Exploring generic principles of compartmentalization in a developmental in vitro model

A hallmark of living matter is the ability to form and replicate well-defined cellular entities that eventually constitute a multicellular organism. In fact, embryogenesis is supposedly one of the most complex self-organization phenomena in nature as systems form dynamically adapting structures across multiple length and time scales, far from thermal equilibrium. Individual molecules self-assemble in a few milliseconds into higher-order complexes on the nanoscale, eventually leading to micron-sized organelles that structure individual cells on length scales of ∼ 10 μm within periods of minutes to hours. Each cell converts about 10⁷ ATP molecules per second to dynamically maintain its ordered state while joining a collective arrangement with neighbors into extended tissues. Ultimately, living animals emerge that are eight to nine orders of magnitude larger than their molecular constituents, and they may exist for a life span of up to 100 years. It appears therefore appropriate to call embryogenesis a true multiscale and non-equilibrium problem that mankind is just about to explore in its whole complexity. In fact, arriving at an in-depth understanding for any one of these two facets, including the development of proper theory and tools, is already a major and still very open challenge in the quantitative natural sciences.

Many developmental model organisms, from worms (e.g. Caenorhabditis elegans) and flies (e.g. Drosophila melanogaster) up to fish (e.g. Danio rerio) and mammals (e.g. Mus musculus), have been instrumental in revealing key steps of embryogenesis. These include, but are not restricted to, pivotal morphogen signaling pathways that direct development and spatial organization via self-organized biochemical gradients, hence bridging length and time scales from the molecular to the organismal level. Relying on the concept of Turing patterns and related mechanisms (Turing, 1952; Kondo and Miura, 2010), morphogen gradients may single out preferred directions and loci, e.g. during the establishment of body axes in C. elegans (Goehring et al., 2011; Gross et al., 2019) or mesoderm segmentation in mice (Lauschke et al., 2013; Sonnen et al., 2018).

Despite their tremendous success, developmental model organisms often come with inherent limitations. Studying the dynamic evolution of an embryo, for instance, may require the use of light microscopy techniques, requiring the specimen to be sufficiently transparent. Although this is mostly given for C. elegans or D. rerio, imaging the development of Drosophila at early stages can be challenging owing to the opaque yolk, requiring advanced imaging modalities (Krzic et al., 2012). In addition, confining envelopes, such as the chitin egg shell of C. elegans, often hamper a controlled perturbation of the embryonic development via the addition of chemicals and pharmaceuticals at specific stages. Moreover, mechanical boundary conditions within the embryo, e.g. an anisotropically varying tissue stiffness, may be key for certain processes (Koser et al., 2016; Franze, 2020), but are hard to access or to manipulate in the intact organism. Last, but not least, any model organism is typically much more complex than the particular problem that is being studied, and insights derived from one model may not be transferable to other organisms. Well-defined and accessible in vitro assays for studying generic developmental processes, preferably minimal systems with greatly reduced complexity, are therefore an attractive approach to complement the work with model organisms.

A high degree of accessibility facilitates, for instance, the exploration of mechanobiology aspects because mechanical cues and boundary conditions become more susceptible to external control. In fact, the mutual action of mechanical and biochemical processes has gained considerable attention in recent years, not least in the context of embryogenesis. Cell positioning during the early development of C. elegans is, for example, strongly guided by mechanical cues between cells and the confining egg shell (Fickentscher et al., 2013, 2016; Yamamoto and Kimura, 2017), and a fluid-to-solid jamming transition has been shown to facilitate vertebrate body axis elongation (Mongera et al., 2018). Going beyond these examples, the fundamental ability of organisms to achieve a highly ordered spatial organization of cellular entities is a stunning and very general phenomenon that invokes mechanobiology. Fly wings and retinae, like many other (embryonic) tissues, generate near-hexagonal cell patterns (Classen et al., 2005; Sugimura and Ishihara, 2013; Hayashi and Carthew, 2004), the syncytium of Drosophila and oocytes of the marine ascidian Phallusia mammillata feature a regularly ordered, not necessarily enveloped, compartmentalization (de Carvalho et al., 2022; Guignard et al., 2020; Khetan et al., 2021), and even monolayers of MDCK culture cells arrange in a near-periodic lattice on large length scales (Kaliman et al., 2021).

