A minimal mathematical model for polarity establishment and centralspindlin-independent cytokinesis Open Access

Polarization and cytokinesis are fundamental processes in organismal development and physiology (Dewey et al., 2015). Cell polarization is encoded by asymmetric distributions of protein molecules, which are shaped by local regulation of binding and diffusion, and especially active transport by cortical flows (Munro et al., 2004; Mogilner et al., 2012; Lang and Munro, 2017). Likewise, cytokinesis involves the formation and constriction of the actomyosin ring in the cell mid-plane, a process driven by a balance of contractility, flow and membrane mechanics (White and Borisy, 1983; White, 1985; Glotzer, 2005). The two processes can be biochemically and mechanically connected, as cell polarity can regulate spindle positioning, which controls the site of contractile ring assembly and, consequently, the division plane (Grill et al., 2001; Maddox et al., 2007, 2012; Davies et al., 2014).

The Caenorhabditis elegans zygote provides a powerful model in which to study both polarization and cytokinesis (Fig. 1). In C. elegans, as in many other animal cells, contractility in polarity establishment and cytokinesis is mediated by the GTPase RHO-1, which activates myosin through its effector Rho kinase. RHO-1 transitions between an active (GTP) state and inactive (GDP) state via interactions with the RhoGEF ECT-2 and RhoGAP RGA-3/4, which activate and inactivate RHO-1, respectively (Fig. 2A; Michaux et al., 2018; Basant and Glotzer, 2018). Contractility in polarization and cytokinesis relies on modulation of this circuit. During polarization, contractility is activated by a nematode specific protein, NOP-1, which appears to globally activate ECT-2 (Tse et al., 2012). Contractility is inhibited by centrosomes at the position of sperm entry, establishing the posterior pole and triggering anterior-directed cortical flows that facilitate the segregation of anterior and posterior PAR proteins into distinct domains (Goldstein and Hird, 1996; Munro et al., 2004; Lang and Munro, 2017; Gross et al., 2019). During cytokinesis, the centralspindlin complex accumulates at the mid-plane of the spindle, where it activates ECT-2 and RHO-1, thus promoting contractility (Glotzer, 2005; Basant and Glotzer, 2018). This pathway combines with a second pathway similar to polarization, i.e. activation of RHO-1 spatially modulated by the centrosomes (Tse et al., 2012). Given the positions of the separated centrosomes, this also biases contractility to the cell equator (White, 1985; Werner et al., 2007; Loria et al., 2012). The goal of this study was to use mathematical models to determine whether the known regulatory pathways are sufficient to explain these highly stereotyped behaviors.

Recent studies have characterized the mechanism by which polarizing and dividing cells pattern the RHO-1 activation pathway. During polarization, Aurora A kinase (AIR-1) associates with the sperm centrosome, which is the sperm-derived structure that promotes polarity establishment (Hannak et al., 2001; Cowan and Hyman, 2004; Klinkert et al., 2019; Kapoor and Kotak, 2019; Zhao et al., 2019; Longhini and Glotzer, 2022). Recent work (Longhini and Glotzer, 2022) showed that AIR-1 impacts cortical dynamics by inhibiting ECT-2. Specifically, ECT-2 dissociates from the posterior membrane in an AIR-1-dependent manner, and it contains consensus sites for AIR-1 that are required for AIR-1 responsiveness. During polarization, ECT-2 exhibits posterior depletion and anterior enrichment, a pattern of accumulation that requires cortical myosin flows. A similar set of events occur upon anaphase onset, coincident with cytokinesis (Longhini and Glotzer, 2022). Although the centrosomes have duplicated, matured, moved farther from the cortex (Fig. 1) and accumulated much more AIR-1 in cytokinesis, there remains a strong, ultra-sensitive dependence between the distance of the centrosome from the nearest cortical domain and the amount of cortical ECT-2 at that site; proximal centrosomes correlate with a reduction in cortical ECT-2 (Longhini and Glotzer, 2022).

Although the qualitative mechanisms by which AIR-1, ECT-2 and myosin interact to generate contractility and flow have thus been well characterized, how the hypothesized pathways could generate the quantitative patterns of myosin and ECT-2 accumulation during polarization and cytokinesis is still unclear. For instance, unlike the anterior PAR proteins, which have residence times on the order of 100 s (Robin et al., 2014), ECT-2 cannot be strongly advected, as it exchanges rapidly between the cytoplasm and the cortex on timescales of a few seconds, appearing to preferentially accumulate on the cortex at myosin-enriched sites (Longhini and Glotzer, 2022). Consequently, it remains unknown whether a short residence time, preferential recruitment by myosin and weak advection by cortical flows could combine to generate the observed asymmetric accumulation of ECT-2 during polarization. More generally, it is not known whether additional mechanisms are required to explain the pattern of ECT-2 accumulation during cytokinesis, as the centrosomes are so much further from the cortex at that stage.

