Previous quantitative modeling of centrosome growth in C. elegans has suggested that centrosomes are autocatalytic droplets growing via phase separation in a limited pool of building blocks (Zwicker et al., 2014; Decker et al., 2011). Autocatalytic growth arises if the centrosome assembly rate increases with centrosome size, creating a size-dependent positive feedback (Figure 1B). To investigate if autocatalytic growth can ensure size equality of centrosomes, we considered a reaction-limited model of centrosome growth via stochastic assembly and disassembly of its subunits. Theoretical estimates indicate that the timescale of diffusion is much faster than the timescales of reactions observed in experiments. For instance, the scaffold formers diffuse over 5 – 10 μm in about 1 s while they have turnover timescale of ∼ 100 s (see Materials and methods section for more details). Although there are multiple essential components involved in PCM assembly (Dobbelaere et al., 2008; Conduit et al., 2010a, Conduit et al., 2014b), we first examined a one-component centrosome model to illustrate the role of autocatalytic growth on size control. The deterministic description for the growth of a centrosome pair is given by

where ${\displaystyle n_i(t)}$ is the number of subunits in ${\displaystyle i^{th}}$ centrosome (${\displaystyle i=1,2}$), ${\displaystyle k_0^{+}}$ and $k_1^{+}$ are the rate constants for non-cooperative and cooperative assembly, respectively, and ${\displaystyle k^{-}}$ is the disassembly rate constant. Equation 1 can be derived from the phase segregation model for centrosome assembly studied by Zwicker et al., 2014 (see Appendix), with ${\displaystyle k_0^{+}}$ and ${\displaystyle k_1^{+}}$ representing centriole activity and the strength of autocatalytic interaction, respectively. In Equation 1, ${\displaystyle \rho (t)}$ is the cytoplasmic concentration of centrosomal subunits, given by ${\displaystyle \rho (t)=(N-n_1(t)-n_2(t))/V_c}$ where ${\displaystyle V_c}$ is cell volume and N is the total amount of subunits in the cell. Centrosome volume is given by ${\displaystyle V_i(t)=n_i(t)\delta v}$, where ${\displaystyle \delta v}$ is the effective volume occupied by a single subunit. As shown before (Zwicker et al., 2014), this model can capture the essential quantitative features of the growth of a single centrosome (Figure 1C), including sigmoidal growth curve, temporal control of size and scaling of centrosome size with cell size. However, this model is unable to ensure the size equality of two identical centrosomes growing from a shared subunit pool. Stochastic simulation of this model, using the Gillespie algorithm (see Materials and methods), shows a significant difference in steady-state size even with a small initial size difference (Figure 1C).

It is instructive to first compare two opposite limits of the model, ${\displaystyle k_0^{+}=0}$ (purely autocatalytic growth) and ${\displaystyle k_1^{+}=0}$ (non-cooperative growth). For ${\displaystyle k_0^{+}=0}$, Equation 1 can be interpreted as assembly and disassembly occurring throughout the PCM volume, with the assembly rate scaling with centrosome size. As a result, the centrosome with a larger initial size would end up growing to a larger steady-state size. Stochastic simulations of this model show that the ensemble-averaged absolute difference in centrosome size (${\displaystyle |\delta V|=|V_1-V_2|}$) increases with the initial centrosome size difference ${\displaystyle (\delta V_0)}$, indicating lack of robustness in size regulation (see Appendix 1 and Figure 2—figure supplement 1). On the other hand, the limit ${\displaystyle k_1^{+}=0}$ corresponds to a model where the assembly rate is size-independent, and material turnover is distributed throughout the PCM volume. This model guarantees size equality of a centrosome pair competing for a limiting subunit pool (see Appendix 1 and Figure 2—figure supplement 2), even in the presence of large initial size differences (Figure 2D), with the steady-state size given by ${\displaystyle V=k^{+}N\delta v/(k^{-}+2k^{+})}$. However, the resulting growth curve is non-sigmoidal, thus fails to capture experimental data in C. elegans (Decker et al., 2011; Zwicker et al., 2014).

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To quantify the robustness of size control, we measured the relative difference in steady-state centrosome size, ${\displaystyle |\delta V|/⟨V⟩}$, starting with an initial size difference ${\displaystyle \delta V_0\sim 0.01⟨V⟩}$, where ${\displaystyle |...|}$ denotes the absolute value and ${\displaystyle ⟨V⟩}$ is the ensemble average of centrosome size at steady-state. For a robust size regulation mechanism, the final size difference is expected to be independent of the initial size difference. The resulting size inequality is controlled by the rate constants ${\displaystyle k_0^{+}}$, ${\displaystyle k_1^{+}}$, ${\displaystyle k^{-}}$ and the pool size N. Our analysis shows that there is a relatively small region of the parameter space where the strength of the autocatalytic feedback is low enough to ensure a small difference in centrosome size (Figure 2A). Through linearization of the rate equations, we derive the analytical condition for size equality to be ${\displaystyle 2k_0^{+}+k^{-}{V}_c>k_1^{+}N}$ (see Appendix 3 for details). However, in this range of parameter values, the growth is essentially non-cooperative and the growth curve is not sigmoidal (Figure 2B). Larger size inequality is associated with higher values of ${\displaystyle k_1^{+}}$, when the growth dynamics is sigmoidal in nature (Figure 2C). For a detailed study of the lack of robustness in size regulation, please refer to Appendix 1 and Figure 2—figure supplement 3.

