The mobility of many molecules varies widely and often during their lifetime within cells, dependent on viscosity, crowding and local interactions. Most often measured is translational mobility, and numerous super-resolution light microscopy approaches have been developed over the past decade suitable for this purpose (see [Huang et al., 2010; Huang et al., 2009] for reviews). Highly localized effects, such as those produced by multiple binding interactions or increases in crowding, often produce small changes in translational mobility (nanometer-scale step sizes in milliseconds) that are very difficult, if not impossible, to detect by these methods. However, these environmental changes can produce significant and detectable changes in rotational mobility. We surmised that the rotational mobility of a probe within the FG-network of NPCs would be strongly influenced by the densities of the FG-polypeptides and other macromolecules, such as NTRs, that increase crowding and decrease mobility of the FG-polypeptides, and therefore, we developed a method that could detect differences in rotational mobility.

Our basic approach was to genetically attach a photoactivatable fluorescent protein to different FG-polypeptides, and at different locations within an FG-polypeptide, in order to determine the rotational mobility of this fluorescent probe in different environments within the FG-network. Fluorescence polarization measurements are often used for probing rotational motion and are readily applied at the single molecule level (Forkey et al., 2000; Forkey et al., 2005; Ha et al., 1999; Harms et al., 1999; Lakowicz, 2006; Loman et al., 2010; Testa et al., 2008). However, single molecule polarization measurements within the NPC pose a special challenge: since the NPC has eight-fold rotational symmetry, any Nup genetically tagged with a fluorescent protein (or chemically tagged with a dye) will be present in numerous copies, and therefore, due to their proximity, the diffraction-limited emission from individual fluorescent tags will overlap significantly, thereby complicating analysis. Consequently, we combined single molecule polarization measurements with PALM (Betzig et al., 2006), in which probe molecules are stochastically and individually photoactivated. In this approach, termed polarization PALM (p-PALM), single fluorescent protein molecules were activated as in PALM, but the emission was split by a polarizer onto separate halves of an EMCCD camera (Figure 1), enabling polarization measurements to be made on individual molecules. As we show, there are significant advantages of this single molecule approach over ensemble fluorescence polarization methods.

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Similar approaches to the p-PALM method described herein were used previously to detect changes in rotational mobility (Gould et al., 2008; Testa et al., 2008). However, a quantitative relationship between rotational diffusion constants and p-PALM measurements has not been reported. In our analysis, we explored the different rotational time regimes and now more fully describe the parameter space, which is essential to interpret the results and provides numerous additional insights into the power of the approach. Rotational random walk simulations were used to determine the effect of imaging speed, fluorescence lifetime, anisotropic rotational diffusion, dipole orientation, thresholding, noise level, and numerical aperture over ~10 orders of magnitude of the average rotational diffusion constant, D_r. These simulations revealed that p-PALM can detect changes occurring on timescales that are largely inaccessible by other means. The details of the experimental approach and simulations are described in the Materials and methods section. An overview of the approach and the general results of our analysis are summarized in the following sections.

The main principle of p-PALM experiments is that rotational mobility information is extracted not from the average bulk polarization, but from polarization measurements obtained from thousands of individual molecules that are pooled to generate polarization frequency histograms (Figure 2). The primary experimental readouts from these data are the average polarization, <p>, and the variance of the polarization distribution, Var(p). In addition, the overall shape of polarization histograms and photon scatterplots can provide additional clues as to the underlying physical constraints on the probe’s rotational mobility. Polarization was defined as p = (gI_p – I_s)/(gI_p + I_s), where I_p and I_s are the fluorescence intensities measured for each single molecule spot in the two polarization channels, and g corrects for the different photon collection efficiencies of these channels (Gould et al., 2008; Harms et al., 1999). We used a measurement timescale (10 ms) comparable to the timescale of protein import by the NPC (Grünwald et al., 2011; Tu and Musser, 2011). For circularly polarized excitation, the average polarization (<p>_cir) as well as the peak of the distribution are theoretically always 0 (for uniformly distributed dipole orientations), providing a convenient check on instrument alignment and calibration (see below and Materials and methods), but providing no information on rotational mobility. Instead, the histogram width, quantified as Var(p), provides an estimate of the rotational mobility, with increasing width corresponding to decreasing rotational mobility (lower D_r values). Since slowly rotating molecules emit from distinct orientations during the data collection period, a wider range of polarization values are obtained for lower D_r values, whereas rapidly rotating probes yield time-averaged polarizations near zero. For linearly polarized excitation, <p>_lin is almost always non-zero and was the parameter used for inferring rotational mobility in this excitation mode. The effects of the D_r on histograms of polarization measurements, <p>, and Var(p) for a spherical particle are shown in Figure 2.

