Experimental Test of Sinai’s Model in DNA Unzipping

The experimental measurement of correlation functions and critical exponents in disordered systems is key to testing renormalization group (RG) predictions. We mechanically unzip single DNA hairpins with optical tweezers, an experimental realization of the diffusive motion of a particle in a one-dimensional random force field, known as the Sinai model. We measure the unzipping forces $F_w$ as a function of the trap position $w$ in equilibrium and calculate the force-force correlator ${\mathrm{\Delta }}_m(w)$, its amplitude, and correlation length, finding agreement with theoretical predictions. We study the universal scaling properties since the effective trap stiffness $m^2$ decreases upon unzipping. Fluctuations of the position of the base pair at the unzipping junction $u$ scales as $u\sim m^{-\zeta }$, with a roughness exponent $\zeta =1.34\pm 0.06$, in agreement with the analytical prediction $\zeta =\frac{4}{3}$. Our study provides a single-molecule test of the functional RG approach for disordered elastic systems in equilibrium.

Figure 1

(a) Experimental setup. (b) Unzipping (red) and rezipping (blue) FDC’s demonstrating equilibrium behavior. The residual hysteresis at the end of the FDC is due to the DNA end-loop that slows down the initiation of stem formation upon reconvolution. (c) Experimental FDC’s, $F_w$, for various salt concentrations. The mean pinning force varies between 12–17 pN, and is nonuniversal.

Figure 2

Variation of the effective stiffness $m^2$ versus $w$ according to Eq. (3). The points correspond to the measured values of $1/m^2$ for the four FDC regions (each one shown with a different colour in the inset). The fit to data (dashed line) and the extrapolation to $w=0$ gives the stiffness of the optical trap, $k_{\mathrm{b}}=0.05\pm 0.01\mathrm{pN}·{\mathrm{nm}}^{-1}$. The inset illustrates the four studied regions in a FDC at 1M NaCl.

Figure 3

Measured ${\mathrm{\Delta }}_{m,T}(w)$ for the first region (red). $1\sigma$ error is shown as a pink strip. Deconvolution (black solid) and extrapolation to $w=0$ (black dot-dashed). The inset shows ${\mathrm{\Delta }}_{m,T}(w)$ at short range with subtraction of the peak at $w=0$, as explained in the main text.

Figure 4

Inset: The function ${\mathrm{\Delta }}_m(w)$ for the four regions changes with the measured $m$ (see Fig. 5). Main: Collapse of ${\mathrm{\Delta }}_m(w)$ according to Eq. (4) with $\zeta =4/3$. In black we show the theoretical ${\mathrm{\Delta }}_m(w)$, with ${\rho }_m=29(3)\mathrm{nm}$ as predicted by the microscopic disorder.

Figure 5

Top: Properties of the force correlator for the four regions in Fig. 2. The correlation length ${\rho }_m=C{\mathrm{\Delta }}_m(0)/{\mathrm{\Delta }}_m^{\prime }(0)$, with $C={\tilde{\mathrm{\Delta }}}_m(0)/{\tilde{\mathrm{\Delta }}}_m^{\prime }(0)=1.36$, see Eq. (7). Bottom: The scaling with $m$ of ${\rho }_m$ (red, solid), $\zeta =1.41\pm 0.10$, and ${\mathrm{\Delta }}_m(0)$ (blue dashed), $\zeta =1.29\pm 0.08$. Their mean $\zeta =1.34\pm 0.06$ agrees with the expected value, $\zeta =4/3$.