4.2.1 Approximate solution

If the disorder of the DNA energies ( $ \epsilon _i$) is neglected, Eq. 4.4 can be rewritten as

$\displaystyle E(x_\mathrm{tot},n)\simeq\frac{1}{2}k(x_\mathrm{tot}-2dn)^2-\mu n$ (4.6)

where $ \mu=\langle \epsilon_i \rangle$. By minimizing this expression with respect to $ x_\mathrm {tot}$, we immediately get the following result for the number of open base pairs:

$\displaystyle n^*(x_\mathrm{tot})=\frac{1}{2d}\left(x_\mathrm{tot}+\frac{\mu}{2dk}\right)$ (4.7)

which allows to express the minimum energy of the system as

$\displaystyle E_m(x_\mathrm{tot})=-\frac{\mu}{2d}\left(x_\mathrm{tot}+\frac{\mu}{4kd}\right)$ (4.8)

and obtain the FDC after calculating the derivative

$\displaystyle f=-\frac{\mu}{2d}$ (4.9)

The three previous expressions capture the dependence of the averaged number of open bps, energy and force on the external parameters. The solutions to this approximation are smooth expressions that collect the average behavior of the system over an ensemble of sequences (i.e., realizations of the disorder). Indeed, the bases open in a continuous fashion, the energy is linear with $ x_\mathrm {tot}$ and the unzipping force is constant. The black curves in Figs. 4.8b,c,d show the approximated solution superimposed on one disorder realization. Interestingly enough, the toy model predicts a plateau of force whose value is only determined by formation energy of the base pairing ($ \mu $) and the properties of the ssDNA ($ d$), and it is independent of the number of bases of the DNA ($ N$).

JM Huguet 2014-02-12