In this talk, we look at combinatorial methods to describe biomolecular processes, namely DNA self-assembly and DNA:RNA interactions. In the first part, we focus on self-assembling graph-like structures using branched junction DNA molecules which are star-shaped molecules that join together through adhesion sites at the end of their arms. We describe a combinatorial representation of these molecules as vertices with labeled half-edges, and consider the problem of optimally building a target graph under different laboratory settings. We show how this question give rises to new graph invariants and how these invariants can be studied using edge-colorings and graph decompositions. In the remainder of the talk, we use formal grammars to describe R-loops, three-stranded structures formed by a DNA:RNA hybrid and a single strand of DNA. A formal grammar is a system to generate words; it consists of a set of symbols, terminals and non-terminals, and a set of production rules. The production rules specify how to rewrite non-terminal symbols to obtain words formed by only terminals. We introduce a "braid grammar" to model R-loops as strings of terminal symbols representing different states of the braided structure and discuss approaches to add probabilities to the production rules, thereby yielding a stochastic grammar and a probabilistic model for R-loop prediction.

Contact Jacob Schach Møller jacob@math.au.dk, for Zoom details.