1
Title: Cellular logics bringing the symmetry breaking in spiral nucleation 1
revealed by trans-scale imaging 2
Authors: Taishi Kakizuka1†, Yusuke Hara2†, Yusaku Ohta2, Asuka Mukai2, Aya Ichiraku2, 3
Yoshiyuki Arai3, Taro Ichimura1,4, Takeharu Nagai1, 3 and Kazuki Horikawa2,* 4
Affiliations: 5
1Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Yamadaoka 2-1, 6
Suita, Osaka 565-0871, Japan 7
2Department of Optical Imaging, Advanced Research Promotion Center, Tokushima University, 8
3-18-15 Kuramoto-cho, Tokushima City, Tokushima 770-8503, Japan. 9
3Department of Biomolecular Science and Engineering, The Institute of Scientific and Industrial 10
Research (SANKEN), Osaka University, Mihogaoka 8-1, Ibaraki, Osaka 567-0047, Japan. 11
4PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan 12
13
14
* Correspondence to: [email protected] 15
† Equal contribution 16 17
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2
Summary: 18
The spiral wave is a commonly observed spatio-temporal order in diverse signal relaying systems. 19
Although properties of generated spirals have been well studied, the mechanisms for their 20
spontaneous generation in living systems remain elusive. By the newly developed imaging system 21
for trans-scale observation of the intercellular communication among ~130,000 cells of social 22
amoeba, we investigated the onset dynamics of cAMP signaling and identified mechanisms for the 23
self-organization of the spiral wave at three distinct scalings: At the population-level, the 24
structured heterogeneity of excitability fragments traveling waves at its high/low boundary, that 25
becomes the generic source of the spiral wave. At the cell-level, both the pacemaking leaders and 26
pulse-amplifying followers regulate the heterogeneous growth of the excitability. At the 27
intermediate-scale, the essence of the spontaneous wave fragmentation is the asymmetric 28
positioning of the pacemakers in the high-excitability territories, whose critical controls are 29
operated by a small number of cells, pulse counts, and pulse amounts. 30
31
Keywords: 32
trans-scale imaging; cAMP; spiral wave; Dictyostelium discoideum; self-organization; symmetry 33
breaking; excitable system; growing excitability; small number control 34
35
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Main 36
Introduction: 37
The spiral wave is spatiotemporal order commonly observed in the diverse range of excitable 38
media1, including the chemical2 and biological systems3,4. In the biological context, the 39
intercellular relay of communication signaling in the form of a rotating spiral is often associated 40
with pathological cases such as neuronal epilepsy5, progressive dermal inflammation6,7, and 41
ventricular fibrillation8,9. The spiral core, also known as the spatial phase singularity (PS), is a self-42
sustaining structure that is highly robust to external perturbations; a deeper understanding of the 43
properties and generation of spiral waves is needed to manage and regulate the diseases associated 44
with spiral waves. 45
While the properties of the mature spirals have been well documented1, the onset mechanism, 46
especially in the spontaneous spiral nucleation, is still poorly understood. Conceptually, the spiral 47
wave arises from a pair of open ends of fragmented waves, but the fragmentation of the excitation 48
waves never develops spontaneously in homogeneous systems1. Thus, experimental induction of 49
the spirals in the homogeneous system requires designed conditions such as a mechanical break in 50
the waves2, geometrical constraints10, or additional wave initiation at the vulnerable region in the 51
preceding wave8,11 to bring the symmetry breaking, i.e., conversion from closed ring waves 52
(symmetric) to fragmented waves (asymmetric). Alternatively, the systems implemented with 53
heterogeneity in their excitability12,13, refractoriness14,15, or coupling strength16 are capable of 54
generating wave fragments. It is still unclear what types of cellular rationales organize these 55
heterogeneities during the spontaneous spiral nucleation in living systems. 56
The social amoeba, Dictyostelium discoideum (D. discoideum), is an ideal model of the self-57
organized pattern formation. During its development, initiated by the nutrient starvation, 103-6 cells 58
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establish spiral-shaped aggregation streams with sub- to few-millimeter wavelengths17. The 59
intercellular relay of the chemo-attractant cyclic adenosine monophosphate (cAMP) drives the 60
process18,19, whose reaction-diffusion dynamics are explained by a simple combination of the local 61
synthesis, degradation, and diffusion20-22. The system is consisted of typical excitable cells having 62
three distinct states including the absolute- or relative-refractory and the excited state, wherein the 63
above-threshold input of extracellular cAMP at the relative refractory state induces the transition 64
to the excited state. The uniqueness of this living excitable system lies in its ability to self-organize 65
the wave patterns even from a quiescent initial state, during which the low-to-high transition of 66
excitability is believed to play a role22,23. The excitability is the cellular ability to relay cAMP 67
pulses that are positively regulated by repeatedly relaying cAMP pulses, which is realized by a 68
gradual increase of the gene expression controlling the cAMP synthesis, release, and 69
degradation18,24,25. Thus, the question is how the symmetry breaking in the form of the wave 70
initiation and spontaneous fragmentation takes place in the growing excitability. This should be 71
ideally addressed by sensitive and large-scale observations of the cAMP pulse dynamics. The 72
