11institutetext:  22institutetext: † 22email<EMAIL_ADDRESS>33institutetext: 1 Faculty of Electrical Engineering, Mathematics and Computer
Science, Delft University of Technology, Mekelweg 4, 2628 CD Delft, the
Netherlands 44institutetext: 2 Leiden Observatory, Leiden University, Niels
Bohrweg 2, 2333 CA Leiden, the Netherlands 55institutetext: 3 Division of
Particle and Astrophysical Science, Graduate School of Science, Nagoya
University, Furocho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan
66institutetext: 4 National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan 77institutetext: 5 SRON – Netherlands Institute
for Space Research, Niels Bohrweg 4, 2333 CA Leiden, the Netherlands
88institutetext: 6 Institute of Astronomy, Graduate School of Science, The
University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan
99institutetext: 7 Research Center for the Early Universe, Graduate School of
Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan 1010institutetext: 8 Kitami Institute of Technology, 165 Koen-cho,
Kitami, Hokkaido 090-8507, Japan

# DESHIMA 2.0: Rapid redshift surveys and multi-line spectroscopy of dusty
galaxies

M. Rybak1,2,† T. Bakx3,4 J. Baselmans1,5
K. Karatsu1,5 K. Kohno6,7 T. Takekoshi8,6
Y. Tamura3 A. Taniguchi3 P. van der Werf2
A. Endo1

###### Abstract

We present a feasibility study for the high-redshift galaxy part of the
Science Verification Campaign with the 220-440 GHz Deshima 2.0 integrated
superconducting spectrometer on the ASTE telescope. The first version of the
Deshima 2.0 chip has been recently manufactured and tested in the lab. Based
on these realistic performance measurements, we evaluate potential target
samples and prospects for detecting the [CII] and CO emission lines. The
planned observations comprise two distinct, but complementary objectives: (1)
acquiring spectroscopic redshifts for dusty galaxies selected in far-
infrared/mm-wave continuum surveys; (2) multi-line observations to infer
physical conditions in dusty galaxies.

###### Keywords:

(sub)mm astronomy, spectroscopy, high-redshift Universe, galaxies, integrated
superconducting spectrometer

††journal: Journal of Low Temperature Physics

##  1 Introduction

Throughout cosmic history, more than half of all the stars form in dust-
obscured galaxies1, 2, 3. Due to their massive dust reservoirs, these dusty
star-forming galaxies (DSFGs) are often invisible in the optical / near-IR
part of the spectrum but bright in the far-infrared (FIR) to (sub)-mm
wavelengths. Consequently, thousands of DSFGs have been identified in wide-
field continuum surveys in the 0.1-2.0 mm regime2, 4, 3.

However, studies of DSFGs suffer from a considerable _redshift bottleneck_.
This is because long-wavelength continuum observations provide only weak
constraints on the redshift of individual galaxies - a critical prerequisite
for further emission line studies and high-resolution imaging with
interferometers such as ALMA5. Similarly, the optical and near-IR
spectroscopic follow-up is often inefficient due to the high extinction in
DSFGs, particularly in the FIR-brightest sources2.

Consequently, (sub)mm-wave spectroscopy has become the key to obtaining robust
spectroscopic redshifts for dusty galaxies at high redshift. This is chiefly
through the rotational emission line of 12CO and the fine-structure transition
of C+, the 158-$\mu$m [CII] line (rest-frame frequency $f_{0}$=1900.5 GHz).
Typically, these are conducted using heterodyne receivers via spectral scans,
requiring multiple instrument tunings.

Alternatively, several dedicated wide-band instruments have been developed:
e.g., grating spectrometers such as the now-defunct Z-Spec6, 7 at the Caltech
Submilimeter Observatory (CSO; 190-305 GHz) or wideband heterodyne receivers
such as the ZSpectrometer8 at the Green Bank Telescope (26.5-40 GHz) and the
Redshift Search Receiver (RSR9, 73-111 GHz) on the Large Millimeter Telescope.
While these have allowed redshift measurements out to $z\simeq 6$10, mainly
using the CO emission lines, due to the relative faintness of these lines,
such observations are limited to the rare, very bright galaxies. Critically,
the bright [CII] fine-structure line - ideal for rapid redshift measurements
due to its brightness - is generally beyond the reach of these instruments.

To properly exploit the [CII] line for redshift measurements, wideband
spectroscopy must be extended to higher frequencies, i.e. the 350-GHz and
400-GHz atmospheric windows. These frequency bands are particularly promising
for spectroscopic confirmation of DSFGs, because the number density of DSFGs
(and thus [CII] emitters) peaks between $z=2-4$11, corresponding to [CII]
being redshifted to 380-600 GHz.

In addition to measuring redshifts, mm-wave spectroscopy provides critical
insights into the physical conditions in DSFGs. Namely, observations of
multiple chemical species (e.g., CO, C+, C, O, dust continuum) can be linked
to the underlying physical conditions (e.g., gas density and temperature,
irradiation, turbulence) using chemical and radiative transfer modelling. For
example, the $J_{\mathrm{upp}}\geq 8$ CO lines are sensitive to non-thermal
gas excitation, such as heating by X-rays and cosmic rays or turbulence, which
might be significant in intensely star-forming DSFGs. Indeed, recent studies
point towards highly excited CO rotational lines in strongly lensed DSFGs12,
13. The octave-wide bandwidth offered by the ISS architecture is ideally
suited to simultaneous observations of multiple high-excitation CO lines in
high-redshift galaxies.

##  2 DESHIMA 2.0: instrument description

Deshima (DEep Spectroscopic HIgh-redshift MApper) is an integrated
superconducting spectrometer (ISS) operating in the mm-wave regime14, 15,
combining on a single chip a superconducting filterbank with an array of
microwave kinetic inductance detectors (KIDs). Several other KID mm-wave
spectrometers are currently under development, e.g., CONCERTO16, 17 (130–310
GHz), and Super-Spec18, 19 (190–310 GHz).

The first version of the instrument - Deshima 1.0 - achieved the first light
in 201715, 20. These demonstrated the spectroscopy of point sources (post-AGB
star IRC+10216, merging galaxy pair VV 114 at redshift $z=0.02$) and the on-
the-fly spectroscopic mapping of extended regions (the Orion KL star-forming
region and the nearby galaxy NGC 253).

In 2022, an upgraded Deshima 2.0 spectrometer will be installed at the ASTE
(Atacama Submillimeter Telescope Experiment) 10-meter telescope21 in the
Atacama desert, Chile, at an altitude of 4,860 metres. Deshima 2.0 will
provide an instantaneous frequency coverage of 220 - 440 GHz at $R\simeq 500$
($\Delta v\simeq$ 600 km/s). Besides the significantly expanded bandwidth,
major upgrades between Deshima 1.0 and Deshima 2.0 include a leaky-lens
antenna22, improved filter design, and a sky-position chopper111For a detailed
summary, see the contribution by Taniguchi et al., in this volume23..
Together, these upgrades result in a factor 4-8 improvement in sensitivity
over Deshima 1.0. Further sensitivity improvements might be achieved
explicitly modelling the instrument and atmospheric noise, rather than simply
subtracting the on- and off-source spectra24.

