1. For example, consider the following PWM for a motif with length 4:

We say that this motif can generate sequences of length 4. PWMs typically assume that the distribution of one position is not influenced by the base of another position. Notice that each position is associated with a probability distribution over the nucleotides (they sum to 1 and are nonnegative).

2. We can also model the background distribution of nucleotides( the distribution found across the genome):

Notice how the probabilities for A and T are the same and the probabilities of G and C are the same. This is a consequence of the complementarity DNA which ensures that the overall composition of A and T, G and C is the same overall in the genome.

3. Consider the sequence (S = GCAA.)

- The probability of the motif generating this sequence is [P(S|M) = 0.4 × 0.25 × 0.1 × 1.0 = 0.01. onumber]
- The probability of the background generating this sequence [P (S|B) = 0.4 × 0.4 × 0.1 × 0.1 = 0.0016. onumber]

4. Alone this isn’t particularly interesting. However, given fraction of sequences that are generated by the motif, e.g. P(M) = 0.1, and assuming all other sequences are generated by the background (P(B) = 0.9) we can compute the probability that the motif generated the sequence using Bayes’ Rule:

[egin{align*} P(M|S) &= frac{P(S|M)P(M)}{P(S)} [4pt] &= frac{P(S|M)P(M)}{P(S|B)P(B)+P(S|M)P(M)} [4pt] &= frac{0.01 imes 0.1}{0.0016 imes 0.9 + 0.01 imes 0.1} = 0.40984 end{align*}]