# Exploiting Sparse Statistics for a Sequence-Based Prediction of the Effect of Mutations
## 2 Materials and Methods  
As described in Reference [2], the sequences of the structures in the Protein Data bank (PDB) [4] in 2018 were downloaded as the file ss.txt from the PDB website, [[https://www.rcsb.org/](https://www.rcsb.org/)]([https://www.rcsb.org/](https://www.rcsb.org/)). Filtering out sequences over 50% sequence identity and fewer than 20 residues resulted in 35,667 sequences, referred to here as the experimental or PDB set. Furthermore, histidine tags were removed before each analysis. The set of sequences of a separate set of structures, deposited after the ss.txt file was also generated from the PDB to be used for the prediction of foldability; it will be referred to as the new PDB set. Two randomly generated sets were used. As described in Reference [2], one set sampled the amino acids from the uniform distribution (referred to as RU set) and the other with the probability of their respective natural propensities (referred to as RW set). The full sequences of proteins in the dataset of mutations [5] were matched to the sequences in the file ss.txt. Finding the correct match required adding the specification of starting residue IDs in the file ss.txt to the dataset--this was done manually. The score developed in Reference [2] for triplets and quadruplets has been applied to pentuplets in the present work, defined as follows. For each sequence and each construct \(p\) (triplet, quadruplet and pentuplet) the logarithmic score \(SC_{p}\) was defined as:  
\[SC_{p}=\left[\sum\
olimits_{i=1}^{N}\ln(PN_{i}/PR_{i})\right]/N, \tag{1}\]  
where \(N\) is the number of constructs in the sequence and \(PN_{i}\) and \(PR_{i}\) are the probabilities of finding the construct \(i\) in the experimental set and in the \(RW\) set, resp. For constructs \(i\) that were missing from the experimental set \(PN_{i}\) was set to 0.5/20\({}^{\prime}\) (\(f\) = 3, 4 and 5 for triplets, quadruplets and pentuplets, resp.). For hexamers and heptamers, where the sequences actually seen in the set of folded proteins used is a small fraction of the total number, a simpler score \(SSC_{p}\) was defined:  
\[SSC_{p}=\left[\sum\
olimits_{i=1}^{N}n_{i}\right]/N, \tag{2}\]  
where \(n_{i}\) is the number of occurrences of the construct \(i\) in the experimental set. This choice was motivated by the fact that the hexuplet and heptuplet counts are so sparse that they can not be considered to be a reasonable approximation of their probability of occurrence. It should be noted that the problem of sparse statistics can be also dealt with using a reduced AA alphabet, as done, e.g., in [6]. For the study of mutations, however, it would be of limited use since a significant number of mutations would not represent change in terms of the reduced representation. While the simple score \(SSC_{p}\) was used only to study mutations, the pentuplet score \(SC_{p}\) was also tested for its ability to predict the foldability of a sequence employing the methods used in Reference [2]. This test consisted of the following steps:  
1. Determine the distribution of scores over the PDB set. 2. Determine the distribution of scores over the RW set, consisting of 100,000 sequences of 200 residues. 3. Calculate the overlap between the two distributions to see how well the scores can distinguish between folding and non-folding sequences. 4. For a given sequence, calculate its score and see where it lies with respect to the score value at the intersection of the two distributions, resulting in the prediction regarding the foldability of that sequence. For the study of a mutation using the \(n\)-tuplet statistics (\(n=3\), 4, 5, 6, 7), the scores of all \(n\)-tuplets that included the mutated residue were averaged before and after the mutation. The change in the scores was then compared with the change in melting temperatures that are diagnostic of the stability of the protein in question.