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The monomeric fluorescent protein mEos3.1 (hereafter, simply denoted mEos3) (Zhang et al., 2012) was used as our probe molecule. Photoactivation of mEos3 by UV (405 nm) light results in conversion (photoactivation) from a ‘green’ (~500–550 nm emission) to an ‘orange’ (~570–650 nm emission) fluorescent state, a process that is irreversible due to polypeptide cleavage (McKinney et al., 2009; Wiedenmann et al., 2004). The mEos3 proteins were successively and individually photoactivated by continuous low level UV irradiation, and deactivated by photobleaching. PALM images and p-PALM polarization histograms were generated from thousands of position or polarization measurements, respectively, from many tens to hundreds of NPCs. For mEos3 tagged FG-polypeptides, the average number of probes detected per NPC was typically ~3–5. This is consistent with the known photoactivation efficiency of ~50% for photoactivatable proteins (Durisic et al., 2014). Narrow-field epifluorescence imaging (Yang and Musser, 2006b) was used for single molecule detection. In narrow-field epifluorescence, a small diameter excitation beam is confined to the center of the objective such that only a small area within the sample plane is illuminated (in our case, a 300 µm pinhole yielded an ~7 µm diameter illumination area; see Figure 1). This approach largely eliminated depolarization effects that normally result from focusing an excitation beam toward the z-axis (optical axis) by a high numerical aperture (NA = 1.46) objective (Ha et al., 1999; Olivini et al., 2001). The p-PALM approach does not require the high spatial precision typically obtained from super-resolution methods, which minimizes the need for high precision image alignment. Rather, polarization measurements are based on intensities, and spatial localizations are only necessary for spot correlation between the p- and s-channels. Thus, relatively low photon counts are acceptable. The excitation intensity was adjusted for all of our experiments so that the average total emission intensity was typically ~300–400 photons (N_photons). While the number of photons collected from an individual molecule in the two imaging channels depends on the 3D orientation trajectory of the probe dipole during image acquisition, we emphasize that our method does not require knowledge of the individual 3D rotational trajectories of each probe. Instead, the average rotational mobility is inferred from the statistical properties of experimental polarization frequency histograms by comparison with the theoretically predicted values calculated from simulated rotational random walk trajectories (Figure 2 and Materials and methods).

Generally, for a given single molecule, the photons collected in a single image correspond to hundreds of excitation/emission cycles, during and between each of which the probe might rotate. Throughout this paper, we have assumed that the excitation and emission dipoles of the fluorescent particle are parallel (hereafter simply referred to as the transition dipole), which is the case for GFP and many fluorophores (Ha et al., 1999; Inoué et al., 2002). Our theoretical model of rotational diffusion (based on rotational random walk simulations – see Materials and methods) revealed five important rotational diffusion regimes in p-PALM experiments (Figure 2; more fully described and illustrated in Figure 2—figure supplement 1). These rotational diffusion regimes are defined by two time parameters, the fluorescence lifetime of the fluorophore (τ_F) and the image integration time (t) (see Figure 2—figure supplement 2). The τ_F determines the time for the molecule to rotate between excitation and fluorescence emission. In combination with the number of photons collected, t determines the average time allotted for the molecule to rotate between potential excitation events (τ). In rotational random walk simulations, τ was calculated from t and the number of rotational walk steps (N_S) as τ = t/N_S, and was typically ~3-7 µs. Photon collection after each rotational walk step was dependent on excitation and emission probabilities of the dipole and the collection efficiency of the microscope channels. N_S was set to approximately yield the experimentally collected number of photons (see Materials and methods). In general, the relationship between D_r and <p> or Var(p) cannot be obtained analytically. However, our simulations of rotational diffusion were verified with an analytical solution for a special case (see Materials and methods and Appendix 2).

Considering the five rotational diffusion regimes (Figure 2), Regimes I and II correspond to the rotational mobility probed in most bulk fluorescence anisotropy experiments, where the increase in anisotropy at lower D_r values is described by the Perrin equation (Figure 2—figure supplement 3) (Lakowicz, 2006). However, as demonstrated in the following sections, experimental polarization measurements from probes within the crowded FG-network of NPCs correspond to the slower rotational mobilities in Regimes III and IV. Linear excitation is primarily useful for molecules with high rotational mobility (Regimes I and II), where <p>_lin varies from 0 to ~0.4 (Figure 2). In single molecule experiments at lower D_r values (Regimes IV and V), the fluctuations in <p>_lin strongly depend on thresholding and noise levels (Figure 2—figure supplement 4). Circular excitation is preferred over linear excitation for Regime IV (Figure 2) due to the larger dynamic range in Var(p)_cir and because this parameter is less sensitive to acquisition parameters than <p>_lin (Figure 2—figure supplement 4). Consequently, we primarily used circular excitation for the results that follow.

