Current studies of phase separation in biology have focused on the behaviors of microscopic molecular assemblies 1, including P granules 2, stress granules 3, and nucleoli 4. Theoretical considerations show that the requirements for phase separation are fairly minimal; it would not be surprising if analogous phenomena were to occur at other scales in biology.

*C. elegans* is a macroscopic animal that exhibits chemotaxis 5, learning 6, and complex social behaviors 7,8. It has been shown that some strains of *C. elegans* feed in clumps at the edges of existing bacterial lawns 9. This social feeding behavior has been attributed to the sensing of local oxygen levels 10,11. However, some strains, including the classic lab strain, N2 Bristol, lack the ability to clump in this fashion 9.

In the course of other studies, we observed what appears to be a particularly simple type of *C. elegans* colony formation: we found that N2 Bristol worms growing on plates to high density (a high number of worms per unit area), so that the bacterial food source was exhausted, often formed colonies (Fig. 1a). Recently, Demir *et al*. reported what appeared to be a related patterning behavior, and showed that it depended on bacteria 12. To test if residual bacteria or a pre-deposited pheromone pattern is required for the colony formation we had observed, we washed and transferred adult N2 worms to a fresh agarose plate. In the absence of added bacteria, a high density (0.18 worms/mm2) of adult worms formed colonies within minutes (Fig. 1c, Supplementary Movie 1), whereas a low density (0.01 worms/mm2) did not (Fig. 1b, Supplementary Movie 2). Even after 12 h of incubation, low densities of worms did not form colonies (Supplementary Fig. 1). We also tested *C. elegans* at other developmental stages, including dauer and asynchronized stages. Notably, to obtain dauer worms, we washed and treated with a low dose of detergent before replating, which should lyse and eliminate any trace amounts of bacteria. As was the case with adults, worms in these developmental stages formed colonies at high density, but not at low density (Supplementary Fig. 2). Thus, a system consisting of worms alone, with no bacteria, can undergo colony formation, and this colony formation depends upon the density of worms. Here we set out to see if this process can be viewed as a phase separation phenomenon.

On plates with colonies, we observed that worms dynamically moved into and out of the colonies (Supplementary Movie 3). In this way, the phenomenon resembled molecular liquid-liquid phase separation, as molecules also dynamically exchange between two phases. We asked if the formation of *C. elegans* colonies might be explained by phase separation theory. Typical theoretical approaches begin with expressions for the equilibrium energy or chemical potential of all of the species involved. However, phase separation phenomena can also be explored with rate equations, a complementary approach where rates, which can be directly determined by tracking experiments, rather than enthalpies and entropies, are the relevant species.

We defined two states or compartments for the worms, worms in a colony, *w**, and dispersed, solitary worms, *w*, and considered the processes that allow a worm to join or leave a colony (Fig. 2a). The worms’ crawling grossly resembled a random walk 13 (Fig. 2b), and a log-log plot of mean squared displacement vs. time yielded a slope of α = 1.11 ± 0.15 (Fig. 2c), again consistent with a random walk. We therefore assumed that the worms approached colonies from random directions, and that the probability of an encounter between a worm and a colony was proportional to the perimeter of the colony. Note that if the worms’ trajectories had been more ballistic (straight lines with few turns and α = 2), the probability should be proportional to the cross-sectional diameter of the colony. In either case, the encounter rate should be proportional to the square root of the area of colonies, and the square root of the number of worms in the colony *w**1/2. The encounter rate should also be proportional to the number of out-of-colony worms, *w*. With these assumptions, the forward rate of the process can be expressed as:

where *k*1 is the rate constant for the process. The departure of worms from a colony should also be proportional to the square root of the area of colonies, *w**1/2. Hence, the reverse rate is:

where

*k*− 1 is the reverse rate constant. The net rate of colony formation can thus be written as

We assume that the total number of worms in the system is a constant *w**tot*, which means that:

Therefore, Eq. 3 can be written as:

with a single time-dependent variable (*w**). At steady state, the time derivative must equal zero:

where W*

ss is the steady-state number of worms in the colony. There are two solutions for W*

ss:

which means that there is no colony and all of the worms are dispersed and solitary, and:

Note that neither W*ss nor *w**tot* can be smaller than zero. This means that there is a single steady state when 0 ≤ *w**tot* ≤ *k*−1/*k*1—the steady state given by Eq. 7—and two steady states when *w**tot* >k − 1/*k*1 (Fig. 2d, e)—the solutions to Eqs. 7 and 8.

To determine which of the steady states is stable, we performed rate-balance analysis (Fig. 2f, g). When 0 ≤ *w**tot* ≤ *k*−1/*k*1, the single steady state, where W*ss = 0 and *w**ss* = *w**tot*, is stable (Fig. 2f). On the other hand, when *w**tot* > *k*−1/*k*1, there are two steady states, which include a stable steady state, where W*ss = *w**tot* - *k*− 1/*k*1 and *w**ss* = *k*−1/*k*1, and an unstable steady state, where W*ss = 0 and *w**ss* = *w**tot* (Fig. 2g). Thus, when *w**tot* is below a threshold value of *w**tot* = *k*−1/*k*1, no colony will form. And, when *w**tot* is above this threshold, a colony will form and increase in size as *w**tot* increases, with the density of the out-of-colony worms remaining at its maximal possible density of *k*− 1/*k*1 (Fig. 2d - g). The transition from one to two steady states that occurs at *w**tot* = *k*−1/*k*1 is called a transcritical bifurcation 14. Transcritical bifurcations are seen in simple models of micelle formation, liquid-liquid phase separation, and precipitation, various condensation processes that occur on a molecular level 15−17.

