Discrete and continuous mathematical models of DNA branch migration

DNA junctions, known as Holliday junctions, are intermediates in genetic recombination between DNAs. In this structure, two double-stranded DNA helices with similar sequence are joined at a branch point. The branch point can move along these helices when strands with the same sequence are exchanged. Such branch migration is modeled as a random walk. First, we model this process discretely, such that the motion of the branch is represented as transfer between discrete compartments. This is useful in analysing the results of DNA branch migration on junction comprised of synthetic oligonucleotides. The limit in which larger numbers of smaller steps go to continuous motion of the branch is also considered. We show that the behavior of the continuous system is very similar to that of the discrete system when there are more than just a few compartments. Thus, even branch migration on oligonucleotides can be viewed as a continuous process. One consequence of this is that a step size must be assumed when determining rate constants of branch migration. We compare migration where forward and backward movements of the branch are equally probable to biased migration where one direction is favored over the other. In the latter case larger differences between the discrete and continuous cases are predicted, but the differences are still small relative to the experimental error associated with experiments to measure branch migration in oligonucleotides.