Article
https://doi.org/10.103
8/s41467-023-41813-6
Macroscopic waves, biological clocks and
morphogenesis driven by light in a giant
unicellular green alga
Eldad A
fi
k
1,2
,ToniJ.B.Liu
1
& Elliot M. Meyerowitz
1,2
A hallmark of self-organisation in livi
ng systems is their capacity to stabilise
their own dynamics, often appearing t
o anticipate and act upon potential
outcomes.
Caulerpa brachypus
is a marine green alga consisting of differ-
entiated organs resembling leaves, s
tems and roots. While an individual can
exceed a metre in size, it is a single multinucleated giant cell. Thus
Caulerpa
presents the mystery of morphogenesi
s on macroscopic scales in the absence
of cellularization. The exp
eriments reported here reveal self-organised waves
of greenness
—
chloroplasts
—
that propagate through
out the alga in antici-
pation of the day-night light cycle. Using dynamical systems analysis we show
that these waves are coupled to a self-su
stained oscillator, and demonstrate
their entrainment to light. Under constan
t conditions light intensity affects the
natural period and drives transition
to temporal disorder. Moreover, we
fi
nd
distinct morphologies depending on l
ight temporal patterns, suggesting
waves of chlorophyll could link biolo
gical oscillators to metabolism and
morphogenesis in this giant single-celled organism.
A free-falling ball follows a trajectory determined by external forces.
When we aim to catch the ball, in contrast, our dynamics follows an
intrinsic predictive model of the world, exhibiting anticipatory
behaviour
1
–
3
. Such behaviour has been observed in living systems
down to cellular levels
4
–
7
.
Time-keeping and synchronisation with the external world, as well
as within the organism, seem to be essential for homoeostasis, the
ability of living systems to maintain essential variables within physio-
logical limits
1
,
8
–
10
. This is one of the ways by which living organisms
manifest self-organisation
11
,de
fi
ned here as networks of processes that
are self-stabilizing far from thermodynamic equilibrium.
Biological oscillators are found throughout the living world,
constituting a mechanism for organismal synchronisation. Their
manifestations, such as pulse rate and pressure, are measures of
homoeostasis. Biological oscillators are amenable to analytic approa-
ches from dynamical systems
9
,
12
,
13
. Here we study active rhythmic
transport in
Caulerpa brachypus
, a marine green alga which presents
complex morphology while being a giant single cell.
Caulerpa
chal-
lenges central paradigms in developmental biology, as it exhibits
pattern formation and morphogenesis without multicellularity
14
,
15
.
While tracking morphogenesis of regenerating algal segments we
observed that the growing tips of stem-like (stolon) and leaf-like
(frond) regions exhibit temporal variations in intensity of their green
colour. During night distal regions typically appear more transparent,
while during day the green coverage is more homogeneous. This has
been attributed to long-distance chloroplast migration
16
. Whether
rhythmic transport in this single cell is light induced, or is of auton-
omous nature, is unknown. Our
fi
ndings show that the waves of
greenness exhibit anticipatory behaviour, usually starting toward the
daylight state before dawn, and toward the night state before dark.
Moreover, the analysis shows that anticipatory behaviour in this
system is explained by entrainment
—
adjustment of the rhythm of an
organismal self-sustained oscillator by interaction with another
oscillator
17
,
18
, the driving illumination in this case. Using a scalable
and affordable methodology, varying two control parameters
—
the
temporal period of the illumination and its intensity
—
we identify
distinct dynamical states of the self-organised waves. The results
are compatible with dynamics of self-oscillations and forced
Received: 14 April 2023
Accepted: 19 September 2023
Check for updates
1
Division of Biology and Biological Engineering, California Institute of Technology, 1200 E California Blvd., Pasadena, CA 91125, USA.
2
Howard Hughes Medical
Institute, Maryland, USA.
e-mail:
eldad.a
fi
k@gmail.com
;
meyerow@caltech.edu
Nature Communications
| (2023) 14:6204
1
1234567890():,;
1234567890():,;
synchronisation
18
. In constant illumination, the intrinsic period is
intensity dependent, and under higher illumination intensities ana-
lysis indicates a transition to states of increased temporal disorder.
Moreover, we
fi
nd that development depends not only on the aver-
age photon
fl
ux but also on its temporal distribution, as manifested
in the resulting morphology.
