# Exploiting Sparse Statistics for a Sequence-Based Prediction of the Effect of Mutations
## 1 Introduction  
There are conflicting reports in the literature addressing the question whether the amino acid (AA) sequence of proteins that fold into well-defined structure is recognizably different from those that are randomly generated, as reviewed by [PERSON] et al. [1]. However, recent work [2] has shown that there is a significant difference in the relative propensities of AA triplets and quadruplets in sequences that represent actually folded proteins with a known structure and sequences that were generated randomly, even if the randomly generated sequences followed the natural amino acid propensities.  
By distilling the propensity statistics into a score for a given sequence this relative propensity difference was then used to make rather reliable predictions about the foldability of sequences not used in the development of the statistics. However, no significant correlation was found between the stability of a protein and its score.  
The previous work was limited to triplets and quadruplets since even for quadruplets some of the \(20^{4}\) sequences were absent in the sequence set of proteins with known structure that was used in the development of the score. The present work is based on recognizing the fact that the absence of an \(n\)-tuple in the sequences of folded proteins is also significant information thus it is worth examining what can be learned from the, obviously limited, statistics of AA pentuplets, hextuplets and heptuplets.  
Accurate prediction of the change in stability of a protein as a result of mutation is of fundamental interest both for the understanding of the mechanisms of disease-causing mutation and for the field of protein engineering. Given the complexity of proteins and their environment, quantitative predictions require statistical-mechanical approaches, namely, free-energy simulations generally requiring long calculations on hundreds of processors (for a review, see Reference [3]). Thus computationally simple approaches, even if of lower accuracy, can be of significant interest.  
The previous work also compared properties of folded proteins and intrinsically disordered proteins (IDP). As only small differences were observed, given the low precision of the data that the present paper is dealing with, IDPs are not considered in this work.  
Furthermore, there is a vast literature on the structural implication of various sequence motifs, including various attempts to predict structure from sequence. Given that the number of relevant protein sequences significantly outnumbers the number of proteins with a known structure, there is merit in studying the properties of protein sequences without including structural information, the aim of the present work.  
It will be shown that the pentuplets statistics, despite not sampling over 30% of the pentuplet space, is able to provide an even better separation of folding sequences from the randomly generated ones than the quadruplet statistics-based score used in Reference [2]. Furthermore, it will be shown that a simplified score, based on the limited hexuplet statistics and on the very limited heptuplet statistics is capable to produce useful prediction for the direction of stability change of a protein upon a mutation.