This PhD dissertation proposes a stochastic analysis of two questions of molecular biology in which ran- domness is a key feature of the processes involved: protein polymerisation in neurodegenerative diseases on the one hand, and telomere shortening on the other hand. Self-assembly of proteins into amyloid aggregates is an important biological phenomenon associated with human diseases such as prion diseases, Alzheimer’s, Huntington’s and Parkinson’s disease, amyloidosis and type-2 diabetes. The kinetics of amyloid assembly show an exponential growth phase preceded by a lag phase, variable in duration, as seen in bulk experiments and experiments that mimic the small volume of the concerned cells. After an introduction to protein polymerisation in chapter I, we investigate in chapter II the origins and the properties of the observed variability in the lag phase of amyloid assembly. This variability is currently not accounted for by deterministic nucleation-dependent mechanisms. In order to tackle this issue, a stochastic minimal model is proposed, simple, but capable of describing the characteristics of amyloid growth curves. Two populations of chemical components are considered in this model: monomers and polymerised monomers. Initially, there are only monomers and from then, two possible ways of polymerising a monomer: either two monomers collide to combine into two polymerised monomers, or a monomer is polymerised by the encounter of an already polymerised monomer. However efficient, this simple model does not fully explain the variability observed in the experiments, and in chapter III, we extend it in order to take into account other relevant mechanisms of the polymerisation process that may have an impact on fluctuations. In both chapters, asymptotic results involving different time scales are obtained for the corresponding Markov processes. First and second order results for the starting instant of nucleation are derived from these limit theorems. These results rely on a scaling analysis of a population model and the proof of a stochastic averaging principle for a model related to an Ehrenfest urn model. In the second part, a stochastic model for telomere shortening is proposed. In eukaryotic cells, chromo- somes are shortened with each occurring mitosis, because the DNA polymerases are unable to replicate the chromosome down to the very end. To prevent potentially catastrophic loss of genetic information, these chromosomes are equipped with telomeres at both ends (repeated sequences that contain no genetic information). After many rounds of replication however, the telomeres are progressively nibbled to the point where the cell cannot divide anymore, a blocked state called replicative senescence. The aim of this model is to trace back to the initial distribution of telomeres from measurements of the time of senescence.