While our theoretical estimates suggest that centrosome growth is primarily reaction-limited, the increasing distance between centrosomes during maturation — especially in certain organisms or depending on cell size—could lead to a diffusion-limited growth scenario. To investigate how diffusion affects centrosome size regulation, we extended our model to include subunit diffusion (see Appendix 4). Our results indicate that diffusion does not qualitatively alter centrosome size regulation. Size inequality can be reduced when the diffusion constant is low or when centrosomes are far apart, though in this regime, the growth curves lose their characteristic sigmoidal shape (see Appendix 4 and Figure 2—figure supplement 4). Crucially, the presence of diffusion does not resolve the issue of robustness in size control; the size difference between centrosomes still increases with larger initial size disparities (Figure 2—figure supplement 4).

Since our model for centrosome growth is limited by a finite amount of subunits, it is capable of capturing centrosome size scaling with cell size (Figure 4A), in excellent agreement with experimental data (Decker et al., 2011; Zwicker et al., 2014). However, the extent of organelle size scaling with cell size depends on the assembly rate and becomes negligible when the assembly rate is not significantly higher compared to the disassembly rate (Figure 4B). In particular, centrosome size scaling is connected to the extent of subunit pool depletion, such that the steady-state cytoplasmic fraction of the subunits is low when centrosome size scales with the cell size and higher otherwise (Figure 4C).

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To understand how size scaling is regulated by the growth parameters, we derived a simplified analytical form (see Appendix 2) for the steady-state centrosome size given by

where $\delta v$ is the volume occupied by a centrosome subunit, ${\rho }_0$ is the total subunit density, and the enzymes are assumed to reach their steady-state abundance $E^{*}$ very fast. From the above expression, we can see that centrosome size $V$ will strongly scale with cell size $V_c$ when $k^{+},k^{*}\gg k^{-}{V}_c$. This result is reflected in the phase diagram of size scaling (measured as the slope $\sim \mathrm{d}V/\mathrm{d}{V}_c$), which shows stronger size scaling with increasing assembly rates (Figure 4D). The subunit pool depletion also increases with the assembly rates, reaching a state of almost complete depletion (i.e. $V\to {\rho }_0{V}_c\delta v$) as we approach the regime of strong size scaling (see Figure 4—figure supplement 1).

It is important to note here that size scaling with cell size reported here is different from the linear size scaling predicted by the canonical limiting pool model (Decker et al., 2011; Goehring and Hyman, 2012). Robust size control for multiple centrosomes requires size-dependent negative feedback and with this feedback, the size scaling with cell size becomes a feature achieved in a range of cell volumes by tuning growth rates. Interestingly, strong size scaling has been observed in C. elegans embryos (Decker et al., 2011), which are smaller in size ($\sim {10}^4\mu {\text{m}}^3$) than Drosophila embryos ($\sim {10}^6\mu {\text{m}}^3$) that do not exhibit size scaling with centrosome number (inferred from intensity data in Wong et al., 2022). This feature can be explained by our model in the regime of weaker size scaling, which is expected for larger system sizes (see Appendix 2 and Figure 4—figure supplement 1). Thus, the parameters of our model can be tuned to capture both sigmoidal and non-sigmoidal growth and strong or weak size scaling, without changing the nature of the molecular interactions that are largely conserved across organisms (Raff, 2019).

An essential aspect of centrosome size regulation is the modulation of centrosome size by centriole activity. In particular, it has been shown that the centrosome associated with a more active centriole will grow larger, resulting in centrosomes of unequal size (Januschke et al., 2013; Conduit and Raff, 2010b). Control of centriole activity-driven centrosome size asymmetry is important as this size asymmetry may play a crucial role in stem cell division as observed in Drosophila neuroblasts (Conduit and Raff, 2010b). We test the effectiveness of size regulation by studying the growth of a centrosome pair with different centriole activities, controlled by the values of the growth rate constants $k_0^{+}$ and $k^{+}$ for the autocatalytic (Equation 1) and the catalytic (Figure 3B) growth models, respectively (Figure 5A). For both the models, we bias the initial size of the centrosomes by assigning a smaller initial size ($V_0-\delta V_0$) to the centrosome with a higher centriole activity (i.e., $k_0^{+(1)}=k_0^{+}+\delta k_0^{+}$ or $k^{+(1)}=k^{+}+\delta k^{+}$). We then simulate the growth of $N_{\text{tot}}$ centrosome pairs and quantify the efficiency ($\epsilon$) of size control as the ratio of the number of cases ($N^{+}$) where the centrosome with higher growth rate ($k_0^{+}+\delta k_0^{+}$ or $k^{+}+\delta k^{+}$) becomes larger, to the total number of simulated pairs, $\epsilon =N^{+}/N_{\text{tot}}$.