In this study, we tested whether a minimal set of interactions can explain the dynamics of ECT-2 and myosin during both polarity establishment and centralspindlin-independent cytokinesis. To do this, we constructed a mathematical model that uses an AIR-1 signal, which diffuses from the centrosomes to the cortex, as an input to a continuum model of contractility (Fig. 2), which is similar to those previously described (Michaux et al., 2018; Gross et al., 2019). We show that our model can explain the initial dynamics of polarization, similar to those observed in the absence of PAR proteins. Furthermore, the same model reproduces the patterns of ECT-2 accumulation observed during cytokinesis, thus demonstrating a quantitative similarity between the two processes.

We used the model to explore the quantitative similarities between polarization and cytokinesis. As discussed in the Materials and Methods section, some of the parameters are constrained using measurements from the polarized wild-type embryo. As such, the statements we make about wild-type polarization are shaped by existing data. Changing parameters, such as the centrosome–cortex distance and ECT-2 residence time, will demonstrate the importance of the centrosome position and relevance of rapid ECT-2 exchange during polarization. Perhaps more important is the model's direct extrapolation to cytokinesis, which concludes this section. Without changing any parameters, we demonstrate that the model can also explain the observed ECT-2 accumulation patterns during cytokinesis (Longhini and Glotzer, 2022), which are cued by larger centrosomes with 30-fold more AIR-1 than in polarization (Fig. 1).

During cell polarization, the newly duplicated centrosomes typically sit very close to the posterior cortex (∼1.9 µm away) and are small (approximate radius of 0.4 µm), as they have not yet accumulated large amounts of pericentrosomal material (Bienkowska and Cowan, 2012; Decker et al., 2011). To explore the effects of centrosome distance and location on AIR-1, we positioned centrosomes at a distance 1.9, 5 and 10 µm from the posterior pole, and measured the resulting AIR-1 signal by solving the diffusion equation (Eqn 2) on the embryo cross-section and extracting boundary values (see schematics in Fig. 2B and Fig. 3A). As expected, there is a significant decrease in the AIR-1 signal as the distance between the centrosomes and the cell boundary increases. Compared to a 1.9 µm distance, centrosomes 5 µm from the cortex exhibit a decrease of 50% in AIR-1, and those 10 µm from the cortex exhibit a further decrease of 50% (Fig. 3B).

Using these profiles of AIR-1 activity, we ran the cortex model forward in time to reach a steady state for polarization. For reference, in the absence of PAR proteins, the centrosomal signal induces a transient clearing of myosin from the posterior pole, and the myosin profile reverts back to a uniform state after the centrosomes move towards the cell equator (Gross et al., 2019; fig. 2E). Fig. 3C shows that our simulations reproduce the initial smaller-scale clearing of both myosin and ECT-2. Notably, the steady state ECT-2 accumulation in our simulation (which is steady because the centrosome positions are fixed) matches experimental data from early establishment phase (4 min before pseudo-cleavage), but diverges from the quasi-steady state that emerges once polarity is established (Fig. 3D), perhaps reflecting the influence of PAR proteins in the network (Gross et al., 2019). The predicted domain of ECT-2 clearance comprises ∼30% of the half-perimeter (∼20 µm on either side of the pole), which is in good agreement with experimental observations in PAR mutants (Gross et al., 2019; fig. 2E), suggesting that the model can effectively capture the transient posterior clearing induced by the AIR-1 signal when PAR proteins are absent.

In the model, changing the distance between the centrosomes and the cortex affects the quantitative values of the asymmetries and the time to reach them (Fig. 3E), but not the myosin peak location. As shown in the kymographs in Fig. 3C, the peak location, which correlates with the position of the pseudo-cleavage furrow (Reymann et al., 2016), is roughly the same across all conditions, because it is controlled by the hydrodynamic length scale (20% of the half-perimeter). Similar to previous observations (Bienkowska and Cowan, 2012; fig. 3F), the time for ‘symmetry breaking’, defined as a local (5%) clearance in myosin from the posterior pole, is tenfold higher when centrosomes are 10 µm away from the cortex than when they are 1 µm away. However, our model predicts an exponential scaling of the time to symmetry breaking at very large distances; this does not match the linear trend up to 10 µm that was previously reported (Bienkowska and Cowan, 2012; fig. 3F).

