Abstract
Background The correct positioning of the mitotic spindle during the asymmetric division of the nematode C. elegans zygote relies on the combination of centering and cortical-pulling forces. These forces, revealed by centrosome anaphase oscillations, are regulated through the dynamics of force generators, related to mitosis progression. Recently, we have presented the control of oscillation onset by the posterior spindle pole position in related species C. briggsae, necessitating a re-evaluation of the role of astral microtubules dynamics.
Results After exhibiting such a positional switch in C. elegans, we mapped the microtubule ends at the cortex and observed a correlation between the proximity of the centrosomes and the density of microtubule contacts. To explore the functional consequences, we extended the “tug-of-war” model and successfully accounted for the positional switch. We predicted and experimentally validated that the control of oscillation onset was robust to changes in cell geometry or maximum number of attached force generators. We also predicted that the final position of the posterior centrosome and thus the spindle has a reduced dependence upon the force generator dynamics or number.
Conclusion The outburst of forces responsible of spindle anaphase oscillations and positioning is regulated by the spindle position through the spatial modulation of microtubule contacts at the cortex. This regulation superimposes that of force generator processivity putatively linked to the cell cycle. This novel control provides robustness to variations in zygote geometry or detailed properties of cortical force generators.
Highlights
Microtubule contacts at the cortex concentrate at regions close to the centrosomes.
This regulates pulling forces and creates a positional switch on oscillation onset.
The onset position is robust to changes in embryo length or force generator dynamics.
The final centrosome position is robust to changes in generator number or dynamics.
eTOC Blurb Observing inhomogeneous MT contact density at cortex, Bouvrais et al. propose that the posterior centrosome position regulates engagement of pulling force generators, creating a positional switch on oscillation onset. This, and thus final centrosome position, is robust to variation in number or dynamics of force generators.
Introduction
Asymmetric cell divisions, in which daughter cell sizes, content and fates differ, are essential to the development of multicellular organisms [1,2]. In the nematode Caenorhabditis elegans [3] as in many other species [4,5], the mitotic spindle contributes to positioning the cytokinesis furrow. It needs to be oriented along the polarity axis [6] and in some cases displaced out of cell center prior to cytokinesis to correspond to cortical polarity cues [6,7]. Pulling forces, exerted on the plus-end of astral microtubules from the cell cortex, are common to most asymmetric divisions and play a key role in positioning and orienting the spindle [6-8].
In the nematode one-cell embryo, cortical forces are generated by the well–conserved trimeric complex, which pulls on astral microtubules, and which comprises a dynein/dynactin complex, a NuMA homolog LIN-5 and a G-protein regulators GPR-1/2, homolog of mammalian LGN [9]. In such an asymmetric division, GPR-1/2 translate polarity cues [10] through their asymmetric localization [11,12], increasing the number of active force generators locally. Prior to the cell division, during the centering phase [13], GPR-1/2 contribute to orienting the spindle along the anteroposterior axis (AP-axis) and displace the pronuclei centrosomes complex (PCC) from the posterior side of the embryo, where pronuclei have met, to a slightly anterior position (overcentration) [12,14]. During prometaphase and metaphase, centering forces independent of GPR-1/2 and putatively due to microtubules pushing against the cell cortex, maintain the spindle in the center [15]. GPR-1/2–dependent cortical pulling forces become dominant in late metaphase and anaphase and displace the spindle posteriorly, make it rock and contribute to its elongation [16-18].
The activity of the cortical force generators is regulated in three different ways: (1), in space through modulation across the various cortical regions in response to polarity cues: the rate of force generators binding to astral microtubules [19] is increased in the posterior half during metaphase and anaphase, leading to twice more active force generators in the posterior cortex compared to the anterior one, in the region marked by the PAR-2 polarity protein [10,16,20]. However, such an active region spans only from 70 % of the anteroposterior axis (AP axis) to the posterior tip of the embryo because the force generators activity is diminished in the middle region (from 40 % to 70 % of the AP-axis) by the LET-99 protein [21,22]. (2), the number of active force generators is regulated in time and increases during cell division [17], putatively depending on the cell cycle [23]. We found that the spindle anaphase rocking and posterior displacement were accounted for by a decrease of force generators off-rate from microtubules throughout the anaphase, resulting in a global increase of pulling forces [18] (SI text § 2.2.1). The original “tug-of-war” physical model proposed there assumed however that astral microtubules were abundant at the cortex during anaphase and that the sole limiting factor was the force generators binding/unbinding dynamics. Such a microtubule abundance, likely true during anaphase, is questionable in earlier phases. (3), microtubule dynamics can play a role in regulating the forces [24]. Kozlowski and co-authors proposed an alternative model in which the limited access of microtubules to the cortex could account for spindle oscillations [25]. Microtubule contact durations at the cell cortex appeared modulated between anterior and posterior sides [24]. Microtubule dynamics could also be a mean of time regulation through its increase along the course of the cell division [26].
Previous studies have underlined the key role of microtubules in microtubule organizing center (MTOC) positioning. Indeed, they are able to “sense” the cell geometry for e.g. to bring MTOC at the cell center [27,28] or to orient the nucleus by exerting pulling forces that scale with microtubule length [29]. Similarly, in HeLa cells, microtubules “integrate” the adhesive pattern, whose cues are cortical, to orient the spindle accordingly [30]. In C. elegans embryo, microtubules may contribute to orient the spindle together with polarity cues reading its oblong shape [31]. We therefore asked whether the modulation of the microtubule contacts in various regions of the cortex could modulate the cortical force generation.
We recently observed that anaphase spindle oscillation onset, and thus pulling forces, were controlled by the position of the posterior spindle pole rather than by the mitosis progression in C. briggsae. Indeed, oscillations start when the posterior pole had reached 70 % of the AP-axis in two nematode species (C. elegans and C. briggsae), which have diverged 100 million years ago. Interestingly, this position was reached 30 s after anaphase onset in C. briggsae and simultaneously to it in C. elegans. Considering that the amount of GPR-1/2 is key to regulate cortical pulling forces [10,16], this robust onset of oscillation was even more striking considering that GPR was duplicated only in C. elegans and that GPR sequences had diverged between the two species [12]. These observations suggested that a positional switch controls pulling forces. We propose here that it relates to microtubule dynamics.
Measuring the residency time of microtubules at the cell cortex along the course of the mitosis was key to explore such an hypothesis. Indeed, if the dynamics of the microtubules in the cytoplasm, and its time-evolution, are quite clear in the nematode embryo [26], the time they spend at the cortex is more elusive since published values range from second [25] to more than ten seconds [24]. Such a discrepancy can be explained by the very fast dynamics of microtubules calling for imaging at high frame rates. In this paper, we carefully measured the space modulation of such a dynamics. We then used it to decipher the regulating role of microtubule dynamics on cortical pulling forces accounting for the decoupling between oscillation and anaphase onsets observed in C. briggsae [12] and here in C. elegans. We extended our original “tug–of–war” model, focused on force generators dynamics, to account for this microtubule–dynamics– related positional switch on cortical force generators. We challenged this model with experiments comparing both predicted and experimental robustness of the switch to embryo shape perturbations or the force generators active region boundary. Reasoning on posterior displacement and using an in silico approach, we explored the consequences on the spindle positioning regulation and especially on the spindle final position, which contributes to setting the cytokinesis furrow position.
Results
Spindle oscillations can start before anaphase onset in C. elegans
We previously reported that the position of the posterior pole of the spindle controls the onset of the spindle oscillations in C. briggsae [12]. We asked whether the simultaneous anaphase and oscillation onset observed in C. elegans were coincidental. We delayed the anaphase using a such-1ANAPC5(h1960) mutant of anaphase promoting complex/cyclosome (APC/C) [32], labeling centrosomes and chromosomes through SPD-2CEP192: :GFP;HIS-58H2B: :mcherry. We tracked the centrosomes [15,18] and observed precocious oscillations with respect to anaphase onset (Table 1). The oscillations started when the posterior centrosome was at 70 % of embryo length, both in control and mutant, as in C. briggsae [12]. In contrast, for both mutant and control, the oscillation die–down happened about 2 min. after anaphase onset, disregarding the timing of their onset and leading to variations of their duration (Table 1). We concluded that a positional switch controls anaphase oscillation onset in C. elegans embryos while its end depends on cell cycle. This assay based on centrosomes tracking will be instrumental to functionally test the positional switch.
