Reconstitution of actomyosin networks in cell-sized liposomes dissects distinct mechanisms of membrane blebbing and symmetry breaking

Abstract

The actin cortex, a thin layer of actomyosin network beneath the plasma membrane, regulates various cell functions by generating active forces and inducing membrane deformations, including blebs. Although upstream signaling is involved in regulating cell shape, the extent to which downstream actomyosin molecules can control the shape remains elusive. Here, using a minimal reconstituted system combined with an agent-based computational model, we show that actin-membrane coupling strength determines the magnitude of membrane deformation, while its balance with actin network connectivity governs the bleb initiation mechanism, either by detachment of the cortex from the membrane or by rupture of the cortex. This balance also modulates single versus multiple bleb formation, thereby regulating symmetry breaking. Furthermore, our results suggest that not only the dense cortical network but also the sparse volume-spanning network actively contributes to the regulation of bleb number. These findings provide mechanistic insights into how cells tune actin network organization to control their shape and polarity.

Actin network organization dictates how and where membrane blebs form, revealed through reconstitution and simulations.

INTRODUCTION

Cells assemble diverse structures of actin networks to operate biological functions. Among these structures, one of the most ubiquitous structures is the actin cortex, a thin layer of an actomyosin network formed beneath the plasma membrane (1). The actin cortex is not only a passive scaffold that stabilizes the cell shape but also plays pivotal roles in regulating various biological functions, such as migration, division, and polarity establishment, by generating active contractile forces (1–4). These forces drive morphological transitions of cells, including the formation of spherical membrane protrusions called blebs. During apoptosis, cells form transient multiple blebs in random directions (5). By contrast, in many other processes, the number and position of blebs are tightly regulated. During cell migration mediated by the actin cortex, primordial germ cells create a single bleb with a size comparable to the cell body at the front end to drive migration (6–8). Similarly, during amoeboid migration under physical confinement, various epithelial and mesenchymal cells such as HeLa cells also form a single bleb (9). In contrast, Walker 293 carcinosarcoma cells (10, 11) and Dictyostelium cells (12, 13) create relatively small multiple blebs at the leading edge. During cell division, cells form multiple blebs near the poles, possibly to relieve intracellular pressure elevated by furrow ingression and ensure successful cytokinesis (14).

How can the same key molecular elements of the actin cortex, namely actin filaments (F-actins) and myosin motors, induce various morphological transitions with tunable properties? This remains an open question. Although the upstream biochemical signaling pathways mediated by Rho-family guanosine triphosphatases and phosphoinositides are involved in the regulations (3, 5, 15), downstream proteins in the actin cortex directly mediate the morphological transition. Thus, it is important to ascertain the extent to which physical aspects of the actin cortex can control the cell shape. One of the promising but challenging strategies is a bottom-up approach: reconstituting the process from a minimal set of proteins and cell-sized liposomes (16, 17). An early study demonstrated that liposomes containing actin and myosin exhibit different patterns of contraction depending on the actin-membrane coupling strength (18). Another study successfully reconstituted shape transformation of cell-sized liposomes, including blebs, providing direct evidence that force generation by actomyosin interacting with the membrane alone is sufficient to drive the morphological transitions (19). These studies imply that actin-membrane coupling strength, actin network connectivity, and actin network distribution influence liposome morphology. Nevertheless, a comprehensive mechanistic understanding remains elusive because these parameters were not independently controlled. Furthermore, these studies primarily focused on stable postcontraction states as the dynamics of actomyosin contraction and morphological transitions were difficult to observe because of limitations in experimental setups, thereby limiting a mechanistic understanding of these processes. A recent study successfully reconstituted the bleb expansion process upon laser ablation of the cortical actin network (20). Experiments suggest that higher myosin contractility promotes bleb formation, whereas stronger actin-membrane coupling suppresses bleb formation and that the balance between coupling strength and contractility determines the retraction speed of the actin cortex. However, how bleb formation is controlled by these mechanical parameters remains unresolved because bleb initiation was externally imposed by laser-induced local disruption of the actin network.

As an alternative approach, agent-based computational models have been used. To recapitulate bleb formation, cells were often simplified as structures consisting of a membrane, actin cortex, and cytoplasm with internal cytoskeleton (21–24). These models could not reproduce large bleb formation because it did not account for membrane fluidity, which permits movement of anchor points between the membrane and the cortex. Although some models have incorporated membrane fluidity (25, 26), the cytoskeletal networks were drastically simplified, limiting the capability and predictability.

Here, we investigate the physical mechanism of morphological transition driven by the actin cortex, combining a minimal reconstituted model with an agent-based computational model. Our reconstituted system, consisting of purified cytoskeletal proteins and cell-sized liposomes, enables independent control of (i) actin-membrane coupling strength, (ii) actin network connectivity, and (iii) actin network distribution inside liposomes. Unlike previous reconstituted systems, (iv) myosin contractility in our system gradually increases over time through phosphorylation by myosin kinase, enabling estimation of the critical active myosin density required for morphological transition. The agent-based computational model, consisting of an actomyosin network and a deformable membrane, enables molecular-scale spatiotemporal analysis, providing mechanical insights inaccessible to experiments. Using these advantages, we systematically investigate how each parameter affects the morphological transition process.

RESULTS

Reconstituting the actin cortex inside a cell-sized liposome

We developed a minimal model of the actin cortex by reconstituting a contractile actomyosin network from purified proteins (fig. S1A) inside a cell-sized liposome (Fig. 1A). Actin monomers were mixed on ice with myosin (unphosphorylated), α-actinin, zipper-interacting protein kinase (ZIPK), adenosine 5′-triphosphate (ATP), and ATP regeneration system. The mixture was immediately encapsulated into a cell-sized liposome by the inverted emulsion method (27–29), and the temperature was elevated to 25° ± 1°C to promote actin polymerization and activate myosin contractility. The membrane was composed of 85% phosphatidylcholine, 10% phosphatidylglycerol, and 5% Ni–nitrilotriacetic acid (Ni-NTA)–conjugated lipid. In cells, myosin contractility is regulated by phosphorylation of the myosin regulatory light chain (MRLC) by myosin kinases (30). Phosphorylated myosin dimers assemble into submicrometer-long filaments that generate contractile forces on F-actin (Fig. 1A, inset) (30). To recapitulate this regulation, we used ZIPK (31, 32), one of the major myosin kinases involved in cell motility (33), cytokinesis (34, 35), and apoptosis (36). We tuned the ZIPK concentration to let the cortex formation and contraction processes take place sequentially (Fig. 1B and movie S1). We confirmed by biochemical assays (37, 38) (Fig. 1, C and D) that 90% of 10 μM actin was polymerized ∼10 min after the encapsulation (Fig. 1C, green line), whereas it took ∼40 min to phosphorylate 90% of 1 μM myosin (Fig. 1C, magenta line). We also confirmed by fluorescence microscopy that myosin self-assembled into submicrometer-long filaments upon its phosphorylation (Fig. 1E and fig. S1B).

Fig. 1. Reconstitution of the active actin cortex inside a cell-sized liposome.

(A) Schematic illustration of the experimental system. (B) Time-lapse confocal images of a liposome at the equator showing actin cortex formation, followed by myosin localization on F-actin, cortex contraction, and bleb formation and expansion (movie S1). A black arrowhead and a white arrowhead indicate the bleb and the rupture point of the actin cortex, respectively. The liposome contains 10 μM actin, 2 μM His-α-actinin, 1 μM SMM (10% Alexa 488 labeled), ZIPK (1.4 × 10⁻⁴ U ml⁻¹), and 1 mM ATP. (C) Biochemical assays showing the kinetics of actin polymerization and myosin phosphorylation. Actin polymerization was monitored using pyrene-labeled actin. Myosin phosphorylation was quantified from gel electrophoresis shown in (D). Three independent experiments were performed. Error bars indicate SDs. (D) The image of urea/glycerol polyacrylamide gel electrophoresis. 1 μM myosin was incubated with ZIPK (1.4 × 10⁻⁴ U ml⁻¹) in the presence of 1 mM ATP for the indicated times at 25°C. Unphosphorylated (0P), monophosphorylated (1P), and diphosphorylated (2P) MRLCs were separated by gel electrophoresis. (E) Total internal reflection fluorescence (TIRF) images of myosin before (left) and after (right) 120 min incubation with ZIPK (1.4 × 10⁻⁴ U ml⁻¹) in the presence of 1 mM ATP and ATP regeneration system. Submicrometer-long minifilaments were formed by phosphorylation (fig. S1B). (F) Confocal images of liposomes at the equator containing only (left) His-tagged or (right) His-tag truncated α-actinin, showing that α-actinin was anchored to the membrane through the His-tag and Ni-NTA interaction. (G) Confocal images of a liposome at the equator containing only actin, showing that F-actin has no specific interaction with the membrane. (H) Schematic illustration of three physical parameters examined in the experiments. For all microscopic images, dashed lines indicate the liposome periphery. Scale bars, 10 μm.