The first version of the Deshima 2.0 on-chip filterbank has been recently
manufactured and tested in the lab23. The filters cover almost the entire
target bandwidth, with a mean peak coupling efficiency of 14%, increasing up
to 30-50% for some channels (target: $\sim$30 %). The main source of
discrepancy between the current and target performance is uneven channel
spacing, which reduces the coupling efficiency of individual channels.

In Fig. 1, we show the current and target Deshima 2.0 performance compared to
Z-Spec25 at the CSO on Mauna Kea, and the current suite of receivers on the
12-m Atacama Pathfinder EXperiment (APEX) telescope (nFLASH230, SEPIA345,
nFLASH460222The APEX sensitivity calculations are based on https://www.apex-
telescope.org/ heterodyne/calculator/ns/index.php), which is just a few
kilometres away from ASTE. Compared to the Z-Spec, Deshima 2.0 has sensitivity
1.5–2.0$\times$ lower for a given pwv value. However, ASTE has more favourable
weather conditions (pwv=0.6 mm corresponds to the top 25th percentile for
ASTE, but only the 10th percentile for Mauna Kea26). Deshima 2.0 is thus
competitive with Z-Spec, with the added advantage of covering the 305-440 GHz
range. Compared to APEX, the current Deshima 2.0 sensitivity is 4–5$\times$
lower; the target sensitivity will be $\sim$2$\times$ higher. At that point,
science objectives that would require four or more APEX tunings will be more
economically achieved with Deshima 2.0. For such applications, Deshima 2.0
will be directly competitive with APEX.

Figure 1: 5$\sigma$ minimum detectable line flux (MDLF) for a 600 km/s wide
line for the current and target Deshima 2.0 performance23, compared to the
Z-Spec25 and the current suite of APEX receivers. For all instruments, we
assume a precipitable water vapour of 0.6 mm, target elevation of 60 deg and
on-source time of 3.6 hr. The instantaneous bandwidth of Deshima 2.0
corresponds to 20 APEX tunings (APEX bandwidth = 8.0-15.8 GHz, depending on
the receiver).

##  3 DESHIMA 2.0: Science Verification Campaign targets

The primary targets for Deshima are DSFGs with large apparent FIR luminosity
(i.e. $L_{\mathrm{8-1100\mu m}}$ of few $\times 10^{13}L_{\odot}$), many of
which are strongly gravitationally lensed. In fact, hundreds of strongly
lensed dusty galaxies were discovered in wide-field continuum surveys at FIR
and sub-mm wavelengths with Herschel27, 28, 29 and Planck satellites30 and the
South Pole Telescope31 (SPT).

Fig. 2 shows the redshift and apparent FIR luminosity distribution of DSFGs
from the SPT and Planck samples (virtually all with spectroscopic redshifts)
and the Herschel-selected high/low-redshift samples. For comparison, we show
the limiting FIR luminosity of sources for which the CO(5–4), (10–9), (13–12)
and the [CII] line can be detected at 5$\sigma$ level in 5-hr on source
($\sim$12 hr total with overheads). These are based on empirical CO–FIR
relations from Kamenetzky et al.32 for CO(5–4), Greve et al.33 for CO(10–9)
and (13–12), and $L_{\mathrm{[CII]}}/L_{\mathrm{FIR}}$ ratio of $10^{-3}$,
typical for DSFGs34, 35. The CO (and [CII]) luminosities in individual
galaxies can deviate from these trends by up to 1 dex33, 36, 32, 12; large-
sample surveys with Deshima 2.0 will further constrain the range of CO
excitation in DSFGs.

Figure 2: Redshift distribution and apparent FIR luminosity of bright DSFGs
observable from the ASTE site from the SPT, Planck and the Herschel DSFGs,
compared to a) the current and b) targetDeshima 2.0 performance (lower). The
black lines indicate the 5$\sigma$ detection limits for the CO(5–4), (10–9),
(13–12), and [CII] emission from individual sources, assuming empirical CO-FIR
ratios32, 33 and a [CII]/FIR ratio of $10^{-3}$. We assume 12-hr observations
at precipitable water vapour of 1.0 mm, i.e. less than two median observing
nights. Galaxies at $z\leq 3.3$ have the [CII] line outside Deshima 2.0
frequency coverage, but will be prime targets for multi-line spectroscopy. We
highlight J1329+2243 - a very bright $z=2.05$ lensed DSFGs, a simulated
spectrum for which is presented in Fig. 3. The current Deshima 2.0 performance
already allows science-grade observations of high-redshift galaxies.

###  3.1 Spectroscopic redshifts for bright Herschel-selected galaxies

One of the primary aims of the Deshima 2.0 Science Verification Campaign is to
demonstrate the rapid redshift acquisition capability. The atmospheric windows
and bandwidth of Deshima offer two promising regions for efficient redshift
searches.

Firstly, at lower-redshift ($z\sim 0.5-2.0$), the large bandwidth of Deshima
offers multiple-line detections of galaxies. Counterintuitively, finding
robust redshifts of lower-redshift dust-obscured galaxies is challenging.
While the spectral lines suffer less from the cosmological dimming, wider
bandwidths are necessary to cover the entire possible redshift space. In
addition, DSFGs at $z\leq 2$ are undetected in SPT and Planck survey, while
most Herschel redshift follow-up prioritises high-$z$ targets. The Herschel
catalogues have thus left $\sim 4000$ $z\sim 1$ targets unexplored. As $z\leq
1$ DSFGs have low lensing probability 37, this sample presents a population of
intrinsically-bright galaxies after the peak of the cosmic star-forming
activity 1, i.e. when galaxy-wide quenching should be in full effect.
Deshima’s 220-GHz bandwidth will allow fast redshift acquisition for these
sources, removing degeneracies in redshift due to the wide spacing of the CO
lines at low redshift 38.

As for the high-redshift end, while most bright DSFGs have secure
spectroscopic redshifts, a large population of DSFGs with lower apparent
luminosities remains unexplored. Namely, the $\geq$1000 deg2 Herschel
footprint39, 40 provides a sample of $\approx$ 2000 “500-$\mu$m risers”: DSFGs
with flux density peaking at/beyond 500-$\mu$m. A 500-$\mu$m rising colour
selection, relative to 250 and 350 $\mu$m, promises to select the highest-
redshift Herschel candidates41, with $z_{\rm phot}\geq 3.5$. A major advantage
of Deshima 2.0 is the wideband spectroscopy in the 385-440 GHz band
(interrupted by the 425-GHz telluric line), corresponding to the lower half of
ALMA Band 8. This enables [CII] observations at $z=3.3-3.9$, the epoch when
the [CII] luminosity function is predicted to peak11.

