Enucleation of the C. elegans embryo revealed dynein-dependent spacing between microtubule asters

Establishment of enucleated C. elegans embryos by genetic manipulation

To characterize the spacing activity between the two centrosomes, an experiment was designed to remove the chromosomes from the C. elegans embryo (“enucleated embryo”). Enucleated C. elegans embryos were produced in classic experiments by Schierenberg and Wood, where the nuclei are removed by penetration of the eggshell using laser microsurgery, followed by pressing the cytoplasm to push the nucleus out of the eggshell (Schierenberg & Wood, 1985). This method often removes the centrosome together with the nucleus and is unsuitable for analyzing centrosome behavior.

To create an enucleated embryo with centrosomes, paternal and maternal chromosomes were removed using emb-27 mutant sperm (Sadler & Shakes, 2000), and by knocking down the klp-18 gene (Segbert et al, 2003) (Fig S1). emb-27 encodes a subunit of an anaphase-promoting complex. This mutation causes chromosome segregation defects and produces centriole-containing fertilization-competent enucleated sperm (Sadler & Shakes, 2000; Kondo & Kimura, 2018 Preprint). klp-18, a member of the kinesin family, is required for oocyte meiosis. The klp-18 knockdown oocyte occasionally extrudes all chromosomes into the polar body, resulting in embryos without maternal chromosomes. By mating the worms with emb-27 mutant sperms and klp-18 knockdown oocytes, we expected to obtain enucleated embryos to characterize the spacing activity of the centrosomes independent of the nucleus and spindle.

In this study, C. elegans strains were used in which the centrosomes (γ-tubulin), chromosomes (histone H2B), and cell membranes (PH^PLCδ1) were visualized using GFP (Fig 1 and Table S1). In control one-cell stage embryos, sperm- and oocyte-derived pronuclei appeared after fertilization (Fig 1A and Video 1 [left]). The two centrosomes associated with the sperm pronucleus move toward the cell center and meet the oocyte pronucleus before the first cytokinesis. When emb-27(g48ts) males were mated with control hermaphrodites, the sperm pronucleus was absent, as previously reported (Sadler & Shakes, 2000) (Fig 1B and Video 1 [middle]). The centrosomes migrate toward the cell center and meet the oocyte pronucleus. The emb-27 mutants affected the number of centrosomes supplied by the sperm (Kondo & Kimura, 2018 Preprint, 2019). Consequently, the presence of 1–4 centrosomes was observed in the one-cell stage emb-27 mutant and in the enucleated embryo. Oocyte pronuclei were not detected in klp-18 (RNAi) embryos, as previously reported (Segbert et al, 2003) (Fig 1C and Video 1 [right]). We designed an experiment to obtain embryos without chromosomes by mating emb-27(g48ts) males with klp-18 (RNAi) hermaphrodites. Enucleated embryos were successfully obtained using this experimental setup (Fig 1D and Video 2). No sperm- or oocyte-derived pronuclei or chromosomes were detected in the embryonic cells at subsequent stages. Chromosomal signals from the polar bodies were detected outside the cytoplasm.

Centrosomes moved dynamically in enucleated embryos, which was the main topic of this study. This indicates that centrosomes can move without requiring nuclei or chromosomes. Interestingly, the centrosomes duplicated periodically for multiple rounds, which appeared to correspond to the cell cycle. Cytokinesis was impaired for at least several cell cycles (Fig 1D), possibly because of chromosomal loss (Bringmann & Hyman, 2005). Most importantly, in this study, the positions of these centrosomes did not overlap, but were spread throughout the cell (Fig 1D), suggesting the existence of spacing activity. Therefore, the enucleated C. elegans embryo is suitable for analyzing the centrosome-spacing mechanism, which is independent of the nucleus and spindles. In summary, sister and non-sister centrosomes share a common cytoplasm and move dynamically in an enucleated embryo.

A repulsive spacing between sister and non-sister centrosomes was observed in the enucleated embryo

To characterize the force acting between the centrosomes, the change in distance between the centrosomes over time was quantified. In the present study, we focused on the time window corresponding to the two-cell stage in control embryos with nuclei (Fig 2A and B). In the enucleated embryo, the cytokinesis failed; therefore, the cytoplasm did not divide into two in the “two-cell stage.” At this stage, four (or more, depending on the number of centrosomes in the one-cell stage, as explained in the previous section) centrosomes of the two sister and non-sister pairs coexist in the common cytoplasm. We focused on this stage because it was the earliest at which potential interactions between non-sister pairs of centrosomes could be tracked. We set the time zero of the time window when we first detected two discrete centrosome (γ-tubulin::GFP) spots for the sister pair after the second centrosome duplication (Fig 2A). The time window ended when the signal of the spot became too weak to be identified or when the spot was duplicated into two in the subsequent round of centrosome duplication.

