Worm Like Chain (WLC) model

The WLC model is used to describe the elastic response of semi-flexible polymers. The main property of these polymers is that some energy is required to bend them. The model assumes that the polymer is a continuous rod with a characteristic bending energy. The persistence length ($ l_p$) is the only parameter that controls the flexibility of the polymer. The other important parameter is the contour length ($ L_0$) that determines the length of the polymer chain.

The exact solution of the model requires to solve a transcendental equation numerically. However, Marko & Siggia [40] proposed the following approximated solution for the Force vs. Extension Curve in the WLC model

$\displaystyle f(x)=\frac{k_B T}{4 l_p}\left( \left( 1- \frac{x}{L_0}\right)^{-2} -1 + 4\frac{x}{L_0} \right)$ (F.1)

where $ f$ is the force applied and the ends of the polymer, $ x$ is the molecular extension, $ k_B$ is the Boltzmann constant and $ T$ is the temperature. This equation was derived after calculating the partition function of the model, computing its free energy and approximating the equation of state. However, this equation of state does not come from the actual free energy. So if we want to know the free energy $ G(x)$ of this expression, we need to calculate its integral which is
$\displaystyle G(x)$ $\displaystyle =$ $\displaystyle \int_{0}^{x} f(x')\,dx =$  
  $\displaystyle =$ $\displaystyle \frac{k_BT L_0}{4 l_p} \left( \left(1-\frac{x}{L_0}\right)^{-1}-1-\frac{x}{L_0}+2\left(\frac{x}{L_0}\right)^2\right)~.$ (F.2)

This expression is consistent with Eq. F.1 because the elastic free energy is $ G=0$ when the extension of the polymer is $ x=0$, i.e., when no force is applied at the ends of the polymer.

There are two main extensions of this model [160]. The first one involves polynomial corrections to Eq. F.1 up to order $ \mathcal{O}(x/L_0)^7$ so that the equation of state is as close as possible to the exact solution. In the case of dsDNA, this correction permits to fit the measured FEC with the same relative error at all range of forces. The second one involves an enthalpic correction that allows the contour length of the polymer to increase at high stretching forces. It enters in Eq. F.1 as a stretch modulus and it converts the vertical asymptote located at $ x=L_0$ into an oblique one.

In our unzipping problem, though, these two corrections are not necessary. We are interested in the elastic energy of the handles, so the total area under the FDC is hardly affected by the polynomial correction. Moreover, the unzipping of DNA occurs below 20 pN, where the effect of the enthalpic correction is small. Indeed, using the enthalpic correction to fit the FEC at low forces (<20-25 pN) induces some instability in the values of the fitting parameters. In particular, the persistence length increases whereas the stretch modulus decreases [161].

JM Huguet 2014-02-12