Inspired by these observations in different models, we strived for a minimal self-organizing system that is capable of showing a similar spatial organization on super-cellular length scales. Furthermore, we aimed to develop a system without a confining envelope to facilitate physical and chemical manipulations. Xenopus egg extracts, which have frequently been used to explore fundamental cell-biological and developmental processes, e.g. the formation and interactions of microtubule asters and mitotic spindles (Good et al., 2013; Mitchison et al., 2012, 2015), nicely satisfy this criterion, and we therefore employed a recently introduced in vitro assay that is based on extracts from unfertilized Xenopus laevis eggs (Cheng and Ferrell, 2019). In contrast to embryonic development, which is driven by replication and inheritance of a fertilized oocyte as priming template, these extracts show a spontaneous de novo formation of ordered arrays of cell-like compartments without a membrane envelope (abbreviated as ‘protocells’ hereafter). Protocells with typical radii of approximately 100 μm emerge spontaneously within about 30 min (see Fig. 1 for an example), even though the extract is arrested in interphase with a chemically abolished protein synthesis and despite a lack of chromatin that could serve as nucleation seed. This spontaneous and dynamic compartmentalization process is a genuine non-equilibrium phenomenon that is disrupted by ATP/GTP depletion, or by blocking the microtubule cytoskeleton dynamics or dynein motors; blocking actomyosin or kinesin motors involved in mitotic spindles had little to no effect (Cheng and Ferrell, 2019). Albeit requiring the activity of microtubules and molecular motors, protocells are not mere microtubule arrays as they accumulate remnant vital organelles, e.g. mitochondria (Cheng and Ferrell, 2019). Notably, supplementing the assay with sperm-derived nuclei and centrosomes, mitosis with defined nucleation seeds has also been reconstituted (Afanzar et al., 2020).

Given the similarity of protocell patterns with organized arrays of cells and cell-like entities in developmental organisms, e.g. syncytial energids (Mitchison and Field, 2019; de Carvalho et al., 2022), we have explored potentially generic aspects of space compartmentalization with this assay at a strongly reduced complexity and improved accessibility, in the absence of priming and pre-patterning template structures. In particular, we investigated which geometrical features protocell patterns have, how these evolve over time, how robust these features are when the cytoskeletal dynamics is perturbed, and how one can capture the pattern with a mesoscopic model that might allow one to conclude and extrapolate on common, generic principles of self-organized space compartmentalization in other systems, e.g. in developing organisms and tissues that involve a syncytial state.

Following previous protocols (Cheng and Ferrell, 2019), we were able to confirm the robust emergence of protocell patterns in slab-like droplets of Xenopus extracts (see Materials and Methods for details). Starting from homogenous droplets of freshly prepared extract, we observed the emergence of fairly regular arrays of protocell compartments within 30-50 min of sealing the sample chamber (Fig. S1) even without priming templates, chromatin assemblies, or the action of acto-myosin (Fig. 1A,B, Movie 1). In line with previous observations (Cheng and Ferrell, 2019), the pattern faded and disintegrated after about 3 h, most likely as a result of progressive limitation of available ATP/GTP molecules that fuel necessary active processes.

Notably, upon pattern emergence we did not observe a successive formation of individual protocells at different spatial locations. Rather, the entire pattern became visible at the same instant of time. Given that each droplet contained several hundred protocells, i.e. the droplet diameter was always much larger than individual protocells, this observation suggests a global coupling of the entire system, akin to a spinodal decomposition or Turing pattern formation. In both of these scenarios, a critical wavelength can be named that couples growing fluctuations across the entire system, eventually leading to a global onset of the formation of periodic patterns with a long-range order.

Boundary lines between protocells had a lower absorbance in brightfield images, and hence a lower density than the interior of protocells. It is likely that radially organized microtubules are responsible for the increased crowding inside protocells as they provide a means for a constant shuttling of mitochondria and other organelles to the center region via dynein motors (Cheng and Ferrell, 2019). In line with this notion, we observed radial arrays of microtubules within individual protocells, with little to no visible overlap of microtubules between neighboring compartments (Fig. S2A,B, Movie 3). Exploiting and highlighting the permanent radial influx, we added minute amounts of accessory tracer beads (diameter 1 µm) to freshly prepared extract droplets, which eventually accumulated in the center of protocells. The addition of these tracers did not appear to perturb protocell formation, but enhanced the contrast for subsequent image analysis.

Unlike the rather large tracer beads, inert fluorescently labeled macromolecules (10 kDa FITC-coupled dextran) were almost excluded from the interior of protocells and instead accumulated at the boundaries (Fig. 1C). This exclusion of inert macromolecules from densely crowded regions is similar to observations inside living culture cells (Donth and Weiss, 2019). Because of the good contrast between bright boundary zones and dark protocell centers, an automatic segmentation of brightfield images via a Voronoi tessellation was possible (see Materials and Methods and Figs S3 and S4 for details), facilitating quantification of the protocells’ geometry over time. A representative example for the tessellation is shown in Fig. 1D. Owing to a lack of a well-defined envelope, we defined the spatial extent of each protocell to correspond to its Voronoi cell for all evaluations.

Using the tessellation approach allowed us to go beyond a qualitative visual inspection. As a first step, we analyzed the local geometry of protocells at two different time points, i.e. right after the first emergence of the pattern and 1-2 h later. To this end, we extracted individual protocell areas A, perimeters L, and vertex numbers n_v from images of different experiments and times. Accounting for varying average protocell areas, we determined normalized areas A_n=A/〈A〉 and perimeters L_n=L/〈L〉 by dividing out the mean values of the respective image (see Fig. S5A,B for exemplary histograms p(A) and p(L) before normalization). As a Kolmogorov–Smirnov test (5% level) did not indicate significant differences of these normalized quantities between different experiments, we combined these for comparable time points into the same set and inspected their probability density functions (PDFs) p(A_n), p(L_n) and p(n_v). In particular, no significant differences were detected between any two experiments on untreated extracts samples or extracts to which a very low amount of taxol (concentration ≤ 0.1 μM) had been added; these data were hence combined for the analysis.

