The worms aggregated to form blobs in a flat environment. An acrylic box (390 × 290 mm; As One, Japan) filled with water (226 ml, 2 mm in height) and scattered worms (60 g) were used.

Each worm initially moved randomly; however, once it contacted nearby worms, they became entangled with one another (Figure 1A, 0–90 min; Supplementary Video 1). Subsequently, many small blobs formed (90 min). These blobs gradually increased in size by merging with the surrounding worms and blobs (90–270 min). The number of blobs gradually decreased. Worms in a blob did not move actively; however, worms at the edge of the blob wiggled their tails outside the blob, with their heads inside the blob (Figure 1B; Supplementary Video 2).

We investigated whether the blobs exhibited a collective escape from aversive stimuli. We used a petri dish with 150 mm diameter filled with water (10 ml, 0.57 mm height). We placed a blob (1g) at the center and added Japanese mustard (0.3g; S & B Shokuhin Co. Ltd., Japan; hereafter referred to as mustard) at the edge of the blob as an aversive stimulus.

The surface of the blob fluctuated significantly when mustard was added (Figure 1C, 0–10 min; Supplementary Video 3). Some worms removed their heads from the blob and moved to the opposite side of the mustard (10 min). Subsequently, the blob escaped the stimulus (20 min). It escaped straight from the mustard while maintaining its hemispherical shape (20–30 min).

We made oval (Figure 2A; Major axis 50 mm, minor axis 20 mm, height 15.5 mm), and dumbbell-shaped cases (Figure 2B; Room: Inside radius 10 mm, height 15.5 mm; Distance between the centers of the two rooms 30 mm; Narrow aisle: Inside width 10 mm, height 15.5 mm), and compared the behaviors of blobs between the two cases. We added water 6.39 ml (7 mm height) to the oval and 5.16 ml (7 mm height) to the dumbbell-shaped cases. Subsequently, we added a blob of 1 g, a diameter of ~15 mm to one side of each case. Here, we note that the diameter of the room in the dumbbell-shaped case was sufficiently wide to hold the 1 g blob; however, the width of the narrow aisle was so narrow that the blob could not pass through while maintaining its hemispherical shape. In the oval-shaped case, the room diameter is the same as that in the dumbbell-shaped case.

Thus, the blob had to deform so that it passed through the narrow aisle of the dumbbell-shaped case, whereas deformation was unnecessary in the oval case. An acrylic lid with ventilation holes was placed on each case to prevent evaporation. The subsequent progress of 10 trials was observed for 66 h using GoPro9 (time-lapse mode, 5 s, 4 K) for each condition. Blob behavior was evaluated quantitatively by counting the number of one-way trips.

Snapshots of a representative trial for each condition (Figure 3) are explained below. For the oval-shaped case (Figure 4A; Supplementary Video 4), the blob started to move after a time, even though we did not provide any stimuli (0–0.4 h). The blob moved straight to the other side (0.4–0.9 h). Subsequently, it returns to its initially placed straight side (4.0–9.1 h). It is unclear why blobs or worms did not remain in a certain location despite the absence of mustard. We speculated that these worms released a substance that functions as a repellent. The trajectory of the blob's center in an oval case trial (Figure 4A) showed an irregular oscillatory pattern. The blob frequently moved back and forth from 0 to 21 h; however, after 21 h, the motion became inactive.

In the dumbbell-shaped case (Figure 3B; Supplementary Video 5), some worms got out of the blob (0–9.6 h). The blob could not pass through the narrow aisle for a certain amount of time (9.6–19.5 h). Subsequently, the entire blob was deformed to fit the shape of the case and moved to another room (19.5–21.2 h). After the blob reached the room, it returned to the initial room using a similar process (23.1–26.0 h). These results suggest that a hemispherical blob can deform and pass through an aisle; however, passing through the aisle by the blob took a long time. The trajectory of the blob's center in the dumbbell case trial (Figure 5B) showed that the blob had stayed in the initial side of the case from 0 to 40 h and then traveled only one-way (41–44 h).

