Catalytic growth in a shared enzyme pool ensures robust control of centrosome size

Reviewer #1 (Public Review):

The work analyzes how centrosomes mature before cell division. A critical aspect is the accumulation of pericentriolar material (PCM) around the centrioles to build competent centrosomes that can organize the mitotic spindle. The present work builds on the idea that the accumulation of PCM is catalyzed either by the centrioles themselves (leading to a constant accumulation rate) or by enzymes activated by the PCM itself (leading to autocatalytic accumulation). These ideas are captured by a previous model derived for PCM accumulation in C. elegans (ref. 8) and are succinctly summarized by Eq. 1. The main addition of the present work is to allow the activated enzymes to diffuse in the cell, so they can also catalyze the accumulation of PCM in other centrosomes (captured by Eqs. 2-4). The authors claim that this helps centrosomes to reach the same size, independent of potential initial mismatches.

A strength of the paper is the simplicity of the equations, which are reduced to the bare minimum and thus allow a detailed inspection of the physical mechanism. One shortcoming of this approach is that all equations assume that the diffusion of molecules is much faster than any of the reactive time scales, although there is no experimental evidence for this.

Another shortcoming of the paper is that it is not clear what species the authors are investigating and how general the model is. There are huge differences in centrosome maturation and the involved proteins between species. However, this is not mentioned in the abstract or introduction. Moreover, in the main body of the paper, the authors mention C. elegans on pages 2 and 3, but refer to Drosophila on page 4, switching back to C. elegans on page 5, and discuss Drosophila on page 6. This is confusing and looks as if they are cherry-picking elements from various species. The original model in ref. 8 was constructed for C. elegans and it is not clear whether the autocatalytic model is more general than that. In any case, a more thorough discussion of experimental evidence would be helpful.

The authors show convincingly that their model compensates for initial size differences in centrosomes and leads to more similar final sizes. These conclusions rely on numerical simulations, but it is not clear how the parameters listed in Table 1 were chosen and whether they are representative of the real situation. Since all presented models have many parameters, a detailed discussion on how the values were picked is indispensable. Without such a discussion, it is not clear how realistic the drawn conclusions are. Some of this could have been alleviated using a linear stability analysis of the ordinary differential equations from which one could have gotten insight into how the physical parameters affect the tendency to produce equal-sized centrosomes.

The authors use the fact that their model stabilizes centrosome size to argue that their model is superior to the previously published one, but I think that this conclusion is not necessarily justified by the presented data. The authors claim that "[...] none of the existing quantitative models can account for robustness in centrosome size equality in the presence of positive feedback." (page 1; similar sentence on page 2). This is not shown convincingly. In fact, ref 8. already addresses this problem (see Fig. 5 in ref. 8) to some extent. More importantly, the conclusion seems to largely be based on the analysis shown in Fig. 2A, but the parameters going into this figure are not clear (see the previous paragraph). In particular, the initial size discrepancy of 0.1 µm^3 seems quite large, since it translates to a sphere of a radius of 300 nm. A similarly large initial discrepancy is used on page 3 without any justification. Since the original model itself already showed size stability, a careful quantitative comparison would be necessary.

The analysis of the size discrepancy relies on stochastic simulations (e.g., mentioned on pages 2 and 4), but all presented equations are deterministic. It's unclear what assumptions go into these stochastic equations, and how they are analyzed or simulated. Most importantly, the noise strength (presumably linked to the number of components) needs to be mentioned. How is this noise strength determined? What are the arguments for this choice? This is particularly crucial since the authors quote quantitative results (e.g., "a negligible difference in steady-state size (∼ 2% of mean size)" on page 4).

Moreover, the two sets of testable predictions that are offered at the end of the paper are not very illuminative: The first set of predictions, namely that the model would anticipate an "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.", are so general that they apply to all models describing centrosome growth. Consequently, these observations do not set the shared enzyme pool apart and are thus not useful to discriminate between models. The second part of the first set of predictions about shifting "size scaling" is potentially more interesting, although I could not discern whether "size scaling" referred to scaling with cell size, total amount of material, or enzymatic activity at the centrioles. The second prediction is potentially also interesting and could be checked directly by analyzing published data of the original model (see Fig. 5 of ref. 8). It is unclear to me why the authors did not attempt this.