In line with the visual impression of a highly regular appearance of protocells, the statistics of vertex numbers, p(n_v), highlighted a predominant occurrence of hexagonal protocells with appreciable probabilities also for pentagons and heptagons (Fig. 2A); polygons with more vertices were rare. This observation suggests a (slightly) disordered hexagonal arrangement of protocells. Despite the lower complexity of the in vitro system, these data are in very good agreement with earlier reports on the spatial organization of epithelial monolayers (Kaliman et al., 2021) and chemically induced aster patterns in Phallusia oocytes (Khetan et al., 2021).

Moreover, the PDF of normalized protocell areas and perimeters, p(A_n) and p(L_n), assumed narrow shapes around a peak at unity (Fig. 2B,C), i.e. protocells in each extract showed very similar areas and geometrical shapes. These results are strikingly similar to comparable quantifications on monolayers of MDCK cells that were grown on substrates of different rigidities (Kaliman et al., 2021), suggesting common principles of space compartmentalization despite the very different nature of these samples. Moreover, using protocell area and perimeter to define the individual geometrical compactness, ⁠, revealed typical values between a circular geometry (C=1) and squares (C=4/π) (Fig. 2D). It is worth emphasizing at this point that all of these PDFs were independent of the time point at which they were acquired (compare blue histograms and symbols in Fig. 2), indicating a stable and well-organized pattern of convex compartments with uniform geometrical properties at all times.

Because protocell formation occurs at the same time as a focusing of microtubules into organized structures (Cheng and Ferrell, 2019), we reasoned that stabilizing these cytoskeletal filaments may affect the protocell pattern. Taxol has been described to stabilize microtubules without inducing major changes to cytoskeletal arrangements (Verde et al., 1991), so we supplemented fresh extracts with this anti-cancer drug at different concentrations (see Materials and Methods). As a result, we observed that an increasing taxol concentration (here tested up to 2 µM) led to patterns with decreasing protocell sizes (see Fig. 3A for example images). Confocal fluorescence imaging also revealed that microtubule arrays with a radial symmetry were still formed in the presence of taxol, albeit appearing slightly less tidy and, owing to the stabilizing action of taxol, being visible already at the earliest time point at which imaging was possible (Fig. S2C,D, Movie 4).

Quantification of the average protocell area 〈A₈₀〉, measured 70-90 min after starting the experiment, confirmed this visual impression: a successive reduction of 〈A₈₀〉 was observed for increasing taxol concentrations (Fig. 3B). Similar observations have been reported for microtubule asters in intact Phallusia oocytes (Khetan et al., 2021), suggesting that the minimalistic in vitro system captures generic aspects of a self-organized spatial organization.

Quantification of the statistics of vertex number, areas and perimeters [ p(n_v), p(A_n), and p(L_n)] in the presence of taxol did not reveal marked changes, irrespective of whether images were evaluated immediately after the emergence of the pattern or 1-2 h later. The mean average fractions 〈φ〉 of pentagons, hexagons and heptagons was virtually unaltered (Fig. 3C), and the PDFs p(A_n) and p(L_n) for high taxol concentrations assumed the same shapes as the data shown in Fig. 2 (see Fig. S5C,D). Thus, taxol treatment maintained all geometrical features of the pattern and only reduced the intrinsic length scale of protocells.

To gain insights into the dynamics of protocell patterns, we monitored the average protocell area 〈A〉 as a function of time for varying taxol concentrations. In all cases, we observed a roughly linear growth, 〈A〉≈γt, albeit with variations in the growth rate γ (see Fig. 4A for representative examples). Because the area of the extract droplet was conserved, an equivalent but alternative signature of this is a power-law decrease ∼ 1/t of the number of protocells (compare with Fig. S6A). The observed growth in area was mainly due to merging events of protocells (Movie 2, Fig. S6B), i.e. due to a coarsening of the pattern.

Our experimental results therefore reveal that protocell patterns feature the same statistical scale invariance as two-dimensional foams (Saint-Jalmes, 2006). Irrespective of any taxol treatment, the PDF of normalized areas, p(A_n), is narrow and assumes a time-independent shape with the average protocell area growing linearly in time while coarse-graining proceeds at a conserved total area.

Moreover, despite some fluctuations of the growth rates of individual assays at the same taxol concentration, the mean growth rate 〈γ 〉 showed a clear decrease for increasing amounts of taxol in the investigated concentration range (Fig. 4B). Therefore, the typical time scales for pattern coarse-graining becomes successively larger when the dynamics of microtubules is compromised by taxol addition.

