Three dimensional mathematical modelling of pronuclei migration for the mouse

Paper in proceeding, 2009

It is still an open question when the orientation of the embryonic-abembryonic axis of the mouse embryo is laid down. The two most explicit symmetry breaking events for the egg are the extrusion of the second polar body and the sperm entry. The main question addressed in this paper is what happens between the sperm entering the egg and fusion of the two pronuclei. Orientation of the apposing pronuclei probably plays a decisive role in the polarity of the developing embryo. In order to shed some lights on this intriguing question, a mathematical model that describes the pronuclei dynamics have been constructed in the form of a stochastic differential equation. The model concerns pronuclei migration from the time of the sperm entry to the fusion and spatial orientation of this fusion. The methodology consists of using stacks of confocal microscopy time-lapse images of the pronuclei migration together with statistical methods to identify realistic parameters in the model. Given different angles between the sperm entry and the position of the second polar body, the final model is then used to produce distributions of orientations of the meeting positions between the pronuclei. However, the main result is the suggested model itself which describes the main features of the migration. The fitted model is based on two forces of attraction. Migration is directed towards the centre but also towards the other pronucleus. Parameter values corresponding to the size of these forces are estimated from data of both eggs treated with a microtubule inhibitor and untreated eggs. Simulations from the model with the different model parameters are accomplished and distributions of meeting positions are plotted. These simulated distributions could for instance be used as initial value distributions for future models of egg cleavage.