Figure 5

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In the absence of any initial size difference (${\displaystyle \delta V_0=0}$), the catalytic growth model shows better control of differential growth-induced size asymmetry (Figure 5B), while the autocatalytic growth model shows wide variations in centrosome size difference (Figure 5C). We find that the catalytic growth model ensures that the centrosome with a larger $k^{+}$ (higher centriole activity) end up being larger, irrespective of the initial size difference (Figure 5D). This illustrates robust control of centrosome size asymmetry by controlling differences in centriole activity. By contrast, in the autocatalytic growth model, the efficiency of size control monotonically decreases with increasing initial size difference, reflecting the lack of robustness in size control (Figure 5E).

One major postulate of the one-component PCM model was that the enzyme pool was shared between the two centrosomes rather than being localized to each. Here, we support this assumption using a more realistic multi-component centrosome model that allows us to model the specific interactions between the enzyme and the centrosome components, making it possible to study the relative dynamics of the two main scaffold formers. While we draw parallels between this model and the interactions observed in Drosophila, the model should be relevant to other organisms where similar pathways are in action via functionally similar proteins.

Based on recent studies (Alvarez Rodrigo et al., 2019; Conduit et al., 2015b), we model the centrosomes with two essential scaffold-forming proteins, $a$ and $b$, whose assembly into the PCM scaffold is regulated by the kinase $E$. The total size of the PCM scaffold, $S$, and the centrosome volume $V$ are given by $S=S(a)+S(b)$ and $V=V_a+V_b$, where $S(a)$ ($S(b)$) and $V_a$ ($V_b$) denote the contribution to the scaffold size (in number of subunits) and the centrosome volume by the component $a$ ($b$). The molecular identities of these key components are listed in Table 2 for different organisms. In particular, for Drosophila, $a$ and $b$ can be identified as the scaffold forming proteins Spd-2 and Cnn, while $E$ represents the kinase Polo. It has been observed that Spd-2 and Cnn cooperatively form the PCM scaffold to recruit almost all other proteins involved in centrosome maturation (Conduit et al., 2014b). To effectively coordinate cooperative growth of the scaffold, Spd-2 proteins recruit the kinase Polo, which in turn phosphorylates Cnn at the centrosome (Alvarez Rodrigo et al., 2019). In the absence of Polo, Cnn proteins can bind to the scaffold but fall off rapidly, leading to diminished centrosome maturation (Alvarez Rodrigo et al., 2019; Woodruff et al., 2015).

We incorporated these experimental observations in our multi-component model as described in Figure 6A. We then test two different models for enzyme spatial distribution: (i) enzyme $E$ (Polo) is activated at each centrosome by the scaffold component $a$ (Spd-2), which then assembles the second component $b$ (Cnn) into the scaffold of that particular centrosome (for details see Appendix 5), and (ii) enzyme $E$ activated by the scaffold component $a$ is released in the cytoplasmic pool, promoting assembly of the $b$-scaffold at both centrosomes (for details see Appendix 5). In the first case, localized enzyme interaction exclusively enhances the growth of the individual centrosomes, creating an autocatalytic feedback that leads to size inequality of centrosomes (Figure 6B). Similar to model (Equation 1), the steady-state size difference between the two centrosomes increases with the increasing initial size difference, resulting in a failure of robust size control (Figure 6—figure supplement 1).

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We then considered the second case where the enzyme-mediated catalysis is shared between the growing centrosome pair. Experimental observations suggest a dynamic enzyme population around the centrosomes (Mahen et al., 2011; Kishi et al., 2009), with a turnover timescale much smaller than the scaffold forming proteins (Conduit et al., 2014b; Conduit et al., 2015a). These findings point towards the possibility that the enzyme is transiently localized in the centrosome during activation and the active enzyme is then released in the cytoplasmic pool that can enhance the growth of both the centrosomes (Figure 6A). We incorporate this shared catalysis mechanism in the second model where $a$ activates the enzyme to $E^{*}$ which then gets released in the cytoplasm, facilitating $b$-scaffold expansion in both the centrosomes (see Appendix 5 for details). This growth mechanism is able to robustly control centrosome size equality (Figure 6—figure supplement 2), giving rise to the characteristic sigmoidal growth dynamics (Figure 6C), where the first scaffold former $a$ is smaller in amount than the second, enzyme-aided component $b$. This difference in the abundances of $a$ and $b$ proteins, when translated into their respective radial spread from the centrosome center ($R\propto V^{1/3}$), bears close resemblance with the relative spread in Spd-2 and Cnn observed in the experiments, where the Cnn spread is twice as large as Spd-2 (Conduit et al., 2014b; Alvarez Rodrigo et al., 2019; Figure 6D). The active enzyme dynamics also resembles the observed pulse in Polo dynamics at the beginning of centrosome maturation (Wong et al., 2022; Figure 6C). Overall, the two-component model provides crucial insights into the role of shared catalytic growth on centrosome size control and lays the theoretical foundation for further investigations into the molecular processes that govern centrosome assembly.

