Figure 1. Framework of the Coevolutionary Model.

(Top row) The independent rate matrix Q_indep is derived from the single amino acid rate matrix. Entries corresponding to both amino acid changes are zeros and entries corresponding to single amino acid changes have the same rates as the single amino acid rate matrix (e.g., Q_indep[HR,HA] = Q[R,A] ≡ q_RA). The coevolutionary rate matrix Q_cov is obtained by reweighting the independent rate matrix. Entries of single amino acid changes are penalized by multiplying ɛ and entries of both amino acid changes are rewarded by replacing zeros with r.

(Second row) Suppose two protein domains M ₁ and M ₂ interact at certain positions. We acquire the homologous domains of M ₁ and M ₂ across four species (S ₁ − S ₄) and align each family of sequences.

(Third row) We acquire the joint phylogenetic tree of the two families of sequences. For each pair of positions, we place the joint sequences on the leaves of the tree as the observed states of the CTMP. The conditional probability of interval t is given by e^Qt.

(Fourth row) The joint likelihood of a CTMP along a tree is the product of prior and conditional probabilities. The marginal likelihood of each pair of aligned positions is obtained by summing over all possible states of internal nodes. It can be efficiently evaluated by dynamic programming.

(Bottom row) The log-likelihood ratio between the coevolutionary and independent models specifies how likely the observed sequences arise from coevolution relative to the null (independent) model.