Mechanical stability of bipolar spindle assembly

We analyze the stable configurations of the two centrosomes when a single chromosome located at the mid-plane between two centrosomes located at $-x$ and $x$. The net force due to the interactions of the MTs with the chromosome is directly proportional of the number of MTs reaching the chromosome arms. Typically, the number of MTs reaching the chromosome arms at a distance $x$ is proportional to $e^{-x/L}$ ferenz ; exp-length where $L$ is mean length of MTs. Accordingly we can write down the expression for the polar ejection force, $F_{pe}=Ae^{-x/L}$, where $A$ is the maximum polar ejection force. The sliding force due to the motors is proportional to the overlapping MTs from the opposite centrosomes raja ; ferenz . Thus for centrosomes separated by a distance $2x$, the force due to the motor sliding is $F_{m}=2Bxe^{-2x/L}$, where $B$ is the net force per unit overlap length of the MTs. The sign of B is positive for net outward forces (when the force due to kinesin motors exceeds the forces of dynein) and it is negative for net inward (attractive) force between centrosomes. The force due to sub-cellular machinery of kinetochore with MTs is attractive f-chromo1 ; f-chromo2 . While in general the attractive force due to the interaction between the kinetochore sub-cellular machinery and MTs depends on the distance of the kinetochore from the centrosome raja2 , for the sake of simplicity of the analysis here it is assumed to be constant ferenz . Finally, the role of the interaction of the MTs with the cell cortex is ignored as this interactions does not significantly impact the steady state pattern of the spindles raja . Adding all the different contributions, the expression for the net force between the centrosomes as a function of their separation distance reads ferenz

We use Eqn.(1) to determine the mechanically stable steady states. For this we first note that the condition for mechanical equilibrium implies that for a separation distance $2x=2x_{p}$, $F(x_{p})=0$. Further for the stable mechanical equilibrium, $\frac{dF}{dx}\big{|}_{x_{p}}<0$, which simply means that on slight deviation from equilibrium separation, the restoring forces are such that the system regains its equilibrium configuration. Thus the conditions; $F(x_{p})=0$ and $\frac{dF}{dx}\big{|}_{x_{p}}=0$ determines the phase boundary, separating linearly stable and unstable regions. In order to simplify the analysis we express Eqn. (1) in terms of dimensionless scaled variables $x_{o}=2x/L$, $F_{o}=F/A$, $C_{o}=C/A$, $B_{o}=\frac{BL}{A}$. Therefore on the phase boundary,

Using Eq. (2) and Eq.(3), we obtain the expression for the equilibrium separation between the centrosomes at phase boundary as,

Substituting Eq. (4) into Eqs. (2),(3), we obtain the equation for the phase boundary in terms of the scaled parameters $B_{o}$ and $C_{o}$

In Fig. 2 we plot the stability diagram in terms of the scaled variables $B_{o}$ and $C_{o}$. As shown in the figure the bipolar spindle solution is always stable. Moreover, for a range of negative values of $B_{o}$ which corresponds to motors exerting a net inward force between the centrosomes, and when $C_{o}$ is less than a critical value, there is a region of bistability, where two different solutions of spindles, corresponding to two different finite separation distance between centrosomes are stable.

When many chromosomes are present, the force acting along the line joining the two centrosomes Eq. (1) can be generalized to:

where $R_{i}=\sqrt{{x}^{2}+r_{i}^{2}}$ is the distance between the centrosome and the $i$th chromosome lying on disc in the mid-plane between the two centrosomes and $r_{i}$ is the distance of the chromosome from the center of the disc. Assuming that $N$ chromosomes are are uniformly distributed on the disc of radius $R_{d}$, the expression in terms of summation can be converted into an integral expression for the force which reads

which upon integration yields,

The net force acting perpendicular to the axis joining the two centrosomes add up to zero due to the radially symmetric arrangement of chromosomes in disc configuration. The radius of the disk $R_{d}$ is set by the number of chromosomes, $N$, in the disc and by the chromosomal packing fraction that ultimately depends on the mutual interaction between chromosomes. In order to estimate $R_{d}$, we make a crude approximation that each chromosome occupy a surface area $\pi r^{2}_{ch}$ within the disc, where $r_{ch}=1\mu m$raja is the radius of a single chromosome. Therefore, $R_{d}\sim r_{ch}\sqrt{N}$.

