original is . Given any particular unit of this amino acid, the probability of being replaced by the th amino acid in the assumed time interval is . Thus, the probability of the alignment is , the numerator within the logarithm. If the proteins are not related, the events that the two aligned amino acids are the th and th amino acids must be independent. The probabilities of these events are and , which means the probability of the alignment is , the denominator of the logarithm. Thus, the logarithm in the equation results in a positive entry if the alignment is more likely due to point accepted mutations, and a negative entry if the alignment is more likely due to chance. Properties of the PAM matrices Symmetry of the PAM matrices While the mutation probability matrix is not symmetric, each of the PAM matrices are. This somewhat surprising property is a result of the relationship that was noted for the mutation probability matrix: In fact, this relationship holds for all positive integer powers of the matrix : {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Generalisation of property to positive integer matrix powers |- | This generalisation can be proven using mathematical induction. Suppose that for a matrix And that for a positive integer By expansion of the matrix product , Using the property we have assumed of the matrix And using the property for the matrix In this case, it is only known at first that the result holds for . However, the above argument shows that the property also holds for . This new knowledge then shows that the property also holds for and this repeats to show that the property holds for all positive integers . |} As a result, the entries of the PAMn matrix are symmetric, since Relating the number of mutated amino acids and the number of mutations The value represents the number of mutations that occur per 100 amino acids, however this value is rarely accessible and often estimated. However, when comparing two proteins it is easy to calculate instead, which is the number of mutated amino acids per 100 amino acids. Despite the random nature of mutation, these values can be approximately related by {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Derivation of relationship between and |- | Mutations in the primary structure of a protein can occur anywhere along the sequence. If it