PHYSICAL REVIEW E 107, 014602 (2023)

Quasi-two-dimensional bacterial swimming around pillars: Enhanced trapping efficiency
and curvature dependence
Yuki Takaha

1,2,*

and Daiki Nishiguchi

1,3,†

1

2

Department of Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan
Department of Basic Science, The University of Tokyo, 3-8-1 Komaba, Tokyo 153-8902, Japan
3
PRESTO, Japan Science and Technology Agency, 4-1-8 Honcho, Saitama 332-0012, Japan(Received 29 March 2022; accepted 16 November 2022; published 9 January 2023)

Microswimmers exhibit more diverse behavior in quasi-two dimensions than in three dimensions. Suchbehavior remains elusive due to the analytical difficulty of dealing with two parallel solid boundaries. The
existence of additional obstacles in quasi-two dimensional systems further complicates the analysis. Combining
experiments and hydrodynamic simulations, we investigate how the spatial dimension affects the interactions
between microswimmers and obstacles. We fabricated microscopic pillars in quasi-two dimensions by etching
glass coverslips and observed bacterial swimming among the pillars. Bacteria got trapped around the circularpillars and the trapping efficiency increased as the quasi-two-dimensionality was increased or as the curvatureof the pillars was decreased. Numerical simulations of the simplest situation of a confined squirmer showedanomalous increase of hydrodynamic attractions, establishing that the enhanced interaction is a universal
property of quasi-two-dimensional microhydrodynamics. We also demonstrated that the local curvature of the
obstacle controls the trapping efficiency by experiments with elliptic pillars.
DOI: 10.1103/PhysRevE.107.014602
I. INTRODUCTION

Microswimmers such as bacteria, algae, and spermatozoa
live surrounded by boundaries in natural, clinical, and experimental environments. Bacteria often colonize epithelial
surfaces of host organisms, and spermatozoa find their way
to an egg inside narrow channels [1]. The presence of such
surfaces gives rise to fascinating behavior of microswimmers
both at the single cell level and at the collective level. Swimming bacteria are attracted to a flat wall, which has been
discussed based on the hydrodynamic interaction between
the bacteria and the wall by considering mirror images of
singularities of the Stokes flow [2,3] and/or on the interplay
between self-propulsion and steric interactions [4,5]. Such
attractions are of crucial importance as an initial process of
biofilm formations. When a dense suspension of bacteria isconfined in circular geometries [6,7] or flown into periodic
pillar arrays [8,9], these boundaries work to rectify turbulent
bacterial collective motion into ordered vortices. Several attempts have been made recently to understand the mechanismof such emergent collective behavior in bacterial turbulencefrom the microscopic hydrodynamic point of view not only inthe bulk [10] but also in the presence of boundaries [9].
When microswimmers are more strongly confined in aquasi-two-dimensional (quasi-2D) fluid layer between two
no-slip plane walls, hydrodynamic flows around microswimmers behave differently from the bulk three-dimensional (3D)flow and thus peculiar swimming behaviors are observed.
Bacteria confined in quasi-2D exhibit collective motion
with long-range nematic order [11], which is explained*†

yukitakaha@g.ecc.u-tokyo.ac.jp
nishiguchi@noneq.phys.s.u-tokyo.ac.jp

2470-0045/2023/107(1)/014602(15)by suppressed hydrodynamic instability and/or stabilizing
nonequilibrium force arising from geometrical confinement[12]. Flow fields created by eukaryotic swimmers such as
puller-type swimming microalgae and pusher-type dinoflagellates were also experimentally measured recently [13]. It was
turned out that they have differences not only in the signs but
also in the symmetries of their flows, which suggested that
there remains diverse phenomena unexplored in quasi-2D active hydrodynamics. However, rigorous theoretical analysis ofquasi-2D hydrodynamics between two no-slip walls has been
precluded due to the necessity of considering infinite seriesof mirror images of higher-order singularities such as force
multipoles and source multipoles [14], which have led to approximated or empirical calculations [15–20] and numericalapproaches [21]. Therefore, unlike the mirror image technique
for the 3D Stokes equations or conformal mapping for 2D
ideal fluid, no easy-to-use tools for analytically calculating theinteractions among multiple objects such as microswimmers
and obstacles in quasi-2D have been established so far. As
these kinds of situations are frequent both in microfluidic
devices and in nature, more experimental inputs and explorations are required for better understanding of the diverse lifeof microswimmers in quasi-2D.
In this paper, we address the question of how pusher-typebacteria in quasi-2D interact with another boundary perpendicular to the two confining no-slip walls. We realized thissituation by fabricating microscopic pillars between two noslip planes by a combination of photolithography and glass
etching technique. To explore how the dimensionality affects
the interactions between bacteria and no-slip pillar walls, we
varied the gap width H from 11.1 µm to as small as 1.9 µm,
which is about twice the diameter of bacterial bodies ∼1 µm.
We found that bacteria get more strongly trapped close to the014602-1©2023 American Physical Society

YUKI TAKAHA AND DAIKI NISHIGUCHIPHYSICAL REVIEW E 107, 014602 (2023)

FIG. 1. (a) Schematic figure of our experiment. The figure is not to scale. (b) 3D surface plot of a part of the glass substrate reconstructedfrom confocal images. The height of the pillars is H = 6.9 µm. The radii of the pillars are 38 µm at the bottom and 33 µm at the top. (c) An
experimental snapshot for H = 6.9 µm (see Supplemental Movies 2, 5, and 7 [24]). Bacteria were detected as bright spots. Yellow circles aredetected pillars. Red trajectories are some 10 s trajectories of the bacteria. The color changes red to white with time. (d) xz slice reconstructedfrom the confocal images of the pillars with R = 14.3 µm and H = 6.9 µm. The black regions correspond to the pillars and some shadowsabove the pillars due to optical deflection.

pillar surfaces as the system became highly quasi-2D. This
was numerically supported by the increase of hydrodynamic
attractive force when the gap widths were decreased. In addition, by varying the diameter of the microscopic pillars,
we found that the local curvature of the surface controls thetrapping efficiency, which was directly demonstrated by anexperiment with elliptic pillars. Our findings highlight thepeculiar behavior and possible control of quasi-2D ways of
microswimmers’ life.
II. EXPERIMENTS
A. Method

