equivalent to the statement that one is setting the mutation rate to 1) and reducing the number of free parameters to eight. Specifically, there are five free exchangeability parameters ( through , which are expressed relative to the fixed in this example) and three equilibrium base frequency parameters (as described above, only three values need to be specified because must sum to 1). The alternative notation also makes it easier to understand the sub-models of the GTR model, which simply correspond to cases where exchangeability and/or equilibrium base frequency parameters are constrained to take on equal values. A number of specific sub-models have been named, largely based on their original publications: There are 203 possible ways that the exchangeability parameters can be restricted to form sub-models of GTR, ranging from the JC69 and F81 models (where all exchangeability parameters are equal) to the SYM model and the full GTR (or REV) model (where all exchangeability parameters are free). The equilibrium base frequencies are typically treated in two different ways: 1) all values are constrained to be equal (i.e., ); or 2) all values are treated as free parameters. Although the equilibrium base frequencies can be constrained in other ways most constraints that link some but not all values are unrealistic from a biological standpoint. The possible exception is enforcing strand symmetry (i.e., constraining and but allowing ). The alternative notation also makes it straightforward to see how the GTR model can be applied to biological alphabets with a larger state-space (e.g., amino acids or codons). It is possible to write a set of equilibrium state frequencies as , , ... and a set of exchangeability parameters () for any alphabet of character states. These values can the be used to populate the matrix by setting the off-diagonal elements as shown above (the general notation would be ), setting the diagonal elements to the negative sum of the off-diagonal elements on the same row, and normalizing. Obviously, for amino acids and for codons (assuming the standard genetic code). However, the generality of this notation is beneficial because one can use reduced alphabets for amino acids. For example, one can use and encode amino acids by recoding the amino acids using the six categories proposed by Margaret Dayhoff. Reduced amino acid alphabets are viewed as a way to reduce the impact of compositional variation and saturation. Mechanistic vs. empirical models A main