Collection of data from phylogenetic trees

For each branch in the phylogenetic trees of the protein families, the number of mismatches that were observed were recorded and a record kept of the two amino acids involved.^[7] These counts were used as entries below the main diagonal of the matrix ${\displaystyle A}$. Since the vast majority of protein samples come from organisms that are alive today (extant species), the 'direction' of a mutation cannot be determined. That is, the amino acid present before the mutation cannot be distinguished from the amino acid that replaced it after the mutation. Because of this, the matrix ${\displaystyle A}$ is assumed to be symmetric, and the entries of ${\displaystyle A}$ above the main diagonal are computed on this basis. The entries along the diagonal of ${\displaystyle A}$ do not correspond to mutations and can be left unfilled.

In addition to these counts, data on the mutability and the frequency of the amino acids was obtained.^[6][7] The mutability of an amino acid is the ratio of the number of mutations it is involved in and the number of times it occurs in an alignment.^[7] Mutability measures how likely an amino acid is to mutate acceptably. Asparagine, an amino acid with a small polar side chain, was found to be the most mutable of the amino acids.^[7] Cysteine and tryptophan were found to be the least mutable amino acids.^[7] The side chains for cysteine and tryptophan have less common structures: cysteine's side chain contains sulfur which participates in disulfide bonds with other cysteine molecules, and tryptophan's side chain is large and aromatic.^[4] Since there are several small polar amino 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.^[6][8]

Construction of the mutation matrix

For the ${\displaystyle j}$th amino acid, the values ${\displaystyle m(j)}$ and ${\displaystyle f(j)}$ 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 ${\displaystyle j}$th amino acid is ${\displaystyle n(j)}$, and ${\displaystyle N}$ is the total number of all amino acids, then

${\displaystyle f(j)={\frac {n(j)}{N}}}$

Based on the definition of mutability as the ratio of mutations to occurrences of an amino acid

${\displaystyle m(j)={\frac {\sum _{i=1,i\neq j}^{20}A(i,j)}{n(j)}}}$

or

${\displaystyle {\frac {1}{Nf(j)}}={\frac {1}{n(j)}}={\frac {m(j)}{\sum _{i=1,i\neq j}^{20}A(i,j)}}}$

The mutation matrix ${\displaystyle M}$ is constructed so that the entry ${\displaystyle M(i,j)}$ represents the probability of the ${\displaystyle j}$th amino acid mutating into the ${\displaystyle i}$th amino acid. The non-diagonal entries are computed by the equation^[7]

${\displaystyle M(i,j)=\lambda A(i,j){\frac {m(j)}{\sum _{i=1,i\neq j}^{20}A(i,j)}}={\frac {\lambda A(i,j)}{Nf(j)}}={\frac {\lambda A(i,j)}{n(j)}}}$

where ${\displaystyle \lambda }$ is a constant of proportionality. However, this equation does not compute the diagonal entries. Each column in the matrix ${\displaystyle M}$ 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

${\displaystyle M(j,j)=1-\sum _{i=1,i\neq j}^{20}M(i,j)}$

which simplifies to^[7]

${\displaystyle M(j,j)=1-\lambda m(j)}$

Calculation of the diagonal entries

${\displaystyle M(j,j)=1-\sum _{i=1,i\neq j}^{20}M(i,j)}$ Substituting in the expression for the non-diagonal entries mutation matrix: ${\displaystyle M(j,j)=1-\sum _{i=1,i\neq j}^{20}{\frac {\lambda m(j)A(i,j)}{\sum _{i=1,i\neq j}^{20}A(i,j)}}}$ ${\displaystyle M(j,j)=1-{\frac {\sum _{i=1,i\neq j}^{20}\lambda m(j)A(i,j)}{\sum _{i=1,i\neq j}^{20}A(i,j)}}}$ Since the values of ${\displaystyle \lambda }$ and ${\displaystyle m(j)}$ are constants that don't change with the value of ${\displaystyle i}$ ${\displaystyle M(j,j)=1-{\frac {\lambda m(j)\sum _{i=1,i\neq j}^{20}A(i,j)}{\sum _{i=1,i\neq j}^{20}A(i,j)}}}$ And thus cancellation reveals that ${\displaystyle M(j,j)=1-\lambda m(j)}$

A result of particular significance is that for the non-diagonal entries

${\displaystyle f(j)M(i,j)={\frac {\lambda }{N}}A(i,j)={\frac {\lambda }{N}}A(j,i)=f(i)M(j,i)}$