Results
Anticipatory green waves and morphogenesis
To study quantitatively the self-organised green wave dynamics in
Caulerpa brachypus
, we developed a set of protocols for algal culture
and propagation, as well as automated illumination, live imaging and
image analysis. Using Raspberry Pi-controlled illumination and
cameras, we track over weeks the morphogenesis of tens of samples
concurrently, while tracing at resolution of tens of seconds the var-
iation of the green coverage. A diagram of the experimental
fl
ow is
shown in Fig.
1
a; snapshots exemplifying the relevant time scales are
presented in Fig.
1
b
–
d. Our observations of samples cultured under
cycles of 12 hours light followed by 12 hours dark (12hL
–
12hD, T =
24h) have allowed analysis of waves of greenness that propagate over
centimetres, within a few hours, at a whole-organism scale; a time-
lapse is presented in Supplementary Movie 1;
Light
and
Dark
denote
reference illumination intensities, the latter is 1
=
200
I
L
, for
I
L
≈
4
:
5
μ
mol
s
1
m
2
. By coarse-graining in space we achieve a
reduced description to a dynamic macroscopic observable. The time
series indicates that the initiation of the waves anticipates the
external change in illumination, as exempli
fi
ed in Fig.
1
e. This has led
us to ask whether the period of the waves follows that of the driving
illumination.
Equivalence of dynamics near 24h driving periods
To quantify the temporal frequency content of the biological
response, we apply power spectral analysis
19
,
20
. While the time series of
individual samples exhibit high variability, the spectral decomposition
using power spectra captures and highlights the similarities among
them. Examples from three samples and their corresponding power
spectra are plotted in Fig.
2
a and b, respectively. The pronounced local
maxima in the power spectra correspond to the response fundamental
frequency
f
r
,0
, centred at 1/24h for these samples, and its higher har-
monics
—
integer multiples of
f
r
,0
. The relative power of the higher
harmonics characterises the non-sinusoidal waveforms shown
in Fig.
2
a.
Under driving periods within 18h to 30h the response of the green
pulses
f
r
,0
matches the driving frequency
f
d
in 1:1 correspondence, as
shown in Fig.
2
d. Moreover, the power spectra suggest an equivalence
of the dynamics when rescaled by the control parameter
f
d
.Thatis,
presenting the power spectra from various
f
d
as a function of
f
r
/
f
d
reveals the similarity in dynamics in this range; see Fig.
2
e. The cor-
responding curves prior to rescaling are shown in Fig.
2
c.
The waves are coupled to an intrinsic oscillator
Subject to driving periods away from the 18h to 30h range, we
fi
nd in
addition to the driving frequency evidence for a distinct circadian
mode
—
an indication that the waves are coupled to, or part of, an
Fig. 1 | Self-organised macroscopic green waves: experimental design and
coarse-graining. a
Schematic of the Cut & Regenerate experimental cycle. A
Caulerpa
cell is cut and segments are let to regenerate in individual wells; custom-
built controllers apply the individually assigned illumination protocols per dish and
control the time-lapse imaging; samples regenerating over weeks provide tens of
segments for the next experiment; thus the experiments provide living material for
the next cycle of experiments.
b
Snapshots from morphogenesis dynamics over the
course of 35 days.
c
A developmental trajectory of a frond, focusing on a sub-
interval of 13 days; these correspond to the frond marked by red rectangles in (
b
).
d
A temporal sub-interval spanning 48h, showing two periods of the green waves;
thegrey-scaleframesrepresenttheintensityintheBluechannelfromtheRGB
images.
e
Coarse-graining of the spatio-temporal dynamics achieved by counting
pixels whose intensity in the Blue channel falls below a preset threshold; the white
and grey shadings correspond to 12h intervals of Light and Dark illumination levels,
respectively.
Article
https://doi.org/10.103
8/s41467-023-41813-6
Nature Communications
| (2023) 14:6204
2
intrinsic nonlinear oscillator. For example, under driving period T = 48
h time series exhibit a steady phase relation to the illumination pattern;
at the same time, at mid-Light intervals the dynamics resemble the day
to night transitions observed under T = 24h; see Fig.
3
a. Under a driving
period T = 3 h, eight times faster than earth
’
s rotation, the time series
show fast undulations attributed to a response to the driving illumi-
nation; these ride over an undulation of a longer period, as shown
in Fig.
3
b.
