Acoustophoresis of Legionella species in water and the influence of collective
hydrodynamic focusing Alen Pavlic,∗a Marjan Veljkovic,b Lars Fieseler,b and
Jürg Duala Legionella are gram-negative, facultative intracellular, and
pathogenic bacteria that pose a risk for human health and cause significant
energy losses due to extensive preventive heating-up of water installations.
We investigate acoustically-driven motion — acoustophoresis — of several
Legionella species, Escherichia coli, Pseudomonas aeruginosa, and Acanthamoeba
castellanii, a common Legionella host in water. All the investigated cells can
be acoustically manipulated in an ultrasonic standing wave in water, as they
possess a non-zero acoustic contrast that is positive for all of the cells,
leading to the focusing into pressure nodes of the standing wave. Multi-body
simulations indicate that an increase in cell concentration could
significantly accelerate the rate of focusing due to hydrodynamic interactions
— a phenomenon that we call collective hydrodynamic focusing (CHF). The
results form a foundation for acoustic manipulation of bacteria and could pave
a path towards acoustically-aided detection of Legionella in water.
---

##

††footnotetext: a Institute for Mechanical Systems, Swiss Federal Institute of
Technology Zurich, Zürich, Switzerland.††footnotetext: b Institute of Food and
Beverage Innovation, Zurich University of Applied Sciences, Wädenswil,
Switzerland.††footnotetext: ∗ E-mail<EMAIL_ADDRESS>
## Introduction

Legionella is a genus of gram-negative bacteria that are harmful for human
health and the causative agent of legionellosis that can appear as
legionnaires’ disease, Pontiac fever, or Pittsburgh pneumonia.1 Due to the
danger of Legionella, there are regulations in place to monitor and minimize
their presence in water installations. A common approach to eradicate
Legionella is periodically heating-up water supplies to above
$60\text{\,}\mathrm{\SIUnitSymbolCelsius}$. However, periodic heating requires
continued energy supply.2 To save energy and to minimize the general impact of
diseases originating from Legionella bacterial monitoring and analysis could
be further improved. Currently, the detection of Legionella from water samples
relies on manual collection of a rather large sample ($1\text{\,}\mathrm{L}$),
followed by labour-intensive and time-consuming culturing (up to $10$ days).3
Low concentration limits stipulated by legislature and the associated low
abundance of Legionella further complicate the process. One way to improve the
monitoring of Legionella presence could be an up-concentration of Legionella
in an at-line fashion, prior to the analysis, combined with the use of novel
and faster detection systems. Such a combined system could accomplish a
sustainable and reliable real-time detection of Legionella, leading to
considerable energy savings and to the reduction of legionellosis risk on a
global scale.

The up-concentration of Legionella could be accomplished through the means of
acoustophoresis,4 which is known to be a non-destructive method for
manipulation of microscopic objects at relatively high throughputs (flow rates
of $>$100\text{\,}\mathrm{\SIUnitSymbolMicro L}\text{\,}{\mathrm{min}}^{-1}$$
per device5). Acoustophoresis is often used in biomedical applications to
manipulate cells without affecting their viability, provided that the
operating power remains below thresholds for the onset of cavitation and for
reaching the temperatures harmful to cells.6, 7 The acoustic field giving rise
to acoustophoresis can for example stem from diagnostic or therapeutic medical
ultrasound, advanced flow cytometers that use standing waves for higher
throughput and accuracy,8 or lab-on-a-chip devices for characterization of
mechanical properties of eukaryotic cells and organisms.9, 10, 11, 12 However,
acoustic manipulation of bacteria is limited, mainly due to the small size of
bacterial cells that leads to a small acoustic radiation force (ARF), which is
the main force in acoustophoresis and scales with the volume of a cell. The
acoustic streaming that is generally present in acoustofluidic devices acts on
the cells through the Stokes drag, which scales with the radius of a cell.
This difference in scaling of the two competing forces can be characterized
with a critical radius, above which the cell motion is dominated by the ARF,
while the smaller cells move with the streaming flow. For a cell-like material
in water, assuming a standing ultrasonic wave with the frequency $f$ of
$2\text{\,}\mathrm{MHz}$, the critical radius is typically
$\approx$0.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and scales with the
frequency as $\sqrt{1/f}$.13

Despite the scaling of forces hindering the acoustic manipulation of bacteria
like Legionella, with the equivalent sphere radius (ESR) in the range of
$\sim$0.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,14 various methods were
developed in the recent years. For example, using larger “seed” particles that
trap bacteria supposedly with acoustic interaction forces,15, 16 using two-
dimensional acoustic fields to avoid the disturbance from the acoustic
streaming,17 or simply scaling down the dimensions of the acoustofluidic
device and increasing the frequency to boost the acoustic radiation force.18
Acoustophoresis of bacteria was so far demonstrated on, for example,
Salmonella Typhimurium,19 Escherichia coli,17, 20, 21 Enterobacter cloacae,18
and even Legionella pneumophila,22 but in a fixed state and with potentially
dominating influence of agglutination.

The fundamental behaviour of biological cells in a standing acoustic wave was
so far characterized for various eukaryotic cells and microorganisms in terms
of the acoustic contrast factor $\varPhi$ that determines the magnitude of the
ARF (ARF $\propto\varPhi$) and the locations of stable equilibria of cells in
the wave at either the pressure nodes ($\varPhi>0$) or pressure antinodes
($\varPhi<0$). 9, 10, 11, 12 To the best of our knowledge, there are no such
characterizations reported for bacteria, apart from some studies qualitatively
and indirectly implying a positive acoustic contrast factor of several
bacteria in water for Salmonella Typhimurium,19 Escherichia coli,17, 20, 21
and Enterobacter cloacae.18

Fig. 1: Experimental setup and procedure. (a) Description of the measurement
setup and the key components. The motion of bacteria, amoebae, and polystyrene
particles inside the water-filled channel is observed through a phase contrast
microscope from above. (b) The structure and dimensions of the glass-silicon-
glass bulk-acoustic-wave (BAW) device, driven by a piezoelectric transducer
(PZT). During experiments, PZT is facing downwards, toward the light source.
(c) A standing $\lambda/2$-wave (solid line) or a $\lambda$-wave (dashed line)
with a pressure amplitude $p_{\mathrm{a}}$ is established in the water-filled
channel across its width. The channel width in experiments is either
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ or
$700\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (d) The experiments of a given
experimental series rely on the measurement of $p_{\mathrm{a}}$ and wavenumber
$k$ by tracking the motion of polystyrene particles of known size and
properties. Measured $p_{\mathrm{a}}$, $k$, and the equivalent sphere radius
(ESR) from high-magnification images (1000x) then serve as an input parameter
for the determination of the effective acoustic contrast factor
$\varPhi_{\mathrm{eff}}$, based on a single-cell tracking. In the case of
bacteria, experimental videos are segmented to aid the tracking; many
individual bacteria are tracked across several experiments, yielding an
average $\varPhi_{\mathrm{eff}}$ of a sample. White scale bar corresponds to
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$, black scale bar corresponds to
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

We study acoustophoresis of several species of Legionella, E. coli, and P.
aeruginosa that are relevant for water safety.1, 23 We experimentally
determine the acoustic contrast factor of 10 Legionella species, specifically
L. pneumophila, L. fraseri, L. pascullei, L. anisa, L. cherrii, L. micdadei,
L. parisiensis, L. israelensis, L. gormanii, L. longbeachae, and of
Escherichia coli. All of the measured species possess a positive acoustic
contrast, leading to the focusing into a pressure node of a standing wave. A
broad variation in the shape and size across the species, as well as within
each species, considerably affects the magnitude of the acoustic radiation
force, resulting in observable differences in the rate of focusing among the
species. Since Legionella are known to infect protozoa, such as Acanthamoeba
species,1 and grow within, we also measure the acoustic contrast of A.
castellanii in cyst and trophozoite states, co-cultured with L. pneumophila.
A. castellanii generally also feature a positive acoustic contrast factor and
thus focus into a pressure node. Encysted A. castellanii exhibit a
significantly higher acoustic contrast factor than the trophozoites.

