Cavity approach for modeling and fitting polymer stretching

The mechanical properties of molecules are today captured by single molecule manipulation experiments, so that polymer features are tested at a nanometric scale. Yet devising mathematical models to get further insight beyond the commonly studied force-elongation relation is typically hard. Here we draw from techniques developed in the context of disordered systems to solve models for single and double-stranded DNA stretching in the limit of a long polymeric chain. Since we directly derive the marginals for the molecule local orientation, our approach allows us to readily calculate the experimental elongation as well as other observables at wish. As an example, we evaluate the correlation length as a function of the stretching force. Furthermore, we are able to fit successfully our solution to real experimental data. Although the model is admittedly phenomenological, our findings are very sound. For single-stranded DNA our solution yields the correct (monomer) scale and yet, more importantly, the right persistence length of the molecule. In the double-stranded case, our model reproduces the well-known overstretching transition and correctly captures the ratio between native DNA and overstretched DNA. Also in this case the model yields a persistence length in good agreement with consensus, and it gives interesting insights into the bending stiffness of the native and overstretched molecule, respectively.

Figure 1

The cavity method solution vs heat bath Monte Carlo and large- and small-$f$ limits. As an example, we fix here $b_B=4.04$ nm, $J_B=4.04$ pN nm and $T=20{}^{\circ }C$ for the value of $\beta$. The main panel shows the force-elongation curve. Our theoretical solution (red solid line) agrees very well with the Monte Carlo elongation (blue filled circles). Both results, in turn, are also captured by the small (green dashed line) and large (orange dot-dashed line) force limits when $f$ goes to zero or to large values, respectively. Such results confirm the cavity approach may be profitably exploited to study the stretching problem. The inset shows the correlation length $\xi \left(f\right)$ as a function of the stretching force. Again the cavity result [Eq. (13), red solid line] reproduces extremely well the Monte Carlo simulations (blue filled circles), ensuring that our approach may also be used to evaluate the correlation length for each value of the stretching force. Note that both results correctly converge to the persistence length ${\xi }_0$, Eq. (2), when $f\to 0$.

Figure 2

Model fit to ssDNA stretching experiments. The main panel shows the experimental force elongation (blue circles), vs the WLC interpolation formula [18, 35] (orange dashed line) and the cavity method solution (11) (red solid line). The parameter values returned by the fit are $L_{\mathrm{WLC}}=2\mu m$ for the WLC contour length and ${\xi }_0^{\mathrm{WLC}}=1.04$ nm for the WLC persistence length and $b_B=0.64$ nm, $J_B=6.13$ pN nm for the cavity solution. $\beta$ was fixed by assuming room temperature $T=20{}^{\circ }C$. The measured elongation is normalized over the known number of bases (3000) times the monomer length $b_B$. The agreement between our theory and the measurements is confirmed by the ${\chi }^2=0.063$; the WLC interpolation fit yields instead ${\chi }^2=0.098$. The inset shows the correlation length $\xi \left(f\right)$ as a function of the stretching force (solid line), computed from Eq. (13). It is a decreasing function departing from the value $\xi \left(0\right)=0.78$ nm at zero force (dashed line), i.e., the persistence length ${\xi }_0$ of the molecule.

Figure 3

Model fit to dsDNA stretching experiments. (a) The experimental elongation normalized over the native monomer length $b_B^{\left(0\right)}$ (blue circles) vs the cavity method elongation (21) (red solid line). To obtain the theoretical curve, we proceeded first to fit the experimental measurements below 50 pN with model (1) and then fixed the values for B-DNA to fit the remaining range of forces with model (17). The parameter values yielded by the fit (we again assumed a temperature $T=20{}^{\circ }C$ to fix $\beta$) are reported in Table 1. (b) the fraction of B-DNA (solid line) and S-DNA (dashed line) as a function of the stretching force $f$. We recall that we do not make any specific assumption on the actual conformation of the S-DNA and that, for us, this is just a generic overstretched phase of B-DNA. We find the overstretching transition to be highly cooperative, in agreement with consensus: S-DNA appears very suddenly around 70 pN and it rapidly picks up to the whole molecule. (c) The correlation length $\xi \left(f\right)$ as a function of the stretching force. It is a decreasing function of $f$ that correctly starts at the persistence length value ${\xi }_0=43$ nm.