Energetic extension of a strongly stretched worm-like chain ($x/L\simeq 1-\frac{1}{2}\sqrt{k_{B}T/(f{L}_p)}+f/K_s$)

An inextensible worm-like chain gains extension by reducing configurational fluctuations. At high force its fractional entropic extension approaches $1-\frac{1}{2}\sqrt{k_{B}T/(f{L}_p)}$. A finite axial stretch modulus $K_s$ adds the leading energetic extension $f/K_s$. This approximation distinguishes suppressed bending fluctuations from actual stretching of the chain backbone; it does not describe an unfolding event that changes contour length.