To complement experimental observations, we introduce a multi-body dynamics
algorithm for investigating the role of the experimentally-observed cell size
distribution and cell concentration in the collective acoustophoresis of
bacteria. The algorithm is based on Stokesian dynamics for modeling the
hydrodynamic interactions,24 supplemented by the Brownian motion,25 acoustic
radiation forces26 and acoustic interaction forces.27 The simulations reveal
that an increase in the cell concentration accelerates the rate of acoustic
focusing due to hydrodynamic interactions. This phenomenon, which we call
collective hydrodynamic focusing (CHF), does not significantly influence our
measurements, but could improve the understanding of many other studies and
applications that feature an externally-forced motion of bacteria-sized
objects, such as acoustic seed-particle trapping of bacteria and
nanoparticles,15 bacteria separation using ultrasound,28 or capturing of
paramagnetic beads using magnetic field gradient.29

## Results

Fig. 2: Acoustic focusing of Legionella species, E. coli, and P. aeruginosa in
a standing acoustic wave. (a) & (b) Time evolution of line widths of all the
investigated species, from two series of focusing experiments;
$p_{\mathrm{a}}=210\pm$8.4\text{\,}\mathrm{kPa}$$ in series 1 and
$p_{\mathrm{a}}=190\pm$13.7\text{\,}\mathrm{kPa}$$ in series 2, while the
frequency was constant across the two series at
$f=$3829\text{\,}\mathrm{kHz}$$. (c) Time evolution of line widths of L.
pneumophila subjected to different conditioning protocols;
$p_{\mathrm{a}}=352\pm$19.7\text{\,}\mathrm{kPa}$$ and
$f=$3941\text{\,}\mathrm{kHz}$$. (d) Snapshots of segmented binary images of
L. pneumophila and L. fraseri at $t=$0\text{\,}\mathrm{s}$$ and
$t=$20\text{\,}\mathrm{s}$$, and comparison of the line widths (shaded areas)
between the two species, defined as two standard deviations
$\sigma\left(y_{i}\right)$ of the $y$-position of all the bacteria in the
observed frame of $$292\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times$164\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ (the snapshots are half
of the actually analyzed length in the $x$-direction). The insets show a large
difference in the morphology of the two subspecies; white scale bar
corresponds to $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$, black scale bar
corresponds to $40\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The line width
curves in (a)-(c) represent a mean of $2\sigma\left(y_{i}\right)$ from 4-8
repetitions of experiments; shaded area surrounding the curves is a standard
deviation of the mean $2\sigma\left(y_{i}\right)$ across the repetitions. The
vertical dashed line in (a) represents a reference timestamp at
$t=$20\text{\,}\mathrm{s}$$, while the horizontal dashed line corresponds to a
reference line width of
$2\sigma\left(y_{i}\right)=$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Fig.
3: Experimentally observed variation in shape and size between investigated
bacterial species. (a) Legionella spp. investigated in experimental series 1
and 2; (b) E. coli and (c) P. aeruginosa investigated in series 2; (d)
conditioned L. pneumophila investigated in series 3. Images are obtained by a
high-magnification (1000x) phase-contrast microscopy. Red spheres next to L.
longbeachae are reference polystyrene particles with the diameter of
$2.21\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Scale bar corresponds to
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

#### Definition of an effective acoustic contrast factor

During acoustophoresis, the motion of an object of mass $m$ is governed by
Newton’s second law of motion

$m\frac{\mathrm{d}\bm{v}_{\mathrm{obj}}}{\mathrm{d}t}=\bm{F}_{\mathrm{S}}+\bm{F}_{\mathrm{ARF}},$
(1)

with the velocity of the object $\bm{v}_{\mathrm{obj}}$, the Stokes drag
$\bm{F}_{\mathrm{S}}$, and the ARF $\bm{F}_{\mathrm{ARF}}$. For length scales
applicable in our work, it is sufficient to approximate the motion of such an
object by assuming it reaches its terminal velocity instantaneously,30
resulting in the force equilibrium

$\bm{F}_{\mathrm{ARF}}=-\bm{F}_{\mathrm{S}}.$ (2)

The ARF on an object featuring a plane of symmetry in a plane standing wave in
an inviscid fluid that is small in size compared to the acoustic wavelength
$\lambda=c_{0}/f$, with the speed of sound in the fluid $c_{0}$, can be
generally written in the following form,31

$\bm{F}_{\mathrm{ARF}}=3V\varPhi
kE_{\mathrm{ac}}\sin\left(2ky\right)\bm{e}_{y},$ (3)

with the object’s volume $V$, the wavenumber $k$, the position of the object
in the wave $y$ along the direction of the unit vector $\bm{e}_{y}$ that is
parallel to the pressure gradient of the wave and normal to a plane of
symmetry of the object, and the acoustic energy density
$E_{\mathrm{ac}}=p_{\mathrm{a}}^{2}/\left(4\rho_{0}c_{0}^{2}\right)$, with the
equilibrium fluid density $\rho_{0}$ and the pressure amplitude
$p_{\mathrm{a}}$. The acoustic contrast factor $\varPhi$ represents the
acoustic scattering at the object and is consequently shape-, orientation- and
material-dependent. For a homogeneous sphere, $\varPhi$ can be expressed as26,
30

$\varPhi^{\mathrm{sph}}=\frac{1}{3}\left[\frac{5\rho_{\mathrm{obj}}-2\rho_{0}}{2\rho_{\mathrm{obj}}+\rho_{0}}-\frac{\kappa_{\mathrm{obj}}}{\kappa_{0}}\right]$
(4)

with the density of the object $\rho_{\mathrm{obj}}$, and the compressibility
of the particle and the fluid, $\kappa_{\mathrm{obj}}$ and $\kappa_{0}$,
respectively. For a long cylinder with its axis parallel to a pressure nodal
plane, $\varPhi$ follows as32

$\varPhi^{\mathrm{cyl}}=\frac{1}{3}\left[\frac{4\rho_{\mathrm{obj}}}{2\rho_{\mathrm{obj}}+2\rho_{0}}-\frac{\kappa_{\mathrm{obj}}}{\kappa_{0}}\right].$
(5)

Considering a polystyrene33 object in water results in
$\varPhi^{\mathrm{sph}}=0.1707$ and $\varPhi^{\mathrm{cyl}}=0.1709$; the
difference between $\varPhi^{\mathrm{sph}}$ and $\varPhi^{\mathrm{cyl}}$ grows
with the density contrast, but remains negligible in the density range
associated with biological cells. With the density contrast, viscous
contributions to the ARF also rise and can be absorbed into $\varPhi$, since
only the overall ARF scaling is affected while the spatial dependency with $y$
remains unchanged.34 For objects of various shapes, sizes, and materials,
$\varPhi$ that accounts for the viscosity can be extracted from a recent
numerical investigation.34

In addition to the ARF, a moving object in an acoustic field is subjected to
the Stokes drag35 due to the acoustic streaming velocity
$\left<\bm{v}_{2}\right>$ and the velocity of the object itself, namely,

$\bm{F}_{\mathrm{S}}=6\pi\eta\mathrm{ESR}\left[\bm{K}\bm{\cdot}\left(\left<\bm{v}_{2}\right>-\bm{v}_{\mathrm{obj}}\right)\right],$
(6)

with the shape-, orientation, and configuration-dependent translation tensor
$\bm{K}$ that depends also on the surrounding objects and walls, the
equivalent sphere radius $\mathrm{ESR}$, while neglecting the contribution
from rotations. $\bm{K}$ is implicitly normalized by $6\pi\mathrm{ESR}$,
compared to the dimensional $\bm{K}$ in Happel and Brenner.35 Since we are
interested only in the force component along the $y$-axis and since we did not
observe significant motion in other directions
($\bm{v}_{\mathrm{obj}}\bm{\cdot}\bm{e}_{x}\approx\bm{v}_{\mathrm{obj}}\bm{\cdot}\bm{e}_{z}\approx
0$), neglecting the streaming velocity
($\left<\bm{v}_{2}\right>\approx\bm{0}$), we can write

$\bm{F}_{\mathrm{S}}\bm{\cdot}\bm{e}_{y}=-6\pi\eta\mathrm{ESR}K_{yy}\left(\bm{v}_{\mathrm{obj}}\bm{\cdot}\bm{e}_{y}\right),$
(7)

where $K_{yy}$ can be interpreted as a correction to the Stokes drag on a
sphere due to the shape, orientation, and configuration of the system (i.e.
$K_{yy}=1$ for an isolated sphere in an unbounded medium). The influence of
walls on $K_{yy}$ of $\text{\,}\mathrm{\SIUnitSymbolMicro m}$-sized particles
in a typical acoustofluidic device was quantified by Barnkob _et al._ 13,
while Happel and Brenner 35 quantified the influence of the nearby objects in
a few representative configurations.

For a plane standing wave, the velocity of an object undergoing
acoustophoresis can be expressed as

$\bm{v}_{\mathrm{obj}}\bm{\cdot}\bm{e}_{y}=\frac{\mathrm{ESR}^{2}\varPhi_{\mathrm{eff}}kp_{\mathrm{a}}^{2}}{6\eta\rho_{0}c_{0}^{2}}\sin\left(2ky\right),$
(8)

with the effective acoustic contrast factor $\varPhi_{\mathrm{eff}}$ defined
as

$\varPhi_{\mathrm{eff}}=\frac{\varPhi}{K_{yy}},$ (9)

which depends on the material properties, orientation and shape of the object,
configuration of the system, as well as viscous contributions to the ARF. As
such, $\varPhi_{\mathrm{eff}}$ is not necessarily constant in $y$, since
distances to walls and other objects vary with the motion of the object.