F-actin in the cortex is interconnected by various cross-linking proteins, including α-actinin, fimbrin, plastin, and fascin. The filaments are also anchored to the plasma membrane by diverse actin-membrane linkers such as ezrin-radixin-moesin proteins and anillin (1). Among the cross-linking proteins, α-actinin is the most abundant actin cross-linker in the actin cortex, as revealed by the proteomic analysis (39, 40), and is involved in diverse morphological transition processes, including cell motility (41, 42), division (43), and mitotic cell rounding (44). We therefore used α-actinin to create a cross-linked network within a liposome. α-Actinin also binds to the plasma membrane through its pleckstrin-homology domain, which has affinity for phosphatidylinositol 4,5-bisphosphate (45, 46). To control this interaction, we instead used a defined linkage between a histidine-tag introduced to the recombinant α-actinin and Ni-NTA–conjugated phospholipids in the membrane (Fig. 1A, inset). We confirmed that, while His-tagged α-actinin (hereafter “His-α-actinin”) binds to the membrane (Fig. 1F, left), His-tag–truncated α-actinin (hereafter “α-actinin”) does not bind to the membrane (Fig. 1F, right). We also confirmed that F-actin shows no specific interactions with the membrane (Fig. 1G).

In subsequent experiments, we fixed the concentrations of actin (C_A) and myosin (C_M), and ZIPK at 10 μM, 1 μM, and 1.4 × 10⁻⁴ U ml⁻¹, respectively. Then, by varying the concentrations of His-α-actinin (C_H) and α-actinin (C_N), we independently controlled actin-membrane coupling strength, defined as R_C = C_H/C_A [Fig. 1H, (i)] and actin network connectivity, defined as R_X = C_X/C_A [Fig. 1H, (ii)], where C_X = C_H + C_N is the total cross-linker concentration. A cosedimentation assay showed that the apparent dissociation constant K_d between His-α-actinin and F-actin is 1.4 μM, with binding to 10 μM actin reaching saturation at ∼5 μM (fig. S1, C and D). Therefore, we set the maximum C_X at 5 μM (R_X = 0.5). Furthermore, the spatial distribution of F-actin was modulated by adding methylcellulose to the inner solution of the liposomes. This crowding agent induces the localization of F-actin beneath the membrane by the depletion effect (47, 48), thereby enabling the formation of both a two-dimensional (2D) cortical network and a 3D volume-spanning network [Fig. 1H, (iii)].

Taking all these advantages, we systematically investigated the effects of (i) actin-membrane coupling strength, (ii) actin network connectivity, and (iii) actin network dimensionality on the morphological transition process of cell-sized liposomes. The size of observed liposomes was 19.9 ± 7.56 μm (mean ± SD; n = 355) in diameter, comparable to the typical cell size.

His-α-actinin concentration determines the morphological transition modes of liposomes

We initially focused on a 2D cortical network created with methylcellulose (Fig. 2A). Only His-α-actinin was used for simplicity (i.e., R_X = R_C). First, we confirmed without myosin and ZIPK that His-α-actinin was colocalized with F-actin (fig. S2A), and the actin density in a 2D cortical network remained nearly constant across the entire range of C_H (Fig. 2, B and C, and fig. S2B). Then, we performed time-lapse microscopy in the presence of myosin and ZIPK to examine how varying C_H affects the morphological transition of liposomes (Fig. 2, D to G). In the absence of His-α-actinin (C_H = 0; R_X = R_C = 0), neither global contraction of cortical actin networks nor detectable deformation of liposomes was observed (Fig. 2E and movie S2). This result is consistent with previous reports as myosin filaments are unable to induce global contraction of actin networks without actin cross-linking proteins (ACPs) at physiological concentration of ATP (maintained at 1 mM in our experiments) (49–51). At C_H = 0.5 μM (R_X = R_C = 0.05), the cortical actin network showed contraction in 58% (14 of 24) of the liposomes, consistent with the lower limit α-actinin density (R_X = 0.05) required for the global contraction in millimeter-scale actomyosin gels (51). More than half of these liposomes (64%) (9 of 14) showed detectable deformation during contraction (Fig. 2F and movie S3). A further increase in C_H increased the fraction of liposomes showing detectable deformation from 38% (C_H = 0.5 μM; R_X = R_C = 0.05) to 100% (C_H = 5 μM; R_X = R_C = 0.5) (Fig. 2G and movie S4).

Fig. 2. His-α-actinin concentration switches the morphological transition modes.

(A) Schematic illustration of the experimental system. Methylcellulose was added to localize F-actin beneath the membrane. (B) Representative confocal images of liposomes at the equator showing F-actin distribution in the presence of various concentrations of His-α-actinin. (C) Quantitative analysis of F-actin distribution. The fluorescence intensity of F-actin beneath the membrane FI_mem divided by the fluorescence intensity of F-actin in the bulk FI_bulk was plotted for each liposome. n.s.: P ≥ 0.05 (Welch’s t test, two-sided). (D) Probability distribution of the morphological transition modes of liposomes. Liposomes were classified into four categories in terms of their shapes: no deformation, weak deformation, single bleb, and multiple blebs. Liposomes exhibiting negative membrane curvature at more than one location during the observation period were considered to have undergone bleb formation. These were further divided into single bleb or multiple blebs based on the number of blebs. Liposomes that showed detectable deformation without regions of negative curvature were classified as weak deformation, while those showing no detectable deformation during the period of observation (2 hours) were assigned to no deformation. (E to G) Time-lapse epifluorescence images of representative liposomes with various concentrations of His-α-actinin. (E) No deformation (movie S2), (F) weak deformation (movie S3), and (G) single bleb (movie S4). A black arrowhead and a white arrowhead indicate the bleb and the rupture point of the actin cortex, respectively. For all microscopic images, dashed lines indicate the liposome periphery. Scale bars, 10 μm.

To elucidate a relationship between C_H and the shape of liposomes, we classified the shape into four categories: “no deformation,” “weak deformation,” “single bleb,” and “multiple blebs” (Fig. 2D, top). Quantitative analysis revealed that increasing C_H shifted the dominant mode from no deformation to weak deformation to single bleb (Fig. 2D, bottom). At the saturated C_H (5 μM; R_X = R_C = 0.5), all liposomes showed deformation, and 81% of liposomes (22 of 27) formed single bleb (Fig. 2D, bottom). Formation of multiple blebs was a very rare event; it was observed in only one case (1 of 17) at C_H = 1 μM (R_X = R_C = 0.1) (fig. S3A and movie S5).

Actin-membrane coupling strength is a primary determinant controlling the morphological transition mode and the magnitude of membrane deformation

We demonstrated that the single parameter, C_H, switched the morphological transition mode of liposomes (Fig. 2). However, changing C_H modulates both actin-membrane coupling strength (R_C) and actin network connectivity (R_X). Therefore, it remains unknown how these two physical parameters contribute to the morphological transition mode and magnitude of membrane deformation. To dissect the role of each parameter, we mixed His-α-actinin and α-actinin at various concentrations to control these two parameters independently (Fig. 3A). Then, we quantified the morphology and magnitude of membrane deformation in various conditions. We confirmed that the density of F-actin beneath the membrane was similar regardless of C_H and C_N (Fig. 3, B and C, and fig. S2, B and C). Therefore, this experimental setup enabled us to independently control the actin-membrane coupling strength and the network connectivity without a noticeable change in F-actin distribution.

Fig. 3. Actin-membrane coupling strength is the key parameter controlling the morphological transition mode and the magnitude of membrane deformation.

(A) The method to independently control actin-membrane coupling strength, R_C, and network connectivity, R_X. (B) Representative confocal images of liposomes at the equator showing F-actin distribution in the presence of (left) α-actinin or (right) His-α-actinin. (C) Quantitative analysis of F-actin distribution. The fluorescence intensity of F-actin beneath the membrane FI_mem divided by the fluorescence intensity of F-actin in the bulk FI_bulk was plotted for each liposome. (D to F) Time-lapse epifluorescence images of representative liposomes under various conditions. (D) The actin cortex was detached from the membrane (white open arrowheads). No clear membrane deformation was observed during the contraction (movie S6). (E) Rupture of the actin cortex (white filled arrowhead), followed by the contraction and blebbing of the membrane (black filled arrowheads) (movie S7). (F) Detachment of the actin cortex from the membrane (open white arrowhead), followed by the contraction and blebbing of the membrane (black filled arrowheads) (movie S8). The cortex was not ruptured until the completion of contraction. (G to J) Effects of R_C and R_X on (G) the morphological transition mode, (H) the magnitude of membrane deformation, (I) the contraction start time, and (J) the critical concentration of myosin required for the contraction, estimated from the median contraction start time in (I) and the time course of myosin phosphorylation (Fig. 1C). (H) The magnitude of membrane deformation is defined as the maximum length of the major axis of the liposome during contraction, L_max, divided by the diameter of the liposome before contraction, D_init. For all microscopic images, dashed lines indicate the liposome periphery. Scale bars, 10 μm. ***P < 0.001, **P < 0.01, *P < 0.05, n.s.: P ≥ 0.05 (Welch’s t-test, two-sided).

