Cell culture, transfection and labellingCell cultureCOS-7 cells (Sigma, 87021302) and HEK293T cells (Beyotime, C6008) were cultured in Dulbecco’s Modified Eagle Medium (Thermo Fisher, 41965-039) supplemented with 10% (v/v) fetal bovine serum (FBS; Sigma, F7524), 1% penicillin–streptomycin (Thermo Fisher, 15140-122), 1% l-glutamine (Thermo Fisher, 25030-081) and 1% sodium pyruvate (Thermo Fisher, 11360-070). U-2 OS-CRISPR-NUP96-mEGFP cells originally obtained from Cytion (300174) and provided by Y. Li were maintained in McCoy’s 5A (Procell, PM150710) supplemented with 10% (v/v) FBS and 1% penicillin–streptomycin. Cells were cultured at 37 °C in a humidified incubator containing 5% CO2 and were passaged every 2–3 days. All experiments were performed using cells in 20 passages.To generate a doxycycline-inducible U-2 OS-CRISPR-NUP96-mEGFP cell line expressing importin-β, lentiviral particles were produced in HEK293T cells by cotransfection of pTRIPZ-importin-β-3×FLAG-ALFA-P2A-tagBFP (derived from the pTRIPZ; Addgene, 206981), psPAX2 (Addgene, 12260) and pMD2.G (Addgene, 12259) at a plasmid ratio of 4:3:1. Viral supernatants were collected 48 and 72 h after transfection and concentrated by polyethylene-glycol-8000 precipitation. U-2 OS-CRISPR-NUP96-mEGFP cells were transduced for 48 h and selected with puromycin (1 μg ml−1; Yeasen, 727136ES03) for 4 days. For MINFLUX imaging, expression of importin-β was induced with doxycycline (0.5 μg ml−1; MCE, HY-N0565) for 24 h before fixation.Cell morphology was routinely monitored during culture. The engineered U-2 OS-CRISPR-NUP96-mEGFP/importin-β cell line was also verified by NUP96-mEGFP fluorescence and doxycycline-inducible importin-β expression. No in-house short tandem repeat profiling was performed. All cell lines were regularly tested for mycoplasma contamination by PCR and with the MycoBlue Mycoplasma Detector Kit (Vazyme, D101-01), with negative results.TransfectionCOS-7 cells were seeded on 35-mm glass-bottom imaging dishes (ibidi, 81158) 24 h before transfection to reach 60–70% confluency. Cells were transfected with plasmids listed in Supplementary Table 1 using jetPRIME (Polyplus) according to the manufacturer’s instructions. At 4–5 h post-transfection, the medium was replaced with fresh culture medium supplemented with 10 mM HEPES and either 50 μM BCN for site-specific labelling or 50 μM BOC for control experiments. Where indicated, 10 μM Thiamet-G (MCE, HY-12588) or 10 μM OSMI-4 (MCE, HY-114361) was added at this stage and maintained throughout the experiment.Live-cell labellingAt 20–24 h following transfection, cells were washed with medium containing 50 μM BOC and 10 mM HEPES and incubated for 2 h at 37 °C to allow excess BCN to clear. Cells were then labelled with 2 μM HD653 in serum-free medium for 30 min at 37 °C. For O-GlcNAc perturbation experiments, 10 μM Thiamet-G or 10 μM OSMI-4 was included during labelling to maintain treatment conditions. Where indicated, SiR-tetrazine (Spirochrome, SC008) or JF646-tetrazine (Tocris, 7279) was used following the same protocol. After labelling, cells were washed with 1× PBS and imaged immediately in FluoroBrite Dulbecco’s Modified Eagle Medium (Thermo Fisher, A1896701) supplemented with 10% (v/v) FBS, 1% l-glutamine, 1% sodium pyruvate and 10 mM HEPES. Imaging was performed at room temperature with the dish sealed to ensure stable conditions during the roughly 2-h acquisition period.Immunostaining of endogenous importin-βCOS-7 cells labelled with HD653 were fixed with 2% paraformaldehyde (PFA) in PBS for 10 min, permeabilized with 0.5% Triton X-100 in PBS for 15 min and blocked with 3% bovine serum albumin in PBS for 90 min. Cells were incubated with an anti-KPNB1 antibody (Abcam, ab2811; 1:1000 dilution) followed by an Alexa Fluor 488-conjugated anti-mouse secondary antibody (Thermo Fisher, A-11001; 1:1000 dilution). Immunofluorescence imaging confirmed localization of endogenous importin-β at the nuclear envelope following live-cell labelling (Extended Data Fig. 1b).Estimation of importin-β concentration at the nuclear poreTo estimate the local concentration of importin-β at the nuclear pore, solutions containing free Alexa Fluor 488 at known concentrations (1 μm–1 mM) were added to immunostained cells. Nuclear pore fluorescence remained distinguishable from the background in the presence of up to 100 μM free dye but became indistinguishable at 1 mM (Extended Data Fig. 2a), indicating that the effective local concentration of importin-β at the nuclear pore is in the millimolar range.Assessment of expression levels of exogenous importin-β by immunoblottingThe doxycycline-inducible U-2 OS-CRISPR-NUP96-mEGFP cell line described above was treated with doxycycline (0.5 μg ml−1) for 0, 6, 10, 15 or 24 h. Non-transduced U-2 OS-CRISPR-NUP96-mEGFP cells without doxycycline treatment were used as a parental control. Cell lysates were collected at the indicated time points and analysed by immunoblotting using an anti-FLAG antibody (Proteintech, 66008-4-Ig; 1:1,000 dilution) to detect exogenous importin-β