We expect to invest a total of 400 hr for both the low- and high-redshift goal
(200 hr per goal); this should yield robust redshifts for $\sim 15-20$
galaxies each.

Figure 3: A simulated spectrum of J1329+2243, assuming 3 hours on-source at
source elevation of 40 deg and pwv=0.6 mm. This very bright source is the best
target for CO ladder observations with Deshima 2.0. The red line shows the
input spectrum based on existing observations12. The grey line denotes the
S/N=5 threshold for each channel; increasing the on-source time will move the
grey line downwards. We expect to detect four CO rotational lines; our
observations will also cover several other potentially bright lines - a key
discovery space for Deshima 2.0.

###  3.2 Multi-line spectroscopy of bright lensed DSFGs

Because of their high apparent brightness, the lensed DSFGs selected by Planck
and SPT are ideally suited for demonstrating multi-line spectroscopy
capabilities of Deshima 2.0. Multi-line spectroscopy is critical for
constraining different physical properties. For example, the [CII] 158-$\mu$m
line is a sensitive probe of the far-UV irradiation. On the other hand, the
excitation of CO rotational lines is primarily driven by gas density42, 43.
Depending on the complexity of the data, the models can range from static,
one-dimensional gas slabs44, 45, 46, 47 to fully three-dimensional models48
and might incorporate time evolution.

Previous studies with wide-band spectrometers such as Z-Spec have been limited
to a handful of very bright sources49, 50, 51; with Deshima 2.0, we will
expand this approach to a much larger sample of DSFGs. In particular, apart
from the very bright Planck-selected sources, the high-excitation CO emission
in DSFGs remains almost completely unexplored. Deshima 2.0 should detect the
high-$J$ CO lines in the brightest Planck and SPT DSFGs in only a few hours
on-source.

As a demonstration of a pilot observation, Fig. 3 shows a simulated spectrum
of J1329+ 2243. J1329+2243 is the most FIR-luminous source at $z\geq 3$ from
the samples considered in Fig. 2, with extensive archival CO observations12.
As for the chip properties, we adopt a nominal R=500 design with 347 frequency
channels spanning the 220–440 GHz range. The response function of individual
filters follows a Lorentzian profile with a peak coupling efficiency of 13.6%
(based on the laboratory measurements of the first Deshima 2.0 chip). Note
that this is a conservative estimate; further sensitivity improvements are
expected from the optimization of the chip manufacturing process. We assume a
precipitable water vapour (pwv) of 0.6 mm, source elevation of 40 degrees, and
a total on-source time of 3 hr. Even with such a short integration, we expect
robust detections of multiple CO lines and [CI](2–1). Moreover, we will cover
the potentially bright H2O and H2O+ lines.

Finally, we note that due to the relatively low spectral resolution
(R$\sim$500), several emission lines might blend together: particularly
CO(7–6) and [CI](2–1) (rest-frame frequency separation $\Delta f_{0}=2.69$
GHz, $\Delta v\sim$ 1000 km/s) and CO(9–8) and OH+ ($\Delta f_{0}=3.86$ GHz,
$\Delta v\sim$ 1100 km/s). The latter pair is particularly susceptible to
blending as OH+ often traces gas that is out- or inflowing at high velocities
and might be seen in absorption13, 52; consequently, the CO(9–8) flux measured
from $R\sim 500$ spectra might be significantly over/underestimated.

##  4 Conclusions

We have presented the high-redshift extragalactic science case for the Deshima
2.0 integrated superconducting spectrometer, which will be mounted at the 10-m
ASTE telescope in 2022. Thanks to its combination of an octave-wide bandwidth,
access to high frequencies and competitive sensitivity, Deshima 2.0 will allow
science-grade observations of high-redshift galaxies. The first integrated
chip has been manufactured and tested in the lab. The current performance is
already comparable with the Z-Spec grating spectrometer, and after further
improvements, should be competitive with APEX.

In the upcoming Science Verification Campaign, we expect to: (1) measure
redshifts for $\sim$30 Herschel-selected galaxies at $z\sim 1$ and $z\geq 4$;
(2) obtain multi-line spectroscopy of 5–10 strongly lensed DSFGs from the
Planck and SPT samples to study the physical conditions in these extreme
sources. These figures are conservative; with further sensitivity
improvements, the campaign can be expanded significantly. These Deshima 2.0
observations will pave the way for future large-scale spectroscopic campaigns
with ultra-wideband, multi-pixel MKID spectrometers53.

###### Acknowledgements.

M. R. is supported by the NWO Veni project “Under the lens” (VI.Veni.202.225).
J. B. is supported by the European Research Council ERC (ERC-CoG-2014 -
Proposal #648135 MOSAIC). Y. T. and T. B. are supported by NAOJ ALMA
Scientific Research (grant No. 2018-09B). T. T. was supported by MEXT Leading
Initiative for Excellent Young Researchers (grant No. JPMXS0320200188). The
ASTE telescope is operated by the National Astronomical Observatory of Japan
(NAOJ).

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# Active beating of a reconstituted synthetic minimal axoneme

Isabella Guido<EMAIL_ADDRESS>Max Planck Institute for Dynamics and
Self-Organization (MPIDS), 37077 Göttingen, Germany  Andrej Vilfan  Max Planck
Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
Jožef Stefan Institute, 1000 Ljubljana, Slovenia  Kenta Ishibashi  Graduate
School of Frontier Biosciences, Osaka University, Osaka 5650871, Japan  Center
for Information and Neural Networks (CiNet), National Institute of Information
and Communications Technology, Osaka 565-0871, Japan  Hitoshi Sakakibara
Advanced ICT Research Institute, National Institute of Information and
Communications Technology, Kobe 651-2492, Japan  Misaki Shiraga  Graduate
School of Life Science, University of Hyogo, Hyogo 678-1297, Japan  Eberhard
Bodenschatz  Max Planck Institute for Dynamics and Self-Organization (MPIDS),
37077 Göttingen, Germany  Institute for Dynamics of Complex Systems, Georg-
August-University Göttingen, 37073 Göttingen, Germany  Laboratory of Atomic
and Solid-State Physics, Cornell University, Ithaca, NY 14853, United States
Ramin Golestanian<EMAIL_ADDRESS>Max Planck Institute for
Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany  Rudolf
Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU,
United Kingdom  Kazuhiro Oiwa  Advanced ICT Research Institute, National
Institute of Information and Communications Technology, Kobe 651-2492, Japan
Graduate School of Life Science, University of Hyogo, Hyogo 678-1297, Japan

###### Abstract

Propelling microorganisms through fluids and moving fluids along cellular
surfaces are essential biological functions accomplished by long, thin
structures called motile cilia and flagella, whose regular, oscillatory
beating breaks the time-reversal symmetry required for transport. Although
top-down experimental approaches and theoretical models have allowed us to
broadly characterize such organelles and propose mechanisms underlying their
complex dynamics, constructing minimal systems capable of mimicking ciliary
beating and identifying the role of each component remains a challenge. Here
we report the bottom-up assembly of a minimal synthetic axoneme, which we call
a synthoneme, using biological building blocks from natural organisms, namely
pairs of microtubules and cooperatively associated axonemal dynein motors. We
show that upon provision of energy by ATP, microtubules undergo rhythmic
bending by cyclic association-dissociation of dyneins. Our simple and unique
beating minimal synthoneme represents a self-organized nanoscale biomolecular
machine that can also help understand the mechanisms underlying ciliary
beating.

