Centrioles generate a local pulse of Polo/PLK1 activity to initiate mitotic centrosome assembly

Abstract Mitotic centrosomes are formed when centrioles start to recruit large amounts of pericentriolar material (PCM) around themselves in preparation for mitosis. This centrosome “maturation” requires the centrioles and also Polo/PLK1 protein kinase. The PCM comprises several hundred proteins and, in Drosophila, Polo cooperates with the conserved centrosome proteins Spd‐2/CEP192 and Cnn/CDK5RAP2 to assemble a PCM scaffold around the mother centriole that then recruits other PCM client proteins. We show here that in Drosophila syncytial blastoderm embryos, centrosomal Polo levels rise and fall during the assembly process—peaking, and then starting to decline, even as levels of the PCM scaffold continue to rise and plateau. Experiments and mathematical modelling indicate that a centriolar pulse of Polo activity, potentially generated by the interaction between Polo and its centriole receptor Ana1 (CEP295 in humans), could explain these unexpected scaffold assembly dynamics. We propose that centrioles generate a local pulse of Polo activity prior to mitotic entry to initiate centrosome maturation, explaining why centrioles and Polo/PLK1 are normally essential for this process.


Mathematical model of PCM scaffold assembly kinetics (Model 1)
We assume that centriolar Spd-2 receptors, , are able to convert cytoplasmic Spd-2, , into an unstable Spd-2 scaffold, * , via the complex ̅ . The on, off, and catalytic conversion rates of this process are on , off , and cat * / , respectively, where * ( ) is the total amount of active Polo at time and ( ) is the number of centrioles in the embryo at time (which will double after every cycle), so that * / describes the amount of active Polo at each centriole. In the first instance, we do not attempt to model * ( ) but instead treat it as a given function which we use as an external stimulus for the system. Although * is unstable, it can recruit cytoplasmic Cnn, , and cytoplasmic Polo, , to form the more stable complex ̅ , which can phosphorylate to convert it into a stable Cnn scaffold form * . The on, off, and catalytic conversion rates of this process are on , off , and cat , respectively. The two scaffold forms of Spd-2 have disassembly rates * dis and ̅ dis , respectively, while the disassembly rate of the Cnn scaffold is given by dis * / , as we assume that this disassembly rate is proportional to the size of the Cnn scaffold. This assumption is based on our previous observation that at the start of S-phase the old mother (OM) centrosome organises a larger Cnn scaffold than the new mother (NM), but the two scaffolds ultimately grow to the same size by the end of S-phase (Conduit et al, 2010). As the Cnn incorporation rate is the same at OM and NM centrosomes (Conduit et al, 2010;S.S.W, unpublished observations) we infer that the rate of loss of Cnn during S-phase must be larger at the OM, indicating that the larger the Cnn scaffold, the larger the rate of Cnn loss.
The previous description can be summarised as a system of reactions For simplicity, we assume that cytoplasmic species diffuse sufficiently fast in the embryo that we may treat these variables as spatially homogeneous, and therefore we neglect spatial effects from the model. By imposing the law of mass action, we derive the following system of four ordinary differential equations (note that the explicit dependence in the dependent variables on time has been dropped) where the PCM quantities, * , ̅ , * , * , and ̅ are defined as the total number of the corresponding species in the embryo (i.e. dimensionless units), and the cytoplasmic quantities, , , and , are defined as the volumetric concentration of the corresponding species (i.e. units m −3 ). We assume, for simplicity, that the embryo is a closed system which implies that the total amount Spd-2 ( 0 ) and Cnn ( 0 ) in the embryo is conserved.
Further, since the total amount of Polo in the system ( 0 ) is large (Casas-Vila et al, 2017) we treat cytoplasmic Polo as a prescribed constant unaffected by absorption into the scaffold. Finally, we assume that the total number of Spd-2 receptors in the embryo is proportional to the number of centrioles. These constraints read ̅ + + * + ̅ = 0 , where 0 is the total number of receptors per centriole and is the volume of the embryo.
