4.2.4 Fit of CUR size distributions

The analytical solution for the CUR size distributions is a problem that has not been solved yet. Although this calculation can be performed in the unzipping at controlled force [29], it is not that straightforward at controlled position. The computation requires extra efforts and it is not the primary goal of this section. Instead, an empirical expression for the CUR size distribution is provided in order to capture the qualitative dependence of the distribution with the tunable parameters of the model.

The CUR size distributions are well fit by an expression both in linear and log-log scale. It is a power law with a super-exponential cutoff:

$\displaystyle P(n) = An^{-B}\exp\left(-(n/n_c)^C\right)$ (4.10)

where $ P(n)$ is the probability of observing a CUR of size $ n$; $ A,B,C$ and $ n_c$ (cutoff size) are positive fitting parameters. All the distributions obtained from the simulations with the toy model are fit to Eq. 4.10 (see black curves in Fig. 4.11).

Figure 4.12 shows the dependence of the fit parameters with $ \sigma _{\epsilon _i}$ and $ k$. The cutoff parameter $ n_c$ gives an idea of the maximum size of the CURs. It has a physical meaning and can be understood as the parameter that indicates the width of the CUR size distribution: the higher the value of $ n_c$ the wider the distribution. As expected, the value of $ n_c$ increases with the amount of disorder $ \sigma _{\epsilon _i}$ (see Fig. 4.12a, panel $ n_c$) because the NN energies are more dispersed and the intermediate states show more variety of open base pairs. However, what is interesting from the simulations is that the value of $ n_c$ can be reduced at will by increasing $ k$. In fact, there is a limit in which $ n_c$ collapses down to 1 when $ k\geq 100$ pN$ \cdot $nm$ ^{-1}$ (see Fig. 4.12b, panel $ n_c$), indicating that all CUR sizes are of size 1. Under this circumstances, the mechanical unzipping of DNA is done one base-pair at a time and establishes and interesting limit for the experimental conditions that would allow to sequence DNA by force.

Figure 4.12: Fit parameters vs. toy model parameters. (a) Dependence on $ \sigma _{\epsilon _i}$ at $ k=60$ pN$ \cdot \mu $m$ ^{-1}$. (b) Dependence on $ k$ at $ \sigma _{\epsilon _i}=3.20$ kcal$ \cdot $mol$ ^{-1}$.

JM Huguet 2014-02-12