Reconstituted axoneme, synthoneme, microtubule pair, axonemal dynein,
molecular nanomachine

##  I Introduction

Eukaryotic cilia and flagella accomplish vital physiological tasks
Satir_Christensen ; Gilpin , ranging from the swimming and feeding of
microorganisms to the clearance of mucous in airways Loiseau.Viallat2020 ,
establishment of left-right body asymmetry and transport of cerebrospinal
fluid in brain ventricles Faubel.Eichele2016  and the spinal cord. Defects in
ciliary assembly or in their function are responsible for diseases that affect
millions of patients worldwide Fliegauf . Each cilium’s structural framework
is the axoneme, which consists of nine microtubule doublets and several
hundred other proteins that maintain its structure, drive and control the
beat, and transport the materials Ishikawa . The beating is powered by
axonemal dyneins that form two arrays (outer and inner dynein arms) along the
side wall of the doublet microtubules. They generate interdoublet shear that
causes the bending of microtubule doublets Lin .

The analysis of Chlamydomonas flagellar mutants Kamiya2014 , biochemical and
biophysical Oiwa99  identification and structural studies Oiwa2003 ,
especially using cryoelectron tomography Lin , have progressed our
understanding on the molecular organization and function of each axonemal
component Mitchison2010 . Recently, axonemal dyneins have been reported to
have mechanical and structural properties so distinct from other protein
motors to be suitable for the fabrication of biomolecular nanomachine Oiwa2012
; Furuta .

How this complex structure drives the beating patterns is not yet well
understood Mitchison2010 ; Sartori ; Lindemann ; Lindemann.Lesich2010 ;
Geyer.Howard2016 ; Sartori.Julicher2016 . A controlled assembly of minimal
systems that reproduce certain aspects of ciliary dynamics with far less
cellular feedback mechanisms can be an important step towards solving the
puzzle of ciliary beating dynamics. To address this issue we use the bottom-up
approach and reconstitute in-vitro a minimal axoneme made of the main axonemal
constituents, outer dynein arms (ODA) and microtubules, and we reproduce the
bending oscillation of the natural cilia (flagella). We call this structure a
“synthoneme” in analogy to the the well described axoneme.

The modularity of the reconstituted structure of this synthoneme, its defined
geometry and the nature of the building blocks allow the system to go beyond
the minimal beating systems reported in the past. Namely, two microtubule
doublets extended from frayed and disintegrated axonemes that exhibit
repetitive buckling Kamiya1986 ; kamiya2005 ; mixtures of microtubule bundles
and kinesin-1 clusters, which self-assemble in cilia-like beating structures
Sanchez_2011 ; Vilfan2019 . The controlled assembly of our synthoneme has the
potential for the development of controllable beating structures that can
serve as a tool for the understanding of the flagellar/ciliary beating and for
eventual technological advance in the field of bio-inspired systems.

## Results

The reconstituted minimal bending system is made of taxol-stabilized
microtubules and ODA extracted from flagella of the green algae Chlamydomonas
reinhardtii (See Supplementary Fig. S1). We chose to use ODA for its ability
to self-organize in arrays on microtubules Owa , the high yield in
preparation, and its robust motility in vitro Kamiya2014 ; Furuta2009 ;
Seetharam .

Microtubules are polymerized in an experimental chamber by flowing porcine
brain tubulin over fragmented Chlamydomonas axonemes (hereafter named seeds).
The seeds represent the joint basal end of the microtubule pair and are
spontaneously attached to the coverslip. This heterologous seeding procedure
provides microtubule pairs with the same polarity in which the filaments grow
at a distance on axoneme scale. With the seeds the relative movement of
microtubules is blocked at the basal end of the pair, but free at the distal
end.

These properties allow the repetitive buckling of the microtubules driven by
dynein. The distance between two polymerized microtubule singlets at the basal
end probably ranges from the minimal distance between two A-tubules of
adjacent doublets (approx. 30 nm) and the maximum corresponding to the
diameter of the axoneme (approx. 200 nm) (see Supplementary Fig. S2). Since
the length of microtubules is longer than 10 $\mu$m, the tips of the filaments
thermally fluctuated unless ODAs formed cross-bridges between them. Therefore,
a precise determination of the inter-microtubule distance in individual cases
is not applicable in our observations.

The ODA is provided to the system by perfusion of extract from axonemes
containing ODAs and ODA-docking complex (ODA-DC) into the flow chamber. Due to
the presence of the docking complex, ODAs are associated with the microtubule
with high cooperativity in an end-to-end fashion and thereby align
unidirectionally along one protofilament Owa . We confirm the reconstitution
of ODAs on microtubules with negative staining electron microscopy (Fig. 1A
and Supplementary Fig. S3) and the longitudinal 24-nm periodicity of ODA/ODA-
DC aligned along the filament is identified by Fourier analysis (Fig. 1B).
This periodicity is characteristic for ODAs in native axonemes Lin  and
reconstituted ODAs systems shown in previous studies Oda .

The EM images and subsequent analysis also provide information about the
occupancy of dynein arrays on microtubules (see SI text and Figs. S3 and S4
for more details). At high mixing ratios ODA arrays occupy most of
microtubules and the length of the arrays are probably underestimated due to
the limited size of the observation field. However, the length distribution of
contiguous dynein arrays shows that the average length of about $\sim 200\,\rm
nm$ (corresponding to approx. eight ODAs) is largely independent of the mixing
ratio (Fig. 1C). We concluded that the ODA alignment is a highly cooperative
assembly process.

Figure 1: A) Negative staining EM image showing a portion of a microtubule
microtubule and an assembly of the ODA complex. Globular blobs are regularly
aligned along the microtubule. Scale bar: 100 nm. B) Fourier transform of EM
images shows a 24 nm periodicity of ODA on microtubules. 1/4 nm-1 layer lines
derived from microtubules are clearly observed (scale unit: 1/24 nm-1). C)
Length distribution of ODA patches at different mixing ratios between ODA and
microtubules. The peak length is around 100–200 nm irrespective of the mixing
ratio.

After adding 1mM Mg–ATP to the system we observe many microtubule pairs with a
joint basal end bending persistently We explain the functional mechanism
behind the observation in the following way. Free microtubule ends fluctuate
in the bulk. Dynein motors assembled on one microtubule attach to the adjacent
microtubule, then Brownian fluctuations are suppressed. By using energy from
ATP hydrolysis, the motors cooperatively produce shearing forces between these
two microtubules (see Fig. 2A). When they reach the buckling instability, the
filaments separate.