and an anti-KPNB1 antibody (1:2,500 dilution) to detect both endogenous and exogenous importin-β. α-Tubulin detected with an anti-α-Tubulin antibody (ABclonal, AC012; 1:5,000 dilution) was used as a loading control. Doxycycline induction resulted in detectable FLAG-tagged importin-β expression, whereas the total KPNB1 signal showed no substantial increase compared with the parental control, consistent with near-endogenous expression of the exogenous protein (Supplementary Figs. 19 and 20).Assessment of cellular O-GlcNAc levels by immunoblottingTo verify modulation of cellular O-GlcNAcylation, COS-7 cells were treated with 10 μM Thiamet-G or 10 μM OSMI-4 as described above and lysed directly in Laemmli sample buffer. Proteins were separated by sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS–PAGE), transferred to nitrocellulose membranes and probed with the anti-O-GlcNAc antibody RL2 (Invitrogen, MA1-072; 1:1,000 dilution). Bound antibody was detected using an IRDye 800CW goat anti-mouse IgG secondary antibody (LI-COR Biosciences, 926-32210; 1:10,000 dilution) and visualized using an Odyssey imaging system (Fig. 4b).Measurement of nuclear-import rates in live cellsLive-cell nuclear-import assays were performed as described previously49 with minor modifications. COS-7 cells expressing NES–mCherry–LINuS were treated with 10 μM Thiamet-G or 10 μM OSMI-4 and imaged 24 h after transfection following Hoechst 33342 staining (10 μg ml−1, 10 min). Imaging was performed on a Zeiss LSM 980 confocal microscope. Nuclear import was induced by 488-nm illumination and monitored by mCherry fluorescence using 561-nm excitation. Images were acquired every 10 s for 820 s. Nuclear regions were segmented using Hoechst fluorescence, and nuclear mCherry intensity was quantified over time and normalized to the pre-activation signal. Apparent nuclear-import rate constants were obtained by fitting fluorescence accumulation curves to y = 1 + (b − 1)(1 − e−kt), where b is the plateau fluorescence intensity and k is the apparent nuclear-import rate constant.Protein expression, purification and labellingExpression and purification of NUP98 FG domainThe human NUP98 FG domain (residues 1–505) was cloned into a pQE-14his-TEV vector. As described previously26,59, the GLEBS domain (residues 157–213) was excluded to avoid misfolding of this structured region on rapid dilution from denaturant during phase separation (see ‘In vitro phase separation assay’ below); intrinsically disordered FG regions, by contrast, readily regain their dynamic ensemble under native buffer conditions. The construct was transformed into Escherichia coli BL21-AI cells and grown in terrific broth supplemented with kanamycin (50 µg ml−1) at 37 °C with shaking at 200 rpm. Protein expression was induced when optical density at 600 nm (OD600) ≈ 0.6 with 0.02% (w/v) arabinose and 1 mM iIsopropyl β-D-1-thiogalactopyranoside, followed by incubation for 16 h at 18 °C.Cells were collected and lysed using a high-pressure cell disruptor (CF1, Constant Systems) in lysis buffer containing 6 M guanidine hydrochloride (GdmCl), 0.2 mM tris(2-carboxyethyl)phosphine (TCEP), 20 mM imidazole and 50 mM Tris-HCl (pH 8.0). Following clarification by centrifugation (12,000g, 1 h, 4 °C), lysates were incubated with nickel-nitrilotriacetic acid (Ni-NTA) resin for 2 h at 4 °C. Bound protein was washed with 2 M GdmCl buffer and eluted with 500 mM imidazole in 2 M GdmCl and 50 mM Tris-HCl (pH 8.0). After dialysis into 0.5 M GdmCl and 50 mM Tris-HCl (pH 8.0), the N-terminal His-tag was removed by overnight TEV protease digestion. The cleaved protein was separated from uncleaved protein and TEV protease by a second Ni-NTA purification step and further purified by size-exclusion chromatography (Superdex 200 Increase 10/300 GL, Cytiva, 28990944) in 2 M GdmCl, 0.2 mM TCEP and 50 mM Tris-HCl (pH 8.0). Purity was confirmed by SDS–PAGE and Coomassie staining. Protein fractions were concentrated to about 15 mg ml−1, quantified using a bicinchoninic acid assay and stored at −80 °C.In vitro O-GlcNAcylation of NUP98 FG domainThe purified NUP98 FG domain was enzymatically glycosylated using recombinant OGT1 and UDP-GlcNAc (Sigma, U4375). The reaction mixture contained 10 µM NUP98 FG domain, 1 µM OGT1 and 1 mM UDP-GlcNAc in glycosylation buffer (50 mM Tris-HCl, pH 7.5, 200 mM NaCl, 20 mM MgCl2, 0.2 mM TCEP and 1% Tween-20). Samples were incubated for 16 h at room temperature with gentle rotation. Following O-GlcNAcylation, His-tagged OGT1 was removed using Ni-NTA resin. O-GlcNAcylated NUP98 FG domain was further purified by size-exclusion chromatography (Superdex 200) in 2 M GdmCl, 0.2 mM TCEP and 50 mM Tris-HCl (pH 8.0) and concentrated to about 40 mg ml−1. O-GlcNAcylation efficiency was assessed by mass spectrometry, confirming successful modification of the NUP98 FG domain (Extended Data Fig. 8c).For experiments requiring higher levels of O-GlcNAc modification, the purified glycosylated protein was subjected to a second round of enzymatic O-GlcNAcylation under identical reaction conditions. Unless