acids, these extremes suggest that amino acids are more likely to acceptably mutate if their physical and chemical properties are more common among alternative amino acids. Construction of the mutation matrix For the th amino acid, the values and are its mutability and frequency. The frequencies of the amino acids are normalised so that they sum to 1. If total number of occurrences of the th amino acid is , and is the total number of all amino acids, then Based on the definition of mutability as the ratio of mutations to occurrences of an amino acid or The mutation matrix is constructed so that the entry represents the probability of the th amino acid mutating into the th amino acid. The non-diagonal entries are computed by the equation where is a constant of proportionality. However, this equation does not compute the diagonal entries. Each column in the matrix lists each of the twenty possible outcomes for an amino acid — it can mutate into one of the 19 other amino acids, or remain unchanged. Since the non-diagonal entries listing the probabilities of each of the 19 mutations are known, and the sum of the probabilities of these twenty outcomes must be 1, this last probability can be calculated by which simplifies to {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Calculation of the diagonal entries |- | Substituting in the expression for the non-diagonal entries mutation matrix: Since the values of and are constants that don't change with the value of And thus cancellation reveals that |} A result of particular significance is that for the non-diagonal entries Which means that for all entries in the mutation matrix Choice of the constant of proportionality The probabilities contained in vary as some unknown function of the amount of time that a protein sequence is allowed to mutate for. Instead of attempting to determine this relationship, the values of are calculated for a short time frame, and the matrices for longer periods of time are calculated by assuming mutations follow a Markov chain model. The base unit of time for the PAM matrices is the time required for 1 mutation to occur per 100 amino acids, sometimes called 'a PAM unit' or 'a PAM' of time. This is precisely the duration of mutation assumed by the PAM1 matrix. The constant is used to control the proportion of amino acids that are unchanged. By