We have shown that long-range redistribution of ECT-2 is possible despite its short residence time. The driver of this redistribution is our assumption of indirect recruitment of ECT-2 by a longer-lived species, which is advected by cortical flows. To explore how this assumption influences polarization kinetics, in Fig. 4 (first two columns) we considered two alternative models for how ECT-2 might segregate during polarization. In the first model, we remove recruitment of ECT-2 (by setting k_ME=0 in Eqn 1a) and observe, at most, 10% clearing at the posterior. Indeed, with realistic flow speeds of at least 5 µm/min (Gross et al., 2019), molecules with residence time of 5 s can move at most 0.4 µm, which is less than 0.5% of the embryo perimeter, indicating that cortical flows alone are insufficient to reproduce typical ECT-2 clearance levels.

An alternative model, albeit inconsistent with measured rates of ECT-2 exchange, is one in which ECT-2 stably associates with longer-lived components on the cortex [for example through oligomerization (Illukkumbura et al., 2023)], so that the lifetime of ECT-2 is four times longer on the cortex. In this case, we observe contractile instabilities not seen in wild-type embryos. As in control conditions, a local maximum in ECT-2 initially forms near ∼30% embryo length. Because of the increased residence time, however, ECT-2 is also advected from the anterior into the peak, which creates a local minimum in myosin and drives the formation of a second, posterior peak. Thus, the rapid exchange of ECT-2 is important for stabilizing unidirectional flows during polarization. This outcome represents an experimentally testable prediction, whereby a version of ECT-2 with more stable binding kinetics should produce contractile instabilities.

The contractility cue that drives polarization establishment has been suggested to result from dynein-dependent removal of myosin (Chapa-Y-Lazo et al., 2020). Although experiments have shown that polarity establishment is dynein independent (Longhini and Glotzer, 2022), we nevertheless considered whether a transient reduction in contractility would be sufficient to trigger polarity establishment. In particular, we performed two simulations: one with the AIR-1 cue active for 5 min, and a second that lacks an AIR-1 cue but in which we acutely remove myosin and ECT-2 at the posterior pole. The resulting dynamics over 10 min are shown in Fig. 4. In both cases, transient cues or initial conditions ultimately relax to a steady state in which ECT-2 and myosin are nearly uniform and flow speeds are near zero.

The main difference between the transient AIR-1 cue and acute removal of myosin is in the intermediate dynamics. For a transient cue, the flow starts at zero and steadily increases over the first minute, peaking with a flow speed of 8 µm/min. Turning off the cue causes the flow speeds to slow below 2 µm/min in less than 1 min. By contrast, unloading myosin and ECT-2 from the posterior at t=0 triggers a similar set of flows towards the anterior, but the flows are maximal at t=1 min and then steadily decrease over time. Experimental data in PAR mutants show flows that reach a maximum velocity almost immediately after polarity triggering, but the magnitude (5 µm/min) of these flows persists throughout polarity establishment phase (3–4 min) (Gross et al., 2019; fig. 2G). Thus, this supports models in which AIR-1 triggers a local and persistent inhibition of contractility, similar to that reported previously (Gross et al., 2019).

The rapid recovery of ECT-2 and myosin in our transient cue simulation comes from the complete removal of the AIR-1 cue at t=5 min. Of course, the cue's removal is more gradual in vivo and corresponds to steady motion of the centrosomes away from the posterior cortex. In Fig. S8, we demonstrate that a much longer timescale of posterior recovery results when we model this (more realistic) case. The modeled recovery of myosin with relocalizing centrosomes compares well with previous results that measured myosin recovery in PAR-depleted embryos (Gross et al., 2019; fig. 2E).

The asymmetric accumulation of ECT-2 on the cortex during cytokinesis is sensitive to the position of the two centrosomes (Longhini and Glotzer, 2022). A plot of ECT-2 accumulation as a function of distance from the anterior or posterior pole closest to the centrosome reveals an ‘S-shaped’ curve; at short and long distances there is a plateau in the ECT-2 accumulation (Longhini and Glotzer, 2022; fig. 7A). By contrast, for distances in the range 10–20 µm, there is an ultra-sensitive dependence of the ECT-2 concentration on the proximity. Note that during cytokinesis, there are two mature centrosomes that contain significantly more AIR-1 (∼30 times; Fig. S2) than the immature centrosomes that trigger polarity establishment.