otherwise stated, all experiments were performed using NUP98 FG domains that had undergone a single round of O-GlcNAcylation. Proteins subjected to two rounds of O-GlcNAcylation were used only for the experiments shown in Supplementary Fig. 18.Labelling of NUP98 FG domainNUP98 FG-domain variants containing a single cysteine substitution (residue 36, 100, 221, 312, 387 or 482) were exchanged into 4 M GdmCl, 1× PBS, 0.1 mM EDTA and 0.2 mM TCEP (pH 7.0). Proteins were labelled overnight at 4 °C with Cy3B maleimide (GE Healthcare, PA63131) or Alexa Fluor 594 maleimide (Thermo Fisher Scientific, A10256) at a protein/dye molar ratio of 1:2. Excess dye was quenched with 10 mM dithiothreitol and removed by ultrafiltration (3 kDa molecular-weight cutoff), followed by size-exclusion chromatography (Superdex 200). Protein concentrations and labelling efficiencies were determined spectroscopically using a Duetta fluorometer (Horiba). Labelled proteins were flash-frozen in liquid nitrogen and stored at −80 °C.Expression and purification of importin-βThe human importin-β coding sequence was cloned into a pTXB3-intein-CBD-12His vector and transformed into E. coli BL21-AI cells. Cultures were grown in terrific broth supplemented with kanamycin (50 μg ml−1) at 37 °C with shaking (200 rpm). Protein expression was induced at OD600 = 0.4–0.6 with 0.02% (w/v) arabinose and 1 mM iIsopropyl β-d-1-thiogalactopyranoside, followed by incubation for 5–6 h at 37 °C. Cells were lysed by high-pressure homogenization in 50 mM Tris, 650 mM NaCl, 5 mM MgCl2, 5 mM imidazole, 1 mM phenylmethylsulfonyl fluoride and 0.2 mM TCEP (pH 7.0). After clarification by centrifugation (12,000g, 1 h, 4 °C), lysates were incubated with Ni-NTA resin for 2 h at 4 °C. Bound protein was washed with lysis buffer containing 10 mM imidazole and 1.5 M NaCl and eluted with 400 mM imidazole. The eluate was dialysed into imidazole-free buffer and subjected to a second Ni-NTA purification step. Importin-β was further purified by size-exclusion chromatography (Superdex 200) in 50 mM Tris, 650 mM NaCl, 5 mM MgCl2 and 0.2 mM TCEP (pH 7.0). Purity was confirmed by SDS–PAGE. Purified protein was concentrated to about 500 μM, flash-frozen in liquid nitrogen and stored at −80 °C.In vitro phase separation assaysPurified NUP98 FG domains stored in 2 M GdmCl were induced to phase separate by rapid dilution into transport buffer (20 mM HEPES, 110 mM KOAc, 5 mM NaOAc, 2 mM Mg(OAc)2, 1 mM EGTA and 2 mM dithiothreitol, pH 7.3). Unless otherwise indicated, 1 μl protein solution was mixed with 24 μl transport buffer on a chambered coverslip (ibidi, 81507) immediately before imaging. Wild-type NUP98 FG domains were used at a final concentration of 10 μM mixed with 2 nM labelled FG domain. The residual GdmCl concentration after dilution was about 80 mM and did not interfere with condensate formation.Because O-GlcNAcylated FG domains exhibited reduced phase-separation propensity, a final concentration of 25 μM was used for the experiments in Fig. 4g–i to generate condensates of comparable size and morphology to those formed by wild-type FG domains.For cophase separation assays, importin-β was diluted into transport buffer to the desired concentration before mixing with FG domains. Wild-type FG domains were used at 10 μM unless otherwise indicated. Glycosylated FG–importin-β condensates were prepared from 25 μM glycosylated FG domains and 10 μM importin-β to obtain condensates comparable to those formed by wild-type FG domains. The final concentration of GdmCl was maintained at 80 mM across all conditions.Quantification of FG-condensate phase concentrationsDense- and dilute-phase concentrations of NUP98 FG condensates were determined using trace amounts of Alexa Fluor 594-labelled NUP98 FG domain. Condensates were formed from mixtures of unlabelled wild-type NUP98 FG domain and labelled tracer protein at molar ratios of 2,000:1, 5,000:1 or 10,000:1, as described above. Immediately after formation, condensates were sedimented by centrifugation at 4,300g for 5 min.Dense-phase fluorescence was measured on a custom-built confocal microscope equipped with a 560-nm pulsed laser. Dense-phase regions were identified from their high fluorescence intensity and characteristic condensate morphology. Fluorescence intensities were converted to protein concentrations using calibration curves generated from known concentrations of free Alexa Fluor 594 measured under matched buffer conditions. Differences in fluorescence quantum yield between free dye and dye-conjugated protein were measured independently using a Duetta fluorometer and used to correct the calculated protein concentrations.Dilute-phase concentrations were measured by focusing well above the condensate layer and applying the same calibration and quantum-yield correction. An unlabelled/labelled protein ratio of 20:1 was used to increase dilute-phase signal while minimizing perturbation of phase behaviour. Small dynamic nanoclusters were occasionally observed in the dilute phase of wild-type NUP98 FG condensates in the