4.2.3 CUR size distribution

Equation 4.7 is useful to describe the average number of open base pairs. However, for each realization of the energies, $ \epsilon _i$, the function $ n^*(x_{tot})$ is a discontinuous function and has to be numerically calculated (see Fig. 4.10a black curve). Each discontinuity represents an opening of base pairs, i.e., a CUR. The size of a CUR is the difference of the number of open base pairs between two states. This way, we can calculate the size of the CUR from the $ n^*(x_{tot})$ function by extracting the step size of the discontinuities (Fig. 4.10a, red curve). A distribution of CUR sizes can be obtained for one realization (Fig. 4.10b, blue curve, inset).

Figure 4.10: CUR size distribution. (a) Black curve shows the number of open base pairs for one sequence of $ 10^3$ base pairs ($ k = 0.06$ pN$ \cdot $nm$ ^{-1}$, $ a = 0.59$ nm, $ \mu =-1.6$ kcal$ \cdot $mol$ ^{-1}$ and $ \sigma _{\epsilon _i}=$0.44 kcal$ \cdot $mol$ ^{-1}$). Red curve shows the size of the CURs. (b) CUR size distribution.
\includegraphics[width=\textwidth]{figs/chapter4/CURtoymodel.eps}

The average distribution of CUR sizes can be obtained by simulating random realizations, calculating the CUR size distribution for each realization of the disorder and averaging over them. By varying the parameters of the model along a wide range we observe how the shape of the CUR size distribution depends on them. Generally speaking, we have observed that the shape of the CUR size distribution is independent of the mean NNBP energy ($ \mu $). It mostly depends on the standard deviation of the NNBP energies $ \sigma _{\epsilon _i}$ and on the trap stiffness $ k$. Figure 4.11 shows these dependencies for a model with the following parameters: $ d=0.59$ nm, $ \mu =-1.6$ kcal$ \cdot $mol$ ^{-1}$ simulated for a $ 10^4$ bp long sequence over $ 10^4$ realizations of the disorder.

Figure 4.11: Average CUR size distributions plotted in log-log scale. (a) For different values of $ \sigma _{\epsilon _i}$ at fixed $ k=60$ pN$ \cdot \mu $m$ ^{-1}$. (b) For different values of $ k$ at fixed $ \sigma _{\epsilon _i}=3.20$ kcal$ \cdot $mol$ ^{-1}$. The black curves show the fits of all the distributions to Eq. 4.10.
\includegraphics[width=\textwidth]{figs/chapter4/CURsigmak.eps}

JM Huguet 2014-02-12