Bounds for the minimum step number of knots confined to slabs in the simple cubic lattice

Abstract

Volume confinement is a key determinant of the topology and geometry of a polymer. For instance recent experimental studies have shown that the knot distribution observed in bacteriophage P4 cosmids is different from that found in full bacteriophages. In particular it was observed that the complexity of the knots decreases sharply when the length of packed genome decreases. However it is not well understood exactly how the volume confinement affects the topology and geometry of a polymer. This problem is the motivation of this paper. Here a polymer is modeled as a self-avoiding polygon on the simple cubit lattice and the confining condition is such that the polygon is bounded between two parallel planes (i.e., bounded within a slab). We estimate the minimum length required for such a polygon to realize a knot type. Our numerical simulations show that in order to realize a prime knot (with up to 10 crossings) in a 1-slab (i.e. a slab of height one), one needs a polygon with length longer than the minimum length needed to realize the same knot when there is no confining condition. In the case of the trefoil knot, we can in fact establish this result analytically by proving that the minimum length required to tie a trefoil in the 1-slab is 26, which is greater than 24, the known minimum length required to tie a trefoil without a confinement condition. Additionally, we find that in the 1-slab not all geometrical realizations of a given knot type are equivalent to each other, suggesting that the topology of a polymer in confined volume alone is not enough to describe the state of a polymer.

Lattice knots in a slab

“…An implementation of the GAS-algorithm for knotted cubic lattice polygons in slabs [11] was used, and we collected data on the entropy and minimal length of knotted lattice polygons up to eight crossings. Our data verify similar results obtained in reference [8]. In section 3 we discuss the first model and present our data, and in section 4 we present data on the second model and discuss our results.…”

“…Since the GAS algorithm can be implemented as a flat histogram method, it is efficient at rare event sampling and thus at finding knotted polygons of minimal length. A comparison of our results with the data in reference [8] makes us confident that our results are exact in almost all cases.…”

“…The data in table 5 agree for all knot types, except for 8 18 , with the data in reference [8]. We improved on the estimate for the minimal length of 8 18 in S 1 by finding states at length n = 70, compared to 72 in that reference.…”

“…In both these models it is necessary to determine the number and length of lattice knots in slabs of width L. Some data of this kind were obtained in reference [8], and we will at the same time verify in most cases, and improve in one case, on their results.…”

“…For example, one may deduce that n 2,31 = 24 from reference [4]. However, simulations show that n 1,31 = 26 [8]. In other words, in a slab of width L = 1, it is necessary to have a polygon of length 26 to tie a lattice trefoil, while if L ≥ 2 then 24 edges will be sufficient (and necessary) [4].…”