absence of importin-β (Extended Data Fig. 3d).The experimentally determined dense- and dilute-phase concentrations were used to constrain FG–importin-β interaction strengths in the coarse-grained molecular-dynamics simulations (Extended Data Fig. 3e).O-GlcNAcylation-dependent selective partitioning of FG domainsTo examine the effect of O-GlcNAcylation on FG-domain partitioning, NUP98 FG domains containing a cysteine substitution at residue 221 were site-specifically labelled with fluorescent maleimide dyes. Unglycosylated NUP98 FG domains were labelled with Alexa Fluor 488 maleimide, whereas unglycosylated, modestly O-GlcNAcylated (single-round glycosylation) and highly O-GlcNAcylated (two-round glycosylation) NUP98 FG domains used as probe molecules were labelled with LD655 maleimide.Preformed condensates were generated from unglycosylated NUP98 FG domains (final concentration roughly 10 μM supplemented with 4 nM Alexa Fluor 488-labelled protein) as described above. Samples were incubated for 30 min at room temperature to allow condensate maturation before probe addition. LD655-labelled probe proteins (unglycosylated, modestly glycosylated or highly glycosylated NUP98 FG domains) were then added to the chamber at a 1:1 volume ratio, yielding a final probe concentration of 20 nM. The incorporation of probe molecules into preformed condensates was monitored by time-lapse confocal microscopy using a Zeiss LSM 980 microscope (Supplementary Fig. 18).TRFA measurementsInstrument configurationTRFA measurements were conducted on a custom-built confocal microscope equipped with picosecond pulsed laser diodes (560 nm: LDH-D-TA-560, PicoQuant; 660 nm: LDH-D-C-660, PicoQuant) controlled by a Sepia II PDL 828 multichannel laser driver26. The excitation beam was delivered through a single-mode polarization-maintaining optical fibre (KineFLEX-P-2-S-405/640-2.5-2.5-p2), passed through a Glan-laser polariser (Thorlabs), directed through a FLIMbee laser scanning system (PicoQuant) and focused onto the sample through a ×60 water immersion objective lens (SR Plan Apo IR, 1.27 NA, Nikon). Fluorescence emission was collected through the same objective, spatially filtered using a pinhole (100 μm for live-cell measurements and 50 μm for in vitro measurements) and separated into parallel and perpendicular polarization components using a polarizing beam splitter. Each component was further separated spectrally by dichroic filters (ZT561 RDC and T647 LPXR, Chroma), passed through bandpass filters (Cy3B: 612/69 BrightLine HC, Semrock; HD653: ET700/75m, Chroma) and detected with hybrid photon-counting detectors (PMA Hybrid 40, PicoQuant). Photon arrival times were recorded with a HydraHarp 400 time-correlated single-photon-counting module at 16-ps resolution. Data were acquired using SymPhoTime 64 software (v.2.6, PicoQuant).Instrument response functions (IRFs) were measured daily using Erythrosine B in saturated potassium iodide solution and used for reconvolution analysis. Detection channels were aligned using the measured IRFs. We note that a small discontinuity around 4 ns is observed in the raw anisotropy-decay curves; this arises from weak optical reflections in the time-correlated single-photon-counting setup and is accounted for in the analysis by reconvolution with the measured IRFs.Live-cell measurementsHD653 fluorescence intensity at the nuclear rim was first verified under 660-nm laser excitation (optimized to 50 μW average power at 40 MHz) to assess the expression level, ensuring comparable expression levels and avoiding overexpression artefacts. Data acquisition was performed using T3 mode, with fluorescence photons collected separately by parallel and perpendicular detectors. Imaging parameters included a pixel size of 100 nm, an image size of 256 pixels × 256 pixels, a pixel dwell time of 150 µs and an imaging time of 5 min. Regions of interest corresponding to the nuclear rim were selected, where thresholding algorithms were applied to identify regions of interest on the basis of pixel intensity and average fluorescence lifetime. The time-resolved intensity curves for parallel and perpendicular channels were extracted for further analysis.In vitro measurementsCondensates were imaged immediately after formation as described above. Fluorescence photons were collected in T3 mode using 560-nm excitation (40 MHz repetition rate). To minimize photobleaching and ensure representative sampling, the imaging field was changed every 5 min to examine different condensates throughout the measurement period. Imaging parameters included a pixel size of 150 nm, an image size of 256 pixels × 256 pixels, a pixel dwell time of 203 μs and a time resolution of 16 ps. Condensates were identified on the basis of fluorescence intensity and morphology, and time-resolved intensity curves were extracted from condensate regions using SymPhoTime 64 software.For measurements of individual FG chains in solution, purified and labelled FG domains were diluted in transport buffer to a final concentration of 10 nM. Data acquisition and analysis were performed as described above for condensate measurements.TRFA decay analysisFitting proceduresTRFA decay, r
