is assumed the distribution of the mutations among amino acid positions is uniform, the problem is analogous to a distribution of "balls into bins", a common problem in combinatorics. In a case where balls (i.e. mutations) are distributed amongst bins (amino acid positions), the number of bins containing at least one ball, has a distribution with a mean given by If the rate of mutation is mutations per 100 amino acids, then And if there are mutated amino acids per 100 amino acids, then it is approximately equal to Now and can be related by For large values of , an assumption that can be reasonably made for typical proteins, this expression is approximately equal to |} The validity of these estimates can be verified by counting the number of amino acids that remain unchanged under the action of the matrix . The total number of unchanged amino acids for the time interval of the PAMn matrix is and so the proportion of unchanged amino acids is An example - PAM250 A PAM250 is a commonly used scoring matrix for sequence comparison. Only the lower half of the matrix needs to be computed, since by their construction, PAM matrices are required to be symmetric. Each of the 20 amino acid are shown down the top and side of the matrix, with 3 additional ambiguous amino acids. The amino acids are most commonly shown listed alphabetically, or listed in groups. These groups are the characteristics shared among the amino acids. Uses in bioinformatics Determining the time of divergence in phylogenetic trees The molecular clock hypothesis predicts that the rate of amino acid substitution in a particular protein will be approximately constant over time, though this rate may vary between protein families. This suggests that the number of mutations per amino acid in a protein increases approximately linearly with time. Determining the time at which two proteins diverged is an important task in phylogenetics. Fossil records are often used to establish the position of events on the timeline of the Earth's evolutionary history, but the application of this source is limited. However, if the rate at which the molecular clock of protein family ticks — that is, the rate at which the number of mutations per amino acid increases — is known, then knowing this number of mutations would allow the date of divergence to be found. Suppose the date of divergence