M. Error estimation in the enthalpy and entropy inference

Here we provide and estimation of the error ( $ \sigma_{\Delta h_i}$) of the 10 parameters $ \Delta h_i^0$ $ (i=1,\dots,10)$ involved in the fit of Eq. 5.4. We simplify the notation by writing the $ \Delta h_i^0$ $ (i=1,\dots,10)$ values given in Table 5.3 in vectorial form according to $ \vec{\Delta h}_m$, where the subscript $ m$ stands for minimum. Note that $ \vec{\Delta h}_m$ minimizes the $ \chi^2(\vec{\Delta h})$ error function (Eq. 5.4). By definition, the first derivatives of $ \chi^2(\vec{\Delta h})$ with respect to $ \Delta\vec{h}$ vanish at the minimum ( $ \vec{\nabla}\cdot\chi^2(\Delta\vec{h}_m)=0$). So we can write a Taylor expansion of $ \chi^2(\vec{\Delta h})$ around $ \vec{\Delta h}_m$ up to second order:

$\displaystyle \chi^2(\Delta\vec{h}_m+\delta\vec{\Delta h})\simeq \chi^2(\vec{\D...
... \mathbf{H} \left( \chi^2(\vec{\Delta h}_m) \right) \cdot \delta \vec{\Delta h}$ (M.1)

where $ \delta \vec{\Delta h}$ is a variation of the $ \vec{\Delta h}_m$ vector and $ \mathbf{H} \left( \chi^2(\vec{\Delta h}_m) \right)$ is the Hessian matrix of second derivaties

$\displaystyle H_{ij}=\frac{\partial^2\,\chi^2 ( \Delta h_1^0,\dots,\Delta h_{10}^0)}{\partial \Delta h_i^0 \, \partial \Delta h_j^0}$ (M.2)

evaluated at the minimum $ \vec{\Delta h}_m$. After fitting Eq. 5.4, the estimation of $ \chi^2(\vec{\Delta h}_m)=1.74$ (in units of squared Celsius degrees $ ^\circ $C$ ^2$) is lower than the typical experimental error in melting experiments, which is 2$ ^\circ $C (i.e., $ \chi^2=4$). So there is a range of $ \vec{\Delta h}$ values around the minimum $ \vec{\Delta h}_m$ that still predict the melting energies within an average error of 2$ ^\circ $C. This range of values is what determines the error in the estimation of $ \vec{\Delta h}_m$.

Following this criterion, we look for the variations around the minimum ( $ \delta \vec{\Delta h}$) that produce a quadratic error of 4$ ^\circ $C$ ^2$. We divide this quadratic error into the 10 fitting parameters ( $ \Delta h_i^0$ $ i = 1,\dots,10$) and the 10 related ones ( $ \Delta s_i^0$ $ i = 1,\dots,10$). So we look for each $ \delta \Delta h_i$ that induces an error of 4 $ ^\circ\mathrm{C}^2/20 = 0.2^\circ$C$ ^2$. Now, introducing $ \chi^2(\vec{\Delta h}_m+\delta \vec{\Delta h})=4$ and $ \chi^2(\vec{\Delta h}_m)=1.74$ (divided by 20 parameters) into Eq. M.1 and isolating $ \delta \vec{\Delta h}$, we get one expression to estimate the errors of the 10 fitting parameters:

$\displaystyle \delta \Delta h_i = \sigma_{\Delta h_i} = \sqrt{\frac{2\cdot(4-1....
...rac{0.226}{H_{ii}\left( \chi^2 (\vec{\Delta h}_m)\right)}}, \qquad i=1,\dots,10$ (M.3)

which gives values between $ 0.3 - 0.6$ kcal$ \cdot $mol$ ^{-1}$ (see Table 5.3). Now, the error in the estimation of the entropies ( $ \sigma_{\Delta s_i}$) can be obtained from error propagation of Eq. 5.5:

$\displaystyle \sigma_{\Delta s_i}=\frac{1}{T}\left( \sigma_{\Delta h_i} + \sigma_{\Delta \epsilon_i} \right) \qquad i=1,\dots,10$ (M.4)

where $ T = 298.15$ K is the temperature and $ \sigma_{\Delta \epsilon_i}$ are the experimental errors of our estimated NNBP energies from the unzipping measurements (Table 5.2). The errors range between $ 1.2 - 2.2$ cal/mol$ \cdot $K (Table 5.3).

JM Huguet 2014-02-12