These equations describe the total amount of each species in the embryo. However, it is useful to describe the model on a per-centriole basis. We do this by defining the auxiliary (lower case) per-centriole variables: = , ̅ = ̅ , * = * , ̅ = ̅ , * = * , and * = * . In terms of these variables, our system reads subject to While equations (14) -(20), subject to the appropriate initial conditions, are sufficient to describe the system, it is convenient for its mathematical analysis to instead formulate the model in terms of "dimensionless" variables. Through this process, we determine the dimensionless parameter groups (e.g. the ratio of the reaction rates to the cell cycle timescale) which govern the dynamics of the system, which in turn enables us to simplify the system and reduce the number of independent variables in the model. We nondimensionalise the system by using the following scalings where is the typical period of the cell cycle, and max is the maximum amplitude of the imposed Polo activity. In terms of dimensionless variables, the model reads subject to Given a solution to this system, the total size of the Spd-2 and Cnn scaffolds and total amount of active Polo surrounding each centriole are given by where tot , tot , and tot are dimensionally scaled with 0 , 0 , and max , respectively.
To allow us to compare accurately the output from our models to the experimental data we first determined reasonable initial conditions, as the centrosomes in our experiments are already initially associated with some PCM (that was acquired in the previous cycle) at the start of S-phase. To do this, we first solve (22) -(28) subject to the initial conditions ̅ = * = ̅ = * = 0. (i.e. no Spd-2 or Cnn scaffold is assembled around the centriole).
Since the system is approximately cyclic, we then use the final values output by this solution, ̅ = ̅ 0 , * = 0 * , ̅ = ̅ 0 , as our new initial values. Since the Cnn scaffold divides and partially breaks away during centriole separation, we cannot impose the cyclic condition on Cnn. However, since the output of * is (1), this suggests that an (1) input is consistent with our model and therefore we set * = 1 as our initial condition. In this way, the centrioles in our model start the cycle already associated with some Spd-2 and Cnn scaffold that they acquired in the previous cycle, as is the case with our experimental data.
In Figure 2B, we plot the incorporation of Spd-2 and Cnn into the PCM, tot and tot , over the duration of a single cycle by solving (22) -(28) subject to the initial conditions ̅ = ̅ 0 , * = 0 * , ̅ = ̅ 0 , * = 1, the parameter values given in Appendix Table S4, and the constraint that the number of centrioles is constant during the cycle, ≡ 1 without loss of generality. We also plot the prescribed Polo activity (i.e. the oscillation in * ( ) that we impose on the system), * ( ) ≔ 1 2 (1 − cos (2 )), and the total Polo at the centriole, tot .
The amplitudes in all the solutions have been normalised to 1.
The initial conditions and parameters used in this model are listed in Appendix Table S4.
Our justification for choosing these parameter values is presented in a later Section.
-We note that, in our model, the dissociation rates of the * and ̅ scaffolds have different functional forms. To investigate if this contributes to the different behaviour of the * and ̅ scaffolds we compare in the graphs below the model output in the case in which the exponent of the * disassembly term ( dis * ) is varied, and the case in which the disassembly rate itself is varied. This shows that * behaviour is primarily determined by the order of magnitude of the disassembly rate, rather than its exponent.

Mathematical model of centriolar Polo activity (Model 2)
Model 1 assumed a given oscillation in Polo. Next, we describe a model for how such an oscillation in Polo activity might be generated by the centriole through the interaction between Polo and its receptors at the centriole surface, such as Ana1 (Alvarez-Rodrigo et al., 2021). We assume that these receptors, off , are initially inactive and unable to bind Polo. To initiate mitotic PCM assembly, the receptors are activated at a rate on due to their phosphorylation by a protein kinase, which is most likely a Cdk/Cyclin, or a kinase that is regulated by the Cdk/Cyclins (such as Polo or Aurora A). This new form, which we denote , is able to bind Polo with on and off rates on and off , respectively, to form the complex ̅ . We assume that the Polo in this complex is active and able to initiate mitotic PCM assembly as described by Model 1. We also assume that this active form of Polo instigates the deactivation of the receptors at a rate off * / . This final form, which we denote ̅ off , is unable to bind or activate Polo. This system likely resets itself between cycles when ̅ off is dephosphorylated to regenerate off , but we do not model this reset here. Finally, we assume that the reactions occurring in the PCM are the same as before, By imposing the law of mass action, we obtain the following system of ODEs, We also assume that the total number of Polo receptors at each centriole, 0 , is conserved, which reads off + + ̅ + ̅ off = 0 .
