Someone estimates the rms end-to-end distance of a length of double-stranded DNA

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Biophysics & Medical Physics · Macromolecules, membranes and cell architecture

Forces, Machines and Signals, Ch The elastic polymer: DNA from random walk to worm-like chain

Problem

Someone estimates the rms end-to-end distance of a 10 μm10\,\mathrm{\mu m} length of double-stranded DNA, using the long-chain (random-walk) limit of the worm-like chain, ⟨R2⟩=2ξpL\langle R^2\rangle = 2\xi_pL. They argue: "The persistence length sets the length scale in the tangent correlation, ⟨t^(s)⋅t^(0)⟩=e−s/ξp\langle\hat{\mathbf t}(s)\cdot\hat{\mathbf t}(0)\rangle = e^{-s/\xi_p}. For double-stranded DNA this length scale is 100 nm100\,\mathrm{nm}, so ξp=100 nm\xi_p = 100\,\mathrm{nm}. Then for L=10 μm=10 000 nmL = 10\,\mathrm{\mu m} = 10\,000\,\mathrm{nm}, ⟨R2⟩1/2=2ξpL=2×100 nm×10 000 nm=1.41 μm."\langle R^2\rangle^{1/2} = \sqrt{2\xi_pL} = \sqrt{2\times100\,\mathrm{nm}\times10\,000\,\mathrm{nm}} = 1.41\,\mathrm{\mu m}.\text{"} Find the error in this reasoning and give the correct rms end-to-end distance.

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