Static and dynamic properties of knotted biopolymers: from bulk to nanochannels and nanopores

Ropes or yarns, especially when disorderly packed, are prone to develop knots. Polymers are no exception to this rule and, in fact, rigorous mathematical results have been proved regarding the "statistical necessity" that sufficiently-long circular chains are knotted. The current surge of scientific interest in knotted polymers, and especially biopolymers, is prompted by the need to understand the profound implications that these forms of entanglement can have on the mechanical, dynamical and conformational properties of polymer chains. For proteins, for instance, a long standing problem is how exactly the knotting properties of naturally-occurring proteins differ from those of general, non-specific, polymer models. This question has, in fact, motivated studies in several directions, from surveying and classifying systematically the repertoire of knots in peptide chains, to establishing the details of the folding route. For genomic DNA instead, it has long been known that it can be highly entangled due to the high packing degree that it attains in all organisms, from viruses to eukaryotic cells. In this case, the advent of single-molecule manipulation techniques, which are routinely applied to DNA filaments of various length, has opened new, and still largely unexplored, perspectives for detecting or controlling the spontaneous knotting properties of DNA. In this thesis, I will use theoretical and computational techniques to tackle various aspects of the aforementioned issues. In Chapter 1, I will provide a primer on knots, which sets a reference for concepts and methods used in subsequent chapters. I will in particular present a small resume of knot theory, focusing mainly on notions useful in our context. Afterwards, I will introduce some of the computational techniques used to detect and pinpoint knots along closed and open chains. In Chapter 2, I will discuss our recent survey of the entire protein data bank, where we searched for all instances of knotted protein chains. The analysis yielded an up-to-date information about the overall knotting probability, the repertoire of knot types, as well as insight on the length and sequence position of knots in peptide chains. In Chapter 3, I will use a general polyelectrolyte chain model, mapped to DNA, to study the dynamical mechanisms governing knot formation when DNA is confined inside a nanopore channel with size compatible with the DNA persistence length. I will shown that the deep looping and back-folding of the chain ends will be responsible for the knot formation and destruction. Upon increasing the chain length, the knotting probability of DNA increases due to the growing time a knot can diffuse alongside the chain. Instead unknotted lifetimes level off to a constant because they are ruled by the backfolding process. In chapter 4, I will present the translocation of flexible and knotted polyelectrolyte chains, parametrized after single-stranded DNA, inside a pore too narrow to allow knot passage. This out-of-equilibrium process, which can affect the polymer translocation in complex and counter-intuitive ways, depends deeply on the knot topology. We tackled the resulting translocation compliance in a simple framework based on how the pulling force, applied only inside the pore, propagates along and past the knot, and how it is related to the structural properties of different knot types. In chapter 5, I will discuss the translocation of double-stranded DNA chains through wide nanopores. The study is motivated by a recent experimental breakthrough, for which we provide key insight and explanations for the observed phenomenology.