Polo kinase phosphorylation determines C. elegans centrosome size and density by biasing SPD-5 toward an assembly-competent conformation

Centrosomes are major microtubule-organizing centers composed of centrioles surrounded by an extensive proteinacious layer called the pericentriolar material (PCM). In C. elegans embryos, the mitotic PCM expands by Polo-kinase (PLK-1) phosphorylation-accelerated assembly of SPD-5 molecules into supramolecular scaffolds. However, how PLK-1 phosphorylation regulates SPD-5 assembly is not known. We found that a mutant version of SPD-5 that is insensitive to PLK-1 phosphorylation (SPD-54A) could localize to PCM but was unable to rescue the reduction in PCM size and density when wild-type SPD-5 levels were decreased. In vitro, purified SPD-54A self-assembled into functional supramolecular scaffolds over long time scales, suggesting that phosphorylation only controls the rate of SPD-5 scaffold assembly. Furthermore, the SPD-5 scaffold, once assembled, remained intact and supported microtubule nucleation in the absence of PLK-1 activity in vivo. We conclude that Polo Kinase is required for rapid assembly of the PCM scaffold but not for scaffold maintenance or function. Based on this idea, we developed a theoretical model that adequately predicted PCM growth rates in different mutant conditions in vivo. We propose that PLK-1 phosphorylation-dependent conversion of SPD-5 into an assembly-competent form underlies PCM formation in vivo and that the rate of this conversion determines final PCM size and density.


INTRODUCTION
Centrosomes are the main microtubule-organizing centers of animal cells, consisting of a pair of centrioles that organize a dynamic protein mass called pericentriolar material (PCM). The PCM consists of a structured, small interphase layer [1,2] around which, amorphous mitotic PCM assembles to achieve maximal microtubule nucleation to facilitate mitotic spindle assembly. Despite the importance of PCM in microtubule organization, the mechanisms behind mitotic PCM formation remain elusive. In C.elegans, large-scale RNAi screens and genetics have revealed a limited number of components required for PCM assembly: SPD-2 [3,4], the polo-like kinase PLK-1 [5] and SPD-5 [6]. It was also shown that similar proteins are involved in PCM assembly in other species, suggesting that a universal assembly mechanism may exist. For instance, PCM assembly in vertebrates and Drosophila requires the assembly of large proteins such as Pericentrin/D-PLP and Cdk5RAP2/Centrosomin, which resemble  in that they contain numerous interspersed coiled-coil domains [6][7][8][9][10][11]. The assembly of these proteins is facilitated by SPD-2/Cep192 and the phosphorylation activity of the conserved Polo-like-kinase Plk1/PLK-1 [3][4][5][12][13][14][15]. However, how these molecular interactions lead to PCM assembly and determine final PCM size and density remain outstanding questions.
We previously hypothesized that PCM is nucleated at centrioles and then rapidly expands via autocatalytic incorporation of cytosolic PCM components [16]. In our model, unassembled PCM proteins exist in a soluble form that can transition into an assembly-competent state within the PCM and then become stably incorporated. Once incorporated, PCM proteins will recruit additional PCM components, which is an autocatalytic event. Consequentially, in our model the kinetics of PCM assembly depend on the rate by which PCM material, after being recruited to the centrosome, converts from the soluble to the assemblycompetent form. The existence of such a conversion is supported by the observations that C. elegans PCM proteins are indeed monomeric in cytoplasm prior to assembly, whereas they interact at the centrosome [17].
We recently reported that purified SPD-5 can form supramolecular PCM-like assemblies in vitro, the formation of which is accelerated by PLK-1 phosphorylation of SPD-5 [12].
Mass spectrometry revealed PLK-1 phosphorylation sites on SPD-5 that, when mutated, prevented PCM assembly in vivo. These results indicate that PCM formation in C. elegans is driven by Polo Kinase-mediated oligomerization of SPD-5 around centrioles. We also found that only SPD-5 assemblies, and not SPD-5 monomers, recruited other PCM proteins, including PLK-1, leading us to propose that this emergent scaffolding property of SPD-5 could be the basis for autocatalytic PCM expansion in vivo [12,17]. Additionally, our vitro experiments revealed that the stability and scaffolding capacity of SPD-5 assemblies were independent of PLK-1 phosphorylation, suggesting that Polo Kinase activity may only regulate the speed of SPD-5 assembly. However, we did not test whether Polo Kinase has additional roles in PCM maintenance or function in vivo. Nor did we test if unphosphorylated SPD-5 can be recruited to existing PCM and then be converted to an assembly-competent state as predicted by our previous model [16].
In this study, we combined in vivo analysis, in vitro reconstitution, and modeling to investigate how Polo Kinase regulates SPD-5 assembly to form PCM in C.
elegans. Our results indicate that Polo Kinase phosphorylation affects the rate of SPD-5 assembly without dramatically affecting SPD-5 recruitment to existing PCM, PCM stability, or PCM function. Additionally, we conclude that a phosphosite binding mechanism cannot explain PCM assembly. Rather, we propose that SPD-5 naturally isomerizes between assembly-incompetent and assemblycompetent states, and that Polo Kinase phosphorylation biases SPD-5 toward the latter state. Furthermore, our results suggest that the conversion rate of SPD-5 into an assembly competent state is the key determinant of final PCM size and density in vivo.

