Topological sorting and self-assembly of knotted molecules: models and simulations

Knots are ubiquitous objects and decorative elements that have been studied since antiquity. During the centuries knots have become important not only for their mysterious and elegant aspects, but also for their practical relevance. Knots in ropes, for example, have always been useful for different practical applications, from climbing to sailing, from fishing to medicine. Chains that are sufficiently long or compactified are prone to develop knots. This is a "statistical necessity" that has been conjectured by Delbruck in 1962 and mathematically proved by Sumners and Whittington nearly 30 years later. In particular, they showed that for a self-avoiding polygon, the knotting probability tends to unity as the polygon length tends to infinity. This statistical necessity makes topological entanglement a genuine characteristic of polymeric systems. In case of linear polymer chains, knots can be untied by a suitable reptation of the polymer in space and therefore the entanglement is referred as physical knots. On the other hand, if the polymer ends are joined by a cyclisation reaction, the geometrical self-entanglement becomes trapped in the form of a proper mathematical knot, whose topology cannot be changed by any geometrical rearrangement of the polymer except by cutting it. Among polymers, double-stranded DNA (dsDNA) provides an ideal system to study the spontaneous occurrence of knots. In fact, differently from proteins and RNA, metric and topological properties of dsDNA are well captured by aspecific polymer models where only the polymer contour length, persistence length and thickness come into play. Studying knots in dsDNA is informative also to understand their biological implication. The presence of knots, in fact, severely affects several cellular processes, such as transcription and replication, with detrimental effects. Fortunately, cellular mechanisms have adopted countermeasures: there exist enzimes, namely topoisomerases, that are capable of simplifying the topological complexity of the DNA entanglement by favouring the selective cross-passage of pairs of DNA strands. The action of topoisomerases has been understood thanks to the topological profiling of DNA molecules realised with gel electrophoresis. This is the typical technique that permits to sort short DNA molecules by knot type. In particular, molecules are electrically driven through the obstacles of an agarose gel, where their mobility depends on the specific knot type. However, this technique can be used to profile only relatively short DNA molecules (10-15 kb). For longer ones, gel electrophoresis resolution would severely degrades, especially for knots with high number of crossings. This raises the problem of developing novel techniques that can be applied to characterise knot types in longer DNA molecules. Here, we will use molecular dynamics simulations and theoretical approaches to discuss the possibility to use spatially modulated nanochannels to sort ring polymer by their knot type. This approach permits, in principle, to separate polymers by their topological complexity, overcoming the aformentioned limits of gel electrophoresis. The spontaneous knotting of DNA is largely controlled by events where, for example, a loop is threaded by one termini; as a result both the complexity and size of the knots, as well as their location along the DNA contour, are stochastic. This is not the case for other types of biomolecules, particularly proteins, where the folding process towards the native state is tightly controlled by their chemical composition (primary sequence) via their intra-molecular interactions. As a result, proteins whose native state is knotted always feature the same knot type in the same sequence location. Mimicking such reproducible molecular knotting processes are, at least in part, the motivation of the ongoing quest of synthetic chemistry to create synthetic molecules tied in specific knot types. In this regard, chemists succeeded in controlling chemical reactions between small building blocks to assemble molecules with a priori desired topology. The chemists who developed this set of techniques, whose contribution opened up the way to a revolutionary chemistry, were awarded with the Chemistry Nobel Prize in 2016. Despite the high interest in the topic, up to recently, only a handful of different knot types have been synthesised. The reason is due to various challenging aspects of the synthesis process. These include the choice of the suitable building blocks, their correct spatial arrangement, and, above all, the selection of the designable target topology. Not every knot type, in fact, is necessarily expected to be equally designable in practice. In this thesis, we performed a computational and theoretical study to explore which designable molecular knots could be accessible for molecular synthesis with current experimental techniques.