Importantly, the single molecule average polarization (<p>_lin) as defined in this paper and the average bulk fluorescence polarization (p_bulk) obtained using linear excitation differ significantly, as illustrated in Figure 2A. There are two reasons for this. First, the intensities collected by a microscope objective are mixtures of parallel and perpendicular intensities. Correcting for these mixed intensities (see Materials and methods) yields <p>_lin,c, which agrees with p_bulk, except at low D_r values (Figure 2A). And second, the remaining difference between <p>_lin,c and p_bulk that occurs at low rotational mobilities results from different weightings for each molecule’s fluorescence emission. That is, <p>_lin,c is calculated by weighting each molecule identically (i.e., the polarization of each molecule counts the same no matter how many photons are emitted), whereas p_bulk weights the contribution of each molecule to the measured polarization depending on the number of photons emitted. Consequently, since most polarizations are recovered in p-PALM experiments under low rotational mobility conditions, <p>_lin (and <p>_lin,c) tends toward 0 at low D_r values, although this decrease is limited by the sensitivity threshold. In contrast, dipoles oriented at large angles relative to the excitation polarization emit few photons, and therefore provide only a small contribution to p_bulk. A mathematical description of the differences between p-PALM and bulk polarization measurements is given in Materials and methods. Notably, Var(p) cannot be obtained from bulk measurements because it requires polarization measurements from individual molecules, and hence, the quantification of slow rotational mobility made possible with p-PALM experiments cannot be obtained in a corresponding bulk experiment. Whereas anisotropy is favored as the measurement parameter for bulk fluorescence measurements, which typically have a parallel and two perpendicular channels (Lakowicz, 2006), polarization is a natural parameter for p-PALM measurements since there are only the two detection channels (p and s), neither of which can be used to directly account for the photons that escape detection.

The approach taken in many of the experiments reported herein was to explore various local environments of the FG-network by covalently attaching an mEos3 probe to different FG-polypeptides and measuring its rotational mobility by p-PALM. Under such conditions, the principal rotational diffusion constants of the probe could be differentially affected relative to a freely diffusing spherical particle due to constraints generated by the local environment and/or by the FG-polypeptide to which it was tethered. We therefore used rotational random walk simulations to determine the effect of varying the relationship between the three principal rotational diffusion constants D_x, D_y, and D_z. Somewhat surprisingly, our results and simulations suggested that the probe’s rotation mobility behavior was at most only slightly anisotropic or the angle between the dominant rotational axis and the transition dipole was near the magic angle of 54.7° (Axelrod, 1989). Consequently, the probe’s behavior largely resembled that of an untethered spherical particle. A more detailed discussion of the effects of rotational anisotropy on p-PALM measurements is given in Appendix 1.

The NA is an important parameter in p-PALM experiments because it directly influences the experimentally measured values of <p> and Var(p) (Equations 7-11). Knowing the NA, <p> can in principle be standardized by converting the experimental value into the corresponding bulk parameter (Equation 21). However, there is no corresponding bulk value for Var(p). More importantly, the effective NA (NA_eff) under the acquisition conditions can be significantly different than the nominal NA of the objective. For example, the spherical aberration that results from the refractive index mismatch when using an oil immersion lens for an aqueous sample yields a reduced NA, which is particularly significant when probing the sample far from the coverslip surface. A more detailed discussion of the effect of NA on p-PALM measurements is given in Appendix 1. Importantly, our major conclusions are not affected by a moderate uncertainty in the NA_eff.

For initial proof-of-concept experiments, mEos3 was fused to the C-terminus of Pom121 (Pom121-mEos3), and a stable HeLa cell line was generated. Pom121 is generally agreed to be a central membrane-integrated Nup (Antonin et al., 2005; Hallberg et al., 1993; Söderqvist and Hallberg, 1994; Söderqvist et al., 1997; Talamas and Hetzer, 2011; Yang and Musser, 2006a) with its N-terminal domain anchored to the NPC scaffold and its C-terminal FG-domain within the FG-network. We used both linearly and circularly polarized excitation beams (ellipticity >100 and <1.1, respectively), and p-PALM data were obtained from permeabilized HeLa cells by focusing on the bottom of cell nuclei. Thus, the sample plane coincided with the plane of the nuclear envelope (NE).

We first examined the as-isolated (wildtype) NPCs in permeabilized cells, and then determined whether addition of the NTR Importin β1 (Imp β1) or the transport inhibitor wheat germ agglutinin (WGA) (Adam et al., 1990; Dabauvalle et al., 1988; Finlay and Forbes, 1990; Wolff et al., 1988; Yoneda et al., 1987) influenced the rotational mobility of the mEos3 probe (Figure 3). Polarization frequency histograms using the collection/integration time t = 10 ms (per image) revealed that most measurements for Pom121-mEos3 were close to p = 0.4 for linearly polarized excitation and near p = 0 for circularly polarized excitation (Figure 3A and D). In the presence of Imp β1 (10 μM), the polarization histogram obtained with linear polarization was virtually unchanged and that obtained with circular polarization appeared slightly broader (Figure 3B and E). In the presence of WGA (1 mg/mL; ~26 μM), the difference between linear and circular excitation was clearly observed, with circular excitation producing a substantially wider polarization histogram (Figure 3C and F). WGA binds to O-GlcNAc-modified Nups, of which there are at least five in humans (Finlay and Forbes, 1990; Hülsmann et al., 2012). Since the WGA dimer has eight GlcNAc-binding sites (Schwefel et al., 2010), WGA molecules can potentially non-covalently ‘crosslink’ the FG-network, and thus severely inhibit the rotational mobility of proteins embedded within it. The data in Figure 3 support this hypothesis (compare with Figure 2). Overall, these data demonstrate that the p-PALM approach can detect changes in rotational mobility within the FG-network. A reduced rotational mobility likely implies an increased number of contacts with surrounding macromolecules (which could be strongly interconnected), and hence, we interpret a reduced rotational mobility as primarily a result of increased molecular crowding.

