A simple Turing reaction-diffusion model can explain how mother centrioles break symmetry to generate a single daughter

  1. Zachary M. Wilmott
  2. Alain Goriely
  3. Jordan W. Raff

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Abstract

Centrioles duplicate when a mother centriole gives birth to a daughter that grows from its side. Polo-like-kinase 4 (PLK4), the master regulator of centriole duplication, is recruited symmetrically around the mother centriole, but it then concentrates at a single focus that defines the daughter centriole assembly site. How PLK4 breaks symmetry is unclear. Here, we propose that phosphorylated and unphosphorylated species of PLK4 form the two components of a classical Turing reaction-diffusion system. These two components bind-to/unbind-from the surface of the mother centriole at different rates, allowing a slow-diffusing activator species of PLK4 to accumulate at a single site on the mother, while a fast-diffusing inhibitor species of PLK4 suppresses activator accumulation around the rest of the centriole. This “short-range activation/long-range inhibition”, inherent to Turing-systems, can drive PLK4 symmetry breaking on a continuous centriole surface, with PLK4 overexpression producing multiple PLK4 foci and PLK4 kinase inhibition leading to uniform PLK4 accumulation—as observed experimentally.

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    Reply to the reviewers

    1. General Statements

    We thank the Reviewers for their detailed and constructive comments. As we describe below, we have now amended the manuscript to address their concerns and suggestions.

    2. Point-by-point description of the revisions

    Reviewer #1

    __*In the first paragraph the reviewer states that our study is well presented and convincing, but that it seems “an incremental advance to the previous ones, which properly accounted for PLK4 symmetry breaking and are based on similar assumptions”. *__We apologise for not explaining properly why our work is an important advance on these previous studies. Although both previous models can account for some aspects of PLK4 symmetry breaking, they both have significant issues. For example, Takao et al. perform no analysis of the robustness of their model, and from the small number of simulations shown it is clear that some very odd behaviours emerge—e.g. the oscillation of the dominant PLK4 site around the 6 compartments (Figure 3C, Example 3) and the bizarre manner in which PLK4 overexpression drives the formation of multiple PLK4 peaks (Figure 4B, first two examples). The authors do not comment on, analyse, or explain these strange phenomena. This model also relies on STIL being added to the system only after PLK4 has already broken symmetry; this is not plausible in rapidly dividing systems such as the fly embryo where Ana2/STIL levels remain constant through multiple rounds of centriole duplication (Steinacker et al., JCB, 2022). The Leda et al. model predicts that inhibiting PLK4 kinase activity will deplete PLK4 from the centriole, but it is now clear that PLK4 accumulates at centrioles when its kinase activity is inhibited (e.g. Yamamoto and Kitagawa, Nat. Comms., 2019). Moreover, this model supposes no spatial relationship between PLK4-binding compartments; this has important implications for the system’s behaviour (see point 1 in our response to Reviewer #2), and is biologically highly implausible. Thus, neither of the previous models can properly account for several important aspects of PLK4 symmetry breaking.

    Moreover, the two previous studies are not based on similar assumptions. It is only through our analysis that we discover that the underlying biological process driving symmetry breaking in both previous models can be described in the same terms: with short-range activation and long-range inhibition causing diffusion-driven instability. This crucial conclusion was not obvious from, nor claimed by, either of the previous publications. We believe this is an important step in model development for these systems.

    __*The reviewer raises a number of minor concerns, the first of which is a previous study from Chau et al. (Cell, 2012), which studies how two component systems break symmetry. Differential diffusion is not essential for symmetry breaking in some of the models considered by Chau et al., and so they wonder if it is really essential in our system. *__We thank the reviewer for pointing us to this study. It can be proven mathematically that differential diffusion is essential for symmetry breaking in the Turing-type framework. In the systems studied by Chau et al., symmetry can be broken without differential diffusion if one of the two components can be depleted from the cytoplasm. Such cytoplasmic depletion does not occur in traditional Turing-type systems, and it is almost certainly not occurring during PLK4 symmetry breaking—e.g. FRAP experiments show that PLK4 continuously turns over at centrioles (Cizmecioglu et al., JCB, 2010; Yamamoto and Kitagawa, Nat. Comms., 2019). We discuss this point (p8, para.3).

    __*The reviewer states that it is unclear which term in equations (3-4) and (5-6) correspond to the self-activation and activation/inhibition of the other component that are indicated in the schematic summary of the models shown in Figure 1C. *__As we now clarify, in general it is not always possible to pinpoint a single term in an equation that corresponds to activation/inhibition. Mathematically, a positive feedback for means that , and a negative feedback for means that . Hence, activation and inhibition can change depending on the values of these derivatives during the dynamics as these inequalities may be achieved with complex expressions that extend beyond the usual proportional relationships. We have amended the manuscript to make this clearer (p10, para.2).

    The reviewer pointed out an error in the arrows in Figure 2 (we believe this is actually Figure 4). We thank the reviewer for pointing this out and have now corrected this mistake.

    Reviewer #2

    Major Comments:

    __* 1. The reviewer points out that in all models of PLK4 symmetry breaking the overexpression of PLK4 should be able to generate multiple PLK4 peaks (as, experimentally, PLK4 overexpression can generate up to 6 procentrioles around the mother centriole). The Reviewer suggests that the two previous models can do this, but we only show examples where PLK4 overexpression generates two peaks, and the reviewer questions whether this is a general limitation that would invalidate our approach. *__We are grateful to the reviewer for pointing this out, and we now expand our analysis and discussion of this important issue (p13-15). It is indeed possible to produce more peaks in our model using different parameters—e.g. decreasing diffusivity leads to thinner peaks, allowing more peaks to form (Figure 3B, Figure 5B). Importantly, however, when diffusion is decreased, the region of the parameter space in which only a single peak will form inevitably becomes smaller—as diffusion can no longer efficiently suppress the formation of additional peaks around the rest of the centriole surface. Hence, in both our original models we struggled to find a parameter regime in which PLK4 robustly formed a single peak, but also formed >3 peaks when PLK4 was overexpressed. As we now discuss in detail, we believe that this is a general problem, as any model of PLK4 symmetry breaking must involve information being communicated around the centriole surface. We now show that a possible solution to this problem is to postulate that increasing PLK4 levels leads to a decrease in PLK4 diffusivity (Figure 3C, Figure 5C)—a biologically plausible possibility (p15, para.2).