In order to determine the stability boundary we first express Eq.8 in terms of dimensionless variables. The choice of scaled dimensionless variables are, $x_{o}=x/L$, $F_{o}=\frac{FR_{d}^{2}}{2NAL^{2}}$, $R_{o}=R_{d}/L$, $C_{o}=C/A$, $B_{o}=\frac{BR^{2}_{d}}{NAL}$. Then in terms of the these scaled variables, the expression for the dimensionless scaled force is,

This expression is of the form $F(x_{o})=x_{o}g(x_{o})$ and thus $x_{o}=0$ is always a solution for Eq.9. From the condition for stability of this particular solution we obtain,

For determining the stability boundary for the case when $x_{o}\neq 0$, we set $F_{o}^{{}^{\prime}}(x_{p})=0$ and $F_{o}(x_{p})=0$, which yields the condition, $g^{{}^{\prime}}(x_{p})=0$ and $g(x_{p})=0$. Using these conditions we obtain,

These equations are numerically solved to obtain the boundary curves (solid lines) shown in the panels of Fig.3. The stability diagrams in Fig.3 show a variety of scenarios. It exhibits regions where there is bistability with two different stable solutions for bipolar spindle, region where both bipolar spindle and monopolar configuration ($x=0$) is stable, region with just one solution for stable bipolar spindle and a region where no bipolar spindle is stable and only monopolar configuration is stable. The corresponding plots for scaled force as function of centrosome separation is plotted in Fig.4 for these four different regions in the stability diagram. We find that for outward motor-MT forces $(B>0)$, the bipolar spindle is stable as long as $C$ (corresponding to kinetochore-MT interaction strength) is not very large.

By integrating the expression of dimensionless scaled force in Eq. 9 it is possible to derive an effective dimensionless potential energy function $V_{o}(x_{o})$

Fig.5 shows the potential energy landscape for different strengths of $B_{o}$ for fixed values of other parameters. In particular Fig.5 illustrates that although in the bistable regions two different spindle configurations are stable, in general one of the two corresponds to a global minima in terms of potential energy. The fact that Eq. (7) can be derived from an effective potential, $V_{o}$ is crucial since the functional form of $V_{o}$ provides insight not only about the stability of the stationary states but also provides the clue about the dependence on the initial configuration of the system and the long term evolution of the system in presence of noise. It also rationalizes many of the observations of the numerical studies in Ref.raja ; ferenz . First of all it was seen in Ref raja ; ferenz that the final configuration of the spindle ( on whether it is bipolar or monopolar) is dependent on the initial separation distance. The potential energy landscape provides an explanation on how the final configuration would be dependent on the zones of attraction of the potential landscape with multiple minima. It was also seen in Ref. raja that for certain situations, the initial configuration eventually relaxed to a fixed configuration after an intermediate metastable configuration. This can understood in terms of the different depths of the potential minima for the two different configurations. Eventually the system would relax to the global minima in presence of noise in the system. In fact the height of the potential barrier would then determine the time scales of relaxation of the system to its final steady state configuration.

In order to quantitatively compare the results of the stability diagram obtained here with the numerical studies in Ref.raja , we note that they had observed that for parameter values $L=4$, $N=46$, $C=10pN$ and $A=125pN$, for $B<0pN/\mu m$, monopolar configuration was stable, whereas for $B>25pN$, bipolar spindle configuration was stable. In order to compare, first we estimate the typical value of $R_{o}\equiv R_{d}/L$. $R_{o}=\sqrt{N}/L\sim 2$. With this specific choice of $R_{1}$, we find that bipolar spindle configuration is unstable, while monopolar configuration is stable for $B=0$ only for $C_{o}>0.4$ (Fig.3b), whereas in Ref. raja even for $C_{o}\equiv C/A=0.08$ bipolar spindle was observed to be unstable. We however find that for lower values of B i.e; $(B=-25pN/\mu m)$, bipolar spindle is unstable like in Ref.raja . The quantitative discrepancy between the analytical prediction of this model and the simulation results reported in Ref.raja can partially be attributed to our rather crude estimation of the radius of the disc $R_{d}$. An higher value of $R_{d}$ would in general result in lower value of $C$ for which stability of bipolar spindle configuration is lost. In general, the parameter range of $A$, $B$ and $C$ along with $L$ and $N$ can be varied over a certain range by altering the physiological conditions within the cell and analogously the scaled variables would span a range of values including $R_{d}$.

