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I need someone to explain to me from the nuts and bolts how algorithms/ maths is used to work out RNA folding. Explain it to me like I am an alien, or a child, an alien-child.

I am looking at this paper - https://eprint.ncl.ac.uk/240069 I understand the science, I am a student, trying to get acquainted with the maths. ((I am a mature student, it's been a long time since I've done any maths.))

All the beginner books on maths, and information theory & mathematical methods in bioinformatics are either too basic or fail to describe adequately what the research paper is doing in a way I can grasp.

I want to understand how ViennaRNA & NUPACK SPECIFICALLY uses maths to describe and work out RNA folding. What do the math symbols mean, what are they representing, and why? why do you write it like that? why cant it be written a different way?

In the paper I mention above, it describes how it calculates single stranded folding as per below? I do not understand. Can someone please clearly explain to me what the purpose of the below is?

(a) What does all this mean? x ∈ {0, 1} |x | = 1 if base i is specified in the design of x to be bound and 0 otherwise. Further, let p x ∈ [0, 1]|x |

^ I know conceptually it is referring to vectors and components, I know that the idea is to represent RNA folding as a graph structure in order to describe mathematically, but I do not know or understand exactly how or what it is describing in terms of the reality of an RNA folding. two vertices connected by an edge represent a base-pair etc etc, that I get, but I cant seem to understand any further than that.

Can someone please provide a clear, answer - everywhere I look seems to leave out some crucial context. Here is the description from the paper:

Single stranded folding. Firstly, we evaluate the ability of a single DNA strand to fold into its specified target structure (as shown in Figure 1 top). Using the secondary structure predictor of the ViennaRNA 2.0 so‰ware suite [13], we calculate the partition function of all secondary structures that x might fold into.1 For any strand x, let |x | denote its length and d x ∈ {0, 1} |x | a vector whose component d x i = 1 if base i is specified in the design of x to be bound and 0 otherwise. Further, let p x ∈ [0, 1]|x | denote a vector whose i-th component p x i denotes the Boltzmann probability (obtained from the partition function) that base i of strand x is paired with another base. The single-stranded folding score Ssf is then defined as the normalized Euclidean distance || · || between d x and p x as Ssf(x) = 1 − 1 |x | ||d x − p x ||. (1) Note that 0 ≤ Ssf ≤ 1 and Ssf(x) = 1 if x folds unambiguously into its target structure.

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