To obtain a more significant, self-contained measure for the spatial arrangement of protocells, we inspected the variance of protocell centers encountered within a circle of radius R around randomly chosen points in the extract. For growing radii, more and more protocells will be within such a circle, yet fluctuations of the actual encountered number will very much depend on the large-scale organization of the pattern. Denoting by μ and σ² the mean and variance, respectively, of the number of center points found within a test circle of radius R, the normalized number variance Σ₂(R)=σ²/μ is indeed a sensitive measure for spatial order. For simple Poissonian random point (PRP) patterns, Σ₂ remains at unity for increasing radii, hence highlighting strong spatial fluctuations. In contrast, a strictly hexagonal point pattern yields an oscillatory decay of Σ₂ to zero for increasing radii (see also Discussion). In fact, patterns for which Σ₂ approaches zero for increasing R are called hyperuniform and have recently gained considerable interest (Torquato, 2018). Although strictly periodic systems are, somewhat trivially, hyperuniform, a wide class of systems with a disordered hyperuniformity has also been found (Torquato, 2018). These lack the characteristic oscillations of the normalized number variance for increasing values of R, but Σ₂ still approaches zero for large radii, indicating a long-range, but non-crystalline, order. Given the high degree of organization observed in protocell arrays and epithelia, suggestive of hexagonal arrays, one might hypothesize that space allocation in biology strives for (disordered) hyperuniformity. This hypothesis is reinforced by observations of hyperuniformity in avian photoreceptor patterns (Jiao et al., 2014) as well as the jamming transition in vertebrate axis elongation (Mongera et al., 2018), as jamming phenomena are linked to hyperuniformity (Atkinson et al., 2016). The similarity of protocell patterns to two-dimensional foams, described in the previous section, leaves this aspect open given that foams can feature both – a long-range order with signatures of disordered hyperuniformity (Ricouvier et al., 2019) but also a non-hyperuniform random pattern (Chieco and Durian, 2021).

We therefore probed protocell patterns on this aspect via the normalized number variance of protocell centers, Σ₂(R). For hyperuniform systems, Σ₂(R) should monotonously decrease for increasing test radii, R. The experimental data showed, however, a rapid and clear saturation at Σ₂≈0.3, immediately after the onset of pattern formation and also 1-2 h later (Fig. 5B). This indicates that protocell patterns are not hyperuniform at any time point, even though a visual inspection may have suggested a near-crystalline order. Assuming values Σ₂(R)<1, the pattern can be viewed instead to have gross geometric properties of a hard-sphere fluid (Torquato, 2018).

Summarizing our experimental results, we have found striking geometrical features in a minimalistic in vitro assay that exhibits a spontaneous spatial compartmentalization. The observed pattern of protocells is characterized by narrow distributions of vertex numbers, areas and perimeters (Fig. 2) with a characteristic length scale that is tunable by the addition of the microtubule-directed drug taxol (Fig. 3). The pattern shows dynamic coarse-graining features similar to those of a two-dimensional foam (Fig. 4) without long-range ordering (Fig. 5). Notably, coarse-graining preserves the pattern geometry, i.e. protocells show self-similar arrangements over time. Given that the observed phenomena compare favorably to previous observations in Phallusia oocytes (Khetan et al., 2021), Drosophila syncytial blastoderms (de Carvalho et al., 2022), and even culture cell monolayers (Kaliman et al., 2021), it is tempting to hypothesize common generic principles of a dynamic space compartmentalization, e.g. similar force fields and optimization functionals, upon which organism-specific features may be superimposed.

To follow up on this idea and to rationalize our experimental findings, we have formulated two simple and generic statistical models of how protocell centers attain their spatial arrangement, hence allowing for a comparison with the geometrical features of protocell patterns. As a reasonable baseline, we also considered a PRP pattern, which is supposedly the most simple process of distributing center points in the plane (see Fig. 6A for a visualization). To formulate these two models, we took the following experimental observations as an input. Protocell patterns were seen to emerge spontaneously all over the droplet by radially arranging microtubules to support an accumulation of organelles at the center. Despite a slow coarsening, all geometric features of the patterns were maintained over time. Although the protocell pattern did not emerge from a single point (as also mentioned by Cheng and Ferrell, 2019), we have assumed for simplicity only point-like seeds for the model, representing the centers of the initial protocell structures. Therefore, we neglected at first instance any coarse-graining dynamics and assumed an instantaneous existence of N center points (seeds) that are placed on the unit square, hence defining a typical length scale (chosen in accordance with Zhu et al., 2001). Using a tunable parameter, α∈[0, 0.7], two-dimensional point patterns were created in both models with the following rules (see sketches in Fig. 6B,C): in model 1, cell centers are chosen randomly from the unit square with the constraint that the minimal distance to neighboring centers is at least αλ; in model 2, cell centers are created by displacing vertices of a triangular lattice by a distance ξλ in a random direction, with ξ∈[0, α] being a uniformly distributed random number.

