| This thesis focuses on an active area in mathematics applied to biology, namely the modelling of genes interactions controlling the differentiation of embryonic stem cells, starting from microarray data. We have developed and implemented novel algorithmic approaches to this question, and tested them on two very large microarray data sets obtained by Austin Cooney's Lab at Baylor College of Medicine.;One of the challenges is the massive size of the microarray data and the huge combinatorial possibilities number of groups of genes which may actually interact. Another statistical and mathematical challenge was to enforce a parameter parsimony principle, to avoid the massive and not very meaningful over-parameterization.;In particular the thesis has focused on the intensive modeling and validation (or invalidation) of more than 5,000 biologically plausible instances of two basic architectural motifs for miRNA interactions with the main genes controlling differentiation in ES cells. These basic motifs are of small size and always involve at least one pair of potentially interacting mRNA and miRNA, in systematic compliance with major biological reference tables listing potential (mRNA, miRNA) interactions.;A key biological question was to exploit microarray data to elucidate, for specific messenger RNA (mRNA) genes, which mRNAs are actually repressed by each microRNA (miRNA) in a specific list of 266 known miRNAs. The algorithmic technique we have developed is to formalize potential genes interactions by explicit chemical kinetics equations (CKEs) parameterized by unknown parameters, and then to compute good estimates of these parameters. An essential technical problem was to derive the adequate types of nonlinear CKEs to model genes interactions, and to estimate the unknown parameters of these CKEs by algorithmic analysis of the time evolutions for expressions data. |