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Pom121 is one of three membrane proteins anchoring the NPC to the NE (Eriksson et al., 2004; Stavru et al., 2006), yet its membrane topology remains unresolved. We considered that the number of transmembrane domains (1 or 2) might be resolved by comparing the effect of WGA on the rotational mobility of the mEos3 probe attached to the N- or C-terminus of Pom121. Most studies describe Pom121 as having a single transmembrane domain (TMD) near its N-terminus, based primarily on a single long sequence of hydrophobic residues (Antonin et al., 2005; Funakoshi et al., 2011; Hallberg et al., 1993; Söderqvist and Hallberg, 1994; Söderqvist et al., 1997; Talamas and Hetzer, 2011). This topology would place the N-terminus of Pom121 in the perinuclear space (between the inner and outer NE membranes), and thus, inaccessible to exogenous reagents such as WGA. However, the hydrophobic transmembrane region of Pom121 is ~44 residues, and thus, it is long enough to span the membrane twice – a possibility that was considered in the initial report identifying this protein (Hallberg et al., 1993). In this alternate scenario, the N-terminus of Pom121 would be accessible from (or embedded within) the FG-network (Figure 4A), and the rotational mobility of an N-terminally attached probe could potentially be influenced by WGA.

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To probe the membrane topology of Pom121, we attached mEos3 to the N-terminus of Pom121 (mEos3-Pom121), and obtained p-PALM measurements under wild-type, +NTR, and +WGA conditions after permeabilization of a stable cell line. WGA had a strong effect on the rotational mobility of the mEos3 probe (Figure 4B). These data support the hypothesis that Pom121 has two TMDs, and that the N-terminus of Pom121 is embedded within the FG-network. Alternatively, WGA could have long-range effects that are mediated across the NE membrane, thus reducing rotational mobility of mEos3 within the ER lumen, but we consider this scenario unlikely. These data demonstrate that the p-PALM approach can address structural questions difficult to ascertain with other methods by using the effects of increased crowding to report on the local environment of (or accessibility to) a probe.

Having established a framework for interpreting p-PALM measurements and demonstrating its utility for addressing structural questions, we then explored the rotational mobility of mEos3 tethered at various locations within the FG-network. We expected that different FG-polypeptide densities and different local concentrations of embedded NTRs and WGA would be reflected in differential effects on rotational mobility, and hence report on different levels of molecular crowding. In addition to the centrally located Pom121, we examined Nup98, a crucial element of the permeability barrier (Hülsmann et al., 2012). To probe the nuclear basket, we used Nup153 (Fahrenkrog et al., 2002; Hase and Cordes, 2003), which plays roles in both protein import and mRNA export and regulates the permeability barrier (Lowe et al., 2015; Makise et al., 2012; Shah and Forbes, 1998; Ullman et al., 1999). To probe the cytoplasmic filament region, we used RanGAP, which is essential for promoting GTP hydrolysis resulting in export complex disassembly (Bischoff and Görlich, 1997; Bischoff et al., 1994; Güttler and Görlich, 2011; Kutay et al., 1997a; Okamura et al., 2015) and which binds in its SUMOylated form to Nup358 (Reverter and Lima, 2005), a major component of the cytoplasmic filaments (Walther et al., 2002; Wu et al., 1995).

p-PALM measurements indicated that WGA had a strong effect on rotational mobility when the mEos3 probe was fused to all four proteins. In contrast, NTRs had a substantially weaker or no effect on probe rotational mobility (Figure 4B). Since both NTRs and WGA were added at ~1 mg/mL (~10 µM and 26 µM [dimer], respectively), these data indicate that exogenous NTRs introduce less molecular crowding in the FG-network structure than WGA.

Different segments of an FG-polypeptide are expected to sample different regions of the FG-network depending on its anchoring point and the polypeptide length from the anchor point. To probe the molecular crowding in the neighborhood of different segments of an FG-polypeptide, we performed p-PALM measurements on mEos3 probes attached at different positions within the disordered region of Nup98. We chose Nup98 because of its important role in forming the permeability barrier (Hülsmann et al., 2012). The domain structure of Nup98 is shown in Figure 5A. With the exception of a 56-residue GLEBS domain, the first ~700 residues of this 920 residue protein are largely disordered, and most of this region contains FG repeats. The C-terminal autoproteolytic (APD) domain of Nup98 binds to Nup88 and Nup96, and is therefore thought to anchor Nup98 onto the NPC scaffold (Griffis et al., 2003; Hodel et al., 2002; Pleiner et al., 2016; Stuwe et al., 2012).