Note that so far we are considering the interplay between dispersed worms and a single colony. However, the analysis can easily be extended to multiple colonies (Supplemental Text), and yields the same prediction of a critical worm concentration *w**tot* = *k*−1/*k*1 below which no colonies will form.

Thus the model predicts that (i) at steady state, there will be a density threshold, above which one or more colonies form, and below which no colony forms; (ii) when the seeding density is above the colony formation threshold, the density of out-of-colony worms should remain constant; and (iii) the critical colony concentration is equal to the ratio of the association and dissociation rate constants, *k*− 1/*k*1. Note that there are only three parameters in the model, the two rate constants and the total density of the worms. In a typical experiment, the density is known, and the rate constants can be directly measured by assessing the rate at which worms enter and leave colonies; therefore, it is possible to quantitatively test the model.

As a first experimental test, we asked whether there is a critical density of worms for colony formation. We placed different initial densities of dauer-stage N2 worms on an agarose plate, gently spread the worms, and took pictures of the plates after 30 min, when the colonies were already stable and no further colonies were forming. In agreement with the model, we found that the out-of-colony worm density increased approximately linearly with the seeding density when the seeding density was below a threshold, and became maximal and constant when the seeding density exceeded the threshold (Fig. 2h, i). The critical density was approximately 1.33 ± 0.25 worms/mm2 (Fig. 2h, i), estimated by fitting a straight line to the data points where there were no colonies (open circles, Fig. 2i), a flat line to those data points where there were colonies (filled circles, Fig. 2i), calculating the intersection between the two lines, and then averaging over three independent experiments. Thus, colony formation has the hallmarks of a simple condensation process with a transcritical bifurcation.

Next, to test whether the colony-forming threshold can be predicted from *k*− 1/*k*1, we took time-lapse videos of worms near existing colonies and measured the association and dissociation rates (Supplementary Movie 3). Note that, according to the model, *k*− 1/*k*1 should be a constant and can be measured in a system that has not yet reached the steady state. From several experiments, we calculated the number of association and dissociation events per unit time (as in Eqs. 1 and 2), and calculated

*k* − 1/*k*1 by dividing the ratio between the dissociation and the association rates by the density of out-of-colony worms. The critical density threshold predicted from these rate measurements was 1.78 ± 0.17 worms/mm2, close to the directly-measured threshold of 1.33 ± 0.25 worms/mm2 (Student’s *t*-test *p*-value = 0.303) (Fig. 2j). Thus, a simple condensation model both qualitatively and quantitatively accounts for the worms’ behavior.

To further explore the mechanism underpinning colony formation, we hypothesized that much like molecules in a condensed phase, worms in a colony moved more slowly due to their interactions with other worms. To test this hypothesis, we tracked movements of fluorescently-labeled worms sparsely mixed with label-free worms (Supplementary Movie 4). Worms outside of the colony moved six times faster than worms in a colony (Fig. 3a) (Wilcoxon *p*-value < 0.001). This spatially distinct behavioral difference could be a result of (i) two behaviorally differentiated populations of worms, or (ii) a single population of worms with two behavioral states. To distinguish between these possibilities, we tracked individual worms before and after they transited into or out of a colony. We found that individual worms promptly accelerated upon leaving a colony and decelerated upon joining a colony, supporting the second hypothesis (Fig. 3b, c).

Additionally, chemotaxis or other mechanisms of attraction could contribute to the formation of colonies by increasing the probability of encounter between worms and colonies. To assess the degree of contribution by attraction, we measured the radial and tangential components of velocities of worms at different distances from the center of a colony. If a worm is attracted to a colony, one would expect the radial component to be greater than the tangential component (Fig. 3d). However, we did not see such trend (Wilcoxon *p*-value = 0.73 without distance bins). We also dissected the radial and the tangential components of the velocity by binning the worms according to their distance to a colony (Fig. 3e). At all distances from the colony there was no apparent increase in the radial velocity relative to the tangential velocity, and no significant difference between the speeds of worms moving towards vs. away from the colony (Fig. 3e). This suggests that the worms join colonies through a simple random walk, and then stay in the colony because their movement slows down.

In many inhomogeneous physical-chemical systems, the small-sized structures shrink over time, and eventually disappear, while the large-sized structures grow. This coarsening process is termed Ostwald ripening 18,19. Ostwald ripening is driven by a greater stability of the larger structures. We have observed a similar phenomenon in *C. elegans* colonies. When there were two or more colonies on the same agarose pad, the small colonies often eventually disappeared. (Fig. 4b). We found that a small modification to the rate equation model (Eq. 5) – adding a colony size-dependent dissociation rate constant – is sufficient to generate Ostwald ripening in the worm model (Fig. 4a; Supplementary Text and Supplementary Fig. 3).

In summary, we have demonstrated that, at high density, *C. elegans* can self-organize and form colonies even in the absence of bacteria. Even though this is a complex behavior exhibited at the level of a group of living organisms, *C. elegans* colony formation can be explained by simple equations like those governing condensation and phase separation. The model predicts a density threshold for colony formation and a constant density of worms out of colonies when the threshold is reached. We found these predictions to be correct through direct experimental observation. With minor adjustments, the model accounts for the phenomenon of Ostwald ripening as well.

Together with other recent work, these observations indicate that biological self-organization and pattern formation, through phase separation, occur across many scales, from molecules 20–23, organelles 2,3, and possibly sub-cellular compartments 24, all the way to a population of organisms.