Autocatalytic feedback in centrosome growth drives centrosome size inequality.

(A) Schematic showing the dynamics of centrosomes during the cell cycle. In the G1 phase, there is a single centrosome with mother (M) and daughter (D) centrioles at the core, surrounded by the pericentriolar material (PCM). The two new centriole pairs with the old mother (oM) and the new mother (nM) separate into two centrosomes in the G2/M phase after centriole duplication. The spatially separated centrosomes then grow via a process called centrosome maturation (red arrow), prior to cell division. (B) Schematic of the autocatalytic growth model for centrosomes, where the assembly rate increases with increasing centrosome size. (C) Autocatalytic growth of centrosomes captures the sigmoidal size dynamics for a single and a pair of centrosomes, but is unable to ensure size equality of a centrosome pair. See Table 1 for a list of parameter values.

Parameter Values.

Lack of robust size control in autocatalytic growth.

(A) The relative difference in centrosome size, |δV|/⟨V ⟩, as a function of the growth rate constants and , with an initial size difference of 0.1 μm3. The light gray and dashed black lines represent the lines |δV|/⟨V⟩ = 0.2 and |δV|/⟨V⟩ = 1.0. (B,C) Size dynamics of a pair centrosomes for (B) weakly cooperative and (C) strongly cooperative growth regimes. (D) Dynamics of centrosome size for a single centrosome and a pair of centrosomes simulated using the non-cooperative growth model. Inset: Schematic of centrosome growth via centriole-localized assembly and disassembly distributed throughout the PCM. The |δV|/⟨V⟩ values in (A) represent an average over 1000 ensembles. The values of and are in the units of ×600 μM−1 s−1. See Table 1 for a list of parameter values. Parameter values for panel D were chosen to obtain typical steady-state centrosome size (∼ 5 μm3) and timescale of growth (∼ 500 s).

Figure 2—figure supplement 1. Purely autocatalytic growth model.

Figure 2—figure supplement 2. Growth via localized assembly and distributed disassembly.

Figure 2—figure supplement 3. Analysis of size inequality in the autocatalytic growth model.

Figure 2—figure supplement 4. Effect of diffusion on centrosome size regulation.

Catalytic growth in a shared enzyme pool leads to robust size control of a centrosome pair.

(A) Schematic of centrosome growth via catalytic activity of an enzyme that is activated by PCM proteins at a rate proportional to PCM size. (B) Reactions describing centrosome growth via catalytic activity of enzyme E. The centrosome (Sn) can activate the enzyme in a state E*, which in turn creates an activated subunit (S*) that binds the PCM. (C) Size dynamics of a centrosome pair (blue, red curves) growing via catalytic assembly and the dynamics of the activated enzyme ([E*]) in time (blue curve). (D) The ensemble average of relative absolute size difference |δV|/⟨V⟩ is insensitive to change in relative initial size difference δV0/V0. Inset: Probability distribution of δV for two different values of initial size difference (δV0/V0 = 0.1 and δV0/V0 = 0.4). (E) Centrosome growth curves obtained from the catalytic growth model (lines) fitted to experimental growth curves (points) measured at different stages of C. elegans development. (F) Degree of sigmoidal growth, measured by Hill coefficient α, as a function of the growth rate constant k+ and the total enzyme concentration [E]. (G) Model of shared catalysis considering a constant concentration of inactive enzyme (E) throughout the growth period. Inset: Schematic of the reactions showing the steady state cycle between S1, and Sn. (H) Centrosome pair growth in the presence of unlimited inactive enzyme pool exhibits size equality as well as cooperative growth dynamics. Inset: Dynamics of S1 and concentrations. See Table. 1 for a list of parameter values. Parameters were chosen to match typical steady-state centrosome size (∼ 5 μm3) and the timescale of growth (∼ 500 s). Parameters for panel E were obtained by fitting the enzyme kinetics.

Figure 3—figure supplement 1. Origin and regulation of the enzyme pulse.

Figure 3—figure supplement 2. Comparison between catalytic and autocatalytic growth models.

Figure 3—figure supplement 3. Effect of subunit diffusion on catalytic growth.

Centrosome size scaling with cell size.

