LET-99-dependent spatial restriction of active force generators makes spindle's position robust

TL;DR

This paper uncovers a novel LET-99-dependent spatial regulation mechanism that restricts active force generators to a posterior crescent, ensuring robust spindle positioning during C. elegans asymmetric cell division.

Contribution

It reveals a new positional control of cortical pulling forces via microtubule dynamics regulated by LET-99, enhancing understanding of spindle positioning robustness.

Findings

LET-99 restricts force generators to a posterior crescent.

Spindle position is robust despite variations in force generator numbers.

Microtubule contact density correlates with centrosome-cortex distance.

Abstract

During the asymmetric division of the Caenorhabditis elegans nematode zygote, the polarity cues distribution and daughter cell fates depend on the correct positioning of the mitotic spindle, which results from both centering and cortical pulling forces. Revealed by anaphase spindle rocking, these pulling forces are regulated by the force generator dynamics, which are in turn consequent of mitotic progression. We found a novel, additional, regulation of these forces by the spindle position. It controls astral microtubule availability at the cortex, on which the active force generators can pull. Importantly, this positional control relies on the polarity dependent LET-99 cortical band, which restricts or concentrates generators to a posterior crescent. We ascribed this control to the microtubule dynamics at the cortex. Indeed, in mapping the cortical contacts, we found a correlation between the centrosome-cortex distance and the microtubule contact density. In turn, it modulates pulling force generator activity. We modelled this control, predicting and experimentally validating that the posterior crescent extent controlled where the anaphase oscillations started, in addition to mitotic progression. Finally, we propose that spatially restricting force generator to a posterior crescent sets the spindle's final position, reflecting polarity through the LET-99 dependent restriction of force generators to a posterior crescent. This regulation superimposes that of force generator processivity. This novel control confers a low dependence on microtubule and active force generator exact numbers or dynamics, provided that they exceed the threshold needed for posterior displacement. Interestingly, this robustness originates in cell mechanics rather than biochemical networks.

Equations

$$q=\frac{\tau }{\frac{d}{v^{+}}+\frac{d}{v^{-}}+\tau }=\frac{1}{1+\alpha d}\text{with }\alpha =\frac{1}{v^{+}\tau }+\frac{1}{v^{-}\tau },$$

$$\rho =\frac{1+\alpha d_2}{1+\alpha d_1}$$

$$p\left(\mathcal{S},\alpha ,\theta ,\varphi \right)$$

$$P\left(\mathcal{S},\alpha \right)$$

$$\mathcal{M}\left(\mathcal{S},\alpha \right)$$

$$I(\bar{p})\ddot{y}+\left(\Gamma -\Xi (\bar{p})\right)\dot{y}+Ky=0,$$

$$\Xi (\bar{p})=2\bar{N}\left\{\frac{\bar{f}}{f_c}\bar{p}\left[\left(1-\bar{p}\right)-\frac{f_c}{\bar{f}}\right]\right\}{f}^{\prime },$$

$$I(\bar{p})=2\bar{N}\left\{\frac{\bar{f}}{f_c}\bar{p}\left(1-\bar{p}\right)\right\}{f}^{\prime }\frac{\bar{p}}{k_{on}},$$

$$K_{a}&=\frac{N_{\text{microtubule--force-generator}}}{N_{\text{free-microtubule-at-cortex}}\,N_{\text{free-force-generator}}}\\ N=N_{\text{Microtubule--force-generator}}+N_{\text{free-force-generator}}\\ M=N_{\text{microtubule--force-generator}}+N_{\text{free-microtubule-at-cortex}}\\ =\mathcal{M}\left(\mathcal{S},\alpha\right)$$

$$K_a=\{\begin{array}{l}\widehat{k_{on}}/\overline{k_{off}}\\ \bar{p}/\left(1-\bar{p}\right)/N_{\text{free-microtubule-at-cortex}}\end{array}$$

$$\begin{array}{rl}\mathcal{N}& \left(\mathcal{M}\left(\mathcal{S},\alpha \right)\right)=N\frac{\varphi -1}{\varphi +1}\\ & \text{with}\varphi ={\zeta }^{-}+\sqrt{1+{{\zeta }^{-}}^2+2{\zeta }^{+}}\\ & {\zeta }^{\pm }=K_a\left(\mathcal{M}\left(\mathcal{S},\alpha \right)\pm N\right)\end{array}$$

$${\left|\frac{X}{a}\right|}^n+{\left|\frac{Y}{b}\right|}^n=1$$

$${\tilde{K}}_a=\frac{\left[\text{Microtubule–force-generator}\right]}{\left[\text{Microtubule-at-cortex}\right]\left[\text{force-generator}\right]},$$

$$\tilde{\mathcal{M}}\left(\mathcal{S},\alpha,\theta,\phi\right)=\frac{M}{4\pi}\frac{1}{1+\alpha\,r_{\mathcal{S}}\left(\theta,\phi\right)}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{1}{r_{\mathcal{S}}\left(\theta,\phi\right)^{2}}}$$

$${\mathcal{F}}^{ante|post}\left(x,\overline{k_{off}}\right)=\frac{2\pi {\int }_{\theta =0}^{{\theta }_0^{ante|post}}p\left(\mathcal{S},\alpha ,x,\theta \right)\cos \theta d\theta }{P\left(\mathcal{S},\alpha ,x\right)}\times \mathcal{N}\left(\mathcal{S},\alpha ,x,K_a^{ante|post}\right)\bar{f}$$

