posts:FRC: add freely rotating chain post.

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+[[!meta  title="Freely rotating chains"]]
+
+[[Velocity clamp force spectroscopy|Force_spectroscopy]] pulls are
+[often fit to polymer models][carrionvazquez99] such as the worm-like
+chain (WLC).  However, [Puchner et al.][puchner08] had the bright idea
+that, rather than fitting each loading region with a polymer model, it
+is easier to calculate the change in contour length by converting the
+abscissa to contour-length space.  While the WLC is commonly used,
+Puchner gets better fits using the freely rotating chain (FRC) model.
+
+Computing force-extension curves for either the WLC or FJC is
+complicated, and it is common to use interpolation formulas to
+estimate the curves.  For the WLC, we use [Bustamante's
+formula][bustamante94]:
+
+\[
+  F_WLC(x) = \frac{k_B T}{p} \left[
+    \frac{1}{4}\left(\frac{1}{\left(1-\frac{x}{L}\right)^2} - 1\right)
+    + \frac{x}{L} \right]
+\]
+
+For the FRC, Puchner uses [Livadaru][livadaru03]'s equation 46.
+
+\[
+  \frac{R_z}{L} \approx
+    \begin{cases}
+      \frac{fa}{3k_B T} & \text{for } \frac{fb}{k_B T} \lt \frac{b}{l} \\
+      1-\left(\frac{fl}{4k_B T}\right)^{-\frac{1}{2}}
+        & \text{for } \frac{b}{l} \lt \frac{fb}{k_B T} \lt \frac{l}{b} \\
+      1-\left(\frac{fb}{ck_B T}\right)^{-1}
+        & \text{for } \frac{l}{b} \lt \frac{fb}{k_B T}
+    \end{cases}\;.
+\]
+
+Unfortunately, there are two typos in Livadaru's equation 46.  It
+should read (confirmed by private communication with Roland Netz).
+
+\[
+  \frac{R_z}{L} \approx
+    \begin{cases}
+      \frac{fa}{3k_B T} & \text{for } \frac{fb}{k_B T} \lt \frac{b}{l} \\
+      1-\left(\frac{4fl}{k_B T}\right)^{-\frac{1}{2}}
+        & \text{for } \frac{b}{l} \lt \frac{fb}{k_B T} \lt \frac{l}{b} \\
+      1-\left(\frac{cfb}{k_B T}\right)^{-1}
+        & \text{for } \frac{l}{b} \lt \frac{fb}{k_B T}
+    \end{cases}\;.
+\]
+
+Regardless of the form of Livadaru's equation 46, the suggested FRC
+interpolation formula is Livadaru's equation 49, which has continuous
+cross-overs between the various regimes and adds the possibility of
+elastic backbone extension.
+
+\[
+  \frac{R_z}{L} = 1 - \left\{
+    \left(F_\text{WLC}^{-1}\left[\frac{fl}{k_BT}\right]\right)^\beta
+    + \left(\frac{cfb}{k_BT}\right)^\beta\right\}^{\frac{-1}{\beta}}
+    + \frac{f}{\tilde{\gamma}} \;,
+\]
+
+where $l=b\frac{\cos(\gamma/2)}{|\ln(\cos\gamma)|}$ (Livadaru's
+equation 22) is the effective persistence length, $\beta$ determines
+the crossover sharpness, $\tilde{\gamma}$ is the backbone stretching
+modulus, and $F_\text{WLC}^{-1}[x]$ is related to the inverse of
+Bustamante's interpolation formula,
+
+\[
+  F_\text{WLC}[x] = \frac{3}{4} - \frac{1}{x} + \frac{x^2}{4} \;.
+\]
+
+By matching their interpolation formula with simlated FRCs, Livadaru
+suggests using $\beta=2$, $\tilde{\gamma}=\infty$, and $c=2$.  In his
+paper, Puchner suggests using $b=0.4$ nm and $\gamma=22^{\circ}$.
+However, when I contacted him and pointed out the typos in Livadaru's
+equation 46, he reran his analysis and got similar results using the
+corrected formula with $b=0.11$ nm and $\gamma=41^{\circ}$.  This
+makes more sense because it gives a WLC persistence length similar to
+the one he used when fitting the WLC model:
+
+\[
+  l = b\frac{\cos(\gamma/2)}{|\ln(\cos\gamma)|} = 0.366\text{ nm}
+\]
+
+(vs. his WLC persistence length of $p=0.4$ nm).
+
+In any event, the two models (WLC and FRC) give similar results for
+low to moderate forces, with the differences kicking in as $fb/k_B T$
+moves above $l/b$.  For Puchner's revised numbers, this corresponds to
+
+\[
+  f \gt \frac{l}{b} \cdot \frac{k_B T}{b}
+    = \frac{\cos(\gamma/2)}{|\ln(\cos\gamma)|} \cdot \frac{k_B T}{b}
+    \approx 122 \text{ pN} \;,
+\]
+
+assuming a temperature in the range of 300 K.
+
+
+[carrionvazquez99]: http://dx.doi.org/10.1073/pnas.96.20.11288
+[puchner08]: http://dx.doi.org/10.1529/biophysj.108.129999
+[bustamante94]: http://dx.doi.org/10.1126/science.8079175
+[livadaru03]: http://dx.doi.org/10.1021/ma020751g
+
+[[!tag tags/theory]]