Can you just give the models for part E and part F (objective function and constraints) ? (Thanks in advance.)

We will denote s as a string of n characters consisting of nucleotides: Adenine (A), Guanine (G), Uracil (U) and Cytosine (C).

s = ACGUCCAUGCAG.

Moreover, it is averted that the stability of the RNA is measured by the number of bonds. Consequently, the most stable structure is the one with the maximum binding strength; i.e., the largest number of bonds. Having all the above information, there are also some important conditions that should be taken into consideration while dealing with the RNA folding problem.

Nucleotide A must be paired with only U in which C must be paired with only G (or vice versa).

There are 2 hydrogen bonds between A and U, 3 hydrogen bonds between G and C. Therefore, binding strength differs from pair to pair.

Each nucleotide must be paired with at most 1 nucleotide.

The pairings are not allowed to cross each other. For example, let i < i < j < j , then (i, j) and (i , j ) cannot be paired at the same time. Figure 2: Arc representation of cross pairings (pseudoknots)

There is a distance limitation that close nucleotides cannot be paired; i.e., the nucleotide cannot pair with any nucleotide that is less than 4 positions away from it on the sequence s.

We try to find an integer programming (IP) model that solves the following questions while considering the above conditions.

E) Now consider pseudo-knots, which occur if and only if there are two stacks, S(i) and S(i ), starting at positions i and i respectively, such that every pair in S(i) crosses every pair in S(i ). Assume that RNA folds both upwards and downwards. It means that when we consider the line representation, it is possible to have pairings both above and the below of the line (see Figure 4 (right)). However, note that cross matching is not still allowed, but pseudoknots are incorporated in the IP model. Considering the information given/obtained in part (d), find a model that gives the minimum total free energy.

F) Until now, we try to predict the secondary structure of the RNA by IP models. Now, consider a simple dynamic programming (DP) algorithm to minimize the total free energy using the information given/obtained in part (b). To construct the algorithm, begin with creating a pair with two specific points, let say (i, j). Consider the cases until two pairs form the stacked pairs. If two different pairs (stacked pairs) are matched, they lead to the energy level, which we try to keep it at minimum. Also note that there is a case that the structure of RNA can bifurcates in such a way that the sum of energies of two substructures is minimized.

	A-U	C-G	G-C	G-U	U-G	U-A
A-U	-1.1	-2.1	-2.2	-1.4	-0.9	-0.6
C-G	-2.1	-2.4	-3.3	-2.1	-2.1	-1.4
G-C	-2.2	-3.3	-3.4	-2.5	-2.4	-1.5
G-U	-1.4	-2.1	-2.5	-1.3	-1.3	-0.5
U-G	-0.9	-2.1	-2.4	-1.3	-1.3	-1.0
U-A	-0.6	-1.4	-1.5	-0.5	-1.0	-0.3

Table 1: energy levels of stacking pairs

	U (A)	G (C)
A (U)	-1.33	-
C (G)	-	-1.45

Table 2: Energy levels for matched pairs