In the experimental setting, we use a glass-silicon-glass bulk-acoustic-wave
(BAW) device shown in Fig. 1(a)-(c), driven by a piezoelectric transducer at
$\text{\,}\mathrm{MHz}$-frequencies, to generate a one-dimensional standing
acoustic wave across the width of the water-filled channel. The see-through
channel design is chosen to avoid the need for any kind of staining the
bacteria, since the influence of staining on the acoustic contrast factor is
unknown. The standing wave is experimentally characterized through
$p_{\mathrm{a}}$ and $k$, which are, as indicated in Fig. 1(d), measured by
tracking the motion of $4.97\text{\,}\mathrm{\SIUnitSymbolMicro m}$
polystyrene (PS) particles with known material properties, and fitting the
theoretical trajectory of a particle36, 13, 30

$y_{i}(t)=\frac{1}{k}\arctan\left\\{\tan\left[ky_{i}(t=0)\right]\exp\left[\frac{\varPhi_{\mathrm{eff}}p_{\mathrm{a}}^{2}}{3\rho_{0}c_{0}^{2}\eta}k\mathrm{ESR}^{2}t\right]\right\\}$
(10)

that is found by integrating eq. (8) in time, to the experimental trajectory,
using $p_{\mathrm{a}}$ and $k$ as the fitting parameters and assuming
$\varPhi_{\mathrm{eff}}=\varPhi^{\mathrm{sph}}$. The effective acoustic
contrast factor of a cell is measured by tracking the motion of individual
cells in the characterized standing wave and then fitting the theoretical
trajectory from eq. (10) to the experimental trajectory, using
$\varPhi_{\mathrm{eff}}$ as the fitting parameter. The $\mathrm{ESR}$ is for
amoebae specific to an individual cell, while for bacteria, the average ESR of
an individual species is used for all $\varPhi_{\mathrm{eff}}$ measurements
within the species. The spatial dependence of $\varPhi_{\mathrm{eff}}$ is
investigated in the supplemental material by measuring $\bm{v}_{\mathrm{obj}}$
of bacteria at various $y$-positions, and inserting the measured velocities
into eq. (8).

### Acoustophoresis of bacteria

Transient behaviour of cells during acoustic focusing is shown in Fig. 2 for
all bacterial species that were investigated. The behaviour is quantified
through the line width $2\sigma\left(y_{\mathrm{i}}\right)$, where
$\sigma\left(y_{\mathrm{i}}\right)$ is the standard deviation of the
$y$-positions of all the bacteria that are within the
$$292\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times$164\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ frame under observation.
The line width is for each species averaged across 4-10 repetitions of the
experiment. The results are split into two series, shown in Fig. 2(a) and
2(b), corresponding to two separate experimental sessions, at
$f=$3829\text{\,}\mathrm{kHz}$$ and with the average pressure amplitudes of
$210\pm$8.4\text{\,}\mathrm{kPa}$$ in series 1 and
$190\pm$13.7\text{\,}\mathrm{kPa}$$ in series 2. In the experiments of series
3 ($f=$3941\text{\,}\mathrm{kHz}$$,
$p_{\mathrm{a}}=352\pm$19.7\text{\,}\mathrm{kPa}$$) in Fig. 2(c), the acoustic
focusing is analyzed for samples of L. pneumophila that were subjected to
different conditioning protocols.

Figures 2(a) and 2(b) indicate that across the tested Legionella species, rate
of focusing varies significantly. Surprisingly, even subspecies L.
pneumophila, L. fraseri, and L. pascullei considerably differ in their
acoustic focusability, reaching for example
$2\sigma\left(y_{\mathrm{i}}\right)=$25\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ at $29\text{\,}\mathrm{s}$, $76\text{\,}\mathrm{s}$, and
$16.5\text{\,}\mathrm{s}$, respectively. The large spread in the observed
behaviour could be due to variation in the volume, shape, material properties,
or cell concentration. Significant differences in cell morphology across the
species and subspecies, observable in Fig. 3, indicate potentially large
influence of the morphology on the rate of focusing. However, an important
observation is that all the Legionella species, as well as E. coli and P.
aeruginosa, experience observable acoustic radiation force that drives the
focusing of cells towards the pressure node, despite the large variability in
the rate of focusing.

Fig. 4: Experimentally determined effective acoustic contrast factor
$\varPhi_{\mathrm{eff}}$ of the bacterial species. (a) Mean
$\varPhi_{\mathrm{eff}}$ for all the bacterial species, averaged over
$N=285-1290$ cell trajectories, obtained from 4-8 experiments per species. (b)
The corresponding equivalent sphere radius (ESR) measured from high-
magnification images (1000x) for $N=19-154$ cells per species. (c) The
corresponding initial cell concentration $\phi_{\mathrm{c}}$. (d) Mean
$\varPhi_{\mathrm{eff}}$ for L. pneumophila conditioned in BYE and tap water,
for 7 days and for 2 months. (e) The corresponding ESR measured from high-
magnification images (1000x) for $N=28-197$ cells per sample. (f) The
corresponding $\phi_{\mathrm{c}}$. Errorbars in the figure represent the
standard deviation from the mean value. Fig. 5: Experimentally determined
effective acoustic contrast factor $\varPhi_{\mathrm{eff}}$ of A. castellanii
co-cultured with L. pneumophila. (a) Mean $\varPhi_{\mathrm{eff}}$ for the
individual amoebae, averaged over $N=1-2$ cell trajectories ($N=6$ for the
mean values). Amoebae were subjected to a standing wave with
$f=$1987\text{\,}\mathrm{kHz}$$. Reference single-cell acoustic contrast
factors denoted with ∗ are obtained from Hartono _et al._ 9, while the
contrast of polystyrene is based on the reported material properties.33 (b)
The corresponding equivalent sphere radius (ESR). (c) Morphology of the three
visually distinguished states of A. castellanii, namely, trophozoite, cyst,
and an intermediate state (undetermined). Red and white spheres next to
amoebae are reference polystyrene particles (diameter of
$2.21\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$4.97\text{\,}\mathrm{\SIUnitSymbolMicro m}$, respectively). Scale bar
corresponds to $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

The trajectories of individual bacteria from experiments in Fig. 2(a)-(b) were
used to measure the effective acoustic contrast factor
$\varPhi_{\mathrm{eff}}$ of the bacterial species, shown in Fig. 4(a). The
mean measured $\varPhi_{\mathrm{eff}}$ varies between $0.28\pm 0.089$ for L.
anisa and $0.076\pm 0.029$ for L. parisiensis. In terms of the rate of
focusing from Fig. 2, L. parisiensis focuses significantly faster than L.
micdadei ($\varPhi_{\mathrm{eff}}=0.23\pm 0.069$), despite the greater
$\varPhi_{\mathrm{eff}}$ of the latter and the lower $p_{\mathrm{a}}$ in
experimental series 2. The faster focusing of L. parisiensis can be attributed
to the larger volume
($\mathrm{ESR}=0.86\pm$0.29\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) compared
to L. micdadei ($\mathrm{ESR}=0.44\pm$0.097\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$), since
$\bm{v}_{\mathrm{obj}}\propto\varPhi_{\mathrm{eff}}\mathrm{ESR}^{2}$. P.
aeruginosa and L. dumoffii were excluded from the
$\varPhi_{\mathrm{eff}}$-analysis, due to excessive swimming motion of the
former and the significantly elongated shape of the latter, as depicted in
Fig. 3. Size-wise, the mean ESR across the bacterial species varies between
$0.44\pm$0.097\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for L. micdadei and
$0.86\pm$0.29\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for L. parisiensis, and
generally decreases with an increase in the measured $\varPhi_{\mathrm{eff}}$.

In the environment (e.g. in water installations), Legionella can enter a so-
called viable but non-culturable (VBNC) state, in which they are still
dangerous for humans, while also very difficult to detect using standard
methods.37 In Fig. 2(c) and Fig. 4(d)-(f), we compare the acoustophoresis
among the same strain of L. pneumophila, but exposed to various conditions
that trigger VBNC state through starvation: cells conditioned in sterilized
tap water, compared to cells conditioned in buffered yeast extract (BYE) and
transferred into the sterilized tap water shortly before the experiment.
Measurements show that cells conditioned in tap water exhibit larger mean ESR
than the cells conditioned in BYE. Analogously to the general relation of
$\varPhi_{\mathrm{eff}}$ from Fig. 4(a)-(b), mean $\varPhi_{\mathrm{eff}}$
decreases with the increase in the mean ESR. Prolonged conditioning (2 months
compared to 7 days) enhances this trend. Overall, the conditioning influences
the magnitude of $\varPhi_{\mathrm{eff}}$, but the cells are in all cases
focusing towards the pressure node, as indicated in Fig. 2(c).

