5.2.1 Elastic response of the ssDNA

The mean unzipping force of DNA depends on the elastic properties of the ssDNA according to the following explanation. Let us assume that the ssDNA is more rigid than usual. At the same force, a more rigid ssDNA is capable of storing less elastic energy ( $ E_b=f^2/2k$, see Eq. 3.16). Therefore, the force needed to acumulate enough elastic energy to break the base-pairs of the duplex increases. Thus, the elastic properties of ssDNA entering in Eq. 3.22 strongly determine the mean unzipping force of the FDC. In general, the elastic response of the ssDNA is basically determined by its stiffness (which is controlled by the persistence or Kuhn length) and the contour length (which is controlled by the interphosphate distance and the number of released bases).

A priori, one might think that the elastic parameters of the ssDNA are well-known and fixed. However, these values depend on the type of buffer at which the experiments are performed. It is crucial to know the elastic response of the ssDNA at forces below 20 pN because this is the range of forces at which unzipping is observed. Precisely at these forces it is more difficult to extract the ideal elastic response because the ssDNA tends to rezip and form secondary structures. So we need to measure a partial property of the experiment (ssDNA elasticity) that is hidden by the proper experiment (DNA unzipping).

The structure of the molecular construct used in our experiments helps to solve this problem. Indeed, the end loop added to the DNA duplex (see Sec. 3.2.3 and Appendix E) prevents the total separation of the two strands of DNA. When all the base pairs are disrupted, the molecular construct looks like a ssDNA molecule that can be pulled to obtain the elastic response. However, it is only possible to measure the elasticity of the ssDNA after the molecule has been completely unzipped, i.e., for force values higher than 20 pN. Yet we face the problem that different models and different values for the elastic parameters are compatible with the same curve above 20 pN, but they show very different properties for forces below 20 pN. So the problem persists.

The easiest solution then is to synthesize a fragment of ssDNA and perform pulling experiments on it. Since this fragment has a random sequence, the formation of secondary structure is minimized and we can observe the elastic response of the ssDNA at low forces. We carried out pulling experiments with a 3 kbp piece of ssDNA from $ \lambda $-DNA (see Appendix I for the details about the synthesis). In contrast to the molecular constructs for unzipping, this 3 kbp sequence is not self-complementary and it forms less secondary structure at low forces. The measured FDCs are converted to Force vs. Extension Curves (FECs) in which the force applied is depicted versus the extension of the ssDNA molecule $ x_m$ (rather than the relative position of the trap $ x_\mathrm {tot}$). The transformation from one to type of curve to another is performed point by point. So each point $ (x_\mathrm{tot},f)$ is converted into a point $ (x_m,f)$ by subtracting the elongation of the bead to the total distance of the system according to

$\displaystyle x_m=x_\mathrm{tot}-\frac{f}{k}$ (5.1)

where $ f$ is the force measured and $ k$ is the stiffness of the optical trap (see Fig. 5.3a,b). The resulting FEC is fit to a FJC or a WLC (Fig. 5.3a,b). The fitting parameters are the Kuhn length ($ b$) or the persistence length ($ l_p$) depending on the model, and the interphosphate distance ($ d$). The FEC is forced to pass through the point $ (x_m, f)=(0,0)$, while the number of bases is fixed to $ n=3000$ (see Appendix I). It has also been checked that the obtained elastic properties of the ssDNA match the last part of the unzipping FDC, when the molecule is fully extended (see Fig. 5.3c,d).

Both the FJC and the WLC models are similar but have some differences at low forces. The WLC model correctly fits the ssDNA elastic response for salt concentrations below 100 mM [NaCl]. Above this value, FDCs develop a plateau at low forces that cannot be reproduced by the ideal models (FJC or WLC). Figure 5.3 shows the fits of the FEC of ssDNA at different salt concentrations. As the salt concentration increases, the matching between the experimental FEC and the model is worse. The problem is the appearance of a force plateau above 100 mM [NaCl]. Such force plateau is related to the formation of secondary structures (self-hybridization) in ssDNA [139]. How can we determine the elastic response of the ssDNA at high salt concentration? At 100 mM [NaCl] we find that FJC fits better than WLC. At salt concentrations higher than 100 mM [NaCl], it is reasonable to assume that the ideal elastic response of the ssDNA is given by the FJC model, without considering the force plateau. So we fit the FEC to a WLC above 15 pN, because no secondary structure survives at this force. Indeed, above 15 pN, the FJC model fits data better.

Summing up, Fig. 5.3e shows the FEC at different salt conditions and the best fit to each of them and Table 5.1 shows the values of parameters for the best model at each salt.

Figure 5.3: Fit of the elastic response of ssDNA. (a,b) Elastic response of the ssDNA at 10 mM [NaCl] and 1 M [NaCl], respectively. The panels show the conversion of FDCs (black curves) into FECs (red curves) for the 3 kb ssDNA molecule (yellow arrows indicate the direction of the conversion). The green curves show the fit of the FEC to the ideal models (WLC or FJC). (c,d) Predicted FDC for the fully unzipped molecule (orange curve) superimposed on the experimental unzipping FDC (blue curve) at 10 mM [NaCl] and 1 M [NaCl], respectively. (e) Elastic response of a 3 kb ssDNA molecule at various salt concentrations. For each salt, the raw data of three different molecules are shown (orange, green and blue curves). Red curve shows the best fit to the elastic model. The models are: FJC for [NaCl]$ \le 100$ mM and WLC for [NaCl]$ >100$ mM.

Table 5.1: Elastic parameters of ssDNA at different salt concentration. $ d$ is the interphosphate distance for each model, $ l_p$ is the persistence length of the WLC and $ b$ is the Kuhn length of the FJC. The mean values were obtained after averaging over 5 molecules for each salt, except for 25 mM and 100 mM that were averaged over 4 molecules and for 50 mM that were averaged over 3.
  WLC model FJC model
  $ d=0.665$ nm $ d=0.59$ nm
Salt [NaCl] $ l_p$ (nm) $ b$ (nm)
10 mM 1.14 (0.1) -
25 mM 0.93 (0.1) -
50 mM 0.88 (0.1) -
100 mM - 1.37 (0.1)
250 mM - 1.25 (0.1)
500 mM - 1.20 (0.1)
1000 mM - 1.10 (0.1)

JM Huguet 2014-02-12