$$I^{post}{\ddot{x}}_{post}+\left(\Gamma -{\Xi }^{post}\right){\dot{x}}_{post}+K{x}_{post}-K_{ante}{x}_{ante}=\eta +{\mathcal{F}}^{post}(x_{post})-{\mathcal{F}}_{ante}$$

$$k_{on}=\widehat{k_{on}}\left(\mathcal{M}\left(\mathcal{S},\alpha ,x_{post}\right)-\mathcal{N}\left(\mathcal{S},\alpha ,x_{post},K_a^{post}\right)\right)$$

$$F_{ante}=\{\begin{array}{ll}{\mathcal{F}}_{ante}& \text{if}\overline{k_{off}}\ge \overline{k_{off}^0}\\ \lambda {\mathcal{F}}_{ante}& \text{if}\overline{k_{off}}<\overline{k_{off}^0}\end{array}$$

$$K_{ante}=\{\begin{array}{ll}K& \text{if}\overline{k_{off}}\ge \overline{k_{off}^0}\\ \lambda K& \text{if}\overline{k_{off}}<\overline{k_{off}^0}\end{array}$$

Full text

1. 1 Introduction

2. 2 Modelling the positional switch on oscillation onset

3. 2.1 Quantity of microtubules reaching the posterior crescent of active force generators

4. 2.1.1 Modelling hypotheses and microtubule dynamics parameter estimates

5. 2.1.2 Microtubule dynamics “measure” the centrosome–cortex distance.

6. 2.1.3 Number of microtubules reaching the cortex

7. 2.2 Towards the expanded tug-of-war model

8. 2.2.1 The initial model

9. 2.2.2 Evolution of the initial model to account for the polarity encoded through force generator on-rate

10. 2.2.3 Number of engaged force generators: modelling the binding of a microtubule to a force generator

11. 2.2.4 The change of regime in the number of microtubules reaching the cortex versus the centrosome position is independent of detailed embryo shape

12. 2.2.5 Sensitivity analysis of the oscillation onset position to embryo geometry and microtubule dynamics

13. 2.2.6 Discussion: number– or density–limited force generator-microtubule binding

14. 2.2.7 The processivity and microtubule dynamics set two independent switches on force generators: the expanded tug-of-war model

15. 3 Simulating posterior displacement and final position

16. 4 Conclusion

1 Introduction

During the division of nematode zygote, the spindle undergoes a complex choreography. Firstly, during prophase, the pronuclei-centrosomes complex (PCC) moves from posterior half of the embryo to its center, during the so-called centring phase (Ahringer,, 2003) and concurrently the two centrosomes align along the antero-posterior-axis. We previously found, in contrast to Caenorhabditis elegans embryo, that this displacement was a bit excessive in C. briggsae reaching a slightly more anterior position, a phenomenon called overcentration (Kimura and Onami,, 2007; Riche et al.,, 2013). The consequence is a delay in spindle posterior displacement for this species with respect to C. elegans. Interestingly, the same proteins that cause the anaphase posterior displacement are needed for this (Riche et al.,, 2013), namely the trimeric compex GPR-1/2${}^{\text{LGN}}$, LIN-5${}^{\text{NuMA}}$ and dynein (Nguyen-Ngoc et al.,, 2007). Later on, during prometaphase and metaphase, the spindle is maintained in the middle by centring forces that are independent of GPR-1/2${}^{\text{LGN}}$ and may be caused by microtubule pushing on the cell cortex (Pecreaux et al.,, 2016). Finally, during late metaphase and anaphase, GPR-1/2-dependent cortical pulling forces become dominant and displace the spindle posteriorly, make it oscillate, and contribute to its elongation (Grill et al.,, 2003; Labbe et al.,, 2004; Pecreaux et al.,, 2006).

We aim here to complement our previously published “tug-of-war” model (Grill et al.,, 2005; Pecreaux et al.,, 2006), later called initial model, which was mainly focused on the dynamics of cortical force generators (f.g.), by including the dynamics of astral microtubules (MTs). Indeed, we mapped the microtubule contacts at the cortex and revealed that they mostly concentrated in cortical regions close to the centrosomes (Bouvrais et al.,, 2018). In consequence, the position of the centrosomes, as microtubule organizing centres (MTOC), regulates the quantity of engaged force generators pulling on astral microtubules and in turn spindle’s anaphase oscillation and posterior displacement.

First, focusing on the oscillation onset, we expanded our initial model of spindle oscillation to account for microtubule dynamics. We detailed the expanded model and then explored how this novel positional regulation combines with the one by force generator processivity previously reported (Pecreaux et al.,, 2006). Second, through a stochastic simulation approach, we looked at the feedback loop created between the position of the posterior centrosome and the pulling forces contributing to spindle displacement.

2 Modelling the positional switch on oscillation onset

2.1 Quantity of microtubules reaching the posterior crescent of active force generators

Recent work suggested that force generators would be active only on a posterior cap instead of the whole posterior half cortex of the embryo (Krueger et al.,, 2010). This means that only the microtubules hitting the posterior crescent of the cortex would contribute to spindle displacement by binding to active force generators. We thus calculated the number of microtubules reaching this so-called active region of the cortex.