Both schemes yield the steady-state pattern when protocells emerge simultaneously, grow over time and stop growing upon contact. Cessation of growth upon contact reflects that coarse-graining, i.e. a fusion of neighboring units, is not considered (see below for a dynamic extension). In model 1, random seed positions are combined with a uniform and isotropic growth rate, whereas model 2 assumes hexagonal cells (owing to global instability of the uniform state with a defined wave vector, as observed in Turing pattern formation or spinodal decomposition), perturbed by spatiotemporal fluctuations of the growth rate. The two models therefore reflect very different ways of how the pattern emerges (local versus global onset).

Although not apparent immediately, the models feature very different long-range organization. PRP patterns are characterized by a strong randomness and hence the normalized number variance remains at unity for all test radii R, whereas for α≈0.5 both, model 1 and model 2 deviate from unity for large radii (Fig. 6D). Whereas model 1 displays a geometry that complies with a hard-disk fluid, indicated by Σ₂→const.<1 for large radii (Torquato, 2018), model 2 bears a resemblance to hexagonal patterns and features an ever-decreasing number variance (Σ₂→0), as expected for systems with disordered hyperuniformity (Torquato, 2018).

Using N=1000 center points (for comparability with experiments), we produced M=50 different realizations for each model and different choices of α. These point patterns were subjected to a Voronoi tessellation and evaluation of PDFs was performed as for the experimental data. As a result, we observed that model 1 and model 2 lead to a very good agreement with the experimental data when choosing α₁=0.55 and α₂=0.45, respectively, whereas PRP patterns were inconsistent with the experimental data. In particular, PDFs for vertex number, cell area and perimeter feature a shape that is empirically well captured by a narrow gamma distribution (Zhu et al., 2001), and all PDFs of models 1 and 2 match the experimental data so well that one cannot really claim a superiority of one, even though the compactness yields a first hint that model 1 might describe the experimental data somewhat better (see Fig. 2).

In line with this hint, the PDF of NN correlations of protocells provides further evidence for model 1 being the more adequate description (see Fig. 5A). Finally, the normalized number variance Σ₂ of model 1 captures the experimental data well, whereas the hyperuniform model 2 shows marked deviations (Fig. 5B). Again, data for PRP patterns are inconsistent with the experimental data. As a formal caveat, we note here that only an asymptotic vanishing of Σ₂(R) can properly reveal hyperuniformity, i.e. the finite sample size and potential inhomogeneities of protocell densities might mask asymptotic hyperuniform signatures (Dreyfus et al., 2015).

As model 1 captures all geometrical features of protocell patterns, we implemented a dynamic coarse-graining of these point patterns as a next step. In line with experiments, we demanded a conservation of the total area of the pattern, i.e. in each time step only a single fusion event of an existing cell with a neighboring cell was allowed, leading to a slow linear increase of the mean cell area. Reasoning that close neighbors have an elevated probability to fuse (see below), we determined at each step for all available points of the pattern the respective nearest neighbor. After sorting this set of pair distances in ascending order, we randomly selected one pair of points from the first one-third subset of distance values for fusion (combining the pair of points into a single new point located halfway between the original points). With this approach, small distances between neighbors are favored without enforcing fusion events to always invoke the smallest cells and distances. As a result, we observed that this simple, area-preserving dynamics resulted in very similar statistics before and after coarse-graining (see Fig. S7 for an example), i.e. the self-similarity of the pattern was preserved, in agreement with our experimental observations.

Based on the very good matching of our experimental data with model 1 (even after applying a simple coarse-graining dynamics), we can narrow down the emergence of protocell patterns to the following set of rules: (1) protocell seeds emerge simultaneously at almost final positions; (2) these seeds grow in an isotropic fashion at very similar growth rates by radial uptake of material from the close vicinity until touching neighboring protocells; (3) at this stage, growth stops and protocells repel each other sufficiently strong to not fuse immediately, i.e. a quasi-stationary, frustrated state of the pattern is reached; and (4) fusion events of close neighbors then lead to a slow coarse-graining without altering the geometry, i.e. a pattern with scale-invariant properties is maintained.

