# M. Error estimation in the enthalpy and entropy inference

Here we provide and estimation of the error ( ) of the 10 parameters  involved in the fit of Eq. 5.4. We simplify the notation by writing the  values given in Table 5.3 in vectorial form according to , where the subscript stands for minimum. Note that minimizes the error function (Eq. 5.4). By definition, the first derivatives of with respect to vanish at the minimum ( ). So we can write a Taylor expansion of around up to second order: (M.1)

where is a variation of the vector and is the Hessian matrix of second derivaties (M.2)

evaluated at the minimum . After fitting Eq. 5.4, the estimation of (in units of squared Celsius degrees C ) is lower than the typical experimental error in melting experiments, which is 2 C (i.e., ). So there is a range of values around the minimum that still predict the melting energies within an average error of 2 C. This range of values is what determines the error in the estimation of .

Following this criterion, we look for the variations around the minimum ( ) that produce a quadratic error of 4 C . We divide this quadratic error into the 10 fitting parameters (  ) and the 10 related ones (  ). So we look for each that induces an error of 4 C . Now, introducing and (divided by 20 parameters) into Eq. M.1 and isolating , we get one expression to estimate the errors of the 10 fitting parameters: (M.3)

which gives values between kcal mol (see Table 5.3). Now, the error in the estimation of the entropies ( ) can be obtained from error propagation of Eq. 5.5: (M.4)

where K is the temperature and are the experimental errors of our estimated NNBP energies from the unzipping measurements (Table 5.2). The errors range between cal/mol K (Table 5.3).

JM Huguet 2014-02-12