A SPD-5 phospho-mutant binds to PCM in C. elegans embryos
We first set out to determine if PLK-1 phosphorylation of SPD-5 is required only for PCM expansion or also for SPD-5 binding to PCM. For this purpose we used spinning disk confocal microscopy to observe PCM assembly in C. elegans embryos expressing GFP-tagged versions of SPD-5 (SPD-5 WT and SPD-5 4A ) as their sole source of SPD-5. SPD-5 4A is mutated in the critical sites for PLK-1 mediated centrosome assembly in vivo ( Figure 1A; [12]). As previously shown, GFP::SPD-5 4A localized to pre-mitotic centrosomes, but, in contrast to GFP::SPD-5 WT , failed to expand the PCM ( Figure 1B; [12]). To test if SPD-5 4A is still capable of binding to existing PCM, we observed PCM assembly in embryos expressing endogenous SPD-5 and GFP::SPD-5 4A . In such embryos, mitotic PCM assembled and GFP::SPD-5 4A localized to the PCM ( Figure 1B).
PCM size and density are reduced in embryos ectopically expressing a GFP::SPD-5 4A transgene.
In addition to being smaller, centrosomes assembled in the presence of the SPD-5 4A mutant were also less dense. Based on the assumption that SPD-5 forms the underlying PCM scaffold [6,12], we used GFP::SPD-5 fluorescence at the PCM to approximate PCM density from the mean pixel intensity of maximum intensity z-projections. In embryos expressing endogenous SPD-5 and GFP::SPD-5 WT , centrosomal GFP signal increased with time after fertilization until the onset of mitosis ( Figure 2A, gray points), indicating an increase in PCM density up to mitosis. In embryos expressing endogenous SPD-5 and GFP::SPD-5 4A , centrosomal GFP signal started at a similar mean intensity shortly after fertilization but remained constant until mitosis ( Figure 2A, red points). A comparison of GFP fluorescence of centrosomes at NEBD revealed 26% higher intensity for centrosomes assembled with GFP::SPD-5 WT (WT = 420 ± 99 a.u. vs. 4A = 332 ± 76 a.u., mean ± STD; Figure 2B). We tested if this difference in GFP intensity resulted from hampered binding of GFP::SPD-5 4A to PCM or generally a lower SPD-5 density at the PCM. Immunostainings against GFP and SPD-5 showed that SPD-5 levels at the PCM were reduced in embryos expressing the 4A mutant ( Figure 2C and D) and that the ratios of transgenic GFP::SPD-5 to total SPD-5 immunostaining signal at wild-type and mutant PCM were very similar ( Figure S2A). These results suggest that the reduction in GFP fluorescence seen in GFP::SPD-5 4A embryos reflects a general reduction of SPD-5 levels at the PCM. Also, immunostainings revealed a similar difference in SPD-2 and PLK-1 levels at wild type and mutant PCM ( Figure 2E and 2F), indicating that concentrations of SPD-2 and PLK-1 correlate with SPD-5 concentration at the PCM. Thus, expression of GFP::SPD-5 4A reduces volume and density of the functional PCM scaffold.

