two states and it has even been implemented as the "JC2" model in the popular IQ-TREE software package (using this model in IQ-TREE requires coding the data as 0 and 1 rather than R and Y; the popular PAUP* software package can interpret a data matrix comprising only R and Y as data to be analyzed using the CFN model). It is also straightforward to analyze binary data using the phylogenetic Hadamard transform. The alternative two-state model allows the equilibrium frequency parameters of R and Y (or 0 and 1) to take on values other than 0.5 by adding single free parameter; this model is variously called CFu or GTR2 (in IQ-TREE). Amino acid substitution models For many analyses, particularly for longer evolutionary distances, the evolution is modeled on the amino acid level. Since not all DNA substitution also alter the encoded amino acid, information is lost when looking at amino acids instead of nucleotide bases. However, several advantages speak in favor of using the amino acid information: DNA is much more inclined to show compositional bias than amino acids, not all positions in the DNA evolve at the same speed (non-synonymous mutations are less likely to become fixed in the population than synonymous ones), but probably most important, because of those fast evolving positions and the limited alphabet size (only four possible states), the DNA suffers from more back substitutions, making it difficult to accurately estimate evolutionary longer distances. Unlike the DNA models, amino acid models traditionally are empirical models. They were pioneered in the 1960s and 1970s by Dayhoff and co-workers by estimating replacement rates from protein alignments with at least 85% identity (originally with very limited data and ultimately culminating in the Dayhoff PAM model of 1978). This minimized the chances of observing multiple substitutions at a site. From the estimated rate matrix, a series of replacement probability matrices were derived, known under names such as PAM250. Log-odds matrices based on the Dayhoff PAM model were commonly used to assess the significance of homology search results, although the BLOSUM matrices have superseded the PAM log-odds matrices in this context because the BLOSUM matrices appear to be more sensitive across a variety of evolutionary distances, unlike the PAM log-odds matrices. The Dayhoff PAM matrix was the source of the exchangeability parameters used in one of the first maximum-likelihood analyses of phylogeny that used protein data and the PAM