Microtubule contacts at the cortex depend upon centrosomes position
To account for this positional switch,we hypothesized that the network of astral microtubules emanating from the posterior centrosome could have a reduced accessibility to the posterior crescent of the cortex,where are located the active force generators [22] and further termed active region. When the spindle is close to the cell center, the density of microtubule contacts in the active region would be very low (graph. abstract,left); this density would increase as the posterior centrosome is displaced toward the posterior (graph. abstract,middle and right). The oscillations,which build up above a threshold number of active force generators, would depend on the position of the centrosome. We challenged our hypothesis by measuring directly the spatial distribution of the microtubule contacts at the cortex. We preserved the embryo shape by using spinning disk microscopy and microtubule dynamics through α-tubulin rather than EB labelling (see Suppl. Exp. Proc.,S1A). Because microtubule dynamics is fast,we imaged microtubule contacts at the cortex at 10 frames per second. Our method correctly recovered an exponential distribution of the residency times (Fig. S1B) consistent with previously published values [25]. We computed the distribution of the microtubule contacts along the anteroposterior axis. To gain certainty, we block–averaged the distribution in 10 regions distributed along the AP-axis and performed a running time–average with a 10 s window. We observed spatial inhomogeneities, in particular ridge lines and an overall increase of the number of contacts between metaphase and anaphase consistent with the increasing nucleation rate measured previously [26] (Fig. S1C). To test whether the ridge lines corresponded to the centrosomes position, we imaged the spindle plane in the same strain and temperature with a wide–field microscope. We tracked the centrosomes as previously described [18]. We combined the results of both experiments and aligned them on anaphase onset (see supp. exp. proc.). We found that the centrosome positions coincided with the ridge lines delineated by the highest microtubule density regions (Fig. 1). Because we initially observed a positional switch on cortical pulling forces in one-cell C. briggsae embryos [12], putatively related to the modulation of microtubules cortical contacts, we performed the same experiments in this species and obtained similar results (Fig. S2). We concluded that the distance of the centrosome to the cortex strongly modulates the number of microtubule end contacts in both species. As a consequence, the number of contacts in the active region increased with the posterior displacement of the the spindle.
A comprehensive model for pulling force regulation and spindle oscillations
We expected the modulation of microtubule contacts density by the centrosome–to–cortex distance to regulate the cortical pulling forces and to create the positional switch that we previously observed. Indeed, we reported on C. briggsae at anaphase onset [12] and observed that when the posterior centrosome was not localised posteriorly enough, it led to a reduced cortical density of microtubules and a number of active force generators below the threshold required for oscillations [18], delaying anaphase oscillations. Such a putative positional control of oscillation onset contrasted with the original tug-of-war hypothesis, which posits that both the build up and die down timings were regulated by the processivity of force generators [18], putatively related to mitosis progression [23]. To challenge this hypothesis, we extended the original “tug–of–war” model to quantitatively capture how the microtubules network could create a positional switch on cortical pulling forces.
Extending the model to account for microtubules dynamics
We modeled the dynamic instability of microtubules considering that they alternately grow and shrink [33] but neglecting the putative force dependent catastrophe rate [34]. Furthermore, we assumed that catastrophes happen only at the cortex (no free end catastrophe) and that once shrinking, microtubules fully depolymerize (negligible rescue rate) [25,26,35] (SI text §2.1.1). We also set a constant number of microtubule nucleation sites at the centrosomes, which were never empty [34], and from where microtubule emanated with an isotropic angular distribution [26,36]. We computed the number of microtubules that reached the cortex in the active region (Fig. 2C, left, purple color) as a function of the position of the posterior centrosome (Fig. 2A, black curve): we noticed a steeper increase at a position consistent with oscillation onset at 70 % of embryo length. We modeled the embryo as an ellipsoid but our result was independent of that hypothesis: we tested various super-ellipse shapes [37] and found the same switch behavior (SI text § 2.1.3 and Fig. S3). We pursued modeling using an ellipsoid to represent the embryo shape. We concluded that microtubule dynamics, by regulating the number of microtubules available to force generators, can implement the positional switch observed experimentally. Furthermore, the large number of microtubules contacting the active region during mid and late anaphase is consistent with the previous assumption of the original “tug-of-war” model that microtubules saturate a limited number of cortical force generators during this period [18,38].
Combining with force generators dynamics
To account for the dynamics of force generators, which sets oscillation frequency and the timings of their peak amplitude and die down [18], we modeled the microtubule binding to the force generators [9], as a first-order chemical reaction using the mass action law (therefore assuming no cooperative binding between force generators when binding) [39] and estimating the association constant from the binding and unbinding rates used in modeling anaphase oscillations [18] (SI text § 2.2.2). For the sake of clarity, we initially assumed a time independent association constant to model the onset of oscillations. It enabled us to compute the number of engaged force generators versus the posterior centrosome position (SI text§2.2). We found that when the centrosome was far from the posterior tip, the scarcity of astral microtubule contacts in the active region of the cortex limited the number of engaged force generators below the previously described threshold for oscillations [18] (Fig. 2B, black curve, 2C, left). We observed a steep increase in engaged force generators upon posterior displacement of the centrosome above 60% of AP-axis (compare Fig. 2AB, black curves), similar to the number of microtubule contacts followed by a saturation starting from 70% of AP-axis. This switch–like behavior was consistent with our positional switch hypothesis. The precise position, at which oscillations started, was dependent on the position of the active region boundary (Fig. 2B). We assumed that this region was set up by LET-99 force generators inhibition [22] and extended from 70 % to 100 % of AP-axis [21] (see experimental validation below). Thus, the positional switch was localized at about 70 % of the AP axis, consistent with our previous experiments. We kept this choice for the positional switch in the further steps. The observed saturation in the number of engaged force generators suggests that their dynamics rather than number becomes the control parameter. This is consistent with the timings of peak and die–down of oscillations being mostly independent of centrosome position but happening after a delay from anaphase onset (Table 1), as accounted for by the original model. In writing the mass action law in force generators number, we have assumed that the diffusion of the cortical anchor of force generators at the cell membrane is fast enough to not be limiting (SI text § 2.2.4). We checked this assumption by computing the number of engaged force generators versus the position of the posterior centrosome, but in this case with the microtubule-force generator binding modeled by the law of mass action in areal concentration and found again a positional switch (compare S4AB, black curves). In conclusion, our model suggest that the oscillation onset is specifically regulated by the posterior centrosome posterior displacement.
Microtubules and force generators dynamics set two independent switches
We next wondered how processivity (reflecting mitosis progression [18,23]) and position of the posterior centrosome combined to set oscillations on. We completed the extended tug-of-war model by making the microtubule–force-generator association constant dependent on time through the off-rate (inverse of processivity) as it was the control parameter in the original model; this parameter decreases along the course of mitosis [18] (SI text § 2.2.5). In contrast with the original “tug-of-war” model, the force generators on-rate is not constant and depends here upon the number of microtubules available at the cortex for binding a force generator (SI text § 2.2.2) on top of its regulation by the polarity [19] (SI text § 3). It suggests that the posterior centrosome needs to be posterior enough to enable oscillations supporting our positional switch experiment (Fig. 2D, blue curve). On top of this positional control, the processivity needs to be in a given range (Fig. 2D, blue region), below and above which the oscillations are dampened out (Fig. 2D, white regions), consistent with the control by a steady increase of processivity [18]. This leads to a dual control of pulling forces. Interestingly, the oscillation onset (Fig. 2D, blue curve) depends much more strongly on the position than its dying down (Fig 2D, green curve with a higher slope than the blue one), as seen experimentally (SI text §2.2.5). In conclusion, we extended the original “tug-of-war” model by adding a positional switch to control oscillation onset and similarly the forces contributing to spindle elongation and posterior displacement.
We then aimed at validating the model through three experiments: we firstly tested whether the boundary of the active region sets the centrosome position corresponding to oscillation onset; we secondly confirmed that this onset position is not controlled by the force generators activity. Finally, we challenged one prediction of the positional switch stating that the position, in contrast to the timing, of oscillation onset weakly depends on the embryo length.
