4.2 Toy model

The cooperative dynamics of DNA unzipping at low pulling regimes has been described as a stick-slip motion elsewhere [24]. The opening of base pairs during the unzipping process is produced through a series of avalanches between stable intermediate states that have different number of open base pairs. We want to describe the most relevant properties of avalanche dynamics in terms of a simple model that reproduces the statistical properties of DNA unzipping. We have developed a toy model, simpler than the one described in section 3.4, that captures the essential behavior of DNA unzipping.

In this toy model (see Fig. 4.8a), the energy of bead in the optical trap is also assumed to be quadratic (see Eq. 3.16). However, no elastic contributions of the molecular construct are taken into account and only the free energy of formation of the DNA duplex is considered (see Eq. 3.17). Summing the two energy contributions of the model we end up with:

$\displaystyle E(x_b,n)=\frac{1}{2}kx_b^2-\sum_{i=1}^n \epsilon_i$ (4.3)

where $ E(x_b,n)$ is the total energy of the system, $ k$ is the stiffness of the optical trap, $ x_b$ is the position of the bead in the optical trap and $ \epsilon _i$ are the NN energies of the DNA hairpin. In the toy model, the energy of the DNA is no longer given by a specific sequence. Instead, we will consider that the NNBP energies are Gaussian distributed $ \mathcal{N}(\mu,\sigma)$, with mean $ \mu=\langle \epsilon_i \rangle$ and standard deviation $ \sigma=\sigma_{\epsilon_i}$. Since the released ssDNA is taken as inextensible, its extension ($ x_s$) is given by $ x_s=2dn$, where $ d$ is the interphosphate distance, $ n$ is the number of open bps and the factor 2 stands for the two strands of ssDNA. By using the relation $ x_b=x_\mathrm{tot}-2dn$, the total energy of the system can be rewritten as:

$\displaystyle E(x_\mathrm{tot},n)=\frac{1}{2}k(x_\mathrm{tot}-2dn)^2-\sum_i^n \epsilon_i~.$ (4.4)

At fixed $ x_\mathrm {tot}$, the system will tend to occupy the state ($ n^*$) that minimizes the total energy of the system, i.e., $ E(x_\mathrm{tot},n^*)\le E(x_\mathrm{tot},n), \forall n$. Therefore, it is possible to define the function $ n^*(x_\mathrm {tot})$ (see Fig. 4.8c). Combining this function with Eq. 4.4 we can obtain $ E(x_\mathrm{tot},n^*(x_\mathrm{tot}))=E_m(x_\mathrm{tot})$ (see Fig. 4.8b), which gives the FDC (Fig. 4.8d) according to,

$\displaystyle f(x_\mathrm{tot})=\frac{\partial E_m(x_\mathrm{tot})}{\partial x_\mathrm{tot}}~.$ (4.5)

The final shape of the FDC depends on the values given to $ \epsilon _i$, ($ i=0,N$) in Eq. 4.4. Therefore, for each realization of $ \{\epsilon_i\}$ there is a univocal FDC. The FDCs obtained from this model reproduces the formal sawtooth pattern that is experimentally observed. Moreover, the opening of base pairs is discontinuous and the size of the opening is not constant, but shows a distribution of sizes.

Figure 4.8: Toy model. (a) Scheme. (b) Minimum energy vs. distance, $ E_m(x_\mathrm {tot})$, for one realization (red curve) of a $ 10^3$ base-pair long sequence. The following parameters have been used: $ k=60$ pN$ \cdot \mu $m$ ^{-1}$, $ d=0.59$ nm, $ \mu =-1.6$ kcal$ \cdot $mol$ ^{-1}$, $ \sigma =3.20$ kcal$ \cdot $mol$ ^{-1}$. (c) Open base pairs vs. distance, $ n^*(x_\mathrm {tot})$, depicted in blue. Inset shows a detailed view of the step function. (d) FDC, $ f(x_\mathrm {tot})$, depicted in green. The average behavior of the system is shown in black in panels b,c,d (see Sec. 4.2.1).

JM Huguet 2014-02-12