These somewhat abstract rules may be put into the biological context as follows. Protocell formation requires the spatiotemporal organization of microtubules into arrays with radial symmetry that serve as the defining mechanical structure and as a means for an inwards transport of material. Therefore, their spontaneous emergence can be related to rule (1): similar to artificial systems made from purified components (Nedelec et al., 1997; Surrey et al., 2001; Roostalu et al., 2018), the initially well-mixed extract will start polymerizing microtubules that interact via (multi-headed) motors when reaching a critical length and overlap. As a result, microtubules (or microtubular bundles) will try to focus into radial arrays at all positions in the assay. Because all loci in the extract have the same, supposedly low, kinetic barrier for microtubule formation, fluctuations will eventually determine where these structures will emerge, akin to fluctuation-induced nucleation in oversaturated solutions. Radially organized microtubule structures will hence form at about the same time at random positions. Once established, they recruit material to their centers, e.g. organelles, hence creating protocell seeds as required by rule (1). Uptake of more material and inwards transport along microtubules let protocells grow at the center of the structures, which is in accordance with rule (2). When neighboring protocells have grown large enough that they are prone to touching, competition for the same local pool of material needs to be considered as it might be a limiting factor. Moreover, without the action of an appropriate subset of (multi-headed) motors and a sufficient overlap of the two microtubule arrays, neighboring microtubule arrays will entropically repel each other like soft balls (Nedelec, 2002), leading to a mechanically frustrated pattern, in accordance with rule (3). Note that a comparable aster repulsion has recently been seen in the Drosophila syncytial blastoderm (de Carvalho et al., 2022).

Subsequent coarse-graining can take place if two neighboring arrays occasionally have a sufficient overlap and an appropriate subset of (multi-headed) motors for driving a fusion event (for detailed simulations, see Nedelec, 2002; Roostalu et al., 2018). This slow coarse-graining by fusion corresponds to rule (4), in which the assumption that close neighbors can fuse more easily (but also randomly) is based on the fact that a proper mutual configuration is needed for an attractive interaction. Given the still considerable complexity of our assays and our current lack of sufficient knowledge on the interaction between microtubule arrays in protocells, we need to defer the implementation of a more refined dynamic model, beyond the statistical model 1 with its simplified coarse-graining dynamics, to future work.

The observed effect of taxol addition, i.e. a decreasing length scale of the pattern, is most likely rooted in a reduced fraction of long microtubules because of taxol-induced suppression of microtubule dynamics (Yvon et al., 1999): In untreated extracts, a lack of translation leads to a limited pool of tubulin and/or microtubule-associated proteins (Ishihara et al., 2021), i.e. the dynamical instability will lead to an outcompetition of small and shrinking microtubules by long and growing filaments. Hence, remaining microtubules grow longer at the expense of unsuccessful smaller ones, which eventually vanish. Stabilization of all seeds by taxol increases the amount of growing filaments that compete for the same pool while growing with similar kinetics, eventually resulting in more but shorter filaments, which may consist of single microtubules or bundles (Ishihara et al., 2021). In support of this reasoning, we observed that the ratio ζ of short versus long microtubules, as extracted from fluorescence images (see Materials and Methods), increased from ζ≈4.25 in untreated extracts to an average of ζ≈9.54 in extracts treated with 1 μM taxol (see Materials and Methods). Fluorescence imaging of protocells in the presence of taxol also indicates shortened microtubules (see Fig. S2C,D in comparison with Fig. S2A,B).

Stabilization and shortening of microtubules (or bundles) by taxol also supports the suggestion that stabilizing microtubules in the initial phase results in smaller radial structures, i.e. the frustrated, quasi-stationary state prior to coarse-graining will feature more, but smaller, protocells. Because only the filament length scale but not the rules of interaction have been changed, the pattern maintains the same geometrical properties as in untreated extracts. Moreover, taxol hampers the dynamic instability of microtubules and hence their ability to explore adjacent regions, so that achieving sufficient microtubule overlaps between neighboring protocells for fusion events will require more time. As a consequence, lower coarse-graining rates γ are expected in the presence of taxol, in line with our experimental observation.

The protocell patterns observed here are not only similar to aster arrays in Phallusia oocytes (Khetan et al., 2021) and the Drosophila syncytial blastoderm (de Carvalho et al., 2022), but also to epithelial cell monolayers (Kaliman et al., 2021). Based on the phenotypical similarities, we speculate that these diverse systems share common principles for their space compartmentalization that are most likely rooted in fairly simple mechanical cues. Although this hypothesis is plausible when considering aster arrays without an envelope, e.g. energids in a syncytium state, the relation to developing tissues is somewhat less straightforward as it invokes successive divisions of membrane-enveloped cells, rather than spontaneous protocell formation and fusion events. Nevertheless, an initially random placement of cells [compare with rule (1)] and an intermediate frustrated arrangement of them in an emerging tissue as a result of competition for space [compare with rule (3)], is conceivable: cells are small and hence are subject to thermal and active fluctuations that perturb cell division axes and cleavage planes, leading to a certain randomness in cell positions and sizes. Adhesion will keep cells together, maybe even with perfect wetting angles, as suggested for the developing retina (Hayashi and Carthew, 2004), but cells will also exert mutual repulsive forces to maintain their individual volumes and positions, potentially blocking each other's motion. As a result, a mechanically frustrated pattern of cells with all properties of a random-packing process of soft spheres may emerge and hence key features of model 1, discussed above, are met. In fact, modeling cells as simple soft-repulsive balls that force each other into positions of least constraints has been surprisingly successful in predicting cell positions in the early embryogenesis of C. elegans (Fickentscher et al., 2013, 2016). Because cell division is typically much slower than local shape adaptions, similar to slow protocell fusion in our assay, the random-packing geometry can be expected to be preserved for long periods. Successive cell divisions could even be interpreted as a time-reversed, coarse-graining dynamics that maintains geometric features of the pattern.

