The Topologies of p53 Single-site and Cluster-site Models. (a) A Profile Hidden Markov Model (PHMM) contains three hidden states for each position in a sequence motif of length n: a match state (green squares), an insertion state (orange diamonds), and a delete state (gray circles). The arrows represent allowed transitions between states and have associated probabilities. The match and insertion states also have associated nucleotide emission probabilities. The first and last insertion states (I-0 and I-n) and associated transitions (in red) are shown for completeness. However, they are not present in the p53 models since they are replaced by FIM and FEM models. (b) The topology of the Finite Emission Module (FEM) of length N allows the ability to model any distribution of spacer-lengths between 1 and N. For the p53 models, the model and background probabilities within the FEM modules are identically uniform so that there is no-cost for spacer-lengths between 1 and N, and are referred to as "no-cost FEMs". (c) The topology of the Free Insertion Module (FIM) allows for the ability to model an exponentially decaying distribution of spacer-lengths. However, by setting the model and background probabilities to identically uniform, the FIM can model any sequence of infinite length with no associated cost to the overall score (hence the word "Free"). (d) The main components of the p53 single-site model are the left and right half-site PHMMs, which potentially contain corresponding positions between them. These two half-site models are separated by a no-cost FEM model that limits the length of any intervening spacer sequence to 20 bp. The half-site models are also wrapped by two FIMs that allow the Viterbi algorithm to find the best matching motifs anywhere in the candidate sequences. (e) The topology of the p53 cluster-site model consists of a single PHMM that models a general half-site, and two back-transitions that allow for modeling an infinite number of half-sites within the cluster-site. The back-transition through the no-cost FEM-14 model limits the spacer-sequence between the half-sites to lengths ≤ 14 bp.