The concentration of phosphorylation-receptive SPD-5 determines PCM size and density in vivo
How can the presence of a mutated transgenic SPD-5 cause a reduction of PCM volume and density? It is possible that GFP::SPD-5 4A acts as a dominant negative mutant that interferes with accumulation of wild-type SPD-5 at the PCM, possibly by occupying and blocking required SPD-5 binding sites in the PCM scaffold. Alternatively, GFP::SPD-5 4A may act as a loss-of-function mutant, and, due to protein level compensation, the phenotype seen in embryos expressing mutant SPD-5 could be a consequence of the reduction of available wild-type SPD-5. We previously observed such compensation of the centrosomal protein SPD-2; however, ectopic expression of a codon-adapted version of SPD-5::GFP did not influence the expression of endogenous SPD-5 [5]. Surprisingly, ectopic expression of transgenic GFP::SPD-5 WT in our current strain led to a reduction in endogenous SPD-5, and selective RNAi against endogenous SPD-5 lead to an upregulation of transgenic SPD-5 ( Figure 2G). These different behaviors could be caused by the sequence differences in the SPD-5 transgenes. The To test if PCM volume and density respond to SPD-5 concentration changes, we fully removed endogenous SPD-5 and then reduced the concentration of GFP::SPD-5 WT using a double RNAi condition targeting the endogenous and transgenic spd-5 transcripts with different strengths. Using this method, we reduced GFP::SPD-5 WT levels to about 63% compared to the control condition, while fully removing the endogenous copy in both cases ( Figure 2H). Reduction of GFP::SPD-5 WT reduced PCM volume by ~40% (3.9 ± 1.2 µm 3 , p < 0.001) and PCM density by ~34% , (p < 0.001) ( Figure 2I and 2J). These changes in centrosome size and density were similar to changes observed when GFP::SPD-5 4A was ectopically expressed (see Figure 1D and 2A). These results are consistent with GFP::SPD-5 4A being a loss-of-function mutant. Furthermore, we conclude that the concentration of available wild-type SPD-5 determines PCM size and density.

PLK-1 phosphorylation of SPD-5 affects the rate of PCM matrix assembly without affecting matrix function in vitro
Our in vivo analysis showed that PCM size and density are reduced in embryos when either the concentration of wild-type SPD-5 is reduced via RNAi or when SPD-5 4A is ectopically expressed in addition to endogenous SPD-5. However, these experiments did not allow us to exclude the possibility that SPD-5 4A could act as a dominant negative mutant. We directly tested this hypothesis using an in vitro assay for PCM assembly developed in our lab [12].
Similar to our observations in vivo, we found that purified SPD-5 4A ::GFP localized to PCM-like networks formed with SPD-5 WT ::TagRFP in vitro ( Figure 3A). Next, we tested the effect of SPD-5 4A ::GFP on wild-type SPD-5 network growth in the presence of PLK-1. We prepared network reactions on ice with equimolar amounts of SPD-5 WT ::GFP and PLK-1, then added either buffer (WT), SPD-5 4A , (WT+4A), or additional SPD-5 WT (WT+WT). We warmed the tubes to 23°C to initiate network assembly, then, after 30 min, we squashed a sample under a cover slip for analysis. Under these conditions, small, nascent networks could be seen in the control sample ( Figure 3B), and we verified that growth had not yet plateaued (unpublished data); thus, our experiments should allow detection of any stimulatory or inhibitory effects.
Total network mass was ~2-fold higher in the sample containing unlabeled SPD-5 WT compared to the control where only buffer was added (WT+WT vs. WT; p = 0.007) ( Figure 3B). In contrast, total network mass in the sample containing SPD-5 4A was only slightly higher than the control sample, and the difference was not statistically significant (p = 0.86) ( Figure 3B). These data indicate that during PCM assembly SPD-5 4A behaves as a loss-of-function mutant rather than a dominant-negative mutant. Furthermore, our in vitro results corroborate our in vivo findings that PCM assembly rate is largely determined by the amount of phosphorylation-responsive SPD-5 available in the system.
We then used this in vitro assay to test if PLK-1 is also required for proper functioning of the PCM scaffold. As observed previously, SPD-5 4A assembled into supramolecular networks at a rate similar to unphosphorylated wild-type protein [12]. After one hour of incubation at 23°C, networks exclusively assembled from SPD-5 WT ::TagRFP or SPD-5 4A ::TagRFP equivalently recruited SPD-2::GFP and PLK-1::GFP ( Figure 3C-E), suggesting that SPD-5 4A and unphosphorylated SPD-5 WT can form functional PCM scaffolds in vitro given sufficient time.

