Abstract:
In this work, we use methods of knot theory to describe and analyze structural elements of a RNA secondary structure by construction of a new generalisation of the classical Kauffman bracket polynomial, which factorisation characterises these structural elements. To this end, we develop a mathematical model of RNA endowed with a topology invariant ($RKB$ polynomial) that allows to determine a type, number and characteristics of standard structural elements that form the RNA secondary structure. In order to define the $RKB$ polynomial, we introduce a new skein relation to smooth hydrogen bonds and a new technique to color nucleotides, and use the known skein relation of the Kauffman bracket polynomial. Note that the proposed technique to color nucleotides allows to take into account the positional relationship of structural elements, which can be used to investigate properties of RNA. Invariance of the $RKB$ polynomial is shown. Computation of the $RKB$ polynomial by the given Dot-Bracket notation is implemented as a small Mathematica program. Using $RKB|$ polynomials calculated by our program, we analyze some RNA secondary structures presented in the bpRNA-1m database. The obtained results agree with the real data.
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