realization of such observations, however, has been challenging due to the technical limitations in 73
fulfilling all of the following conditions: 1) high enough sensitivity to detect faint cAMP pulses 74
even in the low-excitability regime, 2) a large enough observation field to capture the spiral cores 75
at the millimeter-scale, and 3) high enough spatial resolution to identify the pacemaking (PM) cells 76
whose presence is expected to be rare. 77
In this study, we have developed an improved imaging strategy that allows the bird’s eye view 78
analysis of the cAMP pulse dynamics with a field coverage of > cm2 at the single-cell resolution, 79
and have found that the spontaneous wave fragmentation develops in close association with the 80
heterogeneously structured excitability that is highlighted by the fine-mapping of the cumulative 81
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pulse count. The analysis at single-cell resolution also identified the critical phenomena in the 82
temporal evolution of local signaling dynamics at the center of the highly pulsing areas, of which 83
a small number of cells controls the collective behavior. Along with the modeled dynamics, we 84
emphasize the functional importance of the microscopic heterogeneity in the initial excitability, 85
which makes the cellular discreteness as the origin of the symmetry breaking. 86
87
Results: 88
Trans-scale imaging of intercellular communications of 130,000 cells 89
To analyze the onset dynamics of the spiral nucleation, cellular cAMP dynamics were 90
detected by using a fluorescent reporter Red-FL2, a fusion protein of cAMP indicator Flamindo2 91
and mRFPmars, whose ratiometric imaging allows quantitative analysis free from motion artifacts 92
(Fig. 1a). Instead by using conventional microscopes, exceptionally largescale imaging was 93
achieved by the newly developed imaging system. For short, the system is featured by an objective 94
lens with a large diameter (~ 50 mm, magnification = 2, N.A. = 0.12), a hundred megapixel CMOS 95
image sensor with a small pixel size (2.2 µm), that eventually allowed the snapshot of 130,000 cell 96
/ 14.6 × 10.1 mm2 with a spatial resolution of 2.3 µm (Fig. 1a, 1b, the detail of the imaging system 97
is described elsewhere). To minimize any damages on our sample associated with live imaging, 98
cAMP dynamics were imaged every 30 sec, being high enough temporal resolution to dissect the 99
timing of cellular cAMP pulse whose FWHM (full width at half maximum) is ~90 sec at 6-15 min 100
interval (Fig. 1c). After 12 hrs of observation started at 4 hrs-post-starvation, we obtained ~360 101
G-bites of image sequences that were spatially discretized into 12,236 regions of interest (ROIs, 102
100 ×100 pxl each containing ~ 100 cells) and were digitally processed for the analysis of the wave 103
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dynamics. The primary focuses of the analysis were the wavefront, oscillation phase, recovery 104
state, and pulse counts (Fig. 1d, Supplementary Fig. S1). 105
The spontaneous fragmentation of the cAMP waves 106
To find spatial locations of the wave fragmentation, we reconstructed the phase maps from 107
the temporal dynamics of the cAMP pulses in each locality (Supplementary Fig. S1). Because 108
the PS is the singular space in which the entire oscillation phase is encircled, the emergence of the 109
PS in the phase map represents de novo wave fragmentation. The localized pulse in two ROIs 110
(asterisk in Fig. 2a, 7:32) propagated outward as a symmetric, closed ring, and was broke into the 111
wave fragment whose open ends were marked with a pair of PSs (filled circles in Fig. 2a, 7:37). 112
While a part of fragment extending from the counter-clockwise rotating PS (cyan circle in Fig. 2a, 113
7:38) was disappeared through the annihilation with incoming PS (dashed magenta in Fig. 2a, 114
7:38) from right, another fragment from the clockwise rotating PS (magenta) showed a long 115
survival for more than the next 60 min. 116
What makes propagating waves fragmented in this living excitable system? For the 117
simplest expectation, the low excitability and/or low recovery in the refractoriness would locally 118
block the wave propagation, thus causing the circular wavefront to break open. We examined the 119
properties of the signal-receiving region for the presence or absence of the wave propagation and 120
investigated which parameters were low specifically in the wave-blocking region (Figs. 2b–2e). 121
The cumulative number of pulses, i.e., a measure of the excitability (Supplementary Fig. S2; 122
Supplementary Note S1), in the wave-blocking region was found to be approximately one-half 123
of that in the wave-permissive region (Figs. 2b, 2c). The elapsed time from the previous excitation 124
in the wave-blocking region, i.e., an indication of the recovery state, was almost identical to that 125
in the wave-permissive region (Figs. 2d, 2e), suggesting that the local difference in the excitability 126
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but not recovery state causes the fragmentation of the traveling waves. The importance of the 127
spatially different excitability was further supported by a quantitative analysis of the 128
spatiotemporal dynamics of the PSs in the development (7 to 10 hours of development). The spatial 129
distributions of all the emerging PSs (n = 738) were localized at the boundary between the high- 130
and low-pulsing regions in the heterogeneously structured excitability that was highlighted by the 131
high-resolution map of the cumulative pulse counts (Fig. 2f). To further understand the 132
