exercise-sheet-9.Rmd

---
title: "Exercise sheet 9: RNA-RNA interaction"
---


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# Exercise 1 - Double-site accessibility-based RRI prediction

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::: {.question data-latex=""}

So far, we discussed only how to predict via accessibility-based approaches a single RRI site where the respective subsequences are free of base pairing.

Consider now the following steps of a meta approach that predicts and combines two independent sites.

1. RRI prediction of first (most stable) RRI $B_1$
2. constraint RRI prediction of second RRI $B_2$, where the subsequences involved in $B_1$ are constraint to be unpaired
3. combine $B_1$ and $B_2$ into one prediction

Note, step $1$ and $2$ are using different ED value, since step $2$ is a constraint prediction also requiring a constraint ED-value computation.
In the following, we refer to them via $ED(B_1)$ and $ED(B_2 \mid B_1)$. Similarly, we denote respective energy $E$ and hybridization energy $E_{hyb}$ terms.

In step $3$, we are combining both sets of inter-molecular base pairs into one RRI, while keeping track of the respective blocks $B_1$ and $B_2$.
Given that, we propose that the overall energy of RRI can be approximated as follows:

$$
\begin{equation}
E(B_1 \land B_2) = E(B_1) + E(B_2\mid B_1)
\end{equation}
$$

Prove the equation above by converting one side of the equation into the other side.

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##### Solution

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$$
\begin{aligned}
	E(B_1) + E(B_2\mid B_1)
	&= E_{hyb}(B_1) + ED(B_1)
	 + E_{hyb}( B_2 | B_1 )+ ED(B_2 | B_1) && \text{(i)} \\
	&= E_{hyb}(B_1) - RTlog(\mathcal{P}^u(B_1))
	 + E_{hyb}(B_2 | B_1) - RTlog(\mathcal{P}^u(B_2 | B_1))  && \text{(ii)}\\
	&= E_{hyb}(B_1)  + E_{hyb}(B_2 | B_1)
	 - RTlog(\mathcal{P}^u(B_1) * \mathcal{P}^u(B_2 | B_1))  && \text{(iii)}\\
	 %\label{eqRight}
	 &= E_{hyb}(B_1)+E_{hyb}(B_2)-RTlog(\mathcal{P}^u(B_1 \land B_2))  && \text{(iv)}\\
	 &= E_{hyb}(B_1 \land B_2) + ED(B_1 \land B_2)  && \text{(v)}\\
	 &= E(B_1 \land B_2)  && \\
\end{aligned}
$$

To show that the equality holds, we used the following tricks to reformulate the left side of the equation into the right side:

i. The mfe ($E$) can be calculated using $E = E_{hyb} + ED$
ii. We know that we can compute the ED value for sub-sequence $a \cdots b$ by the unpaired probability $ED= -RTlog(\mathcal{P}^u(ab))$
iii. Due to independence of hybridization energies of sites we can use $E_{hyb}(B_2 | B_1)=E_{hyb}(B_2)$, and we can use: $log(a) + log(b) = log(a*b)$
iv. We can use the conditional probability: $P(A \land B) = P(A) * P(B |A)$
v. $ED= -RTlog(\mathcal{P}^u(ab))$

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# Exercise 2 - Problems and drawbacks

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Is the predicted double-site RRI based on the previous question the optimal prediction?
What problems, side effects and special situations are not covered?

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##### Solution

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i. more than two sites
ii. ED of $B_1$ can change when conditional $B_2$ is formed, thus we have to iterate constraint predictions until ED is not changing any more
iii. combination of sites might be sterically not feasible in reality

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