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Effect of interaction shape on the condensed DNA toroid

We investigate how different microscopic interactions between semiflexible chain segments can qualitatively alter the physical properties of the condensed toroid. We propose a general form of the Hamiltonian of the toroid and discuss its analytic properties. For different interactions, the theory predicts different scaling behaviours of the mean toroidal and cross sectional radii, $r_c$ and $r_{cross}$, as functions of the contour length L: $(r_c, r_{cross}) \sim L^{ν(N_c)}$ with $ν=(1/5, 2/5)$ for the van der Waals type, $ν=(-1/3, 2/3)$ for the Coulomb type, $ν=(-1, 1)$ for the delta function type attractions in the asymptotic limit. For the toroids with finite winding number $N_c=100 \sim 400$, we find $ν\simeq 0$ for the Yukawa interaction with screening parameter $κ=0.5 \sim 1.0$, and $ν=0.1 \sim 0.13$ for the van der Waals type interactions. These findings could provide possible explanation for the experimentally well known observation $ν\simeq 0$ of the condensed DNA toroids. Conformational transitions are also discussed.