In the lecture on motif discovery, we developed a generative model that consists of a set sequences each of which contains one and only motif. This is called the OOPS model (One Occurrence Per Sequence). Figure 1 replicates a slide from the lecture. z": 1 mmol'startsa pooltlon [In egg] HHII p(,,_z,u| M) = p(l|z.,, M) P(Zii) (Eq. 1) Figure I: The OOPS modeifor Motif'Dis'covery with MEME We would like to extend the model of Figure 1 to allow for Zero or One Occurrences Per Sequences the ZOOPs model. This requires that we modify the model above to include the probability that a sequence contains a motif. We will do this with a new binary random variable Qi for each of the I sequences, where Qi=1 if the i'ih sequence has a motif, and Qi=0 if it does not. And for every i, P(Qi=1)= 7. This new variable is summarized in Figure 2. 01:1 ' oz=1 ' o,=o ' o,=1 _' Q1= ' P(Oi=1) = 7 Figure 2: He new binary random variable, Q, that we need to add to the mode! a. Update the generative model probability in Equation 1 of Figure 1 to include the new random variable Q. b. Recall that to perform EM, we need to calculate the posterior probability of the motif positions given the data and the model. For the OOPS model, this is given by: P(Z, =1|S,,M) = P(S, [ Z, =1, M) P(Z; = 1) [P(S, 1Z, = 1, M) P(Z, = 1) Update this equation for the posterior probability to be correct for the ZOOPS model. c. Now imagine we want to learn the extra parameter y. Imagine you have fully labeled data - in other words, you know which sequences have motifs and where - how would you calculate y? Justify your answer. d. Now imagine that you lack labels - you do not know where motifs are located or if a sequence has a motif. How would you learn the parameter y in this case. Be as specific and detailed as possible. (Hint, how do we learn the motif model in this case?)