= Solution
The coefficient in the linear <force> law is
$$
\boxed{k_{\rm chain}=\frac{3k_BT}{Nb^2}
=2.205\times10^{-3}\,\mathrm{N\,m^{-1}}
=2.205\,\mathrm{pN\,nm^{-1}}\quad(T=298.15\,\mathrm K).}
$$
Stretching suppresses the many disordered orientations available to the chain. The <Gaussian chain> <free energy> cost $\Delta\mathcal F\simeq3k_BTR^2/(2Nb^2)$ has gradient equal to this restoring <force>. This is <entropic elasticity>; individual <covalent bonds> need not stretch to produce the weak-<force> spring.
At moderate <force>, even an ideal <FJC> becomes nonlinear as its orientations align, so failure of a \i[linear] fit alone need not imply nonideal chemistry. But failure of the full <FJC> curve can reflect backbone <spring constant> and correlated segment angles, nonuniform effective links, remaining folded contacts, <force>-induced conformational changes, energetic bond compliance, or tether/calibration effects. An extensible <FJC> or an appropriate semiflexible model includes some of these effects; their parameters require independent justification. The stated inextensible $14\,\mathrm{nm}$ <polymer contour length> already excludes a literal $15\,\mathrm{nm}$ extension.
The first panel of the original model figure below shows the full orientational response and weak-<force> tangents without tracing the experimental data.
\Image[/past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2018-polymer-and-membrane-response.png]
{title=Entropic chain elasticity, permissive potassium gating and chloride-dependent membrane shunting}
{description=Original model calculations and equivalent-circuit schematic. The chain uses the printed segment parameters; the gating and passive voltage examples are normalized illustrative responses, not numerical fits to the source experiment.}
Back to article page