Using this strategy, we first examined the effects of a variation in R_C while keeping R_X fixed [Fig. 3A, (i)]. In the absence of actin-membrane coupling (R_C = 0), most liposomes showed no detectable deformation despite noticeable contraction of the cortical actin network at both R_X = 0.2 and 0.5 (Fig. 3D, Fig. 3G, left, and movie S6). In the presence of actin-membrane coupling (R_C > 0), liposomes were deformed by network contraction (Figs. 3, E and F, and Fig. 3G, left; fig. S3B; and movies S7 to S9). With an increase in R_C, the dominant morphological transition mode shifted from no deformation to weak deformation and then to single bleb at both R_X = 0.2 and 0.5, which is consistent with the experiment using only His-α-actinin (Fig. 2D). At both R_X = 0.2 and 0.5, the magnitude of membrane deformation increased with higher R_C and reached a plateau at high R_C values (Fig. 3H, left). The onset time of network contraction was delayed with increasing R_C with both R_X = 0.2 and 0.5 (Fig. 3I, left), showing a trend similar to the magnitude of deformation (Fig. 3H, left). Using the time course of myosin phosphorylation (Fig. 1, C and D), we estimated the concentration of phosphorylated myosin at the onset time of contraction (Fig. 3J, left). The result showed that higher myosin activity was required to initiate membrane deformation with increased actin-membrane coupling strength. Reducing the ZIPK concentration significantly delayed the onset time of contraction (fig. S4A and movie S10), but the estimated concentration of phosphorylated myosin at the onset time and the magnitude of membrane deformation were comparable between the two concentrations (fig. S4, B to F). These findings support a mechanism in which bleb formation occurs when the network tension exceeds a critical level.

We next examined the effects of a variation in R_X while keeping R_C fixed [Fig. 3A, (ii)]. We found that, in contrast to a change in R_C, changing R_X did not show a clear shift of the morphological transition modes; approximately 40 to 60% of liposomes showed single blebs, 30 to 40% showed weak deformation, whereas the remaining 10 to 20% showed no noticeable deformation (Fig. 3G, right). The magnitude of membrane deformation was also unaffected by R_X (Fig. 3H, right). Notably, the onset time of network contraction was similar, regardless of R_X (Fig. 3I, right), implying that critical myosin activity required for the membrane deformation was not strongly affected by the network connectivity in this parameter range (Fig. 3J, right). Together, these results demonstrate that actin-membrane coupling strength, rather than network connectivity, is the primary determinant of both the morphological transition mode and the magnitude of membrane deformation.

Bleb formation is initiated by one of two distinct mechanisms, “detachment” or “rupture,” depending on the balance between actin-membrane coupling strength and network connectivity

Time-lapse observation revealed that bleb formation was initiated by two distinct mechanisms: detachment or rupture (Fig. 4A). As for the rupture, bleb formation was initiated by the spontaneous rupture of the actin cortex beneath the membrane (Fig. 3E, filled arrowhead, and movie S7). By contrast, as for the detachment, bleb formation was initiated by the local detachment of the actin cortex from the membrane (Fig. 3F, open arrowhead, and movie S8). Note that, in some cases showing detachment, the cortex was eventually ruptured during the bleb expansion (fig. S3B, filled arrowhead, and movie S9). We found that, at constant R_X, detachment was dominant at low R_C, whereas rupture became dominant with increasing R_C (Fig. 4B, left). On the contrary, at constant R_C, rupture was dominant at low R_X, whereas detachment became dominant with increasing R_X (Fig. 4B, right). These results suggest that the bleb initiation mechanism is determined by how well filaments are connected in the network and how strongly the network is coupled to the membrane. Note that, without actin-membrane coupling (R_C = 0), most liposomes exhibited detachment without detectable deformation (Fig. 3D; fig. S5, left; and movie S6). Even in the presence of actin-membrane coupling (R_C > 0), a subset of liposomes exhibited detachment or rupture with no or only weak deformation (fig. S5).

Fig. 4. Bleb formation is induced by one of two distinct mechanisms, which is determined by the balance between actin-membrane coupling strength and network connectivity.

(A) Two types of bleb initiation mechanisms observed in the experiments. Top: A bleb is induced by the detachment of the cortical actin network from the membrane (movies S8 and S9). Bottom: A bleb is induced by the rupture of the cortical actin network (movie S7). Dashed lines indicate the liposome periphery. Scale bars, 10 μm. (B) Probability distribution of the two types of bleb initiation mechanisms under various conditions. (C) Schematic illustration of the molecular mechanisms underlying the two bleb initiation mechanisms. (D and E) Phase diagrams of the bleb initiation mechanisms. ${\mathrm{\gamma }}_{\mathrm{c}}^{\mathrm{D}\ast }$ and ${\mathrm{\gamma }}_{\mathrm{c}}^{\mathrm{R}\ast }$ indicate the critical cortex tensions required for the detachment and rupture mechanisms, respectively. ${\mathrm{\gamma }}_{\mathrm{c}}^{\mathrm{D}\ast }$ is proportional to the actin-membrane coupling strength R_C, as shown in Eq. 8. By contrast, ${\mathrm{\gamma }}_{\mathrm{c}}^{\mathrm{R}\ast }$ is an increasing function of the network connectivity R_X, but it is independent of R_C. (D) Dependence on actin-membrane coupling strength, R_C. (E) Dependence on network connectivity, R_X. Magenta arrows indicate the pathways taken by the system upon myosin phosphorylation.

To gain physical insight into the bleb initiation mechanism, we developed a simple theoretical model. First, we introduce a theoretical description of the detachment mechanism (Fig. 4C, top). We assume that a hydrostatic pressure difference between the inside and outside of the liposome, $∆P$, is balanced by the elastic restoring force, $F$, generated by the coupling between the actin cortex and the membrane-bound His-α-actinin molecules until the time point just before bleb initiation

$$∆P+F=0$$ (1)

Using the Young-Laplace equation, $∆P$ can be written as

$$∆P=\frac{2({\mathrm{\gamma }}_{\mathrm{m}}+{\mathrm{\gamma }}_{\mathrm{c}})}{R}-\frac{2{\mathrm{\gamma }}_{\mathrm{m}0}}{R_0}$$ (2)

where ${\mathrm{\gamma }}_{\mathrm{m}}$, ${\mathrm{\gamma }}_{\mathrm{c}}$, and $R$ indicate the membrane tension, the actin cortex tension, and radius of the liposome at a certain time point t, respectively. ${\mathrm{\gamma }}_{\mathrm{m}0}$ and $R_0$ indicate the membrane tension and the radius of the liposome right after the liposome formation (t = 0), respectively (14, 20). As myosin phosphorylation proceeds and the internal pressure reaches a level close to the bleb initiation threshold, the contribution of ${\mathrm{\gamma }}_{\mathrm{m}}$ would become much smaller than that of ${\mathrm{\gamma }}_{\mathrm{c}}$ (${\mathrm{\gamma }}_{\mathrm{c}}\gg {\mathrm{\gamma }}_{\mathrm{m}}$), as later confirmed by in silico simulations. Thus, the internal pressure can be approximated as

$$∆P\simeq \frac{2{\mathrm{\gamma }}_{\mathrm{c}}}{R}$$ (3)

Next, we consider the binding and unbinding dynamics of actin-membrane coupling. Although there are two potential molecular pathways for decoupling—either unbinding of His-α-actinin from F-actin or unbinding of the His-tag from Ni-NTA-lipid—we assume that the former is the predominant pathway for simplicity (see text S2, section 7). In the absence of the hydrostatic pressure ($\mathrm{\Delta }P=0$), the actin-bound His-α-actinin density ${\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}$ is governed by the kinetic equation

$$\frac{\mathrm{d}{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}}{\mathrm{d}t}=k_{\text{on}}({\mathrm{\rho }}_{\mathrm{H}}-{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}})({\mathrm{\rho }}_{\mathrm{A}}-{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}-{\mathrm{\rho }}_{\mathrm{N}}^{\mathrm{b}})-k_{\text{off}}{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}$$ (4)

where ${\mathrm{\rho }}_{\mathrm{H}}$, ${\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}$, ${\mathrm{\rho }}_{\mathrm{N}}^{\mathrm{b}}$, and ${\mathrm{\rho }}_{\mathrm{A}}$ indicate the surface densities of His-α-actinin, His-α-actinin bound on F-actin, α-actinin bound on F-actin, and F-actin beneath the membrane, respectively. $k_{\text{on}}$ and $k_{\text{off}}$ are the rate constants for binding and unbinding between His-α-actinin and F-actin, respectively. Considering that $∆P$ tends to separate the membrane from the actin cortex, and assuming that each His-α-actinin-F-actin bond is equally stressed, the force per bond, $f$, becomes $f=∆P/{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}$. Using Bell’s kinetics model (52), the kinetic equation (Eq. 4) is replaced by

$$\frac{\mathrm{d}{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}}{\mathrm{d}t}=k_{\text{on}}({\mathrm{\rho }}_{\mathrm{H}}-{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}})({\mathrm{\rho }}_{\mathrm{A}}-{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}-{\mathrm{\rho }}_{\mathrm{N}}^{\mathrm{b}})-k_{\text{off}}^0{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}\text{exp}(\frac{r_0}{k_{\mathrm{B}}T} \frac{∆P}{{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}})$$ (5)

where $k_{\text{off}}^0$ is the zero-force unbinding rate constant, and $r_0$ is a characteristic distance to overcome the energetic barrier for bond separation (52–54). When a small constant $∆P$ is applied, the unbinding rate (second term on the right-hand side of Eq. 5) transiently increases, leading a slight decrease in ${\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}$, and then the system reaches a new equilibrium ($\mathrm{d}{\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}/\mathrm{d}t=0$). In contrast, when a large $∆P$ is applied, the unbinding rate (second term) becomes much larger than the binding rate (first term), causing ${\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}$ to rapidly approach zero. There will thus exist the critical pressure, $∆P^{\ast }$, that is just sufficient to detach all His-α-actinin molecules from F-actin within a local region (text S1). If it is assumed that F-actin near the membrane retains many free binding sites for α-actinin (${\mathrm{\rho }}_{\mathrm{A}}\gg {\mathrm{\rho }}_{\mathrm{H}}^{\mathrm{b}}+{\mathrm{\rho }}_{\mathrm{N}}^{\mathrm{b}}$), we obtain the critical pressure

$$∆P^{\ast }\simeq \frac{0.7 k_{\mathrm{B}}T{\mathrm{\rho }}_{\mathrm{H}}}{r_0}\text{ln}\left(\frac{{\mathrm{\rho }}_{\mathrm{A}}}{K_{\mathrm{m}}^{2\mathrm{D}}}\right)$$ (6)

where $K_{\mathrm{m}}^{2\mathrm{D}}=k_{\text{off}}^0/k_{\text{on}}$ is the dissociation constant of His-α-actinin from F-actin in a 2D network (text S1). Therefore, using Eq. 3, the critical cortical tension required for the detachment mechanism is estimated as

$${\mathrm{\gamma }}_{\mathrm{C}}^{\mathrm{D}\ast }\simeq \frac{0.35 k_{\mathrm{B}}T {\mathrm{\rho }}_{\mathrm{H}}R}{r_0}\text{ln}\left(\frac{{\mathrm{\rho }}_{\mathrm{A}}}{K_{\mathrm{m}}^{2\mathrm{D}}}\right)$$ (7)

