*4.1. Chemicals*

Agarose (#3810), KCl (#6781), Na2SO4 (#8560), 2-(N-morpholino)ethanesulfonic acid (MES, #4256), sodium dodecyl sulfate (SDS, #0183), tris(hydroxymethyl)-aminomethane (Tris, #AE15), chloroform (#AE54) and ethylene glycol-bis(β-aminoethyl ether)-N,N,N,N - tetraacetic acid (EGTA, #3054) were purchased from Carl Roth GmbH & Co. KG (Karlsruhe, Germany). Hexane (#296090), hexadecane (#296317), palmitic acid (#P0500), stearic acid (#S4751), arachidic acid (#A3631), linoleic acid (#L1376) and arachidonic acid (#A3611), dimethyl sulfoxide (DMSO, #472301), the purine nucleotides adenine and guanine tri-, di-, and mono-phosphate (ATP, #A2383; ADP, #A2754; AMP, #01930; GTP #G8877; GDP, #G7127; and GMP, #G8377), carboxyatractyloside (CATR, #C4992) and bongkrekic acid (BA, #B6179) were purchased from Sigma-Aldrich (Vienna, Austria). 1,2-dioleoyl-sn-glycero-3- phosphocholine (DOPC, #850375P), 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine (DOPE, #850725P) and cardiolipin (CL, #710335P) came from Avanti Polar Lipids Inc. (Alabaster, AL, USA).

#### *4.2. Cloning, Purification and Reconstitution of Murine ANT1*

Cloning, purification and reconstitution of murine ANT1 followed a previously established protocol [26]. The protein concentration in proteoliposomes was measured with the Micro BCATM Protein Assay Kit (Thermo Fisher Scientific, Prod. #23235, Waltham, MA, USA). Protein purity was verified by SDS-PAGE and silver staining (Supplementary Figure S1).

#### *4.3. Exchange Rate Measurements of mANT1*

ANT-mediated exchange of ADP/ATP was measured radioactively using 3H-ATP (Prod. #NET420250UC, Perkin Elmer, Waltham, MA, USA) following the protocol as described elsewhere [26] (Supplementary Figure S2).

#### *4.4. Electrophysiological Measurements of mANT1*

Planar lipid bilayers were formed from proteoliposomes as described previously [31,52]. FAs were added to the lipid phase before membrane formation. Proper membrane formation was verified by measuring membrane capacitance (C = 0.72 ± 0.05 μF/cm2), which is independent of the presence of protein, FA and inhibitor. Current-voltage (I-U) measurements were performed with a patch-clamp amplifier (EPC 10USB, Werner Instruments, Holliston, MA, USA). The specific total membrane conductance (Gm) at 0 mV was obtained as the slope of a linear fit of the experimental data at applied voltages from −50 mV to + 50 mV and normalized to the membrane area in cm2. Purine nucleotides (solved in distilled water, pH = 7.34) and ANT-specific inhibitors BA and CATR (solved in DMSO) were added to the buffer solution before forming bilayer membranes. The concentrations of each substrate are indicated in the figure legends. Membrane conductance expressed in relative units was calculated according to [7].

Measurement and calculation of H+ turnover rate of ANT followed the established protocol [31]. The addition of Tris increased the pH value of the buffer solution on the cis side of the membrane to a value of pH = 8.34.

#### *4.5. Fluorescence Correlation Spectroscopy (FCS)*

The average number of ANT1 per liposome was measured using FCS [32,53]. In brief, proteoliposomes obtained after reconstitution of the ANT1 were extruded using a Mini-Extruder system (Avanti Polar Lipids Inc., Alabaster, AL, USA) with a membrane nanopore filter with a pore diameter of 100 nm (Avestin Europe GmbH, Mannheim, Germany, LFM-100). ANT1 was labeled with ATTO 488-maleimide [54] (Sigma-Aldrich, Vienna, Austria; 28562-1MG-F). We used size exclusion chromatography (Sephadex® G-50 Superfine, Merck, Vienna, Austria; GE17-0041-01) to remove the unbound dye. The average residence time, <sup>τ</sup>D, and ANT1- containing proteoliposomes number in the focal volume were derived from the autocorrelation function (G(τ)) of the temporal fluorescence signal (Supplementary Figure S5). To measure the signal, a commercial laser scanning microscope (LSM 510 META/ConfoCor 3, Carl Zeiss, Jena, Germany) equipped with avalanche diodes and a 40× water immersion objective was used. The standard model for two-component free 3D diffusion was applied [55]:

$$\mathbf{G}(\pi) = 1 + 1/(\mathbf{n}\,(1 + \pi/\tau\_{\rm D})) \tag{1}$$

where the number of fluorescent particles, n, in the detection volume, Veff, was determined as n = Veff C, where C is the particle concentration. The diffusion coefficient (D) was determined as D = <sup>ω</sup>2/4<sup>τ</sup>D, where ω = 0.16 μm is the diameter of the confocal volume cross-section as determined from the calibration experiments.

Dissolving the liposomes with 2% (*v*/*v*) SDS was expected to increase the particle number if liposomes contained more than one ANT1. The average number of ANT1 per liposome, <NANT1*<sup>&</sup>gt;*, was obtained from the ratio of the particle number per confocal volume after and before the addition of SDS.

The protein per lipid ratio (ρ) was estimated by:

$$\rho = \frac{\text{N}\_{\text{ANT1}}}{\text{N}\_{\text{Lipids}}} = \left. \frac{<\text{N}\_{\text{ANT1}}>}{<\text{N}\_{\text{Lipids}}>} \right|\_{\text{Lipois} \text{compones}} \tag{2}$$

The average number of lipids per liposomes (<NLipids>) is calculated by the ratio of the surface area of the liposome with radius r and the average area per lipid (AL ≈ 0.6 nm2) of a membrane containing DOPC, DOPE and CL [56,57]:

$$<\text{N}\_{\text{Lipids}}> = 2\frac{4\pi\text{r}^2}{\text{A}\_{\text{L}}}\tag{3}$$

Thus, Equation (2) gives:

$$\rho = < \mathcal{N}\_{\rm ANT1} > \frac{1}{2} \frac{A\_{\rm L}}{4\pi r^2} \tag{4}$$

#### *4.6. Molecular Dynamics Simulations*

We performed all-atom molecular dynamics (MD) simulations of ANT1 protein in a 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) bilayer. Residues (residue 1 and residues 294–297) missing from the crystal structure of ANT1 (PDB code: 1okc) [40] without CATR were added using Modeller 9 [58] and implemented into the DOPC bilayer using CHARMM-GUI (http://www.charmm-gui.org/ (accessed on 7 January 2021)) [59–61]. Three system setups were prepared—the wild-type ANT1, the wild-type ANT1 with ATP4− bound in the cytosolic-open state (c-state) [43], as well as GTP4− bound in the same position. All simulation boxes contained ANT1 protein (with a total charge of +19), 73 DOPC molecules per leaflet (146 per system), ∼11,500 water molecules, and the necessary number of Cl- anions to neutralize the net charge, depending on whether ATP4− or GTP4− are added to the system. All systems were first minimized and equilibrated in six steps using the CHARMM-GUI protocol [62] and then simulated for a further 100 ns without any restraints with a 2 fs time step in a periodic rectangular box of 7.9 nm × 7.9 nm × 9.4 nm using the isobaric-isothermal ensemble (NPT) and periodic boundary conditions in all directions at T = 310 K, maintained via Nosé–Hoover thermostat [63] independently for the DOPC, water/ions and protein subsystems with a coupling constant of 1.0 ps<sup>−</sup>1. The pressure was set to 1.013 bar and controlled with a semi-isotropic Parrinello-Rahman barostat [64] with a time constant for pressure coupling of 5 ps<sup>−</sup>1. Long-range electrostatics were calculated using the particle-mesh Ewald (PME) method [65] with real space Coulomb interactions cut off at 1.2 nm using a Fourier spacing of 0.12 nm and a Verlet cut-off scheme. All simulated systems were described by the CHARMM36m force field [66]. The electrostatic potential maps of all systems were calculated using VMD's PMEPOT plugin [67]. All simulations were run with the GROMACS 5.1.4 software package [68] and visualized with the VMD molecular graphics program [69].
