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\section{Introduction}  Chemical shifts hold valuable structural information that is being used more and more in the determination and refinement of protein structure and dynamics \cite{20111782} with the aid of empirical shift predictors such as CamShift, Sparta+, ShiftX2, ppmone, CamShift \cite{19739624}, Sparta+ \cite{20628786}, ShiftX2 \cite{21448735}, ppmone ,  and Shaic. shAIC \cite{22293396}.  These methods are based typically based on approximate physical models with adjustable parameters that optimized by minimizing the discrepancy between experimental and predicted chemical shifts computed using protein structures derived from x-ray crystallography. The agreement with experiment is quite remarkable with RMSD values around 1, 0.3, and 2 ppm for carbon, hydrogen, and nitrogen atoms. Chemical shift predictions based on quantum mechanical (QM) calculations (mostly density functional theory, DFT) are becoming increasingly feasible for small proteins (xx) and Villa, Scheraga and co-workers have used gone on to develop a DFT-based chemical shift predictor for C$\alpha$ and C$\beta$ called CheShift-2 \cite{24082119}. Generally, these QM-based methods yield chemical shifts that deviate significantly more from experiment than the empirical methods, with RMSD values that generally are at least twice as large. However, many of these studies have also shown that the empirical methods are less sensitive to the details of the protein geometry and that QM-based chemical shift predictors may be more suitable for protein refinement \cite{Sumowski_2014}. Some of us recently showed \cite{24391900} that protein refinement using a DFT-based backbone amide proton chemical shift predictor (ProCS) yielded more accurate hydrogen-bond geometries and $^\text{3h}$\textit{J}$_\text{NC'}$ coupling constants involving backbone amide groups than corresponding refinement with CamShift. Furthermore, the ProCS predictions based on the structurally refined ensemble where yielded amide proton chemical shift predictions that were at least as accurate as CamShift. This suggests that the larger RMSD observed for QM-based chemical shift predictions may, at least in part, be due to relatively small errors in the protein structures used for the predictions and not a deficiency in the underlying method. However, in order to test whether this is generally true we need to include the effect of more than one chemical shift. In this study we extend ProCS to the prediction of chemical shifts of all back-bone atoms and C$\beta$ in a new method we call ProCS15. We describe the underlying theory, which is significantly different from the previous, amide proton-only, version of ProCS (hence the new name) and test the accuracy relative to full DFT calculations as well as experiment. We also compare the accuracy to CheShift-2 and other commonly used empirical chemical shift predictors using both single structures and NMR-derived ensembles.  
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