Stochastic modelling and analysis of early mouse development

Doctoral thesis, 2011

The aim of this thesis is to model and describe dynamical events for biological cells using statistical and mathematical tools. The thesis includes five papers that all relate to stochastic modelling of cells. In order to understand the development and patterning of the early mammalian embryo, stochastic modelling has become a more important tool than ever. It allows for studying the processes that mediate the transition from pluripotency of the embryonic cells to their differentiation. It is still unclear whether the positions of cells determine their future fates. One alternative possibility is that cells are pre-specified at random positions and then sort according to a already set fate. Mouse embryonic cells are thought to be equivalent in their developmental properties until approaching the eight-cell stage. Some biological studies show, in comparison, that patterning can be present already at sperm entry and in the pronuclei migration. We investigate in Paper I the dynamics of the pronuclei migration by analysing their trajectories and find that not only do the pronuclei follow a noise corrupted path towards the centre of the egg but they also have some attraction to each other which affects their dynamics. Continuing in Paper II and III, we use these results to model this behaviour with a coupled stochastic differential equation model. This enables us to simulate distributions that describe the meeting plane between pronuclei which in turn can be related to the orientation of the first cleavage of the egg. Our results show that adding randomness in sperm entry point is different from the randomness added through the environment of the egg. We are also able to show that data sets with normal eggs and eggs treated with an actin growth inhibitor give rise to considerably different model dynamics, suggesting that the treatment is affecting the migration in an invasive way. Altering the pronuclei dynamics can alter the polarity of the egg and may transfer into the later axis-formation process. Invasiveness of experimental procedures is a difficult issue to handle. The alternative to invasive procedures is not appealing since it means that important developmental features may not be discovered because of individual variability and noise, leading to guesswork of the underlying mechanisms. The embryonic cells are easily affected by treatments performed to make the measuring, made by hand, easier or by the light exposure of the microscope. Treatments as such are used for example for producing flourescent proteins in membranes or slowing processes down. Paper IV and Paper V serve to analyse how light induced stress affects yeast cells and we employ a method for analysing the noisy non-stationary time series, which are a result of the yeast experiments, using wavelet decomposition.