J. Shift function
The experimental FDC measurements are force vs. trap position as measured by the light-levers. The unzipping/zipping curves contain reproducible and recognizable landmarks (i.e., slopes and rips) which indicate the true position of the trap. Therefore it is possible to to take advantage of these landmarks to correct for the instrumental drift. Correction for drift is introduced in terms of a shift function
which is built in several steps. Due to its relevance for data analysis we describe the steps in some detail:
- Step 1. Since the distance
of the FDC is a relative (not absolute) magnitude, we firstly determine the origin of coordinates of the experimental FDC (see blue arrow in Fig. J.1a). In order to do so, we fit the last part of the experimental FDC (black and yellow curves in Fig. J.1a) that correspond to the stretching of the ssDNA when the hairpin is fully unzipped.
- Step 2. Having fixed the origin of coordinates we calculate the predicted FDC by using the Unified Oligonucleotide (UO) NNBP energies. It is shown in red (Fig. J.1a,b). The qualitative behavior is acceptable (all force rips are reproduced). However, the predicted mean unzipping force is higher than the value found experimentally and the force rips are not located at the correct position.
- Step 3. Next we generate a FDC with NNBP energies lower than the UO values until the mean unzipping forces of the predicted and the experimental FDC coincide. What we typically do is multiplying all the 10 UO NNBP energies by a factor (e.g., 0.95). The new NNBP energies have an absolute value 8-10% lower than the UO NNBP energies. The resulting FDC with these new energies is shown in green in Fig. J.1b (in this particular case we took
). Although the mean unzipping force of the green and black curves is nearly the same, there is a misalignment between the rips along the distance axis. Moreover, there are discrepancies between the predicted and measured heights of the force rips. As we will see below, the shift function will correct the horizontal misalignments and the NNBP energies will correct the discrepancies along the force axis.
- Step 4. We now introduce a shift function that uses the slopes of the sawtooth pattern of the FDC as landmark points to locally correct the distance to align the experimental data with the theoretical prediction. First, we want to know the approximate shape of the shift function and latter we will refine it. This is done by looking for some characteristic slopes of the sawtooth pattern along the FDC and measuring the local shift that would make the two slopes (theoretical and experimental) superimpose. Figure J.1c shows zoomed regions of the FDC and the blue arrows indicate the local shift that should be introduced in each slope to correct the FDC. The orange dots shown in the upper panel of Fig. J.1d depict the local shifts vs. the relative distance that have been obtained for the landmark points. These orange dots represent a discrete sampled version of an ideal shift function that would superimpose the predicted and the experimental FDC. Because these dots are not equidistant, we use cubic splines to interpolate a continuous curve every three landmark points. The resulting interpolated function that describes the local shift for any relative distance is shown in violet. Note that the violet curve passes through all the orange dots.
- Step 5. Starting from the cubic splines interpolation of the shift function that we have found (violet curve in upper panel Fig. J.1d) we can define new equidistant points (yellow dots in lower panel Fig. J.1)d) that define the same shift function. The yellow equidistant points are separated 100 nm. We call these yellow points Control points.
- Step 6. We now introduce the shift function into the calculation of the theoretical FDC. The results are shown in Fig. J.1e. Again, the black curve is the experimental FDC, the green curve is the predicted FDC without the local shift correction and the magenta curve is the predicted FDC with the local shift function obtained previously. Note that the magenta and green curves are identical, except for local contractions and dilatations of the magenta curve. The slopes of the magenta and the black curves now coincide. Still, the NNBP energies must be fit to make the height of the rips between the theoretical and experimental curve coincident.
- Step 7. At this point we start the Monte Carlo fitting algorithm. At each Monte Carlo step we propose new values for the 10 NNBP energies and we also adjust the shift function in order to superimpose the theoretical and experimentally measured FDCs. The shift function is adjusted by modifying the values of the control points (yellow dots in lower panel of Fig. J.1)d). The horizontal position of the yellow points is always the same (e.g., the yellow dots located at Relative Distance = -1000 nm will always be located there). What we change when we adjust the shift function is the value of each point (e.g., the yellow dot located at Relative Distance = -1000 nm may change its shift value from -54 to -20 nm). During the Monte Carlo optimizing procedure the shift function modifies its shape as the NNBP energies are modified. The results are shown in Fig. J.1f. Black curves show snapshots of the evolution of the shift function during the optimization process from the initial shape (red curve). The green curve shows the final shift function (the control points are not depicted in Fig. J.1f, only the interpolated shift function). When we finally calculate the theoretical FDC using the optimal shift function and the optimal NNBP energies we get the maximum overlap between the theoretical prediction and the experimental FDC. Black curve in upper panel in Fig. J.1g is the experimental FDC and red curve is the predicted FDC after having fit the NNBP energies and the shift function. The correction for drift has now finished. The optimal shift function is also shown in the right panel in Fig. J.1g.
Fit of the shift function. (a) Steps 1 and 2. (b) Steps 2 and 3. (c) Step 4. (d) Steps 4 and 5. (e) Step 6. (f) Step 7. (g) Step 7.