5.2.2 Drift and shift function

The drift of the instrument is one of the major problems in single molecule experiments. The drift is a low frequency systematic deviation of measurements due to macroscopic effects. In the case of the minitweezers, the drift is mainly due to dilatations or contractions of the instrument produced by local changes of temperature or air flows in the room. The drift might be responsible of several effects such as the change in the relative position between the micropipette and the optical trap or the distortion of the optical path.

The importance of drift depends on the kind of the experiment and the protocol used. The unzipping experiments are performed at very low pulling speeds (typically around 10 nm/s), so measuring a whole unzipping/rezipping FDC may take 10 minutes or longer. Therefore it is useful to model the drift in order to remove its effects and extract accurate estimates for the NNBP energies. In order to do so, we introduce into the model a shift function that locally corrects the position of the trap along the FDC (see Appendix J). The final goal of the shift function is to make the slopes and force rips of the experimental and the theoretical FDC match each other. This leads to an improved match between the theoretical and the experimental FDCs.

The shift function is defined by equidistant control points that define a set of cubic splines. The shape of the shift function can be modified by tunning the location and the local shift of each of these control points. Appendix J describes in further detail the issues related with this function.

The shift function has a very important property: It is completely uncorrelated with the NNBP energies. Indeed, the shift function can be introduced into the mesoscopic model without affecting the NNBP energy values. Therefore, in the complete model, the shift function controls the horizontal matching between both curves (i.e., the distance) and the NNBP energies control the vertical matching (i.e., the force). The result is that an improved agreement between the experimental and the theoretical unzipping FDCs is achieved (see Fig. 5.4).

Figure 5.4: Effect of the shift function on the theoretical FDC. Blue curve shows the experimental FDC measured on a 6 kbp sequence. (a) Red curve shows the FDC predicted by the mesoscopic model without the shift function. (b) The inclusion of the shift function locally shifts each rip and the matching is better.
\includegraphics[width=\textwidth]{figs/chapter5/shiftfunceffect.eps}

Although each molecule has its own shift function, these have similar shapes (see Fig. 5.5). We have checked that the undulations of the shift function are not an artifact of the spline interpolation (see Appendix J). The undulations remain when the number of control points of the shift function is increased. The net shift observed in some curves (around $ \sim$100 nm) might be explained by improper calibration of the distance (around 4%). The undulations observed in the shift function might be due to non-linearities in the light-lever (i.e., trap position) measurements or interference fringes in the lenses or in the pellicle located along the optical path to the PSDs. The undulations in the shift function might also be correlated with the DNA sequence as emerges from the fact that undulations observed in different molecules of the same sequence appear at nearby positions. This might indicate new effects in the unzipping curves not accounted for in the NN model (e.g., the presence of next nearest-neighbor corrections).

Figure 5.5: Shift function. The figure shows the shift function for a few 6.8 kbp molecules at different salt conditions. The inset shows the shift function for different molecules of 2.2 kbp at 500 mM [NaCl] (=red) and 1 M [NaCl] (=green). The gray shaded region corresponds to trap positions where DNA is fully unzipped. In this region, the local shift nearly vanishes.
\includegraphics[width=\textwidth]{figs/chapter5/shiftexamples.eps}

It has also been checked whether the correction introduced by the shift function could be explained by a dependence of the Kuhn length on the contour length. Such dependence has not ever been reported. Yet it is interesting to evaluate the consequences of such hypothetic dependence. By letting the Kuhn length depend on the number of open base pairs, the position of the theoretical and experimental slopes and rips match each other if the Kuhn length increases as the contour length decreases. However this matching occurs at the price of an increasing average mean unzipping force as the molecule unzips and the ssDNA is released, an effect which is not experimentally observed. We conclude that the shift function is probably due to instrumental drift superimposed to imperfect calibration of the distance and non-linear optical effects.

JM Huguet 2014-02-12