Since $R_{\mathrm{C}}\simeq {\mathrm{\rho }}_{\mathrm{H}}/{\mathrm{\rho }}_{\mathrm{A}}$, this tension is proportional to the actin-membrane coupling strength R_C

$${\mathrm{\gamma }}_{\mathrm{C}}^{\mathrm{D}\ast }\propto R_{\mathrm{C}}$$ (8)

We next consider the rupture mechanism (Fig. 4C, bottom). It can be reasonably approximated that the critical cortical tension required for the actin network rupture ${\mathrm{\gamma }}_{\mathrm{C}}^{\mathrm{R}\ast }$ is an increasing function of R_X and independent of R_C in our parameter range (48).

On the basis of the analytical results, we generated phase diagrams predicting which bleb initiation mechanism is selected (Fig. 4, D and E). At constant R_X, the model predicts that the mechanism shifts from detachment to rupture with increasing R_C (Fig. 4D). At constant R_C, the model predicts that the mechanism shifts from rupture to detachment with increasing R_X (Fig. 4E). These predictions are entirely consistent with the experimental results (Fig. 4B). Furthermore, the model predicts that at constant R_X, higher myosin activity is required to initiate bleb formation at higher R_C. This is also consistent with the experimental results (Fig. 3I, left, and Fig. 3J, left). Collectively, this simple theoretical model explains that the mechanism triggering the bleb initiation is regulated by the balance between actin-membrane coupling strength and network connectivity.

3D volume-spanning actin network facilitates the formation of multiple blebs

In cells, F-actin not only forms a dense cortex beneath the plasma membrane but also forms a sparse volume-spanning network throughout the cytoplasm. Therefore, it is important to assess how this coexisting bulk network influences membrane deformation. To address this, we created a volume-spanning 3D network by excluding methylcellulose from the inner solution (Fig. 5A) and examined how the concentration of His-α-actinin (C_H) affects morphological transitions, as analyzed in the 2D network (Fig. 2). As in the 2D network case, only His-α-actinin was used for simplicity, (i.e., R_X = R_C). We first confirmed that an increase in C_H increased the cortical F-actin density, reaching a plateau at C_H = 2 μM (R_X = R_C = 0.2) (Fig. 5, B and C, and fig. S2D). However, the relative cortical F-actin density FI_mem/FI_bulk was lower than that in the cases with methylcellulose (Figs. 2C and 3C and fig. S2E), suggesting that F-actin formed a sparse network in the bulk region even at high C_H. We next performed time-lapse microscopy to examine how C_H affects the morphological transition of liposomes (Fig. 5, D to H). In the absence of His-α-actinin (C_H = 0 μM; R_X = R_C = 0), neither contraction of actin networks nor deformation of liposomes was observed (Fig. 5E and movie S11). At C_H = 0.5 μM (R_X = R_C = 0.05), the actin network showed contraction in 39% (7 of 18) of the liposomes, and some of these liposomes deformed during the contraction (Fig. 5F and movie S12). At C_H = 1 μM (R_X = R_C = 0.1), the actin network showed contraction in all the liposomes (20 of 20). A further increase in C_H induced larger deformation (Fig. 5, G and H, and movies S13 and S14).

Fig. 5. 3D volume-spanning actin network facilitates the formation of multiple blebs.

(A) Schematic illustration of the experimental system. Methylcellulose was excluded from the inner solution to create a 3D volume-spanning actin network. (B) Representative confocal images of liposomes at the equator showing F-actin distribution in the presence of various concentrations of His-α-actinin. (C) Quantitative analysis of F-actin distribution. The fluorescence intensity of F-actin beneath the membrane FI_mem divided by the fluorescence intensity of F-actin in the bulk FI_bulk was plotted for each liposome. n.s.: P ≥ 0.05 (Welch’s t test, two-sided). (D) Probability distribution of the morphological transition modes of liposomes. Liposomes were classified into four categories in terms of their shapes, as in the cases with a 2D cortical network (Figs. 2D and 3G). (E to H) Time-lapse epifluorescence images of representative liposomes under various concentrations of His-α-actinin. (E) No deformation (movie S11), (F) weak deformation (movie S12), (G) multiple blebs (movie S13), and (H) single bleb (movie S14). Black arrowheads and white arrowheads indicate the blebs and the rupture points of the actin cortex, respectively. For all microscopic images, dashed lines indicate the liposome periphery. Scale bars, 10 μm.

Similar to the 2D network (Fig. 2D), an increase in C_H shifted the dominant mode from no deformation to weak deformation to single bleb (Fig. 5D). At the saturated C_H (5 μM; R_X = R_C = 0.5), all liposomes (10 of 10) formed single blebs (Fig. 5H and movie S14). At intermediate C_H (0.5, 1, and 2 μM; R_X = R_C = 0.05, 0.1, and 0.2), some liposomes exhibited multiple blebs (Fig. 5G and movie S13), reminiscent of the morphological transition observed during apoptosis (5). This is a distinct feature from the 2D network conditions (Figs. 2D and 3G).

Furthermore, as demonstrated in Figs. 3 and 4, we systematically investigated the individual contributions of R_C and R_X in the 3D network by varying C_H and C_N independently (Fig. 6). We confirmed that the basic two features revealed under the 2D network conditions (Figs. 3 and 4) are preserved in the 3D network conditions: Increasing R_C shifted the dominant morphological transition mode from no deformation to weak deformation and then to single bleb (Fig. 6, A and C), and increasing R_X shifted the dominant bleb initiation mechanism from rupture to detachment (Fig. 6, B and D). These parametric explorations further corroborate the universality of the regulatory mechanisms identified in the minimal reconstituted system. Moreover, we revealed that the 3D network with intermediate R_C and R_X is the optimal condition for multiple bleb formation (Fig. 6, A and C).

Fig. 6. Phase diagram of morphological transition modes and bleb initiation mechanisms in 2D and 3D networks.

(A and B) Phase diagrams of (A) morphological transition modes and (B) bleb initiation mechanisms in the 2D network conditions (in the presence of methylcellulose), (left) plotted using His-α-actinin concentration (C_H) and α-actinin concentration (C_N) as the parameters, and (right) plotted using the network connectivity (R_X) and actin-membrane coupling strength (R_C) as the parameters. (C and D) The same types of phase diagrams for the 3D network conditions (in the absence of methylcellulose).

Agent-based model reveals the critical role of catastrophic F-actin severing for bleb initiation by the rupture mechanism

Although our theoretical model could predict which mechanism is likely to trigger bleb initiation depending on the two physical parameters, how bleb initiation occurs at a molecular scale remains elusive. To define the molecular mechanism, we developed an agent-based model with an actin network encapsulated by a deformable membrane (Fig. 7A and text S2, sections 1 to 8) (55). Beyond the limits of theoretical models and experiments, our agent-based model can access protein-level information to correlate it with bleb formation. In the agent-based model, the network is composed of F-actin, myosin minifilaments, and ACPs with the geometry analogous to α-actinin. A fraction of ACPs can reversibly bind to the membrane as His-α-actinin used in experiments. The actin concentration was 10 μM in all simulations to be consistent with experiments. Unless specified, the membrane diameter was 8 μm, smaller than the mean diameter used in experiments (∼20 μm) to reduce the computational cost. We confirmed that the results obtained using a system with a diameter of 16 μm were similar (figs. S6 and S7 and movie S15). As in the experiments, we created both a 2D cortical network (Fig. 7B, left) and a volume-spanning 3D network (Fig. 7B, right) and then compared the morphological transition process.

Fig. 7. Agent-based computational model for dissecting the molecular mechanism of bleb formation.

