Question

For an observed sequence IGCIGIGC, three paths (P1, P2 and P3) have been modelled based on the Markov model below, showing the transition between exon (E) and intron (1) regions. Which of the three paths is the most likely under this model? Show your calculations. (3 marks) E (Exon) (Intron) A 0.2 0.5 A 0.3 0.5 0.6 C 0.3 с 0.2 0.3 0.4 G 0.2 T 0.2 T 0.3 • G Do с T G T E G E E T E с F - -- Observed sequence P1 P2 P3 - G E E E - E 1 E E E E E E

Answer 1

We are provided with a state diagram for the different possibilities associated with each nucleotide type i.e. the probability of a nucleotide being an intron or an exon. Additionally, we are provided with the transition probabilties.

There are three different paths given for the sequence TGCTGTGC. We are to determine which of the three paths is most probable.

To do so, we calculate the probability of each of the paths.

Let S(0) be the probability of the first nucleotide being either exon or intron as given in the path. S1 be the first item in the path. S1 N refers to the probability of the given item be at the position given it is an exon or an intron as stated in the path. S1 T refers to the first transition probability as present in the path. The same notation can be extended for S2, S3, and so on.

Taking these probabilities for each of the three paths from the state diagram, we see:

Probability	P1	P2	P3
S(0)	0.5	0.5	0.5
S1 N	0.2	0.2	0.3
S1 T	0.5	0.5	0.6
S2 N	0.3	0.3	0.2
S2 T	0.5	0.5	0.4
S3 N	0.2	0.2	0.3
S3 T	0.6	0.6	0.5
S4 N	0.3	0.3	0.2
S4 T	0.6	0.6	0.5
S5 N	0.2	0.2	0.3
S5 T	0.4	0.6	0.5
S6 N	0.2	0.3	0.2
S6 T	0.5	0.4	0.5
S7 N	0.3	0.3	0.3
S7 T	0.5	0.5	0.5
S8 N	0.3	0.3	0.3

Now the probability of each of the paths can be simply calculated by taking the product of all the probabilities mentioned in the column. Doing so, we get:

$\\P(P1) = 5.832\times 10^{-8}\\\\ P(P2) = 10.4976\times 10^{-8}\\\\ P(P3) = 7.29\times 10^{-8}\\\\$

The path that is more probable is the path that has higher probability. From the three paths given, the highest probability is for path 2. Therfore, the most likely path for the given DNA sequence given the state diagram is P2.

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