(A) Scaling of centrosome size with cell size obtained from the catalytic growth model (line) fitted to experimental data (points) in C. elegans embryo (Zwicker et al., 2014). (B) Centrosome size does not scale with cell size when the assembly rates are much lower compared to disassembly rate (i.e., k*, k+ ≲ k−Vc). (C) Dynamics of the cytoplasmic fraction of subunits (S1 and combined) reveal significantly higher pool depletion in the size scaling regimes. The two curves correspond to the growth curves shown in panels A (blue) and B (black). The dashed lines are theoretical results obtained from the deterministic model. (D) An analytically obtained phase diagram of centrosome size scaling as functions of enzyme-dependent and enzyme-independent assembly rate constants. The color indicates the strength of size scaling (measured by dV /dVc). The dashed gray line indicates the contour dV /dVc = 0.1. Here the slope values are shown in δv units. Insets: Characteristic size scaling behaviours. See Table 1 for a list of parameter values. Parameters for panel B were obtained by tuning enzyme-dependent assembly rate and parameters for panel D were similar to panel A.

Figure 4—figure supplement 1. Centrosome size scaling and pool depletion in the catalytic growth model.

Control of centrosome size asymmetry via differential growth.

(A) Schematic illustrating asymmetric size regulation via differential growth in the (top) catalytic growth model and (bottom) autocatalytic growth model. (B,C) Ten representative trajectories showing the dynamics of centrosome size difference (V1 − V2) for (B) catalytic growth model (δk+/k+ = 0.2), and (C) autocatalytic growth model . The two centrosomes are initially of the same size. (D) Efficiency growth-rate-dependent control of centrosome size asymmetry (ε = N /Ntot) as a function of (normalized) initial size difference (δV0/V0) and (normalized) growth rate difference (δk+/k+), in the catalytic growth model. (E) Efficiency of growth-rate-dependent control of centrosome size asymmetry as a function of (normalized) initial size difference (δV0/V0) and (normalized) growth rate difference , in the autocatalytic growth model. See Table 1 for a list of model parameters. Parameter values for panel B & D were chosen to obtain typical steady-state centrosome size (∼ 5 μm3) and timescale of growth (∼ 500 s).

Two-component growth model across organisms.

Parameter values for two component growth via enzyme activity

Multi-component model for centrosome growth.

(A) Schematic of centrosome growth model driven by two scaffold components a and b, and enzyme E. a can bind the existing PCM independent of b or the enzyme E. The enzyme is activated by a in the scaffold, then released in the cytoplasm as E*. The other scaffold former b binds to PCM in a-dependent manner in an intermediate form bi which can undergo rapid disassembly. The intermediate form bi can get incorporated in the b-scaffold by the active enzyme E* via forming an activated subunit form E*bi. The red arrows indicate the size-dependent positive feedback and the green arrow indicates the catalytic activity of the enzyme. (B) Centrosome size (V1, V2) dynamics for growth with localized enzyme. (C) Centrosome size (V1, V2) dynamics for growth with shared enzyme pool (black and red curve) and the pulse-like dynamics of activated enzyme concentration ([E*], blue curve). (D) Radial spread of the two scaffold former components a and b corresponding to the centrosome growth shown in panel-C. See Table 1 for a list of parameter values. Parameter values for panel B & D were chosen to obtain typical steady-state centrosome size (∼ 5 μm3) and timescale of growth (∼ 500 s).

Figure 6—figure supplement 1. Centrosome growth via localized enzyme activity.

Figure 6—figure supplement 2. Centrosome growth via shared enzyme activity.

Failure of size regulation in purely autocatalytic growth.

(A) A model of autocatalytic growth of a single centrosome from a limited pool of subunits exhibits robust size control and sigmoidal growth. For a pair of centrosomes, this model leads to size inequality of the two centrosomes. (B) Phase portrait analysis shows completely overlapping nullclines (in thick gray and dotted black lines), creating a line attractor on which every point is a valid solution for the ODE system. The orange and green trajectories, obtained from stochastic growth simulations, show the extent of size inequality. The black dot indicates the equal size point (V1 = V2, but not a fixed point here). (C) Size difference between the two centrosomes (|δV|) increases with increasing initial size difference δV0 (i.e., two centrosomes have initial sizes V0 + δV0 and V0). The result presented in terms of the relative quantities and , shows lack of robustness in size regulation. The parameters for purely autocatalytic growth are: k+ = 2μM−1 s−1, ρ0 = 0.0108 μM. All other parameters are the same as the fixed parameters listed in Table 1.

