Non-homogeneous Poisson and renewal processes as spatial models for cancer mutation

Advances in sequencing technology assisted biologists in revealing signatures of DNA cancer mutation process and in demonstrating the mutagenesis behind. However, most of these signatures proposed by majority of work focus only on the type and frequency of mutations, without considering spatial information which is non-negligible in exploring mechanisms of mutation occurrence, e.g., Kataegis. Statistical characterization of the distance between consecutive mutations can give us relative spatial information; however, it ignores location information which is as important as distance information. In this work, we assume that DNA cancer mutations are location-dependent and that integrating the two variables, location and inter-distance, is beneficial to study DNA cancer mutation processes more accurately. Particularly, instead of following a specific distribution over the whole DNA sequence, we found out that the distribution of distance between successive mutations alternates between exponential and power-law distributions. Apart from this, the parameters of either of the two distributions vary with DNA locations. The cancers with kataegis phenomenon, a specific mutation pattern caused by abnormal activity of APOBEC protein family, are more likely to be accompanied by higher parameter values of distance distribution, implying higher occurrence rate of mutation. Therefore, we propose non-homogeneous Poisson and Renewal processes to spatially model DNA cancer mutations and to describe mutation patterns quantitatively and more accurately through a statistical perspective.