    In addition, it is not correct to say that the previous formulations of these models do not have this problem (or, in the case of Leda et al., the model actually has a related problem). This problem must apply to the Takao et al. model, as it also relies on information travelling around the centriole surface. This problem is far from obvious, however, because Takao et al. do not analyse the robustness of their model. This problem does not apply to the Leda et al. model, but this is because their model supposes no spatial relationship between the individual compartments and instead assumes that communication between compartments is instantaneous. This allows their system to overcome this communication problem and so robustly form a single peak at low PLK4 concentrations, while forming multiple peaks at high concentrations (as shown in Figure 6B). However, this requires that diffusion is sufficiently fast that concentration gradients are negligible between centriolar compartments, but not so fast that the relevant species are diluted in the much larger cytoplasm. It seems implausible that both of these effects may be achieved with a single diffusion rate in the real-world physical system.

    __* 2. The reviewer points out that in our modelling any multiple PLK4 peaks formed will tend to be evenly spaced around the centriole surface whereas, in their original formulations, the two previous models predict that any multiple ‘winning’ PLK4 compartments will not have any preferential spatial location with respect to each other. They ask that we address this difference and justify why we think our prediction is a better representation of PLK4 symmetry breaking. *__Although it is not obvious, neither of the previous models makes clear predictions about the spacing of multiple PLK4 peaks. As described above, Leda et al. assume no spatial relationship between PLK4-binding compartments, so relative peak-spacing cannot be assessed. Moreover, from the limited analysis shown, it is not clear that Takao et al. predict random spacing. The authors show only two simulations of PLK4 overexpression (Figure 4B, first two simulations) and the behaviour of PLK4 is very odd: the initial noise in the system fades away before PLK4 levels rapidly and near-simultaneously rise at multiple, reasonably well-spaced, peaks, before fading away to low levels—even after STIL addition. At the end of the simulation the “winning” compartments contain very low levels of PLK4 (often lower than the noise initially introduced into the system), but these compartments are reasonably (simulation 1) or very (simulation 2) evenly spaced.

    Nevertheless, the reviewer is correct that the even spacing of multiple peaks is a feature of our model. Unfortunately, it is not possible to compare this prediction to reality because the spacing of multiple PLK4 peaks in cells overexpressing PLK4 has not been quantified yet. Thus, one has to interpret published images, some of which support equal spacing while others do not (e.g. Kleylein-Sohn et al, Dev. Cell, 2007). Moreover, this analysis is likely to be complicated because CEP152 can form incomplete rings. This can be appreciated in Figure 2C in Hatch et al., (JCB, 2010) where the extra centrioles induced by PLK4 overexpression do not appear to be evenly spaced around the centriole, but are quite evenly spaced around the partial CEP152 ring. Therefore, equal spacing of peaks in ideal conditions is a feature predicted by our model that still needs to be fully explored experimentally. We believe that part of the power and value of our model is to suggest such hypotheses. We now discuss this important point (p26, para.2).

    __* 3. The reviewer questions our attempt to discretise our continuum model (where we convert the continuous centriole surface to a series of discrete compartments on the centriole surface and show that symmetry breaking can still occur). They note that we only show one example (9 compartments), they ask for more information about how the discretisation was done, and they question the independence of the compartments as PLK4 appears to accumulate in compartments adjacent to the dominant compartment. *__We apologise for the lack of clarity here. We now state that our models can break symmetry provided that there are at least two compartments, and we now include simulations showing that this happens for 2 – 10 compartments (Figure S2). The discrete model is a finite-difference discretisation of the continuum model (described in Appendix V). We also now clarify that the compartments are ‘independent’ in the sense that all chemical reactions only occur between components that are within the same compartment. The compartments are still spatially linked via a discretized diffusion (as would likely be the case at the centriole), which explains the observed relationship between neighbouring compartments.

    __* 4. The reviewer asks whether all the parameter values that satisfy the mathematical constraints we calculate for our models will break symmetry. If so, they suggest we are using a circular argument when demonstrating that the models break symmetry as we use parameter values chosen specifically to satisfy these constraints. *__In Turing-systems, one can mathematically calculate parameter constraints that allow symmetry breaking. As we now clarify, all parameters that satisfy these constraints can break symmetry, while any parameters outside these constraints cannot break symmetry. Thus, it was never our intention to claim something new or surprising when we illustrated the symmetry-breaking properties of our models (Figures 2 and 4, and associated parameter space analysis in Figures 3 and 5), so we apologise that our intention on this point was unclear. Rather, these Figures illustrate the detailed behaviour of each system under different conditions—something that is not possible to intuit from the equations alone.

    * 5. The Reviewer requests more information about how we chose the particular parameter values we use to illustrate each model and asks that we convince readers that other sets of values that satisfy the derived mathematical requirements would result in the same qualitative outcomes*. As described in point 4 above, and as we now state more clearly, it is a mathematical fact that parameter values that satisfy the derived mathematical requirements can break symmetry. We now discuss our reasons for choosing specific parameters in more detail (see point 6, below).