(1)
where L1 and L2 are polarization mixing factors caused by the high numerical aperture of the objective lens and G accounts for the differential detection efficiencies between the parallel and perpendicular polarization components:$$G=\frac{{\eta }_{\perp }}{{\eta }_{{||}}}$$
(2)
To determine L1 and L2, the fluorescence decays of a reference sample, yellow fluorescent protein (YFP), were measured. YFP has a known fluorescence rotational relaxation time (ρ = 16 ns) (ref. 61). The fluorescence decays for the parallel and perpendicular components of YFP were globally fitted using the following equations61:$${I}_{||}
(3)
$${I}_{\perp }
(4)
where I0 is the initial fluorescence intensity, τ is the fluorescence lifetime and ρ is the rotational correlation time of YFP. The G factor was determined by measuring the ratio of the average intensities in the perpendicular and parallel channels using a solution of tris(2,2’-bipyridyl)dichlororuthenium(II) chloride ([Ru(bpy)3]Cl2), which exhibits isotropic rotational motion.For IDPs, r
(5)
where r0 is the fundamental anisotropy of the fluorophore, ρfast is the fitted rotational correlation time associated primarily with local dye/linker motion, rp is the residual anisotropy remaining after fast local motion and reflecting slower, larger-scale chain motions and ρslow is the fitted slow-component timescale used here as an apparent segmental-motion parameter. Values approaching the upper fitting bound of 1,000 ns were interpreted as indicating very slow apparent segmental motion and were not considered well resolved.When ρfast is much smaller than ρslow, equation (5) can be approximated and simplified to$$\begin{array}{l}r
(6)
where βfast = 1 − rp/r0 and βslow = rp/r0 represent relative amplitudes of the fast (local) and slow (segmental) components, respectively.The fluorescence lifetime (τ) was first determined by fitting the total fluorescence decay obtained from the parallel and perpendicular emission channels. The resulting τ value was then fixed during the subsequent global fitting of the parallel and perpendicular fluorescence decays using the 2_exp_rot model implemented in TauFit in the PAM package62, where IRFs were included through reconvolution analysis. Here we interpret ρslow primarily as an apparent segmental-motion parameter for relative comparisons rather than as a precise absolute rotational correlation time. Rotational components slower than the fluorescence lifetime can nevertheless influence the measured anisotropy decay and be estimated by model-based fitting, with precision depending strongly on photon statistics and decreasing as the correlation time increasingly exceeds the fluorescence lifetime63,64. Although three-exponential models were also evaluated, the additional component was consistently negligible or degenerate with existing components and therefore did not improve the fits.To calculate steady-state anisotropy (rss), the integrated intensities of the fluorescence emissions collected in the parallel (\({I}_{\parallel }\)) and perpendicular \(({I}_{\perp }\)) channels were used. The steady-state anisotropy was computed using the equation$${r}_{\mathrm{ss}}=\frac{G{I}_{{||}}-{I}_{\perp }}{(1-3{L}_{2})G{I}_{{||}}+(2-3{L}_{1}){I}_{\perp }}$$
(7)
The steady-state anisotropy value provides a single, time-averaged measurement of the overall fluorescence anisotropy, which complements the dynamic information obtained from TRFA decay analysis.Statistical analysisFor live-cell measurements, single-cell anisotropy-decay traces were used for subsequent analysis. In Fig. 1i, the fitted slow-component parameter (ρslow) was calculated on a per-cell basis to capture relative temporal trends in apparent segmental motion. For all other experiments, a non-parametric bootstrap strategy was applied to improve signal-to-noise and ensure robust curve fitting.For each condition, 200 bootstrap replicates were generated by resampling single-cell decay traces with replacement, summing photon counts and fitting the resulting decays in TauFit62. Bootstrap depth was selected on the basis of convergence analysis (Supplementary Fig. 2). Differences between conditions were assessed using two-sided bootstrap tests. P values were calculated as the proportion of null resamples with absolute differences greater than or equal to the observed effect. Percentile confidence intervals were derived from the empirical distribution of independent bootstrap differences.For in vitro measurements, anisotropy decays were analysed every 5 min and fitted individually. Reported values represent the mean ± s.d. from three independent experiments.Validation of HD653 for anisotropy measurementsTo assess the suitability of HD653 as a probe for TRFA, control experiments were performed using a globular protein with known rotational properties. Purified EGFP containing a single BCN at residue 39 was site-specifically labelled with either HD653 or the rigid, well-characterized anisotropy dye Cy3B30,31. TRFA measurements were conducted on both unlabelled and labelled EGFP at 10 nM in 1× PBS. The rotational correlation times were obtained by fitting TRFA decay curves using the 2_exp_rot model function in TauFit62. For globular EGFP, the fitted slow component, ρslow, corresponds to its overall rotational correlation time and was used to determine the hydrodynamic radius:$${R}_{{\rm{h}}}={\left(\frac{3{k}_{{\rm{B}}}T{\rho }_{\mathrm{slow}}}{4\pi \eta }\right)}^{1/3}$$
