Opening rates of DNA hairpins: Experiment and model

We present single-molecule measurements of the opening rate of DNA hairpins under mechanical tension and compare with the results obtained from a reduced-degrees-of-freedom statistical mechanics model. We extract the apparent position of the transition state $s$ and find that the model, with no fitting parameters, reproduces the experimental measurements surprisingly well. Our values for $s$ are different from the ones obtained in previous experiments, where, however, the experimental conditions were different (different force fields, different salt concentrations). Thus it appears that the values of $s$ measured for relatively short hairpins are strongly affected by these experimental conditions.

Figure 1

(Color online) Experimental setup. (a) Evanescent-wave scattering optics. (b) The $1\text{−}\mu \mathrm{m}$-diam bead is tethered to the slide by a single 74-bases-long ssDNA molecule containing the 40-bases DNA hairpin. The average position of the bead moves up and down as the hairpin opens and closes. (c) One DNA hairpin investigated, $H_{\mathrm{AT}}$, is 40 bases long, with a stem length of $10\mathrm{bp}$, and a 20-bases-long loop. The second hairpin in this study $\left(H_{\mathrm{GC}}\right)$ differs only by the substitution $\mathrm{AT}\to \mathrm{GC}$ (arrow).

Figure 2

(Color online) (a) Part of a time series of the bead’s position with respect to the slide, showing the DNA hairpin tether repeatedly switching from the open (larger $h$) to the closed (smaller $h$) state. For each event we measure the duration of the closed state, $\Delta$. The interaction potential corresponding to this time series is shown in Fig. 3b. For this sample, $F_{\text{closed}}\approx 8.9\mathrm{pN}$. (b) Semilogarithmic plots of the opening rate vs the force in the closed state, $F_{\text{closed}}=(d\varphi ∕dx)$ $(x=x_{\mathrm{c}})$, for the hairpin $H_{\mathrm{AT}}$. The slope of the linear fit defines a characteristic length scale (the distance from the closed state to the transition state) $s\approx 0.42\pm 0.06\mathrm{nm}$, and the intercept defines the extrapolated zero-force opening rate $r\left(0\right)\approx (1.36\pm 0.20)\times {10}^{-1}{\mathrm{s}}^{-1}$. Vertical error bars represent the statistical error ($\pm 1∕2$ SD) from the finite sample size (determined by a simulation [4]). Horizontal error bars mainly represent the resolution in determining the position of the closed state on the time traces.

Figure 3

(Color online) (a) The cumulative distribution of the duration of the closed state $\Delta$ obtained from a time trace such as shown in Fig. 2a. With a total of $N\left(0\right)$ observed closing events, $N\left(0\right)-N\left(t\right)$ is the number of time intervals with $\Delta <t$. The slope in a semilogarithmic plot gives the opening rate of the hairpin [$(3.22\pm 0.24)\times {10}^{-1}{\mathrm{s}}^{-1}$ in this case]. The straight line is a linear fit. (b) The bead-slide interaction potential measured from the thermal motion of the bead with the hairpin in the open state. The solid line is a fit using Eq. (3). This potential was obtained from the time series of which Fig. 2a shows a part. The horizontal error bars on the experimental points (not shown) are the size of the bins $\left(0.5\mathrm{pN}\right)$; the vertical error bars are determined from the number of counts in the histogram of the bead’s position (not shown); for $\varphi ∕kT<4$ these error bars are smaller than the points and so are not shown. When the DNA hairpin loop is closed, shortening of the molecular tether pulls the bead up the repulsive part of the potential, increasing the tension on the tether. (c) Different tensions in the DNA give rise to different opening rates. The plot shows the cumulative distribution of the residence time in the closed state, for two different experiments corresponding to measured tensions in the closed state of 4.4 and $12.1\mathrm{pN}$. The slopes give opening rates of $1.87\times {10}^{-1}$ and $4.67\times {10}^{-1}{\mathrm{s}}^{-1}$, respectively.

Figure 4

(a) For the hairpin $H_{\mathrm{AT}}$, the quantity $r\left(0\right)=r\left(w\right)\exp [-w\left(s\right)∕T]$, vs the mechanical tension in the closed state, $F=\left(\partial \varphi \partial h\right)\left(x_{\mathrm{c}}\right)$, for different choices of the parameter $s$. $r\left(w\right)$ is the measured opening rate, $w\left(s\right)=\varphi \left(x_{\mathrm{c}}\right)-\varphi (x_{\mathrm{c}}+s)$ is the work performed by the external force in opening the bond, $\varphi$ is the measured potential energy surface, and $x_{\mathrm{c}}$ is the position of the closed state. The correct choice of $s$ should result in $r\left(0\right)$ (the opening rate at zero force) being the same for all measurements. The figure shows that if $s$ is chosen too small, there is a systematic increase of $r\left(0\right)$ for increasing $F$, while if $s$ is too large, there is a systematic decrease. The solid lines are best linear fits through the data plotted for different $s$. In this manner we determine $s\approx 0.53\pm 0.04\mathrm{nm}$ and $r\left(0\right)\approx (1.22\pm 0.16)\times {10}^{-1}{\mathrm{s}}^{-1}$. (b) Same analysis as in (a), for the hairpin $H_{\mathrm{GC}}$. We obtain the values $s\approx 0.58\pm 0.04\mathrm{nm}$ and $r\left(0\right)\approx (2.08\pm 0.52)\times {10}^{-2}{\mathrm{s}}^{-1}$. The error bars for these data points are similar to the ones in (a).

Figure 5

(Color online) Results from the simulation of the Peyrard-Bishop-Dauxois model. (a) The base-to-base distance in the course of time (MC $\text{steps}\times 1000$) for the first base pair of $H_{\mathrm{AT}}$, at constant force. (b) Semilogarithmic plot of the lifetime of the closed state vs force, for $H_{\mathrm{AT}}$, from the simulation. From the linear fit we extract $s=0.58\mathrm{nm}$. (c) The zero force lifetime of the closed state, $1∕r\left(0\right)$, for homogeneous AT and GC sequences as a function of the number of base pairs.

Figure 6

Force (proportional to MC time) vs displacement for the first bp of $H_{\mathrm{AT}}$ from the simulation at constant pulling rate.

Figure 7

(Color online) Rupture force distributions for the sequence used in [10] for four different loading rates $k_0$: (a) $k_0=5r_0$, (b) $k_0=25r_0$, (c) $k_0=2.5\times {10}^2{r}_0$, and (d) $k_0=2.5\times {10}^3{r}_0$, where $r_0={10}^{-8}\mathrm{pN}$/MC steps. Fitting the theoretical distribution of the single bond dissociation [23] to the numerical data we find both $s$ and zero-force opening rate are independent of $k_0$.