PLK-1 phosphorylation is not required to maintain PCM scaffold stability or function in vivo
Our in vitro results predict that SPD-5 scaffolds, once formed, should function without needing continuous PLK-1 phosphorylation in vivo. To test this idea, we constructed a C. elegans strain expressing GFP::SPD-5 WT and an analogsensitive PLK-1 mutant (PLK-1 AS ) that can be inhibited by the drug 1NM-PP1 (plk1Δ; plk-1 as gfp::spd-5; [21]). We permeabilized embryos using partial knockdown of perm-1 via RNAi [22], then identified pre-mitotic embryos where centrosomes had formed but were not yet full-sized. Addition of 10 µM 1NM-PP1 to these embryos arrested centrosome growth: both centrosome size and GFP::SPD-5 fluorescence remained constant thereafter ( Figure 4A and 4B; n = 10). However, centrosomes continued to grow if DMSO was added instead ( Figure 4C; n = 10). Thus, PLK-1 is not required to maintain SPD-5 at the centrosome; this stands in stark contrast to gamma tubulin, which does require continuous PLK-1 activity for centrosomal localization [12].
To test the functionality of PCM-localized SPD-5 in the absence of PLK-1 phosphorylation in vivo, we treated permeabilized embryos with 10 µM 1NM-PP1 and the proteasome inhibitor c-lactocystin-β-lactone for 20 min, then fixed the embryos and visualized microtubules using immuofluorescence. Centrosomes still nucleated microtubules and formed spindles after PLK-1 inhibition, suggesting that SPD-5 retains its functional capacity for scaffolding in the absence of PLK-1 phosphorylation ( Figure 4D; n = 8). These in vivo results are in agreement with our in vitro data and suggest that PLK-1 phosphorylation is not required for the maintenance or function of SPD-5 scaffolds but instead only controls the rate of SPD-5 scaffold formation, and, subsequently, PCM assembly.