In view of our findings and interpretations, it will be interesting to explore quantitatively which self-organizing patterns of asters, energids, cells and (embryonic) tissues comply with the idea of a random-packing process with conserved geometric properties, and whether such patterns eventually do reach a (disordered) hyperuniform ordering. Our present data suggest that hexagonal patterns with a long-range ordering could be a very rare and supposedly special case of space compartmentalization, i.e. an organization based on random seeds might be the standard case. It is also worth noting that all of the aforementioned pattern formation happens far from thermal equilibrium and most likely relies on mechanically driven self-organization and adaption processes (rather than self-assembly and relaxation towards equilibrium states with minimal free energy). Seemingly, the key to the pattern emergence is not only ATP-fueled deterministic forces but also the ambient non-equilibrium noise that arises from the plethora of nucleotide hydrolysis events. In fact, a variety of stationary states only emerges as a result of multiplicative noise far from equilibrium (see Garcia-Ojalvo and Sancho, 1999). Therefore, minimal in vitro assays, such as the one used here, are valuable tools that can help to reduce the daunting complexity of living matter and to reveal putative common principles for a wider class of systems.

The interphase extract preparation protocol was adapted from Deming and Kornbluth (2006) and Sparks and Walter (2018) with minor modifications (see supplementary Materials and Methods for details of important reagents). A single injection of human chorionic gonadotrophin 16-17 h before the experiment was sufficient to obtain proper egg harvest. We changed the concentration of Marc's Modified Ringer's (MMR) to 0.5× (instead of 0.25×) and washed five times instead of three times. Correcting for a typographical error in the published protocols, we used KCl instead of HCl in the egg lysis buffer (ELB) preparation. Cycloheximide was added to the extract only after egg crushing (not in the ELB) but cytochalasin B was added to the ELB in the centrifuge tube to which eggs were transferred for crushing (50 μg/ml final concentration).

Interphase-arrested cytoplasmic extracts were prepared from freshly laid eggs of Xenopus laevis following standard protocols (Deming and Kornbluth, 2006; Sparks and Walter, 2018). In brief, eggs in the metaphase stage of meiosis II were collected and dejellied. These eggs were washed first with 0.5× MMR, then with ELB containing 1,4-dithiothreitol, and finally packed by centrifugation (200 g for 1 min followed by 600 g for 1 min, both at 18°C) with the excess buffer being removed. Packed eggs were crushed and fractionated into three distinctive layers by centrifugation (13,000 g for 10 min at 18°C) using a Beckman Coulter JS-13.1 swinging bucket rotor and open-top polyclear centrifuge tubes (Seton, 4/6.5 ml). The mid cytoplasmic layer was then carefully isolated and supplemented with 5 µg/ml aprotinin, 5 μg/ml leupeptin, 5 μg/ml cytochalasin B, 50 μg/ml cycloheximide, and stored on ice; these extracts were used within 10 h. The addition of cycloheximide inhibits protein synthesis, including the synthesis of new cyclin. Thus, the extract is arrested in an interphase state. The addition of protease inhibitors (aprotinin, leupeptin) limits protein degradation (Chan and Forbes, 2006), and cytochalasin B suppresses actin polymerization (MacLean-Fletcher, 1980). Some extracts were also supplemented with taxol to enhance microtubule stabilization without inducing major changes to cytoskeletal arrangements (Verde et al., 1991).

For imaging, the interphase extract was supplemented with 1  polystyrene beads (1% by volume of stock solution) and FITC-Dextran (1% by volume of stock solution). Not adding any of these resulted in the same pattern formation but the contrast of protocell centers to the periphery was considerably dimmer. DMSO concentrations of all extracts were maintained at 1% by volume (if needed by compensating with pure DMSO) to limit the influence of microtubule stabilization by DMSO (Wignall and Heald, 2001). Extracts were mixed by gentle flicking, pipetting three or four times with a cut-off tip and carefully inverting the tube.

Fluorinated ethylene propylene (FEP) tapes were used to cover the bottom microscope slide and top glass cover slip, which were separated by a 120 µm double-side tape spacer into which circular 9 mm holes were punched for hosting the extract droplets (see Fig. S1A); FEP coating was done 1 day before the experiment. To each of the holes, 4 µl of the extract was carefully pipetted into the center. Then, the whole chamber was sealed immediately with the coverslip to avoid evaporation. These samples were imaged with one of the following two microscopes in a tile-scan mode: a Leica DMI6000B inverted microscope with an HC PLS-APO 10×/0.30 DRY objective and a Leica DFC360FX camera or a Leica SP5 II confocal laser-scanning microscope with a HCX PL APO CS 10×/0.40 DRY UV objective with 3.03 µm pixel width (512×512 pixel tiles) and open pinhole. In both cases brightfield images were recorded, with an additional fluorescence channel on the SP5 for detecting FITC-Dextran (excitation: 496 nm; emission: 511-550 nm). The extract was imaged for more than 4 h at room temperature (20°C) with a minimal time interval between consecutive tile scans.

