More generally a non-null CUT may be incorporated as follows. Suppose we have the codon GGU. Mutations in the 3rd position are synonymous (all code for gly) and their relative frequencies are controlled by the nucleotide mutation matrix . However we also wish to preserve codon bias as a sequence mutates. Since we are always working from an initial known amino acid, we must use relative (instead of absolute) codon frequencies but each frequency must be multiplied by the degeneracy of the corresponding amino acid, otherwise, for example, (leu) will be a factor of six less probable than (met) simply because there are six codons for leu but only one for met. In addition, codon usage statistics implicitly include any nucleotide bias and, conversely, any nucleotide bias described by will automatically lead to a codon bias. Hence the nucleotide equilibrium frequencies must be factored out by dividing each codon usage value by where are the th, th, th nucleotides in the codon. See Firth & Brown (2005) for scripts to produce appropriate CUTs from standard absolute or relative frequency CUTs.