(A) The model comprises an actin network encapsulated by a deformable membrane. (B) Cross-sectional views of (left) 2D and (right) 3D networks assembled within the membrane. For the 2D network, the network was assembled near the membrane within a space between r = 0.9 × r_MB and r = r_MB, where r_MB is a radial distance from the initial membrane center. The mean length of F-actin in the 2D network and 3D network was ∼4.2 and ∼5.2 μm, respectively. (C) Time courses of actin polymerization and motor activity. These were estimated by measuring F-actin concentration and force generated by each myosin filament, respectively. Myosin motors were activated after network formation in an all-or-none fashion to reduce computational cost. This likely has minimal effects on network contraction and membrane deformation, as substantial time was required to develop network tension before bleb initiation, as shown in (G). (D) Example of single bleb formation observed under the reference condition (R_X = R_C = 0.08, motor density R_M = 0.01) (movie S16). The bleb started emerging from ∼150 s, followed by expansion. The inset at 190 s represents a hole on the network beneath the bleb in a different view (white arrow), showing that this bleb was initiated by a network rupture. (E) The diameter of the bleb measured over time. (F) Time-lapse images showing that catastrophic severing events of F-actins (white arrows) led to the initial formation and expansion of a rupture on the network (movie S17). (G) Global tension acting on the network (red) and the membrane (blue) over time. (H) Local tension acting on the network (red) and the membrane (blue) near the bleb over time. Scale bars, 2 μm.

First, we performed simulations using the 2D network (Fig. 7B, left). Initially, actin nucleation and polymerization took place beneath the membrane in the presence of ACPs and myosin filaments to let them self-organize into the 2D network (Fig. 7C, green line). After the cortex formation, myosin was activated to start generating forces (Fig. 7C, magenta line). In the reference case with R_X = R_C = 0.08, which is close to 1 μM His-α-actinin condition (R_X = R_C = 0.1) in the experiments (Fig. 2), a single bleb started forming at ∼150 s and then expanded substantially over time (Fig. 7, D and E; fig. S6, A to D; and movie S16) as observed in the experiments. This bleb was initiated by a network rupture triggered by successive F-actin severing events induced by large tensile forces (Fig. 7F, fig. S8A, and movie S17). Such rupture-initiated bleb formation was not observed when we did not incorporate F-actin severing in the model, indicating that severing is the essential process for the network rupture (text S2, section 5) (56).

The computational model enables us to measure the network tension and the membrane tension separately for better understanding how tension is spatially distributed before, during, and after bleb formation, which is not accessible in experiments. Global network tension increased and remained at a plateau (∼200 pN μm⁻¹) for ∼70 s, after which it began to decrease upon bleb formation and expansion (Fig. 7G, red line, and fig. S6, E and F). This plateau phase implies that it took considerable amount of time for large tensile forces to accumulate in a few filaments and reach the critical severing threshold (assumed to be 300 pN; text S2, section 4) (57). In contrast, global membrane tension was initially low and then increased during bleb expansion (Fig. 7G, blue line), as the expanding bleb membrane experienced expansile forces resulting from volume conservation. Local network/membrane tension measured near the bleb showed similar tendencies before and after bleb formation (Fig. 7H). The local network tension and local motor density were higher in the region showing bleb formation than those in the region showing only membrane deformation (fig. S9, A and B), implying that locally higher contractility induced by more motors enhanced network tension and thus led to the network rupture. This was verified by examining the impacts of motor density; higher motor density induced more frequent F-actin severing events and more blebs (fig. S10, A to C). The network rupture was also accompanied by a decrease in local membrane-bound ACP density (figs. S9, C to E, and S6G). Together, the agent-based computer simulation revealed that catastrophic F-actin severing events at a local region are required for initiating bleb formation via the network rupture mechanism, and this is triggered by higher local motor density, followed by a local decrease in membrane-bound ACPs.

Tension analysis in simulations confirms that bleb is initiated by the detachment mechanism when actin-membrane coupling is weak

Using this 2D network model, we next examined the effect of the actin-membrane coupling R_C by reducing it from the reference case, with the network connectivity R_X fixed at 0.08 (Fig. 8, A to D, and fig. S10D). Without network-membrane coupling (R_C = 0), the network contracted into a smaller cluster without noticeable membrane deformation (Fig. 8A, left; fig. S10D, left; and movie S18). At low network-membrane coupling (R_C = 0.008 or 0.016), a bleb was initiated by the detachment of the network from the membrane (Fig. 8A, center; fig. S10D, center; and movie S19). With R_C higher than 0.048, bleb formation was driven by the network rupture as in the reference case (Fig. 8A, right; fig. S10D, right; and movie S20). At the intermediate regime (R_C = 0.032), some blebs were induced by detachment, but the rest was induced by rupture (Fig. 8D). These results are consistent with the experiments (Fig. 4B, left). We found that the network tension increased nearly proportional to R_C up to the intermediate regime (R_C = 0.048) (Fig. 8C, red lines), whereas membrane tension remained constant regardless of R_C (Fig. 8C, blue lines). At higher R_C, robust coupling between the membrane and network enhanced the network stability. Thereby, a greater force was required to rupture the network and initiate bleb formation, consistent with the experiments (Figs. 3I, left, and 3J, left). At lower R_C, the network began detaching from the membrane even before tension increased to a sufficiently high level required for F-actin severing (Fig. 8C, red lines, R_C ≤ 0.016), resulting in bleb formation by the detachment mechanism, consistent with the model prediction (Fig. 4D, left).

Fig. 8. Simulations reveal that actin-membrane coupling strength, network connectivity, and network dimensionality govern morphological transition modes and bleb initiation mechanisms.

(A to D) Effects of actin-membrane coupling strength, R_C with the 2D network. R_C was changed with R_X fixed at the reference value (0.08). (E to H) Effects of membrane-bound ACP density (i.e., R_X = R_C) with the 2D network. (I to L) Effects of membrane-bindable ACP density with the 3D network. (A, E, and I) Representative snapshots of the network and membrane under different conditions (movies S18 to S24). Color indicates force magnitude on each F-actin. [(A) left] Without actin-membrane coupling (R_C = 0), the network contracted into a small cluster without noticeable membrane deformation. [(A) center] At R_C = 0.016, bleb formation was initiated by detachment of the network from the membrane, evidenced by the intact network remaining beneath the bleb (inset). [(A) right] At R_C = 0.048, bleb formation was initiated by network rupture, evidenced by a hole in the network beneath the bleb (inset). (B, F, and J) Number of blebs as a function of (B) R_C and (F and J) R_X (= R_C). Error bars indicate SDs. (C, G, and K) Global network/membrane tension before and after bleb formation as a function of (C) R_C and (G and K) R_X (= R_C). Darker colors show tension at later times. (D) Effect of R_C on the bleb initiation mechanism. (H and L) Effect of membrane-bindable ACP density (R_X = R_C) on morphological transition modes, with the (H) 2D or (L) 3D networks.

Comparison between 2D and 3D networks in simulations reveals an active role of the bulk actin network

Our experiments showed that the 3D network induces the formation of multiple blebs more frequently (Figs. 2D and 5D and 6), implying the potential contribution of the sparse bulk actin network to the formation of multiple blebs. To dissect the underlying molecular mechanism by accessing protein-level information, we ran simulations using both 2D and 3D networks (Fig. 7B) with various ACP densities and compared the outcomes (Fig. 8, E to L). As in the case of experiments (Figs. 2 and 5), all ACPs were allowed to bind to the membrane (R_X = R_C), so changes in R_X also changed the density of actin-membrane coupling points.

With the 2D network in the absence of ACPs (R_X = R_C = 0), neither the network contraction nor membrane deformation was observed (Fig. 8H). At low ACP density (R_X = R_C = 0.006), either multiple small blebs or only membrane deformation without any bleb were observed (Figs. 8E, left, and 8, F and H; fig. S10E, left; and movie S21). At intermediate ACP density (R_X = R_C = 0.06), only a single bleb appeared more likely than multiple blebs (Figs. 8, E, center, and 8, F and H; fig. S10E, center; and movie S22). At high ACP density (R_X = R_C = 0.4), blebs were negligibly small, which is attributed to high network connectivity (Figs. 8, E, right, and 8, F and H; fig. S10E, right; and movie S23); although small network ruptures followed by the emergence of tiny membrane bulges smaller than 0.8 μm in diameter were observed, they did not expand enough to initiate bleb formation because the ruptures did not increase in size because of a large number of cross-linking points between F-actins (fig. S8B). While the initial membrane tension was rather independent of ACP density (Fig. 8G, blue lines), network tension increased nearly proportionally with ACP density as more ACPs rendered the network more elastic by increasing both cross-linking points within the network and coupling points to the membrane (Fig. 8G, red lines).

Results obtained with the 3D network (Figs. 7B, right, and 8, I to L; and fig. S10F) were qualitatively similar to those obtained using the 2D network (Fig. 8, E to H, and fig. S10E). The minimal ACP density required for the bleb formation was comparable between 2D and 3D networks (R_X = R_C = 0.004). However, the maximum ACP density permitting multi-bleb formation was higher in the 3D network (Fig. 8, H and L). In addition, even at the highest ACP density (R_X = R_C = 0.4), single blebs were formed in the 3D network in all cases, meaning that weak membrane deformation observed in the 2D network with high ACP densities was not observed in the 3D network. Note that we did not increase R_X beyond 0.4 considering that R_X ∼0.5 is the saturated concentration where F-actin can be fully decorated with cross-linkers (fig. S1, C and D). Consequently, the bleb formation probability was higher in the 3D network at high R_X (=R_C). We compared the densities of F-actin, myosin motors, and membrane-bound ACPs between the 2D and 3D networks for our reference case (R_X = R_C = 0.08) and confirmed that all of these components were present at lower densities beneath the membrane in the 3D network (Fig. 9, A to C), consistent with the F-actin distribution observed in the experiments (Figs. 2C and 5C and fig. S2E).

Fig. 9. Comparison between 2D and 3D networks in simulations reveals an active role for the bulk actin network in regulating bleb number.