In more detail, one microtubule slides towards the seed. When the crossing
angle between the microtubules increases, the forces on the motors increase
and thus an increasing number of motors detaches without the possibility to
rebind. The other microtubule bends only weakly. The motion is stalled when
the force produced by the remaining motors reaches the force of the buckled
microtubules. After that, stochastic fluctuations can tip the motors into
cooperative detachment, akin to a catastrophic failure, which leads to a fast
straightening of the microtubules and a return to the original state. The
straight microtubules again allow the attachment of a larger number of motors.
We observed that this process could repeat itself periodically. This cycle and
the functional mechanism are illustrated in the schematic representation in
Fig. 2B.

The long lag phase and subsequent rapid sliding leading to rapid buckling
indicate that the attachment and movement of motors take place in a
cooperative manner, too. The simplest possible explanation is that the
attachment rate of the first motor between unconnected microtubules is low
because of thermal fluctuations. The first motor capturing the adjacent
microtubule then suppresses the fluctuations and largely accelerates the
attachment of subsequent motors. The mechanism in which microtubule buckling
accelerates the detachment of motors is also at the core of the geometric
clutch hypothesis for ciliary beating Lindemann . On the other hand, the
release of the motors under load is reminiscent of models that propose a
dynamic instability in the sliding motion between microtubule doublets
Camalet.Prost1999 .

We quantify the oscillations by measuring the maximal distance $H$ between the
buckled microtubules. A typical time series representing the bending cycle is
shown in Fig. 3A. This cycle can be divided into four phases consisting of an
active bent state and a relaxed state with variable duration and the two
transitions between them, namely a rapid bending from the relaxed to the
active state and a rapid unbending from the active to the relaxed state. The
four phases are influenced also by the stochastic nature of dynein motors.
Both rapid bending and unbending phases completed within a few hundred
milliseconds. The swiftness of the transitions is a likely clue on the
cooperativity of the motors as the motor recruitment acts collectively to
transition between states.

By analyzing the histogram of distances $H$ in Fig. 3B we can observe a
bimodal distribution. Therefore, the above description of the bending cycle as
a process with two states is a suitable one. We will call them open and close,
and consider stochastic transitions between them. We denote the rates of the
closed-open and open-closed transition with $k_{+}$ and $k_{-}$, respectively.
The discretized variable $H$ then has the character of a telegraph noise
process. A robust way to determine the two rate constants is from the
occupancy ratio ($P_{\rm open}/P_{\rm closed}=k_{+}/k_{-}$) and from the
autocovariance function of $H$, which for the telegraph process has the
dependence $\sim\exp(-(k_{+}+k_{-})\tau)$. The autocovariance functions are
shown in Fig. 3C. They have a bi-exponential form – besides the slow
component, related to the open/closed transitions, a fast one resulting from
noise with a short correlation time. From the slow rate and $P_{\rm
open}/P_{\rm closed}=1.7\pm 0.5$, we obtain the rates $k_{+}=(0.18\pm
0.07)\,\text{s}^{-1}$ and $k_{-}=(0.10\pm 0.04)\,\text{s}^{-1}$.

Figure 2: A) Micrographs of bending cycle of two microtubules forming a
synthoneme. Scale bar: 10 $\mu\rm m$ B) Schematic representation of the
filament buckling. Initially, the filaments (green) are straight and the
dynein motors (red) enter the active state. The force produced by the dyneins
buckles the filament on which they are attached. The buckling changes the
crossing angle between the microtubules, which reduces the number of dyneins
that can bind to both of them. Figure 3: A) Maximal distance between the
filaments in the buckled region ($H$), as defined in Fig. 2, as function of
time. B) Distribution of distances $H$ between the filaments. As the
distribution is clearly bimodal, we designated the two states as open and
closed. C) Normalized autocovariance function
$C_{HH}(\tau)=(\left<H(t+\tau)H(t)\right>-\left<H\right>^{2})/\left<(H(t)-\left<H\right>)^{2}\right>$.

We use the elastic model that is described in the next section to estimate the
force and the energy needed to buckle the filaments. With a typical length of
$l=8\,\rm\mu m$ and $EI=4\times 10^{-24}\rm Nm^{2}$, Eq. (2) gives $F\approx
1\,\rm pN$, showing that a small number of dyneins is sufficient to buckle the
filaments. For a typical distance $H=1.2\,\rm\mu m$, we further calculate the
elastic energy of the buckled state as $F\Delta s\approx 480\times
10^{-21}{\rm Nm}\approx 116\,k_{B}T$. Changes in elastic energy that exceed
the thermal energy by two orders of magnitude prove that the beating is
actively driven by molecular motors, i.e., that we assembled an active system.

###  I.1 Theory and Simulation.

Figure 4: Theoretical description of the active synthoneme. A) Elastic model.
Two filament segments of different lengths are pinned to each other at $x=0$
and clamped at $x=l$. They are parameterized with the functions $y_{1}(x)$ and
$y_{2}(x)$. The $x$-axis represents the direction of the force. The condition
for the pinned end at $x=0$ is $y_{1}(0)=y_{2}(0)=0$. At the clamped end, the
condition is $y_{1}(l)=y_{2}(l)$ and $y_{1}^{\prime}(l)=y_{2}^{\prime}(l)$. B)
Active model: two filaments are clamped in the origin. Filament 1 is decorated
with motors from length $L_{0}$ to length $L$. Motors can bind to filament 2
if the distance between the filaments is smaller than $d$ (grey rectangle). C)
Snapshots from a stochastic simulation. The black lines show two filaments,
red and green circles show bound and free motors, respectively. The upper
panels shows the filaments before binding, the middle panel the onset of
buckling and the lower panel a stalled state before catastrophic detachment of
motors. D) The distance between the filaments $H$ as a function of time from
the simulation. E) Distribution of distances $H$ from the simulation. As in
the experiment, a strongly bimodal distribution is visible. F) Autocorrelation
function of the distance $H$ as a function of time.

The shapes of the two bent filaments (Fig. 4A) and the forces that the dyneins
need to produce to enter the bent configuration can be calculated by linear
elasticity theory if we assume that the shape is quasi-static and that the
deflections are small. In contrast to earlier models Brokaw2009 , we allow
both filaments to bend. Because both ends are force-free, the force in
filament 1 is exactly opposite to that in filament 2. The curvature in each
filament is proportional to the local bending moment

$EI\,y_{i}^{\prime\prime}(x)=F_{i}y_{i}(x)\;.$ (1)

The resulting equation is solved with the ansatz $y_{1}=A\sin kx$,
$y_{2}=B\sinh kx$ with $k=\sqrt{F/EI}$. The boundary condition at $l$ leads to
the equation $\tan kl=\tanh kl$ with the lowest non-trivial solution
$kl=3.9266$ and

$F=1.5622\frac{\pi^{2}EI}{l^{2}}\;.$ (2)

This equation can be used to estimate the force produced by the motors that is
needed to buckle the filaments. Notably, the prefactor in the buckling load of
$1.5622$ lies between the values of $1$ that applies to the classical Euler
problem with two pinned ends and $2.046$ for one pinned and one clamped end
Gere . As expected, clamping one end to a flexible filament leads to a lower
critical load than clamping it to a fixed direction in space.