Position of the active region boundary controls oscillation onset
In building the model, we asserted that the boundary of the active region, in which the active force generators are restricted (active region), controls the position, at which oscillations are set on. Our model predicted that when this region extends more anteriorly, the position at which oscillations start is also displaced anteriorly (Fig. 2B, curve with cold colors). To challenge this prediction, we extended the active region by partially depleting the protein LET-99 by RNAi, which is thought to restrict the force generator regulators GPR-1/2 to a active region [11]. In such a case, the active force generators are thought to extend in the whole posterior half of the embryo [22]. We observed that the oscillations started significantly more anteriorly compared to control (Fig. 3A), in agreement with the model predictions. Interestingly, the oscillations started also earlier with respect to anaphase onset, further supporting that the onset is independent of mitosis progression (Fig. 3B). We concluded that it is likely that the position of the active region boundary controls at which position the oscillation onset happens.
Because the positional switch relies on microtubule dynamics, our extended tug-of-war model predicts that the position of oscillation onset is independent of the total number of force generators (Fig. S6A), if this one is above the threshold required for oscillations. We previously suggested experimentally such a result [12] and could successfully repeat it (SI test § 2.2.3).
Reduced sensibility of oscillation onset position with embryo size
Our extended “tug-of-war” model suggested a weak dependence of the position of oscillation onset upon the length of the embryo, particularly to elongation (Fig. 4A). To challenge such a increased robustness with respect to the original model, we depleted C27D9.1 or CID-1 by RNAi to obtain longer or shorter embryos respectively. In both cases, embryos were viable and showed no other visible phenotypes. We measured the variation of the timing and the position of oscillation onset with respect to the variation of the embryo length. We fitted a linear model in both cases and measured an about ten times larger slope for timing with respect to position (Fig. 4B-C). It further suggests that the position and the timing (with respect to the anaphase onset) at which the oscillations start are not correlated. This result was also perfectly consistent with the reduced sensitivity of the oscillation onset position to the embryo length predicted by the model (Fig. 4A). In contrast, the position at which oscillation die down is impacted by embryo length (Fig. 4D), as expected since it is under the timing control. We concluded that our extended “tug-of-war” model was supported by these experiments.
Positional oscillation onset sensitivity analysis of the extended model
Using our model, we finally performed a thorough sensitivity analysis (Fig. S5). As expected, the number (Fig. S5F) and dynamics (Fig. S5C) of the microtubules, were critical in setting the position of oscillation onset. In a lower extend, the embryo width (Fig. S5D) or its scaling keeping proportions (Fig. S5E) were also influential. Interestingly, as the robustness of the position of oscillation onset versus the embryo length suggested (Fig. 4A), the eccentricity, keeping area constant, has a reduced impact (Fig S5B). Similarly, the number (Fig. S6A) or dynamics (Fig. S4A) of the force generators appears to have only small effects when they reach a threshold to enable oscillations as previously reported [18]. The cortical distribution of the force generators and their restriction to a active region is also key (Fig. 2B). In conclusion, the positional control of oscillation onset relies on microtubules dynamics, while forcegenerators dynamics is related to polarity translation and likely progression through mitotis by the processivity.
The astral microtubule dynamics regulate the final position of the spindle
The microtubule dynamics creates a feedback on cortical pulling forces, which set the spindle final position
The cortical pulling forces, which cause the anaphase spindle oscillations, also cause the posterior displacement of the spindle during the late metaphase and anaphase [17,18]. In our original tug-of-war model, we suggested that the final posterior centrosome position resulted from the balance between the cortical pulling forces and the centering forces modeled by a spring [18]. In contrast, in our current model, the averaged number of engaged force generators does not only depend on their dynamics but also on microtubules availability at the cortex and thus centrosome position. We reasoned that the positional control of the pulling forces generated caused a feedback loop on the final position of the spindle, to which they contributed to. To investigate this hypothesis, we simulated the posterior displacement using our extended “tug-of-war” model using the TR-BDF2 algorithm [41] (SI text § 3.1). To have a proper force balance on the spindle, we also included the anterior centrosome using the extended tug-of-war model with an active region from 0 to 40 % corresponding to the region devoid of LET-99 [21]. We however kept it to a fixed position. Such a simplification is relevant since the tug-of-war model was linearized, thus being limited to consider modest variations of the parameters around their nominal values. We considered on the anterior side a twice lower on-rate of force generators [19], resulting in half less engaged force generators than at posterior, as previously reported [16]. We also assumed that this force is reduced by half after anaphase onset to account for sister chromatids separation [42]. We finally modeled the centering force by a spring according to [15] and the control parameter to advance mitosis was the processivity [18]. In comparison to trajectories reported previously [15], we could reproduce the global kinematics of posterior displacement, with a slow displacement prior to anaphase and an acceleration after (Fig. 5, black curve, S7A5). In particular, in account well for the final position of the posterior centrosome. On this ground, we now aimed to validate this simulated model to explore the consequences on the final spindle position of including microtubule dynamics in the model.
The active region but not the force generator total number dictates the final position of the spindle
Consistent with observations in let-99(RNAi) treated embryos (Fig 3A) and [22], the final position of the posterior centrosome was displaced anteriorly when the boundary of the posterior active crescent moved towards the anterior, provided that this region is large enough to initiate posterior displacement (Fig. S7B). This result contrasted with the original tug-of-war prediction that the larger the cortical forces, the more posterior the displacement. The asymmetry of cortical pulling forces causing the posterior displacement is due to a larger number of active force generators on posterior side [16], initially assumed to reflect an asymmetric total number of generators and recently proposed to be due to an asymmetric on-rate [19]. The original model predicted a linear dependence between the number of active force generators and the final position [15]. The extended tug-of-war model offered a reduced sensitivity to this number in comparison (Fig. S7CD, SI text § 2.2.3), consistent with the robustness observed on oscillation onset position (Fig. S6C). We attributed this robustness to a less pronounced increase of cortical pulling forces when the centrosome crossed the position of the boundary (Fig. S7A4). We concluded that accounting fo the dynamics of microtubule is needed to correctly understand the mechanism setting the final position of the spindle, superseding the original tug-of-war, and accounting for the dependence of the final spindle position on the cortical extend rather than number of active force generators.
The robustness of spindle final position towards the final force generator processivity
In the original tug-of-war model, the final position was predicted to not only depend on the imbalance of force generators number or on-rate, but also on their final processivity and on centering spring. This prediction is poorly consistent with our previous observation that mild zyg-9(RNAi) resulted in a degraded centering (equivalent to a less stiff spring in our model) while not affecting significantly the cortical pulling forces [15]. Similarly in such-1 mutant, which affected mitosis progression and thus likely final processivity, the final posterior centrosome position is not altered (Table 1) [23]. We observed that extended model can account for this robustness on modest final processivity variations (Fig. 5). We suggest that the extended model better recapitulates the robustness of the final spindle positioning to changes in force generator number or dynamics. This is essential to properly position of the cytokinesis furrow, in particular.
Discussion
By measuring the spatial distribution of microtubule contacts at the cell cortex, we found that they are modulated in space and more concentrated in the regions closer to the centrosomes. It is however noteworthy that the total number of contacts scale up (as seen in Fig. 1, 80 s after anaphase onset) due to the increased nucleation and persistence of microtubules, as expected from their regulation along the course of the mitosis [26]. This contact modulation regulates the forces responsible not only for the anaphase oscillations of the spindle and posterior displacement are under the control of the position of the posterior centrosome (the so called positional switch). These forces also contribute to spindle elongation and their positional regulation might create a link with tension-based spindle assembly checkpoint satisfaction [43]. We extended our previous “tug-of-war” model of spindle oscillations and posterior displacement [18] to account for it and validated it experimentally. In particular, we observed that the position of oscillation onset, but not the timing, is robust to variations in embryo length, while the position of oscillation onset is correlated to the size of the posterior active force generators region, putatively bounded by LET-99 [21,22]. In the early stages of mitosis, the spindle lies in the middle of the embryo and the both centrosomes are far from their respective cortex: thus, the imbalance in the number of active force generators [16] results in a slight posterior pulling force and causes a slow posterior displacement [17] (Fig. 2C, left). The closer the posterior centrosome gets to the cortex, the larger the force imbalance is, since more microtubules reach the cortex: the pulling force builds up more rapidly and the posterior displacement accelerates. The number of engaged force generators increases, it exceeds the threshold setting oscillations on [18] (Fig. 2C, middle). Once the posterior centrosome crosses the boundary of the active region, the pulling forces start saturating because only their projection along the anteroposterior axis contributes (Fig. S2C, right): these forces, together with the centering forces [18], balance and set the position at which the posterior centrosome finally stops. They decide of the end–of– mitosis position of the spindle.
