Attractive internuclear force drives the collective behavior of nuclear arrays in Drosophila embryos

The emerging collective behaviors during embryogenesis play an important role in precise and reproducible morphogenesis. An important question in the study of collective behavior is what rule underlies the emerging pattern. Here we use the Drosophila embryo as a test tube to study this question. We focus on the nuclear array without membrane separation on the embryo periphery from the nuclear cycle (NC) 11 to NC14. After live imaging with light sheet microscopy, we extract the nuclear trajectory, speed, and internuclear distance with an automatic nuclear tracing method. We find that the nuclear speed shows a period of standing waves along the anterior-posterior (AP) axis after each metaphase as the nuclei collectively migrate towards the embryo poles and partially move back. And the maximum nuclear speed dampens by 38% in the second half of the standing wave. Moreover, the nuclear density is 35% higher in the middle than the pole region of the embryo during the S phase of NC11-NC14. To find mechanical rules controlling the collective motion and packing patterns of the nuclear array, we use the deep neural network (DNN) to learn the force field from data. We find two potential strong nuclear-age-dependent force fields, i.e., the repulsive or attractive force field. Simulations with the particle-based model indicate that only if the net internuclear force is attractive and increases with distance, the pseudo-synchronous mitotic wave in a nuclear array with lower nuclear density in embryo poles can drive the collective motion with the damped standing wave of the nuclear speed, and the collective nuclear motion, in turn, maintains the non-uniform nuclear density.