The distances between sister pairs of centrosomes in enucleated embryos, controls (i.e., embryos with nuclei), and zyg-12 (RNAi) embryos were compared (Fig 2B and C). At early time points in the time window in the control embryos, sister centrosomes slid along the nuclear surface to position themselves at opposite poles of the nucleus (Fig 2C black, and Video 3), as previously reported (Gönczy et al, 1999). In the enucleated embryo, the sister centrosomes separated at a speed similar to that in the control embryos, indicating that the spacing was independent of the nucleus (Fig 2C red and Video 4). The nucleus-independent spacing was consistent with previous observations for zyg-12 (RNAi), in which the centrosomes were not associated with the nuclei (Malone et al, 2003) (Fig 2C blue and Video 5). At later time points, for enucleated embryos, unlike the control embryos, the separation of sister centrosomes did not pause at the distance of the nuclear diameter (∼10 μm), but continued to increase. This behavior can be explained by the loss of association with the nucleus. In zyg-12 (RNAi) embryos, the separation of the centrosomes slowed as the centrosomes formed a mitotic spindle, until the centrosomes separated again during anaphase. In conclusion, centrosomes have an intrinsic ability to separate from their sister centrosomes, independent of their sliding activity along the nuclear surface. In control embryos, the nucleus tethered the sister centrosomes. Therefore, the centrosomes do not separate further until nuclear envelope breakdown (NEBD).

An advantage of enucleated embryos is that the interactions between non-sister centrosomes that share the cytoplasm can be characterized. Notably, the distance between non-sister centrosomes was always longer than that between sister centrosomes (Figs 2D and S2). Although the centrosomes moved dynamically within the embryo, the distances between non-sisters did not become shorter than the minimal distance between sister pairs at each time point. The results indicated that similar spacing activity existed between sister and non-sister pairs of centrosomes. Therefore, repulsive spacing activity is intrinsic to the centrosome.

Dynein-dependent pulling forces were responsible for the timely spacing activity

To obtain insights into the mechanism of centrosome spacing, we searched for the genes involved in this activity in enucleated embryos. Cortical pulling forces that pull microtubules from force generators located in the cell cortex contribute to centrosome separation in Drosophila (Cytrynbaum et al, 2003). In C. elegans, the knockdown of genes required to generate the cortical pulling force (e.g., gpr-1/2 [RNAi]) impairs spacing in zyg-12 knockdown embryos (De Simone et al, 2016). We knocked down gpr-1/2 in an enucleated embryo to inhibit the cortical pulling force and found that the distance between the centrosomes was shortened (Fig 3A and B orange, and Video 6). A significant difference (P < 0.01 10-min, Wilcoxon rank-sum test) was observed between enucleated embryos and enucleated embryos with gpr-1/2 (RNAi) in the distance between sister centrosomes. The distances for enucleated embryos and enucleated embryos with gpr-1/2 (RNAi) were 15.8 ± 2.5 and 6.9 ± 3.2 μm (mean ± SD), n = 10 and 13, respectively, at the 10-min timing when the distance between centrosomes in enucleated embryos reached near saturation. These results indicated that the cortical pulling force mediated the spacing activity. Centrosomes were partially separated from the enucleated gpr-1/2 (RNAi) embryos. This result is consistent with the partial separation in zyg-12; goa-1/gpa-16 (RNAi) embryos (De Simone et al, 2016). It is unlikely that the remaining separation in enucleated gpr-1/2 (RNAi) was because of the incomplete knockdown of the GPR-1/2 protein. Under our gpr-1/2 (RNAi) conditions, P0 cells were divided equally to produce daughter cells of the same volume. This occurred when the protein level of GPR-1/2 was reduced to almost 0%, but not when the protein level was ∼30% (Pecreaux et al, 2006). These results indicated that factors other than the cortical pulling force were involved in centrosome spacing. The following observations support requirements other than the cortical pulling force.

When dhc-1 was knocked down in the enucleated embryo, spacing was almost completely blocked for ∼20 min, corresponding to the duration of the cell cycle in the control embryo (Fig 3A and B, green, and Video 7). dhc-1 encodes the heavy-chain subunit of cytoplasmic dynein and is responsible for all microtubule-pulling forces in C. elegans embryos (Gönczy et al, 1999; Torisawa & Kimura, 2020). A significant difference was found between enucleated embryos treated with gpr-1/2 (RNAi) and those treated with dhc-1 (RNAi) (P < 0.01 10-min, Wilcoxon rank-sum test). The distance for enucleated embryos with dhc-1 (RNAi) was 3.1 ± 0.4 μm, n = 12. Here, we focused on a near-complete block of spacing for the first 20 min under dhc-1 (RNAi) conditions. Notably, we observed an apparent centrosome movement after 20 min. The mechanism of the latter movement is investigated in the final section of this study. Dynein inhibition impaired timely spacing activity, which occurred almost completely within 20 min. Therefore, these results suggest that factors other than the cortical pulling force, which are dependent on dynein, contribute to the spacing.

The cytoplasmic pulling force depends on dhc-1 but not on gpr-1/2, and drives the centration of centrosomes and pronuclei (Kimura & Onami, 2005, 2007; Kimura & Kimura, 2011). We expected the cytoplasmic pulling force to contribute to this spacing. To test this possibility, the dyrb-1 and gpr-1/2 genes were simultaneously knocked down in enucleated embryos. dyrb-1 encodes a roadblock subunit of the dynein complex, which is not essential for dynein motor activity but is required for organelle transport and centration of the centrosome; therefore, it is necessary for the cytoplasmic pulling force (Kimura & Kimura, 2011). In gpr-1/2; dyrb-1 (RNAi)-enucleated embryos, in which cytoplasmic- and cortical-pulling forces were impaired, centrosome spacing was severely defective, as observed in dhc-1 (RNAi)-enucleated embryos (Fig 3A and B magenta, and Video 8). A significant difference was found between embryos enucleated with gpr-1/2 (RNAi) and embryos enucleated with gpr-1/2; dyrb-1 (RNAi) in the distance between sister centrosomes (P < 0.01 10-min, Wilcoxon rank-sum test). The distance for enucleated embryos with gpr-1/2; dyrb-1 (RNAi) was 3.9 ± 1.1 μm, n = 12. The enucleated embryos with dyrb-1 (RNAi) repressed the spacing compared with the control (P < 0.05 at 10-min, Wilcoxon rank-sum test), but not as severely as gpr-1/2; dyrb-1 (RNAi) (Fig 3A and B blue, and Video 9). The distance for enucleated embryos with dyrb-1 (RNAi) was 12.3 ± 3.5 μm, n = 11. These results showed that the cortical and cytoplasmic pulling forces were sufficient to provide spacing between the centrosomes in the initial 20 min of the two-cell stage.

