Scaling of Linking and Writhing Numbers for Spherically Confined and Topologically Equilibrated Flexible Polymers

Scaling laws for Gauss linking number Ca and writhing number Wr for spherically confined flexible polymers with thermally fluctuating topology are analyzed. For ideal (phantom) polymers each of N segments of length unity confined to a spherical pore of radius R there are two scaling regimes: for sufficiently weak confinement (R ⪢ N(1/3)) each chain has |Wr| ≈ N(1/2), and each pair of chains has average |Ca| ≈ N/R(3/2); alternately for sufficiently tight confinement (N(1/3) ⪢ R), |Wr| ≈ |CA| ≈ N/R(3/2). Adding segment-segment avoidance modifies this result: for n chains with excluded volume interactions |Ca| ≈ (N/n)(1/2)f(ϕ) where f is a scaling function that depends approximately linearly on the segment concentration ϕ = nN/R(3). Scaling results for writhe are used to estimate the maximum writhe of a polymer; this is demonstrated to be realizable through a writhing instability that occurs for a polymer which is able to change knotting topology and which is subject to an applied torque. Finally, scaling results for linking are used to estimate bounds on the entanglement complexity of long chromosomal DNA molecules inside cells, and to show how "lengthwise" chromosome condensation can suppress DNA entanglement.