To test whether the model that accurately predicts polarization recapitulates the behavior of ECT-2 during cytokinesis, we modeled the behavior of ECT-2 and myosin using the centrosome positions measured previously (Longhini and Glotzer, 2022) and repeated in Fig. 5. We considered the case of wild-type embryos and three representative experimental conditions, two with asymmetric centrosome positions [dhc-1(RNAi) and zyg-9(b244)], and one [par-2(RNAi)] with symmetric centrosome positions. As in polarization, we solved the diffusion equation (Eqn 2) on the embryo cross-section with the given centrosome positions, then extract the AIR-1 signal (in arbitrary units) along the cell perimeter (Fig. S2). Despite a much (fourfold) stronger posterior AIR-1 signal in dhc-1(RNAi) embryos, the anteroposterior (A/P) asymmetry in ECT-2 accumulation only changes by ∼25% compared to zyg-1(b244); we used this observation to constrain the AIR-1 saturation level A_sat that appears in Eqn 1a (see Materials and Methods).

We used the computed AIR-1 signals as inputs to the same cortex model (Fig. 2A) that we parameterized under polarization conditions. Without changing any of the parameters, we simulated each experimental condition to steady state, then compared the profile of ECT-2 accumulation with the experimental data taken over a 50 s window beginning at cleavage furrow formation (multiplied by a constant to match the mean model concentration). For the four centrosome positions simulated, the model quantitatively reproduces the ECT-2 accumulation pattern (Fig. 5; see Fig. S9 for AIR-1 depletion with wild-type centrosome positions). In wild-type and par-2(RNAi) embryos, the model qualitatively reproduces the observed profile but tends to underestimate the A/P asymmetry. The shift in the central peak is likely due to centralspindlin-directed ECT-2 accumulation (Longhini and Glotzer, 2022), which is not accounted for in the model. In all experimental conditions, the fit to the data significantly degrades when we remove cortical flows and indirect recruitment (setting v=0 and k_ME=0 in Eqn 1a). Thus, as during polarization, cortical flows amplify weaker AIR-1 signals. Indeed, as shown in Fig. S9, the predicted ECT-2 profile without myosin flows and indirect recruitment matches experimental data with partial myosin depletion, indicating that the model correctly accounts for both the response to AIR-1 signals and flow-based amplification (albeit imperfectly in some cases).

Cell polarization in the C. elegans zygote is dependent on a centrosome-dependent signal that locally inhibits contractility. Similarly, cytokinesis can be influenced by aster positioning (White, 1985; Dechant and Glotzer, 2003; Severson and Bowerman, 2003; Munro et al., 2004). Only recently, however, has a better understanding of the nature of these cues emerged (Gross et al., 2019; Kapoor and Kotak, 2019; Klinkert et al., 2019; Zhao et al., 2019; Longhini and Glotzer, 2022; Manzi et al., 2024). In particular, a recent study (Longhini and Glotzer, 2022) showed that AIR-1, which accumulates and is presumably activated at centrosomes, causes inhibition of the RhoGEF ECT-2 (and, consequently, RHO-1 and myosin) at the proximal cortex. The goal of this study was to determine whether a minimal mathematical model could explain experimental observations in both polarization and cytokinesis.

We introduced a hybrid two- and one-dimensional model (Fig. 2), in which the centrosome positions are obtained from experimental data, and used it to obtain a cross-sectional profile of AIR-1 activation. Assuming diffusion of AIR-1 to the cortex (boundary of the cross-section), we then obtained the profile of AIR-1 as an input to a model of cortical contractility. This cortex model, which we defined on the cross sectional-boundary and consequently made one dimensional, included negative feedback of AIR-1 on ECT-2 accumulation, and positive feedback of myosin on ECT-2 accumulation, both through advection and recruitment by advected species. The coupling of the AIR-1 and cortex models allowed the dynamics of the active cortex to be dictated by the positions of the centrosome(s). We constrained the model using experimental observations (Longhini and Glotzer, 2022) and assuming modest (at most, twofold) effects on cortical ECT-2 by AIR-1 (negatively) and myosin (positively). The parameters arising from these constraints placed the model in a regime that can amplify small signals without yielding to contractile instabilities (Fig. S4).

This minimal model recapitulated many of the pertinent observations from polarization and cytokinesis. It produced transient polarization (in the absence of PAR proteins) with the observed flow speeds and protein ratios, and revealed the need for a persistent AIR-1 cue over time to maintain a polarized state in the absence of PAR reorganization. The picture that emerged from our experiments and modeling is a dynamic ECT-2 molecule that rapidly exchanges with the cortex, being preferentially recruited by longer-lived, flow-coupled molecules to sustain polarization. In this way, it reproduces other examples from cell biology in which stable configurations are mediated by transient interactions (Ladurner et al., 2016). The rapid exchange of ECT-2 and, to a lesser extent, myosin gives the model a quasi-steady nature; changes in the distribution of AIR-1 rapidly establish a new steady state. These dynamics are consistent with previous experimental results (Longhini and Glotzer, 2022; fig. 2A), which showed an acute response of cortical ECT-2 to spindle rocking during anaphase on a timescale of 10 s and could help to explain the rapid repositioning of the cleavage furrow in response to spindle displacements (Rappaport, 1985).

