There are several common ways in which weights can be applied for amino acid differences. Karlin and Ghandour (1985, PNAS 82:8597) proposed a method of weights based on chemical, functional, charge and structural properties of the amino acids. Similarly Doolittle proposed weights based on the structural similarities and the ease of genetic interchange (Feng et al., 1985 J. Mol. Evol. 21: 112). However, by far the most common and most famous way to assign weights is to use Dayhoff's PAM250 matrix. This is a matrix of weights that is derived from how often different amino acids replace other amino acids in evolution (see M.O. Dayhoff, ed., 1978, Atlas of Protein Sequence and Structure, Vol5). This was based on a data base of 1,572 changes in 71 groups of closely related proteins appearing in earlier volumes of this amazing predecessor to electronic databases. PAM stands for percent accepted mutations and these were inferred from the types of changes observed in these proteins. Every change was tabulated and entered in a matrix enumerating all possible amino acid changes.
In addition to these counts of accepted point mutations an idea of the relative mutability of different amino acids were calculated. The information about the individual kinds of mutations and about the relative mutability of the amino acids can be combined into one distance-dependent "mutation probability matrix". The elements of this matrix give the probability that the amino acid in one column will be replaced by the amino acid in some row after a given evolutionary interval. For example, a matrix with an evolutionary distance of 0 PAMs would have ones on the main diagonal and zeros elsewhere. A matrix with an evolutionary distance of 1 PAM would have numbers close to one on the main diagonal and small numbers off the main diagonal. One PAM would correspond to roughly 1% divergence in a protein (one amino acid replacement per hundred). The model of evolution that Dayhoff used assumed that proteins diverged as a result of accumulated, uncorrelated mutations. They treat the PAM-1 matrix as a first order Markov chain transition model. To derive a mutational probability matrix for a protein sequence that has undergone N percent accepted mutations, a PAM-N matrix, the PAM-1 matrix is multiplied by itself N times. This results in a family of scoring matrices.
By trial and error Dayhoff et al. found that for weighting purposes a 250 PAM matrix works well. At this evolutionary distance (250 substitutions per hundred residues) only one amino acid in five remains unchanged and the percent divergence has increased to roughly 80%. However, the amino acids vary greatly in their mutability. According to Dayhoff et al., roughly 55% of the tryptophans, 52% of the cysteines and 27% of the glycines would still be unchanged, but only 6% of the highly mutable asparagines would remain. Several other amino acids particularly alanine, aspartic acid, glutamic acid, glycine, lysine and serine are more likely to occur in place of an original asparagine than asparagine itself at this evolutionary distance.
From this matrix an odds matrix is constructed. This matrix
takes the elements of the previous matrix (
) and divides each
term by the frequency of the replacement residue. Hence, each term
now gives the probability of replacement, j to i per occurrence of
residue j.
By tradition the 
of this matrix is used as weights (this is
because to calculate the odds for the whole matrix requires taking
the product of changes for all sites of the protein. Before
calculators it was easier to find the sum of the 
's rather than
the product sum). This 
odds 250 PAM matrix is shown in Table
1 (also note that normally the equivalent amino acids are placed
adjacent, while I have been non-standard and placed amino acids
alphabetically).
Residue pairs with scores above 1 replace each other more often as alternatives in related sequences than in random sequences. This is an indication that both residues can carry out similar functions. A score exactly equal to one indicates amino acid pairs that are found as alternatives at exactly the frequency predicted by chance. Residue pairs with scores less than 1 replace each other less often than in random sequences and would be an indication that these residues are not functionally equivalent.
Some of the properties that are visible from this matrix and go into its makeup are - size, shape, local concentrations of electric charge, conformation of van der Waals surface, ability to form salt bonds, hydrophobic bonds, and hydrogen bonds. Interestingly, these patterns are imposed principally by natural selection and only secondarily by the constraints of the genetic code. This tends to indicate that coming up with your own matrix of weights based on some logical features may not be very successful because your logical features may have been over-written by other more important considerations.
Some of the problems with this measure of distance is that it assumes that all sites are equally mutable. But this is clearly false. Another problem is that by examining proteins with few differences, the highly mutable amino acids have been stressed. Lastly, due to the collection of proteins known at that time, the matrix is biased because it is based mainly on small globular proteins.