The length difference between the filament segments, $\Delta s$, and the
distance $H$ can be derived as functions of the crossing angle $\varphi$ (see
SI).

#### Dynamical model.

The behavior of the system over time observed during the experiments can be
described by a dynamical model. The two filaments (each of length $L$) are
clamped at the minus end (Fig. 4B). Filament 1 is bearing dynein motors
between length $L_{0}$ and $L$, at a line density $\rho$. Each motor can be in
the attached or detached state. Attachment can take place in the overlap zone
$l_{1}\pm d/\varphi$, where $d$ is a characteristic distance between filaments
that the motors can bridge. In that zone, the attachment rate of the motors is
$k_{\rm on}$. However, we use a lower constant $k^{0}_{\rm on}$ for two
reasons: (i) a reduced attachment rate for the motors that align end-to end
along one filament indicates the cooperativity between the motors; (ii) the
effective attachment rate is strongly reduced by thermal fluctuation when the
motors are simultaneously in the detached state. For a motor loaded with force
$f$ we use a detachment rate $k_{\rm off}=k_{\rm off}^{0}\exp(\alpha f)$,
where $\alpha$ is a parameter representing the inverse characteristic
detachment force.

A motor attached on filament 1 with load $f$ moves towards the minus end of
filament 2 with velocity $v=\mathrm{d}(\Delta s)/\mathrm{d}t$ determined by a
linear force-velocity relationship

$f=f_{s}(1-v/v_{m})$ (3)

with the parameters $f_{s}$ (stall force) and $v_{m}$ (unloaded velocity).
When $n$ motors are attached, they share the load equally such that
$F(l_{2})=nf$.

We explain the oscillatory cycle of the filaments observed in the experiments
through the following stages: First, all motors are detached and the first
attachment takes place randomly with the rate $k^{0}_{\rm on}$. This increases
the binding rate of remaining motors to $k_{\rm on}$. After a short time, the
total motor force $nf_{s}$ surpasses the buckling force $F(l)$, where $l$ is
the distance to the first bound motor. The straight state becomes unstable. By
considering the duty ratio of stalled motors (fraction of time spent in the
attached state)

$\eta=\frac{k_{\rm on}}{k_{\rm on}+k_{\rm off}^{0}\exp(\alpha f_{s})}\;,$ (4)

the condition of buckling is

$\eta f_{s}\rho(L-L_{0})>F(L_{0})=1.5622\frac{\pi^{2}EI}{L_{0}^{2}}\;.$ (5)

The initial buckling instability is followed by the growth of the bulge,
driven by the sliding action of motors in the overlap zone. During this phase,
the crossing angle $\varphi$ increases, which reduces the length and the
number of motors in the overlap zone, leading to a metastable state, where the
force balance is reached at an angle $\varphi$ when $\eta\rho
f_{s}2d/\varphi=F(l)$.

Through random detachment of motors, a “catastrophic failure” can take place
when detachment of motors leads to an increased load on the remaining ones,
further increasing their detachment rate. We assess the stability of a state
with the following consideration. At a constant total force $F$, initially
distributed over $n$ attached motors, the on-off dynamics is given by the
master equation

$\dot{n}=(N-n)k_{\rm on}-nk_{\rm off}e^{\alpha F/n}.$ (6)

Its stability is determined by the derivative

$\frac{\partial\dot{n}}{\partial n}=-k_{\rm on}+\left(\frac{\alpha
F}{n}-1\right)k_{\rm off}e^{\alpha F/n}$ (7)

which is positive when $\alpha f_{s}n_{0}/n>1/(1-\eta)$. As a rough estimate,
we expect an instability when $\alpha f_{s}\gtrsim 1$.

Through full detachment, the filaments finish in the straight position where
new motors quickly re-attach and repeat the cycle. This is compatible to the
cyclic dynamics that we hypothesized for our experimental outcomes.

The elastic model and the motor model described above are combined in a
stochastic simulation that confirms the experimental results (Fig. 4C-F). The
distance between the filaments oscillates with steep transitions between the
open and the closed state (Fig. 4D), closely resembling the experimental
traces shown in Fig. 3A. The histogram of distances shows a clear bimodal
distribution (Fig. 4E), as in the experimental data Fig. 3B. Finally, the
autocorrelation function shows a near-exponential decay (Fig. 4F). In contrast
with the experiment (Fig. 3C), the fast phase is missing as we did not include
thermal noise and measurement uncertainty in the simulation.

## Conclusion

The reconstituted minimal axoneme, i.e. the synthoneme, that we present in
this study is a one-of-a-kind result resembling the natural system with its
beating behavior. All crucial components of our synthoneme are assembled in-
vitro in a bottom-up approach. We show that the cooperativity of the dynein
motors to arranging on the filament as well as to actuating the buckling can
be reconstituted in-vitro. Although the amplitude and frequency of
oscillations are still well below those of natural cilia, we have shown that
the basic functionality can be established with a far smaller number of
proteins. Specifically, while cilia contain several hundred different
proteins, our synthetic axoneme works with only three components (tubulin,
ODA, ODA-DC). The observation that synthetic systems consisting only of
molecular motors and filaments can reproduce the basic movement necessary for
cilia/flagella beating also provides clues about the amount of complexity
needed by the ancestor cilia to first achieve beating motility. Indeed, some
organisms like the male gamete of the parasitic protozoan Diplauxis hatti have
a motile axoneme with only three doublets instead the classical “9 + 2”
structure Prensier . This suggests that bending behavior could be produced by
simpler structures that have filaments and motors. Future refinements will
lead to a convergence between the synthetic and the natural cilium. This will
provide a systematic way of dissecting the elusive beating mechanism.

Besides the potential of this system for answering important questions about
ciliary beating, our synthetic cilium may encourage the technological
development of molecular machines for fluid transport at micro- and nanoscale.
Most previous attempts to build artificial cilia have concentrated on magnetic
Vilfan.Babic2010 ; Dong.Sitti2020 , electrostatic Toonder.Anderson2008  or
optical vanOosten.Broer2009  actuation mechanisms to produce motion similar to
the natural one. These studies showed hydrodynamic entrainment between
adjacent structures and creation of metachronal waves. However, these systems
are unrelated to biological cilia. In the future our system will also be
developed as multi-cilia system by using axoneme seeds as the base and
aligning them and we will achieve the synchronization of axonemal beating.

