A semiflexible polymer of persistence length and contour length obeys the worm-like…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Soft Matter at the Thermal Scale, Ch The size and shape of a polymer chain

Problem

A semiflexible polymer of persistence length lpl_p and contour length LL obeys the worm-like chain result ⟨R2⟩=2lpL−2lp2(1−e−L/lp).\langle R^2\rangle = 2l_pL - 2l_p^2\left(1-e^{-L/l_p}\right). ((a)) Expand the exponential to third order in L/lpL/l_p and show that for L≪lpL\ll l_p (a rigid rod) ⟨R2⟩→L2(1−L3lp)\langle R^2\rangle \to L^2\left(1-\dfrac{L}{3l_p}\right), so that ⟨R2⟩1/2→L\langle R^2\rangle^{1/2}\to L to leading order. ((b)) Take the limit L≫lpL\gg l_p and show that ⟨R2⟩→2lpL\langle R^2\rangle \to 2l_pL, identifying the effective Kuhn length b=2lpb=2l_p so that ⟨R2⟩=Nb2\langle R^2\rangle = Nb^2 with N=L/bN=L/b. ((c)) A synthetic polypeptide helix has persistence length lp=100 nml_p=100\,\mathrm{nm}. Using the exact formula, evaluate ⟨R2⟩1/2\langle R^2\rangle^{1/2} for (i) a fragment of contour length L=20 nmL=20\,\mathrm{nm} and (ii) a fragment of contour length L=8.0 μmL=8.0\,\mu\mathrm{m}, and compare each with the appropriate limiting formula from parts (a) or (b).

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