Sakamoto R , Tanabe M , Hiraiwa T , Suzuki K , Ishiwata S , Maeda YT , Miyazaki M .

	Fig. 1: Cell-sized confinement induces cluster formation and periodic actomyosin waves. a Schematic illustration of the experimental setup. The extract-in-oil droplets were confined in a quasi-two-dimensional space between two polydimethyl siloxane (PDMS)-coated glass slides. The aspect ratio of the height to diameter was fixed at 0.3–0.6. b Magnified view of the droplet boundary. The droplet was surrounded by a single layer of natural phospholipids to mimic the cell boundary. Actin filaments are nucleated by the Arp2/3 complex. Myosin induces actin network contraction. c–e Time-lapse images of F-actin dynamics in the extract-in-oil droplet, showing c initial contraction of the F-actin network, followed by d periodic wave propagation, and e the kymograph (Supplementary Movie 1). The broken line shows the theoretical model of the actomyosin wave (Supplementary Note 1). Actin filaments were visualized by tetramethylrhodamine (TMR)-LifeAct. The droplet temperature was elevated from 0 to 20 °C at 0 s to initiate actin polymerization. Periodic actomyosin waves persisted for more than 90 min (Supplementary Movie 2). We performed >10 independent experiments and confirmed the repeatability. f Fluorescence and the bright-field images of organelles stained with Octadecyl Rhodamine B chloride (R18). Organelles were accumulated by the initial contraction of the actomyosin network, forming a single nucleus-like spherical body (called a “cluster”). g Velocity field of the actomyosin wave visualized by particle imaging velocimetry. The same sample as c and d was analyzed. h Relationship between the cluster diameter Dcluster and the droplet diameter Ddroplet (n = 176). The plot was fitted by Dcluster = 0.25Ddroplet (R2 = 0.86; R-squared value for linear regression). We performed two independent experiments. i, j Initial contraction velocity v and period of the wave T displayed along the droplet diameter Ddroplet (n = 35). The plots were fitted by v = 1.6 × 10−2Ddroplet − 1.3 (R2 = 0.69) and T = 2.5 × 10−2Ddroplet + 39 (R2 = 0.12), respectively. Error bars represent standard deviations from the mean velocities and mean periods averaged over three successive waves and wave periods, respectively. We performed two independent experiments. Scale bars, 100 μm.
	Fig. 2: Cluster position becomes off-centered as the droplet size decreases. a Typical examples of the droplet size dependence. Stable position of the cluster became off-centered in small droplets. Images were acquired 1 h after encapsulation. b Quantification of the cluster positioning symmetry. The polarity parameter DC-ratio is defined as the ratio of d to Rdroplet, where d is the distance from the droplet center to the cluster centroid, and Rdroplet is the radius of the droplet. DC-ratio is classified into two distinct regimes: larger than 0.5 is the “edge” and smaller than 0.2 is in the “center”. c–h Typical examples of the c–e inwardly and f–h outwardly directed motion of clusters after their formation (Supplementary Movies 10 and 11). The time point at which clusters started to move (typically ~1 min after the cluster formation) is defined as 0 min. The clusters either rested near their initial positions or moved away from the initial positions. Only the clusters that moved are shown. Different colors in e and h indicate different clusters. i–k Droplet size dependence of cluster position. i Raw data of DC-ratio showing two-state positioning. The stable position depends on the droplet diameter. We analyzed bright-field images acquired more than 1 h after encapsulation. We performed two independent experiments (n = 176 droplets). j Histogram of DC-ratio for each 50 μm bins summarized from i. DC-ratio >  0.5, 0.5 ≥ DC-ratio ≥ 0.2, and 0.2 >  DC-ratio are colored in red, gray, and green, respectively. k Edge-positioned probability calculated from j. Filled circles and the solid curve represent experimental values and model fitting of Eq. (2), respectively. Experimental data were fitted by Eq. (2) using L = 6.1 μm and τ = 0.46 s. Transition diameter Dc = 85 μm at which edge-probability becomes 0.5 was estimated from the fitting curve. Scale bars, 100 μm.
	Fig. 3: Modulations of the surface property alters the cluster positioning. The interaction between the droplet boundary and actomyosin networks in the bulk space was modulated by changing the surface property. a Schematic illustration showing alternations of the surface property of droplets. (Top) PEG30-DPHS, a polyethylene glycol (PEG)-based surfactant, will decrease physical interactions of bulk actomyosin networks with the droplet boundary. (Bottom) The VCA domain of WASP conjugated with a histidine-tag was anchored to the droplet boundary via Ni-NTA-conjugated lipids. VCA recruits Arp2/3 and activates its actin nucleation activity. Thus, it was expected that interactions of bulk actomyosin networks with the droplet boundary increased. b The edge-positioned probabilities were compared between different surface properties. Cyan circles show the control data (egg PC). (Top) Passivated droplet surface promoted the cluster centering, whereas (bottom) activated droplet surface promoted the edge positioning. Kolmogorov–Smirnov test was applied to the scatter plots of DC-ratio between Ddroplet = 50 μm – 150 µm (around the transition point in the control experiment). The distributions were significantly different from the control in both conditions (p < 0.001). We performed two independent experiments.