Which means that for all entries in the mutation matrix

${\displaystyle f(j)M(i,j)=f(i)M(j,i)}$

Construction of the PAM_n matrices

The Markov chain model of protein mutation relates the mutation matrix for PAM_n, ${\displaystyle M_{n}}$, to the mutation matrix for the PAM₁ matrix, ${\displaystyle M_{1}}$ by the simple relationship

${\displaystyle M_{n}=M_{1}^{n}}$

The PAM_n matrix is constructed from the ratio of the probability of point accepted mutations replacing the ${\displaystyle j}$th amino acid with the ${\displaystyle i}$th amino acid, to the probability of these amino acids being aligned by chance. The entries of the PAM_n matrix are given by the equation^[11][12]

${\displaystyle {\text{PAM}}_{n}(i,j)=log{\frac {f(j)M_{n}(i,j)}{f(i)f(j)}}=log{\frac {f(j)M^{n}(i,j)}{f(i)f(j)}}=log{\frac {M^{n}(i,j)}{f(i)}}}$

Note that in Gusfield's book, the entries ${\displaystyle M(i,j)}$ and ${\displaystyle {\text{PAM}}_{n}(i,j)}$ are related to the probability of the ${\displaystyle i}$th amino acid mutating into the ${\displaystyle j}$th amino acid.^[11] This is the origin of the different equation for the entries of the PAM matrices.

When using the PAM_n 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 ${\displaystyle n}$ point accepted mutations to occur per 100 amino acids.

When the alignment of the ${\displaystyle i}$th and ${\displaystyle j}$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 ${\displaystyle j}$th amino acid is the original. Based on the abundance of amino acids in proteins, the probability of the ${\displaystyle j}$th amino acid being the original is ${\displaystyle f(j)}$. Given any particular unit of this amino acid, the probability of being replaced by the ${\displaystyle i}$th amino acid in the assumed time interval is ${\displaystyle M_{n}(i,j)}$. Thus, the probability of the alignment is ${\displaystyle f(j)M_{n}(i,j)}$, the numerator within the logarithm.
• If the proteins are not related, the events that the two aligned amino acids are the ${\displaystyle i}$th and ${\displaystyle j}$th amino acids must be independent. The probabilities of these events are ${\displaystyle f(i)}$ and ${\displaystyle f(j)}$, which means the probability of the alignment is ${\displaystyle f(i)f(j)}$, 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.

Symmetry of the PAM matrices

While the mutation probability matrix ${\displaystyle M}$ is not symmetric, each of the PAM matrices are.^[6][7] This somewhat surprising property is a result of the relationship that was noted for the mutation probability matrix:

${\displaystyle f(j)M(i,j)=f(i)M(j,i)}$

In fact, this relationship holds for all positive integer powers of the matrix ${\displaystyle M}$:

${\displaystyle f(j)M^{n}(i,j)=f(i)M^{n}(j,i)}$

Generalisation of property to positive integer matrix powers

This generalisation can be proven using mathematical induction. Suppose that for a matrix ${\displaystyle M}$ ${\displaystyle f(j)M(i,j)=f(i)M(j,i)}$ And that for a positive integer ${\displaystyle k}$ ${\displaystyle f(j)M^{k}(i,j)=f(i)M^{k}(j,i)}$ By expansion of the matrix product ${\displaystyle M^{k+1}=M^{k}\cdot M}$, ${\displaystyle f(j)M^{k+1}(i,j)=f(j)\sum _{n=0}^{N}M^{k}(i,n)M(n,j)}$ ${\displaystyle f(j)M^{k+1}(i,j)=\sum _{n=0}^{N}M^{k}(i,n)(f(j)M(n,j))}$ Using the property we have assumed of the matrix ${\displaystyle M}$ ${\displaystyle f(j)M^{k+1}(i,j)=\sum _{n=0}^{N}M^{k}(i,n)(f(n)M(j,n))}$ ${\displaystyle f(j)M^{k+1}(i,j)=\sum _{n=0}^{N}(f(n)M^{k}(i,n))M(j,n)}$ And using the property for the matrix ${\displaystyle M^{k}}$ ${\displaystyle f(j)M^{k+1}(i,j)=\sum _{n=0}^{N}(f(i)M^{k}(n,i))M(j,n)}$ ${\displaystyle f(j)M^{k+1}(i,j)=f(i)\sum _{n=0}^{N}M(j,n)M^{k}(n,i)}$ ${\displaystyle f(j)M^{k+1}(i,j)=f(i)M^{k+1}(j,i)}$ In this case, it is only known at first that the result holds for ${\displaystyle k=1}$. However, the above argument shows that the property also holds for ${\displaystyle k=2}$. This new knowledge then shows that the property also holds for ${\displaystyle k}$ and this repeats to show that the property holds for all positive integers ${\displaystyle k}$.

