# 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 2C (i.e., ). So there is a range of values around the minimum that still predict the melting energies within an average error of 2C. 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 4C. 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  kcalmol (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/molK (Table 5.3).

JM Huguet 2014-02-12