Miyuki K. Shimamura

Finite-size and asymptotic behaviors of the gyration radius of knotted cylindrical self-avoiding polygons

Several nontrivial properties are shown for the mean-square radius of gyration R2(K) of ring polymers with a fixed knot type K. Through computer simulation, we discuss both finite size and asymptotic behaviors of the gyration radius under the topological constraint for self-avoiding polygons consisting of N cylindrical segments with radius r. We find that the average size of ring polymers with the knot K can be much larger than that of no topological constraint. The effective expansion due to the topological constraint depends strongly on the parameter r that is related to the excluded volume. The topological expansion is particularly significant for the small r case, where the simulation result is associated with that of random polygons with the knot K.

Gyration radius of a circular polymer under a topological constraint with excluded volume.

It is nontrivial whether the average size of a ring polymer should become smaller or larger under a topological constraint. Making use of some knot invariants, we numerically evaluate the mean-square radius of gyration for ring polymers having a fixed knot type, where the ring polymers are given by self-avoiding polygons consisting of freely jointed hard cylinders. We obtain plots of the gyration radius versus the number of polygonal nodes for the trivial, trefoil, and figure-eight knots. We discuss possible asymptotic behaviors of the gyration radius under the topological constraint. In the asymptotic limit, the size of a ring polymer with a given knot is larger than that of no topological constraint when the polymer is thin, and the effective expansion becomes weak when the polymer is thick enough.

Characteristic length of random knotting for cylindrical self-avoiding polygons

Abstract We discuss the probability of random knotting for a model of self-avoiding polygons whose segments are given by cylinders of unit length with radius r . We show numerically that the characteristic length of random knotting is roughly approximated by an exponential function of the chain thickness r .

Scattering functions of knotted ring polymers

We discuss the scattering function of a Gaussian random polygon with N nodes under a given topological constraint through simulation. We evaluate the form factor PK(q) of a Gaussian polygon of N = 200 having a fixed knot K for some different knots such as the trivial, trefoil, and figure-eight knots. Here the Gaussian polygons with different knots K have distinct values of the mean-square radius of gyration, R2(G,K). We obtain the Kratky plots of the form factors--i.e., the plots of (qR(G,K))2PK(q) versus qR(G,K)--for the different topological constraints and discuss nontrivial large-q behavior as well as small-q behavior for the scattering functions. We also find that the distinct values of R2(G,K) play an important role in the large-q and small-q properties of the Kratky plots.

Geometrical complexity of conformations of ring polymers under topological constraints

One measure of geometrical complexity of a spatial curve is the average of the number of crossings appearing in its planar projection. The mean number of crossings averaged over some directions have been numerically evaluated for N-noded ring polymers with a fixed knot type. When N is large, the average crossing number of ring polymers under the topological constraint is smaller than that of no topological constraint. The decrease of the geometrical complexity is significant when the thickness of polymers is small. It is also suggested from the simulation that the relation between the average crossing number and the average size of ring polymers should depend on whether they are under a topological constraint or not.

Topological Entropy of a Stiff Ring Polymer and Its Connection to DNA Knots.

We discuss the entropy of a circular polymer under a topological constraint. We call it the topological entropy of the polymer, in short. A ring polymer does not change its topology (knot type) under any thermal fluctuations. Through numerical simulations using some knot invariants, we show that the topological entropy of a stiff ring polymer with a fixed knot is described by a scaling formula as a function of the thickness and length of the circular chain. The result is consistent with the viewpoint that for stiff polymers such as DNAs, the length and diameter of the chains should play a central role in their statistical and dynamical properties. Furthermore, we show that the new formula extends a known theoretical formula for DNA knots.