6.1.2 Unzipping/Rezipping cycles at controlled force

The optical tweezers instrument is capable of producing DNA unzipping and rezipping at CF within an acceptable bandwidth. Figure 6.2 shows a full cycle of unzipping/rezipping of a 2.2 kbp sequence. The experiment is ready to start when the DNA molecular construct is stretched between two beads in the optical tweezers. Depending on how much the DNA is frayed (fraying is the melting of DNA in the extremities), the molecule might have some open base pairs at the beginning. In this particular case, we set the unzipping fork at a region in which the molecule has about $ \sim$100 open base pairs. When the pulling protocol starts, the force increases at the indicated loading rate. As the force increases, the distance increases exhibiting sudden horizontal hops. The resulting FDC monotonically increases and the force never retracts (except for some thermal fluctuations). Once the DNA molecule is fully open, the elastic response of the ssDNA is measured just like in the CP protocol. When the maximum force is reached, the loading rate is inverted and the force starts to decrease (rezipping). The measured elastic response of the ssDNA during the rezipping is almost identical to the unzipping. However, during the rezipping the closing of base pairs starts at lower forces than the opening of base pairs during the unzipping. As the force decreases, the extension of the molecule is reduced (and so the distance) until returning to the original starting point of the unzipping.

Figure 6.2: Full cycle of a CF unzipping/rezipping experiment of a 2.2 kbp sequence performed at a loading rate of 0.05 pN/s. Red (green) curve shows the unzipping (rezipping) at CF. The blue curve shows the expected equilibrium FDC at CF. We have superimposed the FDC at CP (in black) and the elastic response of the ssDNA (in orange).
\includegraphics[width=\textwidth]{figs/chapter6/fcycle.eps}

The most prominent characteristic of these unzipping/rezipping curves measured in one single cycle of pulling is the fact that they do not overlap, so a whole pulling cycle shows hysteresis. The hysteresis is one of the main indicators of irreversibility in thermodynamical processes. The area enclosed between the two unzipping/rezipping curves is equivalent to the work dissipated by the system during the whole cycle. This means that although the process has been carried out at a very low loading rate it is not reversible.

The time spent in a pulling cycle is not enough to equilibrate the system and obtain the equilibrium FDC. According to the Kramers theory [151,152], the time scale $ \tau$ to overcome an energetic barrier of height $ \Delta E$ is given by:

$\displaystyle \tau \approx \tau_0 \, \exp\left(\Delta E /k_BT\right)$ (6.1)

where $ \tau_0\approx 10^{-7}$ s is the microscopic time at which the base-pairs breath (i.e., open and close) [136,23], $ k_B$ is the Boltzmann constant and $ T$ is the temperature. The highest barriers at the coexistence force are of the order of $ 25$-$ 30$ $ k_BT$, which gives a mean passage time between hours and weeks. In principle, if we perform a CF pulling experiment slow enough, we should be able to obtain the equilibrium FDC. However, this is not experimentally feasible, because the time spent in such experiment would be extremely large and the pulling experiment could be ruined by several issues such as the breakage of the tether or the unavoidable long term drift.

Figure 6.2 also shows the predicted theoretical Force vs. Distance Curve at Controlled Force (FDC$ _f$) according to the calculations of Sec. 3.4.2. The difference between the experimental and theoretical FDC$ _f$ is extremely significant. It is another evidence that the CF experiments cannot be performed quasistatically at the laboratory time scale.

There is a major difference between the CF experiments and the CP experiments. For the latter, the equilibrium FDC can be obtained just by reducing the pulling speed down to reasonable values. At first sight, one can see that the FDC at CP (FDC$ _x$) lies between the unzipping and rezipping FDCs at controlled force. Although both curves have radically different shapes, the mean unzipping force is around $ \sim$16.5 pN in both cases. What is more, there seems to be a way to relate one type of curve with the other. Note that the FDC$ _f$ follows the slopes of the FDC$ _x$ until a force rip is achieved. At this point, the FDC$ _f$ increases its extension until it intercepts the next slope of the FDC$ _x$. This process is repeated until no slope is intercepted and the molecule is fully extended showing the elastic response of the ssDNA. Similarly, the rezipping FDC$ _f$ has a symmetric relation with the FDC$ _x$. Somehow, the experimental FDC$ _f$ that we can measure are the envelope curves of the FDC$ _x$.

JM Huguet 2014-02-12