Growth via localized assembly and distributed disassembly.

(A-B) Size dynamics for a centrosome pair shows strong suppression of initial size difference and robust control of centrosome size. (inset) The dynamics of δV shows monotonic decay towards small δV value. (C) Size dynamics plotted on the phase portrait, obtained from an equivalent deterministic description, shows the corresponding evolution of the two cases presented in panels A (orange) and B (green), respectively. The black and red lines indicate the nullclines and , respectively. Probability distribution of size deviation from the mean size V − ⟨V⟩. (inset) Probability distribution of size P (V1) and P (V2). These quantities demonstrate that δV in this case originates from the stochasticity in the size dynamics and it is independent of the initial size difference. Parameters: k+ = 1000μM−1 s−1, ρ0 = 0.1 μM. All other parameters are the same as the fixed parameters listed in Table 1.

Centrosome size inequality and size dynamics in the auto-catalytic growth model in different parameter regimes.

(A-F) Phase portrait from deterministic growth description and growth dynamics from stochastic simulations show the resulting centrosome size inequality during autocatalytic growth. (A,C,E) The phase portrait plot shows the nullclines and in thick gray and black dotted lines, respectively, and the growth trajectory from stochastic simulation is shown in orange. (B,D,F) The size dynamics from stochastic simulations show decreasing size inequality and decreasing sigmoidal growth with increasing non-cooperative growth rate . (G-I) Relative size inequality measured by |δV|/⟨V⟩ as a function of the non-cooperative growth rate and the cooperative growth rate . The parameter values for and are expressed in the units of ×600 μM−1 s−1. Here ρ0 = 0.033 μM and all other parameters are the same as the fixed parameters listed in Table 1.

Diffusion-limited growth mitigates centrosome size inequality but lacks sigmoidal nature.

(A) Schematic of reaction-diffusion simulation showing the two centrosomes that are δR distance apart in the 3D simulation volume. (B) Centrosome volume during autocatalytic growth for two different diffusion constant values. Solid and dashed lines indicate the volume curves for the centrosome pair. (C) Centrosome size inequality, , increases with increasing separation (δR) for different diffusion constants (indicated in different colours). The dashed line indicates 2% inequality, i.e., . (D) Centrosome size inequality as a function of separation distance δR. Size inequality increases with increasing initial size difference (indicated in different colours). The parameter values are and , ρ0 = 0.017 μM, Vc = 729 μm3 and all other parameters are the same as the fixed parameters listed in Table 1. We have chosen a smaller system size and a smaller pool size to reduce the computational cost.

Origin and regulation of the enzyme pulse.

(A) Dynamics of and E* show the decay of E* pulse in the wake of the production from E* and S1. (B-C) The features of the active enzyme pulse (E* dynamics) can be modulated by changing the rates of enzyme activation and subunit activation . (C) The centrosome growth rate (deduced from Sn(t)) changes with parameters regulating the pulse dynamics. The growth rate is reduced for a weaker pulse with a smaller amplitude and larger time period. The parameter values are the same as in Fig. 3C and values are obtained by changing values.

Effect of pool size and correlation between final and initial size difference.

(A-B) Centrosome size inequality as a function of the initial size difference , for different concentrations of the subunit pool. does not change with the increasing subunit pool size in autocatalytic growth model (A), while it decreases in the catalytic growth model (B). (C) Centrosome size difference (δV) in the autocatalytic growth model is positively correlated with the initial size difference (δV0), with a Pearson correlation coefficient R = 0.45. (D) Centrosome size difference (δV) in the catalytic growth model is uncorrelated with the initial size difference (δV0) with Pearson correlation coefficient R ∼ 0. The dashed lines in the panel C & D are obtained by linear fit as a guide to the eye. The parameter values for panels A and C are the same as in Fig. 2C and the parameter values for panels B and D are the same as in Fig. 3D.

Effect of subunit diffusion on catalytic growth.