This proposed positional switch adds to the previously described temporal control via the processivity of force generators [18], in turn reflecting mitosis progression [23]. These two controls act independently, as they relate to two independent components. The positional control is determined by microtubule dynamics while the temporal control is set by the force generators dynamics. Indeed, the number of engaged force generators versus centrosome position curve (Fig. 2B) steeply increases from 60 % of embryo length because of microtubule dynamics, while this number saturates above 70 % due to force generator dynamics. Our model predicts respectively two necessary conditions for the oscillations to start (Figure 2D, blue curve): a large enough number of microtubules contacting the active region of the cortex and a high enough processivity of force generators. Indeed, during anaphase, the temporal evolution of the amplitude of cortical pulling forces is controlled by the dynamics of force generators as proposed previously and as revealed by oscillation die down timing [18]. This dual control of pulling forces was furthermore confirmed by three experiments. Firstly, in let-99(RNAi) treated embryos, where the positional control is disturbed by displacing anteriorly the active region boundary; the final position of the centrosome is strongly altered (Fig. 3A) as predicted by the model (Fig. S7B), but the timing of oscillation die down is not significantly different from control (Fig. 3B). Secondly, in such-1(h1960) mutants, where temporal control is perturbed via a delay in anaphase onset, the duration from anaphase onset to oscillation die-down is the same as in control, implying that the die-down timing is delayed in the same proportion as anaphase onset. In contrast, the positions of oscillation onset and die-down are not altered by delaying anaphase onset (Table 1). Thirdly, we observed a precocious oscillation die-down upon decreasing the number of active force generators by gpr-2 null mutant (Fig. S6B); this number exceeds the threshold during a shorter duration, consistently with the original tug-of-war model prediction (see e.g. fig. 5C of [18]). Overall, these experiments supports that during anaphase, force generators dynamics dominate the control of anaphase oscillations.
We hypothesized that these combined controls, in particular the proposed positional switch, confer some robustness to the final position of the posterior centrosome and consequently of the spindle, by buffering against variations in the initial positions of the centrosomes (Fig. S7E) or the final processivity that determines the final cortical pulling forces (Fig. 5) for instance. In the perspective of the asymmetric cell division, the final position of the spindle contributes to prescribing the cytokinesis furrow position, which is essential to ensure a correct partitioning of cell polarity cues and thus daughter cells fate [1-3]. Above the sole C. elegans nematode, we recently performed a comparative study between two nematode cousins (C. elegans and C. briggsae) [12]. We found in particular an alteration of cortical force generators regulation because of a duplication (GPR-1 and GPR-2) in C. elegans with respect to C. briggsae displaying only GPR-2 [12]. We proposed that this evolution was made possible by the positional switch and the robustness it creates towards force generator number or dynamics. Indeed, C. briggsae microtubule contacts at the cell cortex are modulated as in C. elegans (Fig. S3) and robustness to embryo length variations is also observed [12]. Interestingly, the positional control of anaphase oscillation onset in C. briggsae results in a 30 s delay between oscillation and anaphase onsets (attributed to spindle overcentation [12]), while the die-down is in synchrony with the anaphase onset as predicted by our model. Furthermore, cross species insertion of GPR genes modulates oscillation amplitude but preserves the positional switch consistent with our gpr-2(ok1179) experiment. The robustness in final spindle positioning is likely true beyond these sole two species [44]. In conclusion, the proposed robustness mechanism has enabled changes in the regulation of nuclei/centrosome complex position during the course of evolution despite this function is essential.
At the core of this robustness mechanism is the dynamic instability of microtubules, and more precisely the dependency of the number of contacts with the distance. Indeed, the distance centrosome cortex is measured in “units of microtubule dynamics” (SI text § 2.1.2). This is a quite classic mechanism to create centering [28,45] or other shape dependent mechanisms [29,46,47], although it was always inferred from cell level properties. In contrast, we measured here the distribution of the contacts localizing them directly at the cortex at microscopic level and observed a density ratio of about 2 between the most and less microtubule–contact–dense regions for a given time. This ratio represents the sensitivity to centrosome position (SI text §2.1.2). From a theoretical point of view, considering the ellipsoidal shape of the C. elegans embryo and the measured microtubule dynamics (see above), the predicted maximal ratio is 1.64. Our experimental one is close to this latter value, suggesting that microtubule dynamics parameters are optimal to create the positional control discussed here.
Conclusion
The study of the mechanism that leads to a precise timing and positioning of the transverse oscillation onset in the one-cell embryo of C. elegans has highlighted the key role of the microtubule dynamics to probe the boundary of the active force generator region. This positional control of the spindle rocking comes in complement to the previously established regulation through the dynamics of pulling force machinery (temporal control). They both set independent switches preventing premature force burst and centrosome oscillations and leading to a robust mechanism of asymmetric cell division. Indeed, these spatial and temporal switches control not only the oscillation onset but also the final spindle position. The posterior centrosome position set a feedback on the pulling forces causing the spindle displacement. As in the oscillation onset mechanism, microtubule dynamics contributes also to probing the cell shape and to having a good robustness in setting the final position of the posterior centrosome in C. elegans embryo as a proportion of the AP axis. In particular, this final position is robust to changes in final dynamics of force generators (processivity), putatively linked to the mitosis progression. This is a novel example of microfilaments-based mechanical system providing robustness to perturbation and likely to enabling evolution.
Acknowledgments
The gpr-2(ok1179) backcrossed 10x is a kind gift of Prof. A.A. Hyman. We thank Dr G. Michaux for feeding clones library, for technical support and Drs B. Mercat, A. Pacquelet, X. Pinson, Y. Le Cunff, D. Fairbrass, G. Michaux, R. Le Borgne, S. Huet, F. Argoul and A. Arnéodo for technical help, critical comments on the manuscript and discussions about the project. JP was supported by a CNRS ATIP starting grant and la ligue nationale contre le cancer. Some strains were provided by the CGC, which is funded by NIH Office of Research Infrastructure Programs (P40 OD010440; University of Minnesota, USA), the National Bio-resource Project (Tokyo University, Japan). Microscopy imaging was performed at the MRIC facility, UMS 3480 CNRS / US 18 INSERM / Univ. Rennes 1. Spinning disk was co-funded by CNRS, Rennes métropole and region Bretagne (grant AniDyn-MT), which also funded H.B. fellowship. H.B also acknowledges EMBO for her long term post-doctoral fellowship.
Material and Methods
Culturing C. elegans
C. elegans nematodes were cultured as described in [48] and dissected to obtain embryos. The strains were maintained at 25°C and imaged at 23°C except gpr-2 mutant, such-1 mutant and their controls that were maintained at 15°C and imaged at 18°C. The strains were handled on nematode medium plates and fed with OP50 bacteria.
Strains
The TH65 C. elegans (Ce) YFP: :TBA-2 (α-tubulin) [26] and ANA020 C. briggsae (Cb) GFP: : β-tubulin strains (fluorescent labelling of microtubules) were used as the standards for the “landing assay.” TH27 C. elegans GFP: :TBG-1 (γ-tubulin) [49] and C. briggsae ANA022 TBG-1: :GFP [12] strains (fluorescent labelling of the centrosomes) were the standard used for the “centrosome tracking assay.” C. elegans TH231 SPD-2: :GFP strain with centrosome labeling crossed to OD56 mCherry: :HIS-58 histone labeling was the control used for timing the events (table 1). It was crossed with the KR4012 such-1(h1960) mutant strain [32] to create JEP16. Centrosome tracking upon mutating gpr-2 was performed on JEP14 strain obtained by crossing the TH291 gpr-2(ok1179) backcrossed 10 times strain and TH27 C. elegans GFP: :TBG-1 (γ-tubulin).
Gene inactivation by use of mutants or protein depletion by RNAi feeding
RNAi experiments were performed by ingestion of transformed HT115 bacteria. let99, cid1 and c27d9.1 genes were amplified from AF16 genomic ADN and cloned into the L4440 plasmid. The feeding during 48h (except for let-99, reduced to 16-24h) was performed at 20°C to obtain stronger phenotypes. The control embryos for the RNAi experiments were treated with bacteria carrying the empty plasmid L4440.