(A) Ratio of F-actin density near the membrane (ρ_memb) to F-actin density in the bulk (ρ_bulk) in the 2D and 3D networks under the reference condition (R_X = R_C = 0.08, R_M = 0.01). (B) Density of motors beneath the membrane in the 2D and 3D networks (R_X = R_C = 0.08, R_M = 0.01). (C) Densities of F-actin, ACP, and membrane-bound ACPs beneath the membrane in the 2D and 3D networks (R_X = R_C = 0.08, R_M = 0.01). (D and E) Orientation of myosin filaments and the magnitude of motor forces in the 2D network (movie S25) and the 3D network (R_X = R_C = 0.08, R_M = 0.01) (movie S26) (D) visualized at a single time point and (E) the quantitative analysis results. (F) Schematic representation of the force distribution in the 2D and 3D networks. Arrows indicate force directions, and the lengths represent magnitudes.

With the 3D network, multiple blebs formed over a wider range of ACP densities (Fig. 8, H and L, purple bars, and movie S24), reproducing the trend observed in the experiments (Figs. 2D and 5D). We found that the orientation angle of individual myosin filaments showed a marked difference between 2D and 3D networks, and not only the cortical myosin filaments but also myosin filaments in the bulk region generated substantial forces (Fig. 9, D and E, and movies S25 and S26). While myosin filaments beneath the membrane were oriented parallel to the membrane, which may contribute to the cortical tension development both in 2D and 3D networks (Fig. 9E, blue plots), myosin filaments in the bulk region of the 3D network showed nearly random orientations (Fig. 9E, red plots), which can generate inward pulling force to the membrane as an ensemble (Fig. 9F, left). At the molecular scale, sparse bundled actin networks in the bulk region generate contractile forces that act as antagonistic forces opposing membrane expansion (Fig. 9F, insets). This will trigger the formation of new blebs at different positions to release the elevated pressure in liposomes, resulting in multiple bleb formation more likely (Fig. 9F, right). Overall, although the cortical network is expected to play a central role in bleb formation, our results indicate that the bulk actin network may also play an important role in regulating the number of blebs in cells.

Simulations reveal that a local perturbation on actin-membrane coupling strength or network connectivity controls bleb position

Previous studies in living cells (58) and reconstituted systems (20) have demonstrated that local laser ablation of the actin cortex can trigger bleb formation. However, the key molecular parameters within the actin cortex that govern bleb initiation remain unclear because laser ablation disrupts the actin network nonspecifically. In this study, we have identified both in vitro and in silico that the actin-membrane coupling strength, R_C, and the network connectivity, R_X, are the key parameters regulating bleb formation and the initiation mechanism. Using the agent-based model in which each parameter can be modulated, we tested whether external perturbation, either on R_C or R_X, can induce bleb formation in a specified position. Specifically, we locally reduced R_C (Fig. 10, A and B) or R_X (Fig. 10, C and D) to different extents under the reference condition where bleb formation via rupture was observed. When the perturbation level was low, a bleb was still formed at random positions. When the perturbation level was sufficiently high, the bleb was consistently formed from the perturbed region. We could switch the mechanism from rupture to detachment by inducing decoupling of a large portion of the cortex from the membrane (Fig. 10A). These simulations clarified that the location and mechanism of bleb formation can be controlled by locally manipulating one of the two parameters.

Fig. 10. Local perturbation on actin-membrane coupling strength or network connectivity controls bleb position.

(A and B) Perturbation on actin-membrane coupling strength, R_C. (C and D) Perturbation on network connectivity, R_X. We tested whether a local perturbation could initiate bleb formation from a specified location by selectively preventing ACPs from binding to the membrane (R_C = 0) or binding to F-actin (R_X = 0) within a region defined by a fraction of the membrane surface, f_p [white dashed lines in (A) and (C); t = 1 s], where f_p indicates ratio of the perturbed membrane area to the total membrane area. (A) With f_p = 0.10 (top), a bleb formed in a different location. With f_p = 0.22 (bottom), a bleb emerged from the perturbed region, initiated by the detachment of the network from the membrane, as evidenced by the intact network remaining beneath the bleb (inset). (C) With f_p = 0.10 (top), a bleb formed in a different location. With f_p = 0.22 (bottom), a bleb emerged from the perturbed region, initiated by the rupture of the network, as evidenced by a hole in the network beneath the bleb (inset). (B and D) Dependence of f_p on the probability of bleb formation from the perturbed region. At f_p = 0.16, perturbation on R_C controlled the bleb position with 50% efficiency (B), whereas perturbation on R_X controlled the bleb position with 100% efficiency (D). This difference arises because ACPs contribute to both actin-membrane coupling and network connectivity. In the absence of ACPs (R_X = 0), the network becomes mechanically weaker, making the system more susceptible to bleb formation.

DISCUSSION

The actin cortex is a ubiquitous structure found in various cells. However, the interaction between actomyosin machinery and the upstream/downstream signaling pathways, which are different between cell types and species, makes it difficult to isolate the contribution of actomyosin machinery to the morphological transition process. In this study, we combined an in vitro and in silico systems (Figs. 1 and 7) to dissect how and to what extent actomyosin machinery contributes to the regulation of membrane deformation without any biochemical signaling cue.

We demonstrated that the single parameter, His-α-actinin concentration C_H, can switch the morphological transition mode of cell-sized liposomes (Figs. 2, 5, and 8, E to L). With high actin-membrane coupling strength and high network connectivity with the 3D network organization, the minimal model system in both experiments and simulations always formed a single bleb (Figs. 5D and 8L). This result indicates that the actin cytoskeleton can form cell-scale polarity solely by the mechanics of cytoskeletal proteins and the plasma membrane, even in the absence of spatial biochemical cues. At the initial stage of cell migration mediated by the actin cortex, the cell spontaneously establishes cell-scale polarity to define the front end and rear end by changing a uniform actin cortex to a polarized actomyosin structure (3, 5). Similar polarization is observed in the one-cell stage of Caenorhabditis elegans embryos, in which asymmetric actin network contraction segregates partitioning-defective (PAR) polarity proteins to define the anterior-posterior axis (59). Our findings suggest that the cell tunes the actin network connectivity and cortex-membrane attachment to ensure the robust formation of cell polarity.

There is an ongoing debate about the bleb initiation mechanism (2). Observations in various cells suggest that the bleb initiation process may involve two distinct mechanisms: the local rupture of the actin cortex (13, 60) or detachment of the membrane from the actin cortex (61). The mechanism seems to depend on the cell types and animal species, and thus a comprehensive understanding of the two distinct bleb initiation mechanisms is still missing. In particular, it remains unclear which specific parameters determine the switch between these two mechanisms. Here, we demonstrated both in vitro and in silico that the two mechanisms can be switched by varying the actin-membrane coupling strength R_C and network connectivity R_X (Figs. 4 and 8, A to D). The physical model (Fig. 4, C to E) in combination with simulations (Figs. 7 and 8, A to D) clarified that the bleb initiation mechanism depends on whether adhesion between the actin cortex and the membrane is sufficient to sustain myosin-induced forces until F-actin in the network undergoes catastrophic rupture severing by strong tensile forces. Note that the adhesion level was varied by changing the maximum number of coupling points between the cortex and the membrane, not by changing the stability of individual coupling points. It is expected how easily each coupling point can break would affect the mechanisms of bleb formation; as coupling points are more prone to break, the detachment of the membrane from the actin cortex would play a major role in bleb formation.

Control of the bleb initiation mechanism is important for regulating cell functions. Recent observations on cancer cells (62) and primordial germ cells (63) have revealed that the endoplasmic reticulum (ER) network in the cytoplasm flows into the expanding bleb. This ER network entry stimulates store-operated calcium entry (SOCE)–mediated calcium flux to the expanding bleb through the formation of a complex between stromal interaction molecules on the ER membrane and calcium ion-selective Orai1 channels on the plasma membrane (STIM/Orai complex). When the bleb is initiated by the detachment mechanism, the remaining cortex may serve as a physical barrier that blocks ER network entry into the expanding bleb. During bleb-based motility, the remaining cortex can prevent the nucleus and other organelles from being translocated to the leading edge. In this study, we showed that the rupture mechanism is preferred in the presence of strong actin-membrane coupling. Previously, we demonstrated using cell extract–encapsulated droplets that strong actin-membrane coupling is the key factor for transmitting the force generated by the actomyosin machinery to extracellular environments for motility (64). Therefore, from a physical viewpoint, it is reasonable for motile cells to select the rupture-initiated bleb formation for efficient force transmission. Thus, the two distinct initiation mechanisms likely have specific biological roles, and tight regulation of the bleb initiation process may be crucial for controlling cell functions.

Control of the bleb position is also important. For example, E-cadherin confines the bleb-forming region to a restricted area at the leading edge of migrating cells by exerting frictional forces to impede the backward flow of actomyosin structures (65). This localized bleb formation is crucial for maintaining the front-rear polarity and directional persistence during migration. In this study, using simulations, we showed local reduction of actin-membrane coupling strength or network connectivity can induce bleb formation at the perturbation site (Fig. 10). These observations imply that cells control bleb positions by locally manipulating one of the two physical parameters by phosphorylations (66, 67).