It is instructive to contrast the role of ECT-2 in generating cortical flows to that of the anterior PAR proteins, specifically PAR-3. Although it has long been known that reduced PAR-3 levels correlate with reductions in cortical flows, these changes have only recently been linked to the residence time of PAR-3 molecules on the membrane (Chang and Dickinson, 2022; Illukkumbura et al., 2023). In wild-type embryos, PAR-3 monomers (which have residence time less than 1 s) (Lang et al., 2024 preprint) oligomerize to stably bind the membrane (residence times on the order 100 s), which allows them to both create and be advected by cortical flows (Munro et al., 2004). Consequently, embryos with oligomerization-defective PAR-3 fail to polarize because of a lack of coupling to (weaker) cortical flows. Our analysis indicates that ECT-2 lives in a different part of the stability diagram of chemical–mechanical coupling. In wild-type embryos, the ECT-2 exchange kinetics [as measured by fluorescence recovery after photobleaching (FRAP)] are on the order of a few seconds, yet large-scale cortical flows are generated. Consequently, the model predicted that a longer ECT-2 residence time would produce hypercontractility, specifically a counterflow from the anterior end of the cell that prevents proper polarization. This result represents an important prediction that can be tested experimentally. The differences in stability behavior might be due to the coupling of ECT-2 and PAR-3 to myosin; whereas the role of PAR-3 in generating flows is likely indirect, ECT-2 directly generates flows by activating RHO-1 and myosin. Thus, more tunable control of contractility could be achieved by faster turnover rates in the latter case.

The question that underpins our work is how a persistent flow could affect the distribution of transiently bound proteins, independent of the underlying biochemical circuit in which they operate. For PAR-3, the typical measured diffusivities (0.01 µm²/s) and residence times (200 s) are insufficient to explain the measured asymmetries when only advection and diffusion are assumed to contribute to patterning (Illukkumbura et al., 2023; fig. 6H). This problem only worsens with ECT-2, which is apparently patterned by myosin-mediated flows, despite having a residence time on the order of a few seconds (Longhini and Glotzer, 2022). Here, we found that introducing recruitment of ECT-2 by a longer-lived species (or any species that is advected by and colocalizes with myosin) could result in similar patterning as cortical flows. Given that we previously showed that ECT-2 only segregates when its PH domain is intact (Longhini and Glotzer, 2022; fig. 5A), we speculate that myosin could advect factors that cause PIP2 to concentrate anteriorly (Scholze et al., 2018; Hirani et al., 2019; Nakayama et al., 2009), which would then contribute to ECT-2 recruitment. This model could explain how ECT-2 segregates while turning over rapidly.

Because the model parameters were tuned to match observations during polarization, it was most striking that the model also predicted patterns of cortical ECT-2 accumulation in cytokinesis across multiple experimental conditions solely by modifying centrosome size and positions (Dechant and Glotzer, 2003; Verbrugghe and White, 2007; Longhini and Glotzer, 2022). Similar to recent work on contractility during cytokinesis (Werner et al., 2024), we found that the accumulation of ECT-2 in different conditions could be better reproduced by incorporating mechanochemical feedback to amplify the AIR-1/ECT-2 signal. Historically, inhibition of cortical contractility by asters was thought to be microtubule mediated, given their proximity to the cortex (Dechant and Glotzer, 2003; Motegi et al., 2006). Yet experimental evidence (Klinkert et al., 2019; Kapoor and Kotak, 2019; Zhao et al., 2019; Longhini and Glotzer, 2022) and this mathematical model indicate that centrosome position and embryo geometry (i.e. the ability for AIR-1 to diffuse from the centrosomes to the cortex) serve as the primary determinants of polar relaxation. However, as cortical interactions with astral microtubules frequently control spindle and hence centrosome positioning (Grill et al., 2001; Schaefer et al., 2000; Tame et al., 2014), microtubules nevertheless play a role in this process.

In C. elegans embryos, the AIR-1 signal originates from the centrosomes, which are positioned in the interior of the three-dimensional cell (Fig. 1), while the contractile dynamics occur on the two-dimensional cell cortex (boundary). For our model, we consider a cross-section of the embryo, so that the AIR-1 dynamics occur in two dimensions, and the contractile dynamics occur on the one-dimensional boundary (Fig. 2B). The workflow is to first set the centrosome positions according to experimental data (Fig. 1), then solve a diffusion equation to obtain the AIR-1 profile at the cortex. This becomes an input to a set of one-dimensional reaction–diffusion–advection equations that treat the ECT-2/myosin relationship. In order to study how the cortex responds to the expected AIR-1 signals in polarization and cytokinesis, we assume that the centrosomes (and AIR-1 signal) are fixed (see Fig. S8 for a simulation that relaxes this assumption).

