= Solution
On each branch the <worm-like chain force-extension interpolation> relates <force> to the extension of the already released chain. Fitting a sufficiently long branch estimates its <polymer contour length> $L$ and, if the data constrain it independently, its <persistence length> $L_p$. In practice $L_p$ may be fixed across branches so that the changes in $L$ are better identified. Consecutive fitted <polymer contour lengths> reveal how much additional chain became available at an <protein denaturation> event; using a justified <polymer contour length> per residue can then estimate residue counts. A projected folded length and the pulling geometry must be included before assigning a structural boundary.
Writing $u=x/L$, expansion at small extension gives
$$
f=\frac{k_BT}{L_p}\left(\frac32u+\frac34u^2+\cdots\right),\qquad
k_{\rm eff}=\left.\frac{df}{dx}\right|_0=\frac{3k_BT}{2L_pL}.
$$
Near full extension, $f\sim k_BT/[4L_p(1-x/L)^2]$. Larger $L$ shifts a branch toward larger extension at fixed <force>. The effective <flexural rigidity> is $\kappa=k_BT L_p$, with units of <force> times length squared. This is entropic straightening of a <worm-like chain>; bond stretching and barrier rupture themselves are not included in the formula.
\b[The persistence-length wording requires care.] The three-dimensional model gives $\langle\mathbf t(s)\cdot\mathbf t(0)\rangle=e^{-s/L_p}$. At $s=L_p$, $\arccos\langle\cos\theta\rangle=\arccos(e^{-1})=68.4^\circ$. This is not a statement that $\langle\theta\rangle=68.4^\circ$. For example, angles zero and $90^\circ$ with equal <probabilities> have mean angle $45^\circ$, but $\arccos\langle\cos\theta\rangle=60^\circ$. The distinction prevents a false literal interpretation of the printed definition.
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