Fig 1.
Representative images of the collective motion of the nuclear array of a Drosophila embryo (snapshots from S1 Movie).
These images are the projections from the 3D images taken with light sheet microscopy. Each nucleus is segmented and marked with a different color to show the direction of the nuclear velocity along the AP axis (red and green represent the left (anterior) and right (posterior) direction, respectively. And the intensity indicates the magnitude of the nuclear velocity).
Fig 2.
Characteristic features of the packing pattern and the collective motion pattern of the nuclear array in a representative Drosophila embryo from NC11 to NC14.
(A) A representative heat map of the projected nuclear density along the AP axis of the embryo from NC11 to NC14, i.e., rescaled developmental time from 90 min to 150 min after embryo deposition (AED) at 25°C (see S1 Text). Black triangles label the start time point of chromosome segregation during mitotic phase (M phase) 11, 12 and 13. (B) The dynamics of hexatic bond-orientational order parameter φarray (see Materials and methods). The black arrows 1, 2 and 3 label three time intervals around M phase 11, 12 and 13 showing the minimum order parameter. (C) Bar graph (mean ± standard deviation (SD)) comparing anterior (~5–15% EL) or posterior (~85–95% EL) density and the maximal density in the middle of the embryo during interphase. Student’s t-test results: interphase 12 (anterior, p<0.01; posterior, p<0.01; embryo number n = 4); interphase 13 (anterior, p<0.01; posterior, p = 0.015; n = 4); interphase 14 (anterior, p<0.01; posterior, p<0.01; n = 4). (D) A representative heat map of the nuclear speed projected along the AP axis of the embryo from NC11 to NC14. Positive and negative direction point to the posterior and anterior pole, respectively. (E) Dynamics of the order parameter of the collective motion (φspeed, see Materials and methods) from NC11 to NC14. APi (i = 6–8) corresponds to the ith bin with the width of 10% EL from the anterior pole (for the other bins, see S6 Fig). The markers 1, 2 and 3 label three time windows that show high motion collectivity in (D) and (E). (F) Bar graph comparing the maximal wave crest of the first half period and the second half period of the AP speed standing wave. Student’s t test results: NC12 (p<0.01; n = 4); NC13 (p<0.01; n = 4); NC14 (p<0.01; n = 4). (G) The corresponding dynamics of the nuclear density, speed, smoothed φarray and smoothed φspeed of the nuclear array after the onset of anaphase 12 shown in (A-D).
Fig 3.
Force field functions learned from 1D data via the MLFNN model.
(A-C) The DNN learning results based on the Fa assumption. The data from M phase 13 to interphase 14 in one embryo (3078 data points in total) is used while training the DNN. (A) A representative heat map of the function F(T,r). Here F is the magnitude of the internuclear force (), T is the nuclear age after the onset of anaphase, and r is the internuclear distance. Note that,
and
, here s is the Voronoi area of each nucleus. (B) F has a positive correlation with r as T = 4.4–5.6 min. (C) F has a pulsatile relationship with T as r = 7.9–8.2 μm. (D-F) The DNN learning results based on the Fr assumption as in A-C.
Fig 4.
3D simulation results of the packing pattern and the collective motion pattern of the wild type embryo based on the Fa force assumption (for the simulation movie, see S4 Movie).
(A) Heat map of the nuclear density projected along the AP axis. (B) The dynamics of hexatic bond-orientational order parameter φarray (see Materials and methods). (C) Heat map of the nuclear speed projected along the AP axis. (D) Dynamics of the order parameter of the collective motion φspeed (see Materials and methods). APi (i = 1–4) corresponds to the ith bin with the width of 25% EL from the anterior pole (E) Boxplots (whisker, min/max values, boxes, 25/75 percentiles). The medians (red line) of measured and simulated density ratio are 0.58, 0.65, 0.78, 0.37 and 0.45, respectively. Density ratio is defined as the ratio between the anterior (~5–15% EL) or posterior (~85–95% EL) density and the maximal density in the middle of the embryo during interphase. (F) Boxplots (whisker, min/max values, boxes, 25/75 percentiles). The medians (red line) of measured and simulated wave peak ratio are 0.55, 0.72, 0.62 and 0.58, respectively. Wave peak ratio is defined as the ratio between the maximal wave crest of the second half period and the first half period of the AP speed standing wave. The force field used in this simulation is shown in S16A Fig.
Fig 5.
3D simulation results of the embryos with varied start time of metaphase at two poles (Fa force assumption).
(A) The linear relation between the position of the second node of the standing wave of AP nuclear speed and the division time difference between the anterior and posterior poles of experimental (red) and simulation (black) data. (B) The density ratio of the simulation data with different AP division time difference. Boxplots (whisker, min/max values, boxes, 25/75 percentiles). The medians of the four group data in the panel are 0.51, 0.50, 0.47 and 0.46, respectively. (C) The representative characteristic features of the collective motion pattern and packing pattern of the nuclear array in the simulations with different AP division time difference. The force field used in this simulation is shown in S16A Fig.
Fig 6.
The intrinsic properties of the force fields generate the characteristic features of the collective behaviors of the nuclear array.
(A) Asymmetric distribution of the nuclear density is stabilized by the attractive but not repulsive force field. As the nuclear density is high in the middle of the embryo, for a nucleus (orange circle), its nearest neighboring nuclei on the pole side (dark blue circles) are fewer in number but larger in the internuclear distance than those on the middle side (light blue circles) of the embryo. Since the attractive force increases with distance, the pair-wise force is stronger on the pole side (thicker arrows) than that on the middle side (thin arrows). Hence the net force on a given nucleus is balanced. In contrast, the repulsive force decreases with distance, the pair-wise force is stronger on the middle side, and the net force cannot be balanced. (B) Collective motion is driven by the age-dependent force field. The nuclei on the pole side (dark blue circle) have a greater nuclear age T than those in the middle (light blue circle). This age difference leads to a greater (weaker) attractive force from the pole side in the first (second) half period of the standing wave, hence the nuclei (orange circle) collectively move towards (away from) the pole. (C) The dampening standing wave of the nuclear speed is generated due to the greater net force in the first half period compared with the second half period. In the first half period, the amplitude of the attractive force from the pole side (Fp) is much greater than the force from the middle side (Fm). But in the second half period, the amplitude difference between Fp and Fm is much smaller.
Fig 7.
Molecular basis of the Fa force field from the MFLNN model.
(A) The relationship between biological molecular dynamics and the Fa force. Each nucleus dynamically changes between five typical states during each nuclear cycle: “mitotic furrow state”, “mitotic furrow recovery state”, “flat membrane with lager membrane deformation and less myosin II state”, “actomyosin border formation with small membrane deformation state”, and “actomyosin border formation completion state”. The relationship between the states and the Fa force field is shown in the right panel. The corresponding nuclear motion process is shown in the left panel. (B) Heat maps of the calculated average cell area (the Voronoi area of each nucleus) variation relative to the original cell area (left) and the nuclear age (right) after the onset of anaphase 12.