relationship between the PSs and structured excitability, trajectories of the PSs were investigated. 133
As a result, we found that the spatially dynamic trajectories of the PSs in the early development 134
(Fig. 2g, 7:30) were localized at the edges of the highly pulsing islands, while those in the regions 135
with the homogeneously high excitability at the later development (Fig. 2h, 9:10) were less mobile. 136
The presence of a critical period for the spiral nucleation can explain the temporally different 137
trajectories of the PSs. The developmental changes in the PS density presented a bell-shaped curve 138
with the peak at 7:10 of the development, and both the appearing and disappearing PSs were 139
limited to the early development (until 8.5 hours of the development, Fig. 2i). These observations 140
suggested that the transiently structured excitability in the critical period played an important role 141
in the spontaneous wave fragmentation. 142
The functional importance of the macroscopically structured excitability in the spiral 143
nucleation was tested by the perturbation experiments focusing on the oscillation phases and the 144
landscape of the excitability (Fig. 3). The phase resetting was done by a bath application of cAMP 145
and the spatial resetting was newly introduced as follows; the D. discoideum cells can be easily 146
detached from a culture dish by intensive pipetting, then the cells restart the cAMP pulse dynamics 147
after they settle on the dish with the cellular positions being randomized (Supplementary Note 148
S2, Supplementary Fig. S3). The phase resetting at the post-critical period diminished the existing 149
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spirals allowing only the reappearance of the concentric waves (Figs. 3a, 3c) as had been 150
demonstrated previously28, 31. The phase resetting at the critical period, however, is not effective 151
as we observed >50 PSs to have appeared, indicating that cues for spiral cores are temporally 152
specific to the early stage of development and are not affected by the phase resetting. We then 153
tested the spatial resetting at the critical period and observed that the reappearance of PS was 154
reduced to 1/5 of that for the phase resetting (Figs 3b, 3c). These results demonstrate that the 155
heterogeneously structured excitability with an mm-size of coherency is the cause of the 156
spontaneous wave fragmentation during the critical period, but not in the post-critical period. 157
In a short summary, to answer the question as to how the propagating waves are fragmented, 158
we have demonstrated that the presence of the heterogeneous excitability and its functionality in 159
the critical period are both essential and that the high/low boundary of the excitability is the cause 160
of the fragmentation in the isotropically traveling wavefront, which subsequently becomes the 161
generic source of the spiral cores. 162
The development of the high-excitability territories 163
How does such spatially heterogeneous excitability develop from the quiescent initial 164
state? The self-organization of the heterogeneity could be explained either by some distinct local 165
rules controlling the pulse-dependent increase in the excitability, or more simply, by different 166
initial conditions such as the spatially biased presence of the PM cells. To examine how the 167
heterogeneous excitability develops, we focused on localities showing the most active or inactive 168
development and analyzed the growth dynamics of the local excitability including the PM 169
activities at cellular resolution. The images of the cumulative pulse counts highlighted the growth 170
pattern of the highly pulsing area (9 neighboring ROIs, magenta box in Fig. 4a). The first pulse 171
emerged as early as 4:15 after the starvation, and then the wave-permissive area expanded outward 172
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every pulse, each of which originated from inside the highly pulsing territories. In the following 173
development, this area developed to few millimeters in diameter with a pulse count of 9 at its peak 174
(6:30), while the surrounding areas had fewer than 2 pulses (Fig. 4a). 175
To examine whether or not the local rules control the pulse-dependent development of the 176
excitability, we quantified the probability of the pulsing cells (number of pulsing cells/total cells 177
in the neighboring 9 ROIs) in two representative localities with high and low pulsing activities 178
(Figs. 4a, 4b), hereafter called hot and cold spots, respectively. In the hot spot (magenta in Fig. 179
4a), a cAMP pulse started when 2 out of 107 cells were pulsing (pulse probability of 0.02). As the 180
pulse count increased every 10-20 min, so did the local pulse probability until reaching the plateau 181
(0.95) at 7 hours of the development. In the cold spot (cyan box in Fig. 4a, containing 109 cells), 182
the timing of the pulse initiation for cell- and population-level were delayed 1 and 2 hours, 183
respectively, as compared to that of the hot spot. Re-plotting the pulse probabilities for the growing 184
population (> 0.15) against the pulse counts revealed similar growth rates (the half-maximal pulse 185
probabilities yielded the pulse count of ~5, Supplementary Fig. S2). It indicated that the rule 186
controlling the increase in the excitability as a function of accumulating pulses was conserved 187
among different localities. The major difference leading to the distinct timings of the pulse 188
initiation would be the initial condition. Indeed, we found two PM cells showing 8 rounds of 189
leading pulses at the center of the hot spot (inside 3x3 ROIs) together with two supportive PM 190
