## 4 Discussion  
SHH is a main ligand involved in the canonical activation of the HH-GLI signaling pathway. Mature SHH has cholesterol PTM, which is indispensable for its secretion, spread, and reception [48]. Those processes are accomplished by physical interaction between its cholesterol moiety and two proteins, the membrane-protein-dispatched homolog 1 (DISP1) and a member of the Scube family of secreted-proteins-signal peptide, CUB and EGF-like domain-containing protein 2 (SCUBE2) [49]. Our previous research has provided experimental evidence of a complex formation between the C-terminally cholesterol N-terminal domain of SHH and ER\(\alpha\) and AR proteins [18; 20]. In this study, our main focus was to investigate the potential binding of SHH-N-cholesterol to PR, the only member of the nuclear SHR subfamily not previously associated with SHH-N. Computational methods were utilized to predict the protein-protein binding poses and to assess the stability of the resulting complexes. Docking calculations, followed by robust MD simulations and free energy calculations, predict a stable binding of SHH-N-cholesterol protein with all three SHRs. The values of the PP binding energies suggest that C-terminally cholesterol N-terminal domain of SHH binds most strongly with ER\(\alpha\) (\(-\)51.51 \(\pm\) 7.30 kcal/mol) protein, and equally strongly with AR (\(-\)33.03 \(\pm\) 10.27 kcal/mol) and PR (\(-\)23.25 \(\pm\) 7.66 or \(-\)27.95 \(\pm\) 7.80 kcal/mol) proteins. These results suggest that a non-canonical interaction between PR and the C-terminally cholesterol N-terminal domain of SHH could exist, and consequently, that it could have some biological consequences. During microseconds-long MD simulations, cholesterol, free or attached to the SHH-N C-terminus, stays tightly bound in the SHRs' binding site. In the complex with ER\(\alpha\) or PR it occupies the same position as determined for its natural agonist, estradiol and progesterone, respectively. Interestingly, MM/PBSA calculations (Table 1) predict no significant difference in the binding affinity of PR towards progesterone (\(-\)19.02 \(\pm\) 2.97 kcal/mol), compared to its affinity for SHH-N-cholesterol. On the contrary, in the case of binding with ER\(\alpha\), SHH-N-cholesterol showed a more favorable enthalpy of binding (\(-\)51.51 \(\pm\) 7.30 kcal/mol) with the receptor, compared to estradiol (\(-\)12.87 \(\pm\) 2.68 kcal/mol). Considering the nearly identical binding affinities of ER\(\alpha\) towards estradiol and free cholesterol (\(-\)17.50 \(\pm\) 3.42 kcal/mol), it can be inferred that the additional stabilization observed in the ER\(\alpha\)-SHH-N-cholesterol complex stems from favorable PP interactions formed between SHH-N and the ER\(\alpha\) protein. The per-residue MM/GBSA analysis identified numerous favorable interactions between the unstructured C- and N-termini of SHH-N, along with the more structured central part of the SHH-N protein, and ER\(\alpha\), PR, and AR proteins. The analysis of both native and non-native contacts, along with the examination of hydrogen bonds, highlights a notably greater contribution from the N- and C-terminal residues of SHH-N to the stabilization of the complex with ER\(\alpha\), in contrast to the complexes with PR and especially AR, throughout the course of MD simulations. The MM/GBSA and hydrogen bond analyses reveal the crucial role of hydrogen bonds and non-polar interactions in stabilizing natural agonists and cholesterol moiety in the binding site of the SHRs. Notably, the contribution of hydrogen bonding is more pronounced in complexes with estradiol and progesterone. These interactions persist even during significant structural rearrangements of protein subunits observed in the MD simulation of the ER\(\alpha\)-SHH-N-cholesterol complex. To summarize, our calculations suggest that the formation of PR-SHH-N-cholesterol complex is theoretically plausible, though experimental verification is still pending and it is something that we are currently working on.