Furthermore, we successfully reconstituted the formation of multiple blebs from a minimal set of cytoskeletal elements. A combination of experiments and simulations revealed the active role of the bulk actin network in the formation of multiple blebs (Figs. 2, 5, 6, and 9, A to F). However, the parameter range for the multiple bleb formation was narrow. In in vitro experiments, multiple blebbing was not a predominant event even under optimal conditions, compared to the single bleb formation, which was robust and occurred with high probability (Figs. 2, 5, and 6). This suggests that additional factor(s) may be required to enable the robust formation of multiple blebs. In cells, the actin network in the bulk cytoplasm is physically linked with the microtubule (68) and intermediate filament (69) networks. Future studies will be needed to examine how these composite network organizations contribute to the regulation of bleb number. In addition, the cytoplasm in living cells is considerably more viscoelastic and heterogeneous than in our reconstituted system (70, 71). We confirmed that the encapsulated buffers were one order of magnitude smaller than the viscosity of cytoplasm (Materials and Methods) (72, 73). Therefore, pressure release after bleb initiation is likely to propagate rapidly throughout the liposome (23), thereby suppressing the formation of additional blebs in our reconstituted model. The lack of rapid and continuous turnover of F-actin in our reconstituted model, as observed in living cells, may also limit the emergence of multiple blebbing events. Rapid cortical reassembly of actomyosin networks within expanding blebs promotes bleb retraction (74, 75), which would prevent sustained pressure release and thereby facilitate the initiation of new blebs at other locations. A key future challenge will be to investigate how the F-actin turnover rate, interactions between the actin and distinct cytoskeletal networks and organelles, and the viscoelastic properties and heterogeneities of the cytoplasm influence the number and position of blebs both in vitro and in vivo. In summary, a combination of in vitro reconstitution experiments and agent-based modeling has elucidated the extent to which the mechanical properties of the actomyosin network can control the morphological transitions of cell-sized liposomes, providing physical insights into how the cell tunes actin-membrane coupling strength, network connectivity, and actin distribution to control cell shape and biological functions.

MATERIALS AND METHODS

Buffer

All the experiments were performed using A50 buffer (50 mM HEPES-KOH pH 7.6, 50 mM KCl, 5 mM MgCl₂, 1 mM EGTA) (28, 29, 76, 77), unless stated separately.

Protein preparation

Actin was purified from rabbit white skeletal muscle as previously described (78), snap-frozen in liquid nitrogen, and stored at −80°C in G buffer [2 mM tris-HCl (pH 8.0), 0.05 mM CaCl₂, 2 mM NaN₃, 0.1 mM ATP, and 0.5 mM 2-mercaptoethanol]. Unphosphorylated smooth muscle myosin (SMM) was purified from chicken gizzards following our previous protocol (32, 77) and labeled with Alexa Fluor 488 N-hydroxysuccinimide ester (A20000, Thermo Fisher Scientific) as needed. The samples were snap-frozen in liquid nitrogen and stored at −80°C in A50 buffer containing 1 mM dithiothreitol (DTT) and 1 mM ATP. Recombinant human ZIPK (glutathione S-transferase–tagged) was prepared as previously described (32, 77), snap-frozen in liquid nitrogen, and stored at −80°C in A50 buffer containing 1 mM DTT. Recombinant human α-actinin I (6× His-tagged) was prepared according to our previous report (76). The His-tag was removed using PreScission protease (27084301, Cytiva) and labeled with Alexa Fluor 488–maleimide (A110254, Thermo Fisher Scientific) as needed. The samples were snap-frozen in liquid nitrogen and stored at −80°C in A150 buffer [50 mM Hepes-KOH (pH 7.6), 150 mM KCl, 5 mM MgCl₂, and 1 mM EGTA] supplemented with 1 mM 2-mercaptoethanol.

The purity of proteins was confirmed by SDS–polyacrylamide gel electrophoresis (SDS-PAGE) (fig. S1A). The concentrations were determined from ultraviolet absorbance, using absorption coefficients: $A_{290 \text{nm}}^{1\%}=6.3 {\text{cm}}^{-1}$ for G-actin, $A_{280 \text{nm}}^{1\%}=10.84 {\text{cm}}^{-1}$ for His-tagged α-actinin I, and $9.61 {\text{cm}}^{-1}$ for His-tag–truncated α-actinin I, and molecular weights of 42,000 Da for G-actin, 105,300 Da for His-tagged α-actinin I, and 100,000 Da for His-tag–truncated α-actinin I. The other protein concentrations were determined using Protein Assay Kit (500-0006, Bio-Rad), and molecular weights of 474,000 Da for SMM (hexamer: 2 myosin heavy chains + 2 MRLC + 2 myosin essential light chains), and 79,084 Da for ZIPK.

Preparation of lipid-oil mixture

The lipid-oil mixture was prepared as previously described (28, 29). Briefly, l-α-phosphatidylcholine from chicken egg yolk (840051P, Avanti Polar Lipids), 1,2-dioleoyl-sn-glycero-3-phosphatidylglycerol (840475P, Avanti Polar Lipids), and 1,2-dioleoyl-sn-glycero-3-[(N-(5-amino-1-carboxypentyl)iminodiacetic acid)succinyl] (nickel salt) [DGS-NTA(Ni); 790404C, Avanti Polar Lipids] were mixed at a molar ratio of 85:10:5 and then dissolved in mineral oil (M8410, Sigma-Aldrich) using a bath sonicator at 60°C and the power of 60 W for 90 min. The total lipid concentration in oil was 5 mM.

Protein encapsulation into liposomes

The inverted emulsion method was used to encapsulate purified cytoskeletal proteins into cell-sized liposomes (27–29). Before the experiments, 100 to 150 μl of the lipid-oil mixture was gently placed on 1 ml of the outer solution [A50 buffer containing 110 mM glucose, 2 mM Trolox, 10 mM DTT, and bovine serum albumin (BSA, 0.2 mg ml⁻¹)] in a 1.5-ml sample tube, and incubated on ice for >5 min to allow for the assembly of lipid monolayer at the oil/outer-solution interface. Next, 1.5 μl of ×20 E-mix [20 mM ATP, 200 mM phosphocreatine, creatine phosphokinase (2 mg ml⁻¹), 200 mM DTT] in A50 buffer was mixed with 1.5 μl of 10 μM Alexa Fluor 546–conjugated phalloidin (A22283, Thermo Fisher Scientific) in ×3 A0 + KLC50 buffer [150 mM Hepes-KOH (pH 7.6), 50 mM KCl, 15 mM MgCl₂, and 3 mM EGTA], 3 μl of 1 M sucrose and 20 mM Trolox in A50 buffer, 6 μl of 1% (w/v) methylcellulose (1500 cP; 135-02142, Wako) in A50 buffer, 3 μl of α-actinin I in A150 buffer, and 3 μl of SMM in A50 buffer. Then, 3 μl of ZIPK in A50 buffer was added to initiate myosin phosphorylation. Within a few seconds, this mixture (27 μl) was added to 3 μl of 100 μM G-actin in G buffer and pipetted five times to mix instantly. The actin polymerization was initiated by this mixing, and this timing was defined as t = 0 s. Subsequently, 2 μl of the mixture was put into 100 μl of the lipid-oil mixture preincubated on ice in a 1.5-ml sample tube, and this sample was vortexed for 10 to 15 s to form cell-sized droplets. The emulsion was gently placed on the lipid-oil mixture placed on the outer-solution, and then the tube was centrifuged at 12,000g for 60 s at 4°C to transform droplets into liposomes. In each experiment, the rest of the protein mixture (28 μl) was subjected to the measurement of the osmotic pressure by using a micro-osmometer (Fiske 210, Advanced Instruments). It was confirmed that the pressure difference between the encapsulated protein solution and the outer medium was smaller than ±17 mosmol (less than 5% difference between the inner and outer solutions; the mean difference was −0.23 mosmol, n = 222) to reasonably approximate that the liposomes were settled at isotonic condition.

An observation chamber was assembled by placing two double-sided tapes (0.3 mm in thickness) onto a coverslip (60 mm by 24 mm, thickness No.1, Matsunami) with a coverslip (18 mm by 18 mm, Matsunami) on top. Immediately after the liposome formation, the liposome solution was perfused into the chamber and sealed with Valap to prevent flow and then warmed to 25° ± 1°C to promote actin polymerization and myosin phosphorylation. The inner surface of the chamber was immediately coated with BSA, which was included in the outer solution, to prohibit nonspecific interaction between the liposomes and the coverslips. Note that we added 100 mM of sucrose inside the liposomes and used the nearly equimolar concentration of glucose for the outer solution to create a mass density difference to settle down the liposomes on a bottom coverslip for time-lapse imaging. We confirmed that this mass density difference was enough to sediment the liposomes within 10 min after making an observation chamber and was small enough to keep the liposomes spherical.

The liposomes contained 1 mM ATP, 10 mM phosphocreatine, creatine phosphokinase (0.1 mg ml⁻¹), 100 mM sucrose, 2 mM Trolox, 10 mM DTT, 0.2 or 0% (w/v) methylcellulose (1500 cP), 10 μM actin, 0.5 μM Alexa Fluor 546–conjugated phalloidin, 1 μM SMM, and various concentrations of α-actinin I (±His-tag), and 1.4 × 10⁻⁴ U ml⁻¹ ZIPK. One unit is defined as the amount of ZIPK that phosphorylates 1 μmol of MRLC in SMM per minute at 25°C. We have reported that 5% (mol/mol) of Alexa Fluor 546–conjugated phalloidin does not change the actin polymerization kinetics, monitored by pyrene assay (76).