In search for a dynein-dependent mechanism for the spacing between centrosomes

In humans, Drosophila, and Xenopus, plus-end–directed motors are involved in the centrosome spacing of the mitotic spindle (see Introduction section) and are considered to be involved in chromosome-independent spacing by acting on antiparallel microtubules emanating from the two centrosomes (Deshpande et al, 2022; de-Carvalho et al, 2022). In contrast, in this study, minus-end–directed motor dynein provided the necessary force for centrosome spacing in C. elegans embryos. Therefore, we aimed to determine how pulling forces mediate the repulsive interactions between centrosomes.

Analogous to the finding of the antiparallel pushing mechanism for spacing activity in Drosophila and Xenopus from the mechanisms of spindle elongation, we speculated that we could obtain clues for the spacing mechanism in C. elegans from the mechanisms proposed for spindle elongation in the species. Spindle elongation in C. elegans is dynein-dependent. Most proposed models for spindle elongation in C. elegans embryo assume that the distribution of microtubules differs between the two centrosomes (Grill et al, 2001; Hara & Kimura, 2009). A study by Farhadifar et al proposed a mechanism called “the stoichiometric model of cortical pulling forces,” for the spindle elongation in the C. elegans embryo that is independent on the distribution of microtubules (Farhadifar et al, 2020). In this model, the two centrosomes of the spindle poles compete for force generators in the cell cortex to pull. “Stoichiometric” means that one force generator can pull only the nearest centrosome. This model ensures that anterior and posterior cortexes pull only the anterior and posterior centrosomes, respectively. In the present study, we applied a stoichiometric model to explain the spacing activity of four or more centrosomes in enucleated embryos.

Quantification of the length distribution of the microtubules in the C. elegans embryo

The original stoichiometric model (Farhadifar et al, 2020) assumed long and stable microtubules (i.e., exponential decay with a characteristic length of 20 μm). The length distribution of the microtubules is critical for stoichiometric models; thus, we experimentally quantified the length distribution of the microtubules (Figs 4 and S3). We assumed that the brightness intensity of β-tubulin::GFP signal above its cytoplasmic average was proportional to the number of microtubules and quantified the value (Fig 4A and B). The quantified signal intensity fitted well with a Weibull distribution of S(l) = S₀ × EXP[−{(l − l₀)/ξ}^P], where S(l) is the signal intensity of microtubules with their length over l, l₀ is the radius of the centrosome, S₀ is the intensity at the surface of the centrosome, ξ is the length scale, and P is a parameter for how the distribution is affected by the length (Fig 4B and see the Materials and Methods section). The estimated distribution of microtubule lengths did not change dramatically during the observation period (Figs 4C and S3) or among different samples (Fig S3). Therefore, we calculated the average distribution of all samples at all the time points (Figs 4C and D and S3). Our fitting of the average distribution to the Weibull distribution revealed l₀ = 1.6 μm, ξ = 2.3 μm, and P = 0.79.

Stoichiometric model of cortical and cytoplasmic pulling forces as a mechanism for the repulsive spacing between centrosomes in the C. elegans embryo

Our analyses revealed that both cortical and cytoplasmic pulling forces act on the spacing between centrosomes (Fig 3). Therefore, we added cytoplasmic pulling forces (Fig 5A and B, and Table S2) to the stoichiometric model of the cortical pulling force by Farhadifar et al (2020). This modified version of the stoichiometric model of the cortical and cytoplasmic pulling forces reproduced the major features of our experimental measurements (Fig 5C and D). In the three-dimensional space (ellipsoid), we uniformly placed force generators in the cytoplasm and a thin layer of the cortex (Fig 5B), similar to our previous simulation (Kondo & Kimura, 2019). The centrosomes were positioned at their initial positions in representative experiments (Table S3). The simulation was conducted by iterating the microtubule growth processes, summing the pulling forces calculated as in the original stoichiometric model (Farhadifar et al, 2020) but adding the contributions from the cytoplasmic force generators, and moving the centrosomes.

For the simulation parameters, we followed the original stoichiometric model for cortical pulling forces (Farhadifar et al, 2020). See the Materials and Methods section and Table S2 for details on the parameter values. The simulations (Fig 5D) reproduced centrosome spacing of similar magnitudes and increased rates for the control, dyrb-1 (RNAi)-, gpr-1/2 (RNAi)-, and dhc-1 (RNAi)-enucleated embryos, as shown in Fig 3B. The trajectories of the centrosomes inside the cells were similar between the simulations and experiments (Figs 5C and S4). The trajectories fluctuated more during the experiments, possibly because of random fluctuations in the cytoplasm (Guo et al, 2014). Overall, the numerical simulation results supported the feasibility of the stoichiometric model of the cortical and cytoplasmic pulling forces.

