## 2 Materials and Methods  
The graphene oxide was prepared through the chemical exfoliation of graphite following the modified Hummer's method [31,32]. Briefly, graphite particles (Ito Graphite Co., Ltd., Kuwana, Japan) were oxidized in two steps with peroxydisulfuric acid and potassium permanganate in sulfuric acid, and the oxidized graphite was dispersed into pure water to prepare an aqueous suspension of single-layered GO flakes that were exfoliated from the graphite particles. Residual oxidized graphite particles and multi-layered GO flakes were removed by centrifugation. The GO suspension was deposited by drop-casting on a piranha-cleaned thermally oxidized silicon (SiO\({}_{2}\)/Si) substrate. The details of the preparation of the GO suspension and the SiO\({}_{2}\)/Si substrate are described elsewhere [33]. Figure 1: Schematic comparing the distance dependence of the FRET efficiency between a dye molecule and GO. The abscissa and ordinate axes represent the FRET efficiency (\(E\)) and the distance (\(d\)), respectively. The effective range of the FRET to GO, whose rate of energy transfer (\(R_{\text{eg}}\)) depends on \(d^{-4}\), reaches longer than that to dye molecules, whose \(R_{\text{eg}}\) depends on \(d^{-6}\). The typical thickness of a single lipid bilayer membrane is indicated. 1,2-dioleoyl-\(sn\)-glycero-3-phosphocholine (DOPC), 1,2-dipalmitoyl-\(sn\)-glycero-3-phosphothioethanol (DPPTE), and 1,2-distearoyl-\(sn\)-glycero-3-phosphoethanolamine-\(N\)-[amino(polyethylene glycol)-2000] (PEG-DSPE) were purchased from Avanti Polar Lipids, Inc. (Alabaster, AL, USA) and used without further purification. Chloroform solutions of DOPC, PEG-DSPE, and DPPTE were mixed in a glass vial at a molar ratio of 95:5:5 \(\times\) 10\({}^{-7}\) for SPT or (100\(-\)C\({}_{\text{PEG}}\)):C\({}_{\text{PEG}}\):O (C\({}_{\text{PEG}}\) represents PEG-DSPE concentration, \(C_{\text{PEG}}\) = 5.0 or 7.5 mol%) for AFM. The PEG-DSPE was added to suppress the non-specific adsorption of the QD [30]. The mixed lipid solution was dried in a nitrogen stream followed by vacuum-drying for at least 6 h. The dried lipid film was suspended in a buffer solution (100 mM KCl, 25 mM HEPES, pH 7.4/NaOH) to obtain multilamellar vesicles. The suspension was extruded through 800 nm and 100 nm polycarbonate filters to prepare the unilamellar vesicles. We prepared the SLBs on the SiO\({}_{2}\)/Si substrates with and without GO by the vesicle fusion method [12; 34], following the protocols in the previous study [30]. The substrates were incubated in the suspension of the unilamellar vesicle at 45 \({}^{\circ}\)C for 1 h. Excess vesicles were removed by exchanging the suspension with the buffer solution. A carboxyl-coated QD (Qdot(r) 655 ITK(tm), Life Technologies, Carlsbad, CA, USA) was modified with a maleimide-hydrazide hetero-cross-linker (Quanta BioDesign Ltd., Plain City, OH, USA) and 2-(2-aminoethoxy) ethanol (AEE) using 1-ethyl-3-(3- dimethylaminopropyl) carbodiimide (EDC), and then added to the SLB containing DPPTE [35]. AEE was co-adsorbed with the cross-linker to control the number of efficient maleimide groups on the QD surface for the covalent bond formation with DPPTE in the SLB, and to reduce multivalent conjugation of the QD with the DPPTE [30]. Single particle tracking was performed with an inverted fluorescence microscope (IX-71, Olympus, Inc., Tokyo, Japan) and a 532 nm DPSS laser. The GO/SiO\({}_{2}\)/Si substrate after SLB formation was sealed in a cell made of cover glass slips with a silicone resin spacer. We adopted the diagonal illumination setup [36], which enables single molecule imaging on an opaque silicon wafer [37; 38]. The single particle images of the QDs were recorded at a time resolution of 30 ms (33 frames per second) using an EM-CCD camera (iXon DU-897, Andor Technology, Ltd., Belfast, UK)). The pixel size of the SPT recording was 140.84 nm.