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Fit each pre-unfolding branch with its own polymer contour length $L$ and a persistence length $L_p$, allowing for the measured extension zero. The worm-like chain force-extension interpolation has the limitsThus the low-extension slope and the high-extension curvature constrain the flexible segment's length and bending stiffness. The persistence length is related to the bending modulus by $\kappa =k_{B}T{L}_p$. Successive increases in fitted $L$ measure contour released by unfolding, after subtracting the original folded domain's projected length. With an independently known contour increment per residue, this also estimates how many residues became available. A common fitted $L_p$ across branches supports a common type of unfolded chain; a systematic change can signal altered interactions or a poor model.

For a thermal worm-like chain, $⟨\mathbf{t}(s)\cdot \mathbf{t}(0)⟩=e^{-s/L_p}$. Consequently $s=L_p$ gives $⟨\cos \theta ⟩=1/e$, and $\arccos (1/e)=68.4^{\circ }$. The latter is an angle associated with the mean cosine, not generally the mean angle itself. For example, angles $0$ and $90^{\circ }$ with equal probabilities have mean angle $45^{\circ }$, but the inverse cosine of their mean cosine is $60^{\circ }$. This distinction makes the source's informal angular description precise.

Without Figure B, no numerical fit parameters can be read. The requested physical interpretation of those parameters follows without its numerical data.

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 givesNear full extension, $f\sim k_{B}T/[4L_p(1-x/L{)}^2]$. Larger $L$ shifts a branch toward larger extension at fixed force. The effective flexural rigidity is $\kappa =k_{B}T{L}_p$, 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.

The persistence-length wording requires care. The three-dimensional model gives $⟨\mathbf{t}(s)\cdot \mathbf{t}(0)⟩=e^{-s/L_p}$. At $s=L_p$, $\arccos ⟨\cos \theta ⟩=\arccos (e^{-1})=68.4^{\circ }$. This is not a statement that $⟨\theta ⟩=68.4^{\circ }$. For example, angles zero and $90^{\circ }$ with equal probabilities have mean angle $45^{\circ }$, but $\arccos ⟨\cos \theta ⟩=60^{\circ }$. The distinction prevents a false literal interpretation of the printed definition.

Write $u=x/L$ and $A=k_{B}T{L}_p$. The worm-like chain force-extension interpolation isA fit to an elastic rising branch estimates its contour length $L$ and, if not fixed independently, its persistence length $L_p$. Comparing successive fitted contour lengths gives the length released by a domain and helps distinguish whole-repeat unfolding from partial events. $L_p$ estimates the flexibility of the unfolded chain and the bending rigidity $A$; at a stated temperature, $A=k_{B}T{L}_p$.

For small extension, expansion gives $f\simeq [3k_{B}T/(2L_{p}L)]x$, so the effective entropic spring constant is $3k_{B}T/(2L_{p}L)$. Close to full extension, $f\sim k_{B}T/[4L_p(1-x/L{)}^2]$, giving strong strain 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 unfolding kinetic rate.

Persistence-length convention. In a three-dimensional isotropic worm-like chain the standard definition is $⟨\mathbf{t}(s)\cdot \mathbf{t}(0)⟩=e^{-s/L_p}$. Thus at $s=L_p$, $\arccos ⟨\cos \theta ⟩=\arccos (e^{-1})\simeq 68.4^{\circ }$. This is not literally the mean angle $⟨\theta ⟩$. For a simple counterexample, equally probable angles $0^{\circ }$ and $90^{\circ }$ have mean angle $45^{\circ }$ but $\arccos ⟨\cos \theta ⟩=60^{\circ }$. The source's angle description is shorthand for the tangent-correlation convention.