= Solution
An inextensible <worm-like chain> describes <entropic elasticity> and diverges as $x\to L$. Real <protein> backbones and folded <protein domains> also deform energetically under large <forces>. For an axial stretch modulus $K_s$, a simple extensible-chain correction is
$$
\frac xL\simeq1-\frac12\sqrt{\frac{k_BT}{fL_p}}+\frac f{K_s}.
$$
The last term supplies extension through bond and structural deformation that the inextensible model excludes. It becomes especially noticeable on the high-force ends of rising branches.<Energetic extension of a strongly stretched worm-like chain>[High-force entropic and energetic extension terms]{ref} An unfolding transition itself is also outside a single equilibrium fixed-$L$ branch. These mechanisms explain why the source explicitly distinguishes energetic and <entropic elasticity>.
\b[Source-dependent TODO:] the arrows and actual deviations at the first four peaks in Figure B are unavailable. Energetic stretching is the natural explanation of a high-force excess-extension deviation, but its direction, onset, and uniqueness cannot be checked against the missing curves. No particular arrow-marked feature or numerical stretch modulus is invented. This source-dependent identification remains incomplete; the recoverable model limitation is explained above.
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