## Main

Morphogenesis involves forces that are generated within the actomyosin cortical layer of cells1. Improper cortical organization leads to an impairment of key cellular and developmental processes from as early as meiosis in oocytes to every subsequent cell division12. During meiotic maturation of oocytes, the actomyosin cortex transitions from inactive and non-contractile, to active and tension generating2,3. This transition can generate a spectrum of actomyosin cortical structures and dynamics, including an actin cap in the mouse oocyte13, actin spikes in starfish oocytes14 and waves of Rho activation and F-actin polymerization in Xenopus15. Organizing the first active actomyosin cortex requires the recruitment and assembly of various cortical components as well as the polymerization of actin filaments4. These processes have to be coordinated across the entire cell surface in order to generate a uniform actomyosin cortical layer. Here we ask how the formation of an active and tension-generating actomyosin cortex during meiotic maturation in oocytes is orchestrated.

## Actomyosin cortex activation

The hermaphrodite nematode Caenorhabditis elegans is a prime system for investigating actomyosin cortex formation during oocyte maturation16,17,18. In C. elegans, the onset of meiotic divisions and oocyte maturation coincides with ovulation and fertilization16,18. Oocytes are fertilized inside the hermaphrodite mother as they pass through the sperm-containing organ—the spermatheca16. To understand how the formation of the first actomyosin cortex during oocyte maturation is orchestrated, we visualized F-actin in C. elegans oocytes containing Lifeact::mKate2. We observed that, just before fertilization inside the mother, the oocyte cortical layer appears undeveloped with only sparse amounts of filamentous actin present (Fig. 1a, left). By contrast, shortly after fertilization a highly dynamic and dense actomyosin cortical layer is present below the plasma membrane (Fig. 1a, right and Supplementary Video 1). Importantly, we find that actomyosin cortex activation in the oocyte occurs through an intermediate stage that lasts approximately 10 min, results in a dynamic and contractile actomyosin cortical layer, and ends with the extrusion of the first polar body19 (Supplementary Video 1). Strikingly, this intermediate stage is characterized by the transient appearance of thousands of F-actin-rich condensates at the cortical layer (Fig. 1a). Here we use the term condensate to refer to a dense assembly of specific molecular components maintained by collective molecular interactions. F-actin and its nucleators have previously been shown to form biomolecular condensates and evidence for liquid-like properties has been provided5,6,7,8. The F-actin-rich condensates we observe are highly dynamic and inherently unstable. They appear stochastically and each disappear after approximately 10 s.

We next set out to investigate the nature of these transient F-actin-rich condensates. To better observe their dynamics, we took advantage of the fact that oocytes isolated from the mother can mature in the absence of fertilization20. This allowed us to develop a total internal reflection fluorescence (TIRF) assay21 utilized under highly inclined and laminated optical sheet (HILO) conditions for imaging cellular structures near the cell membrane (Fig. 1b). This enabled a quantitative study of actomyosin cortex formation in isolated oocytes at high spatial and temporal resolution (Supplementary Video 2). F-actin polymerization is organized by nucleation pathway members such as N-WASP and the ARP2/3 complex, as well as the elongator Formin22. We first investigated the presence of these components in cortical condensates. We used three strains labelling F-actin (by expressing Lifeact::mKate2) together with either endogenously labelled N-WASP (WSP-1::GFP), capping protein (CAP-1::GFP) or Formin (CYK-1::GFP)23. In addition, we used a strain that endogenously labels both the ARP2/3 complex (ARX-2::mCherry) and N-WASP (WSP-1::GFP). As well as F-actin, we identified WSP-1, the ARP2/3 complex and the capping protein CAP-124 (Fig. 1c, Extended Data Fig. 1 and Supplementary Videos 3 and 4) as components of cortical condensates, but the Formin CYK-1 was absent19,24 (Extended Data Fig. 1). This demonstrates that cortical condensates contain molecules that mediate branched F-actin nucleation, similar, for example, to CD44 nanoclusters, dendritic synapses and podosomes25,26,27. We also noted that during their approximately 10 s lifetime (Fig. 1f) cortical condensates were enriched first in WSP-1 and ARP2/3, and only then F-actin accumulated before first losing WSP-1 and ARP2/3 and then F-actin (Fig. 1d,e). Given the time at which they appear and the fact that they contain molecules that mediate branched F-actin nucleation, we speculate that dynamic cortical condensates play a role in the formation of the first oocyte cortex.

