Pierre-Étienne Meunier

Programming Biomolecules That Fold Greedily During Transcription

We introduce and study the computational power of Oritatami, a theoretical model to explore greedy molecular folding, by which a molecule begins to fold before awaiting the end of its production. This model is inspired by a recent experimental work demonstrating the construction of shapes at the nanoscale by folding an RNA molecule during its transcription from an engineered sequence of synthetic DNA. An important challenge of this model, also encountered in experiments, is to get a single sequence to fold into different shapes, depending on the surrounding molecules. Another big challenge is that not all parts of the sequence are meaningful for all possible inputs. Hence, to prevent them from interfering with subsequent operations in the Oritatami folding pathway we must structure the unused portions of the sequence depending on the context in which it folds. Next, we introduce general design techniques to solve these challenges and program molecules. Our main result in this direction is an algorithm that is time linear in the sequence length that finds a rule for folding the sequence deterministically into a prescribed set of shapes, dependent on its local environment. This shows that the corresponding problem is fixed-parameter tractable, although we also prove it NP-complete in the number of possible environments.

Binary Pattern Tile Set Synthesis Is NP-hard

We solve an open problem, stated in 2008, about the feasibility of designing efficient algorithmic self-assembling systems which produce 2-dimensional colored patterns. More precisely, we show that the problem of finding the smallest tile assembly system which will self-assemble an input pattern with 2 colors (i.e., \(2\)-Pats) is NP-hard. One crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof for the four color theorem and the recent important advance on the Erdős discrepancy problem using computer programs. In this paper, these techniques will be brought to a new order of magnitude, computational tasks corresponding to one CPU-year. We massively parallelize our program, and provide a full proof of its correctness. Its source code is freely available online.

Non-cooperative Algorithms in Self-assembly

We show the first non-trivial positive algorithmic results (i.e. programs whose output is larger than their size), in a model of self-assembly that has so far resisted many attempts of formal analysis or programming: the planar non-cooperative variant of Winfrees abstract Tile Assembly Model. This model has been the center of several open problems and conjectures in the last fifteen years, and the first fully general results on its computational power were only proven recently (SODA 2014). These results, as well as ours, exemplify the intricate connections between computation and geometry that can occur in self-assembly. In this model, tiles can stick to an existing assembly as soon as one of their sides matches the existing assembly. This feature contrasts with the general cooperative model, where it can be required that tiles match on \emph{several} of their sides in order to bind. In order to describe our algorithms, we also introduce a generalization of regular expressions called Baggins expressions. Finally, we compare this model to other automata-theoretic models.