To measure mutation rate change associated with NSAID use, we used a two epoch model in BEAST [31], where the transition time between the first and second sampling periods is the time of change in NSAID use. We ran BEAST for 10 million Bayesian MCMC iterations that sample the space of genealogies and population parameters to obtain posterior distributions for model parameters that best fit the data. We used uniform prior distributions for SGA rate with lower and upper bounds of 10−5 and 104 SGAs per biopsy genome per year, respectively, for the duration of any of the on-NSAID and off-NSAID periods and estimated SGA rate separately for the first and second sampling periods, where each SGA rate adheres to the molecular clock hypothesis (SGA occur at constant rate for all evolving lineages) for the period duration. We added a 0/1 mutation model in BEAST for the SGA absence/presence character states (Text S1: Equations S3–4) and this model assumed that SGAs do not revert to the normal type, i.e., 1→0 transition is impossible. An SGA character was defined by the breakpoints, which had to occur at the exact same SNPs between samples to be considered the same character. We also modified BEAST's likelihood calculation algorithm to consider a last universal common ancestor (LUCA) that has an unaltered genomic state (zeros for all sites), and that connects to the most recent common ancestor (MRCA), at the root of the tree, creating an extra LUCA-MRCA branch. Thus, the final likelihood of the tree is the product of the likelihood of the tree at the root, calculated with Felsenstein's pruning algorithm [46], multiplied by the probability of the LUCA-MRCA branch length. We used crypt density results (Table S4) and a logistic growth model (Text S1) to estimate the age of the root of the tree and to constrain the internal node coalescence times in the two epoch (off- on- NSAID) BEAST analysis.
Maximum parsimony trees were estimated using Wagner parsimony with delayed transformation (DELTRAN) on the individual-specific phylogenetic matrix with 0/1 SGA states using the PAUP program [47]. For PAUP analyses, we also used a character transition matrix that assumes infinite cost for 1→0 transitions, i.e. SGAs do not revert to normal type. For individuals with off-on NSAID regimens (a–k) all lineages (i.e. phylogeny branches) having any off-NSAID descendant lineages (i.e. leading to off-NSAID biopsies) were considered to have evolved during off-NSAID usage period. Similarly, for individuals l and m, all lineages having any on-NSAID descendant lineages were considered to have evolved during on-NSAID usage period. In all individuals (a–m), lineages having descendant lineages leading to all on-NSAID or all off-NSAID biopsies were considered to have evolved during on-NSAID and off-NSAID usage periods, respectively.