using only alignments of proteins that had at least 85% similarity, it could be reasonably assumed that the mutations observed were direct, without any intermediate states. This means that scaling down these counts by a common factor would provide an accurate estimate of the mutation counts had the similarity been closer to 100%. It also means that the number of mutations per 100 amino acids, the in PAMn is equal to the number of mutated amino acids per 100 amino acids. To find the mutation matrix for the PAM1 matrix, the requirement that 99% of the amino acids in a sequence are conserved is imposed. The quantity is equal to the number of conserved amino acid units, and so the total number of conserved amino acids is The value of needed to be pick to produce 99% identity after mutation is then given by the equation This value can then be used in the mutation matrix for the PAM1 matrix. Construction of the PAMn matrices The Markov chain model of protein mutation relates the mutation matrix for PAMn, , to the mutation matrix for the PAM1 matrix, by the simple relationship The PAMn matrix is constructed from the ratio of the probability of point accepted mutations replacing the th amino acid with the th amino acid, to the probability of these amino acids being aligned by chance. The entries of the PAMn matrix are given by the equation Note that in Gusfield's book, the entries and are related to the probability of the th amino acid mutating into the th amino acid. This is the origin of the different equation for the entries of the PAM matrices. When using the PAMn matrix to score an alignment of two proteins, the following assumption is made: If these two proteins are related, the evolutionary interval separating them is the time taken for point accepted mutations to occur per 100 amino acids. When the alignment of the th and th amino acids is considered, the score indicates the relative likelihoods of the alignment due to the proteins being related or due to random chance. If the proteins are related, a series of point accepted mutations must have occurred to mutate the original amino acid into its replacement. Suppose the th amino acid is the original. Based on the abundance of amino acids in proteins, the probability of the th amino acid being the