Nonequilibrium Thermodynamics of DNA Nanopore Unzipping

Using theory and simulations, we carried out a first systematic characterization of DNA unzipping via nanopore translocation. Starting from partially unzipped states, we found three dynamical regimes depending on the applied force $f$: (i) heterogeneous DNA retraction and rezipping ($f<17\mathrm{pN}$), (ii) normal ($17\mathrm{pN}<f<60\mathrm{pN}$), and (iii) anomalous ($f>60\mathrm{pN}$) drift-diffusive behavior. We show that the normal drift-diffusion regime can be effectively modeled as a one-dimensional stochastic process in a tilted periodic potential. We use the theory of stochastic processes to recover the potential from nonequilibrium unzipping trajectories and show that it corresponds to the free-energy landscape for single-base-pair unzipping. Applying this general approach to other single-molecule systems with periodic potentials ought to yield detailed free-energy landscapes from out-of-equilibrium trajectories.

Figure 1

(a) System setup. Typical configuration of a DNA chain of 500 base pairs (oxDNA2 model) as it unzips during the driven pore translocation of one of its strands. Translocation is driven by the external force $f$, which exclusively acts on the DNA tract in the pore. A magnified view of the pore entrance is shown in the inset, where the random DNA composition is color coded as in the legend. (b) Translocation traces. The traces show the time evolution of the index of the nucleotide at the pore entrance $n(t)$ for different driving forces $f$. The traces of ten individual trajectories are shown for each force. Starting from a partially unzipped state, $n(0)\sim 220$, translocation proceeds forward (unzipping) or backward (rezipping) depending on the driving force magnitude; see sketches in panel (d). The number of translocated nucleotides since the start of the simulation is thus $\mathrm{\Delta }n(t)=n(t)-n(0)$. (c) The traces show the average translocation time $\tau$ ($x$ axis) of $\mathrm{\Delta }n$ nucleotides ($y$ axis) at different forces, normalized to the average translocation duration ${\tau }_{\mathrm{tot}}$.

Figure 2

Drift-diffusion regimes. For $f_0<f<f_1\sim 60\mathrm{pN}$, the average and the variance of the number of translocated nucleotides, $⟨\mathrm{\Delta }n⟩$ and ${\sigma }^2(\mathrm{\Delta }n)$, have an overall linear time dependence, as normal drift-diffusive processes. The crossover to the anomalous regime for $f>f_1$ is best appreciated in the right panels, where $⟨\mathrm{\Delta }n⟩$ and ${\sigma }^2(\mathrm{\Delta }n)$ are divided by $t$ and, additionally, by $f$, to facilitate comparison on a similar scale.

Figure 3

Derivation of the unzipping potential from nonequilibrium data. The data points in panel (a) show the average velocity and diffusion coefficient of the nanopore unzipping process at different forces $f$. The continuous theoretical curves are based on Eq. (1) and correspond to the least-square fit of the data via an optimal parametrization of the washboard potential $U(x)$ of Eq. (2). The potential is shown in panel (b) with and without various tilting forces.

Figure 4

Comparison with unzipping free energy from metadynamics calculations. (a) Heatmap of the 2D free-energy landscape for unzipping of two consecutive base pairs at $f=f_0=17\mathrm{pN}$. The landscape was computed with metadynamics simulations of oxDNA2, using two collective variables, $n_{\text{bp}}$ and $z$; see main text. (b) Thermodynamic reweighting and integration of the 2D landscape yield the potential in panel (b), which is the 1D free-energy profile at zero driving force, $f=0$. The profile (red) shows a good accord with the derived potential (black) tilted appropriately for zero driving force, $U(x)+f_{0}x$. The shaded band reflects the statistical uncertainty on $U_0$, 1.5 pN nm. The linear correspondence of $x$ and $n_{\text{bp}}$ was set with the abscissa of the minima, which are displaced relative to the untilted case. The vertical offset of the potentials, which is immaterial for free-energy differences, was set by best matching the height of the two minima.