    __* 6. The Reviewer asks whether the dimensionless parameters we use in our models have any biological relevance, and requests a biological interpretation of all of them. They also request that we relate the Diffusivity ratios of the Activator and Inhibitor species (____) to the experimental observations made by Yamamoto and Kitagawa. __Relating our dimensionless parameters to biologically-relevant dimensional parameters is a complex issue. For example, one can see from equations (5) and (6) that simultaneously doubling (A), (I), and (a), and decreasing (b)* by a factor of 4 leaves the system unchanged. Since the concentrations of A and I are unknown at the centriole surface, this means that it is not possible to determine the dimensional values of the rate of production of I (a) and its rate of conversion to A (b). This limitation is the root of the mathematical fact that FRAP experiments can reveal “off” rates but not “on” rates. Moreover, to convert the rate of loss of A (c) and I (d) into dimensional parameters it is necessary to know the timescale of symmetry-breaking. This is unknown, but was assumed to be on the order of hours in the previous models. This corresponds to a degradation/loss rate of minutes with our current choice of parameters, which is consistent with FRAP data (e.g. Yamamoto and Kitagawa, Nat. Comms., 2019). Regarding the ratio, the effective diffusion in our model depends on both the bulk diffusion and the binding/unbinding/degradation rates – a complexity also noted by Yamamoto and Kitagawa. This makes it very difficult to relate the “effective” surface diffusivity to the bulk diffusivity. We are currently investigating the form of this dependency, but this is a complex mathematical problem that is beyond the scope of this manuscript. These issues are difficult to discuss succinctly, so we now simply state that we chose specific parameter values based, in part, on the values and ratios used in the previous modelling papers (p10, para.2; p17, para.2).

    Unfortunately, we could not find any experimental measurements of diffusivity in the Yamamoto and Kitagawa paper, as the Reviewer suggests. We now clarify, however, that the ratio we use in both models (2500) is chosen to be between the effective diffusivity ratio (as the previous models used binding/unbinding rates rather than diffusivity) used by Takao et al. (10000) and Leda at al. (200). We also include a phase diagram showing how varying the diffusivity of both factors influences symmetry breaking in both models (Figure 3B, Figure 5B), and we state that we have chosen all remaining parameter values to reflect the parameter values in the original models, when adjusted to the same timescale.

    __* 7. The Reviewer asks for more information about how we normalised time in our simulations and whether the time in different simulations is comparable. *__We now clarify that the simulations run for a single unit of dimensionless time (so they can be compared), and that the reaction/diffusion parameters in the system are sufficiently large by comparison with unity that all simulations achieve steady state within a unit of time (p11, para.2).

    __* 8. The Reviewer asks whether concentrations of ____and ____can be compared between simulations, and also questions our description of ____ being uniformly accumulated in Figure 4D, rather than uniformly depleted. *__We clarify that concentrations can be compared within a model, but not between models. This is because the dimensional values depend on the dimensional reaction rates, which differ between the models. This is not just a theoretical limitation; experimental fluorescence signals are typically compared in relative arbitrary units so the absolute values of different systems cannot be easily compared for the same reason. We agree with the reviewer that it is better to describe Figure 4D as showing uniform depletion of the activator, and we have adjusted the legend accordingly.

    The reviewer makes a number of minor points that are not numbered.

    __*The reviewer asks for clarification of what we mean by “robustness”: does this refer to the ability to produce the same result in multiple simulations, or to the ability to produce the same result when parameter values are varied? If the latter, then the reviewer suggests our models are not very robust. *__We apologise for this confusion and now more clearly define what we mean by robust (p13, para.2). As we discuss in point 1 of our response to this Reviewer, our initial models are indeed not very robust at producing a single PLK4 peak over a range of PLK4 concentrations. We now discuss why this lack of robustness is likely to be intrinsic to any PLK4 symmetry breaking system, and how robustness in all such models can be improved by allowing diffusivity to vary with PLK4 expression levels (p13-p15).

    __*The Reviewer points out that the original models introduce a noise term at every iteration, whereas we only introduce an initial noise term; they ask us to discuss this difference. *__We have run simulations introducing a noise term at every iteration and find that this makes negligible difference (Reviewer Figure 1, attached to the end of this letter). We do not take this approach, however, as this would significantly complicate the mathematical analysis that we perform (the additional noise term turns the system of PDEs into a system of SDEs which do not fit the Turing framework as readily). We now mention this in Appendix V.

    The Reviewer states that the reaction schemes are unnecessarily repeated in Figures 1, 2 and 4. We would like to keep these schematics, as in Figure 1 we show a generic scheme (illustrating the two possible Turing-type reaction diffusion systems) whereas in Figures 2 and 4 we show specific reaction regimes (specifying the relevant species) that we test in each model. We feel this information will be useful to readers in this visual format.

    The Reviewer states that it is confusing that we refer to the specific reaction parameters (k11 and k12) that need to be swapped to convert the Leda et al. model to the Takao et al. model, as this information will not mean anything to readers who are not familiar with the models. We agree and have now removed this information.

    The Reviewer suggests several textual amendments and/or corrections. We thank the reviewer for spotting these and have amended them all accordingly.

    __*Finally, the Reviewer states in their significance summary that although our key conclusions are convincing, they are not new as Takao et al. describe their model as analogous to a “reaction-diffusion system (also known as a Turing model)”. *__We were aware that Takao et al. make this statement, but this does not invalidate the novelty or significance of our work. This is because although Takao et al. described their model as being analogous to a “Turing model”, it is not actually a reaction-diffusion system, and it does not exhibit the property of long-range inhibition that is central to all Turing-systems to produce a single PLK4 peak. Instead, they use lateral inhibition (in which the influence of the inhibiting species does not extend beyond the neighbouring compartments) to reduce the number of potential PLK4 binding sites from ~12 to ~6. A single winning site is subsequently selected when STIL is added to the system—with additional positive feedback (not involving reaction-diffusion) ensuring that the compartment with most PLK4 becomes the dominant site. Their analysis of the reaction-diffusion version of their system is limited to a single supplementary figure (Figure S2D), and they do not perform or refer to any of the relevant mathematical analyses of their model that makes these well-studied systems such powerful tools. We believe that the model presented here is simple enough to draw the attention of the applied mathematics community while robust and complete enough to provide a mechanistic explanation of many interesting features and suggest new possible phenomena. We now discuss these points (p22, para.1).