To confine bacteria in a region as small as their bodylength ∼2 µm, microscopic pillars were fabricated on a glass
substrate by combining photolithography and glass etching
technique (see Appendix A 2). We arranged pillar arrays withdifferent diameters on the same substrate [Fig. 1(c)], whichenabled us to simultaneously obtain curvature dependenceof the trapping efficiency as well as large statistics for eachdiameter. In addition, constructing multiple pillars with thesame diameter in a single field of view and averaging overthese pillars enabled us to obtain the density distributions from
dilute bacterial populations. The use of dilute bacterial suspensions also reduced unwanted collisions between bacteria
which may result in forced escapes from the trapped states orsudden turns in trajectories, and thus we could focus on the
single bacterium behavior. Since the substrates were made byetching the glass coverslips in order from the surface, the pil-lars had larger diameters at the base than at the top [Figs. 1(b)and 1(d)]. The glass substrate was washed by sonication in
20w% aqueous soulution of alkaline detergent (Contaminon,
FUJIFILM, Wako Pure Chemical Corporation) and ethanol,and then rinsed with MilliQ pure water before our experiment.
We used a nontumbling chemotactic mutant strain ofEscherichia coli (RP4979) with a plasmid pZA3R-EYFP expressing yellow fluorescent protein. E. coli were grown to
the mid-exponential phase in Tryptone Broth (TB, 1 wt%
tryptone and 0.5 wt% NaCl) with a selective antibiotic (chloramphenicol 33 µg/ml) at 30◦ C and then this culture wasdiluted 100-fold in 10 ml of fresh TB with the antibiotic at
30◦ C. A droplet of the bacterial suspension with L-serine wasconfined between the glass substrate with pillars and a glass
cover slip which had been immersed in BSA solution for atleast 15 min to prevent adhesion of bacteria to the glasses
beforehand. The addition of L-serine allowed bacteria to swim
actively for several minutes even in a confined environment
with little oxygen [22,23].
We captured 3-min movies at 10 frames per second at
22◦ C by an inverted widefield fluorescence microscope (Leica, DMI6000B) equipped with an objective lens (Leica, PL
APO 10×, NA 0.40 and NA 0.45 Ph for the cylindricalpillars and for the elliptic pillars, respectively), a mercury
lamp, and a CMOS camera (Hamamatsu, ORCA-Flash4.0
V3, 16bit, 2048 × 2048 pixels  1.33 × 1.33 mm2 with the
10× objectives) (see Supplemental Movies 1–6 [24]). Since
this observation is at a lower magnification than, for example, those made in previous studies [5,25], it is difficult to014602-2

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FIG. 2. Densities of cells around pillars with (a) H = 1.9 µm,(b) H = 6.9 µm, and (c) H = 11.1 µm. Bacteria were detected evenat h < 0 since the actual cross-sectional radius of the pillars varies
depending on the z position and can become smaller than R estimatedfrom 2D observations [see Fig. 3(e) and Appendix A 5]. (d) Ratio ofthe densities in the nearby region to that in the far region. Error barsof the density ratio: standard errors. Error bars of the radii R: standard
deviations.

detect the exact orientation of the bacteria. Instead, the lowmagnification observation allows us more reliable statisticalanalysis, since the motion of many bacteria can be observed
simultaneously. We detected the trajectories of the bacteria byusing an ImageJ plugin TrackMate [26] (see Supplemental
Movie 7 [24]). The position and radius R of each circularpillar were detected from the captured bright field images byusing Hough transform. The obtained radius R was almost thesame as the radius Rhalf of the cross-section at the mid-heightof the pillar that was evaluated by 3D imaging with confocal
microscopy (see Appendix A 5). The detected pillars are illustrated in Fig. 1(c) as circles with the radii R. We captured
movies of experiments on three substrates: each substrate has
pillars with the height of H = 1.9 µm, 6.9 µm, or 11.1 µm.
The heights of the pillars H were carefully evaluated from
confocal observations (see Appendix A 5 for more details of
the definitions and evaluations of the geometries of pillars).
B. Density distributions

To evaluate trapping efficiencies of the pillars, we firstmeasured the density distributions ρ as a function of the
distance h from the surface of each pillar [Figs. 1(a) and
2(a)–2(c)]. As the average density inevitably differs from one
experiment to another due to the confinement processes andthe differences in the pillar heights H, the density ρ was normalized by the average density ρ∞ for each experiment. The
average density ρ∞ was estimated from the region 10.0 µm <h < 12.8 µm where the density ρ was well converged but
still the distance h was smaller than the minimal distance d0
between the surfaces of adjacent pillars. The obtained normalized density distributions signify that the increase of thedensity depend on the radius of the pillars and that the trapping

effect only persists up to around h = 6 µm with all H. We note
that, although neighboring pillars are reported to influence
the trapping of bacteria in the case of very small interpillar
distance [27], such influence is negligible in our case. In aprevious experiment [27], the pillar height H = 30 µm wasmuch larger than the interpillar distance d0 = 10 µm, allowing
the flow created by a bacterium on a pillar to reach anotherpillar. However, in our experiments the pillar height H isalways smaller than the interpillar distance d0 = 15 µm in all
the setups, which makes the flow decays fast enough to neglect
the adjacent pillars in terms of hydrodynamic interactions.
Further quantification of the trapping efficiency was performed by looking at the density ratios of the nearby regionh  6.4 µm and the far region 6.4 µm < h < 10.0 µm. This
choice was made because the typical length of the trap was
about h = 6 µm regardless of the heights of the pillars (seeAppendix A 7). As shown in Fig. 2(d), the pillars with larger
R have higher trapping efficiency than those with smaller
R. In previous works, such a tendency for curvature of obstacles was theoretically predicted for the trapping around athree-dimensional sphere by an analytical calculation basedon the far-field approximation and mirror image technique[28] and was experimentally observed for the trapping of
autocatalytic rods around solid spheres [29] and for bacterial trapping around pillars that were 7–40 times higher thanthose in our experiment [25]. Importantly, in our experiments,
while no difference in the trapping efficiency was observed for
H = 6.9 µm and 11.1 µm, about up to five times more bacteria
were trapped in the highly quasi-2D case of H = 1.9 µm.
In other words, the trapping efficiencies of the pillars weresignificantly enhanced in the highly quasi-2D setup than in
the relatively 3D setups, which signifies the peculiarity ofquasi-2D systems.
C. Persistent length

To evaluate the trapping efficiency also at the single celllevel, we calculated persistent time and persistent length. For
simplicity, we calculated the persistent time τR,H by taking
the average of the trap times at each R and H, but it should
be noted that the distribution of the trap times is skewed in
the long-time direction [Figs. 3(a)–3(c)]. Since the bacterial
velocities depend on the heights H of the confinement and canalso differ from one preparation of bacterial culture to another,comparing the values of τR,H estimated from independent experiments requires careful treatment. Therefore, to cancel out
the dependence on the velocity, we multiplied the persistenttime τR,H by the average velocity vH of the bacteria in thefield of view. We used this persistent length lR,H = τR,H vH
as a measure of trapping efficiency. Larger radii R of pillars
resulted in larger persistent lengths lR,H and also the pillars
with H = 1.9 µm had a larger persistent length than thosewith H = 6.9 µm and 11.1 µm [Fig. 3(d)]. Figure 3(d) showslR,H from two independent experiments with the substrateof H = 1.9 µm but with different bacterial densities. Bothbacterial cultures were sufficiently dilute and therefore there
were no discernible differences between the two experimentsthat may arise from the effects of bacterial collisions. Since
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FIG. 3. The violin plots of the trap time with (a) H = 1.9 µm, (b) H = 6.9 µm, and (c) H = 11.1 µm. The circular markers represent the
average trap time with each R. (d) Persistent length lR,H of bacteria trapped around the pillars. The two blue plots represent the results of two
independent experiments with different bacterial culture performed in different experimental areas on the same substrate with H = 1.9 µm.
The plots with H = 1.9 µm in Fig. 2(d) and Figs. 3(a) and 3(f) correspond to the experiment labeled with H = 1.9 µm-1. Error bars: standard
errors. (e) The cross-sectional radius at each z position of the pillars with H = 11.1 µm. Error bars: standard deviations (see Appendix A 5 fordetails). (f) The asymmetry parameter A in the trap direction. Error bars of A: 64% confidence intervals (see Appendix A 10 for details). Error
bars of R in panels (d) and (f): standard deviations.