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In the first set of experiments, 109, 399, 499, or 699 residues were deleted from the N-terminus, and the mEos3 probe was attached to the new N-terminus (Figure 1—figure supplement 1). These ‘tip’ labeled, truncated versions of Nup98 were then examined in p-PALM experiments after transient transfections (Figure 5B). To our surprise, mEos3-^500tipNup98 and mEos3-^700tipNup98 were not retained at the NE in permeabilized cells (Figure 5—figure supplement 1), indicating that the Nup98 C-terminal domain was insufficient to anchor these proteins to the NPC scaffold (it is unclear if any binding to the NE occurs in live cells; see Figure 5—figure supplement 1). In separate experiments, we found that residues 1–500 of Nup98 do not bind to the NPC in either permeabilized or live cells (Figure 5—figure supplement 1), indicating that both the FG and C-terminal domains of Nup98 appear to be required for localization to the NPC.

In a second set of experiments, we added the deleted portion back to the Nup98 ‘tip’ mutants, thus placing the mEos3 probe in the middle of two sections of Nup98. These ‘middle’ labeled Nup98 mutants (Figure 1—figure supplement 1) were all localized to NPCs. In as-isolated permeabilized cells, rotational mobility was reduced when the mEos3 probe was attached near the Nup98 C-terminal folded domain. These data suggest more crowded conditions near the NPC scaffold than at the tip of Nup98 (which is likely found most of the time toward the center of the pore or on the nucleoplasmic and cytoplasmic sides, i.e., near the ‘surface’ of the FG-network). Imp β1 did not further affect rotational mobility relative to the as-isolated cells, yet WGA had a significant effect on the probe at all positions (Figure 5B). Notably, WGA had the strongest effect on the rotational mobility of the probe on mEos3-^400midNup98, indicating that larger WGA-induced effects occurred near the middle of the Nup98 disordered region rather than near its FG-polypeptide tip (N-terminus) or the anchor domain (APD domain). These data indicate that different segments of an FG-polypeptide domain experience different environments within the FG-network.

In all our p-PALM experiments, the mEos3 probe was attached to a single location on the protein of interest. Nonetheless, different probe molecules could potentially be in different environments of the FG-network due to the length and flexibility of the FG-polypeptide to which they were attached and/or variable environments within the same or different NPCs. The presence of a mixed population is not obviously apparent in values of <p> and Var(p), since these are obtained by averaging over the population. However, polarization frequency histograms and scatterplots of the photons recovered in the p- and s-channels (photon scatterplots) both can provide evidence for populations of probes with distinct rotational mobilities. In both cases, the distributions obtained from simulations of molecules with different rotational mobilities can be combined and compared with the experimental data to test/verify the mixed population hypothesis.

There were multiple indications that the mEos3 probes in some of our p-PALM experiments sampled multiple environments distinguishable by different rotational mobilities. One example is shown in Figure 6. The broad wings of the polarization histogram in Figure 3D were not expected (see Figure 2D). Moreover, a photon scatterplot for Pom121-mEos3 reveals a substantially wider distribution of points (Figure 6A) than simulated scatterplots that assume a homogeneous population (Figure 6B and C). In contrast, a model that assumes a mixture of two sub-populations of molecules having distinct D_r values yields good agreement with the experimental data (Figure 6D–F). This example demonstrates that evidence for a mixed population can be obtained from both the polarization histogram and the photon scatterplot of a given dataset. Therefore, for more accurate interpretation, the underlying data should be more carefully examined rather than relying on the summary values <p> and Var(p). Appendix 1 contains further discussion and additional evidence for mixed populations within our p-PALM datasets.

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To assist with the interpretation of the rotational mobility data and verify that the probe-labeled proteins were properly incorporated into NPCs, the locations of the mEos3 probe within the FG-network were examined via PALM. NEs were examined at the nuclear equator, allowing us to obtain localization information vis-à-vis the transport axis (Figure 7A). More than 2100 mEos3 localizations were used to generate a 2D density distribution map for each of the constructs used in Figure 4B (Figure 7B–G). Although previous reports have observed the eight-fold rotational symmetry of NPCs using super-resolution approaches (Löschberger et al., 2014; Löschberger et al., 2012; Ori et al., 2013; Pleiner et al., 2016; Szymborska et al., 2013), we found it difficult and unreliable to obtain radial distribution maps (when the optical axis was aligned with the transport axis) due to the low number of mEos3 probes detected per NPC (average of ~3–5).

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We caution that the lateral dimension was artificially compressed in these PALM density distributions. These maps were generated by determining the position of the NE, and aligning clusters of spots based on their centroids. This procedure positioned the centroids of all clusters on the central axis of the NPC. This approach was necessary since there were at most a few tens of spots per NPC, and we had no independent marker for the NPC scaffold. Thus, while the axial dimension was calibrated based on the NE position, the lateral dimension was artificially squeezed. Consequently, probes on the periphery of the pore may appear to be centrally distributed in these PALM density maps. In particular, while the probes on the N-terminus of Pom121 appear to be in the center of the pore (Figure 7C), we consider this to be a consequence of the alignment procedure.

