Analysis of a one-dimensional random walk with irreversible losses at each step: applications for protein movement on DNA

We analysed a one-dimensional random walk between two points when the migrating particle could be irreversibly lost (dissociated) from the system at each step of the process. We show that in the case of losses at each step the average number of steps made by the particle that reaches the final point does not obey quadratic dependence on the distance between the starting and the final points: for long distances this dependence is linear. This is because losses “select” for shorter pathways between the starting and the final points. We applied this analysis to protein translocations within long DNA molecules.