2.1.1 Modelling hypotheses and microtubule dynamics parameter estimates

We set to explore whether the number of microtubules reaching the cortex, assumed to be in excess during anaphase (Grill et al.,, 2005; Pecreaux et al.,, 2006), could be limiting prior to oscillation onset. Key to assess this possibility was an estimate of the total number of microtubules and their dynamics. Based on previously published experiments, we assessed the following microtubule related parameters:

• •

Total number of microtubules. To assess the number of microtubule nucleation sites at the centrosome (CS), we relied on electron microscopy images of the centrosomes (Redemann et al.,, 2016), which suggested $3000$ or more microtubules emanating per centrosome. This order of magnitude was previously proposed by O’Toole and collaborators (O’Toole et al.,, 2003). More specifically in the figure 3, authors provide a slice of about $0.85\;\mu\text{m}$ thick (as estimated from video 8) displaying 520 astral microtubules, while centrosome diameter was estimated to $1.5\;\mu\text{m}$. Only a slice of centrosome was viewed in this assay, so that the number of microtubule nucleation sites per CS was extrapolated to a least $1800$ considering the centrosome as a whole sphere. In this work, we set the number of microtubules to $M=3000$. Variation of this number within the same order of magnitude does not change our conclusions.

• •

The microtubules are distributed around each centrosome in an isotropic fashion. We hypothesized an isotropic distribution of microtubules around each centrosome following (Howard,, 2006). This was also suggested through electron microscopy (Redemann et al.,, 2016).

• •

Free-end catastrophes are negligible. With the above estimate of the microtubule number and considering a microtubule growing speed in the cytoplasm $v^{+}=0.67\;\mu\text{m/s}$ (Srayko et al.,, 2005) and a shrinking one $v^{-}=0.84\;\mu\text{m/s}$ (Kozlowski et al.,, 2007), we could estimate that about 70 microtubules reach the cell periphery (assumed to be at $15\;\mu\text{m}$ from the centrosome) at any moment and per centrosome, if the free-end catastrophe rate is negligible. This estimate appears consistent with the instantaneous number of force generators in an half-cortex, estimated between 10 and 100 (Grill et al.,, 2003).

Furthermore, it was recently proposed that the catastrophe rate could be as high as $0.25\;\text{s}^{-1}$ in the mitotic spindle (Redemann et al.,, 2016). On the one hand, this might be specific to this organelle since the spindle is much more crowded than the cytoplasm. On the other hand, these authors proposed a total number of microtubules two to three folds larger than our estimate. We asserted that our conservative estimate of the microtubule quantity combined with the negligible free-end catastrophe resulted in similar modelling results, with the advantage of the simplicity over a full astral microtubule model. In other words, we focused on the fraction of astral microtubules not undergoing free-end catastrophe, which was the only one measurable at the cortex.

We next wondered whether the assumption of negligible free-end catastrophe is consistent with our measurement of microtubule contact density at the cortex. After (Redemann et al.,, 2016), the vast majority of microtubules emanating from the centrosome are astral: we thus assumed that the kinetochore and spindle microtubules were negligible in this estimate. Focusing on metaphase and with a residency time of microtubule ends at the cortex $\tau=1.25\;\text{s}$ (Kozlowski et al.,, 2007; Bouvrais et al.,, 2018), this led to about 100 microtubules contacting the cortex per centrosome, at any given time. Using our ”landing” assay (Bouvrais et al.,, 2018), we could estimate the number of contacts in the monitored region at any given time to 5 microtubules. Extrapolating this to a whole centrosome and assuming the isotropic distribution of astral microtubules (§2.1.2), we found 26 cortical contacts of microtubules at any time in metaphase. Although a bit low, likely because of the conservative parameters of the image processing that could led to missing some microtubules, this experimental assessment was consistent with the theoretical estimate based on our hypotheses. Furthermore, it was also consistent with the measurement done by (Garzon-Coral et al.,, 2016). In contrast, a non negligible catastrophe rate would have dramatically reduced that number of contacts at any given time. We concluded that free-end catastrophe rate was safely negligible.

• •

No microtubule nucleation sites are left empty at the centrosomes This is a classic hypothesis (Howard,, 2006), recently supported by electron microscopy experiments (Redemann et al.,, 2016).

2.1.2 Microtubule dynamics “measure” the centrosome–cortex distance.

Probability for a microtubule to be at the cell cortex

Because microtubules spend most of their “lifespan” growing to and shrinking from the cortex, the distance between the centrosome and the cortex limits the number of microtubules residing at the cortex at any given time. We could thus summarize microtubule dynamics in a single parameter $\alpha$ by writing the fraction of time spent by a microtubule at the cell cortex:

[TABLE]

where $d$ is the distance from the centrosome (MTOC) to the cortex (estimated to typically $d=15\;\mu\text{m}$, id est about half of the embryo width). We then found $\alpha=2.15\times 10^{6}\;\text{m}^{-1}$ using the above microtubule dynamics parameters. This meant that the microtubule spent $q=3\%$ of its time at the cortex and the remaining time growing and shrinking. This fraction of time spent residing at the cortex was consistent with the estimate coming from investigating the spindle centering maintenance during metaphase (Pecreaux et al.,, 2016).

Range of variations in the microtubule contact densities at the cortex.

The nematode embryo shape is close to an ellipsoid. Therefore, the centrosome displacement can vary the centrosome-cortex distance by 1.5 to 2 fold. We wondered whether the microtubule dynamics were so that one could observe significant variations in cortical microtubule-residing probabilities $q$. We estimated this sensitivity through the ratio $\rho$ of the probability of reaching the cortex when the centrosome was at its closest position $d_{1}$ (set to half of the embryo width, i.e. the ellipse short radius) divided by the probability when it was at its furthest position $d_{2}$ (chosen as half of the embryo length, i.e. the ellipse long axis).