The observed locations vis-à-vis the transport axis for the mEos3 probe attached to the FG-polypeptides generally agreed with previous published results. The mEos3 probe on the N-terminus of Pom121 yielded a distribution pattern along the transport axis that peaked within the central pore (Figure 7C), consistent with a short polypeptide segment anchoring it to the NPC scaffold. In contrast, the mEos3 probe on the C-terminus of Pom121 was more widely distributed (Figure 7D), consistent with access to a large region of the FG-network due to the length of the FG-polypeptide (Hallberg et al., 1993; Söderqvist and Hallberg, 1994). The mEos3 probe on the N-terminus of Nup98 was widely distributed (Figure 7E), consistent with previous studies that found Nup98 within the central pore and on both the nuclear and cytoplasmic sides of the NPC (Chatel et al., 2012; Frosst et al., 2002; Griffis et al., 2003; Krull et al., 2004; Radu et al., 1995). The N-terminus of Nup153 was predominantly localized to the nuclear basket, whereas its C-terminal end was predominantly localized closer to the central pore (Figure 7F and G). These results are consistent with previous antibody domain mapping studies on Nup153 (Chatel et al., 2012; Fahrenkrog et al., 2002; Krull et al., 2004; Lim et al., 2007).

In contrast to the results with the mEos3 probe on FG-polypeptides, which agreed with previous reports, the PALM map for RanGAP was a bit unexpected. RanGAP was predominantly localized in the cytoplasmic filament region, consistent with it being bound to Nup358 in its SUMOylated form (Hutten et al., 2008; Mahajan et al., 1997; Matunis et al., 1998; Reverter and Lima, 2005; Wälde et al., 2012). However, there were a surprising number of localizations within the central pore and nuclear basket regions (Figure 7B), suggesting that the cytoplasmic filaments penetrate into the central pore, and/or that RanGAP can bind to other parts of the FG-network.

HeLa cells (authenticated via STR profiling by ATCC) were cultured in Dulbecco’s Modified Eagle Medium (GIBCO, Invitrogen, Carlsbad, CA) supplemented with 4.5 g/L glucose, 862 mg/L Gluta-MAX-I, 15 mg/mL phenol red, 100 U/mL penicillin, 100 μg/mL streptomycin, and 10% (v/v) fetal bovine serum (GIBCO, Invitrogen, Carlsbad, CA). Cells were transfected with expression plasmids using Lipofectamine 2000 according to the manufacturer’s instructions (Invitrogen). For Pom121-mEos3, mEos3-Pom121, mEos3-Nup98, and RanGAP-mEos3, a stable cell line was generated from a single cell clone. In all other cases, cells were split ~24 hr after transient transfections, and were examined ~24 hr after splitting. Cell lines were occasionally tested for mycoplasma contamination.

Schematics of the mEos3 fusion proteins produced by the following plasmids can be found in Figure 1—figure supplement 2. Protein expressing inserts of all plasmids were confirmed by DNA sequencing.

mEos3 – mEos3.1 was PCR amplified from the mEos3.1 N1 vector (Zhang et al., 2012) (gift of Dr. Tijana Jovanovic-Talisman, City of Hope, Duarte, CA) using forward primer 5'-GTCGCTAGCAGTGCGATTAAGCCAGACATGAAG-3' and reverse primer 5'-CCAGAATTCTTATCGTCTGGCATTGTCAGGCAATCC-3'. The product was digested with Nhe1/EcoR1 (here and elsewhere, restriction sites within primers are underlined) and ligated into pRSETA-mEos2 (gift of Dr. Jie Xiao, Johns Hopkins University, Baltimore, MD) digested with the same enzymes, yielding plasmid pRSETA-mEos3, which produces N-terminally 6xHis-tagged mEos3.

Pom121-mEos3 – mEos3.1 was PCR amplified from the mEos3.1 N1 vector using forward primer 5'-TGGCAATTGGGAGGAAGTGCGATTAAGCCAGACATG-3' and reverse primer 5'-CTAACGCGTTTATCGTCTGGCATTGTCAGGCAATCC-3'. The product was digested with Mfe1/Mlu1 and ligated into plasmid peGFP-rPom121 (gift of Dr. Jan Ellenberg, EMBL, Heidelberg) digested with the same enzymes, yielding plasmid peGFP-rPom121-mEos3. Rat Pom121 was PCR amplified from the plasmid eGFP-rPom121 using forward primer 5’-TTTGCTAGCATGTCTCCGGCGGCTGCGGC-3’ and reverse primer 5'-GGGCAATTGTAACTTCTTGCGGGTGTGCTGCCTTCG-3', which mutates the stop codon TTA on Pom121 to TTA. The rPom121 PCR product was digested with Nhe1/Mfe1 and ligated into peGFP-rPom121-mEos3 digested with the same enzymes, yielding plasmid prPom121-mEos3, which produces Pom121-mEos3.

mEos3-Pom121 – mEos3.1 was PCR amplified from the mEos3.1 N1 vector using forward primer 5'-GGCGCTAGCATGAGTGCGATTAAGCCAGAC-3' and reverse primer 5'-CGGAGATCTTCGTCTGGCATTGTCAGGCAATC-3', which removes the stop codon on mEos3. The product was digested with Nhe1/Bgl2 and ligated into peGFP-rPom121 digested with the same enzymes, yielding plasmid pmEos3-rPom121(long linker). To remove the long linker between the mEos3 and rPom121 domains, rPom121 was PCR amplified from the plasmid peGFP-rPom121 using forward primer 5’-TTTAGATCTTCTCCGGCGGCTGCGGCGGCTGAC-3’ and reverse primer 5'-CCCACGCGTTTACTTCTTGCGGGTGTGCTGCC-3'. The PCR product was digested with Bgl2/Mlu1 and ligated into pmEos3-rPom121(long linker) digested with the same enzymes, yielding plasmid pmEos3-rPom121, which produces mEos3-Pom121.

