= Solution
For the three-dimensional <worm-like chain>, define the bending coefficient $\kappa_b$ by $F_b=(\kappa_b/2)\int|\partial_s\mathbf t|^2ds$. Its <persistence length> is $l_p=\kappa_b/(k_BT)$. Holding that coefficient and the <polymer contour length> $L$ fixed gives
$$
\boxed{\frac{l_p(T_2)}{l_p(T_1)}=\frac{T_1}{T_2},\qquad
\frac{\Delta l_p}{l_p(T_1)}=\frac{T_1}{T_2}-1=-\frac{T_2-T_1}{T_2}.}
$$
The <temperatures> must be absolute. The <end-to-end distance of a worm-like chain> obeys
$$
\langle R^2\rangle=2Ll_p-2l_p^2(1-e^{-L/l_p}).
$$
For a long flexible chain, $L\gg l_p$, the typical size is $R_{\rm rms}\simeq\sqrt{2Ll_p}$, so \b[warming contracts the chain], with $R_{\rm rms}(T_2)/R_{\rm rms}(T_1)\simeq\sqrt{T_1/T_2}$. The finite-length formula also decreases as $l_p$ decreases, but the rod limit $L\ll l_p$ changes very little. A large segment count alone does not guarantee $L\gg l_p$. If the material's bending coefficient itself depends on <temperature>, replace the first ratio by $[\kappa_b(T_2)/\kappa_b(T_1)]T_1/T_2$; the sign then needs that additional material information. This calculation concerns a generic physical filament, with no biomolecular application.
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