In this article, we examined quantitative models for centrosome growth via assembly and disassembly of its constituent building blocks to understand how centrosome size is regulated during maturation. Although there is no generally accepted model for centrosome size regulation, previous studies Conduit et al., 2014b; Zwicker et al., 2014; Conduit et al., 2014a; Alvarez Rodrigo et al., 2019; Woodruff et al., 2015; Conduit et al., 2015b have suggested that centrosome assembly is cooperative and driven by a positive feedback mechanism. It has been quantitatively shown that an autocatalytic growth model (Zwicker et al., 2014) captures the cooperative growth dynamics of individual centrosomes as well as their size scaling features. However, as we showed here, autocatalytic growth does not guarantee the size equality of two centrosomes growing from a shared subunit pool. The resultant size inequality increases with the initial size difference between the centrosomes, indicating a lack of robustness in size control. This observation remains valid even within models where autocatalysis is not explicitly invoked, but emerges from positive feedback between PCM components (Alvarez Rodrigo et al., 2019). For instance, the positive feedback between Spd-2 and Cnn within Drosophila centrosomes results in the accumulation of more Cnn where Spd-2 is abundant. This, in turn, amplifies the retention of Spd-2 and binding of Cnn, culminating in a size-dependent positive feedback (akin to autocatalytic feedback) in PCM assembly. Given the current molecular understanding, it remains an open question whether localized assembly around the centriole, driven by autocatalytic feedback, is sufficient to furnish a robust mechanism for centrosome size regulation. It is important to note that the results shown in Zwicker et al., 2014 indicate that the Ostwald ripening can be suppressed by the catalytic activity of the centriole, therefore stabilizing the centrosomes against coarsening by Ostwald ripening. However, if size discrepancy arises from the growth process (e.g. due to autocatalysis) the timescale of relaxation for such discrepancy is unclear from the above-mentioned result. We show that for any appreciable amount of positive feedback, the system cannot achieve equal size in a physiologically relevant timescale (Figure 2—figure supplement 3).

Following recent experiments on the molecular mechanisms governing centrosome assembly, we constructed an enzyme-mediated catalytic growth model that not only describes cooperative growth behavior but also ensures robustness in size equality of the two maturing centrosomes. The enzyme Polo-like kinase (PLK1) that coordinates centrosome growth (Conduit et al., 2014a; Woodruff et al., 2015; Ohta et al., 2021; Alvarez Rodrigo et al., 2019), gets phosphorylated in the centrosome and has a much faster turnover rate than the centrosome scaffold forming proteins Spd-2 and Cnn (Conduit et al., 2014b). Experiments (∼ 5 μm² s^-1 Mahen et al., 2011) and theoretical estimates (see Materials and methods) indicate high PLK1 diffusivity such that PLK1 transfer between the centrosome pair (assuming at a distance of ∼ 5 – 10 μm) may occur within a few seconds which is much faster than the timescale of centrosome growth (∼ 1000 s). This indicates that the kinase dynamics is not diffusion-limited, consistent with recent studies reporting negligible gradient in cytoplasmic Polo in C. elegans embryo (Barbieri et al., 2022). These insights led us to hypothesize that the kinase, once activated at the centrosome, could be released into the cytoplasm, becoming part of a shared pool of enzymes. This pool would then catalyze the growth of both centrosomes without any inherent bias. While we theoretically demonstrated that this mechanism of shared catalysis can robustly regulate centrosome size, it is important to acknowledge that the specific predictions concerning enzyme dynamics can only be validated through further experiments.

To further explore the role of enzymes in mediating centrosome growth and predict the consequence of an enzyme pool that is not shared equally by the two centrosomes, we extended our single-component model of catalytic growth to a multi-component model. This extended model incorporates the interactions PCM scaffold-forming proteins (Spd-2 and Cnn in Drosophila) and the enzyme Polo kinase. Using this model, we showed that localized catalysis by the enzyme—indicative of an unshared pool—leads to significqnt size differences in the centrosomes. While direct experimental validation of a shared enzyme pool remains outstanding, it is intriguing to consider the findings that a centrosome-anchored Plk1 construct (Plk1-AKAP) induces anomalous centrosome maturation and defective spindle formation (Kishi et al., 2009).

Our findings reveal that centrosome size increases with increasing enzyme concentration and that centrosome growth is inhibited in the absence of the enzyme (Figure 6—figure supplement 2). Since the activity of the Polo kinase is cell-cycle dependent (Hamanaka et al., 1995; Uchiumi et al., 1997), we further explored the dynamics of centrosome growth with a time-dependent dynamics of the enzyme. We found that centrosome growth can be triggered by switching on the enzyme dynamics and centrosome size was reduced when the enzyme was switched off (Figure 6—figure supplement 2). Importantly, it supported the experimental observation that a continuous Polo activity is required to maintain the PCM scaffold (Mahen et al., 2011; Cabral et al., 2019). Many key features of centrosome growth such as the sigmoidal growth curve and size scaling behavior can be modulated in our model by changing the growth rate constants and enzyme concentration, while conserving the underlying molecular mechanisms for assembly. This opens up the possibility that the catalytic growth model may be broadly relevant to other organisms where homologous proteins (Table 2) play similar functional roles in regulating centrosome growth (Conduit et al., 2015b).

