# Exploiting Sparse Statistics for a Sequence-Based Prediction of the Effect of Mutations
## 3 Results
### Foldability Prediction Using Pentuplet Statistics  
The distributions of the pentuplet scores for the PDB set and for the two randomly generated sets (RW and RU) were first calculated. Figure 1 shows the three distributions. There was an almost complete separation between the experimental and RW sets' distributions and a rather small overlap between the distributions of the experimental and RU sets. To quantitate this observation, the overlaps between the distributions were also calculated. They are 0.096, 0.009 and 0.324 for the PDB-RU, PDB-RU and RU-RW distribution pairs. For comparison, the overlaps between the quadruplets scores of the PDB and the random sets were 0.142 and 0.191 for the PDB-RU and PDB-RW, resp. Unlike the distributions of the quadruplet scores where the RU and RW distributions were nearly identical with 0.786 overlap, the two pentuplet-based random distributions have rather small overlaps. For triplets and quadruplets the PDB-RW overlap was also larger than the PDB-RU overlap, conforming to the expectation that the RW set should be more like the PDB set than the RU set. For the pentuplet score distribution, however, this relation was reversed. This was likely an artifact of the treatment of the missing statistics in the present approach. To test the predictive power of the pentuplet-based score, the scores of the new PDB set (that were not used for the development of the scores) were compared to newly generated RU and RW sets. A score was considered diagnostic of foldability if it was larger than the score where the PDB and random score distributions intersected, otherwise it was considered diagnostic of randomness. Table 1 summarizes the predictions using the quadruplet-based predictions from Reference [2] (using the RW set as reference) and the new pentuplet-based predictions using the both the RU set and the RW set as reference. The comparison is mixed between the quadruplet and pentuplet scores. For randomly generated sets the pentuplet scores were significantly better than the quadruplet scores. However, the pentuplet scores missed predicting foldable significantly more of the PDB set than the quadruplet score when the RU-based pentuplet distribution was used; using the RW-based distribution the number of missed predictions in the PDB set was essentially the same for the quadruplet and pentuplet scores. The fact that predictions on the randomly generated sets gave very few false negatives (i.e., if a sequence was predicted to be foldable it was most likely correct) suggested the combination of the quadruplet scores and pentuplet scores in the following way: whenever the quadruplet score predicts randomness, but the pentuplet score predicts foldability, change the prediction to foldable; otherwise \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Prediction** & **Score Source** & **New PDB Set** & **Uniform Random** & **Weighted Random** \\ \hline Folded & Quadruplets, RW & 3855 & 79.3\% & 3964 & 4.0\% & 4257 & 4.3\% \\ Random & Quadruplets, RW & 980 & 20.7\% & 96,036 & 96.0\% & 95,745 & 95.7\% \\ Folded & Pentuplets, RU & 2874 & 60.7\% & 3628 & 3.6\% & 499 & 0.5\% \\ Random & Pentuplets, RU & 1861 & 39.3\% & 96,372 & 96.4\% & 99,501 & 99.5\% \\ Folded & Pentuplets, RW & 3760 & 79.4\% & 30,822 & 30.8\% & 281 & 0.3\% \\ Random & Pentuplets, RW & 975 & 20.6\% & 69,177 & 69.2\% & 99,719 & 99.7\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the foldability predictions using the quadruplet and pentuplet scores. Figure 1: Pentuplet score distributions for the experimental set (full line), randomly generated from the uniform amino acid distribution (short dashes) and randomly generated with amino acids sampled with their natural propensity (long dashes). keep the quadruplet score prediction. Given the large difference in the distributions of the two randomly generated sets, combination with both reference pentuplet distributions were tested.