The histogram of the number of one-way trips of the blobs shows that the maximum number in the oval-shaped case experiment was 15 trips, and the mode was six trips (Figure 5A; Supplementary Video 6). In contrast, the maximum number in the dumbbell-shaped case is three trips, and mode is on one trip (Figure 5B; Supplementary Video 7, bottom three rows). Statistical analysis showed that the number of one-way trips of the blobs in the dumbbell-shaped case was significantly lower than that in the oval case (p < 0.01, Mann–Whitney U-test).

To examine whether worm blobs could adapt to a confined environment, we created a dumbbell-shaped case in which three pegs (1.5 × 1.5 × 10.0 mm) were anchored to each room (Figure 2C). The anchored pegs were so small that the worms could interact with them. We observed how the behaviors differed between with (Figure 2C) and without (Figure 2B) pegs.

Snapshots of representative dumbbell-shaped cases with anchored pegs (Figure 3C; Supplementary Video 8) show that several worms got out of the blob and expanded their bodies toward the opposite room in a similar manner as the condition of the dumbbell-shaped case without pegs (0–5 h). When several worms reached and entwined the pegs at their destination, other worms got out from the blob and joined a group of worms that had already reached the peg (5–10 h). This resulted in the formation of elongated worm bundles. Hereafter, we refer to this bundle as a “pseudopod.” The remaining blob was probably pulled by the pseudopod and deformed considerably, and the entire group was moved to another room (10.1–10.2 h). The blob was returned to the initial room using the same process (10.6–12.5 h). These results suggest that the blob can perform effective locomotion in a confined environment, such as a peg that is useful for locomotion.

Photographs (Figure 3D) and videos (Supplementary Video 9; taken with S9D, Leica, Germany; Eye lens: 1.0x, objective lens: 0.6x) with magnified view were taken to investigate the movement of the worm during the pseudopod formation in another trial wherein the dumbbell-shaped case with pegs (water: 2 ml, 2.75mm height) was used. Several worms first expand their bodies from the blob to a narrow path (0 s). During this process, the worms were aligned along the wall and occasionally entangled with each other, forming an immature pseudopod (0–60 s). Subsequently, other worms joined the pseudopod for growth. We observed that the elongated worms in the pseudopod were almost completely aligned, whereas those in the blob were oriented in random directions and entangled with one another. Over time, other worms were removed from the blob and joined the pseudopod (60–240 s). The trajectory of another representative trial of the dumbbell-shaped case with anchored pegs exhibited irregularities in the oscillatory period (Figure 4C). The blob first stayed on the initial side of the case from 0 to 17 h. Subsequently, one-way trips (17–21.5 h) were performed. Subsequently, it remained on one side for a while (21.5–47 h). Finally, a one-way trip is shown.

The histogram of the number of one-way trips of the blob in the case with pegs shows that the maximum number was five trips, and the mode was on one trip (Figure 5C; Supplementary Video 7, upper three rows). Statistical analysis showed that the number of one-way trips of the blobs in the case with pegs was significantly larger than that in the case without pegs (p < 0.05, Mann–Whitney U-test).

Summarizing Sections 2.2.1 and 2.2.2, worms tend to form hemispherical blobs in open spaces; thus, it is not feasible to pass them through an aisle with a width shorter than the diameter of the blob (Figure 5B). However, environmental heterogeneity (the existence of pegs) helped the blob deform and feasibly pass through the narrow aisle (Figure 5C).

Worm locomotion was observed in a flat environment. The worm was placed in a 150 mm Petri dish filled with water (20 ml, 0.57 mm height). The worm propelled itself by propagating body waves of contraction and expansion from head to tail (Figure 6A; Supplementary Video 10) in a similar manner as earthworms (Gray and Lissmann, 1938) and blackworms (Ozkan-Aydin et al., 2021). In addition, we observed that the head turned frequently and randomly, similar to blackworms (Ozkan-Aydin et al., 2021).