A PLK-1-dependent SPD-5 conversion model can explain in vivo PCM assembly
Our data, combined with previous studies, allow us to propose a simple mechanism for PLK-1-dependent PCM assembly in C.elegans. Prior to incorporation into the PCM, SPD-5 is mostly monomeric and does not interact with SPD-2 or PLK-1 [17]. We term this the inactive form of SPD-5, which cannot contribute to PCM assembly itself but can localize to centrioles and segregate into existing PCM ( Figure 4E). Since GFP::SPD-5 4A alone is not capable of expanding PCM in vivo ( Figure 1B and D), we assume that SPD-5 4A as well as unphosphorylated SPD-5 WT exist primarily in the inactive form. Secondly, we define an active, assembly-competent form, which can self-assemble into supramolecular structures and contribute to PCM growth ( Figure 4E). Because our in vitro data show that purified SPD-5 can spontaneously self-assemble and that PLK-1 phosphorylation accelerates this assembly process [12], we propose that SPD-5 can transition into the assembly-competent state spontaneously and that this transition is much more likely if SPD-5 is phosphorylated by PLK-1.
Thus, we assume that Polo kinase-phosphorylated SPD-5 exists predominately in the assembly-competent state.
Based on this idea we constructed a mathematical model (see SI) in which the inactive form of SPD-5 can be converted locally at the centrosome into the assembly-competent form through an active process such as Polo kinase phosphorylation [16]. We used this model to fit the accumulation rate of total SPD-5 at centrosomes with an exponential function to describe the rate of PCM growth (see SI). Total amounts of SPD-5 were estimated from centrosome volumes multiplied by SPD-5 densities ( Figure S3A). We fit the accumulation rate of total SPD-5 from initiation of assembly until NEBD ( Figure 4F). When embryos only expressed SPD-5 WT , the PCM growth rate was 0.48 ± 0.08 min -1 . In contrast, when embryos only expressed SPD-5 4A , PCM growth rate was only 0.01 ± 0.05 min -1 . We then used these measured rates to predict the PCM growth rate in the mixed scenario where SPD-5 4A is present in a background of endogenous SPD-5 WT . Based on western blot analysis, we estimated that ~70% of SPD-5 protein is wild-type and ~30% is mutated in these embryos ( Figure   S1A). Using these values, our model predicted that the PCM accumulation rate in these embryos should be 0.34 ± 0.06 min -1 (see SI for calculation details). This value is very similar to the accumulation rate we obtained when fitting the data (0.31 ± 0.11 min -1 ). Taken together, these results suggest that a model based on centrosomal conversion of SPD-5 into an assembly-competent form is adequate to describe the complex process of PCM assembly in C.elegans embryos.
How does PLK-1 change SPD-5 to induce self-assembly? In vitro both unphosphorylated SPD-5 WT and SPD-5 4A assembled into supramolecular networks after a long period of time ( Figure 3 and [12]); but, addition of PLK-1 dramatically accelerated SPD-5 WT self-assembly. These results suggest that SPD-5 naturally isomerizes between the inactive form and the assemblycompetent form, and that PLK-1 phosphorylation of SPD-5 lowers the energy barrier of this transition ( Figure S3B). The role of PLK-1 in PCM assembly, then, is to bias SPD-5 isomerization towards the assembly-competent state. Besides PLK-1 phosphorylation, PCM assembly requires SPD-2, a protein known to control centrosome growth rate and thus size in vivo [5,23]. We have shown previously that SPD-2 accelerates SPD-5 assembly in the presence and absence of PLK-1 in vitro, demonstrating that multiple mechanisms regulate SPD-5 selfassembly [12]. Whether SPD-2 also enhances SPD-5 self-assembly by affecting its isomerization or by some other process remains to be investigated. We conclude that SPD-5 has the intrinsic capability to assemble functional PCM and that PLK-1 phosphorylation and SPD-2 simply accelerate the rate of assembly.
An unexpected observation from our experiments was that PCM density increased over time from fertilization until mitosis in wild-type embryos.
However, when we reduced the speed of PCM assembly by reducing the available pool of phosphorylation-receptive SPD-5, PCM density did not change over time. We do not understand the molecular basis of this phenomenon. One possibility is that this is caused by a buildup of elastic stress as the centrosome grows. If the growth rate is faster than the stress relaxation rate, centrosome density will increase. However, if the growth rate is slowed down, for instance, by reducing the concentration of wild-type SPD-5, then the stress could relax and centrosome density would remain constant. This supports the idea that the PCM is not solid, but rather a viscous, gel-like material, which forms by phase transition of soluble molecules into PCM. This notion is further supported by the observation that GFP::SPD-5 4A localized to PCM without interfering with the expansion of the scaffold.
In D. Melanogaster, PCM assembly is driven by Polo kinase-regulated multimerization of the scaffolding protein Centrosomin [13], suggesting that Centrosomin is the functional homolog of SPD-5. Similar to SPD-5, Centrosomin must be phosphorylated at multiple residues to achieve its full scaffolding potential [13]. In vertebrate cells, Polo kinase and the Centrosomin homolog CDK5Rap2 are also required for mitotic PCM assembly [24,25]. Thus, a common mechanism for PCM assembly is emerging that centers on Polo kinase-mediated phosphorylation of large coiled-coil proteins. It will be of interest to determine the similarities in self-assembly properties and regulation of SPD-5, Centrosomin, and CDK5Rap2.

RNAi treatments
RNAi against endogenous spd-5 was carried out as previously described [27].
Briefly, the following primers were used to amplify nucleotides 501 -975 of endogenous spd-5 from cDNA and cloned into Gateway® pDonor™221 vector via BP reaction to create spd-5-pENTR™ vector: spd-5-rev (GGGACCACTTTGTACAAGAAAGCTGGGTgtgctcaagcttgctacac). The amplified sequence was then transferred to L4440_GW (Addgene) destination vector and used to transform HT115(DE3) bacteria strain for RNA expression.
Using this feeding clone, full knock down of endogenous SPD-5 was achieved typically within 24 hs. Simultaneous RNAi against endogenous and transgenic SPD-5 was carried out using the SPD-5 (F56A3.4) clone from the C. elegans RNAi feeding library constructed by the lab of Dr. Julie Ahringer, available from Source BioScience. To achieve full knockdown of endogenous and partial depletion of transgenic SPD-5, both clones were grown simultaneously and mixed (30% Ahringer, 70% endogenous) prior to plating. Due to the increased knockdown efficiency of the Ahringer feeding clone, incubation times had to be shortend to 12 hs. Quantification of SPD-5 protein knock down was quantified from western blots using the Gel Analyzer function in Fiji [28].