cover glass was pressed down while confining the bacteria,
the slight difference in the gap widths around 1.9 µm significantly affects the trapping efficiency even when the same
substrate was used. Such a sensitive response can also beseen in our numerical simulations that will be discussed later(Fig. 4). Nevertheless, both experiments with the substrate of
H = 1.9 µm consistently showed larger lR,H than those oflarger H. This again signifies the enhanced trapping efficiencies for the larger pillars and the pillars in highly quasi-2D
spaces, which is consistent with the results obtained fromthe static information of the density distributions discussed
in Fig. 2. We note that, for small R, the persistent lengthlR,H does not strongly depend on H. The H-dependence oftrap efficiency appeared in situations where bacteria were nolonger scattered and were trapped by the pillars.
D. Chirality of trapping

As swimming bacteria near a single flat wall exhibit circular trajectories due to their intrinsic chirality [30], it isnatural to question whether the directions of rotations oftrapped bacteria are biased or not in the quasi-2D geometry with two parallel flat walls. To this end, we defined
an asymmetry parameter A in the trap direction as A =(NCW − NCCW )/(NCW + NCCW ), where NCW and NCCW are thenumbers of trapped trajectories in the clockwise and counterclockwise directions, respectively. The asymmetry A foreach pillar height and radius shown in Fig. 3(f) signifies thatin the case of H = 11.1 µm the trapped trajectories became
more biased to the clockwise direction when the radius R was
increased. Since the cross-sectional radii of each pillar arelarger near the bottom than that near the top [see Fig. 3(e)],the trapping strength should be larger near the bottom and thus
the observed bacteria are more likely to be trapped near thebottom. In addition, when viewed from above, E. coli tend to
exhibit clockwise and counterclockwise trajectories near thebottom and the top, respectively [30]. Therefore, the observed
asymmetry can be attributed to the bacteria swimming near
the bottom surface. However, almost vanishing asymmetry
was observed for small radii R with H = 11.1 µm. This isbecause the trajectories of bacteria were more like scatterings
than trapping by the small trapping efficiency. In contrast,in the more quasi-2D cases H = 1.9 µm and 6.8 µm, theasymmetry was not so strong as in the case of H = 11.1 µm.
In addition to the fact that the presence of the two walls in
the quasi-2D setup compensates bacterial circular swimmingand makes them swim straight [31], smaller differences in
the cross-sectional radii of the pillars near the top and near
the bottom due to the smaller etching depths H diminish thedifferences in the trapping efficiencies, resulting in weakerasymmetry. Thus, the asymmetry of the traps can be varied
by tuning both the pillar height H and radius R, which may be
used for controlling microbial systems.
III. NUMERICAL EXPERIMENTS WITH A SPATIALLY
FIXED SQUIRMER
A. Setup

The enhanced trapping efficiency in quasi-2D may partly
be accounted for by increased hydrodynamic attraction.
To gain a universal understanding on the interactions between a microswimmer and obstacles in quasi-2D setups, we

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FIG. 4. (a) Numerical setup of the system in the case of the gap width H = 2.5 µm. A spherical squirmer is placed at the origin. The three
no-slip walls are shown. The depicted mesh is used for the finite element calculations. (b) Force exerted on the squirmer toward the verticalwall (+y direction) as a function of the gap width H . The attractive force is enhanced when the system becomes quasi-2D (H  3 µm). The
dashed lines correspond to experimental parameters (blue: H = 1.9 µm, orange: H = 6.9 µm, and green: H = 11.1 µm). (c, d) The inward
fluid velocity profile in the squirmer’s equatorial plane (the yz slice at x = 0 µm) 0.75 µm (1.5 radius) away from the center of the squirmer.
The velocity profile is presented in a polar plot in panel (d) with the origin and the direction of the angular axis flipped to be consistent withpanel (a). The gap widths H are decreased from blue lines to red lines with the same color code as in panel (e). In panel (c), the strengths of
the inward velocity from the vertical wall (from +y direction, blue circles), from the free boundary (from −y direction, red circles), and fromthe top wall (from +z direction, yellow triangles) are shown. (e) Inward fluid velocity profile in the case of no vertical wall. Other setups are
the same as in panel (d).

considered the simplest situation in which a model microswimmer under quasi-2D confinement was placed near avertical flat wall. We conducted a numerical experiment of
measuring the hydrodynamic forces exerted on a squirmer
spatially fixed by applying external forces with, e.g., opticaltweezers in between two no-slip horizontal walls in the presence of a vertical wall representing a pillar [Fig. 4(a)].
The 3D Stokes equations for this confined squirmer
were solved by using a finite element method software
FreeFEM++ [32]. To focus only on the effect of dimensionality, we analyzed the interaction between the verticalflat wall at y = +1 µm and the 1-µm-diameter sphericalsquirmer at the origin by varying the gap widths H of thetwo horizontal planes. The bottom wall was fixed at z =−1 µm while the position of the top wall was varied fromz = 10.5 µm (H = 11.5 µm, 3D) to z = 0.9 µm (H = 1.9µm, highly quasi-2D). The parameters of the squirmer were
estimated from our experimental measurements and a previous flow field measurement [33]: the diameter 1 µm, theswimming speed 20 µm/s, and the pusher-type swimming
strength α = −3. More detailed descriptions on the numericalmethods, setups, and results are given in Appendix B.

B. Hydrodynamic attraction

Although the attractive force toward the vertical wall wasobserved as expected from the pusher-type flow field withinward fluid flow from all the directions at the equatorial
cross section (the yz plane at x = 0 µm) [30], its strength wassignificantly enhanced when the system became highly quasi2D with H  3 µm. Compared with the more 3D cases withH  4 µm where the attractive force stays almost constant, weobserve more than 25% increase of the attractive interaction.
This signifies that, for this situation of the confined 1-µmdiameter squirmer, the effect of the upper wall sets in only forH  3 µm where the system can regarded as truly quasi-2D.
This is consistent with our experiments in which the trapping
efficiencies were almost the same for H = 6.9 µm and H =11.1 µm but were significantly enhanced for H = 1.9 µm. This
sharp increase of the attractive interaction around H = 2 µm
can be responsible for the difference in the two data sets of
H = 1.9 µm in Fig. 3(d). Since it is difficult to control Hwith a precision of 0.1 µm in our experimental design, there
were some differences in the persistent lengths lR,H in the two
independent experiments with H = 1.9 µm.

