TWIST IN AN EXACTLY SOLVABLE DIRECTED LATTICE RIBBON

We investigate the transition between a twisted regime and a disordered regime in a directed ribbon model on a cubic lattice. A fugacity corresponding to an interaction which models half-twists in the ribbon is introduced and the interacting model is solved exactly. Our results suggest that conformational entropy and a local interaction which induces twist are key ingredients to model qualitatively the crossover behavior between a twisted (helical) regime and a denatured regime in duplex biopolymers such as DNA

A Monte Carlo algorithm for lattice ribbons

A lattice ribbon is a connected sequence of plaquettes subject to certain self-avoidance conditions. The ribbon can be closed to form an object which is topologically either a cylinder or a Mobius band, depending on whether its surface is orientable or nonorientable. We describe a grand canonical Monte Carlo algorithm for generating a sample of these ribbons, prove that the associated Markow chain is ergodic, and present and discuss numerical results about the dimensions and entanglement complexity of the ribbons

KNOT PROBABILITY FOR LATTICE POLYGONS IN CONFINED GEOMETRIES

We study the knot probability of polygons confined to slabs or prisms, considered as subsets of the simple cubic lattice. We show rigorously that almost all sufficiently long polygons in a slab are knotted and we use Monte Carlo methods to investigate the behaviour of the knot probability as a function of the width of the slab or prism and the number of edges in the polygon. In addition we consider the effect of solvent quality on the knot probability in these confined geometries

Entropic exponents of lattice polygons with specified knot type

Ring polymers in three dimensions can be knotted, and the dependence of their critical behaviour on knot type is an open question. We study this problem for polygons on the simple cubic lattice using a novel grand-canonical Monte Carlo method and present numerical evidence that the entropic exponent depends on the knot type of the polygon. We conjecture that the exponent increases by unity for each additional factor in the knot factorization of the polygon

Torsion of polygons in Z(3)

The torsion of polygons and self-avoiding walks in the cubic lattice is a measure of the self-entanglement of these objects. We consider several definitions of torsion in polygons, and introduce a fugacity conjugate to the torsion in our models. We study the thermodynamic behaviour of these models using probabilistic methods anti rigorous methods from statistical mechanics. In particular, we prove that at least one of our models has a non-analyticity in its free energy, corresponding to a transition between phases with high and low torsion

Monte Carlo study of the interacting self-avoiding walk model in three dimensions

We consider self-avoiding walks on the simple cubic lattice in which neighboring pairs of vertices of the walk (not connected by an edge) have an associated pair-wise additive energy. If the associated force is attractive, then the walk can collapse From a coil to a compact ball. We describe two Monte Carlo algorithms which we used to investigate this collapse process, and the properties of the walk as a function of the energy or temperature. We report results about the thermodynamic and configurational properties of the walks and estimate the location of the collapse transition

The shapes of self-avoiding polygons with torsion

We consider self-avoiding polygons on the simple cubic lattice with a torsion fugacity. We use Monte Carlo methods to generate large samples as a function of the torsion fugacity and the number of edges in the polygon. Using these data we investigate the shapes of the polygons at large torsion fugacity and find evidence that the polygons have substantial helical character. In addition, we show that these polygons have induced writhe for any non-zero torsion fugacity, and that torsion and writhe are positively correlated

Interacting self-avoiding walks and polygons in three dimensions

Self-interacting walks and polygons on the simple cubic lattice undergo a collapse transition at the theta-point. We consider self-avoiding walks and polygons with an additional interaction between pairs of vertices which are unit distance apart but not joined by an edge of the walk or polygon. We prove that these walks and polygons have the same limiting free energy if the interactions between nearest-neighbour vertices are repulsive. The attractive interaction regime is investigated using Monte Carlo methods, and we find evidence that the limiting free energies are also equal here. In particular, this means that these models have the same theta-point, in the asymptotic limit. The dimensions and shapes of walks and polygons are also examined as a function of the interaction strength

KNOTTING AND SUPERCOILING IN CIRCULAR DNA - A MODEL INCORPORATING THE EFFECT OF ADDED SALT

We consider a model of a circular polyelectrolyte, such as DNA, in which the molecule is represented by a polygon in the three-dimensional simple cubic lattice. A short-range attractive force between nonbonded monomers is included (to account for solvent quality) together with a screened Coulomb potential (to account for the effect of added salt). We compute the probability that the ring is knotted as a function of the number of monomers in the ring, and of the ionic strength of the solution. The results show the same general behavior as recent experimental results by Shaw and Wang [Science 260, 533 (1993)] and by Rybenkov, Cozzarelli, and Vologodskii [Proc. Natl. Acad. Sci. U.S.A. 90, 5307 (1993)] on the knot probability in circular DNA as a function of added salt. In addition, we compute the writhe of the polygon and show that this also increases as the ionic strength increases. The writhe computations model the conformational behavior of nicked circular duplex DNA molecules in salt solution

LATTICE RIBBONS - A MODEL OF DOUBLE-STRANDED POLYMERS

We introduce a discrete ribbon model for double-stranded polymers (such as duplex DNA) where the ribbon is constrained to lie on the simple cubic lattice Z3. The ribbon is made up of a sequence of plaquettes and can either be open or closed. We investigate the growth of the number of ribbons as a function of the number of plaquettes and use Monte Carlo methods to estimate the dimensions of the ribbon, the writhe of the backbone and, in the case of orientable closed ribbons, the linking number of the two boundary curves