An example of point mutations at an amino acid site coding for lysine. The missense mutations may be classed as point accepted mutations if the mutated protein is not rejected by natural selection.

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 sometimes 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.^[1] 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.^[1][2] 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.^[2][3]

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.^[2] 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.^[4] 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.^[2] Changes like these may severely impact a crucial function in a cell, potentially causing the cell — and in extreme cases, the organism — to die.^[5] 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.^[4] However, these amino acids can be categorised into groups with similar physicochemical properties.^[4] 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.^[6] Because the value ${\displaystyle n}$ in the PAM_n 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 PAM_n 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 'PAM_n matrix' is the PAM matrix corresponding to a time frame long enough for ${\displaystyle n}$ mutation events to occur per 100 amino acids.

Construction of PAM matrices

PAM matrices were introduced by Margaret Dayhoff in 1978.^[7] 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.^[6][8] 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.^[6][7][8] 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.^[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]

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

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

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

Choice of the constant of proportionality

The probabilities contained in ${\displaystyle M}$ 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 ${\displaystyle M}$ 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.^[9][10] 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.^[6] This is precisely the duration of mutation assumed by the PAM₁ matrix.

The constant ${\displaystyle \lambda }$ 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 ${\displaystyle n}$ in PAM_n 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 ${\displaystyle n(j)M(j,j)}$ is equal to the number of conserved amino acid ${\displaystyle j}$ units, and so the total number of conserved amino acids is

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

The value of ${\displaystyle \lambda }$ needed to be pick to produce 99% identity after mutation is then given by the equation

${\displaystyle 0.99=1-\lambda \sum _{j=1}^{20}f(j)m(j)}$

This ${\displaystyle \lambda }$ value can then be used in the mutation matrix for the PAM₁ matrix.

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]

${\displaystyle {\text{PAM}}_n(i,j)=log\frac{f(j)M_n(i,j)}{f(i)f(j)}=log\frac{f(j)M^n(i,j)}{f(i)f(j)}=log\frac{M^n(i,j)}{f(i)}}$Zdroj:https://en.wikipedia.org?pojem=PAM_matrix
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