Two worms were placed in a 150 mm Petri dish filled with water (20 ml, 0.57 mm height). First, each worm explored its surroundings actively. When two worms touched each other, they became entangled (Figure 6B; 0–100 s; Supplementary Video 11). After forming the tangle, the worms continued to actively explore (100~300 sec). The worms moved more actively than the blobs that comprised a large number of worms (Section 2.1.1; Figures 1A, B) in a similar manner as the blackworm (Ozkan-Aydin et al., 2021).

We observed the response of worms to the repellent mustard. A worm was placed at the center of 150 mm Petri dish, and a mustard tip was applied near the worm. When mustard was added, worm movement increased (Figure 6C; Supplementary Video 12), compared with those without the aversive stimulus (Section 2.3.1; Supplementary Video 10). It gradually moved away from the stimulus during wiggling. These behaviors are similar to those of a blackworms against aversive water temperatures (Ozkan-Aydin et al., 2021).

The length l_i of part of the major axis changes based on the number of contacting agents N_i, considering that real worms tend to assume a coiled posture when in contact with other worms (Figure 1B, Equation 1). l̄i denotes the target length of l_i and is expressed by the following equation:

where l_a and l_b are the positive constants. Equation (4) indicates that the length of the major axis l_i becomes shorter as the number of contacting agents N_i increases. When N_i exceeds (l_a−a)/l_b, the lengths of the agent's major and minor axes are equal.

The worms changed their movement depending on the situation (Figures 1B, 6B, C). They became more active as the number of worms in contact with them decreased (Figures 1B, 6B) and as the concentration of aversive stimuli increased (Figure 6C). In this model, we introduce this effect as the activity level G_i, which is defined as

where f(x) denotes the saturation function.

α and β are positive constants and A(r_i) represents the intensity of the aversive stimuli at r_i. We hypothesized that a certain repellent substrate would be released from the worms (Section 2.2.1). Thus, we assumed that the aversive stimuli A(r) were composed of two types: substances released from each agent, denoted by a_s(r), and mustard added to the behavioral experiment, denoted by a_m(r). Thus, A(r) is given by the following equation:

Based on gradual staining of the confined space water (Supplementary Videos 6, 7), the term a_s(r) is described by the following reaction-diffusion equation:

where k_A0 is the decay coefficient, k_A1 is the release rate of the repellent, and D_A is the diffusion coefficient.

The propulsion of an agent along its major axis F_{i, prop} is expressed as follows:

where η is a positive constant and t_i denotes a unit vector representing the direction of the major axis. Here, G_i and l_i are multiplied on the right-hand side because it is natural to consider that the propulsion force is larger when the agent is more active, and its posture is more elongated.

The random rotation term W_{i, rand} in Equation (3) is expressed by the following equation:

where d_R is a positive constant and ζ_i is a uniform random number between −1 and 1, which changes at the time interval T_i, where T_i is a uniform random number between t_min and t_max.

The term W_{i, chem} functions such that the major axis of agent i turns in the direction in which the aversive stimuli decrease (Figure 6C). This is expressed as follows:

where d_C is a positive constant, θ¯i denotes the absolute angle of the direction in which aversive stimuli decrease at r_i; that is, θ¯i=arg(−∇A|r=ri). Here, G_i is multiplied by the right side of Equation (11), assuming that agents turn strongly when active.

Real worms interact physically when in contact with one another. This interaction is very complex because the force generated by entanglement (Figures 1A, B) is not a simple frictional force. Here, the interaction force between agents i and j, F_{ij, phys}, is modeled by a combination of the viscous friction force and the force required to maintain the distance between the agents which denotes the entanglement:

where q₁ is a viscosity friction coefficient, ρ₁, ρ₂, λ₁, and λ₂ are positive constants, r_ij = r_j−r_i, and rij=rj−ri, and r^ij=rij/|rij|. The first term on the right side of Equation (12) denotes the viscous friction force. The second term works such that the distance between agents i and j does not become too large or too small.