Taken together, I think the shared enzyme pool is an interesting idea, but the experimental evidence for it is currently lacking. Moreover, the model seems to make little testable predictions that differ from previous models.

Reviewer #2 (Public Review):

Summary:

In this paper, Banerjee & Banerjee argue that a solely autocatalytic assembly model of the centrosome leads to size inequality. The authors instead propose a catalytic growth model with a shared enzyme pool. Using this model, the authors predict that size control is enzyme-mediate and are able to reproduce various experimental results such as centrosome size scaling with cell size and centrosome growth curves in C. elegans.

The paper contains interesting results and is well-written and easy to follow/understand.

Suggestions:

● In the Introduction, when the authors mention that their "theory is based on recent experiments uncovering the interactions of the molecular components of centrosome assembly" it would be useful to mention what particular interactions these are.
● In the Results and Discussion sections, the authors note various similarities and differences between what is known regarding centrosome formation in C. elegan and Drosophila. It would have been helpful to already make such distinctions in the Introduction (where some phenomena that may be C. elegans specific are implied to hold centrosomes universally). It would also be helpful to include more comments for the possible implications for other systems in which centrosomes have been studied, such as human, Zebrafish, and Xenopus.
● For Fig 1.C, the two axes are very close to being the same but are not. It makes the graph a little bit more difficult to interpret than if they were actually the same or distinctly different. It would be more useful to have them on the same scale and just have a legend.
● The authors refer to Equation 1 as resulting from an "active liquid-liquid phase separation", but it is unclear what that means in this context because the rheology of the centrosome does not appear to be relevant.
● The authors reject the non-cooperative limit of Eq 1 because, even though it leads to size control, it does not give sigmoidal dynamics (Figure 2B). While I appreciate that this is just meant to be illustrative, I still find it to be a weak argument because I would guess a number of different minor tweaks to the model might keep size control while inducing sigmoidal dynamics, such as size-dependent addition of loss rates (which could be due to reactions happen on the surface of the centrosome instead of in its bulk, for example). Is my intuition incorrect? Is there an alternative reason to reject such possible modifications?
● While the inset of Figure 3D is visually convincing, it would be good to include a statistical test for completeness.
● The authors note that the pulse in active enzyme in their model is reminiscent of the Polo kinase pulse observed in Drosophila. Can the authors use these published experimental results to more tightly constrain what parameter regime in their model would be relevant for Drosophila? Can the authors make predictions of how this pulse might vary in other systems such as C. elegans?
● The authors mention that the shared enzyme pool is likely not diffusion-limited in C. elegans embryos, but this might change in larger embryos such as Drosophila or Xenopus. It would be interesting for the authors to include a more in-depth discussion of when diffusion will or will not matter, and what the consequence of being in a diffusion-limit regime might be.
● The authors state "Firstly, our model posits the sharing of the enzyme between both centrosomes. This hypothesis can potentially be experimentally tested through immunofluorescent staining of the kinase or by constructing FRET reporter of PLK1 activity." I don't understand how such experiments would be helpful for determining if enzymes are shared between the two centrosomes. It would be helpful for the authors to elaborate.

Author Response

We are grateful to the editor and the reviewers for recognizing the importance of our theoretical study on the mechanisms of centrosome size control. We appreciate their thoughtful critiques and suggested improvements, all of which we intend to address in the revised manuscript as outlined below. We acknowledge that the experimental evidence supporting the proposed theory is currently incomplete. We anticipate that our study will serve as inspiration for future experiments aimed at testing the proposed theory.

As noted by both reviewers, our model is built on the assumption that the diffusion of molecular components is much faster than any reactive time scales. To explore the impact of diffusion on centrosome size regulation, we are presently working on a spatial model of centrosome growth within a spatially extended system. Our objective is to analyze the influence of diffusion, and we plan to integrate these findings into the revised manuscript.