Microscopy

Epifluorescence and bright-field images of the liposomes were observed by a custom-built inverted microscope equipped with a ×60 objective (PlanApo NA1.45 oil TIRFM or PlanApo NA1.40 oil, Olympus), an electron-multiplying CCD (charge-coupled device) camera (iXon3 DU-897E-CS0-#BV, Andor Technology). Confocal fluorescence images of the liposomes were observed by an inverted microscope (IX73, Olympus) equipped with a ×60 objective (UPlanSApo NA 1.30 sil, Olympus), a confocal scanner unit (CSU-X1, Yokogawa), an electron-multiplying CCD camera (iXon3 DU-897E-CS0-#BV, Andor Technology). For both microscopies, a custom-built heat block connected to a water bath (AB-1600, ATTO) was mounted on the objective lens to maintain the sample temperature at 25° ± 1°C.

Analysis of the distribution of F-actin in liposomes

To quantify the steady-state distribution of F-actin, myosin and ZIPK were excluded from the inner solution of liposomes to eliminate the membrane deformation and contraction of actomyosin networks. The liposomes were observed after >20 min from protein encapsulation, in which time scale it was reasonable to assume a steady state with a balance between polymerization and depolymerization reactions, considering net actin polymerization was almost terminated (Fig. 1C). Using confocal microscopy, we measured F-actin density beneath the membrane and compared the value with F-actin density in the bulk region of the liposome (Figs. 2, B and C, 3, B and C, and 5, B and C; and figs. S2, B to E).

Observation of myosin filaments and the length analysis

One micromolar of unphosphorylated SMM labeled with Alexa Fluor 488 (10% labeling ratio) was incubated with 1.4 × 10⁻⁴ U ml⁻¹ ZIPK in A50 buffer containing 100 mM sucrose, 2 mM Trolox, BSA (0.05 mg ml⁻¹), 1 mM ATP, 10 mM phosphocreatine, creatine phosphokinase (0.1 mg ml⁻¹), and 10 mM DTT at 25°C (ThermoMixer C, Eppendorf) for 120 min in a 1.5 ml sample tube to induce minifilament formation. BSA was included in the reaction mixture to prevent nonspecific adsorption of ZIPK to the tube surface. After the incubation, the sample was immediately diluted 100-fold with the same buffer without ZIPK and BSA. A 4-μl aliquot of the diluted sample was placed onto a coverslip (60 mm by 24 mm, thickness No.1, Matsunami), covered with a smaller coverslip (18 mm by 18 mm, Matsunami), and sealed with Valap. The sample was observed using a custom-built total internal reflection fluorescence microscope equipped with a ×60 objective (UPlanApo NA 1.50 oil, Olympus) and electron-multiplying CCD camera (iXon3 DU-897E-CS0-#BV, Andor Technology) (Fig. 1E). Filament length was measured using fluorescence images (fig. S1B). After background subtraction using a rolling-ball algorithm, binary images were generated, and the length of the major axis of each particle was measured. Overlapping filaments were manually excluded from the analysis. In addition, filaments shorter than two pixels (0.34 μm) were excluded from the analysis to avoid contamination of the data by small protein aggregates and dust, which cannot be reliably distinguished because of the resolution limit. This may result in an overestimation of the mean filament length.

Analysis of the phosphorylation rate of myosin

The phosphorylation rate of SMM by ZIPK was measured using urea/glycerol PAGE (32, 38). One micromolar of unphosphorylated SMM was incubated with 1.4 × 10⁻⁴ or 0.53 × 10⁻⁴ U ml⁻¹ ZIPK in A50 buffer containing 100 mM sucrose, 2 mM Trolox, BSA (0.05 mg ml⁻¹), 1 mM ATP, 10 mM phosphocreatine, creatine phosphokinase (0.1 mg ml⁻¹), and 10 mM DTT at 25°C (ThermoMixer C, Eppendorf) for 120 min (1.4 × 10⁻⁴ U ml⁻¹ ZIPK) or 240 min (0.53 × 10⁻⁴ U ml⁻¹ ZIPK) in a 1.5 ml sample tube to induce phosphorylation. BSA was included in the reaction mixture to prevent nonspecific adsorption of ZIPK to the tube surface. The initial sample volume was 400 μl. Every 10-, 20-, or 40-min incubation, 30 μl of the reaction mixture was taken from the sample tube and the reaction was terminated by the addition of urea (1.1 g ml⁻¹). Unphosphorylated (0P), monophosphorylated (1P), and diphosphorylated (2P) MRLCs were separated by urea/glycerol PAGE (10 mA, 230 min). The gel was stained using the SYPRO Ruby protein gel stain (S12000, Invitrogen), and the band intensities were analyzed using Image Lab (Bio-Rad) (Fig. 1, C and D, and fig. S4, B and C).

Analysis of the polymerization rate of actin

The polymerization rate of actin was measured using a pyrene assay (37, 76). Briefly, 8 μl of 100 μM G-actin (5% pyrene-labeled) in G buffer was put into a 384-well microplate (242764, Thermo Fisher Scientific), and then 72 μl of A50 buffer containing ATP, phosphocreatine, creatine phosphokinase, sucrose, Trolox, and DTT was added and pipetted five times. Then, the fluorescence intensity was measured using a microplate reader (Fluoroskan FL, Thermo Fisher Scientific) (Fig. 1C). The final concentrations were 10 μM actin, 1 mM ATP, 10 mM phosphocreatine, creatine phosphokinase (0.1 mg ml⁻¹), 100 mM sucrose, 2 mM Trolox, and 10 mM DTT.

Analysis of the binding affinity of His-α-actinin to F-actin

To measure the binding affinity of His-α-actinin to F-actin, a cosedimentation assay was performed (79, 80). Various concentrations of His-α-actinin in A50 buffer were mixed with 10 μM of G-actin on ice, incubated at 25°C for 60 min in a water bath (AB-1600, ATTO) to polymerize actin, and then centrifuged at 100,000g for 10 min at 25°C. Identical aliquots of the supernatant (S) and the pellet suspended in A50 buffer with the initial volume before centrifugation (P) were applied to SDS-PAGE. (5 to 20% gradient precast gel, ATTO), and the gel was stained by Coomassie Brilliant Blue (fig. S1C). The fractions of bound His-α-actinin and free His-α-actinin were calculated from the band intensities (79). The dissociation constant was determined from the model fitting (fig. S1D) (79, 80).

Measurement of the buffer viscosity

The buffer viscosities were measured with Ostwald’s viscometer (φ = 0.75 mm; CL2370-02-10, Climbing). The measured viscosities at 24°C were 0.92 mPa·s for A50 buffer and 2.27 mPa·s for A50 buffer containing 0.2% (w/v) methylcellulose, both of which were one order of magnitude smaller than the viscosity of cytoplasm (72, 73).

Agent-based model

We developed an agent-based model for a cell-like structure consisting of an actin network and a membrane. The actin network consists of F-actin, ACPs, and motors, all of which are simplified via cylindrical segments. The membrane is simplified into a triangulated mesh. The displacements of all points defining the segments and the triangulated mesh are updated at each time step via the Langevin equation with consideration of extensional, bending, and repulsive forces. F-actin is assembled via nucleation and polymerization events but do not undergo depolymerization. However, F-actin can be severed into two filaments if they experience a tensile force beyond a threshold level. ACPs connect pairs of F-actins to form cross-linking points, and motors bind to F-actin and then walk toward its barbed end. At the beginning of each simulation, an actin network is created via self-assembly and interactions of cytoskeletal elements with specified cross-linking density (R_X) and motor density (R_M) within a spherical membrane. In the 2D network simulations, a thin actin network is assembled only beneath a spherical membrane to create a cortical network (Fig. 7B, left). In the 3D network simulations, a network is assembled in an entire space within the membrane to create a 3D actin structure (Fig. 7B, right). During the network assembly, the spherical membrane does not undergo deformation, and the network and membrane are coupled via a fraction of ACPs defined by the extent of coupling. The fraction is varied between 0 and 1, and it is equal to R_C/R_X, where R_C is actin-membrane coupling strength. Only after the network assembly, motors start walking to generate mechanical forces, and the membrane also starts deforming by thermal fluctuation and motor-generated forces. The details of the agent-based model and parameters used in simulations are explained in text S2.

Analysis of the orientational angles of motors in simulations

The angle between motor backbones and the circumferential direction was calculated as follows. First, an acute angle between a vector defined by the two ends of the motor backbones, ${\mathbf{r}}_{\text{MB}}$, and a vector defined from the membrane center to the centroid of the motor backbones, ${\mathbf{r}}_{\text{OM}}$, was calculated. Then, the angle was subtracted from 90°

$${\mathrm{\theta }}_{\text{MB}}=90°-{\text{cos}}^{-1}\left(\frac{\mid {\mathbf{r}}_{\text{MB}}·{\mathbf{r}}_{\text{OM}}\mid }{\mid {\mathbf{r}}_{\text{MB}}\mid \mid {\mathbf{r}}_{\text{OM}}\mid }\right)$$

Data analysis

Image analysis was performed using Fiji (National Institutes of Health) and MATLAB (MathWorks).

Statistical analysis

Statistical analysis was performed using MATLAB (MathWorks).

Supplementary Materials

The PDF file includes:

Supplementary Text S1 and S2

Tables S1 and S2

Figs. S1 to S10

Legends for movies S1 to S26

References

Other Supplementary Material for this manuscript includes the following:

Movies S1 to S26