[TABLE]

This curves had a sigmoid–like shape with $\lim\displaylimits_{\alpha\to 0}\rho=1$ and $\lim\displaylimits_{\alpha\to\infty}\rho=d_{2}\left/d_{1}\right.$.

Using our measurement of microtubule contact distribution at the cortex (Bouvrais et al.,, 2018), we calculated an experimental estimate of this sensitivity parameter, $\rho_{exp}\simeq 2$. On model side, because the experimental “landing” assay did not enable us to view the very tip of the embryo (Figure 1), we compared the sensitivity ratio calculated from the density map with a theoretical one that did not use the half embryo length as maximum distance but the largest distance effectively measurable. For untreated embryos viewed at the spindle plane, the measured embryo length was $2a=49.2\;\mu\text{m}$, while imaging at the cortex, the length along anteroposterior (AP) axis (denoted with bars) was $2\bar{a}=38.0\;\mu\text{m}$ for the adhering part to the coverslip. We could calculate the truncation of the ellipse due to the adhesion through the polar angle $\zeta=\arccos\left(\bar{a}\left/a\right.\right)$ of the boundary of the adhering region. We obtained $\zeta=39.4\degree$ which corresponded to a spindle plane to flattened cortex distance of $10\;\mu\text{m}$, using a parametric representation of the ellipse. During metaphase (set as the two minutes preceding anaphase onset), when the spindle is roughly centered (Pecreaux et al.,, 2016), the average spindle length was $11.8\;\mu\text{m}$ ($N$ = 8 embryos). The furthest visible region was thus at $d_{2}=16.5\;\mu\text{m}$ while the closest one was at $d_{1}=10\;\mu\text{m}$, leading to a sensitivity ratio $\rho=1.62$ consistent with the microtubule cortical contact density ratio observed in vivo for C. elegans. We concluded that microtubule dynamics in C. elegans enable the read-out of the posterior centrosome position through the probability of microtubules to be in contact with the cell cortex.

2.1.3 Number of microtubules reaching the cortex

We set to estimate the variation of the total number of astral microtubule contacts emanating from a single centrosome versus the position of this centrosome along the AP axis. We worked in spherical coordinates $(r,\theta,\phi)$ centered on the posterior centrosome that displayed a slow posterior displacement assumed to be a quasi-static motion, with zenith pointing towards posterior. We denoted $\theta$ the zenith angle and $\phi$ the azimuth (Figure 2A). We calculated the probability of a microtubule to reach the cortex in the active region, represented as $\theta\in[0,\theta_{0}]$ and $\phi\in[0,2\pi[$. We integrated over the corresponding solid angle and the number of microtubules reaching the cortex $\mathcal{M}\left(\mathcal{S},\alpha\right)$ came readily (Figure 2B):

[TABLE]

where $r_{\mathcal{S}}\left(\theta,\phi\right)$ is the distance centrosome–cortex in polar coordinates centered on the centrosome, dependent upon the shape of the cortex $\mathcal{S}$ and the boundary of the active force generator region $\theta_{0}$ (Figure 1). We observed a switch-like behaviour as the posterior centrosome went out of the cell centre and closer to the posterior side of the embryo (Figure 2B).

2.2 Towards the expanded tug-of-war model

In the initial model (Grill et al.,, 2005; Pecreaux et al.,, 2006), we made the assumption that the limiting factor was the number of engaged cortical force generators while in comparison, the astral microtubules were assumed to be in excess. It resulted that oscillations were driven by the force generator quantity and dynamics. In the linearised version of the initial model, the persistence of force generators to pull on microtubules (i.e. their processivity) mainly governed the timing and frequency of the oscillations, while the number of force generators drove the amplitude of oscillations (Pecreaux et al.,, 2006). However, since the number of microtubules reaching the cortex could be limiting (Kozlowski et al.,, 2007), we expanded the initial model of anaphase oscillations to account for this possible limitation.

2.2.1 The initial model

We provide here a brief reminder of the initial tug-of-war model (Pecreaux et al.,, 2006). It featured cortical force generators exhibiting stochastic binding to and detaching from microtubules at rates $k_{on}$ and $k_{off}$ ($\overline{k_{off}}$ being the detachment rate at stall force $\bar{f}$), respectively. The force generators were assumed to act close to stall force. The mean probability for a force generator to be pulling on a microtubule then reads $\bar{p}=k_{on}\left/\left(k_{on}+\overline{k_{off}}\right)\right.$. The active force generators were distributed symmetrically between the upper and lower posterior cortices but asymmetrically between anterior and posterior cortices (Grill et al.,, 2003). In the model, we also included two standard properties of the force generators: firstly, a force-velocity relation $f=\bar{f}-f^{\prime}\,v$, with $f$ the current force, $v$ the current velocity, and $f^{\prime}$ the slope of the force velocity relation ; secondly, a linearised load dependent detachment rate $k_{off}=\overline{k_{off}}\left(1-\frac{\bar{f}-f}{f_{c}}\right)$, with $f_{c}$ the sensitivity to load/pulling force, assuming that force generator velocity was low, i.e. they acted close to the stall force (Pecreaux et al.,, 2006). We finally denoted $\Gamma$ the passive viscous drag, related in part to the spindle centring mechanism (Garzon-Coral et al.,, 2016; Pecreaux et al.,, 2016; Howard,, 2006) and $\bar{N}$ the number of available force generators in the posterior cortex.