    Reviewer #3

    __*The Reviewer found our manuscript well-written, and judged it of interest to centriole duplication enthusiasts. *__We interpret this to mean that the Reviewer did not think it of more general interest. This seems a harsh assessment, as the precise one-for-one duplication of centrioles is generally considered to be one of the great mysteries of cell biology. It is now widely appreciated that robustly breaking PLK4 symmetry to form a single PLK4 peak is crucial to this process. Thus, our discovery that this process can be described using a well-studied mathematical framework that has already been applied to a vast range of biological processes is potentially of significance even to non-centriole enthusiasts.

    The Reviewer made a number of specific comments:

    Figure 1. The Reviewer felt the graphic in Figure 1A could be improved by combining it with Figure 1B, and noted that the centrioles look strange. We thank the reviewer for these suggestions and we have now rearranged this Figure. We also now clarify that the schematic depicts Drosophila centrioles, which are simpler than human centrioles.

    __Figure 2. The Reviewer suggests that to make the system depicted in Figure 2A fit as a Type I Turing system we have to assume that (I) must dissociate from the centriole or be degraded at higher rates than (I) converts (A) to (I). They suggest this assumption is implicit in the model and they request further explanation. __The reviewer is correct that, in Model 1, the degradation/dissociation of () is the root of its self-inhibition. However, we do not need to make any assumption about the relationship between the rate at which converts to (b), and the dissociation/degradation rate of (d) for this system to work (as the Reviewer implies). This is because, whatever these rates are, the system will approach a steady-state where the production and degradation terms balance, and it is the stability/instability of this state that determines whether the system can break symmetry. Since the degradation rate of (the - term in equation 4) increases more rapidly than its production rate (the term in equation 4) as increases, this results in a stable (i.e. self-inhibiting) system regardless of the parameter values. We have rewritten the sections explaining these equations to try to make these points more clearly and to point readers to Appendix II where we explain the form of the equations.

    __The Reviewer asks if in Model 1 it is realistic to assume no turnover or loss of PLK4 (A), and will the system still work if this is altered? __This is a good point. In Model 1, we set c=0 as this makes the analysis significantly simpler, enabling us to display the mathematical predictions alongside the numerical simulation. We have now added the (c,d) phase diagram to show the effect of varying these parameters on the symmetry breaking properties of the system (Figure 3D). We find that the value of c has a relatively weak effect on the symmetry breaking properties of the model since it does not affect the function of as an activator.

    __*The Reviewer asks if our 1D model would work in 2D, and notes the PLK4 peaks in our models are broad, likely limiting the number of peaks formed. They also note that in our Model 1 it is the unphosphorylated form of PLK4 that accumulates in the peak, which seems unlikely as it is widely believed that PLK4 must be active to phosphorylate STIL to promote its interactions with SAS6 and CPAP. *__From a mathematical perspective, modelling our system in 2D would produce very similar results. Symmetry breaking is driven by long-range inhibition/short-range activation, and these behaviours will work analogously in 2D. As discussed in our response to Reviewer #2 (point 1), the broad peaks do indeed limit the number of centrioles that can form, but by altering the parameters we can generate more peaks that are less broad (Figures 3 and 5). The Reviewer is correct that Model 1 (based on Takao et al.) predicts that non-phosphorylated PLK4 () accumulates in the peak. This is also true of the original Takao et al. model, although this was not highlighted or commented on by the authors. We now expand our discussion of this point (p25-p26).

    The Reviewer asks if our models can form multiple peaks at higher PLK4 levels. This is again related to Reviewer #2, point 1, and we now show that this is indeed possible under the appropriate parameter regime (Figure 3C and Figure 5C).

    The Reviewer asks for more description of how lateral diffusion works in our system. For example, do we consider that not every molecule of (I) will diffuse laterally (as some will be lost to the cytoplasm), or that the probability of a molecule leaving the surface will increase as distance/time increases. We apologise for our lack of clarity. We now state that the proportion of molecules not rebinding to the surface is accounted for in the reaction components of all our models (p7, para.1). In reality, and as we now state, the relationship between this loss and the diffusion rates (and their relation to distance/time, for example) is complicated. We are investigating this relationship in more detail, but this is beyond the scope of the current paper.

    The Reviewer asks if symmetry breaking might eventually occur if the system in which we reduce the kinase activity of PLK4 (Figure 2D) were given more time. They also ask whether reducing PLK4 levels by half would lead to a failure in site-selection. The kinase inhibited scenario we show here will not break symmetry over any period of time; this can be proven mathematically, and is verified in the numerical simulations (Figure 3A and 5A, bottom left regions of graphs), which we now state more clearly are always run for a long enough period to reach a steady-state (p11, para.2). The effect of reducing PLK4 levels in our models is analysed in the phase diagrams shown in Figure 3 and 5 (and analysed in more detail in Figure S1), where it can be seen that there are multiple PLK4 concentrations that can be halved without a failure in site selection (although, see also our response to Reviewer #2, point 1).

    The Reviewer pointed out some errors in our presentation of Figure 3, (and suggested some improvements in presentation in a point further below) and also asked for more information about the parameters used to generate the data in Figures 2B-D and 4B-D. We thank the Reviewer for these suggestions and have made these changes and provided the additional information requested (e.g. marking the specific parameters used in our simulations on the phase diagrams shown in Figure 3 and Figure 5 with coloured dots).

    The Reviewer points out that when PLK4 levels and activity are both high no centrioles are produced in Model 2, whereas 1 centriole is produced in Model 1—neither of which are consistent with experimental observation. We now show an expanded parameter space (new Figures 3A and 5A) where it can be seen that this is not a problem for Model 1. For Model 2, the region of high kinase levels and activity (dark blue, top right, Figure 5A) corresponds to the uniform accumulation of the activator species. Thus, while there are no peaks, this region might produce multiple centrioles, as it is equivalent to a compartmental model in which all of the compartments are occupied. We now discuss this point (p19, para.1).

