6.3 Scaling properties of unzipping

In 2002, Lubensky and Nelson published an extensive theoretical study of DNA unzipping at constant force [29]. Starting from a mesoscopic model, their work treated the unzipping of DNA as a phase transition. They provided the scaling properties of the system and the expected critical exponents at the coexistence (i.e., critical or mean) unzipping force.

One year latter, Lubensky and Nelson also contributed as co-authors in an experimental work carried out by Danilowicz et al. [23]. The unzipping of DNA was performed with magnetic tweezers and the results were compared with a coarse-grained model that qualitatively predicted the experimental observations. However, the resolution of the instrument was not good enough to compare the data with the theoretical results calculated one year earlier.

According to the calculations of Lubensky and Nelson, the number of open base-pairs as the exerted force approaches the critical force goes like,

$\displaystyle \overline{\left\langle m \right\rangle} = \overline{\left\langle \frac{n}{N} \right\rangle} \sim \frac{1}{(f_c-f)^2}$ (6.2)

where $ n$ is the number of open base-pairs, $ N$ is the total number of base-pairs of the molecule, $ \langle \cdots \rangle$ indicates the average over the statistical ensemble, $ \overline{\cdots}$ indicates the average over realizations of the sequence, $ f_c$ is the critical force and $ f$ is the force exerted on the molecule. This calculation is valid of a random heteropolymer. The 2.2 kbp sequence is a random heteropolymer, because the sequence of base-pairs can be considered random (with 4 different types of bases in the sequence). Lubensky and Nelson obtained a different law for homopolymers (i.e., with only one type of base-pair) which is given by: $ \overline{\left\langle m \right\rangle} \sim (f_c-f)^{-1}$

The detection of metastable states in Sec. 6.2 allows us to determine the number of open base-pairs of each experimental point of the FDC$ _f$. Since we know the total number of base-pairs of the molecule ($ N=2252$) we can obtain an experimental estimation of the curve defined in Eq. 6.2, i.e., a curve of $ \overline{\langle m \rangle}$ vs. $ f$ (see Fig. 6.6).

Figure 6.6: Scaling properties at the unzipping transition. The red curve shows the experimental data obtained from the unzipping of a 2.2 kbp sequence at a loading rate of 0.05 pN/s. The blue curve is the fit of the experimental data to the following function: $ m=A/(f_c-f)^2$, with $ f_c\simeq 19$ pN. The inset shows the same data in log-log scale.
\includegraphics[width=10cm]{figs/chapter6/scaling.eps}

The fit of the experimental data to Eq. 6.2 is satisfactory. A priori, we should not expect a perfect agreement. The reason is that Eq. 6.6 was predicted in equilibrium and averaged over realizations, while the experiments are not performed in equilibrium and we only have one sequence. Besides, the 2.2 kbp sequence is far from being an infinite molecule in which the thermodynamic prediction can be applied. So the fact that the parameter that fits the data ( $ f_c\simeq 19$ pN) is significantly higher than the actual critical force ($ f_c=16.5$ pN) might indicate that we are comparing non-equilibrium data with an equilibrium prediction. This result must be corroborated with more experiments performed on longer sequences.

JM Huguet 2014-02-12