Entrainment of a nonlinear oscillator is predicted to occur at
driving frequencies near rational ratios of its natural frequency and the
driving one, in which case
n
cycles of the response would match
m
cycles of the driving oscillator
17
,
18
.Withinan
n:m
-entrainment range,
the response fundamental frequency
f
r
,0
and its higher harmonics are
predicted to be modi
fi
ed accordingly. To test for entrainment, we have
measured samples subject to
f
d
about rational ratios of 24h, namely T =
48 h, 6 h, 3 h, and 1.5 h, corresponding to 2
1
,2
−
1
,2
−
3
and 2
−
4
multiples of
24h, as well as longer ones T = 94 h and 54 h. The results support a 1:1-
entrainment range, as well as evidence for higher order
n:m
-entrain-
ment regions. The corresponding power spectra are presented as
heatmaps, where the columns correspond to
f
d
, including those from
f
d
near 1/24h; see Fig.
3
c, a visualisation inspired by a study on forced
turbulence and its synchronisation regions
21
. The synchronisation
regions, where
n:m
-entrainment are stable, are known as Arnold
tongues
17
,
18
,
21
.
Dynamical states set by the intensity of light
Finally, we test whether alternating environmental conditions are
necessary to sustain the waves, to complete the argument for an
intrinsic self-sustained oscillator, and measure its natural period. To
this end, we let samples regenerate under constant illumination. While
we indeed
fi
nd self-sustained circadian oscillations, subject to constant
illumination the observations indicate new dynamical states. Examples
of time series regenerating under low driving intensity
I
d
=2
5
I
L
show a reduced amplitude of the oscillations, yet these persist over
weeks; see Fig.
4
a. Samples subject to high driving intensity
I
d
=2
I
L
show signs of intermittent oscillations, presented in Fig.
4
b. Power
spectral analysis shows that: (i) the response fundamental frequency
f
r
,0
increases with decreasing illumination intensity, see Fig.
4
c; (ii)
under higher illumination intensities the local maxima in the power
spectra
‘
drown
’
in a rising continuum of frequencies, and (iii) com-
pared with the entrained states, the higher harmonic content is less
pronounced; these
fi
ndings are presented in Fig.
4
d. Thus, under
constant photon
fl
ux, the energy source for this organism, we
fi
nd that
the system can be driven towards aperiodic dynamical states by high
fl
uxes. This is a manifestation of the nonlinear nature of the dynamics,
and a plausible signature of chaotic dynamics.
Morphology depends on the temporal pattern of light
How does the photon
fl
ux affect the developmental dynamics in this
organism? It is expected that the energy injection rate would be a
limiting factor to growth. Inferring the apparent area of samples from
our macroscopic dynamical variable, we
fi
nd that both the central
tendency and the dispersion of growth rates increase with the illumi-
nation intensity; that is, algal samples regenerating under high photon
fl
ux cover larger areas faster as a population, while being less pre-
dictable individually, compared with those regenerating under a low
Fig. 2 | Dynamical equivalence under driving periods near 24h. a
Time series
from three samples regenerating under 12hL-12hD (T = 24h); area fraction corre-
sponds to the measured green area within a whole well, normalised by the area of
the well.
b
Power spectra corresponding to the time series in (
a
); the arrows
annotate the 1st, 2nd, 3rd and 10th local maxima, corresponding to the response
fundamental frequency
f
r
,0
and higher harmonics, namely integer multiples of
f
r
,0
.
c
Power spectra of samples regenerating under illumination periods within 18h to
30h; the curves result from averaging over samples grouped by driving illumination
frequencies,
f
d
=1/
T
.
d
Response fundamental frequencies
f
r
,0
plotted as function of
the driving frequency
f
d
=1/
T
; the guiding dashed line represents a 1:1 correspon-
dence.
e
Power spectra presented as a function of
f
r
/
f
d
, the response frequency
rescaled by the driving frequency. The power spectra whose averages are shown in
(
c
) and whose
f
r
,0
are in (
d
) can be found in Supplementary Fig. 1.
Article
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Nature Communications
| (2023) 14:6204
3
photon
fl
ux. Moving averages, as estimates of apparent sample area,
are presented in Fig.
4
e. However, it turns out that the temporal pat-
tern of the illumination leads to distinguishable morphological traits,
even when the mean photon
fl
ux is kept the same. For example, we
compare two mean photon
fl
ux levels, 2
1
I
L
and
I
L
, each at constant
illumination and T = 24h, presented in Fig.
