# Table 1 Key concepts and results in this paper

Concept/result Description Main location
Ancestry index An ancestry index is assigned to each site. Sharing of an ancestry index among sites indicates the sites’ mutual homology. As a fringe benefit, the indices enable the mutation rates to vary across regions (or sites) beyond the mere dependence on the residue state of the sequence. Section R2 (1st and 2nd paragraphs),
Fig. 2
Operator representation of mutations This enables the intuitively clear and yet mathematically precise description of mutations, especially insertions/deletions, on sequence states. This is a core tool in our ab initio theoretical formulation of the genuine stochastic evolutionary model. Section R2 (3rd paragraph),
Fig. 3
Rate operator An operator version of the rate matrix, which specifies the rates of the instantaneous transitions between the states in our evolutionary model.
In other words, the rate operator describes the instantaneous stochastic effects of single mutations on a given sequence state.
Section R3,
Eqs. (R3.1-R3.9) (full mutational model),
Eqs. ( R3.2 , R3.6 , R3.11 - R3.15 ) (indel model)
Finite-time transition operator An operator version of the finite-time transition matrix, each element of which gives the probability of transition from a state to another after a finite time-lapse. This results from the cumulative effects of the rate operator during a finite time-interval. Section R3,
Eq. ( R3.17 ), Eq. (R3.18)
Defining equations (differential) 1st-order time differential equations (forward and backward) that define our indel evolutionary model. They are operator versions of the standard defining equations of a continuous-time Markov model. Section R3,
Eqs. (R3.19,R3.21) (forward),
Eqs. (R3.20,R3.21) (backward)
Defining equations (integral) Two integral equations (forward and backward) that are equivalent to the aforementioned differential equations defining our indel evolutionary model. They play an essential role when deriving the perturbation expansion of the finite-time transition operator. Section R4,
Eq. ( R4.4 ) (forward),
Eq. ( R4.5 ) (backward)
Perturbation expansion (transition operator) The perturbation expansion of the finite-time transition operator. It was derived in an intuitively clear yet mathematically precise manner, by using the aforementioned defining integral equations. Section R4, Eqs. ( R4.6 , R4.7 )
Perturbation expansion (ab initio PWA probability) The perturbation expansion of the ab initio probability of a given PWA, conditioned on the ancestral sequence state, under a given model setting. Section R4, Eq. (R4.8) or Eq. (R4.9)
Binary equivalence relation An equivalence relation between the products of two indel operators each. The relations play key roles when defining LHS equivalence classes. Section R5, Eqs. ( R5.2a - R5.2d )
Local-history-set (LHS) equivalence class An equivalence class consisting of global indel histories that share all local history components. The classes play an essential role when proving the factorability of a given PWA probability. Section R5,
below Eq. ( R5.4 ),
(e.g., Fig. 5)
Factorability ( ab initio PWA probability) We proved that, under conditions (i) and (ii) (below Eq. (R6.4)), the ab initio probability of a given PWA is factorable into the product of an overall factor and contributions from local PWAs. Section R6, Eqs. ( R6.7 , R6.8 ), 