    __*The Reviewer questions how the biology fits a Type II Turing system, pointing out that current data suggests that active PLK4 turns over more rapidly at centrioles, whereas in the Type II model we describe (based on the Leda et al. model) it is the phosphorylation state of STIL that determines which species of PLK4:STIL turns over rapidly. They also question the logic of the Model 2 Type II circuit (Figure 3A), questioning how A could drive the dephosphorylation of STIL to promote the production of I. *__We agree that current data is more consistent with phosphorylated species of PLK4 turning-over more rapidly at centrioles, but this is not what Leda et al. proposed, and so this is not what we implemented in trying to reformulate their model (although this is effectively the change we make that turns the Leda et al. model into the Takao et al. model). As to the second point, the Reviewer has correctly spotted a problem with our model that arises because the direction of the arrows linking and were inadvertently flipped in Figure 4A. This mistake has been corrected, and we now explain more clearly how the biology of this system fits a Type II Turing system in the legend.

    __The Reviewer points out that although we can convert the Leda et al. Model (Model 2) to the Takao et al. Model (Model 1) simply by changing the identity of the ____ and ____ species, the underlying assumption of the Takao et al. model (that non-phosphorylated PLK4 promotes its own accumulation) was not an inherent assumption of the Leda et al. model. __We apologise for this confusion. As we now clarify (p20, para.1) the Reviewer is correct that when we make mathematical changes to the Leda et al. model we must also assume changes in the underlying biology—so that non-phosphorylated species of PLK4 are now slow diffusing, rather than non-phosphorylated species of STIL, as originally proposed). As the Reviewer points out, current data suggests that non-phosphorylated species of PLK4 do turnover more slowly, although it is not clear why—for example, liquid-liquid phase separation driving the formation of PLK4 condensates has been postulated, but is far from proven. This remains an interesting problem that will be further probed mathematically and experimentally.

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    Referee #3

    Evidence, reproducibility and clarity

    This manuscript attempts to address a very important question in the field of centriole biology: how does PLK4 symmetry breaking occur to produce a new procentriole in a specific single site. The work is theoretical in nature and does not offer new experimental support. Furthermore, the authors are forced to make multiple assumptions to fit PLK4 symmetry breaking into a Turing reaction-diffusion system. In some instances, these assumptions are not intuitive and don't have a strong foothold in the known behavior of the molecules involved. That notwithstanding, I found the manuscript to be well-written for a general audience and believe it will be of interest to the centriole duplication enthusiasts.

    The following comments should be addressed prior to publication:

    Specific comments:

    Figure 1:

    • The graphic in Figure 1A depicting the centriole duplication process could be more effectively presented. Perhaps combining Figure 1A and 1B with a graphic that places these events in the context of the centriole duplication process coordinated with the cell cycle would provide a better insight to the relevant biology. The centrioles also look very strange, with the procentriole width being equal to the height of the parent centriole.

    Figure 2:

    • I take (I) to be synonymous with kinase-active PLK4 (phosphorylated PLK4 in the authors parlance). If (I) phosphorylates (A) to make more (I), then (I) doesn't strictly inhibit the accumulation of (I). It seems to make this fit a Turing system the authors are assuming that phosphorylated (I) must dissociate from the centriole or be degraded at higher rates than it converts (A) to (I). This is an assumption implicit in the model and should be further explained.
    • Is it realistic to assume no turnover or loss of unphosphorylated PLK4 (A). Will the model still work if this assumption is altered?
    • The centriole surface is modeled in 1-dimensional space, when it is, of course, 2-dimensional. How does this change the model? The site selection also appears weak as the distribution of PLK4 localization is very broad. This likely limits the number of PLK4 sites that can be formed. Finally, the model allows for the accumulation of (A) at a single site. Since (A) is unphosphorylated PLK4, I am left wondering how this species could be proficient in initiating procentriole assembly. I find it unlikely that PLK4 kinase activity is only required for symmetry breaking and not procentriole assembly. Multiple PLK4 phosphorylation sites on STIL promote binding interactions with centriole proteins (SAS6 + CPAP) and are required for procentriole assembly.
    • In Figure 2C, are three peaks possible at higher PLK4 levels? Figure 3A would suggest not, which is inconsistent with the known biology.
    • I think it would benefit the reader to have more description of what lateral diffusion entails and what assumptions are made. When (I) is released from the centriole surface, it can rebind to the centriole at a neighboring site (a PLK4 condensate or CEP152) and thus diffuse laterally or diffuse away from the surface of the centriole. Does the model account for the fact that not all every (I) molecule produced at the centriole will diffuse laterally? Moreover, the probability of (I) leaving the surface of the centriole must increase as distance/time increases.
    • In Figure 2D, would a single site of PLK4 form if a longer period of time was given? In other words, are the kinetics of site selection slowed, or will symmetry breaking never occur in this system? I presume that reducing the overall levels of PLK4 levels by half would not lead to a failure of site selection?

    Figure 3:

    • The figure labels do not match what is described in the text. Figure 3B should be the top right graph and the bottom two graphs for Model 2 should be labelled 3C and 3D.
    • The authors should highlight on the graph which parameters were used to generate the data in the experiments in Figure 2B-D and Figure 4B-D.
    • Model 2 predicts that at high levels of PLK4 protein and high levels PLK4 activity, no centrioles are produced, while Model 1 predicts one centriole would be produced. Neither is consistent with experimental observations.
    • The figure organization could be adjusted to improve clarity. As presented here, the text goes from discussing Figure 3A-B and skipping Figure 3C-3D until after discussing Figure 4. Instead of having the phase diagrams in their own figure, they could be incorporated into the respective figure that they are describing (Figure 3A-B becomes Figure 2E-2F, Figure 3C-D after current Figure 4D). With this adjustment, the figures could follow the order of the text.