(8)
where kB is the Boltzmann constant, T is the absolute temperature and \(\eta \) is the solvent viscosity. Hydrodynamic radii determined for labelled EGFP were consistent with wild-type EGFP and previously reported values65,66, confirming HD653 as a reliable anisotropy probe and supporting its use for segmental dynamics measurements in live-cell and in vitro assays.MINFLUX measurementsSample preparation for MINFLUX imagingDoxycycline-induced U-2 OS-CRISPR-NUP96-mEGFP cells expressing importin-β-3×FLAG-ALFA were prepared as described above. Cells were fixed with 4% PFA for 10 min, washed three times with 1× PBS containing 100 mM glycine and permeabilized and blocked for at least 30 min at room temperature in blocking buffer consisting of 3% bovine serum albumin, 10% donkey serum and 0.3% Triton X-100 in 1× PBS.DNA docking strand-conjugated anti-GFP and anti-ALFA nanobodies (Massive Photonics) were diluted 1:200 in antibody incubation buffer (Massive Photonics) and incubated with cells for 1 h at room temperature. Samples were washed three times with washing buffer (Massive Photonics) and post-fixed with 2.4% PFA for 8 min. Gold nanoparticles (250 nm diameter; BBI Solutions) were then added for 10 min as fiducial markers for active drift correction, followed by extensive washing with 1× PBS.Cy3B-labelled DNA-PAINT imager strands (Massive Photonics) were diluted in imaging buffer (Massive Photonics) to a final concentration of 200–500 pM. Samples were mounted in custom imaging chambers, and imager solution was added immediately before imaging. All MINFLUX measurements were completed within 2 days of sample preparation.Two-colour DNA-PAINT MINFLUX imagingTwo-colour DNA-PAINT MINFLUX measurements were performed on a commercial MINFLUX microscope (Abberior Instruments) based on an Olympus IX83 inverted microscope equipped with a ×100 UPlanXApo objective and controlled by Imspector software. The excitation beam pattern and pinhole position were calibrated daily using fluorescent calibration nanoparticles (Gold 150 nm and 2C Fluor 120 nm; Abberior Instruments).Cells expressing the inducible importin-β construct were identified by tagBFP fluorescence. Imaging was performed at the basal surface of the nucleus adjacent to the coverslip. Transient DNA-PAINT binding events were verified by rapid confocal imaging before MINFLUX acquisition. At least three gold fiducials were present in each field of view and were used by the active stabilization system to correct real-time drift, maintaining subnanometre lateral stability throughout acquisition.Regions of interest were imaged using the standard two-dimensional MINFLUX acquisition sequence with a pinhole diameter of 0.83 Airy units. Following acquisition of the first target, imager strands were exchanged using a custom perfusion system while maintaining uninterrupted MINFLUX acquisition and active stabilization. Buffer exchange was performed at about 10 μl s−1 until DNA-PAINT binding events were no longer detected, indicating complete removal of the first imager strand. The second imager strand was subsequently introduced and imaging resumed. Sequential two-colour imaging and strand exchange typically required roughly 4 h.NPC alignment and radial density analysisNPCs were analysed using custom MATLAB routines adapted from the published MINFLUX_NPC_Tracking workflow14. Candidate NPCs were selected from the NUP96 channel and filtered using geometry-based quality-control criteria to exclude incomplete or poorly resolved pores.For each NPC, the pore centre was determined from NUP96 localizations and used for spatial registration. The eightfold rotational phase was estimated from the angular distribution of NUP96 localizations using the complex order parameter \({z}_{8}=\langle {e}^{i8\theta }\rangle \). Individual NPCs were rotated into a common eightfold reference frame, and the same rigid-body transformation was applied to the corresponding importin-β localizations. Aligned NUP96 coordinates were rendered directly to preserve the native octagonal architecture of the NPC scaffold.Because importin-β localizations did not exhibit a robust eightfold angular organization, their spatial distribution was quantified using radial density analysis relative to the NUP96-defined pore centre. Localizations were binned by radial distance, corrected for annular area, normalized in each NPC and averaged across pores. Individual radial profiles were smoothed before calculation of the population mean and s.e.m. The resulting mean radial density profile was rotationally symmetrized to