8.5: Using HMMs to align sequences with affine gap penalties
- Page ID
- 40964
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We can use HMM to align sequences with affine gap penalties. Recall that affine gap penalties penalizes more to open/ start the gap than to extend it, thus the penalty of a gap of length g is r(g) = -d -(g-1)*e, where d is the penalty to open the gap and e is the penalty to extend an already open gap.
We will look into aligning two sequences with the affine gap penalty. We are given two sequences are X and Y, the scoring matrix S (S(xi,yj) = score of matching xi with yj), gap opening penalty of d and gap extension penalty of e. We can map this problem into an HMM problem by using the following states, transition probabilities and emission probabilities.
States:
There are three states involves: M (matching xi with yj), X (aligning xi with a gap), Y (aligning yj with a gap). Also, alongside each transition, there’s an update of the i,j indices. Whenever we are in state M, (i,j) = (i,j) + (1,1). In state X, (i,j) = (i,j) + (1,0). In state Y, (i,j) = (i,j) + (0,1).
Transition probabilities:
There are 7 transition probabilities to consider as shown in figure 6. P(next State = M | current = M) = S(xi,yj)
P(next State = X | current = M) = d
P(next State = Y | current = M ) = d
P(next State = X | current = X) = e
P(next State = M | current = X ) = S(xi,yj)
P(next State = Y | current = Y) = e
P(next State = M | current = Y) = S(xi,yj)
We can also save the transition probabilities in a transition matrix A = [aij], where aij = P(next State = j | current = i) and \(\sum_{j}\)jAij = 1
Emission probabilities:
The emission probabilities are:
From state M: pxiyi = p(xi aligned to yj)
From state X: qxi = p(xi aligned to gap)
From state Y: qyi= p(yjaligned to gap)
Example:
X = ’VLSPADK’
Y = ’HLAESK’
The alignment generated by the model is: MMXXMYM
Which corresponds to:
X = ’VLSPAD K’
Y = ’HL_ _AESK’
Did You Know?
For classification purposes, Posterior decoding ’path’ is more informative than Viterbi path as it is a more refined measure of which hidden states generated x. However, it may give an invalid sequence of states, for example when not all j $->$ k transitions may be possible, it might have state(i) = j and state(i+1) =k