We next asked if cortical condensates constitute a phase that coexists with its surroundings. Such a phase is characterized by material properties (such as density) that are intensive, that is, independent of volume. We used the strain that simultaneously labels F-actin and WSP-1 to show that, throughout their brief lifetime (Fig. 1e,f), cortical condensates varied over two orders of magnitude in both Lifeact (A) and WSP-1 (W) integrated fluorescence intensities (Fig. 1g). We estimated the volume of cortical condensates from the cross-sectional area determined by segmentation28 (Supplementary Methods), and found that for intensity stoichiometries $$A/\left(A+W\right)$$ between approximately 0.65 and approximately 0.93, they occupied a volume V well described by summing the volume contributions of F-actin $${v}_{A}A$$ and WSP-1 $${v}_{W}W$$, with volume coefficients $${v}_{A}=1.54\times {10}^{-7}\left(\pm 1\times {10}^{-8}\right)\,{\rm{\mu }}{{\rm{m}}}^{3}\,{{\rm{IU}}}^{-1}$$ and $${v}_{W}=2.34\times {10}^{-7}\left(\pm 2\times {10}^{-8}\right)\,{\rm{\mu }}{{\rm{m}}}^{3}\,{{\rm{IU}}}^{-1}$$ (where IU denotes total intensity units, see Methods and Fig. 1i). This provides a relation between molecular content and volume, but does not imply that condensates are densely packed structures of only WSP-1 and F-actin. Whereas cortical condensates varied over two orders of magnitude in integrated fluorescence intensities (Fig. 1g), the respective concentrations of WSP-1 and F-actin within the cortical condensates were significantly more restricted in their variation (Fig. 1h). This is also reflected in the emergence of a preferred pair of F-actin and WSP-1 concentrations maintained on average by the ensemble of cortical condensates (Fig. 1j). We conclude that, on the one hand, cortical condensates are maintained far from equilibrium: they are highly dynamic and each disassemble after approximately 10 s. On the other hand, cortical condensates display signatures of a multicomponent condensed phase: they occupy a volume determined by their molecular content and show an enrichment of WSP-1 and F-actin at concentrations distinct from their external environment29,30. Hence, the ensemble of stochastically appearing, growing and subsequently dissolving cortical condensates effectively forms a chemically active micro-emulsion, which, despite continuous turnover, maintains a steady size distribution that does not coarsen11 (Fig. 1k). Both the properties of a condensed phase and the mechanisms underlying its formation and dissolution can be revealed by a study of growth kinetics.

## Cortical condensate growth laws

To study the growth kinetics of these cortical condensates, we quantified their compositions and volumes over time (Fig. 2 and Supplementary Methods). For a single representative cortical condensate, Fig. 2a,b shows the time evolutions of (1) WSP-1 and F-actin total condensate intensity, (2) stoichiometry and (3) volume (Supplementary Notes 1 and 2). For the example shown, WSP-1 precedes F-actin in both growth and loss, stoichiometry grows monotonically with time, and volume first increases and then decreases, and is well captured by summing volume contributions from F-actin and WSP-1. We noted that neighbouring condensates followed similar trajectories in composition and volume despite forming stochastically and at different times (Fig. 1c–e and Extended Data Fig. 9). Thus, at a given time, neighbouring cortical condensates that share their external environment can be at different stages of their internal life cycle. We conclude that the growth kinetics postnucleation are governed by condensate internal composition.

How does the internal composition of a cortical condensate influence its growth and shrinkage? To answer this question we developed a general method to quantitatively study compositional dynamics in an ensemble of multicomponent condensates based on an analysis of the mass flux into the condensates (mass balance imaging31). For this, we quantified the time rate change of protein amounts within cortical condensates as a function of their internal F-actin and WSP-1 amounts. This time rate change of amounts is represented by a vector field, which defines average trajectories in the space of WSP-1 and F-actin amounts (Fig. 2c). Consistent with the representative example (Fig. 2a), average trajectories form loops that pass through three subsequent regimes: an early growth regime in which condensates first grow in WSP-1 and subsequently in F-actin amounts; a transition regime in which WSP-1 is lost while F-actin amounts still increase; and a disassembly regime with loss of both WSP-1 and F-actin. The nullcline of WSP-1 dynamics (the green line in Fig. 2c), that is, the WSP-1 amounts above which condensates grow and below which they shrink in WSP-1 content, reflects an F-actin-dependent critical WSP-1 amount for WSP-1 growth. Stoichiometry is constant on lines that pass through the origin, and hence the WSP-1 nullcline corresponds to a threshold stoichiometry of approximately 0.85. F-actin growth dynamics change from growth to shrinkage at a similar but slightly higher stoichiometry of approximately 0.9 (the magenta line in Fig. 2c shows the F-actin nullcline). We conclude that cortical condensates become unstable and change from growth to disassembly in the transition regime between the two nullclines.

