## 2 Materials and Methods
### Molecular Docking and Molecular Modeling Details  
Protein-protein (PP) complexes between the C-terminally cholesterol N-terminal domain of SHH (SHH-N-cholesterol; PDB code 6 RVD, chain C, resolution 3.5 A) [34] and the ER\(\alpha\), PR or AR ligand-binding domain (PDB code 2 AM9, chain A, resolution 1.64 A; further in text AR) [35], so-called ER\(\alpha\)-SHH-N-cholesterol, PR-SHH-N-cholesterol and AR-SHH-N-cholesterol complexes (respective order), were also built. The cholesterol molecule attached to the SHH-N C-terminal in the complexes with the ER\(\alpha\) and PR proteins was bound in the same position as determined for cholesterol alone in the aforementioned complexes. The position of the cholesterol moiety bound to the SHH-N C-terminal in the complex with AR was determined by superimposing the structure of AR with ER\(\alpha\) from the complex with SHH-N-cholesterol. Mutual orientation of SHH-N and ER\(\alpha\), PR or AR proteins in the PP complex was determined by performing a docking study with the Hdock server ([[http://hdock.phys.hust.edu.cn/](http://hdock.phys.hust.edu.cn/)]([http://hdock.phys.hust.edu.cn/](http://hdock.phys.hust.edu.cn/)) (accessed on 13 April 2023)) [36]. More specifically, ER\(\alpha\) (chain A from PDB code 1 ERE), PR (chain A from PDB code 1A28) or AR (chain A from PDB code 2 AM9) were selected as receptor molecules, and the N-terminal domain of SHH (chain C from PDB code 6 RVD) was selected as a ligand molecule for predicting the ER\(\alpha\)-SHH-N, PR-SHH-N or AR-SHH-N complex structures (respective order) by applying the Hdock server. The E353 residue of ER\(\alpha\), Q725 residue of PR or Q711 residue of AR and G197 residue of SHH-N (residue covalently bonded to cholesterol moiety) were selected as binding site residues in order to identify PP binding poses that could allow binding of cholesterol moiety at the cholesterol binding site. No complex structure with direct interaction between E353-G197 residues in the ER\(\alpha\)-SHH-N complex, Q725-G197 residues in the PR-SHH-N complex or Q711-G197 residues in the AR-SHH-N complex, was obtained, as the unstructured C-terminus of the SHH-N protein closely interacts with a protein surface (Figure S1). However, in a few (2-3 out of top 10) identified complex poses (Figure S2), the cholesterol moiety at C-terminus of SHH-N could bind in the cholesterol binding site of ER\(\alpha\), PR or AR after slight modification, i.e., after detachment of SHH-N unstructured C-terminus from the adjacent alpha helix (residues Q100-N115). This was accomplished by manually changing the dihedral angles of the amino acid residues from the unstructured SHH-N C-terminus. For the further modelling study, one binding pose for each complex structure was selected (more precisely the one that required the least intervention in the C-terminus of the SHH-N protein to allow binding of cholesterol moiety in a position that most closely resembles the one in the complex with cholesterol alone).  
The complexes were solvated in an octahedron box filled with OPC water molecules [37], ensuring an 11 A thick water molecules buffer around the protein. The Cl\({}^{-}\) ions were added to neutralize the system. All Arg and Lys residues in the structures were positively charged (+1e), and Glu and Asp residues were negatively charged (\(-\)1e) as expected at physiological conditions. The histidines were in a neutral state with a proton at the N\(\varepsilon\) atom, except for the H140 of SHH-N, H398 of ER\(\alpha\) and H689 of AR, which were in a neutral state with a proton at the N8. The obtained complex was parametrized by the AmberTools20 module tleap, using a ff19 SB force field [38]. Cholesterol and estradiol were parametrized within the gaff force field [39] and AM1-BCC charges were assigned. For the zinc cation, non-bonded parameters that we developed were used [40].