Finally we study the case in which $N$ chromosomes are uniformly distributed on a ring of radius $R_{r}$. Such a configuration has been shown to be stable in Ref. raja . In such a scenario the expression for the force reads as,

which upon integration yields,

For estimating the radius of the ring $R_{r}$, we assume that each chromosome occupies $2R_{ch}=2$ $\mu m$ length. Therefore $N$ chromosomes distributed on the perimeter of the ring require $R_{r}\sim N/\pi$ $\mu m$.

The scaled dimensionless variables for this case are, $x_{o}=x/L$, $F_{o}=\frac{F}{NA}$, $R_{o}=R_{r}/L$, $C_{o}=C/A$, $B_{o}=\frac{BL}{NA}$. The corresponding expression for the scaled force is,

For the solution $x_{o}=0$, the condition for stability yields,

that is represented in the panels of Fig. 6 as a dashed-dotted line. For determining the phase boundary for the case when $x_{o}\neq 0$, we again set $\frac{dF}{dx}\big{|}_{x_{o}}=0$ and $F_{o}(x_{o})=0$, to obtain,

These equations are numerically solved to obtain the boundary curves (solid lines) in Fig. 6 separating the stable and unstable region. In particular, Fig. 6.(a) exhibits regions of bistability for which two different solutions for bipolar spindle are stable and regions where both bipolar spindle and monopolar configuration are stable. In Fig. 6(b) we choose a value of $R_{r}=4$, obtained by estimating $N=46$ and $L=4$ as done in Ref.raja . We find that for this ring geometry, even for very low value of $C$, the bipolar spindle becomes unstable as was observed in Ref.raja . As in the previous case, by integrating the expression of dimensionless scaled force in Eq. 15, it possible to associate an effective potential energy function $V_{s}(x_{0})$, as a function of centrosome separation distance in the scaled variable $x_{o}$. This expression reads

In summary, we have studied and analyzed the mechanical stability of mitotic spindle structure within the framework of a minimal model proposed in Ref. raja ; ferenz which incorporates the interactions between Chromosomes with MTs nucleated from the centrosome, the sliding forces generated by motor proteins on overlapping MT filaments, and the interactions of the MTs with kinetochore machinery within the cell. Having assumed that the chromosomes are a priori distributed homogeneously on a disc or a ring in the mid-plane between the two centrosomes, we obtain a closed form analytic expression for the forces acting between pair of centrosomes. The assumption of such an arrangement of chromosomes is consistent with earlier numerical and experimental studies raja ; cell ; toroid . We analyze the stability of the spindle configurations and obtain the corresponding stability diagram by invoking the condition of stable mechanical equilibrium apart from the balance of forces. This stability diagram is expressed in terms of dimensionless parameters which are essentially ratios of the strength of the different interaction forces. This stability diagram allows us to quantify the impact of the different interactions at play in determining the final configuration of the spindle. We find that in general both for the case of chromosomes arranged on a ring and disc, if there is a net outward sliding forces by motors $(B>0)$ then bipolar spindle is stable, whereas for sufficiently high inward sliding forces of motor $(B<0)$, monopolar configuration is stable. We also found that for sufficiently high value of $C$, even for outward motor sliding force $(B>0)$, the bipolar spindle configuration can be unstable. Interestingly, when $B<0$, below a threshold value of $C$ corresponding to kinetochore-MT interaction forces, there are regimes of bistability characterized by coexistence of two different spindle length, and also stability of the bipolar spindle configuration and monopolar configuration. The origin of this bistable behavior can be associated with multiple minima in the effective potential energy function that can be derived from the expression of total interaction force. In fact it also allows us to rationalize the observation in Ref. raja ; ferenz , where it was seen that the final stable configuration was seen to be dependent on the initial separation distance. It would be interesting to investigate whether the prolonged lifetime in the bipolar spindle configuration to the eventual monopolar state observed in Ref. raja can be attributed to transition from shallow potential minima corresponding to spindle configuration to deeper potential energy minima associated with the monopolar configuration.

SM acknowledges DBT RGYI Project No. BT/PR6715/GBD/27/463/2012 for financial support, and S. Dietrich (MPI, Stuttgart) for travel grant and stay during research visit to MPI, Stuttgart.