### Acoustophoresis of amoebae

Measurements of the effective acoustic contrast factor
$\varPhi_{\mathrm{eff}}$ for 14 amoebae are presented in Fig. 5. The amoebae
were classified into trophozoites, cysts, and undetermined (amoeba in
transition from a trophozoite to a cyst), based on a visual observation of the
presence of the cellulose wall that is characteristic for cysts.38 The results
indicate that the cysts have a higher acoustic contrast factor
($\varPhi_{\mathrm{eff}}=0.055\pm 0.018$) than the trophozoites
($\varPhi_{\mathrm{eff}}=0.018\pm 0.003$), while the undetermined are in-
between ($\varPhi_{\mathrm{eff}}=0.029\pm 0.010$). This is reasonable also
from a theoretical perspective, since the wall of a cyst contains cellulose
which is denser
($\rho\approx$1.6\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$)39 and less
compressible ($\kappa\approx 5\times$10^{-11}\text{\,}{\mathrm{Pa}}^{-1}$$)40
than a cell-like material
($\rho\approx$1.1\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$ and
$\kappa\approx 3.3\times$10^{-10}\text{\,}{\mathrm{Pa}}^{-1}$$ for a red blood
cell)9, leading to an increase in the effective density and to a decrease in
the effective compressibility of a cell, increasing its
$\varPhi_{\mathrm{eff}}$.

Fig. 6: Influence of collective hydrodynamic focusing (CHF) in computations
and experiments. (a) Extracted ratio between the multi-body
$\varPhi_{\mathrm{eff}}$ and an assigned single-body $\varPhi$ from the multi-
body numerical simulations in dependence of the initial cell concentration
$\phi_{\mathrm{c}}$, for the experimental particle size distribution of L.
pneumophila. (b) Influence of individual contributions to the computational
model on the extracted $\varPhi_{\mathrm{eff}}$ for $\varPhi=0.1$ (dashed
line) and $\phi_{\mathrm{c}}=4\cdot
10^{8}\mathrm{cells}/$\text{\,}\mathrm{mL}$$ (reference - the mean
$\varPhi_{\mathrm{eff}}$ is given by the dash-dotted line): assigning the mean
ESR of L. pneumophila to all the particles in the simulation (uniform size);
neglecting the Brownian motion (w/o Brownian); neglecting the near-field
lubrication interactions (w/o lubrication); neglecting the many-body
hydrodynamic interactions (w/o $\mathrm{M}^{\infty}$); using a more accurate
FTS algorithm24 of Stokesian dynamics that also accounts for stresslets (FTS);
increasing the temperature that increases the Brownian motion
($T=$350\text{\,}\mathrm{K}$$); including the inviscid acoustic interactions27
(with AIF). (c) Experimental estimation of the influence of
$\phi_{\mathrm{c}}$ on the measured $\varPhi_{\mathrm{eff}}$ for a series of
experiments with L. pneumophila in a $200\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ wide channel, performed at $f=$3930.9\text{\,}\mathrm{kHz}$$, with
$p_{\mathrm{a}}=382\pm$30.1\text{\,}\mathrm{kPa}$$ and
$\mathrm{ESR}=0.69\pm$0.17\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

In our experiments, the cysts with
$\mathrm{ESR}=6.5\pm$1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) appear
marginally smaller than the trophozoites with
$\mathrm{ESR}=7.0\pm$1.0\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Since the
trajectory of an amoeba in the standing acoustic wave depends on the factor
$\varPhi_{\mathrm{eff}}\mathrm{ESR}^{2}$, which amounts to
$0.86\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ for trophozoites and
$2.34\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ for cysts, the rate of
focusing of cysts is significantly higher than that of trophozoites.

### Multi-body effects

The comparison across species of Legionella from Fig. 4 indicates that the
initial concentration of cells in the sample $\phi_{\mathrm{c}}$ could
influence the measured $\varPhi_{\mathrm{eff}}$. However, in the existing
studies of acoustophoresis of bacteria or bacteria-sized particles,17, 41, 42,
18, 43 the multi-body effects on the acoustic focusing are of unknown
magnitude and generally neglected, despite the concentrations of cells or
particles in such experiments often surpassing
$10^{8}\mathrm{cells}/$\text{\,}\mathrm{mL}$$.

In Fig. 6, we investigate the influence of the cell concentration of L.
pneumophila on the measured acoustic contrast $\varPhi_{\mathrm{eff}}$ through
multi-body computations, under conditions corresponding to our experiments
($f=$3829\text{\,}\mathrm{kHz}$$, $p_{\mathrm{a}}=$210\text{\,}\mathrm{kPa}$$,
channel width of $200\text{\,}\mathrm{\SIUnitSymbolMicro m}$). The multi-body
simulations for
$10^{8}\mathrm{cells}/$\text{\,}\mathrm{mL}$\leq\phi_{\mathrm{c}}\leq
10^{9}\mathrm{cells}/$\text{\,}\mathrm{mL}$$ and for the assigned single-cell
acoustic contrast factor $\varPhi$ between $0.01$ and $0.1$, reveal a
significant dependence of the measured $\varPhi_{\mathrm{eff}}$ on
$\phi_{\mathrm{c}}$ in Fig. 6(a). We analyze this relation through the ratio
$\varPhi_{\mathrm{eff}}/\varPhi$ that quantifies the multi-body effects
($\varPhi_{\mathrm{eff}}/\varPhi=1$ means a negligible influence of multi-body
effects), which shows a linear dependence on the cell concentration in
$\mathrm{cells}/$\text{\,}\mathrm{mL}$$:
$\varPhi_{\mathrm{eff}}/\varPhi=4.1\cdot 10^{-9}\phi_{\mathrm{c}}+0.90$. A
higher concentration simulation, at $\phi_{\mathrm{c}}=2\cdot
10^{9}\mathrm{cells}/$\text{\,}\mathrm{mL}$$, fits on the linear relation
relatively well.

To determine the mechanism responsible for the amplification of
$\varPhi_{\mathrm{eff}}$ as $\phi_{\mathrm{c}}$ is increased, we turn off or
modify individual contributions in our computational model. The results shown
in Fig. 6(b) indicate that the influences of the experimentally-determined
cell size distribution, Brownian motion, near-field lubrication interactions,
and the acoustic interaction force (AIF) are negligible compared to the many-
body hydrodynamic interactions ($\mathrm{M}^{\infty}$). Turning the latter off
results in $\varPhi_{\mathrm{eff}}$ that is very close to the single-particle
$\varPhi$ of $0.1$; furthermore, turning off other contributions to the
dynamics of bacteria has negligible influence on the extracted
$\varPhi_{\mathrm{eff}}$.

To relate the simulations to our $\varPhi_{\mathrm{eff}}$ measurements, we
perform a series of experiments with our standard experimental setup for
bacteria that was used in Fig. 4. The results for L. pneumophila with
$6.6\cdot 10^{8}\mathrm{cells}/$\text{\,}\mathrm{mL}$\leq\phi_{\mathrm{c}}\leq
5.2\cdot 10^{9}\mathrm{cells}/$\text{\,}\mathrm{mL}$$, presented in Fig. 6(c),
indicate that cell-concentration-related multi-body effects are negligible in
our experiments.

### Swimmers in acoustophoresis

To contain a bacterium with a given maximal swimming speed of
$v_{\mathrm{bac}}^{\mathrm{max}}$ within the one-dimensional acoustic trap,
the pressure amplitude has to exceed a certain threshold
($p_{\mathrm{a}}^{\mathrm{min}}$). $v_{\mathrm{bac}}^{\mathrm{max}}$ defines a
swimming thrust through the balance with the Stokes drag on the bacterium, and
balancing the swimming thrust with the ARF gives the approximation for
$p_{\mathrm{a}}^{\mathrm{min}}$, namely,

$p_{\mathrm{a}}^{\mathrm{min}}=\sqrt{\frac{3\eta\rho_{0}c_{0}^{3}}{\pi
f\mathrm{ESR}^{2}\varPhi_{\mathrm{eff}}}v_{\mathrm{bac}}^{\mathrm{max}}},$
(11)

resulting in $p_{\mathrm{a}}^{\mathrm{min}}\approx$667\text{\,}\mathrm{kPa}$$
for E. coli ($\varPhi_{\mathrm{eff}}=0.11\pm 0.046$,
$\mathrm{ESR}=0.65\pm$0.047\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) with
$v_{\mathrm{bac}}^{\mathrm{max}}=$25\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$44 and $f=$3.9\text{\,}\mathrm{MHz}$$, assuming
a spherical shape.

In the acoustophoresis experiments, we observed swimming motion in several of
the bacterial species, namely, L. cherrii
($v_{\mathrm{bac}}^{\mathrm{max}}=$15.4\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$), L. dumoffii
($v_{\mathrm{bac}}^{\mathrm{max}}=$6.1\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$), L. parisiensis
($v_{\mathrm{bac}}^{\mathrm{max}}=$14.6\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$), P. aeruginosa
($v_{\mathrm{bac}}^{\mathrm{max}}=$40.1\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$), E. coli
($v_{\mathrm{bac}}^{\mathrm{max}}=$5.4\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$), and L. pneumophila conditioned in sterilized
tap water for 2 months
($v_{\mathrm{bac}}^{\mathrm{max}}=$13.6\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$). In the case of elongated rod-shaped swimmers
(e.g. L. dumoffii), the acoustic torque45, 46 aligned the swimmers parallel to
the pressure nodal line.