Embryos preparation for imaging
Embryos were dissected in M9 buffer and mounted on a 2% w/v agarose, 0.6 % w/v NaCl and 4 % w/v sucrose pads, between a slide and a coverslip and were observed on different microscopic setups depending on the assays. We confirmed that embryos were devoid from photodamage by checking that the rate of subsequent divisions was normal [50]. Fluorescent lines were imaged at 23°C unless stated otherwise.
Imaging of microtubule contacts at the cortex
Embryos were dissected in M9 buffer and mounted on a 2% w/v agarose, 0.6 % w/v NaCl and 4 % w/v sucrose pads. We imaged C. elegans or C. briggsae one–cell embryos at the cortex plane in contact with the glass slide (Figure S1A) from the nuclear envelop breakdown (NEBD) until the end of the cell division. In particular, we aimed at preserving embryo shape at most. Thus, the thickness of the perivitelline space [51] and imposed to use spinning disk microscopy rather than TIRF (Fig. S1A). Therefore, cortical microtubule contacts tracking was performed on a LEICA DMI6000 / Yokogawa CSU-X1 M1 spinning disc microscope, using HCX Plan Apo. 100x/NA 1.4 Oil. Illumination was performed by a white light Fianium laser conveniently filtered around 514 nm by an homemade setup. To account for the fast dynamics of the microtubules at the cortex, images were acquired with a 100 ms exposure time (10 Hz) using an ultra-sensitive EMCCD Roper instrument evolve camera and the Metamorph software (Universal imaging Corp.) without binning. We kept the embryos at 23°C during the experiments. To image embryos at the cortex we moved the focus typically between 12 to 15 μm below the spindle plane (Fig. S1A).
Imaging the centrosomes
For the “centrosome tracking” and the “events timing” assays, embryos were observed at the mid-plane using a Zeiss AxioImager upright microscope modified for long-term time-lapse. Firstly, an extra anti-heat filter was added on the mercury lamp light path. Secondly, to decrease the bleaching and obtain optimal excitation, we used an enhanced transmission 12 nm band-pass excitation filter centered on 485 nm (AHF analysentechnik, Tübingen, Germany). We used a 100x/NA 1.45 Oil plan-Apo objectives. Images were acquired with an Andor Ixon3 EMCCD 512x512 camera at 33 frames per seconds and using the Solis software. We aligned centrosome tracks of individual embryos on the beginning of the spindle abrupt elongation (Fig. S8A) as an accurate landmark of anaphase onset [15] to average them or overlay them to “landing assay.”
Statistics
Averaged values were compared using 2 tails Student t-test with correction for unequal variance except otherwise stated. For sake of simplicity, we encoded confidence level using stars: * meaning p≤0.05, ** p≤0.005, *** p≤0.0005, **** p≤0.00005 and n.s. (for non-significant) meaning p> 0.05. n.s. indication might be omitted for sake of clarity. We abbreviated standard deviation by S.D., standard error by s.e. and standard error of the mean by s.e.m.
Data processing, Modeling and Simulation
All analysis software were developed under Matlab (The Mathworks). Modeling was performed using formal calculus software Mathematica (Wolfram). Numerical simulations were performed using simulink and matlab (The Mathworks).
Supplementary Experimental procedures
“Landing assay”: Pipeline to measure microtubule contacts density and dynamics at the cortex
Motivation for the strains choice
We used nematode strains, where the whole microtubules were labeled using yfp: :α-tubulin or gfp: :tubulin transgenes for C. elegans and C. briggsae embryos, respectively, rather than a labeling of the +TIPs of the microtubule via the EB homolog proteins. The advantage is twofold: (1) this labeling preserved the dynamics of the microtubules unlike the overexpression of EB proteins (Straube and Merdes 2007, Komarova, De Groot et al. 2009) and (2) this labelling enabled us to measure the duration of the residency of microtubules themselves at the cortex, not only the time spent growing there, which substantially differs (Kozlowski, Srayko et al. 2007). We compensated the low SNR present within the biological images by tracking with u-track software (Jaqaman, Loerke et al. 2008) to gain a robust detection of microtubule contacts and computed density from tracks.
Preprocessing of the cortical images
Since microtubule tubulin spots signal was very weak at the cortex, we denoised the images to increase the signal-to-noise ratio. Such a denoising usually relies on the assumption that the noise is non-correlated in space and time and follows a Gaussian or Poisson distribution. We opted for the kalman filtering/denoising (Kalman 1960). To perform Kalman denoising, we applied the following parameters: the gain was set at 0.5 and the initial estimate of the noise was equal to 0.05.
Automated tracking of YFP: :α-tubulin fluorescent spots at the cortex
Because a large number of tracks was present at the cortex, we sought an algorithm with robust linking. We opted for u-track (Jaqaman, Loerke et al. 2008) with parameters reproduced in table below. We validated these parameters by analyzing fabricated images of known dynamics (see simulation section below) and found good colocalization between prescribed tracks in simulation and recovered ones and measured similar lifetimes of the recovered track durations to the prescribed one. Because we were conservative in parametrizing u-track algorithm, it is possible that we missed some of the tracks that were displaying low fluorescent intensities.
Measuring microtubules residency time at the cortex
We computed the histogram of the track durations of microtubule contacts at the cortex. We used a bin size of 100 ms, equal to acquisition time. The exponential fit of this histogram led to microtubule lifetime of about 1 s (Figure S1B), consistent with (Kozlowski, Srayko et al. 2007).
Computing microtubule contacts density at the cortex
The “landing assay” consisted in measuring the cortical microtubule contacts (Figure S1A) during the different phases of the mitosis. The region of the embryo contacting the cover-slip is divided in ten regions of equal width along its long axis (AP-axis). The uTrack algorithm enabled us to follow frame after frame the microtubule contacts at the cortex and to have access to their trajectories (Jaqaman, Loerke et al. 2008). We segmented the embryo cytoplasm to follow changes in the shape of the embryo along the cell division using active contours (Pecreaux, Zimmer et al. 2006) and obtained the length and the area of the embryo during the mitosis. We could then count the number of microtubules contacting the cortex in the ten regions along the embryo length (Fig. S1A). To increase certainty on the results, the distribution of the microtubule contacts was averaged along time over 10 s. Finally, we used the onset of cytokinesis furrow ingression as a time reference to align the different microtubule cortical contacts distribution and then averaged over the embryos to get the final averaged density map.
Timing of furrow ingression onset and overlay assay
We first aimed to get the timing of cytokinesis furrow ingression onset, in both planes, by detecting the contour of the embryo taking advantage of the cytoplasmic fraction of dye. We then obtained the contour of the embryo using active contours (Pecreaux, Zimmer et al. 2006). We set the onset of cytokinesis furrow ingression as the fast increase in embryo shape convexity (ratio of the convex area to active contour area, Fig. S8B), practically when it grew above 1.012. In the mid-plane, we calibrated the average time between anaphase, obtained from the inflexion in spindle elongation (Pecreaux, Redemann et al. 2016) and cytokinesis furrow ingression onset in the mid-plane. We then used it to estimate the anaphase onset from the measuring of furrow ingression onset at the cortex and to match the times in landing and centrosomes tracking assays when plotting the overlay (Fig. 1 e.g.).
Robustness plot (fig. 4)
To assess the robustness of the position and timing at witch the posterior centrosome starts oscillating, and the timing of oscillations die–down with respect to variations in embryo length, we aimed to use dimentionless quantities. To do so, we used the duration T between to mitosis events independent of cell mechanics, id est the nuclear envelope breakdown and anaphase onset. We then computed the variation of timings in both strain normalized by this duration. E.g., in the case of oscillation onset time (subscript o), the normalized shift obtained by subtracting from the current value to, the corresponding value for control and we divided the result by this same value for the control, i.e. . We repeated the same computation for all quantities using embryo length in control for space/positional quantities. Independently of the quantity used for normalizing, using a Student t-test to ask whether the linear fit slope is significantly different from 0 ensures to distinguish whether embryo length has an impact on the considered position or timing, as both value and corresponding standard deviation scale identically with this normalizing factor.
