Profile HMMs

HMMs can be used for aligning a string versus a given profile, thus helping us to solve the multiple alignment problem.
We define a profile $\mathcal{P}$ of length L, as a set of probabilities, consisting of, for each $b \in \Sigma$ and $1 \leq i \leq L$, the probability e_i(b) of observing the symbol b at the $i^{\text{th}}$ position. In such a case the probability of a string $X=(x_{1},\ldots,x_{L})$ given the profile $\mathcal{P}$ will be:

$$P(X \vert {\mathcal{P}}) = \prod_{i=1}^{L}{e_{i}(x_{i})}$$ (6.39)

We can calculate a likelihood score for the ungapped alignment of X against the profile $\mathcal{P}$:

$$Score(X \vert {\mathcal{P}}) = \sum_{i=1}^{L}{ \log\frac{ e_{i}(x_{i})}{p(x_{i})}}$$ (6.40)

where p(b) is the background frequency of occurrences of the symbol b.
This leads to a definition of the following HMM, with the match states $M_{1},\ldots,M_{L}$ which correspond to matches with the profile. All those states are sequentially linked (i.e., each match state M_j is linked to its successor M_j+1) as shown in figure 6.2. The emission probability of the symbol b from the state M_j is of course e_j(b).

  

Figure 6.2: Match states in a profile HMM

To allow insertions, we will add the insertion states $I_{0},\ldots,I_{L}$ to the model. We shall assume that:

$$\forall_{b \in \Sigma} \quad e_{I_{j}}(b) = p(b)$$

Each insertion state I_j has an link entering from the corresponding match state M_j, a leaving link towards the next match state M_j+1 and also has a self-loop (see figure 6.3). Assigning the appropriate probabilities for those transitions corresponds to the application of affine gap penalties, since the overall contribution of a gap of length h to the logarithmic likelihood score is:

$$\underset{\text{gap creation}} {\underbrace{\log(a_{M_{j}I_... ... extension}} {\underbrace{(h-1)\cdot\log(a_{I_{j}I_{j}})}}$$

  

Figure 6.3: Profile HMM with an insertion state

To allow deletions as well, we add the deletion states $D_{1},\ldots,D_{L}$. These states cannot emit any symbol and are therefore called silent (Note that the begin/end states are silent as well). The deletion states are sequentially linked, in a similar manner to the match states and they are also interleaved with the match states (see figure 6.4).

  

Figure 6.4: Profile HMM with deletion states

To model both insertions and deletions, we have to add a link from D_j to I_j and a link from I_j to D_j+1.
The full HMM for modeling the profile $\mathcal{P}$ of length L is comprised of L layers, each layer has three states M_j, I_j and D_j. To complete the model, we add begin and end states, connected to the layers as shown in figure 6.5. This model is due to Haussler et al [5].

  

Figure 6.5: Profile HMM for global alignment