A new discrete distribution arising in a model of DNA replication

During DNA replication, small fragments of DNA are formed. These have been observed experimentally and the mechanism of their formation modelled mathematically. Using the stochastic model of Cowan and Chiu (1992, 1994), we nd the probability distribution of the number of fragments. A new discrete distribution arises. The work has interest as an application of the recent theory on quasi-renewal equations in Piau (2000). 1 Finding water for the town We rstly phrase the problem of this paper in a simple non-biological setting. Consider an exploration team which starts at a town located on the x-axis at the origin O. The team's job is to nd water for the residents of the town. The team commences at O and moves along the positive x-axis at a constant speed r. When a water spring is seen, a member of the team immediately starts to build a pipeline back to the town. He builds at a rate c. The rest of the team continues at speed r to look for more springs. Whenever the next spring is seen, another man starts to build a pipeline toward the town. He will eventually reach the earlier pipeline which started at the rst spring and the pipes are instantaneously joined. This process continues; explorers move away from the town at speed r and pipe-builders construct pipes toward the town at speed c. Pipes are joined whenever constructed segments of pipe meet. The team of explorers is assumed innnite and the location of springs is assumed to follow a Poisson process on the x-axis. For large t, what is the distribution of the number of pipe-builders at work at time t (construction by a builder on any segment having ceased when it joins another segment)? In this paper we answer this problem, but in the biological setting which motivated it, namely the formation of Okazaki fragments during DNA replication. A model for this process was rst presented in Cowan and Chiu (1992,1994) and Cowan (2001), where links to the relevant biological literature are given.