1 Introduction
We aim to complement our previously published “tug–of–war” model (Grill et al., 2005; Pecreaux et al., 2006), mainly focused on the dynamics of cortical force generators, by including the dynamics of astral microtubules. Indeed, we revealed that the microtubule contacts mostly concentrated in cortical regions close to the centrosomes (Fig. 1). In consequence, the position of the centrosomes, as the microtubules organizing centers and through the micro-tubule dynamics, regulates the number of engaged force generators, likely cytoplasmic dynein (Nguyen-Ngoc et al., 2007), pulling on astral microtubules. In turn, it regulates anaphase spindle oscillations and posterior displacement. In a first part, focusing on the oscillations onset, we will neglect the change in force generators processivity along the course of mitosis and will detail the model and then explore how this novel regulation combines to the one by force generators processivity previously reported (Pecreaux et al., 2006). In a second part, we will look at the feedback loop created between the position of the posterior centrosome and the pulling forces contributing to displacing the spindle, through a stochastic simulation approach, accounting for evolution of processivity along the course of mitosis.
2 Modeling the positional switch on oscillations onset
2.1 Number 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 cortex in this region would contribute by binding to pulling force generators. We thus aimed to compute the number of microtubules reaching such a so-called active region / posterior crescent of the cortex.
2.1.1 Modeling hypotheses and parameter estimates
We reasoned that 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 oscillations onset. Keys to assess this possibility were an estimate of the total number of microtubules and their dynamics. Based on previously published in vitro experiments, we assessed microtubules related parameters:
•Total number of microtubules To assess the number of nucleation sites at the centrosome, we relied on electron microscopy images of the centrosomes (Redemann et al., 2016), which suggest 3000 or more microtubules. We kept with this conservative estimate as in (O’Toole et al., 2003), more specifically in the figure 3, authors provide a slice about 0.85 µm thick (as estimated from video 8 and figure 3) displaying 520 astral microtubules, while centrosome diameter was estimated to 1.5 µm. Only a slice of centrosome was imaged in this assay, so the number of microtubule nucleation sites per centrosome was estimated to a least 1800 by estimating the number for a whole centrosome (a whole sphere). In this work, we set the number of microtubule to M = 3000, although variation of this number within the same order of magnitude does not change our conclusions.
•Free-end catastrophes are negligible. With the above estimate of the number of microtubules and with a microtubule growth speed v+ = 0.67 µm/s (Srayko et al., 2005) and shrink speed v− = 0.84 µm/s (Kozlowski et al., 2007), we can estimate that about 70 MTs reach the cell periphery (assumed to be at 15 µm) at each second and per centrosome if the catastrophe rate is negligible. After (Redemann et al., 2016), the vast majority of microtubules emanating of the centrosome are astral: we neglected the kinetochore and spindle microtubules in this estimate. Focusing on metaphase and with a residency time of microtubule ends at the cortex of 1.25 s (this work, (Kozlowski et al., 2007)), this means about 100 MTs contacting the cortex per centrosome, at any given time. Using our landing assay (Fig. S1C, S2A), we can estimate the number of contact in the monitored region at any given time to 5 MTs and extrapolate it to a whole centrosome assuming the isotropic distribution of astral microtubules (see below and §2.1.2), finding 26 MTs. Although a bit low, likely because of the conservative parameters of the methods (see suppl. exp. proc.) likely leading to miss some microtubules, it is consistent with the estimate in (Garzon-Coral et al., 2016). A non negligible catastrophe rate would have dramatically reduced that number of contacts at any given time. We concluded that free–ends catastrophe rate is negligible. Recently, it was proposed that in the spindle, the catastrophe rate could be as high as 0.25 s−1 (Redemann et al., 2016). On top of the fact that the spindle is much more crowded than the cytoplasm, likely enhancing free-ends catastrophe, our conservative estimate of the number of microtubules could combine with the negligible free-ends catastrophe: in other words, it means that we may focus on the fraction of astral microtubules not undergoing free–ends catastrophe.
•The microtubules are distributed around the centrosomes in an isotropic angular fashion Finally, we hypothesized an isotropic angular distribution of microtubules around the centrosome following (Howard, 2006). This is also suggested though electron microscopy (Redemann et al., 2016).
•No microtubule nucleation site is 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 “measures” the centrosome–cortex distance
Probability for a microtubule to reside at the cell cortex. Because microtubules spend most of their “lifespan” growing to and shrinking from the cortex, the distance between the centrosomes and the cortex limits the number of microtubules residing at the cortex at any given time. We summarized microtubule dynamics in a single parameter α by writing the fraction of time during which a microtubule resides at the cell cortex: eqn1 Where d is the distance from the centrosome (microtubule organizing center) to the cortex estimated to d = 15 µm (about half of the embryo width). We then estimated α = 2.15 ×106 m−1 using microtubule dynamics parameters above. 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 is consistent with estimate coming from investigating the spindle centering maintenance during metaphase (Pecreaux et al., 2016).
q can also be seen as the probability for an astral microtubule to reside at the cell cortex. The equation above suggests that it is a measure of the centrosome-cortex distance in “units of microtubule dynamics”.
Range of variation of the microtubule contact density at the cortex. The nematode embryo shape is slightly elongated. Therefore, the displacement of the centrosome can vary the centrosome-cortex distance from a factor 1.5 to 2. We thus wondered whether the dynamics of a microtubule is so that one can observe significant variations of residing probability q. We estimated this sensitivity through the ratio ρ of the probability of reaching the cortex when the centrosome is at its closest position d1 (set to half of the embryo width, the ellipse short radius) divided by the probability when it is at its furthest position d2 (chosen as half of the embryo length).
This curves has a sigmoid–like shape with limα→0 ρ = 1 and limα→∞ρ = d2 /d1.
Using our measurement of microtubule distribution at the cortex (Fig. S1C, S2A), we sought an experimental estimate of this sensitivity parameter. Because our assay did not enabled us to image the very tip of the embryo (Fig S1A), we will have to compare the sensitivity ratio computed from the density map with a theoretical one, not using the half embryo length as maximum distance but the larger distance effectively measurable. When imaging non treated embryo labelled with α-tubulin: :YFP, the embryo length in the spindle plane was 2a = 49.2 µm. In the cortex plane, we measured a width (denoted with bars) 2ā = 38.0 µm for the adhering part to the coverslip when imaging at the cortex (Fig. S1A). We can compute the truncation of the ellipse due to the adhesion through the polar angle ζ = arccos (ā/a) of the boundary of the adhering region. We obtained ζ = 39.4◦ which corresponds to a spindle plane to flattened cortex distance of 10 µm, using a parametric representation of the ellipse. During metaphase (the 2 minutes preceding anaphase onset), when the spindle is roughtly centered (Pecreaux et al., 2016), the average spindle length is 11.8 µm (N=8 embryos). The further visible region is thus at d2 = 16.5 µm while the closer one is at d1 = 10 µm, leading to a sensitivity ratio ρ = 1.62 consistent with the microtubule density ratio observed on cortex images (Fig. 1, S1C) for C. elegans. We concluded that microtubule dynamics in C. elegans enable the read-out of the position of the posterior centrosome through the probability of microtubules to reside at the cell cortex.
2.1.3 Number of microtubules reaching the cortex
Since microtubule dynamics were so that it could”measure” the centrosome–cortex distance as a probability of microtubule contacting the cortex, we set to estimate the variation of the total number of contacts for astral microtubules emanating from a single centrosome with the position of this centrosome along the AP-axis. We worked in spherical coordinates (r, θ, ϕ) centered on the posterior centrosome (animated by a slow posterior displacement and assumed as a quasi-static motion), with zenith pointing towards posterior. We denoted θ the zenith angle and ϕ the azimuth (Fig. S3A). We computed the probability of a microtubule to reach the cortex in the active region, represented as θ ∈ [0, θ0] and ϕ ∈ [0, 2π]. We integrated over the corresponding solid angle. Then the number of microtubules M (S, α) comes readily (Fig. 2A), where rs (θ, ϕ) is the distance centrosome–cortex in polar coordinates centered on the centrosome, depending upon the shape of the cortex S and θ0 the boundary of the”active force generators region” (Fig. 2C). We observed a switch-like behavior as the posterior centrosome gets out of the cell center and closer to the posterior side of the embryo (Fig. 2A).
2.2 Extended “tug–of–war” model
Because the number of microtubules reaching the cortex could be limiting (Kozlowski et al., 2007), we set to extend the original model of anaphase oscillations. In contrast, in the original “tug–of–war” model (Grill et al., 2005), we had made the assumption that the limiting factor was the number of engaged cortical force generators while, in comparison, the astral microtubules were assumed in excess. It resulted that oscillations were driven by the number and the dynamics of force generators. In the linearized version of the original “tug–of– war” model, the persistence of force generators to pull on microtubules (processivity) mainly governed the timing and frequency of the oscillations, while the number of force generators drove the amplitude (Pecreaux et al., 2006).