Antibodies and Stainings
Stainings were done following standard procedure described before [6]. The polyclonal mouse αPLK-1 antibody was generated in house by injecting 1 mg of purified full length PLK-1 into mice, purified from serum, and used in a dilution of 1:300. Endogenous SPD-5 and SPD-2 were detected using the previously Centrosome volume was then calculated using the centrosome radius approximated from centrosome area measurements. Total SPD-5 amounts were approximated from centrosome volumes and densities. We multiplied the volumes with concentration corrected intensity values, assuming that the recorded intensities stemmed from the 30% of the total SPD-5 pool labeled with GFP.

In vitro SPD-5 scaffold assembly
Proteins were purified and assembled into scaffolds in vitro as previously described [12]. For a detailed protocol please refer to [29]. Cover slips were cleaned and made hydrophobic using the following steps. First, cover slips were placed in a Teflon holder and submerged in a 1:20 dilution of Mucasol detergent (Sigma) for 10 min with sonication. Second, the cover slips were transferred to 100% Ethanol and incubated for 10 min with sonication.
Third, the cover slips were incubated in a 50 % solution of Rain-X (diluted in ethanol) for >30 min, then washed in ethanol, then twice in water. Finally, the cover slips were dried using N 2 gas and stored in a desiccation chamber.

PLK-1 inhibition in embryos with permeabilized eggshells
PLK-1 inhibition was performed as previously described [12]. Briefly, L4 worms were seeded onto feeding plates containing bacteria expressing perm-1 dsRNA and incubated at 20°C for 14-20 hours [22]. confirm that the imaged embryo was permeable. Imaging was performed as described above, except that 10s intervals and 8% laser power (4.5 mW) were used.

Analysis of diffusion using fluorescence correlation spectroscopy (FCS)
FCS measurements and diffusion analysis were carried out as described previously [17].