cells (twice and once activity) around the center for 4.0-6.5 hrs of development (Figs. 4c, 4d): No 191
PM cell was present in the cold spot where the population pulse was always triggered by 192
propagating waves from the surrounding regions throughout the development. 193
The logics controlling the critical transition 194
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To further investigate the cellular rationales controlling the local development, we asked 195
whether the pacemakers were sufficient to account for the hot spot development, and presumed 196
that the search for the PM activity was important because the PM activities at the single-cell or 197
population level are essential to trigger the wave propagation in the excitable system. Although it 198
has been long believed that the presence of the PM cells is sparse, their actual density and pulse 199
patterns (random or deterministic) have not been analyzed. Our trans-scale observation 200
successfully identified 84 cells (corresponding to 0.13% of the total population distributed in 67 201
of color-coded ROIs, Fig. 4e) which showed the spontaneous and repeated PM activities during 202
the early development (4:00–6:00). While PM activities of these ROIs were evenly distributed 203
along time (data not shown), their position was found to be biased to the center of developing 204
territories as the single or cluster of ROIs (Fig. 4e). To more correctly understand the relationship 205
between the pacemaking activity and the hot spot development, we detected the wave permissive 206
area for the latest activity before 6:00. As in the case for the above analyzed hot spot, these 207
highlighted areas for the representative of well-developed territories (solid line in Fig. 4f) were 208
found to be associated with ROIs with high PM activities (pulse counts >3). Inversely, when we 209
focused on ROIs with high PM activities (pulse counts >3), several ROIs were found to show poor 210
development (dashed lines in Fig. 4f). In the most extreme case, no wave propagation was 211
observed for ROI with 4 rounds of PM activities by the single cell (arrow in Fig. 4f). Collectively, 212
these results suggest that PM cells are essential but insufficient for the development of the hot spot. 213
214
To address what mechanisms are needed to drive the local development, in addition to PM 215
activities, we focused on a locality showing moderate development (arrowhead in Fig. 4f) and 216
investigated the pulse dynamics at the single-cell resolution (Fig. 5). The pulse probability 217
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remained low during the first five population pulses, then an abrupt leap broke out at the 6th pulse 218
(19/~90 cells at 5:25, Fig. 5a). Other regions including the hot and cold spots also displayed similar 219
leaps, suggesting that this is a general phenomenon. We hypothesized that such leaps, or the critical 220
transitions (CTs), may play an important role in the evolution of the excitability. To gain more 221
insights into the cellular mechanisms of the CT, we functionally categorized the pulsing cells based 222
on the timing of the coupling to the population pulse (before or after the CT) and the transition 223
pattern of the pulse amplitude. It was possible to group the activities into three distinct classes, 224
which would correspond to the different excitability (Fig. 5b): The first group was “leader” 225
characterized by PM activities of the cells (Fig. 5c). The large amplitude (>0.4 of the normalized 226
amplitude) throughout the development was a notable feature reflecting the highest excitability. 227
The opposite extreme was the group named “citizens” containing the majority of the cells (>90%) 228
that were characterized by a delayed start of the cAMP pulses after the CT. The amplitude of the 229
pulses in this group increased from a small value, suggesting the low excitability in the beginning. 230
The remaining groups were “follower” (6 cells), showed a gradual increase in the pulse amplitude 231
like the citizens, but the onset was advanced to the CT. 232
To unveil the mechanisms leading to the CT, we asked the difference in the number of 233
pulsed cells and signal strengths among three groups between pre-CT and CT. In definition, it is 234
natural that both the number and amount of pulsed citizens were increased at CT (Figs. 5d, 5e), 235
but it is not plausible to consider citizens bringing the CT. Since the position and pulse timing of 236
citizens at CT is distal and behind to that of the leader and followers, respectively, indicating that 237
the pulse of citizens was the result of CT rather than the cause. Instead, we observed the increase 238
of pulse amount of followers, while its number was kept constant before and after the CT (Fig. 239
5d). This suggested that the driving force of the CT would be the followers. Supporting this idea, 240
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we observed similar increases in the pulse amount of followers in the other two hot spots (Fig. 5f). 241
Furthermore, the ensemble of the pulse amount from 4 different hot spots for pre- and post-CT 242
suggested that conserved mechanism for CT, whose threshold-like behavior was controlled by 243
estimated pulse amount, being 4.5 µM(AU) of summed pulse/9ROIs (Figs. 5f, 5g). 244
Taken together, these findings have demonstrated that the development of hot spots 245
requires both PM activities of the leaders and effective amplification by the followers, the latter 246
controls timings of the CT leading to the locally distinct excitability. 247
The origin of the symmetry breaking 248
Although the above results have demonstrated the development of the macro-scale 249