##  II Materials and Methods

###  II.1 Preparation of the crude dynein extract

Demembranated axonemes are suspended in 0.6M KCl containing HMDEK solution to
extract crude dynein sample (see SI for more details about preparation). After
spinning down the axonemes for 5 min, the supernatant is desalted with the
overnight dialysis against HMDEK solution (Spectra/Por Dialysis Membrane,
MWCO:100-500D, Biotech CE). The concentration of the crude dynein extract is
measured with Bradford method and the concentration used during the
experiments was 0.02 mg/ml.

###  II.2 Preparation of dynein–MT complexes

The flow chamber was build with Teflon-treated coverslips as previously
described in Torisawa  and spaced with double-coated tape (80mm thick, W-12;
3M, St. Paul, MN) to prevent any nonspecific binding of proteins onto the
surface. Microtubules growing close to each other and with the same polarity
were obtained by using fragments of demembranated axonemes prepared by
rigorous pipetting as polymerization seeds. The ODA–MT complexes are
reconstituted in the flow cell by flowing the components one after the other.
Briefly, in the flow cell a small amount of axonemes are attached on the
bottom of the flow cell. After 5-min incubation, the flow cell is washed with
1% (w/v) Pluronic® F127 in BRB80 (80 mM PIPES, 1 mM MgCl2, 1 mM EGTA, pH 6.8
with KOH) and is incubated for 5 min. After washing the chamber with BRB80,
fluorescently-labeled (Cy3–labeled) porcine tubulin (3% labeling) is
introduced into the flow cell, polymerized in the presence of 1 mM GTP, 50%
DMSO, 1mM MgCl2 at 37∘C for 30 min and stabilized with 7 $\mu$M taxol. After
microtubule polymerization, diluted crude-dynein extract is introduced into
the flow cell and incubated for 5 min. The non-bound protein is eliminated by
washing the chamber with buffer and afterwards 1mM ATP is perfused into the
chamber to trigger the activity.

###  II.3 Imaging and tracking

Fluorescence images of the MT-ODA complex are acquired using an inverted
fluorescence microscope Ti-E (Nikon, Japan) equipped with a 60$\times$ CFI
Apochromat objective (N.A.=1.49, Nikon, Japan) and the confocal unit (CSU-X1,
YOKOGAWA, Japan). The data is recorded with an iXon Ultra EMCCD camera (Andor
Japan, Japan). The images are acquired at a frequency of 10 Hz. The movement
of the filaments over times was tracked manually and a third order polynomial
was fitted to the data by using a purpose-written MATLAB code.

#### Acknowledgements

E.B., I.G., and R.G. acknowledge support from the MaxSynBio Consortium which
is jointly funded by the Federal Ministry of Education and Research of Germany
and the Max Planck Society. E.B. acknowledges support from the Volkswagen
Stiftung (priority call “Life?”). A.V. acknowledges support from the Slovenian
Research Agency (grant no. P1-0099). K.O. acknowledges MEXT/JSPS KAKENHI,
grant numbers 26440089, 17K07376 and JP16H06280, the Takeda Science Foundation
and Hyogo Science and Technology Association that partly supported the
project.

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## Supplementary Materials and Methods

###  II.4 Preparation of Chlamydomonas reinhardtii. Axonemes and their
demembranation.

Axonemes were obtained from Chlamydomonas reinhardtii according to the
dibucaine method Kagami ; Witman1978 . C. reinhardtii wild type strain (137c
mt-) and the outer-arm less mutant (oda1) were cultured in 4 liters of the
liquid TAP medium under continuous illumination at 20 ∘C for 4 days with air
bubbling. Cells were harvested by centrifugation at 1692$\times$g (3000 rpm,
R10A3 rotor, Himac CR22E) for 6 min, and then re-suspended in 40 ml ice-cold
HMDS (30 mM HEPES-NaOH, 5 mM MgSO4, 1 mM dithiothreitol (DTT), 4% sucrose, pH
7.4). All procedures described below were conducted at 4 ∘C or on ice unless
otherwise stated. The cell suspension was moved to a 50-ml conical centrifuge
tube and 2.5 mM (final concentration) dibucaine HCl (Wako, 167-15111) was
added. The suspension was pipetted rapidly in and out of 10-ml Falcon
disposable plastic pipettes for 30 s. Afterwards 200 $\mu$l of 100 mM EGTA was
added and the solution was gently resuspended several times. The cell bodies
were removed by centrifugation at 1670$\times$g (3000 rpm, RS-4 rotor, KUBOTA
6800) for 6 min, the supernatant was transferred to a new 50-ml conical
centrifuge tube and centrifuged again as above. The cell-free supernatant was
centrifuged at 27720$\times$g (15,000 rpm, RSR20-2 rotor Himac CR21) for 12
min. The resulting pellet was resuspended in 1.5 ml of HMDEK (30 mM HEPES-
NaOH, 5 mM MgSO4, 1 mM DTT, 1 mM EGTA, 50 mM potassium-acetate, pH 7.4) and
demembranated by adding 15 $\mu$l of 20% Nonidet P-40 (NP-40, Nacalai Tesque).
After 5 min-incubation, the suspension was centrifuged at 21130$\times$g
(15,000 rpm, FA-45-24-11 rotor, Eppendorf 5452R) for 5 min, and resuspended in
1.5 ml HMDEK. This procedure was repeated twice to remove detergent. Axonemes
were collected by centrifugation as described above. An aliquot of the
demembranated oda1 axonemes was used to prepare the seeds of microtubule
polymerization. Suspension of the axonemes was gently homogenized with a
Teflon pestle and pipetting. The resultant suspension was centrifuged at
3380$\times$g at 4 ∘C for 10 min. The pellet was resuspended with a small
amount of HMEDK solution. Intrinsic dyneins of the seeds were deactivated and
the sliding among doublet microtubules was inhibited with the treatment of
ethylene glycol bis(succinimidyl succinate) (Thermo Fisher Scientific, 21565).

###  II.5 Preparation of of the crude dynein extract.

For the extraction of ODAs and docking complex from the Chlamydomonas
reinhardtii wild-type axonemes we follow the method described in previous
works Haimo ; kamiya2005 . As reference, we prepared high-salt extract from
oda1 (outer-arm less mutant) axonemes. Demembranated axonemes were resuspended
in 0.6M KCl containing HMDEK solution to extract crude dynein sample. After
spinning down the axonemes at 21130$\times$g (15,000 rpm, FA-45-24-11 rotor,
Eppendorf 5452R) for 5 min, the supernatant was desalted with the overnight
dialysis against HMDEK solution (Spectra/Por Dialysis Membrane, MWCO:100-500D,
Biotech CE). The concentration of the crude dynein extract was measured with
Bradford method and it was around 0.02 mg/ml. High-salt extract from the
demembranated axonemes contains various types of proteins Haimo . Therefore,
protein components of the dynein-microtubule complex should be examined with
SDS-PAGEs. Spinning down the crude dynein extract with microtubules in the
presence of 1mM ATP partially pulled down the outer-dynein arms and related
proteins with microtubules. Using oda1 axonemes, we repeated the same
procedure on the axonemes and high-salt extracts. The resultant supernatants
and pellets of the centrifugations were examined with the SDS-PAGE and the
contents of oda1 were compared with those of wild type. Inner arm dyneins,
components of the central pair apparatus and radial-spoke components remain in
supernatant under this condition (Fig. S1). Thus, our procedure provided ODA-
DC-MT complex which contained no inner arm dyneins but only outer arm dyneins.

