C. elegans colony formation as a phase separation phenomenon

Phase separation at the molecular scale affects many biological processes. The theoretical requirements for phase separation are fairly minimal, and analogous phenomena may occur at other scales in biology. Here we have examined colony formation in the nematode C. elegans as a possible example of phase separation by a population of organisms. Experimental data and mathematical modeling indicated that, similar to physical condensation processes like phase separation and micelle formation, the population density of worms determines colony formation in a thresholded fashion, with the threshold correctly predicted by phase separation theory. Furthermore, we found that a phenomenon akin to Ostwald ripening – a coarsening seen in many systems that undergo phase separation – occurs. Our results show that populations of organisms can undergo condensation phenomena and phase separation. of Even though this is a complex behavior exhibited at the level of a group of C. elegans colony formation can be explained by simple equations like 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.

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 de ned 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 rst 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/mm 2 (Fig. 2h, i), estimated by tting a straight line to the data points where there were no colonies (open circles, Fig. 2i), a at line to those data points where there were colonies ( lled 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-ofcolony worms. The critical density threshold predicted from these rate measurements was 1.78 ± 0.17 worms/mm 2 , close to the directly-measured threshold of 1.33 ± 0.25 worms/mm 2 (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 uorescently-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 signi cant 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 modi cation to the rate equation model (Eq. 5) -adding a colony size-dependent dissociation rate constant -is su cient 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.

