*4.3. Permeability Calculations*

We calculated osmotic permeabilities for four distinct UCP2 structures obtained from unbiased MD simulations, closely following the procedure illustrated by Zoonens et al. [20] and based on the algorithm described by Aksementiev and Schulten [50]. In this respect, the osmotic permeability of homologically modeled UCP2 was calculated using structures obtained after 2 μs of unbiased MD simulation, while the same property was determined for three distinct UCP2 NMR structures, i.e., structures obtained immediately after equilibration, after 0.2 μs and 2 μs of unbiased MD simulation. The four chosen systems were propagated in the NVT ensemble, where position restraints were applied on Cα atoms of the protein (500 kcal mol−<sup>1</sup> nm<sup>−</sup>2). The simulations were propagated for 10 ns with a 2 fs time step, with the last 5 ns used in the subsequent analysis. The simulation snapshots were saved every 250 steps, i.e., every 0.5 ps. PBC conditions were applied in all three directions and treated the long-range electrostatics using the PME method (see above for details).

The pore formed in the structure of both UCP2NMR and UCP2h possesses a complex topology; thus, careful choice of the region for the osmotic permeability calculations is necessary. Following the procedure of Zoonens et al. [20], we calculated the permeability taking into account only the central region of the pore. This region was rather wellpreserved during simulations, and its topology relatively simple, i.e., it could be accurately described as being roughly cylindrical in nature. In this respect, the chosen region for the UCP2NMR structures was defined by two roughly coplanar rings, each consisting of six Cα atoms. Each Cα atom was chosen to belong to a different transmembrane helix present in the protein. The chosen Cα atoms form a bottom and a top ring, and belong to residues 34, 85, 137, 181, 239, and 274, and residues 20, 101, 120, 194, 227, 288, respectively. Due to a different topology of the UCP2h protein, we used a similar ye<sup>t</sup> somewhat different choice of Cα atoms to delineate its central pore. In this respect, a bottom and a top ring were described using Cα atoms belonging to residues 34, 82, 137, 181, and 274, and residues 20, 101, 120, 192 and 288, respectively (see Figure S8).

Thus, the region encapsulated between the two chosen rings has the form of a cylinder with bases at the centers of mass of the bottom (R0) and the top ring (R1), respectively. The axis of this cylinder lies along the vector R1–R0. The radius of the cylinder (*r* = 2 nm) was chosen so that it is large enough to enclose all water molecules found in the analyzed pore of UCP2 protein.

Water molecules collective displacement within the protein pore at time *t* + Δ*t* of the MD simulation trajectory can now be calculated using the approach developed by Zhu et al. [74], i.e., via

$$n(t + \Delta t) = n(t) + \sum\_{i \in S(t, t + \Delta t)} \left(\frac{\Delta \mathbf{r}\_i \Delta \mathbf{e}}{L}\right)^i$$

where the union of all subsets of water molecules that are found inside the cylinder at time *t* and *t +* Δ*t* is denoted by *S(t +* Δ*t*), Δ**<sup>r</sup>***i* represents the displacement of *i*-th water molecule in the time window *t* to *t +* Δ*t*, **e** represents the unit vector along R1–R0. *L* denotes the length of a cylindrical region and is approximately equal to 1.95 nm in all considered cases. Importantly, displacements of water molecules that enter or exit the cylindrical region between the two consecutive frames were cut at the boundaries of the region in such a way that only the displacement of such water molecules inside the region is taken into account.

The collective diffusion coefficient of water inside the protein, *Dn*, was calculated using 〈*n(t)*〉<sup>2</sup> = 2 *Dnt*, with the average being obtained over 100 subtrajectories, each being 50 ps in length (5 ns of overall post-equilibration simulation time/100), Figure S9. The osmotic permeability was estimated using *Pf* = *vwDn*, where the average volume of a single water molecule is denoted by *vw*. Finally, thus obtained osmotic permeabilities were scaled by a factor of 1/2.87, since real water possesses larger viscosity compared to the used TIP3P water model [50].

#### *4.4. Binding of ATP in the UCP2 Cavity*

To inspect the geometry of the binding site of ATP in the UCP2 protein cavity, we performed a set of simulations (50 per inspected UCP2 structure). We initially placed the ATP molecule inside the UCP2 cavity for three distinct protein structures, UCP2h, and two UCP2NMR structures. More precisely, to represent UCP2h, we chose the structure obtained after 2 μs of its respective free MD simulation, while two distinct UCP2NMR structures, namely the structures obtained after 20 ns and after 2 μs of their respective free MD simulation, were utilized to inspect the behavior of ATP in the cavity of UCP2NMR. For each UCP2 structure, 50 different MD simulations in the duration of 20 ns each were performed (overall 1 μs per UCP2 structure). The ATP starting position was maintained for each of the 50 simulations, with the initial velocities being randomly generated, following the Boltzmann distribution. Thus, while the starting structure in each 50 simulation sets (for each investigated UCP2 structure) is represented by the same point in the conformational phase space, its position in the momentum phase space is different, representing overall distinct starting structures. To inspect whether the initial conditions biased the obtained results, we performed additional simulations (again 20 ns each) for the UCP2h structure, where we placed ATP molecule in five different spots in the cavity of UCP2h and performed 10 simulations for each starting configuration. In all simulations mentioned above, only the last 10 ns were used in the analysis. The first 10 ns were omitted (equilibration time). CHARMM36m force field parameters were used to describe ATP moiety together with the aforementioned parameters of UCP2 and DOPC lipids. The simulations were performed in the NVT ensemble, with all other MD simulation parameters being identical to the ones applied in the long 2 μs simulations (see Simulation Details).