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C. Hydrodynamic torque and steric interactions

The z component of the hydrodynamic torque is responsible for reorienting the squirmer in the xy plane and thus it
determines whether the squirmer tries to swim toward the vertical wall (y = +1 µm) or not. However, as the z component
is negative [see Fig. 12(b) in the Appendix], the reoriented
squirmer tries to swim away from the vertical wall.
Although this cannot directly explain our experimental results, the steric interactions between a rod-shaped bacterium
and a pillar, which is dismissed in our numerical consideration, have been argued to play a role in keeping bacteriaon the boundary [4,5] or guiding sperms along curved surfaces [34,35]. Therefore, the pillar-swimmer steric interaction
should be playing an important role in stabilizing the bacterial

(a)

(b)

θ

relative number [/s]This increase of hydrodynamic attraction may be intuitively understood from the numerically obtained inward fluid
velocity profiles at the equatorial cross section as a functionof the gap widths H [Figs. 4(c) and 4(d)]. Although in a freespace without any walls the inward velocity profile arounda squirmer at the equatorial cross section is isotropic, thepresence of no-slip walls makes it anisotropic. The decreaseof the gap width H by bringing the top wall closer to thesquirmer weakens the inward flow from the top (from the +z
direction) toward the squirmer. This decrease of incoming flux
was then compensated partly by the enhanced inward flowfrom the horizontal directions (the ±y directions), which is
the most clearly demonstrated in the inward velocity from thefree boundary as the inward flow from the vertical wall stays
almost constant due to the proximity of the vertical wall.
To have a better idea of the change of the flow profiles asa function of the gap width H, we calculated the flow aroundthe squirmer in the absence of the vertical wall. The verticalno-slip wall was replaced with a free boundary and the otherconditions remain the same. This excludes possible effects ofthe vertical wall on the inward velocities on the free boundaryside. As shown in Fig. 4(e), even in the absence of the verticalwall, the inward velocities from the horizontal directions withthe free boundaries (the ±y directions) increased as the gap
width H was decreased. These results are both qualitatively
and quantitatively consistent with the case with the verticalwall shown in Fig. 4(d). Therefore, the increase of the inward
flow from the horizontal direction, which was quantified andverified with the simplest setup, can be concluded as a universal feature of pusher-type microswimmers confined in highly
quasi-2D spaces.
Just to have an idea of the magnitudes of the typical hydrodynamic force calculated here, we estimate the translational
velocity of the squirmer when the external forces are turned
off and the squirmer is let to move freely. By using Stokes’
law, the translational frictional coefficient for a 0.5-µm-radius
sphere, γt , can be evaluated as γt = 6π μ × (0.5 µm) = 3π ×10−3 pN µm−1 s, where we assumed the shear viscosity μ =1 mPas, which is the value of water at 20◦ C. Therefore, thetypical values of the hydrodynamic attractive force in the+y direction shown in Fig. 4(b), 0.11 pN, corresponds tothe translational velocity of ≈12 µm/s. Thus, the hydrodynamic forces can be strong enough to explain the phenomena
observed in our experiments from the viewpoint of hydrodynamics.

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FIG. 5. (a) A phase contrast image of the elliptic pillars on thesubstrate (see Supplemental Movies 8 and 9 [24]). The lengths of
semimajor axes and semiminor axes are 52.5 µm and 34.2 µm,
respectively. (b) The number of events where the bacteria gettingout of the nearby regions divided by the number of bacteria at theangle θ.

orientation in the vicinity of the pillar and thus keeping thebacterium trapped around the pillar as previously discussed
in the case of bacterial accumulations close to a flat wall[36]. However, considering that the height H of the pillar in
our experiment was always more than twice as large as thediameter of bacterial bodies and thus the steric interaction
should hardly be affected by H, the increased hydrodynamic
forces should be playing a more dominant role in the increased
trapping efficiency.
D. Interpretation of the trapping mechanism

In conclusion, the enhanced hydrodynamic attractive force
between a bacterium and a pillar in quasi-2D facilitates the approach of the bacterium toward the pillar, and after it reaches
the surface of the pillar then the steric interaction reorients thebacterium to keep it swim along the pillar surface by dominating the hydrodynamic torque. The hydrodynamic attraction
then makes it difficult for the bacterium to escape from the
nearby region of the pillar, resulting in higher trapping efficiencies in quasi-2D.
To fully confirm this story, more faithful simulations on theswimming dynamics of microswimmers in quasi-2D by using
more sophisticate methods, e.g., boundary element method(BEM) [37–40] or smoothed profile method (SPM) [41,42],
may be ideal for understanding how bacteria approach thepillars and for disentangling the effects of hydrodynamic
and steric interactions. However, as calculations on quasi-2D
setups with many boundaries are computationally quite expensive even with BEM or SPM, these are out of the scope of
our current experimental study.
IV. LOCAL CURVATURE DEPENDENCE

Finally, to investigate whether these traps by pillars are
local interactions or not, we performed additional experimentswith elliptic pillars, whose nonuniform curvature prohibits
the use of standard analytical techniques. We made a glass
substrate with elliptic pillars with the height H  7 µm [seeFig. 5(a) and Supplemental Movies 8 and 9 [24]]. Since
there is no function in OpenCV library of Python CV2 for
directly detecting ellipses like Hough transform for detecting
circles, the image analysis was performed as described in

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Appendix A 9 from the captured phase contrast images. We
defined the nearby region as the internal area of a larger
ellipse that was drawn by extending the semimajor axes andsemiminor axes of each elliptic pillar by 6.4 µm.
As shown in Fig. 5(b), the number of bacteria leaving the
nearby regions divided by the density was nonuniform withthe peaks located at the long-axis directions. Thus, the local
trapping efficiency was larger in the direction with larger local
curvature. Similar tendency was also reported in the experiments with sedimented bimetallic-rod swimmers driven by
self-electrophoresis and teardrop-shaped posts in a 3D region[43]. Because the tendency of the higher escape rate for thelarger curvatures is consistent with the experiments with thecircular pillars with globally constant curvatures, the obtained
nonuniform distribution directly demonstrates that the local
curvature by itself can control the escape rates. This suggeststhat the traps by pillars depend on local curvature of pillars
and these interactions are relatively short-ranged. Such shortrange nature of the bacterium-wall interaction in quasi-2D isalso exemplified in our finite element simulation, in which theattractive force as a function of the system size Lx along thex direction converged at Lx  2.5 µm that is comparable withthe bacterial body lengths (see Fig. 4(a) and Fig. 13 in the
Appendix).
V. CONCLUSION

The quantitative differences between quasi-2D and 3Dinteractions found in our study is expected to lead to a better understanding of both quasi-2D far-field and near-fieldhydrodynamic interactions. Our findings may trigger further
quantification and theoretical development of the peculiar
quasi-2D hydrodynamics that has difficulty in rigorous theoretical analysis. It is also important to investigate in detail thesteric interactions in quasi-2D regions. However, the analysisof the steric interactions requires high magnification that has
a drawback in statistics compared with our low-magnification
measurements, and thus it would be a challenging task.
From the active matter viewpoint, the pillar-swimmer interactions quantified in our experiments can also be translated
into the understandings of interparticle interactions, which
gives new insight on collective behavior in lower dimensions
such as bacterial long-range nematic order in quasi-2D [11],bacterial noncontact cohesive swimming in 2D [44], and colloidal swimmers under confinements [45–48]. In terms ofapplication, the relation between the trap efficiency and thestructure of the space confirmed in our experiments can serve
as a design principle for active microfluidic devices using
microswimmers.

