A force-balance model for centrosome positioning and spindle elongation during interphase and anaphase B

Abstract

A computational model in one dimension is proposed to position a single centrosome using astral microtubules (MTs) interacting with the cell cortex. The mechanism exploits mutually antagonistic pulling and pushing forces arising from the astral MTs’ binding to cortical dynein motors in the actin-rich cell cortex and their buckling while growing against the cell cortex, respectively. The underlying mechanism is extended to account for the elongation and positioning of the bipolar spindle during mitotic anaphase B. Besides astral MTs, the model for bipolar spindle involves interpolar microtubules (IPMTs). The composite model can predict spindle elongation and position under various circumstances. The outcome reveals that the bipolar spindle elongation, weakened by decreasing overlap between the antiparallel IPMTs in the spindle mid-zone, is recovered by the astral MTs. The one-dimensional models are extended in two dimensions to include the effect of cortical sliding of the astral MTs for studying the dynamics of the interphase centrosome and the anaphase B spindles in elongated cells. The results reveal that the dynamics in two dimensions stay qualitatively similar to the one dimension.

Appendices

Appendix I: proportional pulling and buckling forces from cortex is essential for centrosome centering with short astral MTS

Fig. 10

Single CS positioning in a $60 \;\; \mu m$ cell with short astral MTs (average length $40 \;\; \mu m$), plotted for intial CS positions at $5\; \mu m$, $15\; \mu m$, $25\; \mu m$ and $35\; \mu m$ away from the left cortex. (a) The mean centrosomal position for large buckling amplitude and dynein density. (b) Mean centrosomal position for small buckling amplitude and dynein density

Earlier, we reported a single CS position away from the cell center with short astral MTs and a given set of interaction parameters, as shown in Fig. 5(a). This occurred because the force balance in either half of the cell was dominated by forces arising from the cortex within the same half, and contribution from the other half was insignificant. To achieve centering, the net pushing force arising from both the cortex must be comparable. One possible way of achieving this is to increase the buckling amplitude while keeping the dynein density unchanged. This would facilitate more pushing force for the same microtubule length, allowing the CS to counter dynein-mediated pulling force at larger distances from the cell cortex. Another way is to decrease the dynein density on the cortex so that the CS experiences less pull toward the cortex for the same distance from the cortex. Based on these hypotheses, we simulated the model with large buckling amplitude and dynein density (see Fig. 10(a)) and small buckling amplitude and dynein density (see 10(b)). In both cases, we found CS could position at the cell center, suggesting proportional buckling and pulling forces from the cell cortex is essential for CS centering with short astral MTs.

Appendix II: symmetric positioning of the bipolar spindle with initially asymmetric poles

Fig. 11

Symmetric positioning of poles that were asymmetrically placed at $t=0 \; s$. (a) Final positioning and length of the spindle remain unaffected for varying initial positions of individual poles; (b) Pole separation kinetics for short astral MTs indicating that the symmetric positioning of the poles remain unaffected even when short astral MTs are present in the system

Figure 11 shows dynamics of individual poles when placed asymmetrically at $t=0\; s$ for normal and short astral MTs respectively. Asymmetric positioning of the poles refers to the initial spindle’s center shifted away from the cell center. In other words, one of the poles is closer to the cortex than the other at the onset of spindle elongation. In Fig. 11(a) we show spindle elongation dynamics of two sets of poles placed asymmetrically on either side of the cell. Data shows that the pole closer to the cell cortex moves relatively less and is positioned rapidly than the one away from the cortex. However, the final positioning of the spindle is symmetric with respect to the cell center. This occurs due to the uniform dynein density and astral MTs buckling on either side of the bipolar spindle. The spindle separation kinetics is slightly delayed with short astral MTs, as shown in Fig. 11(b). Nevertheless, the final spindle positioning and interpolar distance remain the same as observed with normal astral MTs.

Appendix III: single cs positioning with stoichiometric interaction between astral MTS and cortical dynein motors in absence of buckling mediated pushing force

Fig. 12

Centering of a single CS is achieved only with pulling forces; stoichiometric interaction between astral MTs and cortical dynein motors is enforced by fixing the number of cortical motors per astral MT interacting with the cortex

Stoichiometric interaction between astral MTs and cortical dynein motors was suggested in a recent study [58] to facilitate cortical pulling force that can dominate the positioning of CS in a cell. Here, we explore a similar scenario, including stoichiometric interaction between the astral MTs and the cortical motors for positioning a single CS in elliptic cells. Such interactions call for a fixed number of cortical force generators per astral MT. In our original model, we evaluated the cortical pulling force using Eq. 23, which depends on MT length and the linear density of dynein motors given by $k_{dyn}$. We fix the number of dynein motors that each MT interacts with to ensure stoichiometric relation. The recent study [58] investigated this interaction using a 1 : 1 ratio, meaning that each MT interacts with a single cortical force generator (i.e., dynein motor). We explore the impact of a larger ratio of $1:k_{dyn}$ ($k_{dyn}=5$) on the CS dynamics, meaning that each MT has to interact with $k_{dyn}$ number of motors. This causes the pulling force exerted by each MT evaluated by counting the number of motors $k_{dyn}$ per MT where the force exerted by each motor is $F_{dyn}$. The parameters are stated in Table 1. Therefore, with stoichiometric interaction, the pulling force exerted by each astral MT interacting with the cortex remains the same, irrespective of its sliding length along the cortex. Therefore, even when the CS is placed near the cell cortex, the force on it from the proximal wall does not increase monotonically due to a limited number of dynein motors on each MT. As shown in Fig. 12, the CS, when placed on the cell axis at $5\; \mu m$ and $35\;\mu m$, experiences a net pull toward the proximal cell cortex but eventually gets pulled toward the cell center due to the pull from the distal cell cortex. A similar outcome is observed when the CS is placed slightly off the cell center, at $15\;\mu m$ and $25\;\mu m$. This points to the force contribution from the distal cortex dominating the CS trajectory until a force balance occurs at the cell center.

Initially, the MTs emanating from the CS toward the proximal cell cortex are the primary force contributors that pull the CS towards the proximal cortex (Fig.12). When the CS advances towards the proximal cell cortex, more MTs emanating from it and interacting with the proximal wall become obliquely aligned with the centrosomal axis. This is because the astral MTs that interact with the cortex remain anchored at the cortex while the CS advances toward the proximal wall along the cellular axis. As a consequence of increasing oblique astral MTs, the magnitude of the horizontal force components contributing to the pull on the CS starts decreasing. At the same time, MTs emanating from the CS toward the distal cortex greatly align with the x-axis, increasing horizontal pulling force toward the distal cortex. Once the pulling force acting on the CS from the distal cell cortex becomes stronger than the proximal cortex, CS moves toward the cell center, as shown in Fig. 12. This eventually leads to a force balance, stabilizing the CS at the cell center. For the CS starting at the $20\; \mu m$ mark, the horizontal components of the pulling forces coming from the opposite cortex are comparable, localizing the CS at the cell center. Therefore, our model for the CS positioning allows for dynein-mediated positioning of the CS at the cell center, when a stoichiometric ratio fixes the number of cortical motors per MT.