The binding/unbinding terms in the ECT-2 and myosin equations rely on the following major assumptions:

1. The variable E represents unphosphorylated, active ECT-2 bound to the cortex. We do not consider phosphorylation of ECT-2 in the cytoplasm and instead assume that the effective ECT-2 binding rate (which we fit to experimental data) represents the binding rate of unphosphorylated ECT-2. Consistent with this assumption, the negative flux in ECT-2 represents the combined rate of unbinding and phosphorylation, with the latter being proportional to the AIR-1 concentration, except at high AIR-1 when it saturates (see section ‘Coupled model of AIR-1'). Likewise, we assume that activation of ECT-2 by NOP-1 occurs uniformly throughout the cortex (Tse et al., 2012).

2. It was previously shown that ECT-2 has a small residence time at the cortex (on the order of a few seconds) (Longhini and Glotzer, 2022). Under these conditions, we show in Fig. S4 that the direct transport of ECT-2 by flows contributes negligibly to its steady-state profile. It was previously speculated that ECT-2 could be effectively ‘transported’ by associating with other components that are more stably bound to the cortex (Longhini and Glotzer, 2022). We incorporate this assumption into our model by assuming recruitment of ECT-2 by a species that is advected by cortical flows. For simplicity, in our equations, we assume that the concentration of this species is equal to that of myosin, thus giving the k_MEME_c term in the ECT-2 equation (Eqn 1a). In Fig. S7, we show that explicitly introducing a third species, which recruits ECT-2, into the equations gives similar patterns during polarization.

3. Based on previous work that demonstrated an important role for RhoGAP in setting the size of the anterior domain in polarizing embryos (Schonegg et al., 2007), combined with other work showing a nonlinear relationship between RhoGAP activity and Rho/myosin accumulation (Nishikawa et al., 2017; Michaux et al., 2018), we postulate nonlinear negative feedback in the myosin kinetics, with an inactivation rate proportional to M^k. As long as k>1, this term provides a way of controlling potential instabilities that arise in the simple active gel model (Eqn 1c) (Nishikawa et al., 2017). In the main text, we present results using k=2, but in Fig. S5 we show that model predictions are similar when k=3 instead.

As described in the ‘Details of model parameterization' section, we convert the model equations to dimensionless form, then fit the parameters using a combination of direct experimental measurements and inference based on other experimental data. The overall flow of the parameter fitting process goes as follows:

1. We first assign values to the diffusivities and unbinding rates of each component that come from direct experimental measurements (Gross et al., 2019; Goehring et al., 2011; Michaux et al., 2018).

2. Using previously imaged embryos with myosin and ECT-2 markers (Longhini and Glotzer, 2022; fig. 1), we measure the effective myosin and ECT-2 profiles during pseudo-cleavage (Fig. S3A–C), which we treat as a quasi-steady state.

3. Based on the quasi-steady myosin profile, we fit the velocity parameters in Eqn 1c to match bulk flow speeds in wild-type embryos (which are typically at most 10 µm/min) (Gross et al., 2019) (Fig. S3D).

4. To fit the parameters in the myosin equation (Eqn 1b), we impose the measured ECT-2 profile and adjust k_fb and k_EM until we match the experimentally measured myosin profile (Fig. S3E).

5. We use previous measurements in myosin-depleted embryos (Longhini and Glotzer, 2022; fig. 3A) to infer how AIR-1 impacts ECT-2 in the absence of myosin (i.e. to fit k_AE) (see Fig. S4A).

6. With all other parameters fixed, we increase the rate (k_ME) at which myosin (or a species associated with myosin) recruits ECT-2 until we reach the boundary of contractile instabilities. We choose a value for k_ME that sits near the boundary between the stable and unstable regime, without giving unstable behavior (Fig. S4). The result of this parameter fitting is that ∼40% of the ECT-2 that binds to the cortex is recruited by myosin-associated species.