###  II.6 Negative staining electron microscopy on dynein-MT complex.

To evaluate the formation of ODA arrays on a microtubule, we examined the
structure of dynein-MT complex with an electron microscope. Structural
similarity of this complex to the axoneme has been reported in previous
studies and was examined by negative staining electron microscopy. It showed
that microtubules were periodically decorated with ODA dyneins. The crude
dynein extract of 200 $\mu$g/ml was mixed with taxol-stabilized microtubules
(200 $\mu$g/ml) at the various ratios (10:1, 1:1, 1:2 and 1:10). After 30-min
incubation, the small aliquot of the mixture (5 $\mu$l) was applied on to a
carbon grid and washed with MMEK for EM (30 mM MOPS-NaOH, 5 mM MgSO4, 1 mM
DTT, 1 mM EGTA, 50 mM KCl, pH 7.4). Fixation with 2% glutaraldehyde, washing
with MMEK and staining with 1.4% uranyl-acetate with 100 $\mu$g/ml bacitracin.
The sample was examined with JEM-1400 (JEOL, Japan) with 80kV acceleration
voltage and the images of the samples were collected with 4K camera (US4000,
Gatan). Typical electron micrographs are shown in Fig. S3. Coverage of the
microtubules with ODA-DC complex was evaluated with image analysis. To extract
the 24 nm structural repeat of dynein-ODA-DC from the electron micrographs,
the custom-built band-pass filter was applied to them. The merger of the
original and filtered image clearly shows the presence of dynein-DC complex.
The length of the complex and microtubule length in the image field were
measured.

## Elastic model

In this section we derive the relationships between the crossing angle
$\varphi$, the length difference $\Delta s$ and the distance between filaments
$H$ (Fig. 3A in main text). We start with the solution derived in the main
text, where the filaments shapes are described by $y_{1}=A\sin kx$ and
$y_{2}=B\sinh kx$, with $kl=3.9266$. The amplitude ratio is determined from
the condition

$\frac{A}{B}=\frac{\sinh kl}{\sin kl}=-0.02785\;.$ (S1)

In the linear approximation, the opening angle at the pinned end is

$\displaystyle\varphi$ $\displaystyle=y_{1}^{\prime}(0)-y_{2}^{\prime}(0)=Ak-
Bk=Ak\left(1-\frac{\sin kl}{\sinh kl}\right)$ $\displaystyle=1.02785\,Ak\;.$
(S2)

Also in the linear approximation, the contour length of filament $i$ ($i=1,2$)
can be calculated as

$s_{i}=\int_{0}^{l}\sqrt{1+(y_{i}^{\prime})^{2}}\mathrm{d}x\approx\int_{0}^{l}\left[1+\tfrac{1}{2}(y^{\prime}_{i})^{2}\right]\mathrm{d}x$
(S3)

and the difference as

$\displaystyle\Delta s$
$\displaystyle=\int_{0}^{l}\left[\tfrac{1}{2}(y^{\prime}_{1}(x))^{2}-\tfrac{1}{2}(y^{\prime}_{2}(x))^{2}\right]\mathrm{d}x=\frac{lk^{2}}{4}(A^{2}-B^{2})$
$\displaystyle=\frac{\sinh kl+\sin kl}{\sinh kl-\sin
kl}\frac{l\varphi^{2}}{4}=0.2364\,l\varphi^{2}\;.$ (S4)

Finally, we define the maximum distance between the filaments
$H=\max(y_{1}-y_{2})$. The condition for maximality is
$y_{1}^{\prime}(x_{m})-y_{2}^{\prime}(x_{m})=0$ and leads to

$\frac{\cos kx_{m}}{\cosh kx_{m}}=\frac{\cos kl}{\cosh kl}$ (S5)

with the solution $kx_{m}=1.6458$ and
$H=y_{1}(x_{m})-y_{2}(x_{m})=0.24301\,l\varphi$. These relationships are used
in the main text to estimate the elastic energy of the two-filament system.
They are also needed for the stochastic simulation.

Figure S1: Preparation of the outer-dynein arm-microtubule complex. A)
Procedure of dynein-crude extraction. PPT means a precipitate B) The high-salt
crude dynein extract from Chlamydomonas reinhardtii wild type axonemes, oda1
axonemes and centrifuged dynein-microtubule complex were analyzed by SDS-PAGE
with silver staining. Since only inner arm dyneins exist in oda1 axonemes, the
high salt extract did not contain outer arm dyneins. In contrast, the extract
from wild-type axonemes contains both outer-arm dynein and inner-arm dyneins.
The glycerol cushion in the centrifugation can separate inner-arm dynein from
microtubules in the presence of ATP but outer arm dyneins stayed with
microtubules. Lane 1: molecular weight markers. Lane 2: no dynein exists in
oda1-extract-MT complex. No light chains and Intermediate chains of inner-arm
dyneins were found. Lane 3: Removal of ATP from the complex pulled inner-arm
dyneins of oda1 extract into pellet with microtubules. Lane 4: Inner arm
dyneins were found in the pellet when being centrifuged without cushion. Lane
6: Dyneins found in the pellet even in the presence of ATP and cushion but no
light chains and Intermediate chains of inner-arm dyneins were found. This
observation suggests that only outer arm dyneins remained on microtubules and
these dyneins formed the complex with microtubules via dynein-docking complex.
C) High-molecular region of 3% SDS-PAGE. Lane 1: The microtubule pellet in the
presence of ATP and glycerol cushion. No dynein heavy chain was found in the
pellet. Lane 2: Inner arm dynein heavy chains were found. Lane 5: Even in the
presence of ATP, only outer arm dyneins were associated with microtubules.
These observations suggest that our procedure enabled the ODA-DC-MT complex to
contain no inner arm dyneins but only outer arm dyneins. S and P indicate a
precipiate and supernatant, respectively. Figure S2: Schematic representation
of the assembly of a minimal synthetic axoneme with the minimal (a) and
maximal (b) distance between the polymerized microtubules. Figure S3: Taxol-
stabilized microtubules were mixed with crude dynein extract at various mixing
ratios and observed with negative staining electron microscopy (acceleration
voltage 80kV, magnification 25k). Figure S4: Average length (A) and density
(B) of dynein patches on microtubules as a function of mixing ration of crude
dynein extract (4 - 400 $\mu$g/ml) to microtubules (40 $\mu$g/ml).