## SUMMARY

The interaction between two swimming *Paramecium caudatum* was
investigated experimentally. Cell motion was restricted between flat plates,
and avoiding and escape reactions were observed, as well as hydrodynamic
interactions. The results showed that changes in direction between two
swimming cells were induced mainly by hydrodynamic forces and that the
biological reaction was a minor factor. Numerical simulations were also
performed using a boundary element method. *P. caudatum* was modelled
as a rigid spheroid with surface tangential velocity measured by a particle
image velocimetry (PIV) technique. Hydrodynamic interactions observed in the
experiment agreed well with the numerical simulations, so we can conclude that
the present cell model is appropriate for describing the motion of *P.
caudatum*.

## Introduction

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 *Paramecium*
cells (Machemer, 1974). There
are many Ca^{2+} 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 Ca^{2+} 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.

## Materials and methods

### Materials and experimental setup

Cultures of the unicellular freshwater ciliate *Paramecium caudatum*
Ehrenberg 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. caudatum*
in 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 70μ m, 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 Δ*t*=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.

### Surface velocity of P. caudatum

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.

The experimental results of *u*_{s} were approximated by:
(1)

where coefficients *c*_{i} were determined by the method of
least squares (*c*_{1}=1.707, *c*_{2}=0.2400,
*c*_{3}=0.2472, *c*_{4}=0.1506,
*c*_{5}=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).

### Numerical methods

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.

When there are two squirmers in an infinite fluid otherwise at rest, the Stokes flow field external to the squirmers can be given in an integral form as: (2)

where *A*_{m} is the surface of squirmer *m* and
**K** is the Oseen tensor. The single-layer potential, **q**, is the
subtraction of the traction force on the inner surface from that on the outer
surface. The boundary condition is given by:
(3)

where **U _{m}** and Ω

_{m}are the translational and rotational velocities of squirmer

*m*, respectively.

**x**is the centre of squirmer

_{m}*m*, and

**u**

_{s}_{,}

**is the squirming velocity of squirmer**

_{m}*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).

The boundary element method was employed to discretize Eqn 2. The
computational mesh used in this study is shown in
Fig. 6. A maximum of 590
triangle elements per particle were generated, and the mesh was finer in the
near-contact region. Time-marching was performed by 4th-order Runge-Kutta
schemes. A non-hydrodynamic interparticle repulsive force,
**F _{rep}**, was added to the system in order to avoid the
prohibitively small time step needed to overcome the problem of overlapping
particles. We followed published methods
(Brady and Bossis, 1985) (T.
Ishikawa and T. J. Pedley, manuscript submitted for publication), and used the
following function:
(4)

where α_{1} 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 α_{1}=0.1 andα
_{2}=10^{3}. 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.

## Results

### Experimental results

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 Δ*t*=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.

### Numerical results

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.

## Discussion

We can conclude from Table 1 that 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_{in}
is the angle between **e**_{1,in} and **e**_{2,in},
where **e**_{i,in} is the orientation vector of cell *i*
when the two cell surfaces are at a distance of *L* before the
collision. θ_{out} is the angle between **e**_{1,out}
and **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_{in}2π/3
and under head-head or head-body conditions. AR rarely occurs under a
head-tail condition. This is apparently because there are many Ca^{2+}
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_{in}
for HI for the three contact positions are shown in
Fig. 16. We see that dθ
for 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
*d*θ 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

## Appendix A

### Velocity disturbance due to a force-free particle near a wall boundary

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.

For an unbounded fluid, the Oseen tensor in Eqn 2 is given as: (A1)

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

In order to account for a no-slip wall boundary, we exploit an image system for the Oseen tensor. The image system was derived by Blake (Blake, 1971b), and a general outline of the method is included in standard texts (e.g. Kim and Karrila, 1992). If there is a wall boundary, the Oseen tensor needs to be rewritten as: (A2)

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. 1
in 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.

When there are *N* particles and a wall boundary, the Stokes flow
field around the particles can be given in an integral form as (similar to Eqn
2):
(A3)

The right-hand side of Eqn A3 can be expanded in moments about the centre of each particle and each image particle (see Durlofsky et al., 1987): (A4)

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.

## Appendix B

### A boundary condition of constant swimming power

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).

Throughout this paper, the surface squirming velocity was assumed not to
change during the interactions. Real micro-organisms, however, may well change
their swimming motion in the presence of a nearby micro-organism. Of course,
we did not model a cell's biological response to other micro-organisms, but we
can apply a different primitive boundary condition for the squirmers to see if
the effect is significant. In this boundary condition, the surface squirming
velocity is defined as λ**u _{s}**. Here,

**u**is the squirming velocity for a solitary squirmer andλ is a scalar factor that is chosen to realize constant swimming power, equal to the rate of viscous energy dissipation throughout an interaction. The swimming power consumed by a squirmer is defined as: (B1)

_{s}where τ is the traction force on the surface.

We checked the relative translational-rotational velocities under the
constant-swimming-power condition. However, the effect of changing the
boundary condition was very small. This can be explained as follows. The
surface velocity is proportional to λ, and the lubrication force is
proportional to log(ϵ^{-1}) and λ, where ϵ is the gap
distance between two surfaces [for a detailed discussion about the lubrication
force, see Ishikawa et al. (Ishikawa et
al., 2006)]. The power is the product of these two quantities in
the near-field, thus proportional to log(ϵ^{-1}) andλ
^{2}. In order to maintain constant power, λ should
therefore vary as:
(B2)

Because log(ϵ^{-1}) is a very weak singularity,λ
changes 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.

## Appendix C

### Effect of a wall boundary on the swimming velocity of a squirmer

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 δ*u* log(ϵ^{-1}) to the
leading order, where δ*u* 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, δ*u* log(ϵ^{-1}) diverges to infinity
unless δ*u*=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 δ*u* 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, δ*u*
decays 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}.

## ACKNOWLEDGEMENTS

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.

- © The Company of Biologists Limited 2006