As before, we write the system in per-centriole variables, and non-dimensionalise by setting off = 0 off , = 0 , ̅ = ̅ 0 , and ̅ off = 0 ̅ off , and * = 0 * so that the dimensionless model reads subject to where Note that we have scaled * with 0 in this instance rather than max since the maximum amplitude of the Polo activity is not known a priori., and, since the receptors generate the active Polo in this model, this is the correct scaling for * .
In Figure 6, we plot the total Polo under normal conditions (parameter values given in Appendix Tables S4 and S5) as well as half dose Ana1 ( 0 → 0.5 0 , i.e. → 2 and off → 0.5 off ) and half dose Spd-2 ( 0 → 0.5 0 , i.e. → 2 and on → 0.5 on ). All solutions have been normalised with respect to the wild type solution.

Justification of parameter values
We drew on a number of sources to estimate the relative magnitudes of the dimensionless reaction rate parameters (Appendix Tables S4 and S5) for the models depicted schematically in Figures 2B and 4A.
Due to the rapid fluorescence recovery rates observed in FRAP experiments (Conduit et al., 2010(Conduit et al., , 2014Feng et al., 2017) (Figure 5), it follows that the Polo reaction rates, on,off , Spd-2 reaction rates, on,off,cat , and Spd-2 scaffold disassembly rate, * dis , are large relative to the cell cycle timescale. Furthermore, due to the large size and rapid construction rate of the Cnn scaffold, it follows that on,off,cat , are also large. By contrast, since FRAP data shows that the fluorescence level of the Cnn scaffold fails to fully recover even over an entire nuclear cycle ( Figure 5), it follows that the Cnn scaffold disassembly rate, dis , is small by comparison with the cell cycle timescale. In order to quantify the modelling assumption that the ̅ scaffold is more stable than the * scaffold, we prescribe that 1~̅ dis ≪ * dis .
We expect that the catalytic conversion, i.e. activation and subsequent release, of Spd-2 and Cnn is a more complex process than unbinding alone, so we assume the catalytic conversion rates are slower than the off rates. However, for simplicity, we assume that these rates are all of a similar order of magnitude. To ensure that the Spd-2 scaffold can convert Cnn into a scaffold before it disassembles, we also assume that the Spd-2 disassembly rate is less than the catalytic conversion rate of Cnn.
We make the additional assumption that on is larger than off since the amount of Polo in the embryo is large (Casas-Vila et al, 2017) and on is proportional to 0 . By contrast, we assume that on is smaller than off and cat as the amount of Spd-2 in the embryo is comparatively small (Casas-Vila et al, 2017) and on is proportional to 0 . Our own Fluorescence Correlation Spectroscopy (FCS) data indicate that Polo is present in the cytoplasm at 3-5X higher levels than Spd-2 or Cnn (Thomas Steinacker, personal communication).
Since the Polo receptors are required to both activate and deactivate in a single cycle, it follows that on,off are sufficiently large (i.e. greater than order unity) that the receptors have time to reset, but not so large that the resetting is instantaneous. We therefore suppose that they are ≈ 10 for simplicity.
These assumptions may be combined to read Since the total amount of Spd-2 and Cnn in the embryo likely greatly exceeds the number of Spd-2 receptors at the centriole, we prescribe ≪ 1 . Furthermore, we observe through Fluorescence Correlation Spectroscopy (FCS) analysis that the total amount of Spd-2 and Cnn in the embryo are similar (Thomas Steinacker, personal communication) and therefore ≈ . On the other hand, since the total amount of Polo bound to the centriole cannot exceed the number of receptors, it follows that * > 1. Finally, since we are unable to determine the relative sizes of 0 and 0 , we suppose that = (1) for simplicity. Hence, these parameters satisfy The assumptions outlined above lay the foundation for the parameter values we have chosen. However, it worth noting that the characteristic behaviour of the response curves is unaltered by doubling or halving any of these values, and therefore the particular regime we analyse in this manuscript is robust to variation in the parameters.