Quantitative model of SPD-5 incorporation
We use a theoretical approach to discuss the effects of the nonphosphoraylatable SPD-5 4A mutant on the dynamics of PCM assembly and centrosome growth. We extend the previous physical model of centrosome assembly that is based on the physics of liquid droplets [1]. This model can quantitatively account for the volume growth of centrosomes in vivo. Here, we extend this model to a situation where in addition to the wild-type form SPD-5 WT also the mutant form SPD-5 4A is expressed. The biochemical effects of the mutations are not known. By comparing theoretical predictions to experimental data, we aim to identify the biochemical properties of SPD-5 4A during centrosome assembly.
Our original model is based on the idea that SPD-5 molecules exist in two different forms [1]. We distinguish the form that is soluble in the cytoplasm and the form that tends to aggregate and phase separate from the cytoplasm. This form is thus the basis of the PCM phase that forms the centrosome. The two forms of SPD-5 can be converted into each other by chemical processes such as phosphorylation. We showed that the conversion of to must be autocatalytic to account for the sigmoidal centrosome growth curves and the reliable initiation of PCM accumulation at the centrioles [1].
We extend our original model by distinguishing wild-type SPD-5 WT  Chemical reactions lead to the creation and removal of molecules of form inside the PCM, which is described by [1] for = WT, 4A. Here, the first two terms describe the production of a single molecule, while the last term accounts for the back-reaction → . The rateconstant ! in the first term quantifies the rate of the autocatalytic reaction + → 2 . The second term accounts for the catalytic activity of the centrioles, which is quantified by the reaction flux ! . Note that this catalytic activity is necessary to initiate centrosome growth [1] and to stabilize two centrosomes in the same cell [2]. We consider the case where both this catalytic and the autocatalytic reaction are driven by the same enzyme, e.g., PLK-1. We thus choose the ratio ! = ! / ! of the reaction rates to be the same for the wild-type and the mutant, respectively. Here, ! is proportional to the number of catalytic sites on the centrioles.
For the simple case where centrosomes start growing around bare centrioles without PCM, ! ! ( = 0) = 0, and the concentration ! ! in the cytoplasm remains constant, the number ! ! of moleculs in the PCM evolve as Here, is the PCM growth rate. If we include the depletion of the form in the cytoplasm, ! ! ( ) exhibits sigmoidal growth [1]. Since we do not observe a saturation of ! ! in our experimental data, we focus on the early growth phase where depletion can be neglected.
We compared the predictions of our model to our experimental measurements. In our experiments, we label one of the SPD-5 species with GFP, such that the measured total centrosome fluorescent intensity is a proxy for the total number ! = ! ! + ! ! of molecules of species , where ! ! = ! ! , and is the PCM volume. For the case !" ≪ , the time evolution of can be approximated by Consequently, the total number of molecules is given by We use this functional form to measure the growth rate by fitting the function = ( !" − 1) to our measured intensities multiplied by the centrosome volume as a function of time ( Figure 4F). Here, the prefactor sets the intensity scale. The measured growth rates were WT = 0.48 ± 0.08 min !! and 4A = 0.01 ± 0.05 min !! in the experiments with only SPD-5 WT and only SPD-5 4A , respectively. Since !! ≪ !" , the mutant does not lead to significant PCM growth.
Why does the mutant SPD-5 4A not contribute to PCM growth? According to Equation ( Since the cytoplasmic concentrations ! ! of the fluorescent SPD-5's are similar ( Figure S1) we conclude that the binding coefficients ! = ! ! / ! ! are also similar.
In fact, 4A might even be larger than WT . Consequently, a reduced binding efficiency of the mutant is not consistent with our data.
The only possible explanation for the mutant's reduced contribution to PCM growth is that the conversion rate of the mutant from form to form must be much smaller than that of the wild-type ( 4A ≪ WT ). Since the mutant cannot be phosphorylated by PLK-1, this furthermore suggests that phosphorylation by PLK-1 plays an important role in this conversion.
Our model implies that the PCM growth rate is proportional to the concentration of SPD-5 WT , since 4A ≪ WT . We tested this prediction with two additional experiments. First, we quantified centrosome growth in an experiment where both mutant and wild-type SPD-5 were expressed. Using Equation (3), we express the growth rate both in terms of the already measured growth rate WT as both ≈ ! WT WT /( ! WT + ! 4A ). In our experiment, 71% of the SPD-5 molecules are wild-type ( Figure S1), which implies both = 0.71 WT = 0.34 ± 0.06 min !! .
This value agrees well with the measured rate 0.31 ± 0.11 min !! . Second, we reduced the concentration of SPD-5 WT to a fraction of the wild-type concentration without introducing SPD-5 4A and found that the PCM volume is reduced by a similar fraction ( Figure 2H-J). In summary, our model can capture the growth behavior in all our experiments when the only effect of the mutation of SPD-5 is that it cannot transition from form to form .
In addition to the growth behavior discussed above, we also observed a slight increase in the density of wild-type SPD-5 over time in the experiment without the mutant (Figure 2A). Our current model does not explain this behavior, since it considers phase separation with a constant density ! of SPD-5 in its form.
However, the PCM is not a simple incompressible fluid but a complex polymeric material. Therefore, the incorporation of SPD-5 can lead to a transient buildup of elastic stresses in the PCM, which then relax, e.g., by internal rearrangement of SPD-5 molecules. If this relaxation is slow compared to the SPD-5 incorporation rate , we expect increased SPD-5 densities over time. Conversely, if the relaxation rate is larger than or comparable to , the density would be rather constant. We indeed observe these two cases in our experiments ( Figure 2A).
Our data thus suggests that if only wild-type SPD-5 is present, PCM grows faster than stresses can relax, while they relax during the slower growth in the presence of SPD-5 4A . Note however that the presence of GFP::SPD-5 4A not only affects the growth rate , but might also have an effect on the relaxation rate, which we cannot measure directly. We therefore did not analyze this further.
Nevertheless, these arguments imply a slow stress relaxation rate of less than