In addition, the stoichiometric model of the cortical and cytoplasmic pulling forces reproduced the separation and centration of the centrosomes (Fig 5E) associated with the sperm-derived pronucleus of the one-cell stage embryo (Albertson, 1984; Gönczy et al, 1999). For the simulation with the nucleus, the two centrosomes were connected with an elastic bar equal to the length of the nuclear diameter (8 μm). This setup allows the centrosomes to attract each other when the distance between them exceeds the nuclear diameter. This result supports the feasibility of the model even for cells with nuclei.

Application of the experimentally observed length distribution of the microtubules (Fig 4) to the original stoichiometric model resulted in the elongation of the spindle for almost as long as the cell length (Fig S5). This is likely because the microtubules in the cell are shorter than the distribution assumed in the original simulation by Farhadifar et al (2020). The addition of cytoplasmic pulling forces to the original stoichiometric model enabled reasonable elongation with the microtubule lengths consistent with the experiment (Fig S5), indicating that the stoichiometric model of cortical and cytoplasmic pulling forces accounts for spindle elongation, in addition to the separation and centration of centrosomes in normal embryos with nuclei and chromosomes.

Myosin-dependent movements of the centrosomes in the C. elegans embryo

In this study, large movement of centrosomes was observed ∼20 min after the detection of the two centrosomes in dhc-1 (RNAi)-enucleated embryos (Fig 3B green, Video 7). In dhc-1 (RNAi) embryos with nuclei, the centrosomes did not separate during interphase at the one-cell stage, and a small spindle-like structure was formed near the cortex, suggesting that dynein was responsible for all centrosome movements until the spindle formation stage in normal embryos (Fig 6A). We noticed that the centrosomes moved over a large distance in dhc-1 (RNAi) embryos with nuclei in the later cell cycle phase, indicating that large centrosome movement independent of dhc-1 was not specific to enucleated embryos (Video 7 and Video 10 [left]).

A large movement occurred near the time of cytokinesis in dhc-1 (RNAi) embryos; therefore, we speculated the involvement of actomyosin, which drives the constriction of the contractile ring and the accompanying cytoplasmic flow toward the equatorial plane (Bray & White, 1988; Khaliullin et al, 2018). Although cytokinesis does not occur at this stage in enucleated embryos, the cytoplasmic flow may move the centrosomes. Unfortunately, an enucleated embryo cannot be obtained upon knockdown of nmy-2, which encodes non-muscle myosin II and is required for cytoplasmic flow (Munro et al, 2004). This was possibly because NMY-2 is required for polar body extrusion (Dorn et al, 2010) and nmy-2 (RNAi) impairs oocyte enucleation. To this end, dhc-1 and nmy-2 were simultaneously knocked down in the nucleated embryos. We observed an impairment in dynein-independent centrosome movements, indicating that large movements were driven by cytoplasmic flow (Fig 6A and B, and Video 10 [right]). Differences in the distance between centrosomes 5 min after NEBD were tested using the Wilcoxon rank-sum test between dhc-1 (RNAi) and dhc-1; nmy-2 (RNAi) experiments (P < 0.01). The mean values for dhc-1 (RNAi) and dhc-1; nmy-2 (RNAi) were 14.9 ± 2.0 and 3.9 ± 0.9 μm, n = 6 and 5, respectively. In conclusion, cytoplasmic flow drives centrosome movement during late mitosis, which occurs when dynein is inhibited.

Enucleation of C. elegans embryos

Chromosomes are essential for cellular functions because they carry genetic information and constitute a core component of intracellular organelles. Before NEBD, chromosomes form a cell nucleus, whereas after NEBD, they form a mitotic spindle. Historically, chromosome removal from cells has enabled important modeling in cell biology studies (Goldman & Pollack, 1974). In addition to the mechanical removal of the nucleus (e.g., by centrifugation or microneedles), genetic manipulation can also be used to investigate the function of nuclei. The gnu mutant of Drosophila revealed a semi-enucleated system in Drosophila embryos, in which the nucleus did not divide but formed one giant nucleus, whereas the centrosomes were continually duplicated and separated (Freeman et al, 1986). In gnu mutants, centrosome separation was almost entirely independent of the presence of nuclei (de-Carvalho et al, 2022). In the present study, we established a genetic method to obtain enucleated C. elegans embryos by combining previously established methods to remove chromosomes from the sperm (Sadler & Shakes, 2000) and oocytes (Segbert et al, 2003) (Fig S1). Our established method produces enucleated embryos in a noninvasive manner, whereas classical enucleated C. elegans embryos must be obtained by penetrating the eggshell using laser microsurgery (Schierenberg & Wood, 1985). Unlike the Drosophila gnu mutant, our method completely removed chromosomes from embryos. We expect our established method to be applied in various studies and not limited to centrosome research because C. elegans embryos are widely used model organisms.