    Figure 4:

    • It is unclear to me how the biology fits with the underlying assumptions of a type II Turning reaction-diffusion system. Both (A) and (I) contain phosphorylated (and active) PLK4. Current data suggests active PLK4 turns over more rapidly at the centriole - how does this fit with these assumptions? More importantly, the (A) species contain phosphorylated STIL and represent the complex that initiates centriole assembly. (A) promotes the accumulation of more (A) through phosphorylation of STIL, but how does A also drive the dephosphorylation of STIL to promote the assembly of (I)?
    • In the section 'unifying the models....', the authors propose the Leda et al model can be modified so that phosphorylated PLK4 defines the (I) species and (A) represents unphosphorylated PLK4. This modification now mirrors that of Takeda et al., and it recreates the same issue - inactive PLK4 accumulates at the site of centriole assembly. There also needs to be an assumption for how A (non-phosphorylated PLK4) would promote its own accumulation, and this is not an inherent assumption from the Leda et al. model.

    Significance

    Centrioles are microtubule-based structures that comprise the centrosome, the major microtubule organizing center. In mitosis, centrosomes serve to maintain the bipolar spindle to promote faithful cell division. To ensure that only two centrosomes exist in a mitotic cell, centriole copy number is tightly regulated so that centrioles duplicate once and only once per cell cycle. Centriole biogenesis is initiated by Polo-like kinase 4 (PLK4) on the wall of an existing parent centriole to produce a single new procentriole. While progress has been made in understanding how centriole copy number is regulated by PLK4, it is still unclear how procentriole formation is strictly restricted to a single site in each preexisting parent centriole. In this paper, the authors use mathematical modelling to shed some light on this critical question in the centriole field.

    The prevailing model in the field is that PLK4 is recruited around the circumference of the proximal end of the parent centriole at the beginning of G1 phase, and transitions to accumulate at a single locus that marks the site of procentriole assembly at the beginning of S phase. Two mathematical models have been proposed to explain how this PLK4 symmetry breaking occurs. However, both make predictions that are inconsistent with the current experimental data. In this study, the authors reconceptualize both published mathematical models for symmetry breaking and PLK4 site selection as two-component Turing systems that rely on activator/inhibitor dynamics. The original models were thought to differ in several key assumptions. However, in this study, the authors propose that the essential features of both models can be described by Turing systems. Moreover, the authors assert that the phosphorylation status of PLK4 is the driver for symmetry breaking.

    Turing systems are widely understood and have a well-characterized behavior. The central question here is can the biological observations be adequately fit into this simplified reaction scheme.

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    Referee #2

    Evidence, reproducibility and clarity

    Summary

    The authors present a reformulation of two existing mathematical models describing PLK4 symmetry breaking around the mother centriole at the onset of centriole duplication. Rather than considering PLK4 binding to, and unbinding from, a discrete representation of the centriolar periphery as a defined number of compartments, the authors consider PLK4 to diffuse on a continuous 1D ring. Furthermore, the reaction scheme of each existing model is reinterpreted here as a two-component reaction-diffusion system. These alternative representations of the existing models are shown to reproduce the dynamics of the original descriptions of the models.

    With the two existing models put in a similar framework, the authors describe how a modification of the Leda et al. model can lead to the same dynamics as the Takao et al. model. Moreover, they point out a difference in the prediction of the reformulated versions of the two models (accumulation versus depletion of I in the peak, compare Fg. 2B and 4B). Finally, the authors report that discretization of the 1D continuous line into 9 compartments also predicts the accumulation of PLK4 at a single site, and thus does not alter the predictions of the two existing models qualitatively. From this, the authors conclude that PLK4 symmetry breaking around the mother centriole can be represented as a two-component Turing reaction-diffusion system.

    Major comments

    1D continuous space coordinate

    The key difference between the models in their new formulations and their original descriptions is the representation of the centriole periphery as a continuous 1D representation of the ring, rather than a number of discrete compartments. The authors mention that the binding and unbinding between compartments and cytoplasm effectively act as a one-dimensional diffusion process on a ring, justifying the use of a continuous space coordinate. However, this justification might not be fully warranted. As discussed in the points below, the reformulations of the centriole periphery in a continuum result in strong predictions regarding PLK4 symmetry breaking and accumulation at distinct sites that are fundamentally different from the predictions of the two existing models in their original formalism.

    1. Although the authors repeatedly mention "multiple" peaks, they do not present simulations of overexpression conditions that give rise to the accumulation of PLK4 at more than two sites. Would these predictions lie outside the parameter space explored by the authors or are the reformulations of the models intrinsically not capable of recapitulating the formation of more than two foci? The latter would be in contrast to the original formulations of the models, in which a gradual increase in protein levels leads to the stepwise increase of the number of compartments PLK4 accumulates in (Figure 4B, Takao et al.; Figure 6B, Leda et al.). More importantly, PLK4 overexpression has been repeatedly observed experimentally to induce the formation of up to 6 procentrioles around the mother centriole (e.g., Vulprecht et al. 10.1242/jcs.104109). Given this, how can a model that is by design limited to the formation of a maximum of two accumulation sites be a valid representation of PLK4 dynamics around the centriole? The authors must carefully evaluate this apparently central conundrum and adapt their models if needed.
    2. In the case of PLK4 accumulation at two sites (e.g., in the 2xPLK4 condition), two foci always accumulate on opposite sides of the continuous ring. This is in stark contrast to the models in their original formalism, where two 'winning' compartments do not have any preferential location with respect to each other (Leda et al.), or where a second 'winning' compartment should be at least one compartment away, but then could be located anywhere (Takao et al.). The authors should address these differences and justify why their predictions on a continuous ring are a better representation of PLK4 symmetry breaking than the previous discretization of the centriole into compartments.
    3. When returning from the continuous formalism to a compartmentalized centriole surface (Figure 5), the authors report that the model remains valid if the continuum space is "divided into an arbitrary number of discrete compartments" (p. 17). However, as the authors only present one exemplary simulation of the model for 9 compartments, it is not clear if other compartment numbers indeed reproduce the formation of only one dominant focus. More fundamentally, it is not clear how the model was discretized, what sets of equations are simulated, as well as if and how diffusion between compartments is accounted for. The authors report in the legend of Figure 5 that compartments are independent, but this is unlikely given the slight accumulation of PLK4 levels in the two compartments adjacent to the dominant compartment.