(A) Size dynamics of centrosome pairs (dashed and solid lines) in the catalytic growth for different values of subunit diffusion constant (indicated by colour). The distance between the centrosomes is taken to be fixed at δR = 2 μm. (B) Size dynamics of centrosome pairs (dashed and solid lines) in the catalytic growth model for different values of centrosome separation distance (indicated by colour). The subunit diffusion constant is taken to be fixed at 10 μm2/s. (C) Centrosome growth curves with different values of initial size difference shows no significant effect on the final size difference. The parameter values used are ρ0 = 0.016 μM, [E] = 0.006 μM, Vc = 729 μm3 and k+ = 1 μM−1s−1 and other parameters are the same as in Fig. 3B.

Centrosome size scaling and pool depletion in the catalytic growth model.

(A) A phase diagram of centrosome size scaling, measured by the slope dV/dVc, as functions of assembly rate k* and cell (system) size Vc. The phase diagrams shows weaker size scaling for smaller assembly rate and larger system size. The values of the slope are expressed in units of δv. (B) A phase diagram of subunit pool depletion, measured by the cytoplasmic fraction of subunits (fc), as functionsof k+ and k*. (C-D) Model predictions for cytoplasmic fraction of subunits, as functons of Vc and M (organelle number), for parameters corresponding to C. elegans and Drosophila. The black arrows indicate the direction of embryonic development. Inset: Centrosome size scaling with centrosome number as the development progresses. The parameter values for A&B are same as used in main text Fig. 4D with k+ = 1 μM−1s−1 in A. Parameter values for C are: ρ0 = 0.01μM, E* = 0.01μM, k+ = 1 μM−1s−1, k* = 5000 μM−1s−1 and . Parameter values for D are same as C except ρ0 = 0.03μM and k* = 100 μM−1s−1. The inset results in C and D are obtained at cell volume Vc = 20000 μm3 and Vc = 106 μm3 respectively. All other parameters are the same as the fixed parameters listed in Table 1.

Centrosome growth via localized enzyme activity.

(A) The panel lists all the reactions used to simulate the growth of centrosomes consisting of two scaffolds Sn(a) and Sn(b). The a-scaffold grows independently of b and E with centriole-dependent assembly and disassembly throughout the PCM volume (Reaction. 1). The second scaffold former b can bind to the a-scaffold in a size-dependent manner to form an intermediate bi (Reaction. 2) that can disassemble fast from the scaffold . The enzyme gets activated by the a-scaffold and can activate the intermediate form bi, which can then assemble into the b-scaffold and increase the amount of Sn(b) (Reaction. 3 − 5). The b-scaffold can disassemble at a rate (Reaction. 6). The reactions above describe the growth of centrosome-1 as it is indicated as with i = 1. A similar set of reactions will govern the other centrosome too with all the rate constants being the same. (B) Time evolution of centrosome size dynamics and active enzyme dynamics. (C) The relative size inequality |δV|/⟨V⟩ is a monotonically increasing function of the initial size difference δV0/V0, indicating loss of robust size regulation. See Table 3 and Table 1 for a list of model parameters.

Centrosome growth via shared enzyme activity.

(A) The panel lists all the reactions used to simulate the growth of centrosomes of two scaffolds Sn(a) and Sn(b), where they share the activated enzyme E*. The reactions are the ame as the localized enzyme model, except Reaction. 3 − 4. Here the activated enzyme E* is released in a shared pool rather than being specific to a particular centrosoe. This active enzyme can activate the second scaffold former intermediate bi in any of the two centrosomes. (B) Centrosome size dynamics and active enzyme dynamics show insignificant size inequality but clear sigmoidal trend in growth. The active enzyme concentration exhibits pulse-like dynamics at the beginning of centrosome growth. (C) The relative size inequality |δV|/⟨V⟩ is very small in value and independent of the initial size difference δV0/V0, indicating robust regulation of size. (D) Total enzyme concentration [E] can effectively control the steady-state size of the centrosome, with increasing [E] leading to larger centrosome size. (E) Enzyme kinetics can signal the start and end of centrosome maturation and a continuous activity of enzyme is required to maintain the grown centrosome. We turned the enzyme activity on or off by making non-zero (zero) to see the effect of enzyme on centrosome growth. See Table 3 and Table 1 for a list of model parameters.