Next, we explain the physical interaction term for rotational motion W_{ij, phys}. The elongated worms in the pseudopods aligned their bodies when in contact with one another (Figure 3D), whereas worms inside the blob were randomly oriented (Figure 3D). Based on these findings, W_{ij, phys} can be described as follows:

where d_A denotes a positive constant. The term sin{2(θ_j−θ_i)} represents the nematic interaction; that is, the relative angle between agents i and j approaches either 0 or π so that they are aligned. The terms (l_i−a) and (l_j−a) indicate the extent to which agents i and j are elongated. Therefore, the agents are aligned as they elongate.

In this model, the agents interact with the pegs and walls (Figure 9). The pegs are modeled as circles of radius r_c which do not move. The physical interaction between the agent and environment for translational motion F_{i, env} is expressed by the following equation:

The first term, F_{i, peg} represents the interaction force between agent i and the peg it contacts (F_{i, peg} = 0 when agent i does not contact any peg). This force consists of the viscous friction force and the force from the peg to the agent (Figure 9A); thus, it is described as

where q₂ is a viscous friction coefficient, ρ₃ is a positive constant, r_ip = r_p−r_i, and rip=rp−ri, and r^ip=rip/|rip| with r_p being the position of the center of peg p. The second term in Equation (15) represents the dragging force from agent i to the peg (Figures 3C, D). The term l_i−a in the second term indicates that a stronger force is generated as agent i elongates. This was based on the assumption that an elongating agent entwining a peg produces a strong drag force (Ozkan-Aydin et al., 2021).

The second term F_{i, wall} in Equation (14) represents the repulsion force from the wall with which agent i contacts (F_{i, wall} = 0 when agent i does not contact any wall). This is simply described as a force proportional to the penetration length to the wall (Figure 9B), that is,

where k_W is a positive constant and r_W.i denotes the nearest point of the wall surface from r_i.

The physical interactions with the environment for the rotational motion of agent i, denoted W_{i, env}, are described as

where W_{i, peg} represents the interaction with the peg which agent i contacts, expressed as

where d_P is a positive constant and θ_ip denotes the angle of the center of the contacting peg p viewed from r_i; that is, θ_ip = arg(r_p−r_i). This term operates such that agent i is aligned in the direction of the contact peg (Figure 9C). When agent i does not contact any of the pegs, W_{i, peg} = 0.

The second term of Equation (17), W_{i, wall}, represents the interaction with a wall which agent i contacts, which is expressed by the following equation:

where d_W is a positive constant and θ_u is the absolute angle of the wall that agent i contacts. This term operates such that the agent i aligns with the contact wall (Figure 9D), based on the finding that worms move along walls (Figure 3D). When agent i does not make contact with any walls, W_{i, wall} = 0.

In addition, when the head end of agent i touches the wall, l_i is forcibly changed so that it does not penetrate the wall (Figure 9E).

To imitate the blob-formation experiment in a box (Figure 1A), we set a square-shaped wall of length 0.183 m on one side. As the initial condition, 1, 000 agents were randomly deployed throughout the field, and their angles were randomly distributed. No external aversive stimulus was added in this experiment; that is, a_m(r) = 0.

The agents elongated their major axes and attracted the neighboring axes to form small blobs (Figure 10A, 0–0.056 h; Supplementary Video 13). The blobs grew by absorbing the surrounding agents and attracted one another to form larger blobs (0.056–0.39 h). Therefore, the number of blobs decreased. These results are consistent with biological findings (Figure 1A).

We simulated collective escape from the repellent mustard (Figure 1C). Simulations were performed using open boundaries. Two hundred and fifty agents, whose angles were randomly set, were initially located within a circle of radius 0.0125 m centered on the origin, such that they formed a blob at the beginning. The concentration gradient of real mustard was simulated by setting the stimulus term a_m(r) to

where D and E are the positive constants.

The agents at the edge of the blob begin to expand their bodies and move their heads out of the blob because they have a small number of contacting agents N_i (Figure 10B, 0–0.056 h; Supplementary Video 14). They turned their heads in the negative direction of the x axis, to which the aversive stimulus decays (0.056–0.56 h). After that, they dragged the other agents so that the blob moved straight toward the direction in which the stimulus decays (0.56–4.4 h). Therefore, the simulation qualitatively reproduced responses to an aversive stimulus.