A quasi-static linearised model of the spindle posterior displacement reads:

[TABLE]

with

[TABLE]

and

[TABLE]

with $K$ the centering spring stiffness and $I$ the inertia resulting from stochastic force generator binding and unbinding. The spindle oscillations develop when the system becomes unstable, meaning when the negative damping $\Xi(\bar{p})$ overcomes the viscous drag $\Gamma$.

2.2.2 Evolution of the initial model to account for the polarity encoded through force generator on-rate

When we designed the initial model, it was known that the spindle posterior displacement was caused by an imbalance in the number of active force generators (Grill et al.,, 2003), i.e. the number of force generators engaged in pulling on astral microtubules or ready to do so when meeting an astral microtubule. However, the detailed mechanism building this asymmetry was elusive. We recently investigated the dynamics of dynein at the cell cortex (Rodriguez Garcia et al.,, 2017) and concluded that the force imbalance rather resulted from an asymmetry in force generator attachment rate to the microtubule. This asymmetry reflects the asymmetric location of GPR-1/2 (Park and Rose,, 2008; Riche et al.,, 2013). More abundant GPR-1/2 proteins at posterior cortex could displace the attachment reaction towards more binding/engaging of force generators. Therefore, to simulate the posterior displacement of the posterior centrosome (§3), we rather used the equations above (Eq. 6-8) with distinct on-rates between anterior and posterior sides and equal quantity of available force generators.

2.2.3 Number of engaged force generators: modelling the binding of a microtubule to a force generator

Force generator–Microtubule attachment modelling

To account for the limited number of cortical anchors (Grill et al.,, 2005; Pecreaux et al.,, 2006), we modelled the attachment of a force generator to a microtubule (Nguyen-Ngoc et al.,, 2007) as a first order process, using the law of mass action on component quantity (Koonce and Tikhonenko,, 2012) and combined it to the equations of quantity conservation for force generators and microtubules. It corresponded to the pseudo-chemical reaction:

\ce

Microtubule + Force-generator -¿ MT––Force-generator

and the equilibrium equation came readily:

[TABLE]

where $N$ is the total number of force generators present in the active region.

We could relate the association constant $K_{a}$ to our initial model (Pecreaux et al.,, 2006) (§2.2.1) by writing

[TABLE]

with the on-rate $k_{on}=\widehat{k_{on}}N_{\text{free-microtubule-at-cortex}}$, and the off-rate $\overline{k_{off}}(t)$ thought to depend on mitosis progression. Time dependences were omitted for sake of clarity. It was noteworthy that $k_{on}$, used in the initial model as force generator binding rate (assuming microtubules in excess), became variable throughout mitosis in the expanded model as it depends on the number of free microtubule contacts at the cortex, thus on the centrosome position. In contrast, $\widehat{k_{on}}$ appeared constant in the expanded model representing the on-rate of the first order reaction above.

Related parameter estimate

In modelling anaphase oscillation onset, we assumed that the off-rate dependence on mitosis progression was negligible (§2.2.7 and 3 for full model without this assumption). The positional switch modelled here led to a limited number of engaged force generators at oscillation onset. At this time, the force generator quantity just crossed the threshold to build oscillations (Pecreaux et al.,, 2006) and we estimated that typically 70% of the force generators were thus engaged, consistent with the quick disappearance of oscillations upon progressively depleting the embryo from GPR-1/2 proteins. We observed that the oscillation started when the centrosome reached 71% of embryo length (Bouvrais et al.,, 2018, Table 1). At that moment, 52 microtubules were contacting the cortex (§2.1.1). We set the total number of force generators to 50 and got a number of engaged ones consistent with previous reports (Grill et al.,, 2003). We thus estimated the association constant $K_{a}^{0}\simeq 0.1$ (denoted with 0 superscript to indicate that we assumed negligible its variation throughout mitosis). In turn, we estimated $\widehat{k_{on}}\simeq 0.025\;\text{s}^{-1}$ assuming that the detachment rate at that time was about $4\;\text{s}^{-1}$ (Rodriguez Garcia et al.,, 2017). If 70% of the force generators were engaged at oscillation onset, it would correspond to $k_{on}\simeq 0.375\;\text{s}^{-1}$, thus comparable to the estimate of this parameter in the initial model (Pecreaux et al.,, 2006).

Modelling the number of engaged force generators in the posterior crescent

In mitosis early stages, when the spindle lays in the middle of the embryo ( C. elegans) or slightly anteriorly (C. briggsae), both centrosomes are far from their respective cortex and thus the imbalance in active force generator quantity due to embryo polarity results in a slight posterior pulling force and causes a slow posterior displacement. The closer the posterior centrosome gets to its cortex, the larger the force imbalance (because more microtubules reach the cortex), and the posterior displacement accelerates to (potentially) reach an equilibrium position during metaphase resulting in a plateau in posterior centrosome displacement located around 70% of the AP axis. Once anaphase is triggered, the decreased coupling between anterior and posterior centrosomes results into a sudden imbalance in favour of posterior pulling forces so that the posterior displacement speeds up (Bouvrais et al.,, 2018).