2.2.1 The original “tug–of–war” model
We provide here a brief reminder of the “Tug–of–war” model (Pecreaux et al., 2006). This model features cortical force generators (stall force ) exhibiting stochastic binding to and detaching from microtubules at rates kon and the detachment rate at stall force), respectively. The probability for a force generator to be pulling on a microtubule then reads . The active force generators are distributed symmetrically between the upper and lower posterior cortices but asymmetrically between anterior and posterior cortices (Grill et al., 2003) (see §2.2.2). We here provide the computation for the symmetric case (along the transverse axis). In the model, we had also included two standard properties of the force generators: firstly, a force-velocity relation , with f the current force and v the current velocity and f! the slope of the force velocity relation; secondly, a linearized load dependent detachment rate with fc the sensitivity to load/pulling force, assuming that force generators velocity is low i.e. they act close to the stall force (Pecreaux et al., 2006). We finally denoted Γ the passive viscous drag, related in part to the spindle centering mechanism (Garzon-Coral et al., 2016; Howard, 2006; Pecreaux et al., 2016) and N¯ the number of available force generators in the posterior cortex.
A quasi-static linearized model of posterior displacement reads: with and Oscillations develop when the system becomes unstable, meaning when so-called negative damping a overcomes the viscous drag.
2.2.2 Evolution of the original “tug–of–war” model to account for polarity encoded through on-rate
When we designed the original model, it was know that the posterior displacement of the spindle 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. The detailed mechanism building this asymmetry was elusive. We recently investigated the dynamics of dynein at the cell cortex (Rodriguez Garcia et al., 2016) and concluded that an asymmetry in force generators attachement rate (indistinguishably either assembling the trimeric complex (Nguyen-Ngoc et al., 2007) or attaching to microtubule) was the ground of the imbalance, in response to the asymmetric localization of GPR-1/2 (Park and Rose, 2008; Riche et al., 2013). Therefore, to simulate the posterior displacement of the posterior centrosome (see §3), we rather used the equation above with distinct kon on anterior and posterior but with equal number of available force generators.
2.2.3 Number of engaged force generators: modeling of microtubule–force-generator binding
To account for the limited cortical anchors (Grill et al., 2005; Pecreaux et al., 2006), we modeled 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 numbers (Koonce and Tikhonenko, 2012) and combined it to the number conservation equations for force generators and microtubules.
Force generators–Microtubules attachement modelling where N is the total number of force generators present in the posterior crescent (active region).
We could relate the association constant Ka to our previous model (Pecreaux et al., 2006) (see §2.2.1) by writing with the off-rate thought to depend on mitosis progression. Time dependences were omitted for sake of clarity. It is noteworthy that kon used in the original model as force generator binding rate to microtubules is now variable along the course of mitosis as it depends on the number of free microtubule contacts at the cortex, thus to the position of the centrosome. In contrast, is constant representing the real on-rate of the first order reaction.
Related parameters estimate In modeling anaphase oscillations onset, we neglected the variable off-rate along the course of anaphase/mitosis progression (see sections 2.2.6 and 3 for full model without this assumption). The positional switch modeled here limits the number of engaged force generators at oscillations onset. At the time, the number just crosses the threshold to permit oscillations (Pecreaux et al., 2006). We estimated that 70 % of the force generators are engaged at that time, consistent with the quick disappearance of oscillations upon progressively depleting the embryo from GPR-1/2 protein. We estimated above the number of microtubules contacting the cortex when the centrosome reaches 71 % of embryo length, where we observe the oscillations onset (Table 1) to 52. We also set the total number of force generators to 50, so that we get a number of engaged ones between 10 and 100 as previously reported (Fig. S6A) (Grill et al., 2003). We thus estimate the association constant (denoted with 0 superscript to indicate that we neglected its variation along the mitosis). In turn, we estimated assuming that the detachment rate at that time is about 4 s−1 (Rodriguez Garcia et al., 2016). If 70 % of the force generators are engaged at oscillations onset, it would correspond to kon, ≃ 0.375 s−1, thus comparable to the estimate used in the original model (Pecreaux et al., 2006).
Modeling the number of engaged force generators in the posterior crescent In the early stages of the mitosis, 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 force generators number due to the polarization of the embryo 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 imbalances (because more micro-tubules reach the cortex), and the posterior displacement accelerates slightly 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 favor of posterior pulling forces and posterior displacement speeds up.
We quantitatively modeled this phenomenon by combining the law of mass action above with the number of microtubule reaching the posterior crescent (eq. 5) and we obtained:
To challenge our model, we tested the switch behavior in a broad range of association constants Ka (Fig. S4A). When the posterior centrosome is between 50 % and 70 % of embryo length, we observed that the number of engaged force generators was increased up to a threshold that enables oscillations, consistently with (Pecreaux et al., 2006). When the centrosome is posterior enough, practically above 70 %, the number of engaged force generators saturated, suggesting that their dynamics is now the control parameter (during anapahase then), as proposed in the original tug–of–war model. We also observed that a minimal binding constant is required to reach the threshold number of engaged force generators required for oscillations. Interestingly, above this minimal Ka, further increase of the binding constant does not alter significantly the positional switch (Fig. S4A). This suggests that this positional switch operates rather independently of the force generators processivity. This will be further discussed below (§2.2.6).
The positional switch is independent of the total number of force generators, while it 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 computed the corresponding prediction in our model (Fig. S6A) and seek for an experimental confirmation. Up to recently, it was thought that the number of force generators contributing to the posterior displacement was controlled by GPR-1/2 proteins (Colombo et al., 2003; Grill et al., 2003). To keep above the threshold needed for oscillations (Pecreaux et al., 2006), we only decreased partially the number of active force generators in a controled fashion through a mutation of one of their redundant regulators, GPR-2, using a strain carrying both GFP: :α-tubulin transgene and mutation gpr-2(ok1179). Oscillations amplitude was decreased to 7.1 0.9 % of embryo width (N=8, p=1.4710-5) with respect to control, whose amplitude was 19.2 0.9 %, confirming a reduction of the number of active force generators. In these conditions, we observed that the oscillations still started when the posterior centrosome reached 70 % of embryo length (Fig. S6C) and slightly later than the control (Fig. S6B). This result supported our model proposing that when the total number of force generators is above the threshold of the original ?tug-of-war? model, the position of the centrosome sets the moment of oscillations onset. We recently proposed that the asymmetry in active force generators could be an asymmetry of force generators association rate (called on-rate) to form the trimeric complex that pulls on microtubules (Rodriguez Garcia et al., 2016). GPR-1/2 would increase this on-rate. In our extended model, a decreased on-rate (through gpr-2 mutant) would result in a decrease association constant Ka. Similarly to the case with an asymmetry in number, above a certain threshold of Ka, the position at which oscillations are set on is not significantly modified (Fig. S4A). In conclusion, independently of the details used to model the polarity, the mild depletion of GPR-1/2 experiment, causing a reduced number of active force generators, supports our extended model.
To further understand how the various parameters impacts this behavior, we performed a sensitivity analysis using this model (Fig. 2B, S6A, 4A, S4A, S5).
2.2.4 The switch-like behavior of the number of microtubules reaching the cortex versus centrosome position is independent of detailed embryo shape
The above result was 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 modeling the embryo shape by a super-ellipsoid of revolution, based on super-ellipses (Lam´e curves) (Edwards, 1892) 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), positive values towards the posterior side. We obtained a similar switch-like behavior. (Fig. S3). We concluded the the switch–like behavior observed was robust to changes of the detailed shape and thus performed the remaining investigations with an ellipsoid shape, for sake of simplicity.