Methods:
Strains and maintenance. C. elegans strains, N2 Bristol and isogenic GFP-labeled GA631, and E. coli OP50 were obtained from the Caenorhabditis Genetics Center, University of Minnesota. C. elegans were maintained and cultured routinely on nematode growth (NG) plates according to standard procedures 25 .
Liquid culture. To obtain su cient numbers of C. elegans, we cultured C. elegans in liquid before experiments. C. elegans from a 100 mm plate that had been just depleted of food were washed and transferred to 250 mL S Medium 26  To obtain dauer-stage worms, the liquid culture was maintained similarly as for adults except cultured for 10 days. Dauer stage was induced by the high population density in the culture. Worms were transferred to conical 50 mL tubes, and collected from the bottom of tubes after centrifugation at 300 g for 3 min. To remove worms in other developmental stages, worms were then resuspended and incubated in 1% SDS for 40-60 min at room temperature. To separate dauer worms from debris, material was then collected by centrifugation at 300 g for 3 min and followed by centrifugation at 16000 g for 1 min at 4 °C in a tube containing cooled M9 buffer on top and 30% sucrose in M9 buffer at the bottom (sucrose cushion oatation). Dauers were enriched at the interface between the two density layers after centrifugation. These worms were quickly collected with a wide-bore Pasteur pipette and washed twice with M9 buffer at room temperature. Culture density was estimated by counting 20 µL droplets of the puri ed cultures (or a larger volume if the count is smaller than 50 worms) under a stereoscope.
Colony formation assay on agarose pads. Agarose pads were generated by molding melted 2% agarose (A9539-500G, Millipore-Sigma) in S Medium between two clean glass plates spaced with 1 mm-thick spacers. We used a 7 mm diameter biopsy punch (Queens Surgical) to cut out discs of agarose pads. The pads were then transferred to a 60 mm petri dish (351007, Corning) with a pair of tweezers with carbon ber tips (159C-RT, Excelta). Worms of desired number were transferred to the top of the agarose pads. Extra liquid was removed using the tip of a piece of Kimwipe (06-666A, Kimberly-Clark) twisted between ngers. The dish was then covered and imaged under a DMi8 uorescence microscope (Leica).
Colony formation assay on dishes. To make an agarose plate, we dispensed 5 mL 2% agarose with S Medium in a 60 mm petri dish and allowed to solidify. Afterwards, we deposited worms of desired number on to the agarose plate with a pipette. We removed excess liquid and gently dispersed the worms with a soft PVA sponge (40400-8, BVI). The dish was then covered with a lid layered with 2% agarose to reduce condensation and placed on a atbed scanner (B11B198011, Perfection V600, Epson) for image capture. Scanned photos were taken at 1200 dpi and regular time intervals with a custom Python script. Image processing and motion tracking. Before segmentation and tracking, a 2D Gaussian kernel was applied to individual images to reduce local uctuations. The outlines of worms and colonies were generated by thresholding on the edge intensities created by convolution with a Roberts kernel. Object masks for worms and colonies were generated by inversing the binary map of a background ood on the thresholded edge map. Object mask openings of size less than 1,000 µm 2 were removed. We tracked objects by nding the nearest centroid Euclidean neighbor in the previous frame. If the nearest neighbor in the last or the next frame was far enough away that a speed greater than 250 µm/sec would have been required to account for the displacement, the object was considered to be an orphan. Depending on the event sequence and the distance to the edge of the eld of view, orphans were classi ed as entering or exiting a colony or entering or exiting the eld of view. Given the uniformity of size of dauer worms, objects were categorized into a worm or a colony by its area of mask opening. Objects containing two worms in two consecutive frames, or algorithmically greater than 1.75 times the area of median opening in the eld of view (1.75 times the typical size of a solitary worm) were classi ed as a colony. Manual inspection was performed to ensure successful implementation.
Declarations Figure 1 C. elegans form colonies at high density in the absence of bacteria. a N2 C. elegans formed colonies on an NG plate upon consumption of bacteria. Orange arrowheads: colonies; blue arrowheads: dispersed worms. b, c Washed adult N2 C. elegans seeded at a low concentration (0.01 worms/mm2) (b) did not form stable colonies on an agarose plate, but those seeded at high density (0.18 worms/mm2) (c) did. C. elegans colony formation is a condensation process. a Schematic of the rate equation model for colony formation. b Sample trajectory of a single worm. c, d The modeled steady-state densities of worms out of colonies (w), and in one or more colonies (w*), as a function of the total density of worms wtot, based onto Eqs. 7 and 8. The system has a single stable steady state until the concentration of worms reaches a critical value of wtot = k-1/k1 (TC, transcritical bifurcation point). Beyond the critical density, the system bifurcates, and has an unstable steady state (dashed) and a stable steady state (solid) with a constant density of the dispersed worms w. e, f Rate balance analysis of a case with wtot smaller than the critical density (wtot = 0.5k1/k-1), and one with wtot greater than the critical density (wtot = 1.5k1/k-1). The association (green) and dissociation (blue) rates intersect at the steady states. In (e), the single steady state is stable (SSS), as small perturbations make dissociation faster than association, making the system return toward the steady state. In (f), the steady state at w* = 0 is unstable (USS), as small perturbation would make association faster than dissociation, and drive the system away from that steady state. g Dauer stage worms were seeded at various densities on agarose pads and the system was allowed to equilibrate for 30 min. Colonies appeared on pads with higher seeding density (arrows) but not with lower. h Measured densities of dispersed worms as a function of seeding density. Dauer stage worms were seeded at various densities on agarose plates and the system was allowed to equilibrate for 30 min. Plates where colonies formed are indicated with lled circles, and plates with no colonies with open circles. Data are from three independent experiments. i Critical density measured directly as in (h) (mean ± S.E.M., n = 3) and compared to predicted critical density values obtained by measuring k-1 and k1 from time-lapse movies of dauer stage worms (not necessarily at steady state) (n = 7). The p-value was calculated using Student's t-test. shown as an orange disc, and a worm as a blue curl. The radial component of the velocity is shown as a blue arrow, and tangential shown as magenta arrow. e Measured radial and tangential velocities for worms at various distances from a colony. Most of the worms in the rst bin were in colonies. Ten colonies from two independent experiments were included in the analysis. Velocity components were averaged for each colony at each distance before statistical analysis. Error bars are standard errors of the mean.