The three regimes (growth, above the WSP-1 nullcline; transition, between the two nullclines; disassembly, below the F-actin nullcline in Fig. 2c) are also visible when plotting WSP-1 and F-actin growth rates as a function of stoichiometry (Fig. 2d). Because the stereotypical compositional trajectories (Fig. 2a,b) involve a monotonic increase in stoichiometry with time, the x axis of Fig. 2d also represents a progression through time. The dependence of growth rates on stoichiometry reveals the mutual regulation of WSP-1 and F-actin, and can be depicted by the reaction motif shown in Fig. 2g. Processes I and II are mediated by WSP-1, whereas processes III and IV are mediated by F-actin. Process I corresponds to WSP-1 self-recruitment, evidenced by the fact that at low stoichiometry, corresponding to condensates consisting of mainly WSP-1, the WSP-1 growth rate is largest (Fig. 2d). Process II denotes WSP-1 dependent F-actin growth, reflected by a decrease of the F-actin growth rate as stoichiometry increases. This is most evident in the transition regime of Fig. 2d. Process III denotes F-actin-dependent loss of WSP-1, reflected by the fact that WSP-1 growth rates decrease with increasing stoichiometry. This suggests that F-actin counteracts the ability of WSP-1 to self-recruit, similar to previously reported negative feedback of F-actin on its nucleation via Rho15,32. Finally, process IV denotes F-actin depolymerization, reflected by the fact that F-actin is lost fastest at the highest stoichiometry (Fig. 2d). Further support for this reaction motif is provided by an analysis of WSP-1 and F-actin growth rates at constant WSP-1 and F-actin amounts (Extended Data Fig. 9), and an analysis of the impact of RNA interference (RNAi) of proteins involved in regulating F-actin (Extended Data Fig. 4).

The shape of the measured phase portrait (Fig. 2c) and the shape of the growth rates $$\dot{W}$$ for WSP-1 and $$\dot{A}$$ for F-actin as a function of stoichiometry (dots denote time derivatives; Fig. 2d) suggest the following empirical growth laws that define a non-linear dynamical system33 (Fig. 2c, Extended Data Fig. 3 and Supplementary Note 3):

$$\begin{array}{l}\dot{W}={k}_{{\rm{r}}}W-{k}_{{\rm{l}}}\frac{AW}{V}\\ \,\dot{A}={k}_{{\rm{b}}}\frac{AW}{V}-{k}_{{\rm{d}}}A.\end{array}$$

Here WSP-1 self-recruitment depends linearly on W through the recruitment rate kr, consistent with the ability of WSP-1 molecules to dimerize34,35 (process I). Interactions between F-actin and WSP-1 result in ARP2/3 mediated branched nucleation and a subsequent increase in the amounts of F-actin36,37. This behaviour is captured by the term $${k}_{{\rm{b}}}\frac{AW}{V}$$, where kb is a kinetic coefficient describing branching and condensate volume $$V={v}_{A}A+{v}_{W}W$$ depends on molecular amounts (see above and also Supplementary Information; process II). Branched nucleation coincides with a loss in WSP-1; this loss is captured by $${k}_{{\rm{l}}}\frac{AW}{V}$$ with the kinetic coefficient kl describing the branching-dependent loss of WSP-1 (ref. 38; process III). Finally, F-actin is lost with rate kd, consistent with severing and depolymerization39 (process IV) (see also the simplified depiction in Extended Data Fig. 6). Note that these four coefficients together capture all the relevant molecular processes inside the condensates. This may include processes not discussed above. The mathematical form of all four terms is determined by the observation that the relative growth rates $$\dot{W}/W$$ and $$\dot{A}/A$$ are linear functions of the effective F-actin volume fraction $$\varphi =\frac{{v}_{A}A}{V}$$ (Fig. 2e,f; see also discussion in Supplementary Note 3). Figure 2e,f also allow us to estimate kr, kl, kb and kd. With these estimates, the simple growth laws describe the experimental data well, and capture the entire mass flux phase portrait together with the composition-dependent critical sizes as reflected by nullclines (Fig. 2c).

