Aligning Sequences to a Profile HMM

To align the string $X=(x_{1},\ldots,x_{m})$ against a profile $\mathcal{P}$ of length L, we will use a variant of the Viterbi algorithm. For each $1 \leq j \leq L$ and $1 \leq i \leq m$ we use the following definitions:

• Let v^M_j(i) be the logarithmic likelihood score of the best path for matching
  $X=(x_{1},\ldots,x_{i})$ to the profile HMM $\mathcal{P}$, ending with x_i emitted by the state M_j.
• Let v^I_j(i) be the logarithmic likelihood score of the best path for matching
  $X=(x_{1},\ldots,x_{i})$ to the profile HMM $\mathcal{P}$, ending with x_i emitted by the state I_j.
• Let v^D_j(i) be the logarithmic likelihood score of the best path for matching
  $X=(x_{1},\ldots,x_{i})$ to the profile HMM $\mathcal{P}$, ending with the state D_j (without emitting any symbol).

The initial value of the special begin state is:

v_begin(0) = 0 (6.41)

To calculate the values of v^M_j(i), v^I_j(i) and v^D_j(i) we use the same technique as in the Viterbi algorithm. There are however two major differences:

• Each state in the model has at most three entering links (see figure 6.5).
• The deletion states are silent - they cannot emit any symbol.

The three predecessors of the match state M_j are the three states of the previous layer, j-1:

$$v^{M}_{j}(i) = \log\frac{e_{M_{j}}(x_{i})}{p(x_{i})} + \max ... ... \\ v^{D}_{j-1}(i-1)+\log(a_{D_{j-1},M_{j}}) \end{cases}$$ (6.42)

The three predecessors of the insertion state I_j are the three states of the same layer, j:

$$v^{I}_{j}(i) = \log\frac{e_{I_{j}}(x_{i})}{p(x_{i})} + \max ... ...j}}) \\ v^{D}_{j}(i-1)+\log(a_{D_{j},I_{j}}) \end{cases}$$ (6.43)

The three predecessors of the deletion state D_j are the three states of the layer j-1. Since D_j is a silent state, we should not consider the emission likelihood score for x_i in this case:

$$v^{D}_{j}(i) = \max \begin{cases} v^{M}_{j-1}(i)+\log(a_{... ...}) \\ v^{D}_{j-1}(i)+\log(a_{D_{j-1},D_{j}}) \end{cases}$$ (6.44)

We conclude by calculating the optimal score:

$$Score(X,\Pi^{\ast}) = \max \begin{cases} v^{M}_{L}(m)+\lo... ...L},end}) \\ v^{D}_{L}(m)+\log(a_{D_{L},end}) \end{cases}$$ (6.45)

Complexity: We have to calculate $O(L \cdot m)$ values, while calculating each value takes O(1) operations (since we only need to consider the scores at most three predecessors). We therefore need $O(L \cdot m)$ time and $O(L \cdot m)$ space.

  

Figure 6.6: Profile HMM for local alignment

We can use a similar approach for the problem of local alignment of a sequence versus a profile HMM. This is achieved by adding four additional states (the lightly shaded states in figure 6.6) corresponding to the alignment of a sub-string of X to a part of the profile.