Emission probabilities of the HMM.

Subscript denotes the -th methylation pattern, and the -th CpG site. Note that denotes the association (1) or non-association (0) state of DNMT1, parent-strand DNMT3s, and daughter-strand DNMT3s, respectively. Also, denotes the methylated (1) or unmethylated (0) state of CpG on the parent strand and daughter strand, respectively. Additionally, denotes the probability of the -th CpG site being methylated before DNA replication, which is equivalent to the methylation density of the -th site. and are the probability of the maintenance activity of DNMT1 and the DNMT3s, respectively, at associated daughter-strand CpG, whereas and are that of the de novo activity of DNMT1 and the DNMT3s, respectively. The measurement error rates (see Materials and Methods and Table 4) are assumed to be 0 here. This HMM allows for estimation of the hemi-preference ratio for both DNMT1 and the DNMT3s, although additional constraints are needed for this simultaneous estimation. See text for details.