Chromosome-independent and dynein-dependent spacing between the centrosomes

Using an enucleated C. elegans embryo, we demonstrated that spacing activity, independent of the chromosome, exists in the C. elegans embryo before and after NEBD and between sister and non-sister centrosomes. Nucleus-independent spacing between sister centrosomes before NEBD has been previously observed in the zyg-12 mutant, in which the association between the nucleus and centrosomes is impaired (Malone et al, 2003; De Simone et al, 2016). In contrast, as the mitotic spindle forms and most cells divide in zyg-12 mutants, the chromosome-independent interaction after NEBD and that for non-sister centrosomes have not been previously addressed. In addition, even in zyg-12 mutants, nuclei remain in the cytoplasm between the centrosomes during interphase, which acts as an obstacle to microtubule growth in these regions. Therefore, the enucleated embryo is a good model system for studying the interactions between centrosomes in an intracellular space without physical obstacles.

We demonstrated that the spacing between sister and non-sister centrosomes until cytokinesis completely impaired the knockdown of dynein (dhc-1). This result indicated that kinesin-dependent pushing between centrosomes, as revealed in Drosophila and Xenopus (Telley et al, 2012; Nguyen et al, 2014, 2018), did not occur in C. elegans embryo. Our observations in C. elegans is consistent with previous research showing that centrosomes rarely moved in dhc-1 (RNAi) embryos (Gönczy et al, 1999) and that the molecules involved in pushing did not impair the elongation of the mitotic spindle in C. elegans embryo (Powers et al, 2004; Saunders et al, 2007; Lee et al, 2015). Our study using enucleated embryos demonstrates the direct requirement of dynein (dhc-1) for centrosome spacing.

Stoichiometric model of cortical and cytoplasmic pulling forces

The major involvement of dynein indicates that the centrosome spacing in C. elegans is driven by a pulling force outside the centrosome pairs. The separation of centrosomes by outward pulling forces commonly occurs before NEBD (Gönczy et al, 1999; Cytrynbaum et al, 2003; De Simone et al, 2016) or spindle elongation (anaphase B) (Grill et al, 2001). However, it has not yet been determined why the two adjacent centrosomes are pulled in opposite directions. In mitotic spindles, the spindle itself exhibits bipolarity, and this difference is established during spindle formation. In other cases, the cell nucleus may amplify the asymmetry by positioning itself between the centrosomes and obstructing the growth of microtubules toward the nucleus (Donoughe et al, 2022). Cortical flow has also been proposed to separate the centrosomes (De Simone et al, 2016); however, the mechanism that ensures centrosome movement in the opposite direction has not yet been clarified.

Here, we extend the idea of the stoichiometric model of cortical pulling forces proposed by Farhadifar et al (2020) for spindle elongation to explain the spacing activity independent of nuclei and spindles. An important modification was the addition of a cytoplasmic pulling force. The importance of the cytoplasmic pulling force was demonstrated in this study using two independent supports. First, DYRB-1, a molecule involved in the generation of the cytoplasmic pulling force, was required for centrosome separation after knocking down the cortical pulling force via gpr-1/2 (RNAi) (Fig 3). The numerical models with the cytoplasmic pulling force, but not the models with only the cortical pulling force, accounted for the centrosome spacing, and the centrosome separation and spindle elongation, with the experimentally obtained distribution of microtubule length (Figs 4 and S5). The cytoplasmic pulling force contributes to the centrosome spacing even when the microtubules are short and not reaching the cortex. Therefore, the cytoplasmic pulling force may be important for the rapid placement of centrosomes during development, and for cell types with asters that are formed only of short microtubules.

A stoichiometric model is proposed as the underlying mechanism for spindle elongation (Farhadifar et al, 2020). The idea that a force generator can pull only one microtubule among multiple microtubules, potentially reaching the force generator, is in line with a previously proposed force-generator–limited model (Grill et al, 2003; Grill & Hyman, 2005). We noticed that during spindle elongation, the microtubules from the two spindle poles rarely overlap. This observation indicated that the competition assumed in the stoichiometric model may not be critical for spindle elongation. In contrast, in enucleated embryos, there was no apparent bias in the direction of microtubule elongation. We demonstrate that a stoichiometric model of cortical and cytoplasmic pulling forces is critical for centrosome spacing in C. elegans embryos. Because we demonstrated the existence of spacing activity and a stoichiometric model of cortical and cytoplasmic pulling forces as the underlying mechanisms, the model ensured spindle elongation even without the formation of mitotic spindles (Fig S5).

The stoichiometric model of cortical and cytoplasmic pulling forces corroborated the spacing dynamics of centrosome pairs in control and gene knockdown-enucleated embryos (Figs 5 and S4). Moreover, the model explained the separation and centering of the normal embryo with the nucleus when the centrosomes were tethered to the nuclear surface (Fig 5E). Therefore, the stoichiometric model of cortical and cytoplasmic pulling forces is promising for centrosome spacing in C. elegans embryos and can be applied to other cell types and species.