## Discussion

Bacteria are known to have a cell volume-dependent volumetric percentage of
biomass,47 which has a density larger than water at
$1.35\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$ for E. coli.48 This
implies that the density of bacterial cells increases with the decrease in the
volume of the cells, which is proportional to $\mathrm{ESR}^{3}$. Since the
deviation of the cell density from the density of water increases the acoustic
contrast factor, this relation between the cell density and ESR could offer an
explanation for the observed dependency of $\varPhi_{\mathrm{eff}}$ on the ESR
in Fig. 4(a)-(b) and Fig. 4(d)-(e).

In the future, it would be important to analyze $\varPhi_{\mathrm{eff}}$ of
Legionella that are grown in amoebae, as the growth process in amoebae has
been shown to significantly influence the morphology of the bacteria,49 which
could influence the acoustic focusability of Legionella.

The difference in the acoustic contrast factor of the amoebae in different
stages of the encystment cycle could be exploited to acoustically sort the
different states of amoebae. This could be realized by combining an on-the-fly
analysis of the acoustic contrast factor of an amoeba, similar to the device
used by Wang _et al._ 50, and a subsequent step that would sort the amoeba
into a state-specific outlet based on the readout of the acoustic contrast
factor. An applicable acoustic sorting device was already developed, for
fluorescent activated sorting of particles.51

Literature suggests that the difference in the size of the two states of A.
castellanii should be more significant, for example, Bowers and Korn 38
reported mean diameters of $26.5\pm$0.17\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ for trophozoites and $16.2\pm$0.13\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ for cysts, but could also be affected by the culturing process.52 The
relatively small sizes of the amoebae that were present in our experiments
could stem from the co-culturing with L. pneumophila, which are known to
destabilize and lyse the host amoebae.1 Some of the destabilized amoebae could
be lysed during the sample handling steps and during the pumping of the sample
into the BAW device; Moffat and Tompkins 53 for example drew a sample through
a 27-gauge needle with an inner diameter of
$210\text{\,}\mathrm{\SIUnitSymbolMicro m}$ three times to lyse the
destabilized amoebae; for comparison, the capillaries used to pump-in the
sample into our BAW device have an inner diameter of
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. During such processes, the larger
amoebae would experience higher shear stresses in the narrow sections of the
tubing and BAW device, and therefore exhibit a higher probability of lysis,
explaining the small size of amoebae in our analysis.

The theoretically demonstrated acceleration of the acoustic focusing and thus
higher apparent $\varPhi_{\mathrm{eff}}$ explains some previously reported
experimental findings. For example, the influence of the E. coli concentration
on the feasibility of acoustically trapping them has been observed by
Hammarström _et al._ 15. Similar concentration dependencies of bacteria-sized
objects were observed already before,28 but no study so far, to the best of
our knowledge, provided a generally-accepted explanation of the underlying
mechanism. Our finding agrees also with a theoretical study in
magnetophoresis, where focusing of paramagnetic particles due to a magnetic
field was found to accelerate with an increase in particle concentration.29 We
call this phenomenon that stems from the many-body hydrodynamic interactions
collective hydrodynamic focusing (CHF). CHF could apply to many existing
studies on acoustophoresis of bacteria and bacteria-sized particles that
neglected multi-body effects.17, 41, 42, 18, 43 The phenomenon could be
significantly influenced by the presence of walls in the experimental devices,
analogous to the paradoxical difference in sedimenting particle suspensions of
low volumetric concentration that sediment faster when they do not fill the
whole container, which allows the fluid flow to go around the cluster of
particles.35 A similar effect could be caused by local variation in cell
concentration or spatial variation of the force field. In addition,
specifically to acoustophoresis — the presence of the acoustic microstreaming
that can affect the ARF54, 34 as well as the AIF55 — could contribute. These,
so-far unexplored influences diminish the CHF in our system for the
measurement of $\varPhi_{\mathrm{eff}}$, which we showed experimentally.
However, in larger channels and with more complex acoustic modes, past
experiments15 already demonstrated the CHF and its potential benefits. In the
future, studying CHF for differently-sized particles and with an extended
computational model that accounts for the walls56 and other influences could
be used to quantify the multi-body effects in past and future experimental
studies. Furthermore, CHF could be exploited to control and improve methods
that rely on the presence of larger functionalized57 or non-functionalized15
particles to up-concentrate bacteria.

To acoustically focus all the Legionella in an at-line system, including the
swimmers, the pressure amplitudes of $\gtrsim$0.7\text{\,}\mathrm{MPa}$$ would
be required, assuming $f=$4\text{\,}\mathrm{MHz}$$ and an appropriate focusing
length within a device for a given flow rate. In the literature,
$p_{\mathrm{a}}$ of above $1.5\text{\,}\mathrm{MPa}$ have been reported for
similar devices and operating conditions,58 making such an application
feasible in the near future.

## Methods and theory

### Devices and fabrication

The lab-on-a-chip devices, an example of which is shown in Fig. 1, were
produced through cleanroom processing and feature a three-layered structure
made of silicon and glass. A blank silicon wafer
($200\pm$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ thickness) was first
anodically bonded to a glass wafer ($500\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ thickness). The channel designs with
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$700\text{\,}\mathrm{\SIUnitSymbolMicro m}$ width of the main channel were
then patterned on the silicon through photolithography (resist: S1828/S1818,
Shipley, $4000\,\mathrm{rpm}$; developer: AZ351B, Microchemicals) and then
etched with an inductively coupled plasma deep reactive ion etching (ICP-DRIE)
machine (Estrellas, Oxford instruments) through the entire depth of the
silicon $\sim$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Afterwards, a glass
wafer ($200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness) was anodically
bonded onto the etched silicon-glass wafer. The wafer was then diced into
individual $11\times$50\text{\,}\mathrm{mm}$$ chips with a wafer saw (DAD3221,
Disco corporation). Fused silica capillaries
($164\pm$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ outer diameter,
$100\pm$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ inner diameter, Molex) were
inserted into the inlets and outlets of the chips and fixed with a two-
component glue (5 Minute Epoxy, Devcon). A piezoelectric transducer (PZT)
(length$\times$width$\times$thickness $=$ $25\times
1.55\times$1\text{\,}\mathrm{mm}$$ for $700\text{\,}\mathrm{\SIUnitSymbolMicro
m}$-channel width device and $20\times 4\times$0.5\text{\,}\mathrm{mm}$$ for
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$-channel width device, Pz26,
Meggitt Ferroperm) was glued to each device with a conductive epoxy (H20E,
EPO-TEK). PZT was attached to the bottom side of the device that was not
facing the microscope lens, in order to allow for the use of a high-
magnification (100x) solid immersion lens. Copper cables
($0.15\text{\,}\mathrm{mm}$ diameter) were connected to the PZT using a
conductive silver paste and glued to a device with instant glue for mechanical
stability. To fix the devices under a microscope, a chip-holder with the outer
dimensions corresponding to a standard microscope glass slide
($$75\text{\,}\mathrm{mm}$\times$25\text{\,}\mathrm{mm}$$) was designed and
laser-cut from acrylic glass (PMMA).

### Sample preparation

Bacterial strains used in the study: Legionella pneumophila Philadelphia-1
(DSM 7513), Legionella pneumophila subsp. fraseri (DSM 7514), Legionella
pneumophila subsp. pascullei (DSM 7515), Legionella anisa (DSM 17627),
Legionella cherrii (DSM 19213), Legionella micdadei (DSM 16640), Legionella
dumoffii (DSM 17625), Legionella parisiensis (DSM 19216), Legionella
israelensis (DSM 19235), Legionella gormanii (DSM 16641), Legionella
longbeachae (DSM 10572), Pseudomonas aeruginosa (PAO1), and Escherichia coli
(ATCC25922).

All Legionella spp. were grown in buffered yeast extract (BYE) according to
Chatfield and Cianciotto 3 overnight at
$37\text{\,}\mathrm{\SIUnitSymbolCelsius}$. P. aeruginosa and E. coli were
grown in lysogeny broth (LB). For acoustophoresis experiments,
$1\text{\,}\mathrm{mL}$ of overnight culture was used. Cells were washed twice
by centrifugation at $10000\times g$ and resuspension of the pellet in
$1\text{\,}\mathrm{mL}$ autoclaved tap water. Cell concentrations were
determined by optical density ($600\text{\,}\mathrm{nm}$) measurements and
Neubauer counting chambers (Bioswisstec). For starvation and conditioning of
Legionella in BYE and sterile tap water, cells were kept in the media during 5
days or two months before washing, resuspension in sterile tap water, and use
in acoustophoresis experiments.