To address the concerns raised by both the reviewers regarding the applicability of our model to various organisms, we plan to revise the manuscript to clearly delineate the parameter ranges within which our model could be relevant for different organisms such as C. elegans or Drosophila. While centrosomal components may vary among different organisms, the underlying pathways of interactions exhibit similarities. Leveraging the generality of our theory, it has the capability to capture diverse centrosomal growth behaviors contingent on the parameter choices. Our objective is to emphasize these distinctions, illustrating how the modulation of growth cooperativity and enzyme concentration can influence size regulation and size scaling behaviors. Given the limited availability of quantitative experimental data across diverse organisms, we recognize the challenge in directly comparing our theory with data. Nevertheless, we are committed to presenting a thorough motivation for such comparisons to prevent any confusion or readability issues.

We acknowledge the reviewers' concerns regarding the limited details provided on the simulation methods and the rationale behind the choice of model parameters. To address this, we will provide detailed explanations on the stochastic simulations, how the model parameters were calibrated, accompanied by appropriate references for the selected parameter values. Additionally, we thank reviewer 1 for the excellent suggestion to incorporate a linear stability analysis of the ordinary differential equations underlying the model. This analysis will offer valuable insights into how the physical parameters of the model influence the tendency to produce equal-sized centrosomes, and we are committed to including this in the revised manuscript. Additionally, we thank reviewer 2 for proposing the use of Polo pulse dynamics to more precisely constrain the parameter regime for centrosome growth dynamics in Drosophila. We will strive to incorporate this into the revised manuscript, recognizing the challenge of quantitatively interpreting centrosome size or subunit concentration values from experimental data on fluorescence intensities. We also plan to discuss enzyme pulse dynamics in C. elegans in the revised manuscript, as it presents a valuable prediction from our model.

We disagree with reviewer 1's assertion that Reference 8 (Zwicker et al., PNAS 2014) effectively addresses the robustness of centrosome size equality in the presence of positive feedback. The linear stability analysis presented in Figure 5 of Reference 8 demonstrates stability of centrosome size around the fixed point, leading to the inference that Ostwald ripening can be inhibited by the catalytic activity of the centriole. In our manuscript (see Supplementary Figure 3), we demonstrate that the existence of the stable fixed point does not necessarily give rise to equal-sized centrosomes due to the slow dynamics of the solution around the fixed point. With an appreciable amount of positive feedback in the growth dynamics, the solution moves very slowly around the fixed point (similar to a line attractor), and cannot reach the fixed point within a biologically relevant timescale leaving the centrosomes at unequal sizes. Therefore, we argue that the model in Reference 8 lacks a robust mechanism for size control in the presence of autocatalytic growth. Additionally, we wish to emphasize that the choice of initial size difference in our model does not qualitatively alter the results for robustness in centrosome size equality, as shown in Supplementary Figure 3. Nevertheless, we acknowledge the need for a quantitative analysis of the dependence of size regulation on the initial discrepancy in centrosome size. We will incorporate such an analysis into the revised manuscript to strengthen our conclusions. Reviewer 2 has questioned the dismissal of the non-cooperative growth model, suggesting that minor adjustments in that model, such as incorporating size-dependent addition or loss rates due to surface assembly/disassembly, could potentially maintain equally sized organelles with sigmoidal growth dynamics. However, this conclusion is inaccurate. Any auto-regulatory positive feedback would result in size inequality, unless the positive feedback is shared between the organelles. The introduction of size-dependent addition rates due to surface-mediated assembly, would result in auto-regulatory positive feedback, leading to unequal sizes. We have explored a similar scenario of growth dynamics involving assembly and disassembly throughout the pericentriolic material volume in Supplementary Section II, demonstrating significant size inequality in that model and a lack of robustness in size control. We will provide a detailed response to this point in our reply, along with an explicit examination of the surface assembly model.

In addition to the aforementioned modifications, we will revise the section discussing the predictions of the proposed model in the revised manuscript to rectify any lack of clarity in testable model predictions. We aim to provide clearer demonstrations of how our model predictions differ from those of previous models.