffected by ϵ to the leading order, i.e.δ u log(ϵ^-1)=constant. Thus, δudecays as 1/log(ϵ^-1). As mentioned in Appendix B,log(ϵ^-1) is a very weak singularity [-log(10^-4)=9.2 and -log(10^-5)=11.5, for instance]. We can conclude that the effect of a wall boundary on the translational velocity of a squirmer parallel to the wall is small if the distance between the two surfaces ϵ is larger than 10^-4 or 10^-5. In the present experiment, ϵ is about 0.4, which is much larger than 10^-4.

The authors are grateful for helpful discussions with Prof. T. J. Pedley of DAMTP, University of Cambridge. The authors also appreciate Prof. Akitoshi Itoh of Tokyo Denki University for his kind help in dealing with the micro-organisms.