heterogeneity of the excitability, this did not explain the mechanisms of the spontaneous wave 250
fragmentation. Mesoscopically, one of the key factors leading to the wave fragmentation is the off-251
center positioning of the wave initiation loci in the high-excitability territory, which should 252
originate from microscopic asymmetry. To examine the origin of the symmetry breaking, we 253
performed the numerical simulations based on cellular automata (CA). The genetic feedback 254
model was originally introduced by Levine23. The schemes with slightly different assumptions 255
from the first model were later developed by several groups28,32. The model with these schemes 256
describes the two-dimensional reaction-diffusion dynamics of the extracellular cAMP, where the 257
spatially heterogeneous excitability evolves through the pulse-dependent positive feedback on the 258
pulsing potential. Although the model of these schemes well explains the spontaneous wave 259
fragmentation in the presence of densely distributed PM cells, we found that this was not the case 260
when the density of the pacemakers was altered so that it was lower than the level observed in real 261
systems (Figs. 4e, 4f). The schemes assume the homogeneous initial condition of the excitability 262
(Eini), which allows the spatially isotropic development of the local excitability. The asymmetric 263
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positioning of PM activities in the wave-permissive territory can be realized only when a few of 264
these territories are closely packed to each other. We overcame the inability of the schemes to 265
produce spiral waves at low pacemaker density by introducing the distributed Eini instead of the 266
homogeneous one. As shown in Fig. 6, the fixed ROI with PM activities yielded the anisotropic 267
growth of the high-excitability territory. Also, the simulation was able to reproduce the 268
spontaneous wave fragmentation in the high-excitability territory (Figs. 6a, 6b) as observed in the 269
real system (Figs. 6c, 6d). The variations of the size of the high-excitability territories and spiral 270
wave formation for the distributed Eini were reproducible (Figs. 6e, 6f), whereas the homogeneous 271
excitability was not (Supplementary Fig. S4). The reproducibility further confirmed the validity 272
of our assumptions. The pattern of the distribution was not critical since the wave fragmentation 273
itself broke out with a variety of non-homogeneous Eini, including the long-tailed, random, and 274
uniform distributions (Supplementary Fig. S4). Although the Eini is not directly measurable in 275
real systems, the good agreement with the observed and simulated wave dynamics in both the 276
unperturbed and perturbed conditions (Supplementary Fig. S4) has demonstrated that the 277
distributed Eini is the origin of the symmetry breaking in the self-organized spiral nucleation of the 278
excitable system in our settings. 279
280
Discussion: 281
Our trans-scale analysis leads the three major insights at each scale into the mechanisms of 282
the self-organized spiral nucleation. At the macro-scale, the traveling front of the excitation wave 283
is fragmented at the edge of the heterogeneously structured excitability. The micro-scale analysis 284
has revealed that, for its growth, the involvement of both the PM activities of the spatiotemporally 285
non-random, rare leader cells and effective amplification by the follower cells is crucial. 286
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Importantly, these findings themselves do not explain the central mechanism of the symmetry 287
breaking. That is because the key concept of the spontaneous wave fragmentation lies in the off-288
center positioning of the pacemakers in the high-excitability territory. The meso-scale analysis 289
together with the mathematical modeling has demonstrated that such symmetry breaking is seeded 290
by the non-random activities of the fixed pacemakers. They anisotropically amplify subtle 291
differences in the cellular excitability that eventually cause the spontaneous wave fragmentation 292
autonomous to the high-excitability territory. 293
We emphasize that, through the use of the trans-scaling imaging techniques for cAMP 294
dynamics, we have unified the mechanisms of the spiral nucleation at distinct scalings (i.e., micro-295
meso-macro). Specifically, the sensitive detection of the pulse dynamics even at the low-296
excitability regime has allowed us to utilize the cumulative number of pulses as an indirect but 297
valuable measure of the local excitability, for which a suitable molecular marker has not been 298
found. The biochemical and genetic studies showed the pulse-dependent increase of the 299
excitability in D. discoideum cells24,25,33,34. We have also observed similar dependency as 300
summarised in Supplementary Notes S1. The increase in both the pulsing ability at the single-301
cell level and the number of pulsing cells at the population level explains the gradual increase in 302
the amplitude of the population-ensemble cAMP pulses (Supplementary Fig. S2). A clear 303
correlation between the oscillation period and the pulse count also supports our hypothesis that the 304
cumulative number of pulses denotes the local excitability (Supplementary Fig. S2). The 305
identified geometry of the excitability with local coherency (sub-millimeter-scale) and global 306
heterogeneity (millimeter-scale) is of special significance. This is the first visualization of the 307
spatiotemporal evolution in the growth dynamics of the cellular excitability whose importance is 308
uniquely attributed to the biological self-organization, but not to the chemical models (i.e., BZ 309
