This corresponds to a model of evolution in which amino acids mutate randomly and independently from one another but according to some predefined probabilities depending on the amino acid itself. This is a Markovian model of evolution and while simple, it is one of the best models. Intrinsic properties of amino acids, like hydrophobicity, size, charge, etc. can be modeled by appropriate mutation matrices. Dependencies which relate one amino acid characteristic to the characteristics of its neighbors are not possible to model through this mechanism. Amino acids appear in nature with different frequencies. These frequencies are denoted by fi and correspond to the steady state of the Markov process defined by the matrix  M., i.e., the vector f is any of the columns of or the eigenvector of M whose corresponding eigenvalue is 1 (Mf=f). This model of evolution is symmetric, i.e., the probability of having an i which mutates to a j is the same as starting with a j which mutates into an i.

The following is a list of amino acid substitution models which use matrices.

Empirical substitution models

In contrast to DNA substitution models, amino acid replacement models have concentrated on the empirical approach. Dayhoff and coworkers developed a model of protein evolution which resulted in the development of a set of widely used replacement matrices (Dayhoff et al. 1978). In the Dayhoff approach, replacement rates are derived from alignments of protein sequences that are at least 85% identical; this constraint ensures that the likelihood of a particular mutation being the result of a set of successive mutations is low. One of the main uses of the Dayhoff matrices has been in database search methods where, for example, the matrices P(0.5), P(1) and P(2.5) (known as the PAM50, PAM100 and PAM250 matrices) are used to assess the significance of proposed matches between target and database sequences. However, the implicit rate matrix has been used for phylogenetic applications.

PAM matrices

In the definition of mutation the matrix M implies certain amount of mutation (measured in PAM units). A 1-PAM mutation matrix describes an amount of evolution which will change, on the average, 1% of the amino acids. In mathematical terms this is expressed as a matrix M such that

 The diagonal elements of M are the probabilities that a given amino acid does not change, so (1-Mii) is the probability of mutating away from i. If we have a probability or frequency vector p, the product Mp gives the probability vector or the expected frequency of p after an evolution equivalent to 1-PAM unit. Or, if we start with amino acid i (a probability vector which contains a 1 in position i and 0s in all others), M*i (the ith column of M) is the corresponding probability vector after one unit of random evolution. Similarly, after k units of evolution (what is called k-PAM evolution) a frequency vector p will be changed into the frequency vector M^k p. Notice that chronological time is not linearly dependent on PAM distance. Evolution rates may be very different for different species and different proteins.

Dayhoff matrices

Dayhoff et al. (1978) presented a method for estimating the matrix M from the observation of 1572 accepted mutations between 34 superfamilies of closely related sequences. Their method was pioneering in the field. A Dayhoff matrix is computed from a 250-PAM mutation matrix, used for the standard dynamic programming method of sequence alignment. The Dayhoff matrix entries are related to M²⁵⁰ by

. JTT matrices

Recently, Jones et al. (1992) and Gonnett et al. (1992) have used much the same methodology as Dayhoff, but with modern databases. The Jones et al. model has been implemented for phylogenetic analyses with some success. Jones et al. (1994) have also calculated an amino acid replacement matrix specifically for membrane spanning segments. This matrix has remarkably different values from the Dayhoff matrices, which are known to be biased toward water-soluble globular proteins.

Other empirical models

Adachi and Hasegawa (1995, 1996) have implemented a general reversible Markov model of amino acid replacement that uses a matrix derived from the inferred replacements in mitochondrial proteins of 20 vertebrate species. The authors show that this model performs better than others when dealing with mitochondrial protein phylogeny.

Blosum (Block substitution matrices)

A different approach was used by Henikoff and Henikoff (1992). They used local, ungapped alignments of distantly related sequences to derive the BLOSUM series of matrices. Matrices of this series are identified by a number after the matrix (e.g. BLOSUM50), which refers to the minimum percentage identity of the blocks of multiple aligned amino acids used to construct the matrix. It is noteworthy that these matrices are directly calculated without extrapolations, and are analogous to transition probability matrices P(T) for different values of T, estimated without reference to any rate matrix Q. The BLOSUM matrices often perform better than PAM matrices for local similarity searches, but have not been widely used in phylogenetics.

Poisson models

A simple, non-empirical model of amino acid replacement was proposed by Nei (1987). This model implements a Poisson distribution, and gives accurate estimates of the number of amino acid replacements when species are closely related.