5
. The comparison hints that
the temporal distribution of photons, analogous to time-restricted
feeding, impacts development as a control parameter which is inde-
pendent of the mean photon
fl
ux.
Discussion
This study reveals self-organised macroscopic waves in a unicellular
organism. Using coarse-graining analysis and experimental perturba-
tionswehaveidenti
fi
ed distinct dynamical states of the waves. This
report provides evidence for coupling to an intrinsic self-sustained
oscillator, which is entrained by the time-dependent driving illumina-
tion, thus explaining the anticipatory behaviour of the waves. Under
constant conditions we
fi
nd that the natural frequency depends on the
photon
fl
ux, and that the dynamics exhibits a transition to temporal
disorder. Furthermore, our
fi
ndings of distinct morphologies tie the
discovered waves to one of the mysteries of development in macro-
scopic single cells, morphogenesis.
Caulerpa
feeds on photons, so metabolism is likely to be a key to
understanding the observed entertainment by light, via photosynth-
esis.ThisisthecaseintheKaiABCsystem
—
a molecular circadian
oscillator of certain cyanobacteria
22
,
23
. Moreover, considering photo-
synthesis sheds new light on our identi
fi
cation of distinct dynamical
states, subject to the two control parameters
—
illumination intensity
and period. Constant illumination by itself does not drive the system to
temporal disorder. Two of the driving protocols we have studied here
are equivalent in their average photon
fl
ux, namely 12hL-12hD and
constant
I
d
=2
1
I
L
. The latter leads to increased temporal disorder
dynamics, hinting that relaxation in the dark is essential for the typical
organismal dynamics.
How do tissues and organs emerge on centimetre scales in the
absence of cellularization? Active transport has been hypothesised to
play a key role in
Caulerpa
regeneration
14
,
15
. The observation of the self-
organised waves leads us to postulate that their synchronisation with
light-driven metabolic switching may hold the answer. This hypothesis
assumes two key elements: (i) spatial distribution of chloroplasts
would result in local variation in light-harvesting ef
fi
ciency, and (ii)
light-dependent metabolic states of chloroplasts. Under these
assumptions, redistribution of chloroplasts in anticipation of the
temporal changes in photon
fl
ux could create and stabilize metabolic
sub-environments, akin to localised organs. An additional contribution
of the waves to the dynamics may come in the form of an effective
pump: cortical mass migration of chloroplasts could result in bulk
fl
ow, which in turn drives cytoplasmic streaming; dynamics of this kind
has been demonstrated in other macroscopic cells
24
.
Ashby
11
has proposed a paradigm for self-organisation, where a
subset of variables in a dynamical system can undergo abrupt transi-
tions between two values. Such transitions would result in dynamical
switching among distinct behaviours of the whole system. In this light
we propose that biological oscillators, morphogenesis, and metabo-
lism are interconnected sub-systems within a network of processes;
their dynamics and inter-relations lead to the emergence of self-
stabilisation far from thermodynamic equilibrium in this natural sys-
tem which is alive
—
a generalised homoeostasis.
Methods
To quantitatively study the green wave dynamics in
Caulerpa brachy-
pus
, we have developed an affordable and scalable experimental
Fig. 3 | The waves are coupled to, or part of, an intrinsic nonlinear oscillator, in
addition to the driving illumination for periods far from 24h. a
Time-series from
three samples regenerating under driving illumination of T = 24h (12hL-12hD)
switching to T = 48h (24hL-24hD); while each is distinct, the curves reveal a steady
phase relation to the illumination pattern.
b
Time series from three samples
regenerating under T = 24h (12hL-12hD) switching to T = 3h (1.5hL-1.5hD); note that
the fast undulations, which follow the driving illumination, are riding on a longer
period one, close to 24h.
c
Power spectra presented as heatmaps; each column
results from averaging over samples, grouped by driving illumination frequencies,
f
d
=1/
T
; the guiding dashed line represents a 1:1 correspondence; the columns near
f
d
of 1/24h overlap with those in Fig.
2
c, presented here to demonstrate 1:1-
entrainment range; away from this region, in addition to the spectral content
associated with the driving frequency
f
d
, local maxima are found near 1/24h (grey
horizontal line), an indication of an intrinsic oscillator of circadian frequency. The
power spectra whose averages are shown in (
c
) can be found in Supplementary
Figs. 1 to 3.