As a result, the entries of the PAM_n matrix are symmetric, since

${\displaystyle {\text{PAM}}_{n}(i,j)=log{\frac {f(j)M^{n}(i,j)}{f(j)f(i)}}=log{\frac {f(i)M^{n}(j,i)}{f(i)f(j)}}={\text{PAM}}_{n}(j,i)}$

Relating the number of mutated amino acids and the number of mutations

The value ${\displaystyle n}$ 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 ${\displaystyle m}$ 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^[13]

${\displaystyle {\frac {m}{100}}=1-e^{-{\frac {n}{100}}}}$

Derivation of relationship between ${\displaystyle m}$ and ${\displaystyle n}$

Mutations in the primary structure of a protein can occur anywhere along the sequence. If it 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 ${\displaystyle K}$ balls (i.e. mutations) are distributed amongst ${\displaystyle N}$ bins (amino acid positions), the number of bins containing at least one ball, ${\displaystyle M}$ has a distribution with a mean given by[14] ${\displaystyle E(M)=N-N(1-{\frac {1}{N}})^{K}}$ ${\displaystyle {\frac {E(M)}{N}}=1-(1-{\frac {1}{N}})^{K}}$ If the rate of mutation is ${\displaystyle n}$ mutations per 100 amino acids, then ${\displaystyle {\frac {n}{100}}={\frac {K}{N}}}$ And if there are ${\displaystyle m}$ mutated amino acids per 100 amino acids, then it is approximately equal to ${\displaystyle {\frac {m}{100}}={\frac {E(M)}{N}}}$ Now ${\displaystyle m}$ and ${\displaystyle n}$ can be related by ${\displaystyle {\frac {m}{100}}=1-(1-{\frac {1}{N}})^{\frac {nN}{100}}}$ For large values of ${\displaystyle N}$, an assumption that can be reasonably made for typical proteins, this expression is approximately equal to ${\displaystyle {\frac {m}{100}}=1-e^{-{\frac {n}{100}}}}$

The validity of these estimates can be verified by counting the number of amino acids that remain unchanged under the action of the matrix ${\displaystyle M}$. The total number of unchanged amino acids for the time interval of the PAM_n matrix is

${\displaystyle \sum _{j=1}^{20}n(j)M^{n}(j,j)}$

and so the proportion of unchanged amino acids is

${\displaystyle {\frac {\sum _{j=1}^{20}n(j)M^{n}(j,j)}{N}}=\sum _{j=1}^{20}f(j)M^{n}(j,j)=1-{\frac {m}{100}}}$

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.^[13] 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 for two related proteins, taken from organisms living today, is sought. The two proteins have both been accumulating accepted mutations since the date of divergence, and so the total number of mutations per amino acid separating them is approximately twice that which separates them from their common ancestor. If a range of PAM matrices are used to align two proteins that are known to be related, then the value of ${\displaystyle n}$ in the PAM_n matrix which results in the best score is most likely to correspond to the mutations per amino acid separating the two proteins. Halving this value and dividing by the rate at which accepted mutations accumulate in the protein family provides an estimate of the time of divergence of these two proteins from their common ancestor. That is, the time of divergence in myr is^[13]

${\displaystyle T={\frac {K}{2r}}}$

Where ${\displaystyle K}$ is the number of mutations per amino acid, and ${\displaystyle r}$ is the rate of accepted mutation accumulation in mutations per amino acid site per million years.

Use in BLAST

PAM matrices are also used as a scoring matrix when comparing DNA sequences or protein sequences to judge the quality of the alignment. This form of scoring system is utilized by a wide range of alignment software including BLAST.^[15]

Comparing PAM and BLOSUM

Although the PAM log-odds matrices were the first scoring matrices used with BLAST, the PAM matrices have largely been replaced by the BLOSUM matrices. Although both matrices produce similar scoring outcomes they were generated using differing methodologies. The BLOSUM matrices were generated directly from the amino acid differences in aligned blocks that have diverged to varying degrees the PAM matrices reflect the extrapolation of evolutionary information based on closely related sequences to longer timescales.^[16] Since scoring information for the PAM and BLOSUM matrices were generated in very different ways the numbers associated with the matrices have fundamentally different meanings; the numbers for PAM matrices increase for comparisons among more divergent proteins whereas the numbers for the BLOSUM matrices decrease.^[17] However, all amino acid substitution matrices can be compared in an information theoretic framework^[18] using their relative entropy.