Although it is tempting to equate the model's myosin-driven, contractile instabilities with pulsatile RHO-1/myosin excitability (Nishikawa et al., 2017), the latter are actually myosin independent (Michaux et al., 2018), which illustrates that the wild-type embryo does not sit in the model's fully unstable regime. In fact, the overall speed of bulk flows does not depend on pulsatility of RHO-1 (Michaux et al., 2018; fig. 7), affording a biological justification for modeling the simpler case in which these well-characterized pulses of RHO-1 activation are absent (or, more precisely, the case in which these randomly positioned pulses are averaged over many cross-sections to yield a steady signal). It was recently shown that knockdown of the downstream effectors of RHO-1 could cause contractile instabilities, which implies that the wild-type C. elegans embryo likely sits near, but not within, the unstable regime (Yao et al., 2022). In this way, a comparison can be drawn to the Xenopus model system, where normally quiescent oocytes can exhibit excitable dynamics through overexpression of ECT-2 and RGA-3/4 (Michaud et al., 2022; Chen et al., 2024).

The diffusion equation Eqn 2a also contains a basal rate of inactivation of AIR-1 (phosphatase activity). This introduces another parameter – the inactivation rate relative to the diffusion coefficient in the cytoplasm (⁠⁠, units µm⁻²). As shown in Fig. S1, low levels of phosphatase activity give high global AIR-1 levels, which translate to low ECT-2 levels everywhere. Such levels were shown to block pseudo-cleavage in centralspindlin-independent cytokinesis, owing to low contractility (Afshar et al., 2010; Kotak et al., 2016). We choose the phosphatase activity level such that centrosomes close to the posterior pole (in polarization) negligibly impact the anterior domain (<1% of the posterior concentration) (see Fig. S1).

To infer the profile of AIR-1 during cytokinesis for the four experimental conditions shown in Fig. 5, we use the corresponding centrosome positions, set ⁠, and solve Eqn 2. The resulting AIR-1 profile along the embryo perimeter is shown in Fig. S2. Despite large (fourfold) differences in posterior zyg-9 and dhc-1 AIR-1 signal, the A/P asymmetry between the two conditions increases by only 25% (Fig. 5). Because of this, we conclude that the saturation level of AIR-1 activity, A_sat in Eqn 1a, must lie near the zyg-9 posterior levels. As such, we set A_sat=0.25.

We use a standard first-order finite element method to solve Eqn 2a. In brief, the elliptical domain of the embryo is meshed into nodes and triangles (Persson and Strang, 2004), and the finite element matrix equation becomes ⁠, where M is the mass matrix and K is the stiffness matrix for finite elements, which are assembled using standard techniques (Gockenbach, 2006). Solving for a(x) everywhere gives a profile on the embryo perimeter (cortex), which we substitute into Eqn 1 as the one-dimensional profile A(x), using linear interpolation to map from the (irregular) finite element boundary nodes to a regular grid. Once it reaches the cortex, AIR-1 inhibits ECT-2 by increasing its cortical dissociation rate, in correspondence with experimental data in myosin-depleted embryos (Longhini and Glotzer, 2022; fig. 3A).

We use standard numerical methods to solve Eqn 3. We discretize the one-dimensional domain at N points with spacing 1/Δx, and define the centered differentiation matrix D and standard three-point Laplacian differentiation matrix Δ. Given the myosin profile at time step n, we first compute the velocity v⁽ⁿ⁾ by solving ⁠. Once the velocity is computed, the ECT-2 and myosin equations are solved by combining a first-order upwind finite volume scheme for the advection terms (Hundsdorfer et al., 2003; sec. 1.4) with implicit treatment of the diffusion terms (using the standard three-point Laplacian). The reaction terms are all treated explicitly, and time stepping is first-order accurate.

Parameter estimation for the dimensionless equations (Eqn 3) can be performed in three steps: first, we directly assign quantities that have already been measured experimentally. Second, we freeze the ECT-2 profile and assign parameters to the myosin equation (Eqn 3b) to match experimental data. Third, we choose the parameters in the ECT-2 equation (Eqn 3a) based on stability considerations.

Some of the parameters can be determined directly from experimental measurements, as follows:

1. The embryo cross-section is an ellipse with approximate radii 27 µm and 15 µm, which gives a perimeter L=134 µm (Goehring et al., 2011).

2. The variable is the hydrodynamic length scale. In dimensional units, this was measured to be ℓ≈13 µm (Mayer et al., 2010), which means that in Eqn 3c (10% domain perimeter).

3. The myosin-bound lifetime is ∼15 s, according to measurements in the anterior of wild-type embryos, or in Par mutant embryos, which do not polarize (Gross et al., 2019; fig. 1B). Because we typically model polarity establishment, where embryos are initially unpolarized, we set ⁠.

4. We assume that all species have a dimensional diffusion coefficient D_E/M=0.1 µm²/s (Goehring et al., 2011; Gross et al., 2019; Robin et al., 2014). Rescaling length by L and time by gives a dimensionless coefficient ⁠. This dimensionless coefficient is sufficiently small as to render diffusion relatively unimportant in shaping the concentration fields. If we assume instead, for instance, that myosin cannot diffuse in the membrane, while ECT-2 has a tenfold larger diffusion coefficient, the steady-state profiles of ECT-2 and myosin are changed by, at most, 5% (see Fig. S6).

