Double-stranded RNA (dsRNA) behaves as a worm-like chain with persistence length (a…

PHYSICS · Multi-part · 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

Double-stranded RNA (dsRNA) behaves as a worm-like chain with persistence length

\xi_p=63\,\mathrm{nm}

(a typical measured value, somewhat stiffer than duplex DNA's

50\,\mathrm{nm}

). A dsRNA molecule of contour length

L=3.0\,\mathrm{\mu m}

is held at tension

f=5.0\,\mathrm{pN}

, above the low-force entropic regime, with

k_BT=4.11\,\mathrm{pN\,nm}

T=298\,\mathrm{K}

. ((a)) A transverse displacement away from the pulling axis, decomposed into Fourier modes

\mathbf u_q

of a chain under bending rigidity

\kappa=\xi_pk_BT

and tension

f

, has energy

E=\tfrac12\sum_q\left(\kappa q^4+fq^2\right)|\mathbf u_q|^2

. Using equipartition on each mode and converting the sum over

q

to an integral, derive the fractional shortfall in extension

1-\frac{z}{L}=\frac12\sqrt{\frac{k_BT}{f\xi_p}}.

((b)) Evaluate the crossover force

k_BT/\xi_p

for this dsRNA and confirm that

f=5.0\,\mathrm{pN}

lies well above it, so the high-force limit applies. ((c)) Find the fractional extension

z/L

and the extension

z

in micrometres at

f=5.0\,\mathrm{pN}

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