We quantitatively modelled this phenomenon by combining the law of mass action above (Eq. 9a) with the number of microtubules reaching the posterior crescent (Eq. 5) to obtain the number of engaged force generators in the posterior cortex as following:

[TABLE]

To challenge our expanded model, we tested the switch-like behaviour in a broad range of association constants $K_{a}$ (Figure 3A). When the posterior centrosome was between 50% and 70% of embryo length, we observed that the number of engaged force generators was increased up to a threshold that enabled oscillations, consistently with (Pecreaux et al.,, 2006). When the centrosome was posterior enough, practically above 70% of AP axis, the number of engaged force generators saturated, suggesting that their dynamics were now the control parameters, as proposed in the initial model during anaphase. We also observed that a minimal binding constant was needed to reach the threshold number of engaged force generators required for oscillations. Interestingly, above this minimal $K_{a}$, further increase of the binding constant did not alter significantly the positional switch (Figure 3A). This suggested that this positional switch operates rather independently of the force generator processivity. This will be further discussed below (§2.2.7).

The positional switch is independent of the total number of force generators, as soon as this quantity is above a threshold

As we previously suggested that the total number of force generators should not impact the positional switch (Riche et al.,, 2013), we calculated the corresponding prediction in our expanded model (Bouvrais et al.,, 2018, Figure S2B) and compared it with experimental prediction (Bouvrais et al.,, 2018, Figure S2C). The good match supports our expanded model. In modelling gpr-1/2 mutant through the total number of force generators $N$, we followed the common thought that asymmetry of active force generator was due to an increased total number of force generators on the posterior side.

We recently proposed that the asymmetry in active force generators could be an asymmetry of force generator association rate to form the trimeric complex that pulls on microtubules (Rodriguez Garcia et al.,, 2017). GPR-1/2 presence would increase this on-rate. In our expanded model, a decreased on-rate (through gpr-2 mutant) would result in a decrease association constant $K_{a}$. Like is the previous case, above a certain threshold of $K_{a}$, the position at which oscillations were set on was not significantly modified (Figure 3A). In conclusion, independently of the details used to model the force imbalance consequence of the polarity (i.e. the total number or the on-rate), the mild depletion of GPR-1/2 experiment, causing a reduced number of active force generators, supported our expanded model.

2.2.4 The change of regime in the number of microtubules reaching the cortex versus the centrosome position is independent of detailed embryo shape

The above results were obtained by assuming an ellipsoidal shape for the embryo (an ellipsoid of revolution around the AP axis, prolate or oblate). We wondered whether a slightly different shape could alter the result. We thus repeated the computation, modelling the embryo shape by a super-ellipsoid of revolution, based on super-ellipses (Lamé curves) (Edwards,, 1892) defined as:

[TABLE]

with $a$ and $b$ the half length and width, $n$ the exponents, and $(X,Y,Z)$ the cartesian axes with $X$ along the AP axis (long axis), and positive values towards the posterior side. We obtained a similar switch-like behavior (Figure 2). We concluded that the switch-like behaviour was resistant to changes of the detailed embryo shape and thus we performed the remaining investigations with an ellipsoid shape, for sake of simplicity.

2.2.5 Sensitivity analysis of the oscillation onset position to embryo geometry and microtubule dynamics

The expanded model offers a regulation of cortical pulling forces, as revealed by oscillation onset, by the position of the centrosome. We therefore investigated how the shape of the embryo could impact the switch. Indeed, various species of nematode display different long and short axes, resulting in variation of scale and eccentricity (Farhadifar et al.,, 2015). In (Bouvrais et al.,, 2018, Figure 4), we reported that embryo length has a reduced impact on the switch. In contrast, the embryo width is more influential over the switch (Figure 5A). It is noteworthy that embryo length undergoes a stronger selection in genetic studies in comparison with embryo width (Farhadifar et al.,, 2016).

Then, we investigated the sensitivity of the oscillation onset position to parameters describing embryo shape in a different representation. We found a robustness of the position of oscillation onset versus the eccentricity, i.e. variations in embryo length keeping area constant (Figure 5CD), while embryo scale was more influential (Figure 5B). This is perfectly consistent with the positional control, which measures the distances in units of microtubule dynamics (§2.1.2). Consequently, the position at which oscillation starts is highly dependent on microtubule dynamics (Figure 5E).

2.2.6 Discussion: number– or density–limited force generator-microtubule binding

By writing the law of mass action in protein quantity (Eq. 9a), we assumed that the force generator-microtubule binding reaction was rate-limited but not diffusion-limited. We recently investigated the dynamics of cytoplasmic dynein (Rodriguez Garcia et al.,, 2017) and observed that dynein molecules were abundant in cytoplasm, thus 3D diffusion combined to microtubule plus-end accumulation brought enough dynein to the cortex. Therefore, diffusion of dynein to the cortex was not likely to be a limiting factor in binding force generators to the microtubules. However, another member of the force-generating complex, GPR-1/2, essential to generate pulling forces (Nguyen-Ngoc et al.,, 2007; Grill et al.,, 2003; Pecreaux et al.,, 2006), may be limiting. GPR-1/2 is likely localised at the cell cortex prior to assembly of the trimeric complex (Riche et al.,, 2013; Park and Rose,, 2008), and in low amount, leading to a limited number of cortical anchors (Grill et al.,, 2003, 2005; Pecreaux et al.,, 2006). We thus asked whether a limiting areal concentration of GPR-1/2 at the cortex could alter our model predictions. In the model proposed here, we considered force generator as a reactant of binding reaction. This latter included the molecular motor dynein but also other member of the trimeric complex, as GPR-1/2. Therefore, a limited cortical areal concentration in dynein or GPR-1/2 was modelled identically as a limited areal concentration of force generator. We wrote the corresponding law of mass action in concentration:

[TABLE]

with $\left[\text{force-generator}\right]=\frac{N_{\text{force-generator}}}{S_{activeRegion}}$, $\tilde{K}_{a}=K_{a}S_{activeRegion}$ and $S_{activeRegion}$ the posterior crescent surface (active region), whose boundary is considered at $70\%$ of embryo length. Modelling the embryo by a prolate ellipsoid of radii $24.6\;\mu\text{m}$ and twice $15.75\;\mu\text{m}$, we obtained $S_{activeRegion}\simeq 0.147\,S_{embryo}=610\;\mu\text{m}^{-1}$, while the whole embryo surface was $S_{embryo}\simeq 4100\;\mu\text{m}^{2}$.

The probability of a microtubule to hit the cortex (Eq. 3 and 5 ) was modified as follow:

[TABLE]

We then calculated the number of engaged force generators as above (Eq. 11 ) and found also a positional switch (Figure 3B compared to 3A). We concluded that this alternative modelling of force generator–microtubule attachment was compatible with the positional switch that we observed experimentally.

In contrast with the law of mass action in quantity, when the centrosome was further displaced towards the posterior after the positional switch, we did not observe any saturation in engaged force generators but a decrease (Figure 3B). This may suggest that the centrosome position could control the oscillation die-down, if diffusion of member(s) of the trimeric complex in the cortex was the limiting factor. In such a case, one would expect that die-down did not intervene after a fixed delay from anaphase onset, but at a given position. This contrasted with experimental observations upon delaying anaphase onset (Bouvrais et al.,, 2018, Table 1). Therefore, the law of mass action in quantity appeared to better model our data.

On top of this experimental argument, we estimated the lateral diffusion of the limited cortical anchors, likely GPR-1/2, and calculated a corresponding diffusion limited reaction rate equal to $k_{on}^{D}\simeq 1.2\times 10^{5}\;\text{s}^{-1}$ after (Freeman and D.,, 1983; Freeman and Doll,, 1983). We considered the parameters detailed previously, a diffusion coefficient for GPR-1/2 similar to the one of PAR proteins $D=0.2\;\mu\text{m}^{2}/\text{s}$ (Goehring et al.,, 2011), and a hydrodynamic radius of $5.2\;\text{nm}$ (Erickson,, 2009). Compared to the on-rate value proposed above (§2.2.3), i.e. $k_{on}\simeq 0.375\;\text{s}^{-1}$, this suggested that lateral diffusion was not limiting. In contrast, it was proposed that in such a case, lateral diffusion may even enhance rather than limit the reaction (Adam and Delbruck,, 1968). We concluded that the process was limited by reaction, not diffusion, and we considered action mass in quantity (Eq. 9a) in the remaining of this work.

2.2.7 The processivity and microtubule dynamics set two independent switches on force generators: the expanded tug-of-war model

We next asked whether a cross-talk exists between the control of the oscillation onset by the processivity, as previously reported (Pecreaux et al.,, 2006), and the positional switch explained above. To do so, we let $K_{a}$ varying with both the processivity $1\left/\overline{k_{off}}\right.$ and the centrosome position. In the notations of the initial model, since we kept $\widehat{k_{on}}$ constant, it meant that $k_{on}$ varied because of changes in the number of microtubule contacts in the posterior crescent, in turn depending on the centrosome position. We then computed the pairs $(\overline{k_{off}^{c}},x^{c})$ so that Eq. 6 was critical, i.e. $\Xi^{c}=\Gamma^{t}$ (Eq. 7), with $x^{c}$ the critical position of the centrosome along the AP axis and $\overline{k_{off}^{c}}$ the critical off-rate. Because we considered the transverse axis and a single centrosome, we used $\Gamma^{t}=140\;\mu\text{N.s/m}$ after (Garzon-Coral et al.,, 2016) and obtained the diagram reproduced in (Bouvrais et al.,, 2018, Figure 5A) that could be seen as a stability diagram. When the embryo trajectory (the orange arrow) crosses the first critical line (collection of $(\overline{k_{off}^{c}},x^{c})$, depicted in blue) to go into the unstable region (blue area), the oscillations start and develop. Since this line is diagonal, it suggests that such an event depends upon the position of the posterior centrosome (ordinate axis) and the detachment rate (abscissa), suggesting that two control parameters contribute to making the system unstable and oscillating. Interestingly, when the embryo continues its trajectory in the phase diagram, it crosses the second critical line (depicted in green), which corresponds to the moment the system becomes stable again, and oscillations are damped out. This critical line is almost vertical indicating that this event depends mostly on the detachment rate, i.e. the inverse of processivity, consistent with the experimental observations (Bouvrais et al.,, 2018, Table 1). Interestingly, this behaviour is maintained despite modest variations in the range of processivity and centrosome position explored during the division (i.e. the precise trajectory of the embryo in this stability diagram). Note that large values of detachment rate are irrelevant as they do not allow posterior displacement of the spindle (Bouvrais et al.,, 2018, Figure 7C, orange curve). We concluded that two independent switches control the onset of anaphase oscillations and broadly the burst of pulling forces contributing to spindle elongation and posterior displacement.