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Fig. 4 | Free-running at constant-in-time illumination reveals intensity depen-
dent dynamical states. a
Time series from three samples regenerating under
driving illumination of T = 24h switching to constant intensity
I
d
=2
5
I
L
.
b
Time-
series from three samples regenerating under driving illumination of T = 24h
switching to constant-in-time intensity
I
d
=2
I
L
.
c
Response fundamental fre-
quencies
f
r
,0
plotted as function of the driving intensity,
I
d
; the data indicate an
intensity dependence of the response fundamental frequency when the conditions
are constant-in-time.
d
Power spectra inferred from samples regenerating under
driving illumination intensities within 2
5
I
L
to 2
+1
I
L
; the curves result from
averaging over samples grouped by driving illumination intensities,
I
d
; under
higher driving intensities the data reveal an increase in power distribution across a
continuum of frequencies.
e
Moving averages of the time-series for samples
regenerating under constant illumination; in addition to intensity-dependent
growth rates, the data reveal a sample dispersion which increases with the driving
intensity. The power spectra whose
f
r
,0
are shown in (
c
) and whose averages are in
(
d
) can be found in Supplementary Fig. 4.
Fig. 5 | Developmental effect of photon
fl
ux intensity and temporal distribu-
tion.
Snapshots from four driving illumination conditions, comparing mean pho-
ton
fl
ux levels 2
0
I
L
and 2
1
I
L
, at T = 24h and constant illumination. Samples
regenerating under time dependent illumination at T = 24h share qualitative
morphological features, which are distinguishable from those under constant
illumination, for example, the typical frond size and aspect ratio. The panels are
cropped to show three wells out of eight from each culture dish; following 13 days
subject to T = 24h, 2
1
I
L
(12hL-12hD) the new driving illumination was applied;
panels correspond to 25 days after this transition. The developmental effect dis-
tinguishing between the temporal distributions compared above
—
T=24hand
constant illumination
—
are not as apparent in growth curves; corresponding mov-
ing averages of the time series can be found in Supplementary Fig. 5.
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system for the algal culture, propagation and imaging, summarised in
Fig.
1
a. A Raspberry-Pi-Zero-W is mounted on a custom designed 3d-
printed apparatus, such that its camera module is focused at an 8-well-
culture-dish. Using multiple replicas, we concurrently track over
weeks the morphogenesis of tens of samples, while tracing at resolu-
tion of tens of seconds the variation of the green coverage. The
same computers additionally control LED strips, assigning distinct
time-dependent illumination per culture dish that provides both for
photosynthesis and imaging. Samples regenerating over weeks can
provide tens of segments for the next experiment; hence the experi-
ments provide living material for their follow-ups in a cyclic manner. A
time-lapse of samples regenerating under 12hL-12hD, 24h Light-Dark
cycles, is presented in Supplementary Movie 1.
Towards reproducibility, controlling for the variability in initial
conditions of the living samples, we use the regenerative properties of
Caulerpa brachypus
. Individual algal samples are cut in segments of 1
–
3
centimetres in length; within 1
–
2 months a regenerating sample can
provide for about 20 new segments. This approach contributes to the
homogeneity of the observed individuals, both in their genetic back-
ground, as well as the developmental state and geometry set as initial
conditions. A visual summary of the experimental setup is presented
in Fig.
1
a.
The nature of the data presents both conceptual and computa-
tional challenges. Conceptually it is desired to identify a coarse-
graining procedure, to reduce the high-dimensionality of the spatio-
temporal dynamics of the green waves, and at the same time preserve a
minimal set of essential degrees of freedom. Images are taken every
150 s over weeks. This amounts to datasets of about 400 GiB/month/
dish of RGB (Red-Green-Blue) images. Running several experiments
concurrently leads to high-throughput data generation. It is therefore
challenging for human detailed inspection, introducing computational
challenges: for computer vision and for computational resources.
In this report, our dynamic observable is the apparent green area.
Its time series are inferred from applying a threshold to the blue
channel of the RGB images. Chlorophyll absorbs strongly in the blue.
Therefore regions rich in Chlorophyll are expected to show relative
darker values in the Blue channel. An example is summarised in
Fig.
1
b
–
e, where the time series correspond to a region of interest
tracking a single leaf-like organ. Everywhere else in this report the time
series have been inferred from regions of interest corresponding to
wells; hence these represent the apparent fraction of green area, where
the area of the well is the normalisation factor.