    Model parameters

    The authors define their reaction-diffusion system of equations starting from the mathematics, leading them to a set of requirements that the parameters in their equations need to fulfill in order for the system to be able to break symmetry and resolve in a steady state with a single site of PLK4 accumulation.

    1. It is not clear whether all the parameter values that satisfy the mathematical constraints wil lead to symmetry breaking. In other words, is satisfying these constraints sufficient for symmetry breaking? If yes, then it would seem that the authors use a circular argument when demonstrating that their models break symmetry using certain values for a,b,c and d, since these values were chosen in the first place to satisfy the mathematical requirement that will lead to symmetry breaking. If no, then the authors should investigate and report which parameter values that fulfill the mathematical constraints do not lead to symmetry breaking, and why. Thus, in Figure 3, the authors should clarify if regions of the parameter space where the models predict no symmetry breaking (e.g., Figure 3B, left panel, a=b=250) fulfill the mathematical constraints. If so, how can one end up with a uniform distribution -i.e., without symmetry breaking, if the mathematical constraints require this state to be unstable?

    these parameters can have a steady state in the absence of diffusion, at the onset of the simulation, as well as upon diffusion, at the end of the simulation, yet without symmetry breaking.

    turns into another steady state that does not involve symmetry break. that turns unstable in presence of diffusion, but not break symmetry.

    This is an important point to clarify.

    1. Of all the combinations of parameter values that would satisfy the requirements for symmetry breaking, the authors mention that the reason for specifically choosing the values of a,b,c and d presented in the manuscript is their simplicity (p. 11, 15). It remains however unclear why this specific set of parameter values is preferred over other combinations of values. If this set is merely an arbitrary choice, then the authors should discuss this further and convince the reader that indeed any other set of values that satisfies the derived mathematical requirements would result in the same qualitative outcomes. Alternatively, potential empirical reasons why these values are preferred should be mentioned.
    2. Related to the previous point, it is unclear if the parameters presented have much biological relevance. A biological interpretation should be made even for dimensionless parameters. Moreover, this comment is not limited to the a,b,c and d parameters. Concretely, in the reformulation of the model by Takao et al.,D_I is chosen to be 200-fold higher than D_A, whereas for the reformulation of the model by Leda et al., D_I is even 1000-fold higher than D_A. As in both models I and A refer to different species of PLK4 depending on their phosphorylation state, the authors should relate the D_I/D_A ratios chosen to experimental observations of the diffusivity of PLK4 as a function of phosphorylation (Yamamoto and Kitagawa 10.1038/s41467-019-09847-x).
    3. As all simulations are presented to run from t=0 to t=1, the authors must clarify what stopping criterion they used to determine the simulation time, and if they normalized the time for each simulation. At present, it is not clear if the simulation time can be compared between different simulations of parameter sets.
    4. Moreover, it is not clear how the concentrations of A and I are compared between simulations. In both Figure 2D and Figure 4D, the authors report a uniform accumulation of PLK4 on the ring. However, the total level of PLK4 is 30 in Figure 2D and only 2 in Figure 4D. Here, the authors must clarify why in the case of Figure 4D the outcome should not be interpreted as a uniform depletion, rather than a uniform accumulation.

    Minor comments

    • It is unclear what exactly is meant when the "robustness" of the reformulated model is discussed. Robustness could be interpreted as the ability of the model to reproduce the same result in repeated simulations but with the same model parameters, or else as the ability to reproduce the same result under varying model parameters. If the latter is concluded here, then it is questionable how robust the models are given the parameter regime analyzed in Figure 3, where two-fold changes in parameter values lead the model to fail to predict symmetry breaking.
    • The authors mention that an initial stochastic noise in the binding of PLK4, randomly-generated only at the onset of the simulation, will be reinforced and eventually lead to the formation of a single focus. However, in the original descriptions of the models, this noise term is randomly generated and updated every iteration. What would be the consequence of such a continuous noise in the system for symmetry breaking and maintenance of a single site of PLK4 accumulation in the reformulated model simulations presented here? This must be discussed.
    • The diagrams of the reaction schemes are unnecessarily repeated in multiple figures (Figure 1, Figure 2 and Figure 4).
    • It is confusing that the authors use the original notations k_11 and k_12 to refer to specific rate parameters of the Leda et al. paper (p. 17). For readers not familiar with the Leda et al. paper, this is too detailed and this information should be put in an appendix if not omitted.
    • The authors write that PLK4 is recruited in a "poorly understood process" (Introduction). Although the process is indeed incompletely understood, describing the process as "poorly understood" is an overstatement given the ample literature available on this question.
    • The authors refer to the existing models as being "recently" proposed (Introduction). This term may be regarded inappropriate for 5-year-old publications.
    • 'Takao' is misspelled as 'Takeo' on several occasions (p. 9,10,14,16,19).
    • The Takao et al. paper is referenced from the year 2018 instead of 2019 (p. 9 and in the legend of Figure 2).

    Significance

    Although the key conclusions of the manuscript are convincing, they are not new.

