THESIS FOR THE DEGREE OF LICENTIATE OF ENGINEERING Three dimensional mathematical modelling of pronuclei migration for the mouse

The main question addressed in this thesis is what happens between the moment when the sperm enters the egg and the fusion of the male and the female pronuclei. Orientation of the apposing pronuclei most likely plays a decisive role in the polarity of the developing embryo. The migration and the dependence between the pronuclei have been investigated through three different measures of correlation. It was concluded that a measure based on the projection of the movement onto the axis between the pronuclei’s centres was preferred. Two mathematical models that describe the pronuclei dynamics have been constructed in the form of stochastic differential equations. The models concern pronuclei migration from the time of the sperm entry to the fusion and spatial orientation of this fusion. First, a basic model was created. This was then developed into a refined model. 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 models. Given different angles between the sperm entry and the position of the second polar body, the final models are then used to produce distributions of orientations of the meeting positions between the pronuclei. However, the main result is the suggested models themselves which describe the main features of the migration. The basic 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. The refined model is also based on centring and attraction to the other pronucleus, but the centring is modelled into two mechanisms of pushing and pulling of the microtubule exerted forces. Simulated distributions from the models could for instance be used as initial value distributions for future models of egg cleavage.