Acanthamoeba castellanii was cultured in peptone-yeast-glucose (PYG) medium
for 3 days at $30\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and resuspended in
sterile tap water for acoustophoresis experiments. To induce cyst formation
cultures were kept at $4\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 7 days
prior to resuspension. Co-cultures with L. pneumophila Philadelphia-1 were
prepared according to Jarraud _et al._ 59. Briefly, amoebae were resuspended
in “Pages’ Amoeba Saline Solution” (PAS) and the concentration was adjusted to
$10^{5}\mathrm{cells}/$\text{\,}\mathrm{mL}$$. The amoebae were incubated at
$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$ after addition of L. pneumophila
Philadelphia-1 at a multiplicity of infection $\geq 1$ ($\geq
10^{5}\mathrm{cells}/$\text{\,}\mathrm{mL}$$). Acoustophoresis experiments
were performed with infected amoebae harvested from $2\text{\,}\mathrm{mL}$
co-culture and resuspended in $500\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of
sterile tap water.

### Experimental procedure

For visual observation of phenomena within the channels of the lab-on-a-chip
devices, we used a phase-contrast microscope (Leica) with a camera (Leica MC
120 HD) recording at $30$ frames per second. To visualize and characterize the
acoustic field inside a device,36 polystyrene (PS) beads (white
$4.97\pm$0.06\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and red
$2.21\pm$0.03\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, microParticles GmbH)
were used. A regular syringe was connected to the inlet capillary of a device,
for supplying the particles/cells dispersed in water, as well as for cleaning
the chip in-between experiments. A function generator (JDS-2900, JUNTEK) was
used to drive the PZT, monitored through an oscilloscope (UTD2025CL, UNI-T).
Devices were additionally characterized by measuring the admittance of the PZT
with an impedance analyzer (Sine Phase Z-Check 16777k).

Cleaning in-between experiments with different species was performed by a
flushing sequence of $\geq$0.1\text{\,}\mathrm{mL}$$ of
$10\text{\,}\mathrm{\char 37\relax}$ sodium hypochlorite solution,
$\geq$0.05\text{\,}\mathrm{mL}$$ of $70\text{\,}\mathrm{\char 37\relax}$
ethanol solution, and $\geq$0.32\text{\,}\mathrm{mL}$$ of sterilized tap water
from the same batch as the water used for suspending the cells.

The trajectories of PS particles that were used for estimation of the acoustic
pressure amplitude ($p_{\mathrm{a}}$) in the channel were obtained through a
particle tracking velocimetry plugin TrackMate60 in Fiji,61 and post-processed
with a custom Matlab62 code that fits the theoretical trajectory from eq. (10)
to an experimental trajectory, using $p_{\mathrm{a}}$ and wavenumber $k$ as
the fitting parameters. The trajectories of amoebae were obtained the same way
as those of PS particles, while an additional cell-segmentation step was
performed for bacteria, using iLastik.63 Lastly, the acoustic contrast factor
($\varPhi_{\mathrm{eff}}$) of cells was determined by using a modified version
of the Matlab code that fits the theoretical trajectory to an experimental
cell trajectory, using a measured mean equivalent sphere radius (ESR) of
individual species as an input parameter and $\varPhi_{\mathrm{eff}}$ as the
fitting parameter. Depending on the rate of focusing, the number of frames for
the analysis of trajectories of bacteria was reduced by a factor of 5 to 15,
to minimize the processing time. To minimize the influence of multi-body
interactions, only sufficiently isolated PS particles and amoebae were
considered.

The size of individual bacteria species was measured from high magnification
(1000x) phase-contrast microscopic images, by measuring the maximum and
minimum Feret diameters in a software package Fiji, and using a methodology
introduced by Saccà 64 to compute the mean cell volume $V$, which was used to
extract the ESR, specifically,

$\textrm{ESR}=\left(\frac{3V}{4\pi}\right)^{1/3}.$ (12)

The cell concentration of four bacterial samples was determined using Neubauer
counting chambers (Bioswisstec); for the rest of the samples, we determined an
average conversion factor $3.255\cdot
10^{6}$\text{\,}\mathrm{mL}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}\text{\,}{\mathrm{}}^{2}$$ that transforms the initial number of
bacteria in a $$292\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times$164\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ video frame into the
cell concentration in $\text{\,}\mathrm{mL}$.

The mean swimming speeds of individual bacterial species were determined by
visually identifying swimmers and measuring the distance and time that the
swimmers travelled when actively swimming in Fiji.

### Acoustic radiation force

In the underlying theory, basic equations are linearized using the
perturbation approach.30 Accordingly, the physical fields are expanded in a
series $\square=\square_{0}+\square_{1}+\square_{2}+\dots$, where $\square$
represents the field, while the subscript denotes the respective order.

The time-averaged ARF on an object in an acoustic field is the mapping of a
stress tensor $\bm{\sigma}$ onto the outward pointing surface normal
$\bm{n}(t)$, integrated over the object’s oscillating surface $S(t)$, namely,

$\bm{F}_{\mathrm{ARF}}=\left\langle\int_{S(t)}\bm{\sigma}\bm{\cdot}\bm{n}(t)\mathrm{d}S\right\rangle.$
(13)

It has been proven that the second-order expression for the ARF can also be
written as 65

$\bm{F}_{\mathrm{ARF}}=\int_{S_{0}}\left[\left\langle\bm{\sigma}_{2}\right\rangle-\rho_{0}\left\langle\bm{v}_{1}\bm{v}_{1}\right\rangle\right]\bm{\cdot}\bm{n}_{0}\mathrm{d}S,$
(14)

with the velocity field $\bm{v}$, and the equilibrium density $\rho_{0}$. The
difference between the mean second-order stress tensor
$\left\langle\bm{\sigma}_{2}\right\rangle$ and the Reynolds stress
$\rho_{0}\left\langle\bm{v}_{1}\bm{v}_{1}\right\rangle$ is mapped onto the
normal $\bm{n}_{0}$ pointing out of the arbitrary static surface $S_{0}$
enclosing the object, and integrated over $S_{0}$. The mean second-order
stress tensor is defined as

$\left\langle\bm{\sigma}_{2}\right\rangle=-\left\langle
p_{2}\right\rangle\bm{I}+\eta\left(\bm{\nabla}\left\langle\bm{v}_{2}\right\rangle+\left(\bm{\nabla}\left\langle\bm{v}_{2}\right\rangle\right)^{T}\right),$
(15)

with the pressure field $p$ and the dynamic viscosity $\eta$. The
microstreaming and other viscous effects at the second-order are contained in
the stress tensor $\left\langle\bm{\sigma}_{2}\right\rangle$, while the first-
order viscous scattering effects are also part of $\bm{v}_{1}$.

Equation (14) reduces to the inviscid ARF in eq. (3) by neglecting the
microstreaming velocity $\left\langle\bm{v}_{2}\right\rangle$, and neglecting
$\eta$, which simplifies $\left\langle p_{2}\right\rangle$, as demonstrated by
Bruus 30. In our system, $f\approx$3.9\text{\,}\mathrm{MHz}$$ for bacteria,
which corresponds to the viscous boundary layer thickness of
$\delta=\sqrt{\eta/\left(\pi\rho_{0}f\right)}\approx$0.286\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$, in water, corresponding to $0.33\leq\delta/\mathrm{ESR}\leq 0.65$.
Recently, it was shown by Pavlic _et al._ 34 that for a cell-like material at
$\delta/\mathrm{ESR}\approx 0.7$, the viscous contributions to the ARF can
reach up to $\approx$10\text{\,}\mathrm{\char 37\relax}$$ of the inviscid ARF.
For amoebae, the viscous contributions are lower due to the larger size. In
our approach, this contribution is implicitly included in the measured
$\varPhi_{\mathrm{eff}}$. The inviscid and viscous acoustic interaction force
between bacteria are expected to be small compared to the ARF, except in the
vicinity of a pressure node.55

The shape of individual cells, under the condition that it does not deviate
considerably from a sphere (as for example L. dumoffii), has a negligible
viscous and inviscid contribution to the ARF, as long as the actual volume of
the cell is taken into account.34 However, the overall dynamics could be
influenced by the shape-, orientation-, and configuration-dependent drag.35,
45 Both these effects are inherently accounted for in the measured
$\varPhi_{\mathrm{eff}}$, as defined in eq. (9). The correction to the drag
coefficient $K_{yy}$ from eq. (7) due to the presence of the walls is for
bacteria-sized particles in our channels below $1\text{\,}\mathrm{\char
37\relax}$, while for amoebae it could go above $10\text{\,}\mathrm{\char
37\relax}$.11 In the measurements, the influence is minimized, since the
reduced drag due to the presence of the walls is to a similar degree affecting
$p_{\mathrm{a}}$ that is measured by tracking the
$4.97\text{\,}\mathrm{\SIUnitSymbolMicro m}$ polystyrene particles at the same
focal plane. The correction to the ARF due to the presence of the walls is,
however, negligible.66