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reaction) with non-growing excitability. The anisotropic growth of the high-excitability territory 310
and functional classification of the cells by their distinct excitability have collectively 311
demonstrated the effectiveness of our trans-scale analysis. It should also be effective in the studies 312
of other growing excitable media such as the segmentation clock35 and neuronal circuits. 313
Notably, the so-called “the law of the few” drives the self-organized pattern formation in 314
the discrete system41. In each excitable territory, the wave dynamics is initiated by a small number 315
of the PM cells. To our knowledge, this is the first study to estimate the density of such cells (one 316
per 770 cells). The collective behavior of the cell population arises when the critical threshold of 317
~4.5 µM of pulses (AU)/~100 cells/0.13 mm2 is reached. For the wave fragmentation, the 318
difference in the effective pulse count between 0 and 10 has functional importance, but not in the 319
count above 10. Such singular properties of the CT governed by the law of the few should apply 320
to the dynamical systems ranging from the cells to the organisms, the eco-systems, and the climates. 321
More efforts in seeing the forest for the trees, or even for the leaves will help to predict and control 322
the system dynamics in the real world42,43. 323
324
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45. Fey, P., Kowal, A.S., Gaudet, P., Pilcher, K.E. & Chisholm, R.L. Protocols for growth 424 and development of Dictyostelium discoideum. Nat Protoc 2, 1307-1316 (2007). 425
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430
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Figures and Legends: 431
432
Figure 1. Large-scale Imaging of cAMP Pulse with 1-cell Resolution. 433
(a) One-shot-imaging of ~130,000 cells with 1-cell resolution (top). cAMP pulse detected by the 434 ratiometry of mRFPmars and Flamindo2 for a population (bottom). Average of ~100 cell data in 435 the white box). (b) Close-up view of box in (a), color-coded by the pulse timing. (c) Representative 436 cAMP pulses for cells in (b). (d) Still images of [cAMP]i, oscillation phase, and cumulative pulse 437 counts. Phenotypic features in this culture condition were shown on the top. See also 438 Supplementary Fig. S1. Scale bars, 0.5 mm. 439
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440
441
Figure 2. Spontaneous Fragmentation of cAMP Wave. 442
(a) Phase representation of the wave propagation. Green traces represent the wavefront. The 443 asterisk at 7:32 (post starvation) is the point source of the outward propagating wave. A pair of 444
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rotating PSs emerges at 7:37. (b–e) Comparison of the local property in the wave-permissive 445 (solid) and -blocking regions (dashed border). The cumulative pulse counts (b and c) and the time 446 after pulse (d and e) are measures of the excitability and recovery state, respectively. Error bars 447 represent s.d. (f) Spatial distribution of all the emerging PSs during 7:00–8:30 imposed on the map 448 of the cumulative pulse counts at 7:45 of the development. (g and h) Trajectories of PSs during 449 7:30–7:40 (g) and 9:10–9:20 (h) imposed on the map of the cumulative pulse counts at 7:30 and 450 9:10 of the development, respectively. Maximum, minimum, and average of the cumulative pulse 451 counts are 15, 0, and 5.50±2.15 (s.d.), respectively, for (g) and 25, 8, and 14.71±2.12 (s.d.), 452 respectively, for (h). (i) Temporal change in the number of PSs. The line graph is the average 453 density (#/whole view) in 10 min data. The histogram presents the numbers of emerging (black) 454 and disappearing (gray) PSs. Error bars represent s.d. Scale bars are 0.5 mm. 455 456
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457 458
Figure 3. Effects of the Spatial and Phase Resetting on the Spiral Nucleation. 459
(a and b) Phase images of the wave dynamics before and after the resetting. Phase (a) and spatial 460 (b) resetting performed at the critical period (top) or post-critical period (bottom), respectively. 461 (c) The average number of emerging PSs after the resetting obtained from 2 independent 462 experiments. Scale bars, 2 mm. 463 464
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465
Figure 4. Growth Dynamics of the Highly Pulsing Territory. 466
(a) Spatiotemporal growth of the pulse dynamics. (b) Pulse probability of the hot spot (9 467 neighboring ROIs, magenta box in a) and cold spot (cyan box in a) plotted against time. (c and d) 468 Non-random PM activities of fixed cells. Position (c) and pulse pattern (d) of two PM cells (orange 469
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and magenta) and 7 and 12 subsidiary cells for 5:03 and 5:27, respectively (white circles in c) in 470 the hot spot. Temporal flow of the pulse timing, starting from PM cells followed by cells with 471 arrows in this sequence. Grid size is 120 × 120 µm2. Asterisks in (d) are the PM pulses. (e) 472 Location and activity of PM cells in the analyzed field (~ 65,000 cells/7.3 × 10.1 mm2), imposed 473 on the map of the cumulative pulse counts at 6:00 of the development (gray). Box represents the 474 area analyzed in (a). (f) Wave propagated area at 6:00 for representatives of well- (solid) and poor-475 (dashed) developing territories, imposed on the map of location and activity of PM cells. Scale 476 bars are 0.5 mm. 477 478
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479
Figure 5. Critical Transition in the Local Pulse Dynamics. 480
