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exercise-sheet-4.Rmd
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exercise-sheet-4.Rmd
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---
title: "Exercise sheet 4: McCaskill"
---
---------------------------------
# Exercise 1
<!--- --------------------------------- -->
::: {.question data-latex=""}
You are given the following matrix $C$ for sequence $S$ = `AUCCAU`:
```{r, include=knitr::is_html_output(), echo=FALSE, fig.align='center', out.width='35%'}
knitr::include_graphics("assets/figures/exercise-sheet-4/e1-1.svg")
```
It was calculated using the following recursion:
$$
C_{i,j} = \textbf{1}(i,j) \cdot C_{i+1,j-1} \cdot 1 + \sum_{i\leq k < j} C_{i,k} \cdot C_{k+1,j}, \text{ with } \textbf{1}(i,j) = \begin{cases} 1 & \text{if } S_{i} S_{j} \text{ compl.} \\ 0 & \text{else} \end{cases}
$$
We assume a **minimum loop length of 0**.
:::
<!--- --------------------------------- -->
### 1.1
::: {.question data-latex=""}
Calculate the value of $C_{1,6}$. What does it represent?
:::
#### {.tabset}
##### Hide
##### Solution
::: {.answer data-latex=""}
$C_{1,6}$ is the number of possible secondary structures for the sequence $S$.
```{r, include=knitr::is_html_output(), echo=FALSE, fig.align='center', out.width='35%'}
knitr::include_graphics("assets/figures/exercise-sheet-4/e1-1_sol.svg")
```
:::
<!--- --------------------------------- -->
### 1.2
::: {.question data-latex=""}
Does this result make sense? Justify your answer!
:::
#### {.tabset}
##### Hide
##### Solution
::: {.answer data-latex=""}
No, this result does not make sense. The number of possible secondary structures for the sequence $S$ is much lower than 83.
Several structures are counted multiple times.
:::
<!--- --------------------------------- -->
### 1.3
::: {.question data-latex=""}
Recalculate the matrix $C$ using the following recursion:
$$
C_{i,j} = C_{i,j-1} + \sum\limits_{\substack{i \leq k < j \\ S_{k}, S_{j} \text{ compl.}}} C_{i,k-1} \cdot C_{k+1,j-1}
$$
```{r, include=knitr::is_html_output(), echo=FALSE, fig.align='center', out.width='35%'}
knitr::include_graphics("assets/figures/exercise-sheet-4/e1-2.svg")
```
:::
#### {.tabset}
##### Hide
##### Solution
::: {.answer data-latex=""}
```{r, include=knitr::is_html_output(), echo=FALSE, fig.align='center', out.width='35%'}
knitr::include_graphics("assets/figures/exercise-sheet-4/e1-2_sol.svg")
```
:::
<!--- --------------------------------- -->
### 1.4
::: {.question data-latex=""}
Why does the result of this recursion differ from the result of the first recursion?
:::
#### {.tabset}
##### Hide
##### Solution
::: {.answer data-latex=""}
There is no ambiguity, the first part of the recursion only counts cases where $j$ is unpaired and the second part only counts cases where $j$ is paired. Both parts of the recursion are disjoint. Whereas the recursion from exercise 1.1 is ambiguous in the second part of the recursion.
:::