cellular logics bringing the symmetry breaking in spiral ... · 29/06/2020 · 3 36 main 37...

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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

.CC-BY-NC-ND 4.0 International license(which was not certified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprintthis version posted June 29, 2020. . https://doi.org/10.1101/2020.06.29.176891doi: bioRxiv preprint

https://doi.org/10.1101/2020.06.29.176891

http://creativecommons.org/licenses/by-nc-nd/4.0/

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

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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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22. Lauzeral, J., Halloy, J. & Goldbeter, A. Desynchronization of cells on the developmental 367 path triggers the formation of spiral waves of cAMP during Dictyostelium aggregation. 368 Proc Natl Acad Sci U S A 94, 9153-9158 (1997). 369


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23. Levine, H., Aranson, I., Tsimring, L. & Truong, T.V. Positive genetic feedback governs 370 cAMP spiral wave formation in Dictyostelium. Proc Natl Acad Sci U S A 93, 6382-6386 371 (1996). 372

24. Iranfar, N., Fuller, D. & Loomis, W.F. Genome-wide expression analyses of gene 373 regulation during early development of Dictyostelium discoideum. Eukaryot Cell 2, 664-374 670 (2003). 375

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31. Lee, K.J., Goldstein, R.E. & Cox, E.C. Resetting wave forms in dictyostelium territories. 389 Phys Rev Lett 87, 068101 (2001). 390

32. Geberth, D. & Hutt, M.T. Predicting spiral wave patterns from cell properties in a model 391 of biological self-organization. Phys Rev E Stat Nonlin Soft Matter Phys 78, 031917 392 (2008). 393

33. Schulkes, C. & Schaap, P. cAMP-dependent protein kinase activity is essential for 394 preaggregative gene expression in Dictyostelium. FEBS Lett 368, 381-384 (1995). 395

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36. Weijer, C.J., Duschl, G. & David, C.N. Dependence of cell-type proportioning and 402 sorting on cell cycle phase in Dictyostelium discoideum. J Cell Sci 70, 133-145 (1984). 403

37. Gomer, R.H. & Firtel, R.A. Cell-autonomous determination of cell-type choice in 404 Dictyostelium development by cell-cycle phase. Science 237, 758-762 (1987). 405

38. Ohmori, T. & Maeda, Y. The developmental fate of Dictyostelium discoideum cells 406 depends greatly on the cell-cycle position at the onset of starvation. Cell Differ 22, 11-18 407 (1987). 408

39. Maeda, Y., Ohmori, T., Abe, T., Abe, F. & Amagai, A. Transition of starving 409 Dictyostelium cells to differentiation phase at a particular position of the cell cycle. 410 Differentiation 41, 169-175 (1989). 411

40. Muramoto, T. & Chubb, J.R. Live imaging of the Dictyostelium cell cycle reveals 412 widespread S phase during development, a G2 bias in spore differentiation and a 413 premitotic checkpoint. Development 135, 1647-1657 (2008). 414


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41. Gladwell, M. The Tipping Point: How Little Things Can Make a Big Difference., Edn. 415 1st. (Little Brown, New York; 2000). 416

42. Scheffer, M. et al. Early-warning signals for critical transitions. Nature 461, 53-59 417 (2009). 418

43. Liu, R., Chen, P., Aihara, K. & Chen, L. Identifying early-warning signals of critical 419 transitions with strong noise by dynamical network markers. Sci Rep 5, 17501 (2015). 420

44. Veltman, D.M., Akar, G., Bosgraaf, L. & Van Haastert, P.J. A new set of small, 421 extrachromosomal expression vectors for Dictyostelium discoideum. Plasmid 61, 110-118 422 (2009). 423

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

46. Gaudet, P., Pilcher, K.E., Fey, P. & Chisholm, R.L. Transformation of Dictyostelium 426 discoideum with plasmid DNA. Nat Protoc 2, 1317-1324 (2007). 427

47. Mehta, P. & Gregor, T. Approaching the molecular origins of collective dynamics in 428 oscillating cell populations. Curr Opin Genet Dev 20, 574-580 (2010). 429

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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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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