5.4 Salt dependence and thermodynamic properties of NNBP interactions

Up to now, we have seen how to extract the NNBP energies from an unzipping FDC. This section describes how this process can be repeated for several molecules and several salt conditions in order to extract conclusions about the salt dependence of the NNBP free energies, enthalpies and entropies.

The unzipping experiments have been performed at 7 different salt conditions for the 6.8 kbp sequence and at two salt conditions for the 2.2 kbp sequence. All them are quasistatic pulling experiments from which the equilibrium FDC can be obtained (see Fig. 5.11). At each salt condition, the experiment has been repeated for several molecules. The Monte Carlo optimization method has also been applied to fit all the collected experimental data to the model. The result is a collection of 10 NNBP energies (plus the loop contribution) for each salt condition and for each molecule. The error can be estimated from the averages between the individual molecules measured at each salt.

The best-fit energy parameters reduce the error between the measured FDC and the theoretical prediction (see Fig. 5.12a,b). That is, they give improved agreement between the experimental and the theoretical unzipping FDCs as compared with the UO values. Figures 5.12c,d show the average value and the standard error of the NNBP obtained for the 6.8 kbp sequence and the UO prediction for the NNBP energies at 10 mM and 1 M [NaCl]. It is interesting that some of the new values are in good agreement with the results given by SantaLucia [28] (e.g., CA/GT and AT/TA motifs) while others differ significantly (e.g., AA/TT and GA/CT at 10 mM NaCl and AC/TG and CC/GG at 1 M NaCl).

According to the UO salt correction, the NNBP energies are extrapolated homogeneously (i.e., the same salt correction is taken for all base-stack combinations) from standard salt conditions (1 M [NaCl]) down to lower salt concentrations (e.g., 50 mM) (see Sec. 3.3.3). However, such correction does not predict the observed unzipping force at low salt, especially for certain NNBP such as AA/TT or GA/CT.

Figure 5.11: FDCs at various monovalent salt concentrations ([Mon$ ^+$]=[Na$ ^+$]+[Tris$ ^+$], [Tris$ ^+$]=18 mM). Black curve [Na$ ^+$]=10 mM, red curve [Na$ ^+$]=25 mM, green curve [Na$ ^+$]=50 mM, blue curve [Na$ ^+$]=100 mM, magenta curve [Na$ ^+$]=250 mM, cyan curve [Na$ ^+$]=500 mM, orange curve [Na$ ^+$]=1 M.

Figure 5.12: Salt dependencies. (a,b) FDCs for the 6.8 kbp sequence at 10 mM NaCl (panel a) and 1 M NaCl (panel b). Black curve, experimental measurements; blue curve, UO prediction; red curve, our fit; magenta curve, elastic response of the fully unzipped molecule. The theoretical FDC is calculated in equilibrium, which assumes that the bandwidth is 0 Hz and the experimental data is filtered at bandwidth 1 Hz. If data is filtered at higher frequencies ($ >$1 Hz), hopping between states is observed and the experimental FDC does not compare well with the theoretical FDC at equilibrium. If data is filtered at lower frequencies ($ <$1 Hz), the force rips are smoothed and hopping transitions are averaged out. (c,d) NNBP energies and comparison with UO values at 10 mM [NaCl] (panel c) and 1 M [NaCl] (panel d). The following notation is used for NNBP: AG/TC denotes $ 5'$-AG-$ 3'$ paired with $ 5'$-CT-$ 3'$. Black points, UO values; red points, values for the 6.8 kbp molecule; blue points, values for the 2.2 kbp molecule. The values for the 6.8 kbp and the 2.2 kbp molecules have been obtained after averaging over six molecules. Error bars are determined from the standard error among different molecules.

A heterogeneous (sequence specific) salt correction could provide consistent results with the experiments. Such deviations are not unexpected, given the differences in solvation between specific nucleotides and salt ions [141,142]. However the effect has never been quantified. With this goal in mind, we apply the fitting algorithm to extract NNBP energies for data taken at many salt concentrations (see Fig. 5.13 red points). We find compatible NNBP energies between the two molecules (Fig. 5.13 blue dots).