Caulerpabrachypus
culture and propagation
Caulerpa brachypus
algal samples in this report were all derived from a
single origin (shipped from
Inland Aquatics
,Indiana).
As culture medium we use ErdsSchreiber
’
sMedium(
UTEX Culture
Collection of Algae at The University of Texas at Austin
). It is a variation
over the one in previous reports on
Caulerpa
25
,
26
.
Prior to cutting, samples were triple washed using
fi
ltered arti
fi
cial
sea water (35% Instant Ocean in deionized water;
fi
ltered using a Nal-
gene
™
Rapid-Flow
™
with 0.1
μ
m PES Membrane
fi
lters, Thermo Fisher
Scienti
fi
c Inc.). Algal segments were cut using a surgical blade, and let
to regenerate in 8-well-dishes (Non-treated Nunc
™
Rectangular Dishes,
Thermo Fisher Scienti
fi
cInc.),eachwell
fi
lled 7ml of the culture
medium.
Illumination and imaging setup
Custom made apparati were 3D-printed using translucent neutral
colour Polylactic Acid (PLA and 3D Printers access courtesy of Caltech
Library Techlab). These were designed to hold the culture dish, as well
as the LEDs, Raspberry-Pi and camera module.
Raspberry-Pi-Zero-W and its 8MP Camera v2 module (Adafruit
Industries) were used for time-lapse image acquisition (using the
RPi-
Cam-Web-Interface
).Exposuretimeandgainwereautomaticallysetby
the software to allow consistent imaging at various illumination inten-
sities. The Raspberry-Pi also cont
rolled the illumination protocols,
allowing the assignment of time-dependent photon-
fl
ux per culture
dish. This was done by pulse-width modulation (using the Python
interface of the
pigpio
library) of LED strips (5000K cool-white
SMD5050 12V, L1012V-502-1630 from
HitLights), two triplets per dish.
Reference illumination intensity is estimated at
I
L
≈
4
:
5
μ
mol
s
1
m
2
,
measured at the culture medium leve
l covered by the culture dish lid
(photodiode power sensor connecte
dtoanopticalpowermetre,S120C
and PM100D from Thorlabs). Under these conditions, the measured
illumination spectrum shows two main modes, one at 545nm, half-max
extends 505nm-614nm, and the other at 440nm, half-max extends
428nm-449nm (optic spectrometer
USB2000+ from OceanOptics).
The experiments were conducted in incubators (DT2-MP-47L
from Tritech Research, Inc., and VWR 2015 from Sheldon Manu-
facturing Inc.) at temperatures maintained within 22.5°C to 24.5°C.
Spatial coarse-graining, post-processing and inference of power
spectra
Images have been split into regions of interest, each corresponding to
a well in the 8-well-dish. For each region of interest, time series cor-
respond to the fraction of pixels whose value does not exceed a
threshold of 70 out of 255 in the blue channel.
Post-processing of the time series consists of outlier detection
and replacement. Outliers have been de
fi
ned as those exceeding 75%
of the well, or lying outside the 1.5 × Interquartile-Range of a rolling
window spanning 1h40m. Where there are no more than three con-
secutive outliers, these have been replaced by linear interpolation.
Table 1 | Sample sizes per illumination protocol organised by
the
fi
gure to which these contributed
Fig.
2
and Fig.
3
T
sample size
18h
5
21h
5
22.5h
7
24h
3
25.5h
8
27h
8
30h
5
Fig.
3
Tsamplesize
1.5h
6
3h
7
12h
6
24h
27
48h
8
54h
4
94h
5
Fig.
4
I
d
sample size
2
1
6
2
0
8
2
−
1
8
2
−
2
6
2
−
3
6
2
−
4
5
2
−
5
7
In Fig.
3
c, the 27 samples subject to T = 24h are from 4 culture dishes, from 3 distinct
experiments.
Article
https://doi.org/10.103
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Power spectra have been inferred based on 35 day intervals,
starting two days after transitioning from 12hL-12hD to the new driving
illumination protocol. Following the above post-processing procedure
for outlier detection and replacement, time series consisting of more
than 2% unreplaced outliers within these intervals were excluded from
this report.
Power spectra have been estimated by Welch
’
smethod
19
, applying
Hann taper to segments of 8.5 days (4896 time-points), 50% overlap
between segments, zero-padded for interpolated frequency resolution
of 1/18 × (24h)
−
1
, without detrending. Any remaining detected outliers
were replaced by forward and backwards propagation of nearest
valid data.