Aside from capturing the existing data on the dynamics of centrosome growth, our catalytic growth model makes specific predictions that can be tested in future experiments. Firstly, our model posits the sharing of the enzyme between the two centrosomes. This can potentially be experimentally tested through immunofluorescent staining of the kinase or by constructing FRET reporter of PLK1 activity (Allen and Zhang, 2006), where it can be studied if the active form of the PLK1 is found in the cytoplasm around the centrosomes indicating a shared pool of active enzyme. Another possible future experiment can be performed based on photoactivated localization microscopy (PALM; Sillibourne et al., 2011) where fluorescently tagged enzyme can be selectively photoactivated in one centrosome and intensity can be measured at the other centrosome to find the extent of enzyme sharing between the centrosomes. It is important to to acknowledge that while we exclusively focused on Polo kinase as the sole enzyme, this shared catalytic activity might also involve other molecular players that interact with Polo, such as cyclin B/Cdk1 (Kishi et al., 2009). Moreover, our model provides explicit predictions regarding the enzyme’s role in influencing centrosome size and growth. These predictions encompass the anticipated increase in centrosome size with increasing enzyme concentration, the ability to modify the shape of the sigmoidal growth curve, and the manipulation of centrosome size scaling patterns by perturbing growth rate constants or enzyme concentrations. Additionally, the model suggests inducing a shift from strong size scaling to weak size scaling through the reduction of PCM assembly rate or via cytoplasmic subunit pool depletion.

Secondly, an implication of our model is the robust regulation of centrosome size through catalytic PCM assembly during maturation. One direct avenue for testing this result is to observe the dynamics of two initially unequal-sized centrosomes during the early maturation phase. The catalytic growth model predicts that the final size difference of the centrosomes is uncorrelated to their initial size disparity while they are strongly correlated according to the autocatalytic growth model (Figure 3—figure supplement 2). The catalytic model also predicts the final size inequality will increase with decreasing subunit pool size. These predictions can be experimentally examined by inducing varying centrosome sizes at the early stage of maturation for different expression levels of the scaffold former proteins. It is important to note here that the initial size difference has to be induced while keeping the centrioles unaffected otherwise it may create size difference due to differences in centriole activity (Januschke et al., 2013; Conduit and Raff, 2010b; Zwicker et al., 2014). Experimentally validating these predictions will play a pivotal role in building a quantitative understanding of centrosome size regulation during mitosis and in clearly distinguishing the catalytic growth mechanism from the autocatalytic growth.

We use the Gillespie, 1977 algorithm to simulate the stochastic growth of one or multiple structures from a common pool of subunits. At any time $t$ the Gillespie algorithm uses two random variables drawn from an uniform distribution ($r_1,r_2\in 𝒰(0,1)$), and the instantaneous propensities for all of the possible reactions to update the system in time according to the defined growth law. The propensities of the relevant reactions, that is the assembly and disassembly rates of the i^th structure are given by ${\displaystyle K_i^{on}}$ and ${\displaystyle k_i^{off}}$, respectively. For example, for the autocatalytic growth model described in Equation 1, these propensities are functions of subunit pool size (N) and structure size (n_i),

where we are considering growth of M structures from a shared pool. The Gillespie algorithm computes the time for the next reaction at $t+\tau$ given the current state of the system (i.e. the propensities for all reactions) at time $t$ where $\tau$ is given by-

where ${ℛ}_i$ is the propensity of $i^{th}$ reaction and $C$ is the total number of all possible reactions. The second random variable $r_2$ is used to select the particular reaction ($j^{th}$ reaction) that will occur at $t+\tau$ time such that

The condition for the first reaction ($j=1$) is $0\le r_2<\frac{{ℛ}_1}{{\sum }_{i=1}^C{ℛ}_i}$. The two steps defined by Equation 8 and Equation 9 are used recursively to compute the growth dynamics in time.

We used the Gillespie algorithm to find the stochastic trajectories of the above discussed deterministic (mass action kinetics) dynamics of the autocatalytic growth model and its various limits. See Catalytic growth in a shared enzyme pool ensures robust control of centrosome size for the corresponding chemical master equations. Similarly for the catalytic growth and two-component model, we find the stochastic trajectories via the Gillespie algorithm from the reactions given in Figures 3B and 6A.

Although we use single subunit and two subunit models of growth, we have used same value for the volume occupied by the subunit $\delta v$. We estimate the value of $\delta v$ from the molecular weight of SPD-5 which is 135 kDa (Hamill et al., 2002). Taking the protein mass density to be $1.4{\text{g}\text{cc}}^{-1}$(Fischer et al., 2004) and the PCM volume fraction to be ∼ 0.1 (Mahen et al., 2011), we estimate the volume occupied by SPD-5 in PCM to be $0.1\times 162\times {10}^{-7}\mu {\text{m}}^3\sim 2\times {10}^{-4}\mu {\text{m}}^3$.

