A force balance model of early spindle pole separation in Drosophila embryos

Abstract

The formation and function of the mitotic spindle depends upon force generation by multiple molecular motors and by the dynamics of microtubules, but how these force-generating mechanisms relate to one another is unclear. To address this issue we have modeled the separation of spindle poles as a function of time during the early stages of spindle morphogenesis in Drosophila embryos. We propose that the outward forces that drive the separation of the spindle poles depend upon forces exerted by cortical dynein and by microtubule polymerization, and that these forces are antagonized by a C-terminal kinesin, Ncd, which generates an inward force on the poles. We computed the sum of the forces generated by dynein, microtubule polymerization, and Ncd, as a function of the extent of spindle pole separation and solved an equation relating the rate of pole separation to the net force. As a result, we obtained graphs of the time course of spindle pole separation during interphase and prophase that display a reasonable fit to the experimental data for wild-type and motor-inhibited embryos. Among the novel contributions of the model are an explanation of pole separation after simultaneous loss of Ncd and dynein function, and the prediction of a large value for the effective centrosomal drag that is needed to fit the experimental data. The results demonstrate the utility of force balance models for explaining certain mitotic movements because they explain semiquantitatively how the force generators drive a rapid initial burst of pole separation when the net force is great, how pole separation slows down as the force decreases, and how a stable separation of the spindle poles characteristic of the prophase steady state is achieved when the force reaches zero.

FIGURE 1

A diagram of the principle structures involved in spindle morphogenesis in interphase-prophase. The nucleus (pink disc) is relatively isolated by the surrounding furrows (brown curve). Along the inner surface of the membrane is a layer of actin (green strands) which binds to the minus end directed motor, dynein (red, see magnified region on the left) through dynactin. MTs (blue lines) extend outward from each of two centrosomes. A few astral MTs interact with dynein on the surface of the actin cap. Interpolar MTs are cross-linked by Ncd motors (yellow, see magnified region on the right).

FIGURE 2

Geometry of the model (cross section). S is the arclength separation of the centers of the centrosomes, L is the radius of the inner surface of the cylindrical actin cap, h is the depth of the furrow, and g is the distance between the nucleus and the actin cap. In calculating the forces applied to a particular centrosome, we project them onto the tangent plane to the nuclear surface (dashed line). For example, the MT in the top half of the figure exerts a force F_dyn on one of the centrosomes. This force must be decomposed into components, one perpendicular and one parallel to the nuclear surface. It is the parallel force that contributes to the movement of the centrosome. Once the parallel component is calculated, it can be added to the F_pol and F_ncd to find the total driving force.

FIGURE 3

Approximating the polymerization force. The centrosome on the right nucleates MTs that run into and get stalled by the centrosome on the left. In estimating the total polymerization force generated by these MTs, we calculate the area of a disc through the middle of the centrosome to approximate the surface area of MT-centrosome interaction. When the centrosomes are sufficiently far apart, the nucleus blocks some of the MTs that would otherwise exert a force. To account for these, a portion of the area of the disc is eliminated according to the relative positions of the centrosomes and the nucleus, as shown in the diagram.

FIGURE 4

(A) Forces as a function of separation distance. The calculated dynein (F_dyn), Ncd (F_ncd), and polymerization (F_pol) forces are plotted as dotted curves. The total spindle forming force in the wild-type embryo (dashed curve) is calculated by adding all three forces (F_dyn + F_ncd + F_pol). Notice a stable steady state (F = 0) at S ≈ 6 μm. The Ncd-null embryo (solid curve) is simulated by adding the dynein and polymerization forces (F_dyn + F_pol). Eliminating the Ncd force increases the steady-state separation distance to ≈8 μm. (B) Same as A except the solid curve shows the total force when dynein is inhibited. In simulating the effect of inhibiting dynein (by exposing the embryo to either p50 dynamitin and anti-DHC), we reduced the calculated dynein force to a fraction of its original amplitude. Here, we plot the force resulting from a reduction in dynein activity to 25% of its original value (F = 0.25F_dyn + F_ncd + F_pol). Notice the reduction in steady-state separation distance to ≈3 μm. (C) To simulate the combined Ncd-null and dynein inhibited embryo, we eliminate the Ncd force and reduce the dynein force to 25% of its original value. The total force in this case, F = 0.25F_dyn + F_pol, is shown by the solid curve. In this situation, the steady separation distance is ≃8 μm as it was in the Ncd-null case. Also, the amplitude of the force for separation distances between 3 μm and 8 μm is much lower than in the Ncd-null case leading to a much slower separation of the poles, as noted in the experiments (compare Fig. 5, A and C).