A similar pulling force-based mechanism has been proposed for the spacing of nuclei in the syncytium embryo of the cricket Gryllus bimaculatus (Donoughe et al, 2022), despite the observation of a pushing-based mechanism for similar nuclear spacing in Drosophila syncytium embryos (Deshpande et al, 2022). The proposed pulling-based mechanism in crickets supports the generalizability of the mechanism proposed in this study to C. elegans embryos. However, further studies are necessary to clarify pulling-based mechanisms in crickets. The involvement of dynein and other pulling force generators in crickets has not yet been demonstrated. The current argument against the pushing-based mechanism in crickets is that numerical simulations (Dutta et al, 2019) do not reflect certain aspects of nuclear migration (Donoughe et al, 2022). It is possible that kinesin-5, PRC1, or kinesin-4 is required for spacing in crickets. The pulling-based model proposed for crickets (Donoughe et al, 2022) is similar to that used in the present study. Unlike our model, which is independent of the nucleus, the model for the cricket assumed occlusion of the microtubule “cloud” by the nucleus as the primary driving force. In Drosophila, centrosome spacing in the syncytium is independent of the nucleus (de-Carvalho et al, 2022), which may also be the case in crickets. In this scenario, a stoichiometric model of the cortical and cytoplasmic pulling forces that does not require nuclei for centrosome spacing would be more suitable, even for crickets. Finally, in contrast to the cricket model, in which an occlusion between the microtubule “clouds” was assumed without mechanistic bases, the stoichiometric model of cortical and cytoplasmic pulling forces assumes competition based on the reasonable length distribution of the microtubule (i.e., longer microtubules are rare). In this regard, we believe that the stoichiometric model of the cortical and cytoplasmic pulling forces is more widely applicable. The pulling-based mechanism may be used in combination with repulsive pushing between centrosomes using antiparallel microtubules in other cell types and organisms.

Myosin-dependent centrosome movement

We observed a large movement of centrosomes in dynein (dhc-1)-knockdown embryos (Fig 6). Previous studies have focused on some of the earliest phenotypes (defects in centrosome separation, pronuclear migration, and spindle elongation in the one-cell stage embryo) of dhc-1 RNAi or mutant embryos (Gönczy et al, 1999; Kimura & Onami, 2005; Schmidt et al, 2005), but have not focused on the later movements of the centrosomes. The timing of the large centrosome movement in dhc-1 (RNAi) embryos coincides with that of cytokinesis. This suggests the involvement of cytoplasmic flow coupled with cytokinesis (White & Borisy, 1983; Khaliullin et al, 2018). This idea is supported by our RNAi experiment on non-muscle myosin nmy-2, a gene responsible for generating cytoplasmic flow (Shelton et al, 1999). We confirmed that cytoplasmic flow occurred in dhc-1 (RNAi) cells.

The involvement of the cytoplasmic flow in centrosome movement (with or without nuclei) has been previously reported. In the one-cell stage C. elegans embryo, soon after symmetry breaking, the centrosomes moved along the cortex in zyg-12 (RNAi) embryos in an nmy-2-dependent manner (De Simone et al, 2016). This dependence on nmy-2 suggests that cytoplasmic flow is the driving force for this movement. However, an alternative interpretation is possible. Because nmy-2 (RNAi) impaired cortical pulling forces (Redemann et al, 2010), and defective cortical pulling forces impaired movement along the cortex (Figs 5C and S4, compare control-versus-gpr-1/2 [RNAi]), the centrosome movement behavior in zyg-12; nmy-2 (RNAi) (De Simone et al, 2016) can be explained by defects in the cortical pulling force.

Another example of centrosome movement by cytoplasmic flow is from the one-cell stage C. elegans embryo, but occurs earlier than symmetry breaking. Cytoplasmic flow, driven by kinesin and microtubules (McNally et al, 2010; Kimura et al, 2017), moves the sperm-derived pronucleus together with the centrosome, affecting the formation of the anterior–posterior axis of the embryo (Kimura & Kimura, 2020). In Drosophila syncytium embryos, nuclear movement via myosin-dependent cytoplasmic flow is important for nuclear positioning and synchronized cell division (von Dassow & Schubiger, 1994; Deneke et al, 2019). Our findings of centrosome movement by cytoplasmic flow may provide insights into how cytoplasmic flow moves the nucleus and centrosomes in future studies.

Worm strains and RNAi

The C. elegans strains used in this study are listed in Table S1. The DE90 strain (tbg-1::GFP; GFP::histone H2B; GFP::PH^PLCδ1) was used to obtain embryos with nuclei (zyg-12, dhc-1 and dhc-1; nmy-2 RNAi experiments). The strains were maintained under standard conditions (Brenner, 1974). Knockdown of klp-18, zyg-12, gpr-1/2, dhc-1, dyrb-1, and nmy-2 was performed using the injection RNAi method, as previously described (Kimura & Kimura, 2011). For double- or triple-RNAi experiments, RNA was mixed at a 1:1 or 1:1:1 ratio and injected into the worms. The dsRNA concentrations were 18 or 21 μg/μl for klp-18, 15 or 19 μg/μl for gpr-1/2, 19 μg/μl for dhc-1, and 13 μg/μl for dyrb-1. To efficiently obtain the klp-18 phenotype (enucleated embryos), observations were started ≧24 h after injection. Observations were also conducted at ≧26 h after double knockdown and ≧30 h after triple knockdown. The worms were incubated at 25°C for ≧16 h before observation (zyg-12, dhc-1 and dhc-1; nmy-2 RNAi experiments).

Production of enucleated embryos

The enucleated embryos were produced as follows (Fig S1). First, two young adults each of the CAL0051, CAL0181, and CAL2741 strains were transferred to a new plate 5 d before day 1 and preculture was initiated. On day 1, 15 CAL0051 and 10 CAL0181 or CAL2741 young adults were separately transferred onto new 6-cm (diameter) NGM plates with the OP50 E. coli, and the plates were cultured at 16°C, a nonrestrictive temperature. After 24 h, on day 2, the adult worms were removed from the plate, and only embryos that had been laid in the last 24 h remained on the plate. Cultures were maintained at 16°C. On day 3, 24 h after the procedures on day 2, the plates were moved to 25°C, which is the restrictive temperature. On day 4 and 24 h after the procedures on day 3, 25 CAL0181 or CAL2741 L4, or young adults were selected (hermaphrodites with vulva) and injected with klp-18 dsRNA. After injection, culturing on the NGM plate was continued with five times the number of CAL0051 males added (25 hermaphrodites and 125 males). A 3.5-cm NGM plate was used for mating. Finally, on day 5 and 24 h or more after injection, the worms were dissected, and the embryos were observed under a confocal microscope.