    In fact, Takao et al. describe their model explicitly as a "reaction-diffusion system (also known as a Turing model)" (p. 3539, Takao et al. 10.1083/jcb.201904156) and their model already consists of two components, representing an "active" and "inactive" form of PLK4. The conclusion that a two-component Turing reaction-diffusion model can explain how mother centrioles break PLK4 symmetry to generate a single daughter is thus already evident from Takao et al.'s work.

    On the other hand, the original description of the model presented by Leda et al. includes more than two components and is not explicitly labeled as a Turing-inspired reaction scheme, although this might be obvious for people familiar with Turing models. For the Leda model, the authors' reformulation in a two-component reaction-diffusion system could be of potential interest, if the reformulated models lead (the authors) to new interpretations of previous data or generate unanticipated predictions that are testable in experiments.

    At present, however, the provided material fails to demonstrate the significance of the reformulation of the models, and therefore seems better suited as review or commentary piece on reaction-diffusion systems explaining PLK4 symmetry breaking.

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    Referee #1

    Evidence, reproducibility and clarity

    The study by Wilmott and colleagues is the design and test of a novel model that accounts for the symmetry break of PLK4 around centrioles prior to duplication. Two previous models have been proposed by xx et al and xx et al and both are described in the details in the introduction. According to the authors, the two previous models are similar but differ in some key points. The model by Leda et al is discrete since PLK4 can accumulate on nine competing points, which represent the nine triplets of microtubules. It is based on numerous chemical interactions and notably the positive feedback of PLK4 on itself. But somehow it does not account very well for the effect of the inactivation of PLK4 phosphorylation on the accumulation of PLK4 all around the centriole. The model by Takao et al is more continuous since it relies on the dynamics of a condensate formed by PLK4 which consumes adjacent PLK4 and thus leads to the concentration of PLK4 in the condensate. In addition to a positive feedback of PLK4 on itself it takes into account the negative effect of PLK4 on the adjacent recruitment of PLK4. But this model is not very robust to variations in the initial conditions. Here the authors proposed a continuous model based on the equations of a Turing model. It is claimed to unify the two previous model in a more generic one that is easier to implement and to study. It accounts for all known impacts of the modulation of PLK4 phosphorylation on PLK4 symmetry break. I am not skilled enough in biochemical modeling to assess properly their description of other models, neither their own model. However, the present study is very well presented and convincingly highlights the conditions for the symmetry break to occur. It seems to be an incremental addition to the previous ones, which properly accounted for PLK4 symmetry break and it is based on similar assumptions. However, the continuous description is certainly easier in terms of computation and the Turing-like morphogenesis is an interesting novel way to think about symmetry break around the centriole.

    I have few minor concerns:

    • A preceding study by Chau and Lim in Cell in 2012 studied all the interactions patterns between two components that could lead to a symmetry break and the polarization of one of the components. They also studied the robustness of the polarizing patterns. It would be relevant to discuss this study and mention which of these patterns are considered here. In addition, Turing morphogenesis is not used in this study by Chau and Lim. I am not a specialist but it might means that the difference of diffusion rates between the two components might not be essential to the polarization. It would be interesting to test how critical it is in this study. It is somehow studied in the two right phase diagram in Figure 3. But it is unclear to me if the conclusion is that a robust polarization could not appear if the system is not driven by a genuine Turing-like mechanism. It is somehow obvious that if the inactivator diffuse faster than the activator, the activator will aggregates more easily, but it is unclear to me whether this is a requirement. It doesn't seem to be the case in the study by Chau and Lim.
    • The study by Chau and Lim proposed a way to test the robustness of the polarizing pattern to variations of the interaction parameters and concentrations of the two species. It would be a great addition to this study.
    • It is unclear which term of the equations (3-4) and (5-6) correspond to the self-activation and activation/inhibition of the other component. In model1, the positive feedback of the inactivator on itself is drawn in the scheme (Figure 1) but the corresponding term in equation 4 (a positive term depending only on the concentration of the inactivator) seems to lack. In model 2, the positive feedbacks on both the activator and the inactivator, drawn in the scheme (figure 2), are also absent from equations 5 and 6.
    • The two arrows between A and I seem to be inverted in the scheme in Figure 2. I understood from the text and the equations that A must act negatively on I, and not positively, and that I must act negatively A, and not positively.

    Significance

    The study by Wilmott and colleagues is the design and test of a novel model that accounts for the symmetry break of PLK4 around centrioles prior to duplication. Two previous models have been proposed by xx et al and xx et al and both are described in the details in the introduction. According to the authors, the two previous models are similar but differ in some key points. The model by Leda et al is discrete since PLK4 can accumulate on nine competing points, which represent the nine triplets of microtubules. It is based on numerous chemical interactions and notably the positive feedback of PLK4 on itself. But somehow it does not account very well for the effect of the inactivation of PLK4 phosphorylation on the accumulation of PLK4 all around the centriole. The model by Takao et al is more continuous since it relies on the dynamics of a condensate formed by PLK4 which consumes adjacent PLK4 and thus leads to the concentration of PLK4 in the condensate. In addition to a positive feedback of PLK4 on itself it takes into account the negative effect of PLK4 on the adjacent recruitment of PLK4. But this model is not very robust to variations in the initial conditions. Here the authors proposed a continuous model based on the equations of a Turing model. It is claimed to unify the two previous model in a more generic one that is easier to implement and to study. It accounts for all known impacts of the modulation of PLK4 phosphorylation on PLK4 symmetry break. I am not skilled enough in biochemical modeling to assess properly their description of other models, neither their own model. However, the present study is very well presented and convincingly highlights the conditions for the symmetry break to occur. It seems to be an incremental addition to the previous ones, which properly accounted for PLK4 symmetry break and it is based on similar assumptions. However, the continuous description is certainly easier in terms of computation and the Turing-like morphogenesis is an interesting novel way to think about symmetry break around the centriole.

  5. Version published to 10.1101/2023.02.02.526828v2 on bioRxiv
  6. Version published to 10.1101/2023.02.02.526828v1 on bioRxiv