Point accepted mutation

A point accepted mutation — also known as a PAM — is the replacement of a single amino acid in the primary structure of a protein with another single amino acid, which is accepted by the processes of natural selection. This definition does not include all point mutations in the DNA of an organism. In particular, silent mutations are not point accepted mutations, nor are mutations which are lethal or which are rejected by natural selection in other ways.
A PAM matrix is a matrix where each column and row represents one of the twenty standard amino acids. In bioinformatics, PAM matrices are regularly used as substitution matrices to score sequence alignments for proteins. Each entry in a PAM matrix indicates the likelihood of the amino acid of that row being replaced with the amino acid of that column through a series of one or more point accepted mutations during a specified evolutionary interval, rather than these two amino acids being aligned due to chance. Different PAM matrices correspond to different lengths of time in the evolution of the protein sequence.
Biological background.
The genetic instructions of every replicating cell in a living organism are contained within its DNA. Throughout the cell's lifetime, this information is transcribed and replicated by cellular mechanisms to produce proteins or to provide instructions for daughter cells during cell division, and the possibility exists that the DNA may be altered during these processes. This is known as a mutation. At the molecular level, there are regulatory systems that correct most — but not all — of these changes to the DNA before it is replicated.
One of the possible mutations that occurs is the replacement of a single nucleotide, known as a point mutation. If a point mutation occurs within an expressed region of a gene, an exon, then this will change the codon specifying a particular amino acid in the protein produced by that gene. Despite the redundancy in the genetic code, there is a possibility that this mutation will then change the amino acid that is produced during translation, and as a consequence the structure of the protein will be changed.
The functionality of a protein is highly dependent on its structure. Changing a single amino acid in a protein may reduce its ability to carry out this function, or the mutation may even change the function that the protein carries out. Changes like these may severely impact a crucial function in a cell, potentially causing the cell — and in extreme cases, the organism — to die. Conversely, the change may allow the cell to continue functioning albeit differently, and the mutation can be passed on to the organism's offspring. If this change does not result in any significant physical disadvantage to the offspring, the possibility exists that this mutation will persist within the population. The possibility also exists that the change in function becomes advantageous. In either case, while being subjected to the processes of natural selection, the point mutation has been accepted into the genetic pool.
The 20 amino acids translated by the genetic code vary greatly by the physical and chemical properties of their side chains. However, these amino acids can be categorised into groups with similar physicochemical properties. Substituting an amino acid with another from the same category is more likely to have a smaller impact on the structure and function of a protein than replacement with an amino acid from a different category. Consequently, acceptance of point mutations depends heavily on the amino acid being replaced in the mutation, and the replacement amino acid. The PAM matrices are a mathematical tool that account for these varying rates of acceptance when evaluating the similarity of proteins during alignment.
Terminology.
The term "accepted point mutation" was initially used to describe the mutation phenomenon. However, the acronym PAM was preferred over APM due to readability, and so the term "point accepted mutation" is used more regularly. Because the value formula_1 in the PAMn matrix represents the number of mutations per 100 amino acids, which can be likened to a percentage of mutations, the term "percentage accepted mutation" is sometimes used.
It is important to distinguish between point accepted mutations (PAMs), point accepted mutation matrices (PAM matrices) and the PAMn matrix. The term 'point accepted mutation' refers to the mutation event itself. However, 'PAM matrix' refers to one of a family of matrices which contain scores representing the likelihood of two amino acids being aligned due to a series of mutation events, rather than due to random chance. The 'PAMn matrix' is the PAM matrix corresponding to a time frame long enough for formula_1 mutation events to occur per 100 amino acids.
Construction of PAM matrices.
PAM matrices were introduced by Margaret Dayhoff in 1978. The calculation of these matrices were based on 1572 observed mutations in the phylogenetic trees of 71 families of closely related proteins. The proteins to be studied were selected on the basis of having high similarity with their predecessors. The protein alignments included were required to display at least 85% identity. As a result, it is reasonable to assume that any aligned mismatches were the result of a single mutation event, rather than several at the same location.
Each PAM matrix has twenty rows and twenty columns — one representing each of the twenty amino acids translated by the genetic code. The value in each cell of a PAM matrix is related to the probability of a row amino acid before the mutation being aligned with a column amino acid afterwards. From this definition, PAM matrices are an example of a substitution matrix.
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. These counts were used as entries below the main diagonal of the matrix formula_3. 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 formula_3 is assumed to be symmetric, and the entries of formula_3 above the main diagonal are computed on this basis. The entries along the diagonal of formula_3 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. 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. 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. Cysteine and tryptophan were found to be the least mutable amino acids. 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. 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.
Construction of the mutation matrix.
For the formula_7th amino acid, the values formula_8 and formula_9 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 formula_7th amino acid is formula_11, and formula_12 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 formula_16 is constructed so that the entry formula_17 represents the probability of the formula_7th amino acid mutating into the formula_19th amino acid. The non-diagonal entries are computed by the equation
where formula_21 is a constant of proportionality. However, this equation does not compute the diagonal entries. Each column in the matrix formula_16 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
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 formula_16 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 formula_16 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 formula_21 is used to control the proportion of amino acids that are unchanged. By 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 formula_1 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 formula_31 is equal to the number of conserved amino acid formula_7 units, and so the total number of conserved amino acids is
The value of formula_21 needed to be pick to produce 99% identity after mutation is then given by the equation
This formula_21 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, formula_37, to the mutation matrix for the PAM1 matrix, formula_38 by the simple relationship
The PAMn matrix is constructed from the ratio of the probability of point accepted mutations replacing the formula_7th amino acid with the formula_19th 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 formula_17 and formula_44 are related to the probability of the formula_19th amino acid mutating into the formula_7th 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:
When the alignment of the formula_19th and formula_7th amino acids is considered, the score indicates the relative likelihoods of the alignment due to the proteins being related or due to random chance.
Properties of the PAM matrices.
Symmetry of the PAM matrices.
While the mutation probability matrix formula_16 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 formula_16:
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 formula_1 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 formula_67 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
The validity of these estimates can be verified by counting the number of amino acids that remain unchanged under the action of the matrix formula_16. 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 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 formula_1 in the PAMn 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
Where formula_74 is the number of mutations per amino acid, and formula_75 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.
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. 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. However, all amino acid substitution matrices can be compared in an information theoretic framework using their relative entropy.