[[!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.

I've written an `inverse_frc` implementation in
[[crunch.py|Comparing_velocity_clamp_experiments/crunch.py]] for
[[comparing velocity clamp experiments]].  I test the implementation
with [[frc.py|Comparing_velocity_clamp_experiments/frc.py]] by
regenerating [Livadaru et al.'s figure 14][livadaru03].

[[!img Comparing_velocity_clamp_experiments/figure-14.png
  alt="Inverse FRC test matching Livadaru et al.'s figure 14"
  title="Inverse FRC test matching Livadaru et al.'s figure 14"]]


[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]]