# In-Silico Characterization of von Willebrand Factor Bound to FVIII
## 2 Materials and Methods
### Protein Contact Networks  
We built the protein contact networks (PCNs) on FVIII complexes starting from the.pdb files, as previously described [27]. In a PCN, protein residues are the network nodes. Links between nodes are active contacts between residues, e.g., when the inter-residue distance lies between 4 and 8 A, to account for non-covalent residue-residue interactions. The mathematical representation of the PCN is given by the adjacency matrix, defined as:  
\[A_{ij}= \begin{cases}1&if\ 4\ <d_{ij}<8\\ 0&otherwise\end{cases}  \tag{1}\]  
where \(d_{ij}\) is the Euclidean distance between the \(i\)-th and \(j\)-th residue, defined as:  
\[d_{ij}=\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}+\left( z_{i}-z_{j}\right)^{2}} \tag{2}\]  
where \(P_{i}=\left\{x_{i},\ y_{i},\ z_{i}\right\}\) and \(P_{j}=\left\{x_{j},\ y_{j},\ z_{j}\right\}\) are the coordinates in a cartesian space of the i-th and j-th residues, respectively (represented by the coordinates of their \(\alpha\)-carbons). Once the PCN was built, we computed the node degree \(k_{i}\) for the \(i\)-th node, defined as the number of its links with other nodes, computed as the sum of the elements on the \(i\)-th row of the adjacency matrix \(A\):  
\[\mathbf{k}_{i}=\sum_{j}\mathbf{A}_{ij} \tag{3}\]  
In order to characterize the topology of the protein-protein interactions, for each protein-protein interface we identified links between nodes belonging to the different interfacing chains and, accordingly, we introduced the inter-chain degree of each node \(\mathbf{k}_{i}^{IC}\) as the number of links it shares with residues belonging different protein chains. Nodes (residues) endowed with high inter-chain degree are defined as network hotspots of the protein complex interface, addressing their significant role in protein-protein interactions. The energy of a graph \(E\) is defined as the sum of the absolute values of the adjacency matrix \(A\) eigenvalues. Although this is a purely topological descriptor, it captures some physical energy properties of the protein molecular structures [40], particularly oligomers interactions [41]. Focusing on a given interface between two chains, \(A_{i}\) and \(A_{j}\), the overall inter-chain degree for a given interface \(\sum\mathbf{k}_{A_{i}A_{j}}\) is computed as the sum of the inter-chain degree of residues belonging to a single chain, characterizing the overall interface strength. We defined the average inter-chain degree as the average inter-chain degree value over the number of residues participating in the interface. We adopted the geometrical descriptors of protein interfaces according to the method of [PERSON] et al. in [42] for each interface between two chains in the complex: 1. the total number of residues \(Q\) for each chain in the interface; this number is in general lower than the total interface degree, due to multiple links between residues participating to the interface; 2. the length of the chain involved in the interface \(R\); 3. the interface \"roughness\" \(Q\)/\(R\) (previously introduced [9]); 4. the interface amino acid range, \(IAR=R\)/\(N\) being \(N\) the total number of residues in the chain. For a given interface between two chains, \(A_{i}\) and \(A_{j}\), the average value of the inter-chain degree is simply given as:  
\[<k_{A_{i}A_{j}}>=\frac{\sum k_{A_{i}A_{j}}}{Q_{A_{i}}+Q_{A_{j}}} \tag{4}\]We introduced energy descriptors, including the topological description provided by the PCNs method. Considering that the interaction energy is higher if the contact distance is smaller, we introduced a weight for each contact:  
\[e_{ij}=\frac{1}{d_{ij}} \tag{5}\]  
which is also the generic element of the interface energy matrix \(E\), defined as:  
\[E=E_{ij}= \begin{cases}e_{ij}=\frac{1}{d_{ij}}&if\ 4\ <d_{ij}<8\ \ and\ the\ residues\ belong\ to\ dif\ f\ f\ chains\\ \ 0&otherwise\end{cases}