= Solution
Write $u=x/L$ and $A=k_BT L_p$. The <worm-like chain force-extension interpolation> is
$$
f=\frac{k_BT}{L_p}\left[\frac1{4(1-u)^2}-\frac14+u\right].
$$
A fit to an elastic rising branch estimates its <Polymer contour length>[contour length] $L$ and, if not fixed independently, its <persistence length> $L_p$. Comparing successive fitted <Polymer contour length>[contour lengths] gives the length released by a domain and helps distinguish whole-repeat <Protein denaturation>[unfolding] from partial events. $L_p$ estimates the flexibility of the unfolded chain and the <Flexural rigidity>[bending rigidity] $A$; at a stated <temperature>, $A=k_BT L_p$.
For small extension, expansion gives $f\simeq[3k_BT/(2L_pL)]x$, so the effective entropic <spring constant> is $3k_BT/(2L_pL)$. Close to full extension, $f\sim k_BT/[4L_p(1-x/L)^2]$, giving strong strain <Elasticity (physics)>[stiffening]. These regimes constrain the fit differently and correlate its parameters if the measured interval is short. The fit does not by itself identify a unique folding barrier or <Protein denaturation>[unfolding] kinetic rate.
\b[Persistence-length convention.] In a three-dimensional isotropic <worm-like chain> the standard definition is $\langle\mathbf t(s)\cdot\mathbf t(0)\rangle=e^{-s/L_p}$. Thus at $s=L_p$, $\arccos\langle\cos\theta\rangle=\arccos(e^{-1})\simeq68.4^\circ$. This is not literally the mean angle $\langle\theta\rangle$. For a simple counterexample, equally probable angles $0^\circ$ and $90^\circ$ have mean angle $45^\circ$ but $\arccos\langle\cos\theta\rangle=60^\circ$. The source's angle description is shorthand for the tangent-correlation convention.
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