We assume reaction-limited dynamics for centrosome maturation, meaning that the cytosolic diffusion of scaffold-forming proteins and the enzyme is much faster than their reaction rates. Here, we quantitatively discuss the timescales of protein diffusion and reaction based on their mass and fluorescent recovery after photobleaching (FRAP) data. The scaffold-forming proteins have a mass range of 100–150 kDa, while the enzyme mass is approximately 50–70 kDa. Using the Stokes-Einstein relation, which predicts that the diffusion constant scales inversely with protein radius (R), that is D ∼ R^-1 ∼ M^-1/3 where M is the protein mass, we estimate their diffusion constants. Based on the cytosolic diffusion constant of 30 μm²s^-1 for GFP (mass 30 kDa; Milo et al., 2010), we estimate diffusion constants of 17–20 μm²s^-1 for the scaffold-forming proteins and about 24 μm²s^-1 for the enzyme.

The separation distance between centrosomes (d) during maturation depends on the developmental stage of C. elegans and Drosophila embryos, but in later stages, it ranges between 5 and 10 μm (Decker et al., 2011; Alvarez Rodrigo et al., 2019). Using the diffusion timescale ${\displaystyle {\tau }_D=\frac{L^2}{6D}}$, we estimate diffusion times of about 1 s for scaffold-forming proteins and 0.1–0.5 s for the enzyme. These diffusion times are significantly shorter than the turnover times observed in FRAP experiments, which are around 100 s for scaffold-forming proteins and 10 s for the enzyme in Drosophila (Conduit et al., 2014b; Conduit and Raff, 2010b) and C. elegans (Woodruff et al., 2017). This discrepancy suggests that diffusion of the relevant proteins and enzyme is considerably faster than their reaction rates, supporting the use of a reaction-limited model for studying the self-assembly of the PCM during centrosome maturation. Further experiments to directly measure diffusion constants of these proteins are necessary for a more detailed understanding of the role of diffusion in centrosome size regulation.

Here we present the derivation of the autocatalytic growth model of centrosomes developed by Zwicker et al., 2014, where centrosomes are described as phase-segregated liquid droplets. We develop an equivalent kinetic model and study the centrosome size evolution using the corresponding stochastic description. Specifically, we consider PCM droplet growth in the limit of strong phase segregation and reaction-limited growth (i.e., fast diffusion of PCM components). The growth of the centrosome volume V is given by

where ${\displaystyle {\varphi }^A}$ and ${\displaystyle {\varphi }^B}$ are the volume fractions for the soluble (${\displaystyle A}$) and the phase segregated (${\displaystyle B}$) forms of the PCM components, respectively, and ${\displaystyle {\psi }_{-}}$ is the volume fraction of $B$ inside the droplet. Here ${\displaystyle k_{AB}}$, ${\displaystyle k_{BA}}$ and ${\displaystyle k}$ are the reactions rates for ${\displaystyle A\to B}$, $B\to A$ and ${\displaystyle AB\to B}$ respectively. The bulk volume fraction of the ${\displaystyle A}$ and ${\displaystyle B}$ forms, away from the droplet, is given by ${\displaystyle {\varphi }_0^A}$ and ${\displaystyle {\varphi }_0^B}$ respectively and ${\displaystyle {\varphi }_1^A}$ is the volume fraction of ${\displaystyle A}$ inside the droplet. The chemical activity of the centriole, centrosome number, and volume of the cell are given by ${\displaystyle Q}$, ${\displaystyle n_c}$, and ${\displaystyle V_c}$, respectively.

To model the growth of a pair of centrosomes, we write the equation for size kinetics in terms of the number of incorporated subunits. As assumed by Zwicker et al., 2014, we neglect spontaneous production of phase segregated form ${\displaystyle B}$ away from the centriole (${\displaystyle k_{AB}=0}$), and the volume fraction of ${\displaystyle B}$ inside the droplet is considered to be unchanged, i.e., ${\displaystyle {\psi }_{-}}$= constant. The volume fractions of $A$ in the bulk and inside the droplet are given by

where ${\displaystyle \overline{\varphi }}$ is the average volume fraction of the total PCM material and $V_{1,2}$ are size of the two centrosomes/droplets. The average volume fraction is given by ${\displaystyle \overline{\varphi }=\frac{V_A+V_B}{V_c}=\frac{N\delta v}{V_c}}$, where ${\displaystyle N}$ is the total number of PCM subunits that remains constant during the droplet growth and ${\displaystyle \delta v}$ is the volume occupied by a single subunit. The volume fraction of subunits inside the droplet is given by ${\displaystyle {\psi }_{-}=\frac{n_B^i\delta v}{V_i}}$, where ${\displaystyle n_B^i}$ is the number of ${\displaystyle B}$ subunits inside the ${\displaystyle i^{th}}$ droplet (centrosome) and ${\displaystyle V_i}$ is the volume of that centrosome. We can now rewrite ${\displaystyle {\varphi }_0^A=\frac{N_{av}\delta v}{V_c}}$ and ${\varphi }_1^A=(1-{\psi }_{-})\left(\frac{N_{av}\delta v}{V_c}\right)$ where $N_{av}=N-n_B^1-n_B^2$ is the available amount of subunits that can contribute to the droplet growth. Finally using the definition of ${\psi }_{-}$ we get the $n_B^i$ dynamics given by