### Multi-body computational model

The multi-body computational model is based on the force-torque (FT) Stokesian
dynamics algorithm proposed by Durlofsky _et al._ 24, which computes the
velocity of an individual particle based on the positions of all the particles
($\bm{x}$), their sizes, the properties of the fluid, and the external forces
$\bm{F}_{ext}$ acting on the particles. The change in the position of the
particles $\varDelta\bm{x}$ can be computed as67

$\varDelta\bm{x}=\bm{\mathrm{R}}_{FU}^{-1}\bm{\cdot}\bm{F}_{ext}\varDelta
t+k_{B}T\bm{\nabla}\bm{\cdot}\bm{\mathrm{R}}_{FU}^{-1}\varDelta
t+\bm{X}(\varDelta t),$ (16)

with the time step of the numerical integration $\varDelta t$, Boltzmann
constant $k_{B}$, temperature $T$, random displacement due to Brownian motion
$\bm{X}(\varDelta t)$, and with the resistance matrix $\bm{\mathrm{R}}_{FU}$
that describes the many-body hydrodynamic interactions, as well as the near-
field lubrication effects.24, 67, 25 We neglect the Brownian motion term in
eq. (16) involving $\bm{\nabla}\bm{\cdot}\bm{\mathrm{R}}_{FU}^{-1}$, as we are
working with relatively low volumetric concentrations of bacteria. The random
displacement $\bm{X}(\varDelta t)$ is the only contribution of the Brownian
motion to our multi-body system, and it is computed by enforcing a zero mean
value and a covariance given as $\text{cov}\left(\bm{X}(\varDelta
t),\bm{X}(\varDelta t)\right)=2k_{B}T\bm{\mathrm{R}}_{FU}^{-1}\varDelta t$.
For external forces, we consider the inviscid acoustic radiation force from
equation (3),26 and the inviscid acoustic interaction forces (AIF) from Silva
and Bruus 27. Unless stated differently, we compute and update the many-body
hydrodynamic interactions ($\mathrm{M}^{\infty}$ matrix) every $20\varDelta
t$, in line with Bossis and Brady 67.

The computational domain that we consider has a cross-section of
$$120\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\times$120\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and the width of
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$, corresponding to the channel
width in the experimental device. The number of randomly dispersed particles
$N_{\mathrm{p}}$ in each simulation is determined by the initial cell
concentration $\phi_{\mathrm{c}}$, ranging from $N_{\mathrm{p}}=288$ for
$\phi_{\mathrm{c}}=1\cdot 10^{8}\mathrm{cells}/$\text{\,}\mathrm{mL}$$ to
$N_{\mathrm{p}}=5760$ for $\phi_{\mathrm{c}}=2\cdot
10^{9}\mathrm{cells}/$\text{\,}\mathrm{mL}$$. The temperature used in
simulations is $300\text{\,}\mathrm{K}$, unless stated otherwise.

### Material properties

For water, we assume the density of
$1\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$, the speed of sound of
$1500\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$, and the dynamic viscosity
of $1\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$. For polystyrene (PS) and the
associated measurement of $p_{\mathrm{a}}$, we use
$\varPhi_{\mathrm{eff}}=\varPhi^{\mathrm{sph}}=0.165$, which is based on the
material properties reported by Selfridge 33. The compressibility of a solid
object is computed as the inverse of the bulk modulus $K_{\mathrm{obj}}$
through the relation

$\kappa_{\mathrm{obj}}=\frac{1}{K_{\mathrm{obj}}}=\frac{1}{\rho_{\mathrm{obj}}\left(c_{\mathrm{P}}^{2}-\frac{4}{3}c_{\mathrm{S}}^{2}\right)},$
(17)

with the speed of sound of primary and secondary waves, $c_{\mathrm{P}}$ and
$c_{\mathrm{S}}$, respectively. The compressibility of a fluid is computed as
$\kappa_{0}=1/(\rho_{0}c_{0}^{2})$. Density of the cytoplasm of a prokaryotic
cell for the computational model is estimated to be
$1.156\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$, based on the mass
fractions given by Rojas 68, namely $\sim 70\%$ water
($1\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$), $\sim 15\%$ proteins
($1.36\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$ from 69), $\sim 7\%$
nucleic acid ($1.89\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$ from 70),
and other $\sim 8\%$ comprised of sugars and other molecules (assuming
$1.5\text{\,}\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$).

## Author Contributions

All authors designed the research; A.P. designed and built the experimental
devices; M.V. prepared all biological samples; A.P. and M.V. conducted the
experiments; A.P. developed the code for post-processing the experimental
results; A.P. and M.V. post-processed all the experimental results; A.P.
established the computational code, carried out the simulations, and analyzed
the data; A.P. prepared the initial manuscript draft; all authors contributed
to scientific discussion; all authors commented on and contributed to the
writing of the manuscript.

## Conflicts of interest

There are no conflicts to declare.

## Data availability

Data is available from the corresponding author upon reasonable request.

## Acknowledgements

We would like to acknowledge funding from InnoSuisse through the contract
number 49528.1 for the project Contactless separation and concentration of
Legionella species in drinking water installations, and the support of the
project partners at Georg Fischer JRG AG: Stephan Buerli, Philippe Cachot,
Enrico Camelin, Antonio de Agostini, Steffen Lehmann, Hop Nguyen, and Simon
Obrist. We thank Mingzhen Fan and Mathias Horn for providing A. castellanii.
We thank Thierry Baasch for supporting the implementation of the algorithm for
computing hydrodynamic multi-body interactions. We thank Bernhard Zybach for
building the temperature monitoring system.

