Protein-DNA interactions are fundamental not only for gene activity and transcription but also for the formation of different complexes that organize the genetic material. In order for the formation of these complexes or transcription to take place it is first necessary that the protein can find its target sequence in the midst of the nuclear environment that is heavily crowded with different molecules and presents tightly packed DNA. Our understanding of this search process has been extensively improved over the course of the last 40 years but open questions about the microscopic mechanism of target search or even how different structures in the DNA impact the search process still remain.

The chapters of this thesis are committed to develop models for very different protein search processes. Since single-molecule experiments have been able to measure transition rates for sliding, associating and unbind from DNA, it is straightforward to adopt a discrete state stochastic model to obtain the target search time distribution for those cases and understand how these different scenarios impact the search dynamics. First, after recent investigations have completely explained the dynamics of the DNA loop formation process, we describe how the emergence of this protein-DNA structure happens when a fixed crowder molecule or obstacle that prevents protein sliding through or binding on that site is present in our system. Using this model, we are able to show that the obstacle causes the average target search time to diverge exponentially in one of the protein search regimes. We also connected the presence of this obstacle with an increase in stochastic noise. Second, we start analyzing how a fixed loop structure can accelerate the search process if we suppose that there is a probability of transition between its intersection sites. After that we increase the complexity of our system by introducing DNA spatial configuration transition rates and we explain the complex behaviour obtained with this model. Lastly, based on recent experimental measurements of association and unbinding dynamics of different transcription factors in nucleosomal and free DNA, we were able to develop a new model that accounts for how molecules known as pioneer transcription factors are able to invade and find their target sequences much faster than normal transcription factors in a compacted nucleosomal DNA region. This allows us to better understand how even silenced genes, inaccessible to most transcription factors, can still be expressed even when located in dense regions of the chromatin.