We shall drop the index $B$ and write $n_B^i\to n_i$ and rewrite the above equation as

This above description then can be rewritten as

where $k_0^{+}=(1-{\psi }_{-})Q$, $k_1^{+}=(1-{\psi }_{-})k\delta v$ and $k^{-}=k_{BA}$. This final equation (Equation 15) can be recognized as the main text Equation 1.

The stochastic growth corresponding to the above discussed deterministic growth dynamics can be described by a chemical master equation for the joint probability $P(n_1,n_2,t)$:

where $P(n_1,n_2,t)$ is the probability of having centrosomes with size $n_1\delta v$ and $n_2\delta v$ at time $t$. We numerically simulate the growth dynamics using Gillespie, 1977 first algorithm , with the transition probabilities described in the master equation (Equation 17).

Studying the phase portrait of the centrosome pair dynamics reveals the origin of size inequality (Figure 2—figure supplement 3A-D). In the regime of low $k_0^{+}$ values, the growth is strongly autocatalytic and the phase portrait shows a quasi line-attractor where the solutions which are away from the $V_1=V_2$ line get trapped and cannot reach the fixed point in a biologically relevant timescale (Figure 2—figure supplement 3A, C). With increasing $k_0^{+}$ value (weakly cooperative or non-cooperative growth) the quasi line-attractor goes away and solutions reach the equal size fixed point quickly (i.e., size inequality is small) but the sigmoidal nature is absent in the size dynamics (Figure 2—figure supplement 3E, F) in this regime.

Here we consider the two limits of the autocatalytic growth model (Equation 15): purely autocatalytic limit ($k_0^{+}=0$) and purely non-autocatalytic limit ($k_1^{+}=0$). First, we shall consider the purely autocatalytic limit where the resulting growth dynamics can be written as

where $k^{+}=k_1^{+}$ is the assembly rate constant. The concentration of available subunits is given by $\rho =\left(N-{\sum }_{i=1}^2{n}_i\right)/V_c=\frac{N_{av}}{V_c}$. This resulting growth model can be described as growth via the assembly and disassembly of subunits throughout the volume. The steady-state solution to the above equations (i.e., ${\dot{n}}_1^{*}=0$ and ${\dot{n}}_2^{*}=0$) constitute a system of underdetermined equations that give rise to a line attractor (a line of fixed points) given by $n_1^{*}+n_2^{*}=N-\frac{k^{-}{V}_c}{k^{+}}$.

The stochastic description of the growth can be given by the following chemical master equation for the joint probability

where $P(n_1,n_2,t)$ is the probability of having centrosomes with sizes $n_1\delta v$ and $n_2\delta v$ at time $t$. In this purely autocatalytic limit, the size of a single centrosome is well regulated but two centrosomes growing from a shared subunit pool exhibit large size inequality (Figure 2—figure supplement 1A). The phase portrait for a growing pair of centrosomes reveals the line-attractor where the solutions get trapped away from the equal size point (Figure 2—figure supplement 1B). This leads to a lack of robustness in size regulation as size inequality ($|\delta V|$) increases with increasing initial size difference $\delta V_0$ (Figure 2—figure supplement 1C).

The purely non-cooperative growth dynamics can be described by

where $k^{+}=k_0^{+}$ is the assembly rate constant. We can calculate a unique stable fixed point of equal size ($V_1=V_2=n^{*}\delta v$), given by $n^{*}=\frac{k^{+}N}{k^{-}{V}_c+2k^{+}}$. This model describes centrosome growth via localized assembly (i.e., centrosome size-independent assembly) and distributed disassembly throughout the volume. The stochastic description of the growth can be given by the following chemical master equation for the joint probability

This growth model exhibits robust size control for a centrosome pair, giving rise to centrosomes of similar size even in the presence of large initial size differences (Figure 2—figure supplement 2A, B). Unlike the case of purely autocatalytic growth, there is no line-attractor in this model and the solutions leads to the fixed point with equal-sized centrosomes (Figure 2—figure supplement 2C). Though this model leads to robust size control, there is no cooperativity in growth and the resulting size dynamics is non-sigmoidal. Steady-state probability distribution of size can be analytically obtained (as a finite sum) for these two growth models given by Equation 18 and Equation 19 and shows that the size distribution is approximately uniform in the case of purely autocatalytic growth and narrowly distributed in the case of purely non-autocatalytic growth Banerjee and Banerjee, 2022.