To estimate dominant frequencies, local maxima in the power
spectra are detected by a two step process:
fi
rst, local peaks are
fi
ltered
by requiring a prominence of at least twice the level of their baseline;
second, these are re
fi
ned by a local
fi
t to a parabola on logarithmic
scale, applied to 5 points centred at the peaks detected by the
fi
rst step.
The trends in Fig.
4
e have been estimated by a rolling average,
using gaussian weighting window of
σ
corresponding to 2 days
00h55m45s, and spanning 4
σ
.
Sample size
. Experiments were designed to result in at least 3 samples
per illumination protocol. Illumination protocols were applied to at
least one culture dish, each consisting of 8 wells. Time-series exclusion
procedure based on outlier detection is detailed above. The resulting
sample sizes per illumination protocol are listed in Table
1
, organised
by the
fi
gure to which these contributed. A degree of sample disper-
sion within illumination protocols can be appreciated from Fig.
2
d
and Fig.
3
c.
Reporting summary
Further information on research design is available in the Nature
Portfolio Reporting Summary linked to this article.
Data availability
The datasets generated and analysed during the current study are
available in the
fi
gshare repository,
https://doi.org/10.6084/m9.
fi
gshare.23797020
27
.
Code availability
All programming and computer aided analysis has been done using
open-source projects, primarily tools from the Scienti
fi
cPython
ecosystem
28
:
SciPy
29
,
pandas
30
,
IPython
and
JupyterLab
31
;imagepro-
cessing has been distributed using
dask
and
xarray
; visualisation has
been done using
HoloViz
and
Matplotlib
. Custom computer codes
used to analyse the results reported in the manuscript are available in
the
fi
gshare repository, 10.6084/m9.
fi
gshare.23797020
27
,andfrom
the corresponding authors on reasonable request.
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Article
https://doi.org/10.103
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Nature Communications
| (2023) 14:6204
7
Acknowledgements
The authors thank A. Kornfeld for the computer-aided design of the 3d-
printed apparatus, as well as Caltech Library Techlab and staff for 3D
Printers access, materials and support. We are grateful to L.J. Schulman,
M. Mussel and T. Li for stimulating discussions through various stages of
the research, as well as to L. Michaeli
, A.I. Flamholz, G. Manella and S.A.
Wilson for helpful discussions and
critical comments on the manuscript.
E.A. is thankful for fruitful discussions at the SRBR2022 and EBRS2022
meetings. Distributed image proce
ssing was conducted in the Resnick
High Performance Computing Center, a facility supported by Resnick
Sustainability Institute at the Cali
fornia Institute of Technology. The
laboratory of E.M.M. is supported by the Howard Hughes Medical Insti-
tute. E.A. has been awarded the Zuckerman Israeli Postdoctoral Scholar,
Zuckerman STEM Leadership Progra
m, and the Biology and Biological
Engineering Divisional Fellowship, Caltech. T.J.B.L. has been awarded
the Summer Undergraduate Research Fellowship (SURF), Caltech. This
article is subject to HHMI
’
s Open Access to Publications policy. HHMI lab
heads have previously granted a nonexclusive CC BY 4.0 license to the
public and a sublicensable license to HHMI in their research articles.
Pursuant to those licenses, the author-accepted manuscript of this
article can be made freely available under a CC BY 4.0 license imme-
diately upon publication.
Author contributions
Conceptualisation: E.M.M proposed studying morphogenesis in
Cau-
lerpa
; E.A. designed the study; Method
ology: E.A. designed the experi-
mental system and analysis; Inve
stigation: E.A. performed the
measurements; E.A. and T.J.B.L. performed computational analysis;
Visualisation: E.A. and T.J.B.L.; Funding acquisition: E.M.M.; Writing
—
original draft: E.A.; Writing
—
review & editing: E.A. and E.M.M.; All authors
discussed and commented on the manuscript.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary information
The online version contains
supplementary material available at
https://doi.org/10.1038/s41467-023-41813-6
.
Correspondence
and requests for materials should be addressed to
Eldad A
fi
k or Elliot M. Meyerowitz.
Peer review information
Nature Communications
thanks Hanspeter
Herzel and the other anonymous reviewe
r(s) for their contribution to the
peer review of this work. A peer review
fi
le is available.
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