Single-stranded DNA (ssDNA) is well modelled as a freely jointed chain with a much…

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

Single-stranded DNA (ssDNA) is well modelled as a freely jointed chain with a much shorter Kuhn length than duplex DNA. A ssDNA linker used to tether a bead in an optical-trap assay has

N=200

Kuhn segments, each of length

b=1.5\,\mathrm{nm}

, so its contour length is

L=Nb=300\,\mathrm{nm}

. Each segment points in a direction independent of its neighbours (the freely jointed chain idealisation), and thermal energy is

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

T=298\,\mathrm{K}

. ((a)) Starting from

\mathbf{R}=\sum_{i=1}^N\mathbf{b}_i

, derive

\langle R^2\rangle=Nb^2

, stating why the cross terms vanish. ((b)) State, using the central limit theorem, the Gaussian form of

P(\mathbf R)

for large

N

in three dimensions. ((c)) Use

F(\mathbf R)=-k_BT\ln P(\mathbf R)

to derive the entropic spring constant relating the mean force needed to hold the ends a distance

R

apart to

R

. ((d)) Evaluate the rms size and the spring constant for this linker, and find the extension produced by a force of

0.50\,\mathrm{pN}

. Check that this extension is small compared with

L

, as the Gaussian spring requires.

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