Why is the average PRE ratio of a trajectory not calculated as average of the individual ratios?

[AlexanderJussupow]

Hi,

I have a question about the code, particularly related to the following lines in PRE.py:

gamma_2 = np.ma.MaskedArray(gamma_2, mask=np.isnan(gamma_2)) gamma_2_av[self.measured_resnums] = np.ma.average(gamma_2, weights=self.weights, axis=0).data i_ratio = self.r_2 np.exp(-gamma_2_av self.delay) / ( self.r_2 + gamma_2_av ) np.savetxt(self.outputprefix+'-{}.dat'.format(self.residue),np.c[self.resnums,i_ratio,gamma_2_av],header='residue i_ratio gamma_2')

By calculating the average of gamma_2, the i_ratio of a trajectory does not equal the average i_ratio of all individual frames (or the average of i_ratio of multiple trajectories), as exp(average(X)) != average(exp(X)).

As an illustration, here are the PRE results of a (from two individual trajectories) combined trajectory: vs. the results of the two individual trajectories (same amount of frames):

Does this part of the code work as intended? It is consistent with the description in the paper, but I am not sure if this is correct from a theoretical standpoint.

Best regards

Alexander Jussupow

[gitesei]

Hi Alexander,

Thanks for your question. The code works as intended as the observable that the forward model (the “model-free” formalism) predicts is primarily the per-frame Gamma_2 (https://doi.org/10.1021/cr900033p), which is proportional to <r^-6> where the ensemble average is taken over the rotamers of the paramagnetic probe. Assuming that the motion of the paramagnetic probe is much faster than the protein conformational changes, we calculate as the ensemble average over the trajectory frames. The predicted can be directly compared with the observed Gamma_2 obtained from multiple-time-point measurements, which provide a quantitative estimate of the PRE. On the other hand, the Gamma_2 extracted from the intensity ratio measured using a single-time-point method is considered to be a less accurate estimate because of the assumptions involved, e.g. (i) that the decay in the magnetization of the amide protons is exponential and due only to transverse relaxation and (ii) that the decay occurs during a "total INEPT time" estimated from the pulse sequence of the HSQC experiment. Therefore, we use the forward model to predict and, in order to compare with the experimental intensity ratios, we apply the same approach that is commonly used to interpret single-time-point measurements (https://doi.org/10.1021/bi000060h). I hope this answers your question.

Best regards, Giulio

[AlexanderJussupow]

Hi,

Thanks for the comprehensive answer.

Best regards

Alexander Jussupow