No. JPMJPR21O8, Japan. Y.T. performed experiments with
D.N. and analyzed the experimental data. D.N. conductednumerical simulations. Both Y.T. and D.N. interpreted theresults and wrote the manuscript.
APPENDIX A: EXPERIMENT
1. Washing coverslips

First, coverslips were washed by sonication in 20w% aqueous solution of alkaline detergent (Contaminon, FUJIFILM
Wako Pure Chemical Corporation) for 20 min. The washedcoverslips were rinsed with running tap water 15 times andthen with deionized water 3 times, followed by sonicationin deionized water for 2 min twice and then in ethanol oncefor 15 min. After rinsing the coverslips three times in deionized water, they were again washed by sonication twice indeionized water for 2 min and then in 8 × 10−2 mol/l NaOH
aqueous solution for 10 min. Then, the coverslips were rinsed
three times in deionized water and sonicated twice in deionized water for 2 min each. Finally, the washed coverslips weredried at 140◦ C for 40 min in a dry heat sterilizer.
2. Microfabrication by photolithography and etching

Chromium was deposited on the washed coverslips byusing a thermal evaporator to enhance the adhesion of photoresist to the coverslips. The chromium-coated coverlips
were then heated at 100◦ C for 5 min on a hot plate. After thecoverslips were cooled to the room temperature, 500 µl of thephotoresist AZ 1500 (Merck Performance Materials GmbH.)
was spread on the coverslips by using a spin coater (500 rpm
for 5 s and then 2000 rpm for 30 s). Then the photoresistcoated coverslips were heated at 100◦ C for 1 min. We used amaskless aligner (μMLA, Heidelberg Instruments) for exposure and wrote down the desired patterns on the photoresist.
The designed radii of pillars were 15, 20, 30, 40, and 50 µm.
After the exposure, the photoresist were developed. We heatedthe coverslips at 100◦ C for 2 min and then dipped them in
NMD-3 2.38% (Tokyo Ohka Kogyo Co., Ltd.), deionized water, Cr-etchant (MPM-E350, DNP Fine Chemicals Co., Ltd.)and then deionized water in order.
The exposed cover slips were etched in 50% buffered hydrofluoric acid aqueous solution (110-BHF, Morita Chemical
Industries Co., Ltd.) at 23◦ C. The height of the fabricatedpillars were controlled by tuning the etching time. After thecoverslips were moved into deionized water to stop the etching reaction, they were rinsed in acetone, Cr-etchant, anddeionized water in order.
3. Surface treatment

ACKNOWLEDGMENTS

We thank Takuro Shimaya for kind technical support anddiscussions, Matthew S. Turner for discussions, Kazumasa
A. Takeuchi for frequent discussions, and Kenta Ishimotofor valuable and stimulating discussions and critical readingof our earlier manuscript. We thank Yusuke T. Maeda and
Hanna Salman for sharing plasmid DNA. D.N. was supported by JSPS KAKENHI Grants No. JP19K23422, No.
JP19H05800, and No. JP20K14426, and JST, PRESTO Grant

The microfabricated substrates were washed by sonicationin 20w% aqueous solution of alkaline detergent (Contaminon,
FUJIFILM, Wako Pure Chemical Corporation) for 20 min.
The substrates were rinsed with running tap water 15 times
and then with deionized water three times, followed by sonication in deionized water for 2 min twice and then in ethanol
once for 15 min. After rinsing the substrates three times indeionized water, they were again washed by sonication twice
in deionized water for 2 min. After blowing off the water

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PHYSICAL REVIEW E 107, 014602 (2023)

(b)

34
33

radius( m)

35

32

37.5

38
radius( m)radius( m)

36

(c)

36
34

1

2

3

(d)

0

30.0

2

4
z( m)

6

8

0

5

10

15

z( m)

(e)

50

15.0

40

12.5
H( m)

Rave ( m)

32.5

27.5
0

z( m)

30
20

10.0
7.5
5.0

10
0

35.0

2.5
0

10

20
30
40
50
designed radius( m)

60

0.0

0

50
100
150
etching time(min)200

FIG. 6. The radius at each z position of pillars with the designed radius of 40 µm for (a) H = 1.9 µm, (b) H = 6.9 µm, and (c) H = 11.1 µm.
Error bars describe standard deviation at each height of 4 pillars. (d) The relation of Rave and designed radius of pillars with H = 6.9 µm. Orange
line is a linear fit. Error bars: standard deviations. (e) The relation of H and the etching time with 40 µm designed radius. Error bars: standard
deviations.

with a blower, a drop of 1w% BSA (bovine serum albumin,
Sigma-Aldrich) aqueous solution was put on the substrate andthen we placed another coverslip without any fabrications,which would be used as a lid later in the experiment, to ensurethat the BSA solution was spread over the whole substrate.
We waited for at least 15 min to allow the BSA molecules toadhere to the surfaces of both the substrate and the coverslip.
Right before each experiment, the substrate and the coverslip
were rinsed with deionized water and the remaining water
was blown off with a blower. The BSA coated surfaces ofthe substrate and the coverslip were used for confining thebacteria in between them subsequently.
4. Confinement of E. coli into microstructures

We used a nontumbling chemotactic mutant strain ofEscherichia coli (RP4979) with a plasmid pZA3R-EYFP expressing yellow fluorescent protein. Bacteria taken from afrozen stock were grown overnight in Tryptone broth (T broth,
1 wt% tryptone, and 0.5 wt% NaCl) with a selective antibiotic(chloramphenicol 33 µg/ml) at 30◦ C at 200 rpm and then thisculture was diluted 100-fold in 10 ml of fresh T broth withthe antibiotic at 30◦ C at 200 rpm. For the experiments withthe pillar heights H = 6.9 µm and 11.1 µm, when the opticaldensity (OD600 ) of the bacterial culture reached OD600 = 0.1
Abs, the bacterial suspension was confined in between the microfabricated substrate and the coverslip that were precoatedwith BSA. In the case of H = 1.9 µm, bacteria were grown
until the optical density reached OD600 = 0.3 Abs or 0.6 Abs[they correspond to data 1 and 2 in Fig. 3(d), respectively].
The plots labeled with H = 1.9 µm in the other figures represent the data with OD600 = 0.3 Abs.) to enable the observationof the sufficient number of bacteria. To sustain the motility of

the confined bacteria, we mixed fresh T broth containing 2.5
mol/l L-serine and the bacterial suspension in a ratio of 1:9.
Several microliters of the prepared mixture was put onan observation area of the microfabricated substrate and thenthe substrate was covered with the BSA-coated coverslip. To
ensure the contact of the coverslip and the top of the fabricatedstructures so that the bacteria do not swim above the pillars,
we pressed the coverslip on the substrate.
5. Characterization of fabricated pillars by confocal microscopy

Due to the etching procedure, the radii of the fabricatedpillars become smaller than the original designs. Therefore,
we used an inverted confocal microscope (Leica SP8) toevaluate the actual heights and radii of the fabricated pillars
on the etched substrates. We observed the pillars with theoriginal designs of the radii 15, 20, 30, 40, and 50 µm on thesubstrates with different etching times, 40, 120, and 200 min.
We poured 10 µmol/l rhodamin solution on the substrates
and captured confocal images at every 0.1 µm over the range
covering both the bottom of the substrates and the top of thepillars. We defined the z axis in the vertical direction, and theflat surfaces of the substrates and the coverslips were placedin the xy plane as in Fig. 1(a) in the main text.
First, we defined the centers and the radii of the pillars in
the each z slice of the confocal images. Each pillar in the slice
was detected as a circle by Hough transform. Figures 6(a)–6(c) show that the radii become smaller at higher z positions
due to the etching process. The longer etching time resulted
in the larger difference in the radii of the pillars at the top andthe bottom.
The obtained positions of the centers and the radii of thepillars at each z slice were then averaged to define the actual

