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In living organisms, Ribonucleic acid (RNA) is an important matter that

has essential roles. RNA molecules play an active role in different processes

such as encoding and decoding. Subsequently, RNA molecules fold themselves to fit inside cells and perform specific functions in order to achieve

these processes. Accordingly, finding the secondary structure by using primary structure helps to determine the function of RNA. Further, RNA is

said to be highly important for diagnostic and therapeutic design. Hence,

it can be said that the findings of RNA studies could provide a better understanding to develop treatments for RNA-related diseases 1

. Within this

perspective, RNA folding studies aim to clarify the relationship among sequence, tridimensional structure, and biological function 2

. In particular, the

purpose of RNA folding problem is to predict the secondary structure of an

RNA molecule while only its nucleotide sequence is known.

To understand the concept of RNA folding problem, we will denote s as

a string of n characters consisting of nucleotides: Adenine (A), Guanine (G),

Uracil (U) and Cytosine (C). For instance,

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

1January 15, 2021 — B. A. M. (2023, February 22). New videos show RNA as it’s

never been seen. Northwestern Now.

2Shaw, E., St-Pierre, P., McCluskey, K., Lafontaine, D. A., & Penedo, J. C. (2014).

Using SM-fret and denaturants to reveal folding landscapes. Methods in Enzymology,

313–341.

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Figure 1: Arc representation of pairings

Table 1: Energy levels of stacking pairs

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.

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We try to find an integer programming (IP) model that solves the following

questions while considering the above conditions.

Figure 3: A line representation of a nested pairing

(a) Given the nucleotide sequence s (each group will have different sequence

s; provided in data.xlsx) of an RNA molecule, find a nested pairing

that achieves the maximum number of pairs.

Table 2: Energy level of matched pairs

(b) Now consider that we try to find a nested pairing that gives the minimum total free energy associated with pairs in the sequence (please

use Table 2 for energy levels of matched pairs in your model). Make

necessary changes in the IP model and compare your results with part

(a). What are the differences in terms of solution time and the number

of pairs?

(c) Now consider that there must be at least 7 nucleotides (rather than 4)

between pairing ones while keeping the total free energy at minimum

as in part (b). What are the differences in terms of solution time and

the number of pairs?

(d) A matched pair (i, j) in a nested pairing is defined as a stacked pair

if either (i + 1, j − 1) or (i − 1, j + 1) is also a matched pair in the

nested pairing. In other words, there is a stacked pair if and only

if both (i, j) and (i + 1, j − 1) are in the nested pairing. It is also

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known that a stack with k matched pairs adds more to stability than

k individual matched pairs, which do not belong to any stacks. The

stack pair correspond to an energy level, which is shown in Table 1. For

instance, if the structure of the sequence s is s = ACGAAAACGU,

then the stacked pairs’ energy level will be (-2.1) +(– 2.2) = – 4.3,

since we have A-U, C-G, and G-C as nested pairs. That is to say, there

has to be at least 2 stacked pairs to lead the energy level. If there is

one pair, it is not included in the energy level calculation. We now

try to incorporate a count of the number of stacked pairings to obtain

more stable structure. Include the given information of stacked pairs

and make necessary changes in the IP model in part (b) that finds the

minimum total free energy correspond to stacked pairs.

Hint: Define a new decision variable to incorporate the stacked pairs

into the model.

(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)).

Figure 4: Line representation of pseudo-knots

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

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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.

Instructions

Please read the following instructions carefully:

(1) Formulate the models in each part separately.

(2) Solve the models using Gurobi or any other solver (CPLEX, Xpress,

GAMS etc.)

(3) Prepare a written document including your precise mathematical models. Explain your objective values, constraints, decision variables and

parameters explicitly.

(4) Submit your report (including members full names and ID’s) as well as

your Gurobi model (or your choice of solver) and all of your codes as

a .zip file. The name of the .zip file should be your group number (Do

not add names, ID’s etc. to the file name).

(5) There will be a presentation session where you will be asked questions

about your models and the project.

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