3 Simulating posterior displacement and final position

Because the cortical pulling forces involved in the anaphase spindle oscillations are also causing the posterior displacement, and because they depend on the position of the posterior centrosome, it creates a feedback loop on the posterior centrosome position. Resistance to changes of some parameters revealed by the sensitivity analysis of the oscillation onset suggests that these same parameters may have a reduced impact on the final position of the centrosome. In turn, this final position is essential as it contributes to determine the position of the cytokinesis cleavage furrow, a key aspect in an asymmetric division to correctly distribute cell fate determinants (White and Glotzer,, 2012; Rappaport,, 1971; Knoblich,, 2010).

To simulate the kinematics of posterior displacement, we considered the expanded model (§2.2) and a slowly-varying binding constant $K_{a}$ due to the processivity increasing throughout mitosis (§2.2.3). We calculated the posterior pulling forces, assuming an axisymmetric distribution of force generators. The projection of the force exerted by the cortical pulling force generators implied a weakening factor because only the component parallel to the AP axis contributes to displace posteriorly the spindle. To calculate it, we made the assumption that any microtubule contacting the cortex in the active region has an equal probability to attach a force generator. Therefore, we obtained the force weakening due to AP axis projection by writing the ratio of the forces exerted by each microtubule contacting the cortex weighted by the probability of a contact and integrated over the active region, over the number of microtubule contacts calculated using Eq. 14. This weakening ratio was then multiplied by the number of bound force generators previously obtained (Eq. 11). The weakened of the pulling force along AP axis $\mathcal{F}$ then reads:

[TABLE]

with $\theta_{0}$ the polar angle of the active region boundary positioned at $x_{ante}^{0}$ and $x_{post}^{0}$, obtained assuming an ellipsoidal shape for the embryo. $p\left(\mathcal{S},\alpha,x_{ante|post},\theta\right)$ was defined at Eq. 3 and $P\left(\mathcal{S},\alpha,x_{ante|post}\right)$ at Eq. 4. The Eq. 15 was used to calculate both anterior and posterior forces, with their respective parameters. After Rodriguez Garcia et al., (2017), the force asymmetry was due to an asymmetry of f.g.-MT affinity, under the control of GPR-1/2. We accounted for this asymmetric on-rate through an asymmetric attachment constant writing $K_{a}^{ante|post}=\beta^{ante|post}\widehat{kon}\left/\overline{k_{off}}\right.$.

We put the above quantities into Eq 6 to finally get:

[TABLE]

with $\eta$ a white noise modelling the force generator stochastic attachment and detachment (Pecreaux et al.,, 2006; Nadrowski et al.,, 2004). In particular, we used

[TABLE]

and also applied a weakening of anterior force to account for the uncoupling of spindle poles at anaphase onset (Mercat et al.,, 2017; Maton et al.,, 2015). With $\lambda$ the weakening factor and $\overline{k_{off}^{0}}$ the force generator off-rate at anaphase onset, we wrote:

[TABLE]

Similarly, the centering force (Pecreaux et al.,, 2016; Garzon-Coral et al.,, 2016) was also weakened:

[TABLE]

We solved this system numerically using trapezoidal rule and backward differentiation formula of order 2 (TR-BDF2 algorithm) (Hosea and Shampine,, 1996). Since we linearised the equations and kept the anterior centrosome at a fixed position, we could explore only reasonable parameter variations when performing the final position parameter sensitivity analysis (Bouvrais et al.,, 2018, Figures 6A, 7A-C, 8 ) (Figure 6). As a sanity check, we observed that modest variations in the force generator on-rate, thought to translate polarity cues (Rodriguez Garcia et al.,, 2017), modulated the final position (Bouvrais et al.,, 2018, Figure 7A) as expected from experiments (Grill et al.,, 2001; Colombo et al.,, 2003). To ensure that our simulation correctly converged to the final position, we varied the spindle’s initial position and observed no significant change in its final position (Figure 6C).

4 Conclusion

We previously proposed that the final centrosome position was dictated both by the centering force stiffness and by the imbalance in pulling force generation, i.e. mainly the active force generator number in active region and their processivity (Pecreaux et al.,, 2006). In contrast, in the expanded model, when the posterior centrosome enters into the active region, more microtubules are oriented along the transverse axis than parallel to the AP axis (Bouvrais et al.,, 2018, Figure 9, middle and right panels) because of the isotropic distribution of the microtubules around the centrosome. Then, it limits the pulling forces on the posterior centrosome (Bouvrais et al.,, 2018, Figure S3D). As a consequence, the boundary of the active region sets the final position as seen experimentally (Krueger et al.,, 2010; Bouvrais et al.,, 2018). In contrast, the force generator quantity and dynamics become less important and the final position even shows some resistance to changes in these two parameters (Bouvrais et al.,, 2018, Figure 7A-C).

We noticed that when the active region boundary was located at 80% of embryo length or more posteriorly, and the spindle was close to the cell centre, the number of microtubules reaching this region was so reduced that it prevented a normal posterior displacement. Together with the observation that when the region extended more anteriorly the final position was anteriorly shifted, it appeared that a boundary at 70% was a value quite optimal to maximise the posterior displacement. Because this posterior displacement is a key to asymmetric division, it would be interesting (but out of the scope of this work) to see whether a maximal posterior displacement is an evolutive advantage, which would then cause a pressure on the active region boundary.