This package also relies on the openssl package Ooms for sequence and alignment comparisons using the MD5 hash algorithm.

## Bioinformatics: Sequence Alignment and Markov Models

These are predominantly created for the purposes of succinct console-printing. HMMs and PHMMs are explained in more detail throughout the following sections using aphid package functions to demonstrate their utility. The examples are borrowed from Durbin et al , to which users are encouraged to refer for a more in-depth explanation on the theory and application of these models.

Book chapter numbers are provided wherever possible for ease of reference. A hidden Markov model is a probabilistic data-generating mechanism for a sequence or set of sequences.

It is depicted by a network of states each emitting symbols from a finite alphabet according to a set of emission probabilities , whose values are specific to each state. The states are traversed by an interconnecting set of transition probabilities , that include the probability of remaining in any given state and those of transitioning to each of the other connected states.

An example of a simple HMM is given in Durbin et al chapter 3. An imaginary casino has two dice, one fair and one weighted. If the dealer has the fair dice, he may secretly switch to the loaded dice with a probability of 0.

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Alternatively, if he has the loaded dice, he will switch back to the fair dice with a probability of 0. This example can be represented by a simple two-state hidden Markov model.

Figure 1: A simple hidden Markov model for the dishonest casino example. The plot. HMM method depicts the transition probabilities as weighted lines, and emission probabilities as horizontal grey bars. For a sequence of observed rolls, we can establish the most likely sequence of hidden states including when the dice-switching most likely occurred using the Viterbi algorithm.

In the example given in Durbin et al chapter 3. In the following code, the Viterbi algorithm is used to find the most likely sequence of hidden states given the model. Figure 2: Posterior state probabilities for the dice rolls. The line shows the posterior probability that the dice was fair at each roll, while the grey rectangles show the actual periods for which the loaded dice was being used.

See Durbin et al chapter 3. Figure 3: A simple HMM derived from the sequence of dice rolls. As in Fig. This appears to be fairly close to the actual model, despite the fact that the training data consisted of just a single sequence. One would typically derive an HMM from a list of many such sequences hence why the input argument is a list and not a vector but this example is simplified for clarity.

### 1. Introduction

A profile hidden Markov model is an extension of a standard HMM, where the emission and transition probabilities are position specific. That is, they can change at each point along the sequence. These models typically have many more parameters than their simpler HMM counterparts, but can be very powerful for sequence analysis.

The precursor to a profile HMM is normally a multiple sequence alignment.

## Hidden Markov Models and their Applications in Biological Sequence Analysis

Figure 4 shows the three state types listed above as circles diamonds and rectangles, respectively. The states are linked by transition probabilities shown as weighted lines in the graph. Consider this small partial alignment of amino acid sequences from Durbin et al chapter 5. When tabulating the frequencies it is also prudent to add pseudo-counts, since the absence of a particular transition or emission type does not preclude the possibility of it occurring in another unobserved sequence.

Pseudo-counts can be Laplacean adds one of each emission and transition type , background adjusts the Laplacean pseudo-counts to reflect the background frequencies derived from the entire alignment , or user defined, which can include more complex pseudo-count schemes such as Dirichlet mixtures Durbin et al.

Figure 4: Profile HMM derived from a partial globin sequence alignment.

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Match states are shown as rectangles, insert states as diamonds, and delete states as circles. Grey horizontal bars represent the emission probabilities for each residue in the alphabet in this case the amino acid alphabet at each position in the model. Numbers in the delete states are simply model module numbers, while those in the insert states are the probabilities of remaining in the current insert state at the next emission cycle.

Lines are weighted and directed where necessary to reflect the transition probabilities between states. Note that there are only 8 internal modules excluding the begin and end states , while the alignment had 10 columns. The derivePHMM function decided using the maximum a posteriori algorithm that there was not enough residue information in columns 4 and 5 of the alignment to warrant assigning them internal modules in the model.

We can show this by calculating the optimal path of that sequence through the model, again using the Viterbi algorithm:.

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The path can be expressed more intuitively as characters instead of indices as follows:. Sequences do not need to be aligned to produce a profile HMM. Create an account now.

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## Hidden Markov Model

The username and password you entered did not match any accounts in our file. Please try again. Sign In. It provides in-depth coverage of a wide range of autoimmune disorders and detailed analyses of suffix trees, plus late-breaking advances regarding biochips and genomes. Featuring helpful gene-finding algorithms, Bioinformatics offers key information on sequence alignment, HMMs, HMM applications, protein secondary structure, microarray techniques, and drug discovery and development.