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PHYSICAL REVIEW E 107, 014602 (2023)

centers and radii Rave of the pillars. We excluded the z slices
where the number of the detected circles was not equal tothe actual number of the pillars from the calculations of theaveraged centers and the averaged radii Rave .
Next, we defined the z position of the surface of the substrates. The intensities at the largest and the smallest z slices,where the whole regions are filled with the bright fluorophores
and the dim glass, respectively, were averaged to define thethreshold intensity. This threshold was used to distinguishwhether each spatial position was inside or outside of the glass
substrates. The z position of the surface of the substrates asfunctions of x and y were obtained by finding the z positions
at which the intensity exceeded the threshold value. Since thesurface thus obtained exhibited salt-and-pepper noise due tothe noise in the original image, we blurred this surface byapplying a median filter to obtain a smooth surface profile.
Finally, we defined the heights H of pillars as the differenceof the averaged z positions of the bottom of the substrates andthe top of the pillars. The z positions of the bottom and the
top were determined by calculating the averaged√ z positionsin the outside of circles with the radii of√ 2Rave and theinside of circles with the radii of Rave / 2 whose centers
coincided with those of the pillars defined above. The error
σbottom (σtop ) of the z position of the bottom (top) was definedas the standard deviation of the z positions of the pixels in thebottom (top) region defined above. The estimated heights H of
the pillars with the same designed radii of 40 µm but with thedifferent etching times were 1.9√
± 0.2 µm, 6.9 ± 0.3 µm, and11.1 ± 0.4 µm, where ± means σtop2 + σbottom2 . The etchingrate was almost constant during 200 min and was about 20
min/µm [see Fig. 6(e)]. Except for the case of H = 1.9 µm,
the height of the pillar was almost equal to the difference
between the radius Rave calculated above and the designed
radius [see Fig. 6(d)].
Since the cross-sections of the pillars can deviate fromperfect circles due to the effect of etching, we evaluated theshapes of the cross-sections at half the height of the pillars byimage analysis. We detected the edges of the cross-sections of
the pillars by Canny method. For each pillar, the distancesfrom its center defined above to each point on the edges wereaveraged to define the radius of this single pillar at half height.
The standard deviations σhalf of the distances from its centerto each point on the edges were calculated for this single
pillar. Then, we calculated the ensemble average among thepillars with the same designed radii, which gave the mean
radii Rhalf . The standard deviations σhalf were smaller than
0.25 µm, which is two orders of magnitude smaller than Rhalf .
This assures that the cross-sections of pillars can be regarded
as almost perfect circles. These σhalf were sufficiently smaller
than the differences of the radii depending on the heights.
The radii of pillars R used in the main text were definedby Hough transformation of the bright field images of theexperiments. The three definitions of the radii, R, Rhalf , andRave , gave almost the same values and consistent results.
6. Definition of the density ratio

In defining the density in the near region, we had to pay
attention to the following two points. First, because the pillars
were not perfect cylinders and also due to errors in the image

analysis, bacteria could be sometimes detected in the region
with h < 0. Second, a few bacteria got sandwiched between
the tops of the pillars and the cover glass used as a lid, andthus bacteria could be detected in the h < 0 region. Therefore,
the average density of the near region was defined as the number of bacteria detected in the region −6.4 µm < h < 6.4 µm
divided by the area of the region 0 µm < h < 6.4 µm. The
position h = −6.4 µm (10 pixels) is a lower bound to avoid
accounting for bacteria in the h < 0 region.
7. The definition of the near region

The definition of the near region was determined by thedensity of bacteria and its uncertainty. First, we divided
the region −3.2 µm  h < 12.8 µm into 100 subregions withthe width of 0.16 µm and labeled them as I0 , I1 , . . . , I99
in the increasing order of the distance hn from the center ofthe pillar to the middle point of the nth subregion. Next, we
defined the average density ρn in the subregion In , and then we
took the weighted moving average over In , In+1 , . . . , In+10 todefine the smoothed average density ρn and the weighted
standard deviations σn at h = 0.16n − 2.4 µm,
n+10(R + hk )ρkρn = k=n,n+10
k=n (R + hk ) n+10(R + h )(ρ − ρn )2.
σn =  k=nn+10 k k
k=n (R + hk )

(A1)

(A2)

Note that, farther subregions occupy larger areas than closer
subregions, and the weight R + hk is selected to properlyreflect these area ratios.
Figure 7 shows the densityρ and its uncertainty σ . In h <5–7 µm, the density minus its uncertainty ρ − σ is bigger thanthe density ρ∞ defined in the main text. Therefore, we defined
the near region as 0 µm < h < 6.4 µm.
We confirmed that the slight difference in the definition of
the near region does not affect the behavior of our results somuch (see Fig. 8).
8. Definition of the persistent time and length

In the main text, the average values of the trap times areused as the definition of the persistent time. Here we notethat the persistent time cannot be well characterized by otherdefinitions. For example, we can alternatively define the persistent time as the inverse of the decay constant if we regard
the escape of a bacterium from a pillar as a Poisson process.
In reality, however, the number of events that a bacterium kept
staying in the nearby region was not decaying exponentially
except in the short time scale (see Fig. 9), indicating thatthe mechanism of the escape has a memory. This memory
may be associated with the orientatioal degrees of freedomof the bacteria’s bodies, which play a role in trapping but
were not accessible in our low-magnification observations.
Considering the complexity of the mechanism of the trapping
phenomenon, the simple average of the trap times was chosen
as the definition of the persistent time.

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YUKI TAKAHA AND DAIKI NISHIGUCHI

(b)

1.2

1.1

1.1

1.1

1.0

1.0

1.0

0.8

/

1.2

0.9

0.9
0

(d)

5
h( m)

0.8

10

0.9
0

5
h( m)

(e)

0.8

10

1.2

1.1

1.1

1.1

1.0

1.0

1.0

0.8

/

1.2

0.9

0.9
0

5
h( m)

10

0

5
h( m)

10

(g)

0.8

0

5
h( m)

10

0

5
h( m)

10

(f)

1.2

/

/

(c)

1.2

/

/

(a)

PHYSICAL REVIEW E 107, 014602 (2023)

0.9
0

5
h( m)

0.8

10

1.2

/

1.1
1.0
0.9
0.8

FIG. 7. Bacterial density and its fluctuations. (a) H = 6.9 µm, R = 8.9 µm. (b) H = 6.9 µm, R = 13.5 µm. (c) H = 6.9 µm,
R = 23.4 µm. (d) H = 6.9 µm, R = 33.6 µm. (e) H = 6.9 µm, R = 43.1 µm. (f) H = 1.9 µm, R = 34.4 µm, OD600 = 0.3 Abs, and (g)H = 11.1 µm, R = 29.1 µm. Error bars were defined by the Eq. (A2).

