Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 71 / 5 / c / 2

The extension must be compared with the actual contour length, $L=100\mathrm{m}$. For the illustrative $Q=0.2$ force and the persistence-length-corrected chain, $z/L\simeq 2.15\times 10^{-3}$ and $F{b}_K/(k_{B}T)\simeq 6.44\times 10^{-3}$. Both are small, so the small-force Langevin function expansion and entropic-extension approximation are consistent. Although the mean extension greatly exceeds the unforced millimetre-scale coil size, it remains far below the maximum contour length; that does not violate the model.

The material axial elongation at this force would be only $F/(EA/L)\simeq 5.0\times 10^{-5}\mathrm{m}$, illustrating that most of the predicted coil extension is orientation and straightening rather than stretching the backbone. Thus the order-of-magnitude entropic response is internally reasonable within the ideal-chain assumptions, but its numerical optical-force prediction is not unique. Solvent-dependent excluded volume, the unusually large stated contour length, actual trap geometry, and surface interactions all matter to an experimental interpretation.