5. The ECT-2 lifetime was measured using FRAP to be on the order of a few seconds (Longhini and Glotzer, 2022; fig. 3D). In cytokinesis, we set ⁠, for a 3 s lifetime. Rescaling gives ⁠. The data show slightly faster recovery during polarization, so we increase by 20% for those simulations.

To infer the myosin parameters ⁠, K_EM, k and K_fb, our strategy is to impose the ECT-2 profile measured experimentally at the quasi-steady state of pseudo-cleavage, then solve for the myosin parameters required to match the experimental myosin profile. As shown in Fig. S3, we utilize previously imaged embryos with ECT-2 and myosin reporters (Longhini and Glotzer, 2022; fig. 1) for this purpose. For each set of images, we use MATLAB's built-in algorithms to segment the embryo boundary and compute an arclength parameterization of the boundary curve. Following this, we use Fourier filtering to filter the result and obtain a smoothed boundary with a normal vector at each point (see Fig. S3A,B). For each point on the arclength curve, we draw a 30 pixel (3 µm) line inward and compute the maximum intensity along this line. Averaging over all frames in which pseudo-cleavage is present, then repeating for three embryos to generate error bars, gives the curves shown in Fig. S3C. Fitting these curves with a Fourier interpolant then gives smoothed representations that can be used for fitting. We note that the punctate myosin profile during establishment phase can somewhat confound the myosin measurements, but our data show a clear general trend that is captured by the Fourier interpolant.

To transition the target curves to model inputs, we scale by the expected amount of bound protein. Experimental data (Longhini and Glotzer, 2022; fig. 1) show that ∼10% of ECT-2 is bound to the cortex. This estimate is based on Fig. S3C, which shows the average ECT-2 intensity in the cortical region to be ∼1.5 times that in the cytoplasm. If the embryo is an ellipsoid with radii 27×15×15 µm, and the cortex has thickness 400 nm, then the cortex is 6.7% of the total volume. Multiplying by 1.5 gives 10% of the total ECT-2 bound. Although the specific number (10%) is of little consequence to our model, the relative abundance of ECT-2 in the cytoplasm demonstrates that cytoplasmic depletion will not play a role in the dynamics. A similar set of data (Gross et al., 2019; Figs S2 and S3) show that ∼30% of myosin is bound to the cortex. With these parameters, we scale the smoothed curves to obtain target curves for the model (the imposed ECT-2 curve is the experimentally measured curve, but scaled to have mean 0.1).

There are now four parameters remaining: K_AE, A_sat, and K_ME. The saturating AIR-1 threshold A_sat cannot be fit from polarization conditions because AIR-1 signals are low; as such, we set A_sat=0.25 according to cytokinesis measurements (see Materials and Methods). The parameter K_AE can be set by simulating AIR-1 activity in the absence of myosin. In myosin-depleted embryos, it was previously shown that the ECT-2 asymmetry during early polarization is 1.2 (Longhini and Glotzer, 2022; fig. 3A), which implies that the local AIR-1 activity induces a 20% depletion of AIR-1 on the posterior. We therefore infer K_AE by simulating polarization (as in Fig. 3) with no myosin activity (setting K_ME=0 and ⁠). As shown in Fig. S4A (black lines), the steady state ECT-2 asymmetry is 1.2 when K_AE=1.3.

With the AIR-1 parameters set, there is effectively one parameter remaining: the strength of indirect recruitment of myosin on ECT-2 (there is also K_E^on, the basal ECT-2 binding rate, but this is set to maintain 10% bound ECT-2 on the cortex). We take a systematic approach to setting this parameter: as shown in Fig. S4, setting K_ME=0, so that the only interaction of ECT-2 with myosin comes via flows, gives a negligible change in the ECT-2 profile from the AIR-1-only case (no myosin). Thus, indirect recruitment must be responsible for shaping the ECT-2 profile. To fit a value, we increase K_ME until we trigger locally oscillatory patterns of ECT-2 accumulation. As shown in Fig. S4, these patterns occur upon the transition to the regime where the uniform ECT-2 profile is linearly unstable (see the next section for an analysis). The values we use are K_ME=2.5 (for k=2) or K_ME=1.5 (for k=3).

The authors thank Ed Munro (University of Chicago) for helpful input.

The peer review history is available online at https://journals.biologists.com/jcs/lookup/doi/10.1242/jcs.264093.reviewer-comments.pdf