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Table 1 Evolutionary models compatible with pair and profile HMMs

From: Parameterizing sequence alignment with an explicit evolutionary model

Microscopic model

Macroscopic model

Evolutionary model

Total # free parameters

Rates

Geometric parameters

# States minimal pair HMM

Other properties

Single-residue models(Fig. 2, Additional file 1: Figure S1 and S2)

aali

6

\(\lambda,\mu,\mu _{A}^{\{\text {\textit {M,D,I}}\}}\)

p

3

affine ancestral residues; linear inserts

li

4

λ,μ,μ A

p

1

linear; particular case of aali

lr

2

λ,μ A ,(μ=λ+μ A )

(p lr=λ/μ A )

1

reversible; particular case of li

tkf91

2

λ,μ

(p tkf=λ/μ)

2

quasi linear; [5]

Fragment models(Figs. 3 and 4)

afg

9

\(\lambda,\mu,\mu _{A}^{\{\text {\textit {M,D,I}}\}}\)

r M ,r D ,r I ,p

3

fragment-derivative of aali

afr

4

λ,μ A ,(μ=λ+μ A )

r M ,r D =r I ,(p lr)

3

reversible; compatible with Smith-Waterman

tkf92

3

λ,μ

r, (p tkf92)

3

fragment-derivative of tkf91; [6]

Compatible with profile HMMs(Fig. 5)

aif

7

\(\lambda,\mu,\mu _{A}^{\{\text {\textit {M,D,I}}\}}\)

r I ,p

3

inserts-only fragment-derivative of aali

aga

7

\(\lambda,\mu,\mu _{A}^{\{\text {\textit {M,D,I}}\}}\)

s I ,p

3

time-independent geometric inserts

  1. From a microscopic perspective we have three types of models: (1) single-residue models in which residues are inserted and deleted instantaneously one at a time; (2) fragment models that can insert/delete/replace several residues at the time, but where residues created simultaneously act as an indivisible unit (thus the name fragments); (3) The aga model where in one single event inserts can appear or disappear, but they cannot grow or shrink. The aali model (and its particular cases the li and lr models) as well as the tkf91 model belong to the first category of single-residue models. Fragment models can be built starting from any of the single-residue models. In the aga model, the distribution of inserts length is geometric but it does not change with time. From a macroscopic perspective, the li model (and its particular case the reversible lr model) fit into a one-state linear HMM, but the similar model tkf91 requires at least a two-state HMM. The aali model requires a three-state HMM because it is affine with respect to the fate of ancestral residues. The number of states of the minimal HMM does not include the customary begin (B) and end (E) states; we assume in all cases that \(\mu _{A}^{\textsc {b}}=\mu _{A}^{\textsc {m}}\). All fragment models fit into a standard three-state HMM. Parameters that are not independent are given in parentheses. The distribution of ancestral sequences for the fragment model tkf92 is approximately a geometric distribution: the expression of p tkf92 in terms of the free parameters of the model can be found in [6]. The affine fragment reversible (afr) model is a particular case of the afg model obtained as a fragment derivative of the lr model such that in order to preserve reversibility deleted and inserted fragments are drawn from the same geometric distribution (r D =r I ). In the afr model insertions and deletions have identical treatment. There are two models compatible with profile HMMs of Krogh’s form [32]: the aif model and the aga model. The aif model is a particular case of the afg fragment model (with r M =r D =0). The aga model assumes the simplification that inserts are geometrically distributed with a time-independent (but position specific) Bernoulli parameter. All evolutionary models have been implemented in the alignment program e2msa