The Knot Spectrum of Confined Random Equilateral Polygons

Abstract

It is well known that genomic materials (long DNA chains) of living organisms are often packed compactly under extreme confining conditions using macromolecular self-assembly processes but the general DNA packing mechanism remains an unsolved problem. It has been proposed that the topology of the packed DNA may be used to study the DNA packing mechanism. For example, in the case of (mutant) bacteriophage P4, DNA molecules packed inside the bacteriophage head are considered to be circular since the two sticky ends of the DNA are close to each other. The DNAs extracted from the capsid without separating the two ends can thus preserve the topology of the (circular) DNAs. It turns out that the circular DNAs extracted from bacteriophage P4 are non-trivially knotted with very high probability and with a bias toward chiral knots. In order to study this problem using a systematic approach based on mathematical modeling, one needs to introduce a DNA packing model under extreme volume confinement condition and test whether such a model can produce the kind of knot spectrum observed in the experiments. In this paper we introduce and study a model of equilateral random polygons con_ned in a sphere. This model is not meant to generate polygons that model DNA packed in a virus head directly. Instead, the average topological characteristics of this model may serve as benchmark data for totally randomly packed circular DNAs. The difference between the biologically observed topological characteristics and our benchmark data might reveal the bias of DNA packed in the viral capsids and possibly lead to a better understanding of the DNA packing mechanism, at least for the bacteriophage DNA. The purpose of this paper is to provide information about the knot spectrum of equilateral random polygons under such a spherical confinement with length and confinement ratios in a range comparable to circular DNAs packed inside bacteriophage heads.

A statistical approach to knot confinement via persistent homology

“…Similar ideas have previously been considered in e.g. [1][2][3][4], often with the main goal of understanding the mechanism of DNA, or, more generally, polymer packing in a confined volume. Prominent instances of such studies are concerned with tight DNA packing in viral capsids [5][6][7][8] and DNA packing in cells (e.g.…”

New Stick Number Bounds from Random Sampling of Confined Polygons

“…Since more complicated knots tend to be more condensed [43,40,20,19], we expect that sampling polygons in rooted spherical confinement -meaning that the entire polygon is required to be contained in a 2 At least in high dimensions, a single hit-and-run step is unlikely to move very far, so it is preferable to do several steps at once, though not too many since hit-and-run steps are much more expensive than sampling from the torus. Our experience is that 10 steps provides a good balance.…”

“…These knots, together with 8 19 , 9 29 , 10 16 , and 10 79 , appeared on Rawdon and Scharein's list of knots for which they could not find an equilateral stick knot achieving a known bound on stick number, and thus were potential examples of knots for which stick number and equilateral stick number differ. Millett [38] found an example of an equilateral 8-stick 8 19 , proving that eqstick (8 19 ) = 8, which was already known to be the stick number, and our 10-stick 10 16 (see Figure 2) beats the previous best bounds on both stick number (11) and equilateral stick number (12). In turn, our observed equilateral 10-stick examples of 10 107 , 10 119 , and 10 147 match the best previous bound on stick number, leaving only 9 29 and 10 79 from Rawdon and Scharein's list.…”

The symplectic geometry of closed equilateral random walks in 3-space

“…• Now that we can sample confined polygons quickly, with solid error bars on our calculations, what frontiers does this open in the numerical study of confined polymers? We take our cues from the pioneering work of Diao, Ernst, Montemayor and Ziegler [22][23][24][25], but are eager to explore this new experimental domain. For instance, sampling tightly confined n-gons might be a useful form of "enriched sampling" in the hunt for complicated knots of low equilateral stick number, since very entangled polygons are likely to be geometrically compact as well.…”