The WSP-1 nullcline $${W}_{{\rm{c}}}(A)=A({k}_{{\rm{l}}}-{k}_{{\rm{r}}}{v}_{A})/{k}_{{\rm{r}}}{v}_{W}$$ specifies a critical amount of WSP-1 above which WSP-1 amounts grow and below which WSP-1 amounts shrink. Notably, this critical amount is similar to a critical droplet size for nucleation and growth, but here it stems from biochemical reactions and not from condensation physics. The resulting growth laws exhibit a fixed point at $$\left(A,W\right)=\left(0,0\right)$$ with a stable (F-actin) and an unstable (WSP-1) direction. After nucleation, condensate dynamics follow a homoclinic orbit, initially growing rapidly in the unstable direction before turning and eventually undergoing disassembly while moving along the stable direction back towards the fixed point. Together, this represents a dynamic instability of condensates that shares similarities with the dynamic instability of microtubules10: cortical condensates transition from unstable growth to shrinkage, which limits their size, and can display stochastic rescue events (Extended Data Fig. 7).

## Transition to unbounded growth

To understand how the transition from condensate growth to condensate disassembly is orchestrated, we used RNAi to perturb the interplay between WSP-1 and F-actin. RNAi of upstream signalling molecules that regulate F-actin assembly, such as RHO-1 (Rho GTPase), CYK-1, CDC-42 and CHIN-1 (CDC-42 GAP), as well as multivalent adaptors VAB-1 (Ephrin receptor) and NCK-1 (Nck) did not affect condensate dynamics40,41,42 (Extended Data Fig. 4 and Supplementary Notes 4 and 7). This suggests that cortical condensate dynamics are governed by feedback structures independent of the major signalling pathways that regulate the actomyosin cortex43. WSP-1 mediates branched F-actin nucleation through the ARP2/3 complex4,35, and we thus performed RNAi against ARX-2 (ARP2 in the ARP2/3 complex in C. elegans44). Oocytes showed reduced numbers of cortical condensates for less than 20 h of arx-2 RNAi, whereas cortical condensates were absent for more than 20 h of RNAi (Fig. 3a,b and Extended Data Fig. 7A), consistent with a general reduction of F-actin branched nucleation45.

We used the reduction in the number of cortical condensates as a measure of the strength of the perturbation, and distinguished between mild (30 to 70 condensates per oocyte), moderate (70 to 120 condensates per oocyte) and strong arx-2 RNAi (no dynamic cortical condensates). Mass balance imaging of cortical condensates in the mild and moderate conditions revealed that, in comparison with the unperturbed case, compositional trajectories are progressively tilted towards the WSP-1 axis (Fig. 3c). The mathematical form of the growth laws is maintained, but the associated coefficients are changed (Fig. 2e,f). Mild and moderate arx-2 RNAi reduced the rate of WSP-1 self-recruitment kr by 17 ± 4% and 15 ± 4%, respectively, and increased the coefficient kl for moderate arx-2 RNAi by 20 ± 6% (1 ± 5%). The branching coefficient kb remained essentially unchanged, indicating that ARP2/3 amounts are not rate limiting for ARP2/3 mediated branching inside cortical condensates. The dominant effect of mild and moderate arx-2 RNAi is an approximately 1.7- and 2.7-fold increase, respectively, of the F-actin loss rate kd. This is consistent with previous findings that the ARP2/3 complex protects F-actin from depolymerization in vitro46. In addition to the changes of coefficients, mild and moderate arx-2 RNAi both reduced the average F-actin concentration by a factor of approximately 1.4 and 1.5, respectively, and increased the average WSP-1 concentration by a factor of approximately 1.5 and 1.8, respectively (Fig. 3d). In all three cases, no RNAi control, and mild and moderate arx-2 RNAi, the pairs of average concentrations fall on the line of constant total density of WSP-1 and F-actin together (Fig. 3d, yellow dashed line; see Supplementary Information). We conclude that the ARP2/3 complex, largely through its impact on F-actin disassembly, governs the transition from condensate growth to condensate disassembly and determines the ensemble-averaged pair of internal concentration along the line of constant total density.