FIGURE 4

(A) Forces as a function of separation distance. The calculated dynein (F_dyn), Ncd (F_ncd), and polymerization (F_pol) forces are plotted as dotted curves. The total spindle forming force in the wild-type embryo (dashed curve) is calculated by adding all three forces (F_dyn + F_ncd + F_pol). Notice a stable steady state (F = 0) at S ≈ 6 μm. The Ncd-null embryo (solid curve) is simulated by adding the dynein and polymerization forces (F_dyn + F_pol). Eliminating the Ncd force increases the steady-state separation distance to ≈8 μm. (B) Same as A except the solid curve shows the total force when dynein is inhibited. In simulating the effect of inhibiting dynein (by exposing the embryo to either p50 dynamitin and anti-DHC), we reduced the calculated dynein force to a fraction of its original amplitude. Here, we plot the force resulting from a reduction in dynein activity to 25% of its original value (F = 0.25F_dyn + F_ncd + F_pol). Notice the reduction in steady-state separation distance to ≈3 μm. (C) To simulate the combined Ncd-null and dynein inhibited embryo, we eliminate the Ncd force and reduce the dynein force to 25% of its original value. The total force in this case, F = 0.25F_dyn + F_pol, is shown by the solid curve. In this situation, the steady separation distance is ≃8 μm as it was in the Ncd-null case. Also, the amplitude of the force for separation distances between 3 μm and 8 μm is much lower than in the Ncd-null case leading to a much slower separation of the poles, as noted in the experiments (compare Fig. 5, A and C).

FIGURE 4

(A) Forces as a function of separation distance. The calculated dynein (F_dyn), Ncd (F_ncd), and polymerization (F_pol) forces are plotted as dotted curves. The total spindle forming force in the wild-type embryo (dashed curve) is calculated by adding all three forces (F_dyn + F_ncd + F_pol). Notice a stable steady state (F = 0) at S ≈ 6 μm. The Ncd-null embryo (solid curve) is simulated by adding the dynein and polymerization forces (F_dyn + F_pol). Eliminating the Ncd force increases the steady-state separation distance to ≈8 μm. (B) Same as A except the solid curve shows the total force when dynein is inhibited. In simulating the effect of inhibiting dynein (by exposing the embryo to either p50 dynamitin and anti-DHC), we reduced the calculated dynein force to a fraction of its original amplitude. Here, we plot the force resulting from a reduction in dynein activity to 25% of its original value (F = 0.25F_dyn + F_ncd + F_pol). Notice the reduction in steady-state separation distance to ≈3 μm. (C) To simulate the combined Ncd-null and dynein inhibited embryo, we eliminate the Ncd force and reduce the dynein force to 25% of its original value. The total force in this case, F = 0.25F_dyn + F_pol, is shown by the solid curve. In this situation, the steady separation distance is ≃8 μm as it was in the Ncd-null case. Also, the amplitude of the force for separation distances between 3 μm and 8 μm is much lower than in the Ncd-null case leading to a much slower separation of the poles, as noted in the experiments (compare Fig. 5, A and C).

FIGURE 5

(A) The predicted time course of pole-pole separation fit to the experimental data of Sharp et al. (2000a). The p50 dynamitin data is fit assuming dynein activity is reduced to 50% of the wild-type amplitude and the anti-DHC data is fit assuming a reduction to 25% of the original amplitude. (B) The Ncd-null case. Notice the rapid approach to a steady separation distance of ≃8 μm. (C) The dynein inhibited, Ncd-null case. Notice that the slower approach to the same steady state seen in the Ncd-null case (≃8 μm) gives the impression that the combined dynein inhibited-Ncd-null experiment reestablishes the original separation time course.

FIGURE 5

(A) The predicted time course of pole-pole separation fit to the experimental data of Sharp et al. (2000a). The p50 dynamitin data is fit assuming dynein activity is reduced to 50% of the wild-type amplitude and the anti-DHC data is fit assuming a reduction to 25% of the original amplitude. (B) The Ncd-null case. Notice the rapid approach to a steady separation distance of ≃8 μm. (C) The dynein inhibited, Ncd-null case. Notice that the slower approach to the same steady state seen in the Ncd-null case (≃8 μm) gives the impression that the combined dynein inhibited-Ncd-null experiment reestablishes the original separation time course.

FIGURE 5

(A) The predicted time course of pole-pole separation fit to the experimental data of Sharp et al. (2000a). The p50 dynamitin data is fit assuming dynein activity is reduced to 50% of the wild-type amplitude and the anti-DHC data is fit assuming a reduction to 25% of the original amplitude. (B) The Ncd-null case. Notice the rapid approach to a steady separation distance of ≃8 μm. (C) The dynein inhibited, Ncd-null case. Notice that the slower approach to the same steady state seen in the Ncd-null case (≃8 μm) gives the impression that the combined dynein inhibited-Ncd-null experiment reestablishes the original separation time course.

FIGURE 6

The region of dynein-MT interaction. MTs emanating from a centrosome extend only into the half space above the tangent plane to the nucleus because the surface of the nucleus blocks any that extend below this plane. The shadowed region represents the domain in which dynein on the actin cap wall interacts with MTs and is the region over which the dynein force is integrated.