## 2 Materials and Methods
### FOD Model  
Fuzzy oil drop (FOD) model has been described in detail in [31; 32; 33], hence only its key features are reminded in this paper (with example figures in the Supplementary Materials). FOD requires tertiary structure of a protein as the source for its calculation. The results of this calculation are two \"main\" hydrophobicity distributions:  
* \(O\)--observed hydrophobicity distribution based on pairwise hydrophobic interactions between residues, calculated using [PERSON]'s polynomial [34];
* \(T\)--theoretical hydrophobicity distribution calculated using 3D Gauss capsule fit to the molecule (Figure S1), based on location of residues in respect to this \"drop\". An \(i\)-th residue in the sequence (of length \(N\)) is assigned two values: \(T_{i}\) and \(O_{i}\). The protein is then analyzed on the basis of comparison between these \(T\) and \(O\) distributions. A perfectly soluble molecule would have \(T\equiv O\) (\(T_{i}\) = \(O_{i}\) for all \(i\) between 1 and \(N\)), but it is the differences between those distributions that reveal the structural and functional properties as seen from the perspective of hydrophobic core formation [35]. Since \(T\) and \(O\) are normalized to enable their comparison, differences between them are measured using Kullback-Leibler divergence entropy (D\({}_{\text{KL}}\)) [36]. The result of this comparison is called the \(RD\) coefficient (relative distance) which expresses the \(T\) vs. \(O\) \"distance\" on a scale between 0 and 1 (Figure S2). The closer the value of \(RD\) is to 0, the higher accordance between \(T\) and \(O\) is registered, with 0 meaning that \(T\equiv O\). On the other hand, value of \(RD\) closer to 1 denotes higher discordance. Use of \(RD\) as a measure is possible due to introduction of \(R\) (\"random\")--a distribution opposite to 3D Gauss function, representing situation where all residues have uniform hydrophobicity regardless of their location or their amino acid (\(R_{i}\) = \(1/N\) for all \(i\) between 1 and \(N\)), with \(RD\) = 1 meaning that \(O\equiv R\). \(RD\) = 0.5 is therefore the accordance vs. discordance threshold, marking a tipping point between observation of the development of a stable hydrophobic core and the lack of such stability. \(T\), \(O\) and \(R\) hydrophobicity profiles (and subsequently the \(RD\) value binding them together) can be calculated for any structural unit, typically a chain, domain or complex, but also for helices, sheets or for any number of residues (in sequence or scattered), as long as all \(T_{i}\) and all \(O_{i}\) are above 0. In each of these cases, the 3D Gauss capsule (the \"drop\" ellipsoid) is fitted to the effective atoms of selected residues. Once \(T\), \(O\) and \(R\) profiles are obtained for the input structure, \(RD\) can also be used to analyze its sub-domain fragments. This is useful to determine the status of a specific part in respect to the whole, for example a sheet in the chain or a chain in the complex. By calculating \(RD\) only for an extracted subset of \(T\), \(O\) and \(R\) distribution values, one can express the participation of this selection in stability of the hydrophobic core [37]. There is also a fourth distribution--\(H\)--which denotes the (also) normalized intrinsic hydrophobicity of residues. It is based purely on their amino acid and matching value of hydrophobicity parameter (\(H\) parameter in short), acquired by default from the FOD hydrophobicity scale (\(0.0<\text{KEDQRNPSTGAHYLVMWIFC}\leq 1.0\)). Calculation of \(H\) is, actually, necessary to obtain \(O\). It is used for hydrophobicity-based analysis from the sequence perspective, which is especially useful in the case of amyloids [38]. In addition to \(RD\), correlation coefficients (CC) can be calculated to express another type of relation between \(T\), \(O\) and \(H\), yielding three normalized values labeled \(HvT\), \(ToO\) and \(HvO\) (i.e., \(ToO\) is the correlation coefficient of \(T\) vs. \(O\)).