Fig 1.
3D reconstruction of oocyte during the SCW.
A: Maximum intensity projections of an oocyte during the SCW. The oocytes were filled with fluorescently labelled dextran to mark the cell volume. A video is given as S1 Video. B: The surface reconstruction shows rotational symmetry of the SCW. A video is given as S2 Video. C,D: Surface area and volume extracted from the segmentation shown in B. Both show a slight decrease during the SCW. E: Slices of the cell surface containing the AV-axis and the polar body show the typical shape changes during the SCW.
Fig 2.
A: We consider 2D slices through the cell volume containing the AV-axis. The SCW takes about 7 min to run over the surface. Different intermediate stages during the SCW are indicated by numbers. B: The cell contour (blue rim) was tracked with image analysis methods from transmission light microscopy images. It was then described in polar coordinates with the contour centre of mass as coordinate origin. To smooth the curve, higher Fourier modes were discarded (fft). The symmetry of the radius function was used to determine the orientation of the AV-axis and the signal was symmetrised around it (symm fft). C: The diameter of the oocyte at the AV-axis develops from a minimum via a maximum to a minimum, whereas the diameter on the perpendicular axis towards the equator shows the opposite behaviour, as expected from A. D: The radius function was defined with respect to the centre of mass (cyan cross). The AV-axis is marked as an orange line and the animal pole by a star. The surface of the oocyte shows high symmetry during different stages of the SCW. The difference between the smoothed radius function (green line, fft) and its symmetrised version (blue line, symm fft) is very small. In stages 1 and 4 there are local minima at the animal pole whereas in stage 2 we found a local maximum. A video of the SCW is given as S3 Video.
Fig 3.
Dynamics of surface displacements and curvatures during the SCW.
A: Cartoon of a cell with invaginated shape. For each direction at each point on the surface, a curvature (the inverse of the local curvature radius) can be defined. Here the two principal curvatures of the rotationally symmetric surface at the equator are shown as K2 in the yellow plane and K1 in the blue plane. At the same point they have opposite signs (red section). B: The starfish oocyte is assumed to be rotationally symmetric around the AV-axis. This makes it possible to calculate the three-dimensional shape from two-dimensional images. Thus we have access to the full set of curvatures. C,D: The normalised deformation from a perfect spherical shape (C) and the radial surface velocity (D) clearly define the SCW-speed. E-H: The kymographs of mean curvature (H), Gaussian curvature (K) and the two curvatures in φ-direction (K1) and in θ-direction (K2) confirm that the SCW can be described as a band of local flattening running over the oocyte. Evaluation of the Gaussian curvature K demonstrates that the surface transiently evolves towards a saddle shape.
Fig 4.
An analytical hydrodynamic calculation predicts flows from surface displacements.
A: The flows inside the oocyte are modelled as Stokes flows of an incompressible Newtonian fluid. We have constructed an analytical solution for the flows inside a moving sphere-like boundary. We applied it to the flows inside a rotationally symmetric object. The problem reduces to the calculation of three fields ϕ, χ and p from the surface movement as given in the Methods section. B: The model predicts the flows for arbitrary radial surface movement, exemplified here by a surface movement where there is influx at the poles and outflux at the equator. C: Arbitrary tangential surface movement can be included, exemplified by constant surface movement from one pole to another. D: Due to linearity of the Stokes equation different contributions to the flows can be separately calculated and added up so that rigid body movement can be predicted from adding up radial and tangential flows. The model is extended using perturbation theory to also deal with non-spherical shapes. This is exemplified by rigid body movement of a deformed sphere (blue). An undeformed sphere is shown as reference (red).
Fig 5.
A contractile surface model predicts shape changes during the SCW.
A: For each time step a surface Hamiltonian for a spherical shape with varying surface tension is minimised. The surface tension follows a Gaussian profile that is moved over the surface with changing strength. The predictions of the model in B show similar features as a zoom-in to the experimental curvatures during the SCW in C. D: The contractile model can also predict tangential flows in the cortex, that in turn can couple to the hydrodynamic flow, in addition to flows caused by the radial deformations as explained in Fig 4. The tangential surface movement due to the Gaussian-shaped local contraction of a viscous medium is analytically calculated. The tangential surface movement is used as an input for the hydrodynamic model. The effect of different contraction positions on the flow pattern is small. This can be seen from the flows (arrows) due to a rotationally symmetric surface contraction (red colours) around the AV-axis (star).
Fig 6.
Experimentally measured cytoplasmic flows.
The flows inside the starfish oocyte can be measured using a standard PIV-algorithm. A: Only small irregular flow patterns (red arrows) can be detected as long as no surface movement is present. B: The flows point mainly from the vegetal pole to the animal pole. C: The flows now come from the equator and point in direction of both poles. D: The flow goes back towards the vegetal pole. A video of the flows for the full SCW is given in S4 Video. E-G: Quantification of the flows on the AV-axis (E) and on the perpendicular axis (F, ±1 correspond to equator right and left). The colour intensity encodes the flow velocity while the colour tone indicates the flow direction as explained by the polar colour map in G.
Fig 7.
Comparison of the hydrodynamic model with experimental flows.
A: The experimentally measured flows (red arrows) are compared with flows predicted by the hydrodynamic model (black arrows). The hydrodynamic model also predicts the internal pressure field (purple to green colours) that is experimentally not accessible. The model is based only on the radial surface movement obtained from experiment (red to blue rim). Whole body movement parallel to the AV-axis is fitted. B: The model’s accuracy can be improved by taking into account tangential surface movement as an effect of local surface contraction. Now the position and the strength of the contraction band are fitted for each time step individually. Videos of the SCW predicted by the radial and tangential displacement models, respectively, are given as S5 and S8 Videos. C,D: The quality of the tangential displacement model quantified by the normalized residuals is significantly better during the wave (C whole time course, D zoom in to wave).
Fig 8.
Mixing hypothesis and polar body formation.
A: A particle starting at the dot follows the shown line during the SCW. Most trajectories end approximately at their starting point and follow similar paths as their neighbouring ones. Therefore mixing of the cytoplasm can be excluded as a function of the SCW. B,C: It has been suggested that the polar body is pushed out by the SCW due to build up of high internal pressure. Our hydrodynamic model shows that the SCW leads to a locally higher pressure at the animal pole of several hundred μPa. If the polar body was pushed out by a blebbing mechanism, the pressure difference between inside and outside would need to be of the order of several hundred Pa at the animal pole. Therefore the pressure due to the SCW can be excluded as a mechanism for polar body extrusion. D: Both the deformation speed and the internal flows of a blebbistatin-treated cell are much weaker than in the wild type cell. E: A polar body is formed even with blebbistatin-treatment. Full videos are given as S9 and S10 Videos. F: A potential mechanism for polar body extrusion is local actin polymerisation. The actin of the oocyte was labelled with utrophin which makes a local accumulation of actin visible at the position of the polar body (S11 Video).