The winding angle of planar self-avoiding walks

Winding angle and maximum winding angle of the two-dimensional random walk

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10m steps, for m = 2, 3, 4, 5, 6, 7.

Winding angle and maximum winding angle of the two-dimensional random walk

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10 m steps, for m = 2, 3, 4, 5, 6, 7.

Linking topology of tethered polymer rings with applications to chromosome segregation and estimation of the knotting length

The Gauss linking number (Ca) of two flexible polymer rings which are tethered to one another is investigated. For ideal random walks, mean linking-squared varies with the square root of polymer length while for self-avoiding walks, linking-squared increases logarithmically with polymer length. The free-energy cost of linking of polymer rings is therefore strongly dependent on degree of self-avoidance, i.e., on intersegment excluded volume. Scaling arguments and numerical data are used to determine the free-energy cost of fixed linking number in both the fluctuation and large-Ca regimes; for ideal random walks, for |Ca|>N;{1/4} , the free energy of catenation is found to grow proportional, variant|Ca/N;{1/4}|;{4/3} . When excluded volume interactions between segments are present, the free energy rapidly approaches a linear dependence on Gauss linking (dF/dCa approximately 3.7k_{B}T) , suggestive of a novel "catenation condensation" effect. These results are used to show that condensation of long entangled polymers along their length, so as to increase excluded volume while decreasing number of statistical segments, can drive disentanglement if a mechanism is present to permit topology change. For chromosomal DNA molecules, lengthwise condensation is therefore an effective means to bias topoisomerases to eliminate catenations between replicated chromatids. The results for mean-square catenation are also used to provide a simple approximate estimate for the "knotting length," or number of segments required to have a knot along a single circular polymer, explaining why the knotting length ranges from approximately 300 for an ideal random walk to 10;{6} for a self-avoiding walk.