  Genetic stability is a key factor in maintaining, survival and reproduction
of biological cells. It relies on many processes, but one of the most important
is a {\it homologous recombination}, in which the repair of breaks in
double-stranded DNA molecules is taking place with a help of several specific
proteins. In bacteria this task is accomplished by RecA proteins that are
active as nucleoprotein filaments formed on single-stranded segments of DNA. A
critical step in the homologous recombination is a search for a corresponding
homologous region on DNA, which is called a {\it homology search}. Recent
single-molecule experiments clarified some aspects of this process, but its
molecular mechanisms remain not well understood. We developed a quantitative
theoretical approach to analyze the homology search. It is based on a
discrete-state stochastic model that takes into account the most relevant
physical-chemical processes in the system. Using a method of first-passage
processes, a full dynamic description of the homology search is presented. It
is found that the search dynamics depends on the degree of extension of DNA
molecules and on the size of RecA nucleoprotein filaments, in agreement with
experimental single-molecule measurements of DNA pairing by RecA proteins. Our
theoretical calculations, supported by extensive Monte Carlo computer
simulations, provide a molecular description of the mechanisms of the homology
search.