2.2.5 Discussion: number– or density–limited force generator – microtubule binding
By writing the law of mass action in protein number (eq. 9), we have assumed that the force generator–microtubule binding reaction is rate limited but not diffusion limited. We recently investigated the dynamics of cytoplasmic dynein (Rodriguez Garcia et al., 2016), the molecular motor likely pulling on microtubules from the cortex (Nguyen-Ngoc et al., 2007) and observed that dyneins are abundant in cytoplasm, thus 3D diffusion combined to microtubule plus-ends accumulation bring enough dynein to the cortex. Therefore, diffusion of dynein to the cortex is not likely to be a limiting factor in binding force generators to the microtubules. However, another member of the trimeric force generating complex, GPR-1/2, essential to generate pulling forces (Grill et al., 2003; Nguyen-Ngoc et al., 2007; Pecreaux et al., 2006), may be limiting. GPR-1/2 is likely localized at the cell cortex prior to assembly of the trimeric complex (Park and Rose, 2008; Riche et al., 2013), and in limited amount leading the 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. We wrote the corresponding law of mass action: with , and SactiveRegion the surface of posterior crescent, whose border is considered at 70 % of embryo length. Modeling embryo by a prolate ellipsoid of radii 24.6 µm and 15.75 µm, SactiveRegion ≃ 0.147 Sembryo = 610 µm−1, with Sembryo ≃ 4100 µm2 the embryo surface.
The probability of a microtubule to hit the cortex (see eq. 3 and 5) is modified as follow (modification highlighted in blue):
We then computed the number of engaged force generators as above and found also a positional switch (Fig. S4B compared to S4A). We concluded that this alternative modeling of force generator–microtubule attachement is compatible with the positional switch that we observe experimentally.
In contrast with the law of mass action in number, there was no saturation in engaged force generators when the centrosome is further displaced towards the posterior after passing the posterior switch but a decrease. This may suggest that the position of the centrosomes could control the die-down of the oscillations. In such a case, one would expect that die-down did not intervene after a fixed delay after anaphase onset, but at a given position. This contrasts with experimental observations upon delaying anaphase onset (Table 1). Therefore, the law of mass action in number 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. We estimated the corresponding diffusion limited reaction rate to after(Freeman and D., 1983; Freeman and Doll, 1983) considering the parameter detailed previously, a diffusion coefficient for GPR-1/2 similar to the one of PAR proteins D = 0.2 µm2/s (Goehring et al., 2011), a hydrodynamic radius of 5.2 nm (Erickson, 2009). Compared to the value proposed above (see §2.2.3) kon,.. 0.375 s−1, it suggest that lateral diffusion is not limiting. Lateral diffusion may enhance rather than limits the reaction (Adam and Delbruck, 1968). We concluded that the process is limited by reaction, not diffusion, and we considered action mass in number (eq. 9) in the remaining of this work.
2.2.6 The processivity and microtubule dynamics set two independent switches on force generators: extended “tug–of–war” model
We now implement the effect of microtubule dynamics on the original “tug–of–war.” To do so, we let Ka varying with both the processivity and the position of the centrosome. In the notations of the original model, since we kept constant, it meant that kon varies because of varying number of microtubule contacts in the posterior crescent, in turn depending on the position of the centrosome. We then sought the pairs so that eq. 6 is critical, i.e. Ξc = Γt (eq. 7), with xc the critical position of the centrosome along the antero-posterior axis. Because we are on the transverse axis and considered a single centrosome, we used Γt = 140 µN.s/m after (Garzon-Coral et al., 2016) and obtained the diagram reproduced in fig. 2D. It could be seen as a phase diagram. When the embryo trajectory (the orange arrow) crosses the blue line to go into the blue area, the oscillations set on. Since this line is diagonal, it suggests that such an event depend upon the position of the posterior centrosome (ordinate axis) and of the detachment rate (abscissis), suggesting that two control parameters contribute to make the system unstable and oscillating. Interestingly, when the embryo continues its trajectory in the phase diagram, it crosses the green line, which correspond to the moment the system becomes stable again, and oscillations are damped out. This critical line is almost vertical indicating that this event depend almost only from the detachment rate i.e. the inverse of processivity, consistent with the experimental observation. Interstingly, this behavior is maintained despite modest variations of the range of processivity and centrosome position explored during the division (i.e. the precise trajectory of the embryo in this phase diagram). Note that large values of detachment rate are irrelevant as they does not allow posterior displacement of the spindle (Fig. 5). We concluded that two independent switches control the onset of anaphase oscillations and broadly the burst of force 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 position of the posterior centrosome. Robustness to some parameters revealed by the sensitivity analysis of the oscillations onset may also have a reduced impact on the final position of the centrosome. This final position is essential as it contributes to determine the position of the cytokinesis furrow, a key aspect in an asymmetric division to correctly partition cell fate determinants (Knoblich, 2010; Rappaport, 1971; White and Glotzer, 2012).
3.1 Modeling posterior displacement
To simulate the kinematics of posterior displacement, we considered the “tug–of–war” extended model (§2.2) and a slowly varying binding constant Ka due to the processivity increasing along mitosis progression (§2.2.3). We computed the posterior pulling force, assuming an axi-symmetric 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 compute it, we assumed that any microtubule contacting the context in the active region has an equal probability to attach a force generator. Therefore, we computed the force weakening due to projection by computing the ratio of the force exerted by each microtubule contacting the cortex weighted by the probability of contact and integrated over the active region. We then divided by the number of contacts computed above. This weakening ratio was then multiplied by the number of bound force generators previously computed (see 11). It reads: with θ0 is the polar angle of the boundary of the active region positioned at and , obtained assuming an ellipsoidal shape for the embryo. p(S, α, xante|post, θ) is defined at eq. 3 and P(S, α, xante|post) at eq. 4. This equation was used to compute both anterior and posterior forces, with their respective parameters. After Rodriguez Garcia et al. (2016), the force asymmetry is due to an asymmetry of force-generator–microtubule affinity, under the control of GPR-1/2. We accounted for this through an asymmetric attachement constant writing .
We then computed the quantities corresponding to the original model based posterior quantities (see §2.2.1) and put them in the main equation: with η a white noise modeling the stochastic attachment and detachment of force generators (Nadrowski et al., 2004; Pecreaux et al., 2006). In particular, we used and also applied a weakening of anterior force to account for the uncoupling of spindle poles at anaphase onset (Maton et al., 2015; Mercat et al., 2017). We wrote: Similarly, centering force (Garzon-Coral et al., 2016; Pecreaux et al., 2016) is weakened
We solved this system numerically using trapezoidal rule and backward differentiation formula of order 2 (TR-BDF2 algortihm) (Hosea and Shampine, 1996). Since we linearized the equations and kept the anterior centrosome at a fixed position, we were bound to explore resonable variation of the parameters when performing the parameters sensitivity analysis (Fig. 5, S7). As a sanity check, we observed that modest variations in the force generators on-rate, thought to translate polarity cues (Rodriguez Garcia et al., 2016), do modulate the final position as expected from experiments (Colombo et al., 2003; Grill et al., 2001). To ensure that our simulation correctly converges to the final position, we varied the spindle initial position and observed no significant change in the final position (Fig. S7E).
3.2 Result and discussion: robustness of the final position to changes in force generators number or dynamics
We previously proposed that the final centrosome position is dictated both by the centering force stiffness and the imbalance in pulling force generation, mainly the number of active force generators on posterior and their processivity (Pecreaux et al., 2006) (Fig. 5, S7CE dashed lines). In contrast, in the extended tug–of–war model, when the centrosome enters into the region corresponding to the posterior crescent, more microtubules are close to transverse and less are close to parallel to the AP-axis (Fig. 2C middle and right panel). This is because microtubules are isotropically distributed around the centrosome. Then it limits the force pulling on the posterior centrosome (Fig. S7A4). As a consequence, the boundary of the active region set the final position (Fig. S7B) as seen experimentally (Fig. 3A) and (Krueger et al., 2010). In contrast, the force generators number and dynamics become less important and the final position even shows some robustness to the variations in the number and dynamics of the force generators (Fig. 5, S7C).
We noticed that when the posterior crescent boundary is localized at 80 % of embryo length or more posteriorly, the number of microtubules reaching this region when the spindle is close to the cell center is so reduced that it prevents a normal posterior displacement. Together with the observation that when the region extends more anteriorly the final position is anteriorly shifted, it appears that a boundary at 70 % is a value quite optimal to maximize 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.
4 Parameters used in modeling and simulations
In this section, we detail the parameters used in the computation of the number of engaged force generators when validating the extended tug–of–war model, and also used when simulating the posterior displacement. We based the parameters estimated on the oiginal tug–of–war and on experiments performed elsewhere.
Footnotes
2 Equation reads , with t the time, ttherm = 50 s the thermatization time, t0 = 175 s the transition time, T = 70 s the time width and the variation amplitude.