## 3 Results
### SHRs in Complex with SHH-N-Cholesterol (PP Complexes)
During the last 0.6 us of the MD simulation, the structure of the ER\(\alpha\)-SHH-N-cholesterol complex remains mostly unchanged (Figures S3 and S7). Interestingly, during this timeframe, the relative orientation of SHH-N to ER\(\alpha\) closely resembles its position in complexes with AR and PR (Figure S8). According to the R\({}_{\rm g}\) profiles (black curves in Figure S3, middle), complexes formed between AR or PR and SHH-N-cholesterol are slightly more globular than the complex formed between ER\(\alpha\) and SHH-N-cholesterol (see the final complex structures in Figure S8). This is further evident from the reduced number of amino acids from the central, more structured, part of the SHH-N protein that contribute to the stabilization of the complex with ER\(\alpha\), compared to AR and PR (as discussed below). To conclude, MD simulations suggest the possibility of a stable complex forming between the C-terminally cholesterol N-terminal domain of SHH and all three SHRs (ER\(\alpha\), PR and AR). In addition, MM/PBSA energies (Table 1) indicate that the binding of the N-terminal domain of SHH-N with covalently bound cholesterol to ER\(\alpha\) is found to be more \begin{table}
\begin{tabular}{c c c} \hline \hline
**Complex** & **Time Interval/ns** & **(\(\Delta H_{\rm bind}\pm\) SD/(kcal/mol)** \\ \hline ER\(\alpha\)–estradiol & & \(-12.87\pm 2.68\) \\ ER\(\alpha\)–cholesterol & & \(-17.50\pm 3.42\) \\ PR–progesterone & & \(-19.02\pm 2.97\) \\ PR–cholesterol & & \(-19.48\pm 3.63\) \\ \hline PR–SHH-N–cholesterol & & \(0\)–\(500\) \\  & 300–\(500\) & \(-20.99\pm 7.14\) \\  & 500–\(1000\) (a) * & \(-27.95\pm 7.80\) \\  & 500–\(1000\) (b) * & \(-23.25\pm 7.66\) \\ \hline AR-SHH-N–cholesterol & 0–\(500\) & \(-33.03\pm 10.27\) \\  & 300–\(500\) & \(-29.73\pm 11.09\) \\ \hline  & 0–\(300\) & \(-20.70\pm 7.28\) \\  & 300–\(500\) & \(-40.12\pm 9.25\) \\  & 600–\(800\) & \(-45.56\pm 9.5\) \\ ER\(\alpha\)–SHH-N–cholesterol & 800–\(1000\) & \(-36.29\pm 8.66\) \\  & 1100–\(1350\) & \(-42.40\pm 8.45\) \\  & 1400–\(2000\) & \(-51.51\pm 7.30\) \\ \hline \hline \end{tabular}  
* Two separate 1 us-long MD simulations of the PR–SHH-N-cholesterol complex are designated as a and b.  
\end{table} 
Table 1: The enthalpy contribution to the binding free energies, calculated based on MD trajectories of the simulated systems by using the MM/PBSA approach. The specific time intervals subjected to analysis are indicated. exothermic (\(\Delta H_{\rm bind}=-51.51\pm 7.30\) kcal/mol for the last 0.6 us), when compared to binding with AR (\(\Delta H_{\rm bind}=-33.03\pm 10.27\) kcal/mol) and PR (\(\Delta H_{\rm bind}=-27.95\pm 7.80\) kcal/mol for the last 0.5 us). Notably, when considering standard deviations, the binding enthalpies for AR and PR are similar. During MD simulations, cholesterol, whether free or attached to the C-terminus of SHH-N, remained tightly bound within the SHRs binding site, occupying a similar binding position as in the initial structure (illustrated in Figure S4), which was determined through the binding of its natural agonists. Ligands' RMSF profiles indicate higher mobility only for the cholesterol's aliphatic side chain atoms (atoms C25-C27, see Figure 2). According to per-residue MM/GBSA analysis, cholesterol gonane core is within the SHR binding site mostly stabilized by the same amino acid residues as identified for estradiol or progesterone in the complex with ER\(\alpha\) and PR, respectively (see Figures 3 and S10), wherein cholesterol's aliphatic side chain atoms are stabilized by M343, M421, I424, H524 and L525 residues in the ER\(\alpha\)-(SHH-N-)cholesterol complex, and L715, L797, L887, Y890 and C891 residue in the PR-(SHH-N-)cholesterol complex (depicted in Figures 5c and S11). The significance of van der Waal's interactions in ligand stabilization is emphasized by the substantial energy contribution from the H524 residue of ER\(\alpha\) and the C891 residue of PR to cholesterol binding.