(a) Burst in the number of pulsing cells for 9 ROIs containing arrowheded PM in Fig. 4f. Pulsed 481 cell number plotted against the number of population pulse. Gray and orange masks discriminate 482 before and after the critical transition. (b) Pulse dynamics of leader, follower, and citizen cells. 483 Representative 1-cell pulse and the temporal change of the peak amplitude along with the N-th 484 population pulse. (c) Location and the pulse timing by the cell group at pre-CT and CT. Grid is 485 120 x 120 µm. (d, e) The number(d) and quantified cAMP pulse(e) summed by the cell type. (f) 486 Changes in the pulse amount of leader and followers in 3 hot spots at pre-CT and CT. (g) The 487 ensemble of pulse amount of 4 hot spots before and after the CT, sorted by the magnitude. Asterisk 488 is the pulse amount at CT. The dashed line is the estimated threshold for the CT. See 489 Supplementary Fig. S1 for the location of the analyzed area. 490
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491
492
Figure 6. CA Modelling with Few Pacemaker Activities. 493
(a–d) Comparison of simulated (a and b) and observed (c and d) development of the single high-494 excitability territory (a and c) and wave patterns (b and d). Off-center positioning of the fixed 495 pacemaker (asterisk) and high-excitability territory (dashed green line). Arrowheads show the 496 spontaneous wave fragmentation in the single territory. (e) High-excitability territories with 497 variable shapes and sizes developed by 20 symmetrically positioned pacemakers in a larger-scale 498 simulation. (f) Spontaneous generation of the spiral wave from (e). An exponentially distributed 499 Eini (mean value, 0.2) is considered. Asterisks denote the positions of pacemakers. These results 500 were replicated on three different data sets of Eini. Scale bars, 0.5 mm. See also Supplementary Fig. 501 S4. 502 503
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Methods: 504
Molecular biology. cDNA encoding mRFPmars-Flamindo2 fusion protein (Red-FL2) whose 505
codon usage was optimized for D. discoideum was constructed by using inFusion cloning system 506
(TAKARA), then was cloned into pDM304 and pDM35844. Resulting plasmids, pDM304_Red-507
FL2, and pDM358_Red-FL2, were deposited to the Dicty Stock Center. 508
509
Cell culture. The axenic strain Ax2 was cultured and transformed as described elsewhere45,46. For 510
transformation, the cells were washed and suspended with ice-cold EP buffer (6.6 mM KH2PO4, 511
2.8 mM Na2HPO4, 50 mM sucrose, pH 6.4) to 1 × 107 cells/ml. A total of 800 µl of cell suspension 512
mixed with 10 µg of pDM304_Red-FL2 in a 4-mm-width cuvette was subjected to electroporation 513
(two 5 sec separated pulses with 1.0 kV and a 1.0 msec time constant) using a MicroPulser (Bio-514
Rad). These cells were plated on 4–6 pieces of 90-mm plastic dishes with HL5 medium and 515
incubated at 22°C for 18 hours under the non-selective conditions and then cultured in the presence 516
of 10 µg/ml G418 (Wako). After 4–7 days, colonies with high expression of Red-FL2 were 517
manually picked. Some of these were subsequently transformed with pDM358_Red-FL2, then 518
were cultured in the presence of 35 µg/ml hygromycin (Wako) and 15 µg/ml G418. Clones 519
showing higher expression of Red-FL2 with lower heterogeneity were screened. 520
Imaging. The cells expressing Red-FL2 were maintained in HL5 medium on a 90-mm plastic at a 521
density of < 1 × 106 cells/dish. The development was initiated by the 3× washing of the cells with 522
development buffer (5 mM Na2HPO4, 5 mM KH2PO4, 1 mM CaCl2, 2 mM MgCl2, pH 6.4). These 523
cells were plated on a 35-mm plastic dish at a density of 750 cells/mm2. The live-cell imaging was 524
performed by using a custom-built imaging system equipped with the single CMOS image sensor 525
and LEDs illumination (unpublished). 526
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Numerical simulations. The simulations were performed on a 114 × 136 mesh where Dx = Dy = 527
0.048 mm (Figs. 6e, 6f, and Supplementary Fig. S4) using the explicit Euler method at Dt = 0.1 528
min with synchronous updating in MATLAB. The average of the initial excitability (Eini) was set 529
to 0.2 for both the homogeneous and distributed cases. 530
531
Equations. To explain the territory-autonomous wave fragmentation, one minor and two major 532
modifications are introduced into the genetic feedback model based on our experimental 533
observations. 534
Briefly, the genetic feedback model23,28,32 is a hybrid cellular automaton (CA) in which the 535
reaction-diffusion dynamics of the extracellular cAMP (c) and the pulse-dependent increase in the 536
excitability (E) obey the following: 537
c˙ij = Crel*sij(t) − Cdeg*cij + D*∇2*cij (Eq. 1) 538
E˙ij = η + β*cij (Eq. 2) 539
where Crel is the release rate of cAMP, Cdeg is the degradation rate of extracellular cAMP, 540
D is the diffusion constant of cAMP, β is the feedback strength of the cAMP concentration on the 541
excitability and η is the autonomous increase in the excitability over time. Cellij is treated as a CA 542
whose three discrete states (excited, absolute refractory, and relative refractory) are controlled by 543
the binary operator s (s = 1 for excited state and s = 0 for absolute and relative refractory states). 544
The residence times for the states are Tex = 1 min, Tabs = 2 min and Trel = 7 min. Except for the PM 545
activity, a cell receiving an above-threshold cAMP at the relative refractory state is excited where 546
the decreasing threshold CT (from Cmax to Cmin) obeys the following: 547
CT ij (t) = [Cmax − A*t/(t + Tabs)]*(1 − Eij) (Eq. 3) 548
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where A = (Trel + Tabs) *(Cmax − Cmin)/Trel and t measures the elapsed time in the relative 549
refractory state (0 < t < Trel). After Trel, CT is kept to Cmin. For the PM activity, a spatially fixed 550