Strong depletion of ARX-2 by RNAi (more than 20 h of RNAi feeding at 20 °C) resulted in a loss of dynamic cortical condensates, and a considerably altered cortical architecture with large persistent patches of F-actin and WSP-1 (Fig. 3b, right, Supplementary Video 5 and Extended Data Fig. 8). We asked if this phenotype can be understood given the condensate growth laws above. It is not possible to determine the four growth law coefficients for strong arx-2 RNAi using mass balance imaging, because there are no dynamic condensates. However, the systematic change of both the F-actin loss rate kd and the condensate number per oocyte for increasing strength of arx-2 RNAi enabled us to provide a lower-bound estimate of kd for the strong RNAi condition (Extended Data Fig. 7). We find that at and above this estimated value of kd the system crosses a critical point at which the two nullclines switch their position, with the F-actin nullcline now above the WSP-1 nullcline (Fig. 3c, right). This causes a notable change in the growth dynamics of cortical condensates, with a complete loss of the homoclinic orbits that transition from growth to shrinkage. Instead, condensates exhibit unbounded growth consistent with the emergence of large persistent patches of F-actin and WSP-1 (Fig. 3b, right; note that we expect unbounded growth to ultimately become limited by effects we have not considered in our description, such as the depletion of the monomer pool). In conclusion, our analysis suggests that a switching of nullcline positions in strong arx-2 RNAi leads to uncontrolled F-actin growth and impaired cortical activation in the oocyte, and therefore impaired later development22,47 (Supplementary Notes 5 and 6).

## Ensemble properties

How do the growth kinetics lead to a specific pair of average internal concentrations and therefore a specific stoichiometry? To address this question, we change variables from F-actin amount A and WSP-1 amount W to effective F-actin volume fraction $$\varphi ={v}_{A}A/V$$ and condensate volume V. Figure 4a shows that the calculated phase portrait in the $$\varphi$$$$V$$ plane obtained by a change of variables of the empirically determined growth laws is consisted with the experimental one determined from measured WSP-1 and F-actin amounts and measured condensate volumes. Figure 4b shows that the transition from condensate growth (red) to shrinkage (blue) occurs at an effective F-actin volume fraction of approximately 0.8, corresponding to a stoichiometry of approximately 0.86. Notably, at this stoichiometry the rate of change of condensate volumes and stoichiometry is slowest (orange dotted lines in Fig. 4a,b,d), implying that the ensemble of dynamic condensates is governed by this slowly varying, thus dominant, stoichiometry. Hence, the peak of the concentration histograms (Figs. 1j and 3d) occurs at the point at which the line of dominant stoichiometry intersects with the line of constant total density (Fig. 4d).

## Intensive chemical reaction dynamics

We also recognized that the time evolutions of the effective F-actin volume fraction $$\dot{\varphi }$$ and the WSP-1 and F-actin concentrations are independent of condensate volume (Supplementary Notes 810). Thus, condensate dynamics are intensive, which is consistent with mass action kinetics in well-mixed systems. However, conventional mass action kinetics change reactant concentrations at constant volume, but usually do not involve assembly and disassembly as is the case here. Note that intensive condensate dynamics are not consistent with the conventional kinetics of nucleation and growth of liquid-like condensates, in which assembly rates depend on condensate size48,49. This reveals that cortical condensates exhibit an unconventional chemical kinetics in which mass action governs assembly and disassembly, and therefore the effect of mass action dynamics on concentrations in condensates is modified (Supplementary Notes 810). Note, however, that even though cortical condensates do not assemble via classical nucleation and growth, the intensive condensate dynamics show that the condensate material behaves as a well-mixed phase with size-independent properties. Intensive reaction dynamics are expected to arise in situations in which the time for diffusion across the condensate is shorter than the typical time associated with a chemical reaction. The condensate dynamic instability limits cortical condensate size. Therefore, reaction dynamics remain intensive and the resultant chemically active micro-emulsion maintains a steady-state size distribution (Fig. 1k)11.

## Discussion

To conclude, cortical condensates represent a new type of non-equilibrium biomolecular condensate that assembles and disassembles via a non-linear dynamic process governed by mass action chemical kinetics. They recruit molecules that drive branched nucleation of F-actin and support the activation of the actomyosin cortex. The dynamics of the growth and shrinkage of cortical condensates are similar to the dynamic instability of growing and shrinking microtubules10, but arises in a bulk assembly that forms a phase. We suggest that the formation and subsequent dissolution of cortical condensates via a condensate dynamic instability serves to control autocatalytic F-actin nucleation and prevents runaway growth during the activation of the first cortical actin meshwork in the C. elegans oocyte.

## Methods

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.