	Fig. 4: Actomyosin bridges pull the cluster toward the droplet boundary. a, b Oscillatory motion of the cluster (Supplementary Movie 12). a Snapshot images of the droplet showing the oscillatory motion. (Top) The direction of motion of the cluster is indicated by the red double-sided arrow. (Bottom) The bridges connecting the cluster and the boundary are indicated by yellow arrow heads. b Time course of the cluster motion shown in a. c Actomyosin bridges formed between the cluster and the droplet boundary were cut by a UV pulsed laser. After the laser ablation, the cluster began to move back to the droplet center. After 4 min has passed, the cluster began to move to the edge again. In the same time, actomyosin bridges reassembled (white arrow). The original position of the cluster is indicated by the red broken circle (Supplementary Movie 14). d The tug-of-war model. (Top) Actomyosin waves push the cluster to the droplet center in every wave period. A magnified view showing F-actin polymerization nucleated by the Arp2/3 complex. (Bottom) Between periodic actomyosin waves, if actomyosin bridges are formed, network contraction pulls the cluster to the edge. A magnified view showing the dynamics of bulk actomyosin between periodic actomyosin waves. If inter-connected actomyosin bridges are formed between the cluster and the droplet boundary, the cluster is contracted toward the edge. Scale bars, 50 μm.
	Fig. 6: Theoretical modeling of the two-state cluster positioning. a The proposed mechanism underlying droplet size-dependent two-state cluster positioning. The red dotted curve and blue dashed line represent the bridge maturation time τp and the wave period T, respectively. Actomyosin waves transport the cluster toward the droplet center every period of T, while stochastically formed actomyosin bridges (mean maturation time τp) transport the cluster towards the droplet boundary. Actomyosin bridges are stochastically formed in the time period T between actomyosin waves. If bridges are formed, the network contraction pulls the cluster toward the droplet boundary until the subsequent wave collides with the cluster. Since the characteristic time-scales of two antagonistic forces have different size dependences, the transition radius Rc is determined by one unique crossover point. The resultant stable position is determined by the matured network faster than the other. b The percolation model of actomyosin bridge formation. We describe the stochastic binding/unbinding dynamics as the percolation process, wherein crosslinking sites are occupied by crosslinkers with a probability of 1/2 in every turnover time step τ. If all sites (the total number N) are occupied by crosslinkers, an actomyosin bridge is formed between the cluster and the droplet boundary, and the cluster is transported toward the droplet boundary via actomyosin bridge contraction. The mean percolation time τp is defined as the number of time steps necessary to form the actomyosin bridge on average, and kon and koff indicate the binding and unbinding rates of crosslinkers, respectively. c An active gel model of periodic actomyosin wave formation. The wave period is determined by the sum of three sequential processes: (i) actin network formation by F-actin growth, (ii) stress generation by myosin binding, and (iii) the ring starts to contract toward droplet center.
	Fig. 7: Numerical simulations and the analytical solution of the tug-of-war model reproduce the two-state cluster positioning. Both numerical simulations and the analytical solution Eq. (2) of the tug-of-war model (Fig. 6) reproduce the results of perturbation experiments (Fig. 5). (Top) In numerical simulations, the time averaged DC-ratio (1/𝑇0)∫𝑇00[𝑑(𝑡)/𝑅]d𝑡(1/T0)∫0T0⁡[d(t)/R]dt (T0 = 1800 s) for each droplet was determined to statistically evaluate percolation dynamics, displayed as purple dots. (Bottom) Thereafter, the edge probability was determined from the DC-ratio, where a DC-ratio > 0.8 and <0.2 was classified as the “edge” and the “center”, respectively. a F-actin length L was changed with the fixed crosslinker concentration (C0 = 1 μM). Purple circles, blue triangles, and orange squares represent numerical simulations with L = 4 μm, L = 8 μm, and L = 12 μm, respectively. Solid curves are the corresponding analytical solutions. b Crosslinker concentration C0 was changed with the fixed F-actin length (L = 8 μm). Purple circles, blue triangles, and orange squares represent numerical simulations with C0 = 0.1 μM, C0 = 1 μM, and C0 = 10 μM, respectively. Solid curves are the corresponding analytical solutions. We defined different DC-ratio thresholds for the edge region from the experiments, because numerical simulations were performed for clusters without their finite volume, in which the centroid of the clusters is possible to reach the droplet boundary. For details, see Supplementary Note 4.

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