generate the final importin-β density map. To assess the contribution of radial geometry, a uniform random two-dimensional distribution was generated in the analysed pore radius and processed using the same radial binning, annular-area correction, normalization, smoothing and symmetrisation as the experimental localizations (Fig. 3g).Molecular-dynamics simulationsResidue-based coarse-grained model of NPC and importin-βWe adapted a previously established coarse-grained model of the NPC26 to incorporate importin-β proteins. In this model, each amino acid (p) of NPC and importin-β proteins is represented by a single bead, the nuclear envelope is modelled using fixed membrane particles (m), and the solvent is treated implicitly. Protein residues are divided into three sub-categories: scaffold residues (sc) and FG residues (FG) of NPC proteins and importin-β residues. The potential energy of the system is given by$$\begin{array}{l}U={U}_{{\rm{LJ}}}+{U}_{{\rm{FENE}}}+{U}_{{\rm{C}}}\\ \,=\,4{k}_{B}T\sum _{i {R}_{c}}{({r}_{{xy},i}-{R}_{c})}^{2}\end{array}$$
(9)
Non-bonded interactions between particles i and j are described by Lennard–Jones potentials, whose strength \({\widetilde{{\epsilon }}}_{{ij}}\), length σij and cutoff values \({r}_{{\rm{c}}}^{{\rm{\alpha }}}\) are provided in Supplementary Table 2. The Lennard–Jones potentials were shifted by their values at the corresponding cutoff radii: that is, \({U}_{{\rm{LJ}}}({r}_{c}^{\alpha })\). The bond potential between adjacent beads of the FG-protein was modelled using the finitely extensible nonlinear elastic potential67 with kFE = 30kBT and RFE = 1.5σ. The radiall positions of importin-β were confined by a half-harmonic potential with stiffness kc = 100kBT/σ2, limiting their radial distance to r ≲ Rc = 81σ ≡ 48.6 nm.Simulations were performed in LAMMPS (Release Sept. 2021)68,69. Time is reported in units of \(\tau =\sqrt{m{\sigma }^{2}/{k}_{{\rm{B}}}T}\), where m is the bead mass (identical for all bead types). Following earlier work26, we determined the reduced length scale of Lennard–Jones potential obtained as σ ≡ 0.6 nm by matching the root-mean-square inter-residue distance (RE) of a single NUP98 FG chain (amino acids 1–499) in both the homopolymer model and the corresponding Martini representation in the weakly interacting regime70. We found that \(\widetilde{{\epsilon }}\)FG–FG = 0.5 reproduces the RE of a single NUP98 FG chain measured by single-molecule FRET, whereas \(\widetilde{{\epsilon }}\)FG–FG = 0.42 best matches in situ FLIM–FRET for the NUP98 FG domain in the NPC26, reflecting in particular the effects of PTMs.The membrane particles representing the nuclear envelope were constructed from a 100 × 100 nm2 coarse-grained POPC bilayer using the Python script insane.py71,72, with a reduced area per lipid (0.3 nm2) to prevent solvent permeation (command: insane.py -l POPC -x 100 -y 100 -z 100 -a 0.3 -o bilayer.gro). A half-toroidal pore was then generated using BUMPy71 (command: bumpy.py -s double_bilayer_cylinder -f bilayer.gro -z 10 -g l_cylinder:10 r_cylinder:430 r_junction:120 l_flat:1920). To reduce computational cost, only phosphate beads were retained. NPC scaffold and membrane particles were fixed throughout the simulations. To prevent FG domains and importin-β molecules from crossing the membrane, a Lennard–Jones potential with a large repulsive range was applied (Supplementary Table 2).We used the constricted NPC model II26 including the following FG-NUPs: NUP54, NUP58, NUP62, NUP98, POM121, NUP214, NUP153 and NUP358. Importin-β was modelled using PDB ID 3W5K (ligand Snail1 removed)73. Importin-β molecules were treated as rigid bodies using a rigid-body integrator with a Langevin thermostat and a damping coefficient 10τ (LAMMPS command: fix rigid langevin molecule). FG beads were integrated using the Verlet algorithm74. NUP98 FG chains were thermalized using Langevin thermostats with damping coefficient 10τ (ref. 75).Preparation and equilibration of importin-β inside the NPCImportin-β molecules were added to systems of size 320σ × 320σ × 400σ. FG-NUPs were first equilibrated at \(\widetilde{{\epsilon }}\)FG–FG = 0.5 for 106τ, during which they formed a surface condensate on the NPC scaffold. These compact conformations allowed us to efficiently introduce NImp-β = 100 importin-β molecules76 with random initial positions and orientations around the NPC. With FG-NUPs frozen and weak interactions (\(\widetilde{{\epsilon }}\)FG–Imp-β = 0.1), importin-β molecules were allowed to diffuse for 3.4 × 104τ with thermostat damping coefficient 1,000τ. Subsequently, the full system was equilibrated with \(\widetilde{{\epsilon }}\)FG–FG = 0.42 and \(\widetilde{{\epsilon }}\)FG–Imp-β = 0.1 for 105τ using a time step δt = 10−3τ and thermostat damping coefficient 10τ. Production conditions were then applied by setting \(\widetilde{{\epsilon }}\)FG–FG and \(\widetilde{{\epsilon }}\)FG–Imp-β to their target values, followed by simulations of at least 106τ. Three independent configurations from the equilibration phase were selected as starting