Fig. 3

Operator representation of mutations. a A substitution operator, \( {\widehat{M}}_S\left(5,\kern0.5em T\mapsto C\right) \). The residues before and after the substitution are in boldface in blue and red, respectively. b An insertion operator, \( {\widehat{M}}_I\left(6,\kern0.5em 3\right) \), and a fill-in operator, \( \widehat{F}\left(6,\kern0.5em \left[T,A,C\right]\right) \). The inserted sites are shaded in cyan. (Note that “A” at the top of the rightmost inserted column means the ancestry index of 10, not the residue state of A). c A deletion operator, \( {\widehat{M}}_D\left(2,\kern0.5em 4\right) \). The sites to be deleted are shaded in magenta. In this figure, the extended sequence states were used for illustration. The bra-vector below each array denotes the state. The extended state, \( \breve{s} \), is identical to that in Fig. 2b. Each vertical arrow indicates the action of the mutation operator beside it. Note that the first arguments of all operators and the second argument of the deletion operator specify positions along the sequence, and not ancestries (specified at the top of the sites)