mEos3-Nup153 – Human Nup153 was PCR amplified from the plasmid peGFP3-Nup153 (gift of Jan Ellenberg, EMBL, Heidelberg) using forward primer 5'-TTAAGATCTGCCTCAGGAGCCGGAGGAGTCG-3' and reverse primer 5'-CGGACGCGTTTATTTCCTGCGTCTAACAGCAGTC-3'. The product was digested with Bgl2/Mlu1 and ligated into plasmid pmEos3-Pom121 digested with the same enzymes, yielding plasmid pmEos3-Nup153, which produces mEos3-Nup153.

Nup153-mEos3 – Human Nup153 was PCR amplified from the plasmid peGFP3-Nup153 using forward primer 5'-TATGCTAGCATGGCCTCAGGAGCCGGAGGAGTCG-3' and reverse primer 5'-GGGACGCGTTTTCCTGCGTCTAACAGCAGTCTTTATCTTG-3'. The product was digested with Nhe1/Mlu1 and ligated into plasmid prPom121-mEos3 digested with the same enzymes, yielding plasmid pNup153. Then, mEos3.1 was PCR amplified from the mEos3.1 N1 vector using forward primer 5'- TTTACGCGTGGAGGAAGTGCGATTAAGCCAGACATG-3' and reverse primer 5'- CTAACGCGTTTATCGTCTGGCATTGTCAGGCAATCC-3', which includes a stop codon at the end of the coding sequence for mEos3. Plasmid pNup153 was digested with Mlu1 and the digested product was dephosphorylated by shrimp alkaline phosphatase (rSAP, New England Biolabs, Ipswich, MA). The mEos3.1 PCR product was digested with Mlu1, and ligated into the dephosphorylated pNup153 fragment, yielding plasmid pNup153-mEos3, which produces Nup153-mEos3.

mEos3-Nup98 – Human Nup98 was PCR amplified from the plasmid peGFP-Nup98 (gift of Jan Ellenberg, EMBL, Heidelberg) using forward primer 5'-CCGAGATCTTTTAACAAATCATTTGGAACACCCTTTGG-3' and reverse primer 5'-TATACGCGTTCACTGTCCTTTTTTCTCTACCTGAG-3'. The product was digested with Bgl2/Mlu1 and ligated into plasmid pmEos3-Pom121 digested with the same enzymes, yielding plasmid pmEos3-Nup98, which produces mEos3-Nup98. Plasmid pmEos2-Nup98, which produces mEos2-Nup98, was made identically.

Mutant Versions of mEos3-Nup98 – Forward primer 5'-TTTAGATCTGCACAAAATAAACCAACTGGCTTTGGC-3' and reverse primer 5'-TATACGCGTTCACTGTCC TTTTTTCTCTACCTGAG-3' were used to obtain a PCR fragment of Nup98 encoding amino acids 110–920, which was digested with Bgl2/Mfe1 and then ligated into plasmid pmEos3-Nup98 digested with the same enzymes, yielding plasmid pmEos3-^110tipNup98, which produces mEos3-^110tipNup98. Forward primer 5'-CCCGCTAGCATGTTTAACAAATCATTTGGAACACCC-3' and reverse primer 5'-TTTGCTAGCAAAGGCATTGTTTTGGGATGAGAAGAG-3' were used to obtain a PCR fragment of Nup98 encoding amino acids 1–109, which was digested with Nhe1, dephosphorylated as described above, and then ligated into plasmid pmEos3-^110tipNup98 digested with the same enzyme, yielding plasmid pmEos3-^110midNup98, which produces mEos3-^110midNup98. The 400tip, 400mid, 500tip, 500mid, 700tip, and 700mid (see Figure 1—figure supplement 2 for nomenclature) mutant expression plasmids were constructed in a similar manner.

RanGAP-mEos3 – Mouse RanGAP was PCR amplified from the plasmid pET11d-RanGAP (gift of Jan Ellenberg, EMBL, Heidelberg) using forward primer 5'-TTGGCTAGCATGGCCTCTGAAGACATTGCC-3' and reverse primer 5'-GGGCAATTGGATGTTGTATAGCGTCTGCAGCAG-3'. The product was digested with Nhe1/Mfe1 and ligated into plasmid prPom121-mEos3 digested with the same enzymes, yielding plasmid pRanGAP-mEos3, which produces RanGAP-mEos3.