Here we present the deterministic description of the catalytic growth model presented in the main text. We begin by describing the dynamics of a single centrosome of $n$ subunits, whose size is denoted as $S_n$. The subunits in the cytoplasmic pool can be either in the active form $S_1^{*}$ or the inactive form $S_1$. The enzymes can also be in two forms - an active enzyme pool of abundance $E^{*}$, and an inactive enzyme pool of size $E$, respectively. We can completely describe centrosome growth from the dynamics of $S_n$, $S_1^{*}$ and $E^{*}$ by substituting $S_1$ and $E$ with the constraints arising from limited pools: $S_1=N-S_n-S_1^{*}=N_{av}$ and $E=N_E-E^{*}-S_1^{*}=N_{av}^E$. Here $N={\rho }_0{V}_c$ is the total amount of subunits, $N_E={\rho }_E{V}_c$ is the total amount of enzymes and $V_c$ is the volume of the cell. The dynamic rate equations are given by

The analytical steady-state solutions of the above equations are cumbersome and not very insightful. Assuming that all centrosomes attain equal sizes (using results presented in the main text), then the above equations can be easily extended to describe the growth of $M$ centrosomes, given by

where $N_{av}=N-M{S}_n-S_1^{*}$.

To understand how centrosome size scales with cell size and centrosome number, we make a simplifying assumption of fast enzyme activation dynamics, i.e., the active enzyme concentration reaches steady state very fast. This will enable us to obtain useful analytical solutions for steady state centrosome size. Solving for $S_n$ and $S_1^{*}$, assuming a steady state enzyme abundance $E^{*}$, we obtain the steady-state centrosome size

With $M=1$, we derive the expression shown in the main text. Centrosome size scaling with cell size can be obtained when $k^{+}M\gg k^{-}{V}_c$ or $k^{*}M\gg k^{-}{V}_c$. To obtain a quantitative measure of size scaling, we evaluate the slope of centrosome size with cell size given by

These results show that the extent of size scaling will become weaker for larger system size (or cell/organism size) if other growth rates are similar (Figure 4—figure supplement 1A). We can estimate the extent of pool depletion using the cytoplasmic fraction of subunits at steady-state (combined amount of $S_1$ and $S_1^{*}$), given by

The subunit pool depletion is directly connected to the extent of size scaling with stronger size scaling occurring at higher pool depletion (Figure 4—figure supplement 1B).

Here consider a simple example of the growth of $M$ structures in a shared pool of $N$ subunits in volume $V_c$. We assume a linear size-dependent negative feedback to growth rate. Using the prior knowledge of robust size control in this case, we can write down the size ($n$ in subunits) dynamics in terms of a single structure:

Notice that the second term in the RHS $\frac{k^{+}Mn}{V_c}$ embodies the pool depletion rate. We can define a bare rate of pool depletion as $𝒟=k^{+}M/V_c$. The steady-state size is given by:

where $N={\rho }_0{V}_c$. It thus becomes clear that we obtain strong size scaling with system size and structure number in the regime $𝒟=\frac{k^{+}M}{V_c}\gg k^{-}$, when the depletion rate is much higher that the disassembly rate ($k^{-}$) or pool replenishing rate. We can identify the two pool depletion rates in catalytic growth from $S_n$ and $S_1^{*}$ dynamics in Equation 20 as ${𝒟}_1=\frac{k^{+}M}{V_c}$ and ${𝒟}_2=\frac{k^{*}M}{V_c}$. Thus, the condition for strong size scaling comes from the condition of strong pool depletion $\frac{k^{+}M}{V_c},\frac{k^{*}M}{V_c}\gg k^{-}$, similar to the example case described above.

Our analysis indicates that the size scaling of intracellular organelles is a result of fine-tuning of growth parameters rather than due to the physical constraint of having a limited pool of building blocks. Structures growing in a shared limited pool of subunits with size-dependent negative feedback (which is the case in centrosome growth) require information of the system size ($V_c$) and number of structures ($M$) to scale with these quantities. This information is encoded in the depletion rate, i.e., $𝒟=k^{+}M/V_c$. Hence, when strong depletion of the subunit pool sets the structure size, it enables the sensing of the system size and structure number, resulting in strong size scaling.

During the embryonic development of C elegans, as the centrosome number increases with the progression of the development, centrosome size decreases as $\sim 1/M$, where $M$ is the centrosome number Decker et al., 2011. Interestingly, during the development of D Melanogaster centrosome size scaling with centrosome number is negligible in the cycles 11 to 12 (Wong et al., 2022) during which centrosome number increases by $\sim 1000$. This apparent disparity in behaviour of centrosome growth can be understood from the difference in size between the two embryos. The Drosophila embryo is much larger in size at $500\mu m\times 180\mu m$ (L x W) compared to the C elegans embryo which is $50\mu m\times 30\mu m$ (L x W). Our theory predicts strong centrosome size scaling in the initial cycles of C elegans embryo and almost no size scaling for Drosophila embryo during cycles 11 to 12 (Figure 4—figure supplement 1C-D). Thus, explaining how distinctly different quantitative features of growth can emerge from the same underlying mechanisms.