Velocity clamp force spectroscopy pulls are often fit to polymer models such as the worm-like chain (WLC). However, Puchner et al. 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:

(1)${F}_{\mathrm{WLC}}\left(x\right)=\frac{{k}_{B}T}{p}\left[\frac{1}{4}\left(\frac{1}{{\left(1-\frac{x}{L}\right)}^{2}}-1\right)+\frac{x}{L}\right]$

For the FRC, Puchner uses Livadaru's equation 46.

(2)$\frac{{R}_{z}}{L}\approx \left\{\begin{array}{ll}\frac{\mathrm{fa}}{3{k}_{B}T}& \text{for}\frac{\mathrm{fb}}{{k}_{B}T}<\frac{b}{l}\\ 1-{\left(\frac{\mathrm{fl}}{4{k}_{B}T}\right)}^{-\frac{1}{2}}& \text{for}\frac{b}{l}<\frac{\mathrm{fb}}{{k}_{B}T}<\frac{l}{b}\\ 1-{\left(\frac{\mathrm{fb}}{{\mathrm{ck}}_{B}T}\right)}^{-1}& \text{for}\frac{l}{b}<\frac{\mathrm{fb}}{{k}_{B}T}\end{array}\phantom{\rule{thickmathspace}{0ex}}.$

Unfortunately, there are two typos in Livadaru's equation 46. It should read (confirmed by private communication with Roland Netz).

(3)$\frac{{R}_{z}}{L}\approx \left\{\begin{array}{ll}\frac{\mathrm{fa}}{3{k}_{B}T}& \text{for}\frac{\mathrm{fb}}{{k}_{B}T}<\frac{b}{l}\\ 1-{\left(\frac{4\mathrm{fl}}{{k}_{B}T}\right)}^{-\frac{1}{2}}& \text{for}\frac{b}{l}<\frac{\mathrm{fb}}{{k}_{B}T}<\frac{l}{b}\\ 1-{\left(\frac{\mathrm{cfb}}{{k}_{B}T}\right)}^{-1}& \text{for}\frac{l}{b}<\frac{\mathrm{fb}}{{k}_{B}T}\end{array}\phantom{\rule{thickmathspace}{0ex}}.$

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.

(4)$\frac{{R}_{z}}{L}=1-{\left\{{\left({F}_{\text{WLC}}^{-1}\left[\frac{\mathrm{fl}}{{k}_{\mathrm{BT}}}\right]\right)}^{\beta }+{\left(\frac{\mathrm{cfb}}{{k}_{\mathrm{BT}}}\right)}^{\beta }\right\}}^{\frac{-1}{\beta }}+\frac{f}{\stackrel{˜}{\gamma }}\phantom{\rule{thickmathspace}{0ex}},$

where $l=b\frac{\mathrm{cos}\left(\gamma /2\right)}{\mid \mathrm{ln}\left(\mathrm{cos}\gamma \right)\mid }$ (Livadaru's equation 22) is the effective persistence length, $\beta$ determines the crossover sharpness, $\stackrel{˜}{\gamma }$ is the backbone stretching modulus, and ${F}_{\text{WLC}}^{-1}\left[x\right]$ is related to the inverse of Bustamante's interpolation formula,

(5)${F}_{\text{WLC}}\left[x\right]=\frac{3}{4}-\frac{1}{x}+\frac{{x}^{2}}{4}\phantom{\rule{thickmathspace}{0ex}}.$

By matching their interpolation formula with simlated FRCs, Livadaru suggests using $\beta =2$, $\stackrel{˜}{\gamma }=\infty$, 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:

(6)$l=b\frac{\mathrm{cos}\left(\gamma /2\right)}{\mid \mathrm{ln}\left(\mathrm{cos}\gamma \right)\mid }=0.366\text{nm}$

(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 $\mathrm{fb}/{k}_{B}T$ moves above $l/b$. For Puchner's revised numbers, this corresponds to

(7)$f>\frac{l}{b}\cdot \frac{{k}_{B}T}{b}=\frac{\mathrm{cos}\left(\gamma /2\right)}{\mid \mathrm{ln}\left(\mathrm{cos}\gamma \right)\mid }\cdot \frac{{k}_{B}T}{b}\approx 122\text{pN}\phantom{\rule{thickmathspace}{0ex}},$

assuming a temperature in the range of 300 K.

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