Microscopic observation

The localization of fluorescent proteins was observed using a spinning-disk confocal microscope (CSU-MP; Yokogawa Electric) (Otomo et al, 2015; Kamada et al, 2022) equipped with an EM-CCD camera (iXon; Andor) mounted on an IX71 microscope (Olympus) and controlled using NIS-elements software (Nikon). Details of the system and the examination of phototoxicity will be published elsewhere (in preparation). Dissected worm embryos were attached to a poly L-lysine–coated cover glass, mounted under a microscope, and observed using a 40× objective lens at 2× intermediate magnification.

To analyze the centrosome distance, two-photon excitation with a 920-nm laser (ALCOR920-2; Spark Lasers) was used with 96-ms exposure. For Figs 1–3, and related supplemental figures and movies, 61 or 71 images were taken at 0.5-μm intervals on the z-axis. Time-lapse images were collected at 1-min intervals for less than 2 h. For Fig 6, 41 images were taken at 0.5-μm intervals along the z-axis. Time-lapse images were collected at 10-s intervals during the one-cell stage.

To analyze the distribution of microtubules and centrosomes (Figs 4 and S3), dissected worm embryos were attached to a 2% agarose-coated cover glass and mounted on a microscope. A single focal plane was captured using single-photon excitation with a 488 nm laser with 804-ms exposure. The focal plane was manually adjusted to the target aster during this interval. Time-lapse images were collected at ∼1-min interval for 30 min.

Under these microscopic conditions, C. elegans embryos were confirmed to hatch. Captured images were analyzed using ImageJ/Fjii or Imaris software (Oxford Instruments).

Measurement of centrosome distance

The distance between the centrosomes was quantified using Imaris 3D analysis software. Centrosome signals were manually tracked using the spot-tracking mode. The centroid coordinates of the centrosome signals were calculated using a software. The distances between the centrosomes were calculated using the calculated coordinates. For the control, zyg-12 (RNAi), and enucleated embryos (Figs 2C and D and 3B), the centrosomes (Fig 2A), which split into two at the two-cell stage, were tracked until the signal became undetectable or until the next duplication occurred. As shown in Fig 6, centrosome signals during the one-cell stage were tracked until they became undetectable.

Analysis of microtubule distribution

For β-tubulin::GFP images (Fig 4A), the center coordinates of the centrosomes were quantified using the SpotTracker plugin in ImageJ (Sage et al, 2005) (http://bigwww.epfl.ch/sage/soft/spottracker/). The fluorescence intensity of soluble β-tubulin was defined as the peak intensity of the cytoplasmic signal. The fluorescence intensity of polymerized β-tubulin (i.e., microtubules) was defined as the captured intensity subtracted by the soluble β-tubulin intensity. The mean and SEM of the subtracted intensities were calculated for the ring-shaped region for every four-pixel thickness. The mean intensity was then multiplied by the average ring circumference. We assumed that the summed intensity of the ring regions, S(R), was proportional to the number of microtubules reaching the rings, and plotted S(R) against the average radius of the ring, R (Fig 4B). The plot was fitted with a combination of two functions: S(R) = a × R (for R < R₀) and S(R) = a × R₀ × EXP[−{(R − R₀)/ξ}^P] (for R ≧ R₀), where a, R₀, ξ and P are the fitting parameters. Fitting was conducted using the maximum likelihood method (Yesbolatova et al, 2022), assuming that the error was normally distributed, with its mean summed-intensity and variance as the square of the SEM multiplied by the circumference length, using the solver function of Microsoft Excel (Microsoft Corporation). The function S(R) = a × R₀ × EXP[−{(R − R₀)/ξ}^P] represents the Weibull distribution. The Weibull distribution was used to model the distribution of the lifespan, whose death rate is proportional to the power of time. We confirmed that the simulated microtubule length distribution with constant growth/shrinkage velocity and catastrophe/rescue frequency fit well with the Weibull distribution. Therefore, we determined the length (l) distribution of the microtubule as S(l) = a × l₀ × EXP[−{(l − l₀)/ξ}^P] (for l ≧ l₀, where l₀ is the radius of the centrosome).

To calculate the average distribution of microtubule length at every time point and sample, we first fitted the microtubule length distribution at each time point for each sample using the Weibull distribution. After fitting, the average value of the fitted distribution is calculated to obtain the average distribution. This average distribution was further fitted to a Weibull distribution to obtain function parameters. The microtubule length distribution in our simulation was obtained using a Weibull distribution.

Statistical analysis

The distances between centrosomes were statistically compared using the Wilcoxon rank-sum test, which was performed for two experimental groups of interest. Calculations were performed using the “ranksum” function of MATLAB software (The Mathworks).