## References

  * Bartram _et al._ 2007 J. Bartram, Y. Chartier, J. V. Lee, K. Pond and S. Surman-Lee, _Legionella and the prevention of legionellosis_ , World Health Organization, 2007.
  * Van Kenhove _et al._ 2019 E. Van Kenhove, K. Dinne, A. Janssens and J. Laverge, _American Journal of Infection Control_ , 2019, 47, 968–978.
  * Chatfield and Cianciotto 2013 C. H. Chatfield and N. P. Cianciotto, _Legionella_ , Springer, 2013, pp. 151–162.
  * Bruus _et al._ 2011 H. Bruus, J. Dual, J. Hawkes, M. Hill, T. Laurell, J. Nilsson, S. Radel, S. Sadhal and M. Wiklund, _Lab on a Chip_ , 2011, 11, 3579–3580.
  * Undvall _et al._ 2022 E. Undvall, F. Garofalo, G. Procopio, W. Qiu, A. Lenshof, T. Laurell and T. Baasch, _Physical Review Applied_ , 2022, 17, 034014.
  * Apfel 1982 R. Apfel, _The British Journal of Cancer. Supplement_ , 1982, 5, 140\.
  * Wiklund 2012 M. Wiklund, _Lab on a Chip_ , 2012, 12, 2018–2028.
  * Goddard _et al._ 2006 G. Goddard, J. C. Martin, S. W. Graves and G. Kaduchak, _Cytometry Part A: The Journal of the International Society for Analytical Cytology_ , 2006, 69, 66–74.
  * Hartono _et al._ 2011 D. Hartono, Y. Liu, P. L. Tan, X. Y. S. Then, L.-Y. L. Yung and K.-M. Lim, _Lab on a Chip_ , 2011, 11, 4072–4080.
  * Wang _et al._ 2018 H. Wang, Z. Liu, D. M. Shin, Z. G. Chen, Y. Cho, Y.-J. Kim and A. Han, _Microfluidics and Nanofluidics_ , 2018, 22, 1–7.
  * Baasch _et al._ 2018 T. Baasch, P. Reichert, S. Lakämper, N. Vertti-Quintero, G. Hack, X. C. i Solvas, A. deMello, R. Gunawan and J. Dual, _Biophysical Journal_ , 2018, 115, 1817–1825.
  * Jiménez _et al._ 2022 A. V. Jiménez, D. C. O. Cabezas, M. Delay, I. G. Gómez and M. Camacho, _Ultrasound in Medicine & Biology_, 2022.
  * Barnkob _et al._ 2012 R. Barnkob, P. Augustsson, T. Laurell and H. Bruus, _Physical Review E_ , 2012, 86, 056307.
  * Rodgers 1979 F. Rodgers, _Journal of Clinical Pathology_ , 1979, 32, 1195–1202.
  * Hammarström _et al._ 2012 B. Hammarström, T. Laurell and J. Nilsson, _Lab on a Chip_ , 2012, 12, 4296–4304.
  * Habibi and Neild 2019 R. Habibi and A. Neild, _Lab on a Chip_ , 2019, 19, 3032–3044.
  * Antfolk _et al._ 2014 M. Antfolk, P. B. Muller, P. Augustsson, H. Bruus and T. Laurell, _Lab on a Chip_ , 2014, 14, 2791–2799.
  * Ugawa _et al._ 2022 M. Ugawa, H. Lee, T. Baasch, M. Lee, S. Kim, O. Jeong, Y.-H. Choi, D. Sohn, T. Laurell, S. Ota _et al._ , _Analyst_ , 2022.
  * Schwarz and Dual 2012 T. Schwarz and J. Dual, AIP Conference Proceedings, 2012, pp. 779–782.
  * Gutiérrez-Ramos _et al._ 2018 S. Gutiérrez-Ramos, M. Hoyos and J. Ruiz-Suárez, _Scientific Reports_ , 2018, 8, 1–8.
  * Zhao _et al._ 2020 S. Zhao, M. Wu, S. Yang, Y. Wu, Y. Gu, C. Chen, J. Ye, Z. Xie, Z. Tian, H. Bachman _et al._ , _Lab on a Chip_ , 2020, 20, 1298–1308.
  * Jepras _et al._ 1989 R. Jepras, D. Clarke and W. Coakley, _Journal of Immunological Methods_ , 1989, 120, 201–205.
  * Organization _et al._ 2017 W. H. Organization _et al._ , _Guidelines for drinking-water quality, 4th edition, incorporating the 1st addendum_ , World Health Organization, 2017\.
  * Durlofsky _et al._ 1987 L. Durlofsky, J. F. Brady and G. Bossis, _Journal of Fluid Mechanics_ , 1987, 180, 21–49.
  * Brady 1993 J. F. Brady, _The Journal of Chemical Physics_ , 1993, 98, 3335–3341.
  * Yosioka and Kawasima 1955 K. Yosioka and Y. Kawasima, _Acta Acustica united with Acustica_ , 1955, 5, 167–173.
  * Silva and Bruus 2014 G. T. Silva and H. Bruus, _Physical Review E_ , 2014, 90, 063007\.
  * Miles _et al._ 1995 C. Miles, M. Morley, W. Hudson and B. Mackey, _Journal of applied bacteriology_ , 1995, 78, 47–54.
  * Mikkelsen and Bruus 2005 C. Mikkelsen and H. Bruus, _Lab on a Chip_ , 2005, 5, 1293–1297.
  * Bruus 2012 H. Bruus, _Lab on a Chip_ , 2012, 12, 1014–1021.
  * Sepehrirahnama _et al._ 2021 S. Sepehrirahnama, S. Oberst, Y. K. Chiang and D. Powell, _Physical Review E_ , 2021, 104, 065003.
  * Wei _et al._ 2004 W. Wei, D. B. Thiessen and P. L. Marston, _The Journal of the Acoustical Society of America_ , 2004, 116, 201–208.
  * Selfridge 1985 A. R. Selfridge, _IEEE transactions on sonics and ultrasonics_ , 1985, 32, 381–394.
  * Pavlic _et al._ 2022 A. Pavlic, P. Nagpure, L. Ermanni and J. Dual, _Physical Review E_ , 2022, 106, 015105.
  * Happel and Brenner 2012 J. Happel and H. Brenner, _Low Reynolds number hydrodynamics: with special applications to particulate media_ , Springer Science & Business Media, 2012, vol. 1.
  * Barnkob _et al._ 2010 R. Barnkob, P. Augustsson, T. Laurell and H. Bruus, _Lab on a Chip_ , 2010, 10, 563–570.
  * Dietersdorfer _et al._ 2018 E. Dietersdorfer, A. Kirschner, B. Schrammel, A. Ohradanova-Repic, H. Stockinger, R. Sommer, J. Walochnik and S. Cervero-Aragó, _Water Research_ , 2018, 141, 428–438.
  * Bowers and Korn 1969 B. Bowers and E. D. Korn, _The Journal of Cell Biology_ , 1969, 41, 786–805.
  * Sun 2005 C. C. Sun, _Journal of Pharmaceutical Sciences_ , 2005, 94, 2132–2134.
  * Quesada Cabrera _et al._ 2011 R. Quesada Cabrera, F. Meersman, P. F. McMillan and V. Dmitriev, _Biomacromolecules_ , 2011, 12, 2178–2183.
  * Gerlt _et al._ 2022 M. Gerlt, A. Paeckel, A. Pavlic, P. Rohner, D. Poulikakos and J. Dual, _Physical Review Applied_ , 2022, 17, 014043.
  * Devendran _et al._ 2020 C. Devendran, K. Choi, J. Han, Y. Ai, A. Neild and D. J. Collins, _Lab on a Chip_ , 2020, 20, 2674–2688.
  * Hawkes _et al._ 1997 J. Hawkes, M. Limaye and W. Coakley, _Journal of Applied Microbiology_ , 1997, 82, 39–47.
  * Wadhwa and Berg 2021 N. Wadhwa and H. C. Berg, _Nature Reviews Microbiology_ , 2021, 1–13.
  * Yamahira _et al._ 2000 S. Y. S. Yamahira, S.-i. H. S.-i. Hatanaka, M. K. M. Kuwabara and S. A. S. Asai, _Japanese Journal of Applied Physics_ , 2000, 39, 3683\.
  * Schwarz _et al._ 2015 T. Schwarz, P. Hahn, G. Petit-Pierre and J. Dual, _Microfluidics and Nanofluidics_ , 2015, 18, 65–79.
  * Loferer-Krossbacher _et al._ 1998 M. Loferer-Krossbacher, J. Klima and R. Psenner, _Applied and Environmental Microbiology_ , 1998, 64, 688–694.
  * Robertson _et al._ 1998 B. Robertson, D. Button and A. Koch, _Applied and Environmental Microbiology_ , 1998, 64, 3900–3909.
  * Cirillo _et al._ 1994 J. D. Cirillo, S. Falkow and L. S. Tompkins, _Infection and Immunity_ , 1994, 62, 3254–3261.
  * Wang _et al._ 2019 H. Wang, Z. Liu, D. M. Shin, Z. G. Chen, Y. Cho, Y.-J. Kim and A. Han, _Lab on a Chip_ , 2019, 19, 387–393.
  * Jakobsson _et al._ 2014 O. Jakobsson, C. Grenvall, M. Nordin, M. Evander and T. Laurell, _Lab on a Chip_ , 2014, 14, 1943–1950.
  * Stratford and Griffiths 1978 M. P. Stratford and A. J. Griffiths, _Microbiology_ , 1978, 108, 33–37.
  * Moffat and Tompkins 1992 J. F. Moffat and L. Tompkins, _Infection and Immunity_ , 1992, 60, 296–301.
  * Baasch _et al._ 2019 T. Baasch, A. Pavlic and J. Dual, _Physical Review E_ , 2019, 100, 061102\.
  * Pavlic _et al._ 2022 A. Pavlic, L. Ermanni and J. Dual, _Physical Review E_ , 2022, 105, L053101.
  * Durlofsky and Brady 1989 L. J. Durlofsky and J. F. Brady, _Journal of Fluid Mechanics_ , 1989, 200, 39–67.
  * Lee _et al._ 2019 S. Lee, B. W. Kim, H.-S. Shin, A. Go, M.-H. Lee, D.-K. Lee, S. Kim and O. C. Jeong, _Micromachines_ , 2019, 10, 770.
  * Qiu _et al._ 2022 W. Qiu, T. Baasch and T. Laurell, _Physical Review Applied_ , 2022, 17, 044043.
  * Jarraud _et al._ 2013 S. Jarraud, G. Descours, C. Ginevra, G. Lina and J. Etienne, _Legionella_ , Springer, 2013, pp. 27–56.
  * Tinevez _et al._ 2017 J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte and K. W. Eliceiri, _Methods_ , 2017, 115, 80–90.
  * Schindelin _et al._ 2012 J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch, C. Rueden, S. Saalfeld, B. Schmid _et al._ , _Nature Methods_ , 2012, 9, 676–682.
  * Mat 2019 The Mathworks, Inc., Natick, Massachusetts, _MATLAB version R2019b_ , 2019\.
  * Berg _et al._ 2019 S. Berg, D. Kutra, T. Kroeger, C. N. Straehle, B. X. Kausler, C. Haubold, M. Schiegg, J. Ales, T. Beier, M. Rudy _et al._ , _Nature Methods_ , 2019, 16, 1226–1232.
  * Saccà 2016 A. Saccà, _PLoS One_ , 2016, 11, e0151955.
  * Doinikov 1994 A. Doinikov, _Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences_ , 1994, 447, 447–466.
  * Baasch and Dual 2020 T. Baasch and J. Dual, _Physical Review Applied_ , 2020, 14, 024052\.
  * Bossis and Brady 1987 G. Bossis and J. F. Brady, _The Journal of Chemical Physics_ , 1987, 87, 5437–5448.
  * Rojas 2020 E. R. Rojas, _Physical Microbiology_ , Springer, 2020, pp. 1–14.
  * Quillin and Matthews 2000 M. L. Quillin and B. W. Matthews, _Acta Crystallographica Section D: Biological Crystallography_ , 2000, 56, 791–794.
  * Spahn _et al._ 2000 C. M. Spahn, P. A. Penczek, A. Leith and J. Frank, _Structure_ , 2000, 8, 937–948.