points for independent production runs. Structural visualization was performed using VMD77.Martini model of the NPCTo examine how O-GlcNAcylation affects FG-NUP dynamics, we performed molecular-dynamics simulations of the human NPC using the Martini 2.2 coarse-graining scheme. We used the constricted NPC structure (PDB ID: 7R5K, ref. 39). NUPs were grouped into scaffold proteins and disordered FG-NUPs and converted to Martini CG models using the martinize.py script50,78. FG-NUPs included NUP54, NUP58, NUP62, NUP98, POM121, NUP214 and NUP358 (Supplementary Table 3). All scaffold proteins and grafting-anchored ordered domains of FG-NUPs were restrained with a harmonic spring (1,000 kJ mol−1 nm−2).We generated O-GlcNAcylated NPC models at three modification levels (fgly = 0.18, 0.3 and 0.6), where fgly denotes the fraction of O-GlcNAcylated Ser/Thr sites (Supplementary Fig. 11). Potential O-GlcNAcylation sites were predicted using YinOYang79, which assigns each residue a probability p ∈ [0,1]. Residues with p ≥ pth were considered glycosylated. The resulting number of modified sites Ngly determined fgly = Ngly/Ntotal, where Ntotal = 58,768 is the total number of Ser/Thr residues across all FG-NUPs.Simulations were performed using GROMACS (v.2020.6 and v.2025.2)80,81. Interaction strengths for protein–protein, protein–sugar and sugar–sugar interactions were rescaled by parameters α, λ and γ, respectively70,82. The optimal α = 0.78 was determined by matching RE of an untreated NUP98 FG chain to single-molecule FRET measurements (Supplementary Fig. 12) and the dense-phase density of FG condensates (Supplementary Fig. 14 and Supplementary Tables 4 and 5). For O-glycosylated FG-NUPs, interactions involving sugars used λ = 0.15 and γ = 0.5, consistent with experimental osmotic second virial coefficients of aqueous sugars and protein–sugar mixtures83,84.Equilibration of the Martini NPC systemsA cubic box (170 nm per side) was constructed to accommodate the NPC. Secondary-structure restraints were assigned using DSSP85. Elastic networks in ordered domains were generated using Elastic Network in Dynamics86 (default parameters; a distance range of 0.9 nm; force constant 500 kJ mol−1 nm−2). All FG domains were modelled as coils except for locally ordered regions in NUP358 (Supplementary Table 3). The system was solvated with Martini water containing 10% antifreeze beads87 and neutralized with NaCl to 150 mM. The total number of solvent particles was about 4 × 107.Initial energy minimization used steepest descent with maximum step size 0.01 nm and force tolerance 1,000 kJ mol−1 nm−1. During this step, position restraints (1,000 kJ mol−1 nm−2) were temporarily removed to allow global relaxation. Lennard–Jones and Coulomb potentials were first softened using the GROMACS free-energy framework (parameters: couple-lambda0=vdw-q; sc-alpha = 0.1; sc-coul= yes; sc-power = 2; sc-sigma = 0.3), followed by a second minimization using full potentials.Equilibration proceeded as follows. First, NVT simulations at 310 K were performed for 100 ns (velocity-rescaling thermostat88, time constant 1 ps, time step δt = 0.015 ps) using reduced interaction scaling (α = γ = λ = 0.3) to relax chains into coil-like conformations. Next, NPT equilibration at 310 K and 1 bar was performed for 100 ns (velocity-rescaling thermostat88, semi-isotropic Berendsen barostat89, time constant 12 ps, compressibility 3 × 10−4 bar−1). Interaction parameters were then gradually increased. For untreated NPCs, α was raised from 0.3 to 0.5 and then 0.6. For O-glycosylated NPCs, α was increased from 0.3 to 0.6 and then 0.7, with γ = 0.5 and λ = 0.15. This stage lasted 1.5 μs. Production simulations were conducted for at least 2 μs at target α values, using the same thermostat and barostat parameters as in the final equilibration step and a time step δt = 0.02 ps.Statistics and reproducibilitySample sizes were not determined by formal statistical power calculations but were based on previous experience with the respective imaging and biophysical assays26, practical considerations and the requirements for robust and reproducible measurements. For molecular-dynamics simulations, replicate numbers and trajectory blocks were selected on the basis of convergence and reproducibility criteria. Exact sample sizes, numbers of independent experiments and independent sample preparations are reported in the corresponding figure legends.For experiments in which representative images are shown, including those in Figs. 1b, 2a, 3e and 4b, Extended Data Figs. 1a,b, 2a and 8e and Supplementary Figs. 18–20, experiments were independently repeated at least three times and yielded similar results. Representative images were selected from these independent experiments. Randomization was not applicable because treatments were applied to entire cell cultures or biochemical samples. Experimental variability was minimized by using matched culture, sample-preparation and acquisition conditions.Reporting summaryFurther information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