Numerical simulation of the stoichiometric model of cortical and cytoplasmic pulling forces

The settings of our previous simulation (Kondo & Kimura, 2019) were modified to model the embryo as a 3D ellipsoid with the measured lengths of the long axis and the two short axes (Table S3). As in the previous simulation, we distributed “force generation points” throughout the cytoplasm and the cortex (3-μm thick layer) at the vertices of a simple cubic lattice with 1-μm intervals. When a force generation point was associated with a microtubule elongating from the centrosome, the centrosome was pulled with a defined force (Table S2).

The probability of the point attaching a microtubule from the i-th centrosome was defined by the distance between the point and the centrosomes, as assumed in the stoichiometric model of cortical pulling forces proposed by Farhadifar et al (2020): Pi=Ω(di)∑jNΩ(dj)+κ, where Ω(d_i) is the rate microtubules from i-th centrosome contact a force-generator at a distance of d_i, and κ is the rate a force generator detach from a microtubule. There are three notable differences between the original stoichiometric model (Farhadifar et al, 2020) and the stoichiometric model of the cortical and cytoplasmic pulling forces in this study. First, the model was extended to simulate the behavior of more than three centrosomes. Second, the force generators pull microtubules in the cortex and cytoplasm, based on our experimental results showing that simultaneous knockdown of gpr-1/2 and dyrb-1 but not either, is required for spacing defects comparable with those of dhc-1 (RNAi) (Fig 3). Third, we used the distribution of microtubule lengths based on our own experimental measurements of signal intensity, reflecting the microtubule distance, d, from the center of the centrosome: S(d) = a × d₀ × EXP[−{(d − l₀)/ξ}^P] (for d ≧ l₀), where l₀ is the radius of the centrosome (Fig 4 and Table S2). Finally, we defined Ω(d) as Ω(d) = (γ/4)(r/d)² × EXP[−{(d − l₀)/ξ}^P] (for d ≧ l₀), where γ is the rate of microtubule nucleation at the centrosome and r is the force-generator capture radius. For the case where the force generator is located inside the centrosome (d < l₀), we assumed that all the nucleated microtubules reach the distance, and thus defined Ω(d) as Ω(d) = (γ/4)(r/d)² (for d < l₀).

Once we calculated the probability P_i, for each force generator to pull the i-th centrosome, the force that pulls the i-th centrosome was calculated as f₀P_i, where f₀ is the force generated by each force generator (Farhadifar et al, 2020). In this study, we defined f_{0_cort} and f_{0_cyto} as the forces generated by the cortical and cytoplasmic force generators, respectively. The total force vector acting on each centrosome was calculated by summing all the force vectors of the force generators and pulling the centrosome in the direction of each force generator. After summing the forces acting on each centrosome, the velocity of the movement was calculated as v→=F→/η, where F→, η, and v→ are the force, drag coefficient, and velocity vector, respectively. The positions of the centrosomes in the next step were calculated as ct+Δt→=ct→+v→×Δt, where ct→ and ct+Δt→ are the position vectors of the centrosomes at times t and t+Δt, respectively, and Δt is the time interval. This calculation was repeated for the defined steps starting from the initial positions of the centrosomes.

When the centrosomes were tethered to the surface of the nucleus, we added an additional process after each step to apply an elastic force, F1→=−ks(c1→−c2→) if L>|c1→−c2→| to maintain the distance between the centrosomes at L or shorter. Here, c1→ and c2→ are the position vectors of the centrosome and other centrosomes, respectively. k_s is the elastic constant and L is the diameter of the nucleus (8 μm).

The simulation was coded using MATLAB, and the codes are available upon request.

Parameter values of the numerical simulation

The parameter values used for the numerical simulations are summarized in Tables S2 and S3. We followed the embryo geometry based on the experimental measurements and the simulation setup in our previous study (Kondo & Kimura, 2019). The parameters for force generation were essentially identical to those used by Farhadifar et al (2020). In our setup, the number of cortical force generators (N_cort) was 12,410 under the control condition. According to Farhadifar et al (2020), the pulling force produced by a force generator (f_{0_cort}) was 0.08 pN, the force generator capture radius (r) was 0.1 μm, and the microtubule force generator detachment rate (κ) was 4.4 × 10⁻⁴/s.

To determine the cortical pulling force, cytoplasmic pulling force, and force reduction by RNAi, we compared the simulated results for the distance between sister centrosomes under different conditions (control, reduced cortical pulling force, and reduced cytoplasmic pulling force) with the corresponding experimental results (control, gpr-1/2, dyrb-1; Figs 3B and S6). First, we searched for appropriate values of the force produced by a cortical force generator (f_{0_cort}) and a cytoplasmic force generator (f_{0_cyto}) that reproduced the maximum rate of increase in distance for dyrb-1 (RNAi)-enucleated embryos (i.e., defective cytoplasmic forces) and gpr-1/2 (RNAi)-enucleated embryos (i.e., defective cortical forces) (the number of cytoplasmic force generators (N_cyto) in our setup was 18,318 under control conditions). The average values of the optimized force parameters for the three pairs of representative embryos were 0.034 pN for f_{0_cort} and 0.014 pN for f_{0_cyto}. Using these parameter values, we simulated centrosome movement in the control, dyrb-1 (RNAi)-, gpr-1/2 (RNAi)-, and dhc-1 (RNAi)-enucleated embryos. To mimic the low-level spacing in dhc-1 (RNAi)-enucleated embryos, we assumed that RNAi treatment reduced the force (f_{0_cort} and f_{0_cyto}) to 5%, but not to 0%.