  Many fundamental biological processes are regulated by protein-DNA complexes
called {\it synaptosomes}, which possess multiple interaction sites. Despite
the critical importance of synaptosomes, the mechanisms of their formation
remain not well understood. Because of the multi-site nature of participating
proteins, it is widely believed that their search for specific sites on DNA
involves the formation and breaking of DNA loops and sliding in the looped
configurations. In reality, DNA in live cells is densely covered by other
biological molecules that might interfere with the formation of synaptosomes.
In this work, we developed a theoretical approach to evaluate the role of
obstacles in the target search of multi-site proteins when the formation of DNA
loops and the sliding in looped configurations are possible. Our theoretical
method is based on analysis of a discrete-state stochastic model that uses a
master equations approach and extensive computer simulations. It is found that
the obstacle slows down the search dynamics in the system when DNA loops are
long-lived, but the effect is minimal for short-lived DNA loops. In addition,
the relative positions of the target and the obstacle strongly influence the
target search kinetics. Furthermore, the presence of the obstacle might
increase the noise in the system. These observations are discussed using
physical-chemical arguments. Our theoretical approach clarifies the molecular
mechanisms of formation of protein-DNA complexes with multiple interactions
sites.