For imaging of microtubules, rhodamine-labeled tubulin was first resuspended in 1 μl of 1 mM GTP-containing general tubulin buffer and 8 μl of fresh interphase extract. Samples for imaging were subsequently prepared by mixing 3 μl of this solution with 50 μl premixed interphase extract, hosted by a modified sample chamber for high-resolution imaging (see Fig. S1B). Fluorescence imaging was performed with a Leica SP5 II confocal laser scanning microscope, using an excitation wavelength of 561 nm and a detection range of 575-650 nm. Images (512×512 pixels; pixel size chosen according to Nyquist's theorem) were acquired with mono-directional scanning (scan frequency 400 Hz) within a maximum period of 2 h, using an HCPL APO 100×/1.40 OIL or a HCXPL APO 63×/1.4 OIL CS2 objective.

As a first step, shading correction and merging of images from a tile scan was performed with Leica LAS X software using the options auto-stitching, smoothing, and linear blending, taking only the brightfield channel as a reference. Images from the Leica DMI6000B were resized by a factor of 1/4 in each direction to arrive at a pixel dimension of 3 µm/pixel for all microscopes.

Using the time series of these reconstructed large-field images, the start and end times between which protocells were visible was determined visually for each droplet preparation. For these periods, the following operations were performed on the large-field images (see also Fig. S3 for illustration): manual marking of the droplet boundary, defining a rectangular region of interest in each image for further processing; automatic segmentation of the droplet boundary by detecting the ring of darkest pixels that is present at the droplet border; and binarization of the droplet image using the lowest of two thresholds, which are obtained by Otsu's method (three-pixel classes with minimal intra-class variance, implemented via the ‘multithresh’ function in MATLAB).

Subsequently, further filtering was performed to achieve a single-connected component of dark pixels within each protocell, defining the cell center. To this end, a sequence of classic morphological operations was applied (see Fig. S4 for illustration): hole-filling (connectivity: four), image opening (using a 3×3 square), image dilation, hole-filling, and image erosion, with the same structural elements as before. Finally, connected sets of center pixels (highlighted in blue in Fig. S4D) were filtered according to their area histogram (using a bin width according to Scott's rule; connectivity for area: eight), i.e. the areas within the first histogram bin were rated as erratic pixels and hence removed (compare with examples before and after in Fig. S4D,E). The centroid positions of these filtered connected components were then used for a Voronoi tessellation (Fig. S4F).

For the quantitative analyses in Figs 2 and 5, we used the data from five large droplets from different repeats of the assay, yielding a total of 9971 protocells (immediately after the first emergence of the pattern). The average number of 2000 protocells per droplet provided sufficient statistics also for long-range measures such as Σ₂. Although smaller droplets were insufficient for this (and were hence not included), their local measures were in full agreement with the results shown in Fig. 2. For high-taxol conditions (Fig. S5), we used three large droplets with an initial number of 10,049 protocells. Owing to the progressive coarse-graining, statistics for later time points (1-2 h after pattern emergence) relied on about 50% of the initial number of protocells. Because data shown in Figs 3 and 4 only report on local measures, we considered here a total of 13 droplet repeats, i.e. several thousand protocells were available for determining areas and vertex numbers at each indicated taxol concentration.

To extract microtubule lengths from fluorescence images, we used SOAX, a freely available analysis software for biopolymer networks (Xu et al., 2015). Because very short filaments in semi-dilute and dense systems are difficult to assess with light microscopy without missing significant portions of this pool (unless fluorescent filaments are very sparse), we only considered microtubules with a length of at least 5 μm. From all detected microtubules of all images (see Fig. S8 for examples), we calculated the fraction of filaments with a length smaller than 20 μm, f_s, and the complementary fraction of longer filaments, f_ℓ. The ratio ζ=f_s/f_ℓ was taken as a self-normalizing measure for the frequency of long microtubules. From two replicates per condition, we obtained ζ₁=4.06 and ζ₂=4.46 (without taxol) and ζ₁=11.5 and ζ₂=8.14 (1 µM taxol). Using |ζ₁−ζ₂|/(2〈ζ 〉) for the relative error, this resulted in an average ratio 〈ζ 〉=4.25±5% without taxol versus 〈ζ 〉=9.54±17% at 1 µM taxol. Thus, the fraction of shorter filaments was significantly increased in the presence of taxol. We also would like to note that it is likely that long filaments actually consist of microtubule bundles that may emerge by an autocatalytic growth of individual microtubules to a mean length of about 16 µm (Ishihara et al., 2021).

We thank A. C. Ramos and O. Stemmann (University of Bayreuth, Genetics) for providing Xenopus eggs and access to their equipment, and A. Hanold for supporting the extract preparation.