5.4.1 Inference of NNBP enthalpies and entropies

The unzipping experiments provide direct measurements of the free energies ( $ \epsilon _i$) and the salt corrections ($ m_i$) at $ T = 298$ K for all the NNBP motifs ( $ i = 1,\dots,10$), but no information about the enthalpy ( $ \Delta h_i$) and the entropy ( $ \Delta s_i$) is provided. In order to infer these two magnitudes, we should perform more unzipping experiments at different temperatures an apply Eq. 3.5. At present this cannot be achieved with the minitweezers experimental setup because the temperature cannot be controlled at will. The changes in temperature dramatically affect the optics of the instrument, which introduces undesirable drift effects that compromise the resolution of the experiment.

However, combining the results from unzipping experiments with the measurements of melting temperatures of several oligos obtained by optical melting experiments [143] we can infer the enthalpies and the entropies. In order to do so, we define an error function $ \chi^2$ that accounts for the mean squared error between the experimental melting temperatures [143] ( $ T_i^{\textrm{exp}}$) and the predicted ( $ T_i^{\textrm{pred}}$) ones for $ N=460$ different oligos and salt conditions,

$\displaystyle \chi^2 (\Delta h_1,\dots,\Delta h_{10})=\frac{1}{N}\sum_i^N \left...
...^{\textrm{exp}} - T_i^{\textrm{pred}}(\Delta h_1,\dots,\Delta h_{10}) \right)^2$ (5.4)

where $ \Delta h_i$ $ (i=1,\dots,10)$ are the NNBP enthalpies and $ T_i^{\textrm{pred}}$ are obtained according to Eq. 3.14 derived from the NN model. The NNBP entropies are fixed by 10 constraints that relate the free energies, the enthalpies and the entropies according to

$\displaystyle \epsilon_i=\Delta h_i - T\Delta s_i \qquad \Longrightarrow \qquad \Delta s_i=\frac{\Delta h_i- \epsilon_i}{T}$ (5.5)

where $ i = 1,\dots,10$; $ \epsilon_i\,(=\Delta g_i)$ are the experimentally measured free energies with unzipping experiments and $ T = 298$ K. Here, the enthalpies are fitting parameters that fix the entropies. Therefore the enthalpies and the entropies are fully correlated (their correlation coefficients are equal to 1).

The error function (Eq. 5.4) is minimized with respect to the 10 enthalpies using a steepest descent algorithm that rapidly converges to the same solution when starting from different initial conditions. Table 5.3 contains the best fit values for the enthalpies and the entropies which are compatible with the UO values. Appendix M describes how the error of the fitting parameters was estimated.

Table 5.3: Free energies and enthalpies given in kcal$ \cdot $mol$ ^{-1}$; entropies given in cal$ \cdot $mol $ ^{-1}\cdot $K$ ^{-1}$. Left block of columns show the UO values at standard conditions (25$ ^\circ $C and 1 M [NaCl]). Right block of columns show the thermodynamic values inferred from our unzipping experiments. The values in parenthesis indicate the estimation of the error (see Appendix M).
\begin{tabular}{\vert c\vert cccc\vert cccc\vert}
....3 & 0.114 & -0.84 & -8.31 (0.6) & -25.06 (2.1) & 0.091 \\ \hline

JM Huguet 2014-02-12