As mentioned in Sec. 3.3.3 the UO prediction uses a non-specific salt correction for the different NNBP energies. Equation 3.13 gives the entropy correction of the NNBP for salt conditions different from 1 M [NaCl], which can be combined with Eq. 3.5 to write and expression for the free energy of formation of the NNBP at any salt concentration according to:

$\displaystyle \epsilon_i([\textrm{Mon}^+])=\epsilon_i^0-m\cdot \ln ([\textrm{Mon}^+])$ (5.3)

where $ \epsilon_i([\textrm{Mon}^+])$ is the energy of formation of the $ i\,$th NNBP ($ i$=1,$ \dots $,10) at a monovalent salt concentration of $ [\textrm{Mon}^+]$ (expressed in molar units), $ \epsilon _i^0$ is the NNBP energy at 298 K, 1 M monovalent salt and $ m$ is the non-specific prefactor equal to $ m=0.110$ kcal$ \cdot $mol$ ^{-1}$ at 298 K (see Fig. 5.13, green lines) [28,118]. The monovalent salt concentration $ [\textrm{Mon}^+]$ accounts for the total amount of ions with charge $ +1$ that are in the buffer. In the case of the TE buffer at which the unzipping experiments are performed, the total concentration of monovalent ions is given by two contributions: the $ [\textrm{Na}^+]$ ions and the $ [\textrm{Tris}^+]$ ones, which is $ [\textrm{Tris}^+]=18$ mM at $ \textrm{pH}=7.5$.

Figure 5.13: Salt corrections of the NNBP energies. Each panel shows the energy of a different NNBP parameter. Red (blue) points are the experimental results for the 6.8 kbp (2.2 kbp) sequence; green curve, UO non-specific salt correction; black curve, fit to Eq. 5.3 with adjustable parameters $ m_i$ ($ i$=1,$ \dots $,10,loop) and $ \epsilon _i^0$.

To define a heterogeneous salt correction within this scheme, it is only necessary to establish 10 sequence-specific prefactors $ m_i$ to be used with the same logarithmic dependence as shown in Eq. 5.3. Thus we fit all NNBP energies using NNBP-dependent parameters $ m_i$ ($ i$=1,$ \dots $,10,loop) and $ \epsilon _i^0$. The fits to each NNBP are shown in Fig. 5.13 (black lines) and the resulting fit parameters ( $ \epsilon _i^0$ and $ m_i$) are listed in Table 5.2. There we observe that the salt dependence of some NNBP parameters is well described by the UO non-specific correction (e.g., AT/TA and CA/GT) but most of them are better fit with some correction in parameters $ \epsilon _i^0$ and $ m_i$ (e.g., AA/TT,AC/TG,AG/TC).

Interestingly enough, a fit of the previous data with homogeneous salt correction (i.e., one single value $ m$ for all the NNBP energies) gives a value of $ m=0.104$ kcal$ \cdot $mol$ ^{-1}$, which is very similar to the value reported by SantaLucia [28]: $ m=0.114$ kcal$ \cdot $mol$ ^{-1}$. However, the root mean square error of a homogeneous fit is twice the error of a heterogeneous fit. Already with naked eye, it is possible to observe clear discrepancies in the slopes of some NNBP motifs (such as AA/TT, AG/TC, GA/CT) with a homogeneous salt correction (green curves in Fig. 5.13). Therefore, the use of a heterogeneous salt correction is worth, given the improvement in the fit. This is discussed in Sec. 5.5.

Table 5.2: 6.8 kbp (2.2 kbp) are the NNBP energies (in kcal$ \cdot $mol$ ^{-1}$) obtained from the averaged results for the two sequences (standard error in parenthesis) at 1 M [NaCl]. UO are the values extracted from ref. [28]. $ \epsilon _i^0$ and $ m_i$ are the standard energies and prefactors obtained from the fits shown in Fig. 5.13. These values are used in Eq. 5.3 to extrapolate the NNBP energies to other salt concentrations.
\begin{tabular}{\vert c\vert ccccc\vert}
...op & 2.30 (0.06) & 2.46 (0.09) & 2.68 & 2.43 (0.05) & - \\ \hline

JM Huguet 2014-02-12