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 WLC(x)=k BTp[14(1(1xL) 21)+xL]

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

(2)R zL{fa3k BT for fbk BT<bl 1(fl4k BT) 12 for bl<fbk BT<lb 1(fbck BT) 1 for lb<fbk BT.

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

(3)R zL{fa3k BT for fbk BT<bl 1(4flk BT) 12 for bl<fbk BT<lb 1(cfbk BT) 1 for lb<fbk BT.

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)R zL=1{(F WLC 1[flk BT]) β+(cfbk BT) β} 1β+fγ˜,

where l=bcos(γ/2)ln(cosγ) (Livadaru's equation 22) is the effective persistence length, β determines the crossover sharpness, γ˜ is the backbone stretching modulus, and F WLC 1[x] is related to the inverse of Bustamante's interpolation formula,

(5)F WLC[x]=341x+x 24.

By matching their interpolation formula with simlated FRCs, Livadaru suggests using β=2, γ˜=, and c=2. In his paper, Puchner suggests using b=0.4 nm and γ=22 . 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 γ=41 . This makes more sense because it gives a WLC persistence length similar to the one he used when fitting the WLC model:

(6)l=bcos(γ/2)ln(cosγ)=0.366 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 BT moves above l/b. For Puchner's revised numbers, this corresponds to

(7)f>lbk BTb=cos(γ/2)ln(cosγ)k BTb122 pN,

assuming a temperature in the range of 300 K.

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

Inverse FRC test matching Livadaru et al.'s figure 14