Sequence context-specific profiles for homology searching: Supplementary Information

Generation of context profile library N = 1 million training profiles of length l = 2d+1 were generated as described in the main text and Figure 2. Each training profile is represented by a count profile cn(j, x), which specifies the counts of amino acid x ∈ {1, . . . , 20} at position j ∈ {−d, . . . , d}. These counts are obtained by multiplying the sequence profile tn(j, x) by the effective number of sequences Nn(j) at position j in the alignment from which training profile tn(j, x) was calculated: cn(j, x) = Nn(j)tn(j, x) (see next section for details). Here, we describe how these N profiles are clustered in order to obtain a set of K context profiles which recur frequently among the training profiles and which together can describe all training profiles. More precisely, we seek to determine context profiles p = (p1, . . . , pK) and their prior probabilities α = (α1, . . . , αK) that maximize the likelihoodP (c|p, α) that the training profile counts c = (c1, . . . , cN ) were generated by the context profiles. We model the distribution of counts cn(j, x) in each column j by a multinomial distribution. Since cn(j, x) can be real-valued, however, we replace the factorials in the multinomial distribution by Gamma functions (n! = Γ(n + 1)). The probability for context profile pk to have emitted counts cn(j, x) (j ∈ {−d, . . . , d}, x ∈ {1, . . . , 20}) is