9. Elliptic pillars

Since there is no function in OpenCV library of Python
CV2 for directly detecting ellipses like Hough transform fordetecting circles, the image analysis was performed sequentially as follows. The phase-contrast image was blurred andbinarized and the contours in the image were detected by using the findCounter function in CV2. Although the contours
were detected both outside and inside the pillars, the external
contours were taken as the surface of the elliptical pillars.

(b)

20

H=6.9 m
H=11.1 m

10
0
0

20
R( m)

40

(c)

30
20

H=1.9 mdensity ratio

30

H=1.9 mdensity ratio

density ratio(a)

We determined the lengths of the major and minor axes, thepositions and the orientations of the elliptic pillars by applying
the fitEllipse function in CV2. Then, we defined the nearregion of each pillar as the internal region of an ellipse whose
semimajor axis and semiminor axis were 6.4 µm longer thanthose of the ellipse of the pillar.
As in the case of the circular pillars, we calculated thedensity of bacteria around elliptical pillars and the number of
events that bacteria got out of or went into their near regions,

H=6.9 m
H=11.1 m

10
0
0

20
R( m)

40

30
20

H=1.9 m
H=6.9 m
H=11.1 m

10
0
0

20
R( m)

40

FIG. 8. Ratios of the density of the near region to that of the far region with different definitions. The near regions were defined as(a) 0 µm < h < 4.0 µm, (b) 0 µm < h < 6.4 µm, and (c) 0 µm < h < 8.0 µm. The figure (b) is used in the main text. Error bars of density
ratio: standard errors. Error bars of R: standard deviations. The optical density of the bacterial culture was OD600 = 0.3 Abs with H = 1.9 µm
in these figures.
014602-10

QUASI-TWO-DIMENSIONAL BACTERIAL SWIMMING …

(a)

(b)

R=9.2 m101

R=14.3 m
R=24.2 m
R=34.4 m
R=43.9 m

0

10
t(s)

102
R=8.9 m101

R=13.5 m
R=23.4 m
R=33.6 m

10

20

103
number of events

102

0

(c)

103
number of events

number of events

103

10

PHYSICAL REVIEW E 107, 014602 (2023)

R=43.1 m

0

0

10
t(s)

20

102
101

R=10.1 m
R=19.4 m
R=29.1 m

10

R=39.0 m

0

0

10
t(s)

20

FIG. 9. The number of events that a bacterium keep staying in the nearby region for longer than time t with (a) H = 1.9 µm, OD600 =
0.3 Abs, (b) H = 6.9 µm, and (c) H = 11.1 µm. Shaded error bars: standard errors.

which are shown in Fig. 10. In this figure, the density in acertain direction represents the average density in the nearregion with an angular width of θ = π /10 in that direction.
The average density was defined as the time-averaged numberof bacteria divided by the elliptic arc length l θ , which wasthen averaged over 115 pillars. Here, the line element l for an
ellipse x 2 /a2 + y2 /b2 = 1 is given byl=ab(a4 cos2 θ + b4 sin2 θ )1/2.
(a2 cos2 θ + b2 sin2 θ )3/2

(A3)

Note that the angle θ is the angle from the geometrical centerof the ellipse to a point on the ellipse [see Fig. 5(a) in the maintext], not the eccentric anomaly φ which is usually used in
the parametric representation of the ellipse x = a cos φ, y =b sin φ [see Fig. 10(c)]. The numbers of outgoing and incoming bacteria were defined as the number of events per
second divided by the length of the elliptic arc length l θ
and averaged over 115 pillars.
Figure 10(a) shows that the density of the bacteria showed
weak, if any, nonuniformity with the peaks located at the longaxis directions. Figure 10(b) shows that the number of events
that the bacteria got out of the near region is anisotropic,
although the number of events that the bacteria got into thenear region is relatively isotropic.
10. Asymmetry in the direction of rotation

Whether the direction of bacterial rotations around eachpillar is biased to a certain direction or not is a natural question
from the view point of the intrinsic chirality of bacterial swim-

(a)

2.0
1.5
1.0
0.5
0.0

APPENDIX B: FINITE ELEMENT CALCULATION
OF A SQUIRMER UNDER CONFINEMENT
1. Numerical setup

To account for the increased trapping efficiency of bacteria in the quasi-two-dimensional setup in our experiment,
we conducted numerical experiments to estimate the hydrodynamic forces and torques exerted on a microswimmer inthe confined geometry as shown in Fig. 4 in the main text.
Specifically, we numerically solved the 3D Stokes equations for a single pusher-type spherical squirmer whose yand z positions and orientation were fixed by applying external forces and torques with, e.g., optical tweezers in

(b)
×10 3

number of events(/ m/s)

density(/ m)

2.5

ming behavior. To this end, we obtained from the bacterial
trajectories how many rotations each bacterium made arounda pillar while the bacterium kept staying in the near regionof the pillars. The distribution of the angle for each pillar
set with the height H and the radius R is shown in Fig. 11.
We defined an asymmetry parameter A in the direction ofrotation as in the main text, and the error of A was definedby the bootstrap-like method as follows. Suppose that the dataobtained from the experiment consisted of N trajectories andthe number of clockwise rotations was NCW . When a new
sample consisting of N trajectories is obtained by sampling
again from the obtained sample with duplicates allowed, thenumber of clockwise rotations of the new sample follows abinomial distribution B(N, p) with p = NCW /N. Using thisprobability distribution, the 64% confidence interval of A is
used as the errors on Fig. 3(f).

0

90

180
270
(degree)

360

(c)
8

×10 4
into
out

6

P

4
2
0

0

90

180
270
(degree)360

FIG. 10. (a) The density of the bacteria around the elliptical pillars. (b) The number of bacteria per second per arc length that got out of or
into the near region. (c) The angle θ and the eccentric anomaly φ of a point P on the ellipse.
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YUKI TAKAHA AND DAIKI NISHIGUCHIPHYSICAL REVIEW E 107, 014602 (2023)

(f)

(k)

5

5

5

2

1

probability density

3
2
1

2

1

4
3

1

H=1.9 µm
R=24.2 µm

1
2

1

(d)

0
rotation1

H=1.9 µm
4
R=34.4 µm
3
2
1
0

2

1

(e)
5
4
3

0
rotation11
10
rotation

4
3

1

2

probability density
0
rotation11
10
rotation

1

3
2
1

5
4
3

0
rotation

2

1

4
3

1

2

1

2

0
rotation

1

2

1

2

1

2

1

2

H=11.1 µm
R=19.4 µm

2
1
2

1

5
4
3

0
rotation

H=11.1 µm
R=29.1 µm

2
1
2

1

(n)

H=6.8 µm
4
R=33.6 µm

1

5

0

2

5

2

1

(m)

2

2

2

0

2

H=6.8 µm
R=23.4 µm(j)

H=1.9 µm
R=43.9 µm

2

5

0

2

2

0

1

3

H=11.1 µm
R=10.1 µm

(l)

1
2

4

0

2

2

(i)

5

1

3

0

2

0
rotation

H=6.8 µm
4
R=13.5 µm

(h)

2

0

1

5

0

2

probability density

50rotation2(g)

