  The structure of an RNA molecule plays a significant role in its biological
function. Predicting structure given a one dimensional sequence of RNA
nucleotide bases is a difficult and important problem. Many computer programs
(known as in silico) are available for predicting 2-dimensional (secondary)
structures however 3-dimensional (tertiary) structure prediction is much more
difficult mainly due to the far greater number of feasible solutions and fewer
experimental data on the thermodynamic energies of 3D structures. It is also
challenging to verify the most likely three dimensional structure even with the
availability of sophisticated x-ray crystallography and nuclear magnetic
resonance imaging technologies. In this paper we develop three dimensional RNA
folding predictions by adding penalty and reward parameters to a previous two
dimensional approach based on Quadratic Unconstrained Binary Optimization
(QUBO) models. These parameters provide flexibility in the amount of three
dimensional folding allowed. We address the problem of multiple near-optimal
structures via a new weighted similarity structure measure and illustrate
folding pathways via progressively improving local optimal solutions. The
problems are solved via a new commercial QUBO solver AlphaQUBO (Meta-Analytics,
2020) that solves problems having hundreds of thousands of binary variables.
