Substitution Model - GTR: Generalised Time Reversible

GTR: Generalised Time Reversible

GTR is the most general neutral, independent, finite-sites, time-reversible model possible. It was first described in a general form by Simon Tavaré in 1986.

The GTR parameters for nucleotides consist of an equilibrium base frequency vector, giving the frequency at which each base occurs at each site, and the rate matrix

Because the model must be time reversible and must approach the equilibrium nucleotide (base) frequencies at long times, each rate below the diagonal equals the reciprocal rate above the diagonal multiplied by the equlibrium ratio of the two bases. As such, the nucleotide GTR requires 6 substitution rate parameters and 4 equilibrium base frequency parameters. Since the 4 frequency parameters must sum to 1, there are only 3 free frequency parameters. The total of 9 free parameters is often further reduced to 8 parameters plus, the overall number of substitutions per unit time. When measuring time in substitutions (=1) only 8 free parameters remain.

In general, to compute the number of parameters, you count the number of entries above the diagonal in the matrix, i.e. for n trait values per site, and then add n-1 for the equilibrium frequencies, and subtract 1 because is fixed. You get

For example, for an amino acid sequence (there are 20 "standard" amino acids that make up proteins), you would find there are 208 parameters. However, when studying coding regions of the genome, it is more common to work with a codon substitution model (a codon is three bases and codes for one amino acid in a protein). There are codons, resulting in 2078 free parameters, but when the rates for transitions between codons which differ by more than one base are assumed to be zero, then there are only parameters.