H=1.9 µm
4
R=14.3 µm(c)probability density

2

5

0

probability density

1

1
0

probability density

probability density(b)0rotation2

probability density

1

3

probability density2

H=6.8 µm
4
R=8.9 µm

probability density

30

probability density

probability density

H=1.9 µm
4
R=9.2 µm

probability density

probability density

(a)

5
4
3

0
rotation

H=11.1 µm
R=39.0 µm

2
1
0

2

1

0
rotation

H=6.8 µm
R=43.1 µm

2
1
0

2

1

0
rotation

FIG. 11. Histograms of the rotation number for different H and R. The optical density of the bacterial culture was OD600 = 0.3 Abs with
H = 1.9 µm in these figures.

between two horizontal planes with no-slip boundary conditions. We placed a flat vertical wall representing a pillar in our
experiment and let the squirmer to swim toward the x direction. Note that we do not consider any external force
in the x direction so that the squirmer can only swim
straight in that direction. We estimated the strength of thehydrodynamic forces and torques as a function of the gap
width H between the two horizontal walls. To solve the

Stokes equations, we used a finite element method software
FreeFEM++ [32].
The detailed numerical setup is as follows. A sphericalsquirmer with the diameter of 1 µm is fixed at the origin andtries to swim in the +x direction by giving the surface slip
velocity,

α
3 
(B1)
vs (θ ) = U sin θ + sin 2θ ,22014602-12

QUASI-TWO-DIMENSIONAL BACTERIAL SWIMMING …

along the angular unit vector,

PHYSICAL REVIEW E 107, 014602 (2023)

⎞− y2 + z 2
1⎝x cos arctan yz ⎠,eθ = 
(B2)
x 2 + y2 + z2 x sin arctan z
y
√2 2
y +z
is the angle from the x axis, U iswhere θ = arctan x
the swimming speed of the squirmer in a bulk fluid without
any walls, and α represents the strength of the force dipoleof the swimmer. Here, the parameters were set to be U =20 µm/s and α = −3 so that they reproduce our experimental
measurements of bacterial swimming speed and the previous
measurements of the bacterial flow field [33]. We placed thelower horizontal wall at z = −1 µm, and we changed theheight H of the system by changing the position of the upperhorizontal wall from z = +0.9 µm to z = +10.5 µm, whichcorresponds to our experiments with the varying gap widths
H from 1.9 µm to 11.6 µm. The vertical wall is placed at
y = +1 µm with the no-slip conditions, and other boundaries at y = −1 µm, x = +2.5 µm, and x = −2.5 µm are freeboundaries without any constraints on the velocity. The actual
mesh on the solid boundaries used for tetrahedralization in
the finite element method is shown in Fig. 4(a). The systemsize was restricted by the requirement of the RAM > 800 GB
for the larger systems even with the use of relatively large
sizes of the mesh and tetrahedra, which resulted in erroneous
variations seen in the data of Fig. 4(b).
To solve the equation in the laboratory frame, the actual
swimming velocity U0 ex of the squirmer in the presence ofthe boundaries should be subtracted from the surface slip
velocity, where ex is the unit vector along the x axis. The actual
swimming velocity is usually calculated to satisfy both theforce-free and torque-free conditions of a squirmer, but here,
as we consider a situation in which the squirmer is both positionally and orientationally fixed except for the x direction, theforce free and the torque free conditions do not hold. Thus, weset U0 ≈ U as a first approximation. This approximation will
later be justified by the fact that the obtained hydrodynamic
force in the x direction is almost zero within the range ofnumerical errors [see Fig. 12(a)]. Therefore, the given slip
velocity at (x, y, z) on the surface of the squirmer is
⎛  2
⎞ ⎛ ⎞
− y + z2Uvs (θ )
⎝x cos arctan yz ⎠ − ⎝ 0 ⎠.
vs (θ )eθ − U ex = 
x 2 + y2 + z2 x sin arctan z
0
y
(B3)
⎛

After obtaining the full velocity field u(r), we calculated thehydrodynamic force F and torque T exerted on the squirmer
by integrating the stress field on the surface of the squirmer asF=

σ · ndS,(B4)

r × σ · ndS,

(B5)

S

T =

(a)
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
-0.10

0

1

2

3

4

5

6

7

8

9

10

11

12

2

3

4

5

6

7

8

9

10

11

12

10 -3

(b)
2.00
1.50
1.00
0.50
0.00
-0.50
-1.00
-1.50
-2.00
-2.50
-3.00
0

1

FIG. 12. The numerically obtained hydrodynamic (a) force and(b) torque exerted on a squirmer as functions of the gap width H of
the two confining planes. Blue circles: x component. Red squares:
y component. Yellow triangles: z component. The three dashedlines in panel (a) correspond to experimental parameters: blue for
H = 1.9 µm, orange for H = 6.9 µm, and green for H = 11.1 µm.

matrix. The indices i and j represent the spatial coordinates.
We set μ = 1 mPas, which is the value of water at 20◦ C. All
the x, y, and z components of the force and the torque areshown in Fig. 12. In Figs. 4(b) and 13, only the y components
of F are shown for visibility.

S

σ = μ(∇u + ∇uT ) − pI,(B6)

where S represents the surface of the squirmer, σ is the stresstensor, n is the unit normal vector of the squirmer surface
pointing outward, dS is the surface element, μ is the shear viscosity of the fluid, p is the static pressure, and I is the identity

2. Strengths of hydrodynamic forces and torques

Here we evaluate the translational and angular velocitiesof the squirmer when the external forces and torques areturned off and the squirmer is free to move as we did in themain text (Sec. III B). Stokes’ law gives the translational and014602-13

YUKI TAKAHA AND DAIKI NISHIGUCHIPHYSICAL REVIEW E 107, 014602 (2023)

rotational frictional coefficient for a 0.5-µm-radius sphere,γt and γr , respectively, as γt = 6π μ × (0.5 µm) =3π × 10−3 pN µm−1 s and γr = 8π μ × (0.5 µm)3 = π ×10−3 pN µm s. By using these formulas, the typical values of
the hydrodynamic attractive force in the +y direction and thez component of the torque, 0.11 pN and −2 × 10−3 pN µm,
correspond to the translational velocity of ≈12 µm/s and theangular velocity of ≈ − π2 rad/s, respectively, both of which
are strong enough for hydrodynamics to play a role.

0.16
0.14
0.12
0.1
0.08
0.06

3. System size dependence of the numerical results

FIG. 13. Numerical results on the hydrodynamic force acting on
the squirmer toward the vertical wall (+y direction) as a functionof the system size in the x direction Lx . The force can be regarded
converged for Lx > 2.5 µm.

As the choice of the system size can affect systematicallythe numerical results, we have checked the system-size dependence of the hydrodynamic attractive force by performing
the calculations by changing the system size in x directions
with the top wall fixed at z = +1 µm. The other setups are
the same. The vertical wall was placed at y = +1 µm, andthe free boundaries are at y = −1 µm, x = +Lx and x = −Lx .
We increased Lx from 1 µm to 10 µm, which confirmed thatthe estimated hydrodynamic force can be assumed to be converged above Lx = 2.5 µm. This is the reason of our choice of
Lx = 2.5 µm in our simulations with variable gap widths.

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