General Polygonal Length Dependence of the Linking Probability for Ideal Random Polygons(Statistical Physics and Topology of Polymers with Ramifications to Structure and Function of DNA and Proteins)

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The linking probability that two closed random walks (i.e., random polygons: RPs) are mutually entangled, P_<link>, is investigated numerically quite accurately. Here, P_<link> is a function of the distance between two RPs, R, and the number of polygonal segments, N. In a previous paper, we numerically estimated P_<link> precisely in the wide region of 0≤ξ≤3.0 where ξ denotes the normalized distance, i.e., the ratio of R to the radius of gyration R_g:ξ=R/R_g. We have also shown that the N- and R-dependence of P_<link> can be well approximated by a simple function: P_<link>(ξ;N)=exp(-κ_1ξ^<ν_1>)-Cexp(-κ_2ξ^<ν_2>), where κ_1, ν_1, κ_2, ν_2 and C are fitting parameters which depend on N. In this paper we extract the general N-dependence of P_<link>. We evaluate numerically the five fitting parameters as functions of N, i.e., κ_1(N), ν_1(N), κ_2(N), ν_2(N) and C(N). Considering physical requirements of P_<link>, we impose constraints on these functions. By taking account of both the numerical data from N=32 to 512 and the constraints, we propose good approximate functions of N for the above fitting parameters. We find that they are valid from N=32 to 512. We expect that they should also be effective for approximating the linking probability at least in some region of N>512, although it is not clear how effective it is for large values of N. This result enables us not only to estimate P_<link> for an arbitrary N at least roughly, but also to predict the possible asymptotic behavior of P_<link> at N=∞.
2011-12-16

General Polygonal Length Dependence of the Linking Probability for Ideal Random Polygons(Statistical Physics and Topology of Polymers with Ramifications to Structure and Function of DNA and Proteins)

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