cell is autonomously pulsed depending on the firing probability pfire, as assumed in ref32. 551
To reproduce the observed wave dynamics, we modified the above model to 552
c˙ij = Crel*sij(t) *Eij − Cdeg*cij + D*∇2*cij (Eq. 1’) 553
CT ij (t) = Cmax − A*t/(t + Tabs) (Eq. 3’) 554
where cAMP is synthesized proportional to E (in Eq. 1’). The threshold CT is freed from E (in Eq. 555
3’), and Cmax and Cmin are tuned to small enough values to allow the pulse dynamics even at the 556
low-excitability regime. Essentially, these changes do not affect the central dynamics of the model 557
and only increase the dynamic range of c and E. The major two modifications were introduced to 558
the PM activity and Eini. The spatial23, 28, or temporal32 randomness in the PM activity, being the 559
source of the symmetry breaking in the previous schemes, was eliminated by assuming a small 560
number of fixed PM cells (0.6 cells/mm2) with almost identical pulse intervals (20 ± 0.45 min). 561
Finally, the distributed Eini rather than the homogeneous one was considered, which plays the 562
central role in the symmetry breaking in our modeling scheme. To focus on the functional 563
importance of Eini, a regular arrangement of the 20 PM cells on the square lattice with a spacing 564
of 1.1 mm was considered. 565
Collectively, our modifications were minimally considered in (Eq. 1’) and (Eq. 3’). The 566
parameters changed are Cmax = 40 and Cmin = 1, the number of PM cells is 20 (4 × 5 array/5.4 × 567
6.5 mm2) with pfire = 0.0005 ± 0.0001, and all the considered Eini have a mean value of 0.2, while 568
the other parameters are identical to those of ref. 28. (Emax = 0.93, Crel = 300, Cdeg = 8 min-1, η = 569
0.0005, b = 0.005 and D = 2.3 × 10-7 cm2/sec). 570
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571
Image analysis. The ratio images of the background-subtracted and spatially smoothed channels 572
for RFP and Flamindo2 were subjected to the image enhancement of pulsed cells assisted by the 573
supervised machine learning by using AIVIA software (DRVISION tech.). The image field was 574
subdivided into 12,236 ROIs (133 × 92 matrices of 100 × 100 pxl), then the peak detection was 575
performed on time series data for every ROI by using Mathematica (Supplementary Fig. S1). The 576
oscillation phase, time after pulse, cumulative pulse counts, and pulse counts were analyzed in 577
each ROI containing ~10 cells (Supplementary Fig. S1). The image reconstruction of these data 578
was performed on the custom-built analysis pipeline using Excel, Mathematica, and Fiji software. 579
For peak detection, the detection sensitivity was tuned to detect > 2 pulsing cell/ROI. After 580
obtaining a peak table for 900 frames of 12,236 ROIs, it was manually corrected to detect 1 pulse 581
cell / ROI with DR(Red/FL2) > 7.5 %. 582
583
Quantification of the cAMP pulse at 1-cell. To quantitatively compare the amount of cAMP 584
pulse among cells, we performed a two-step normalization for the emission ratio change of 585
Red/FL2. Considering the identical stoichiometry of red and yellow signals of Red-FL2 among 586
cells, we pre-normalized the baseline ratio values to 1.0. This cancels the different baseline ratio 587
of distantly positioned cells affected by a slight imbalance of illumination strength over the image 588
field. For the pre-normalized ratio data, maximum values at the highest peak were found to reach 589
around 4.1 for >300 cells, that is consistent with the previously reported signal change of 590
Flamindo2 (4-fold). We finally normalized the pulse data of all the examined cells by considering 591
a minimum (1.0) and maximum ratio values (4.1) to 0 and 1, respectively. The sum of the 592
normalized ratios of pulsing cells was utilized to estimate the pulsing activity in a given ROI by 593
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using Hill equation: [cAMP]n = Kd*DR /(1-DR), where Hill coefficient (n) of 1.0 for Flamindo2 594
(in vitro) was employed for a simplicity. 595
596
Statistics. As the distribution of data in this study is not normal, nonparametric statistics were used. 597
All statistical tests were two-tailed. The analysis was performed in the R statistical environment 598
version 3.3.2. 599
600
Data availability. Raw data and codes are available from the corresponding author (K.H.) upon a 601
request. 602
603
Acknowledgments: 604
We thank Y. Yoshihara, J. Kajiwara, K. Morimoto, and other members of the K.H’s 605
laboratory for technical assistance. We also acknowledge the Dicty Stock Center for providing us 606
the pDM304, pDM358, and Ax2 cell lines. This work was supported by a Grant-in-Aid for 607
Scientific Research on Innovative Areas “Singularity Biology (No.8007)” (18H05408 to T.N., 608
18H05415 to K.H.), a Grant-in-Aid for Scientific Research on Innovative Areas “Spying minority 609
in biological phenomena (No.3306)” (23115003 to T.N., K.H.), the Grant-in-Aid for Young 610
Scientists (A) (18687014 to T.N.), and the Research Program of "Five-star Alliance" in "NJRC 611
Mater. & Dev." (T.N., K.H). 612
613
614
615
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32
Author Contributions: 616
T.K., T.I, and T.N developed the fluorescence imaging apparatus. T.K, A.M., A.I., and K.H. 617
performed experiments. Y.H., T.K., A.M.,Y.A., and K.H. analyzed data. Y.O. and K.H. performed 618
the modeling. Y.H., T.N., T.I. and K.H. designed and conducted the project and wrote the 619
manuscript. 620
Competing Interests: 621
The authors declare that they have no competing interests. 622
623
Supplementary Information: 624
Supplementary Information includes two Supplementary Notes and four Figures. 625
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