The mEos3, Imp β1 (Kutay et al., 1997b), and Imp β2 (Izaurralde et al., 1997) proteins all contain a 6xHis-tag and were purified by NiNTA and size exclusion chromatography. Plasmids were transformed into Escherichia coli BL21(DE3), and protein production (1 L total culture) was induced with 1 mM isopropyl β-D-1-thiogalactopyranoside (IPTG). After overnight growth at 18°C, cells were centrifuged at 5000 g for 10 min at 4°C. Pellets (~2 g) were resuspended on ice in 20 mM Tris, 1 M NaCl, 10 mM imidazole, 2 mM β-mercaptoethanol, pH 8.0 with protease inhibitors (1 mM phenylmethylsulfonyl fluoride, 2 µg/mL pepstatin, 2 µg/mL leupeptin, and 20 µg/mL soybean trypsin inhibitor). Resuspended cells were lysed via French press (3X at 1000 psi). The lysate was centrifuged at 50,000 g for 15 min. The supernatant was added to a 1 mL Ni-NTA column (Qiagen, Germantown, MD). The Ni²⁺ beads were washed with 40 mL 20 mM Tris, 1 M NaCl, 10 mM imidazole, 0.1% Triton X-100, pH 8.0 with protease inhibitors followed by 40 mL 20 mM Tris, 50 mM NaCl, 20 mM imidazole, pH 8.0 with protease inhibitors. Proteins were eluted with 10 mM Tris, 250 mM imidazole, 50 mM NaCl, pH 8.0. Major protein fractions were combined, concentrated with Microsep 10K or 30K Omega centrifuge filters (Pall Corp., NY), and then further purified by size-exclusion chromatography (Superdex 200; GE Healthcare, Wauwatosa, WI) using 20 mM Hepes, 110 mM KOAc, 5 mM NaOAc, 2 mM MgOAc, 1 mM EGTA, pH 7.3.

Cells were permeabilized and prepared for microscopy as previously described (Izaurralde et al., 1997; Lyman et al., 2002; Yang et al., 2004). In short, HeLa cells were grown on coverslips overnight, and an ~20 µL flow chamber was constructed from high-vacuum grease and a top coverslip. Cells were permeabilized by incubation with 40 mg/mL digitonin in import buffer (20 mM Hepes, 110 mM KOAc, 5 mM NaOAc, 2 mM MgOAc, 1 mM EGTA, 1 mM dithiothreitol, pH 7.3) for 2 min. Permeabilized cells were washed twice with import buffer containing 1.5% (w/v) polyvinylpyrrolidone (~360,000 g/mol). When used, WGA (Vector Laboratories, CA), Imp β1, or Imp β2 were incubated with cells for 10 min before imaging.

Cells were imaged using a Zeiss 200M inverted microscope, equipped with an alpha plan-apochromat 100X, 1.46 NA oil-immersion objective (Zeiss). A 405 nm laser (100 mW, CUBE, Coherent, Santa Clara, CA) was used to activate the fluorescent protein mEos3 (30–35 W/cm²). The green fluorescence of mEos3 was obtained with an ArKr mixed-gas ion laser (2.5 W all lines, Stabilite 2018-RM, Spectra-Physics, Mountain View, CA) at 488 nm and the orange fluorescence of activated mEos3 was obtained with a solid-state laser (150 mW, Excelsior One, Newport, Santa Clara, CA) at 561 nm with an excitation density of 2 kW/cm². Both activation and excitation beams passed through a λ/4 wave plate (ThorLabs, Newton, NJ), generating circularly polarized light. A λ/2 wave plate was used to rotate the angle of linear polarization of the excitation lasers. The sample was illuminated via narrow-field epifluorescence (Yang et al., 2004), that is, a 300 µm pinhole was placed within a specimen-conjugate plane in the activation/excitation beam path, thereby restricting the specimen illumination area to ~7 μm diameter. After passing through a quad-bandpass filter (FF01-446/523/600/677-25, Semrock, Rochester, NY), the fluorescence emission was collected with an EMCCD camera (Evolve 128, Photometrics, Tucson, AZ). Image acquisition was controlled by MetaMorph (Molecular Devices, Sunnyvale, CA).

Single mEos3 molecules were localized using an algorithm written in Matlab (The MathWorks, Inc., Natick, MA). Fluorescent spots were fit by a symmetric two-dimensional (2D) Gaussian function, whose center was assumed to be the particle's position. Particles in consecutive frames were considered to belong to the same trajectory when they were within a user-defined distance r. Considering the NPC size, we set r = 200 nm.

Coverslip adsorbed 1 µM TetraSpeck microspheres were used for image alignment. Spot centers were determined by 2D Gaussian fitting. The coordinates of n spots were summarized as:

where x₁, x₂, …, x_n and y₁, y₂, …, y_n are the coordinates of the spot centers in the p-polarization channel, and the primed values correspond to the coordinates of the spot centers in the s-polarization channel. (X, Y) and (X', Y') are related as follows:

where ${\displaystyle M_r}$ is a rotation, ${\displaystyle f}$ is zoom factor, and ${\displaystyle B}$ is a translation. Fit parameters were determined by using the Matlab non-linear least squares fitting function ‘lsqcurvefit’, yielding ${\displaystyle \mathrm{\Omega }\approx 0}$ and ${\displaystyle f\approx 1}$. Once ${\displaystyle M_r,f,\text{and }B}$ were determined, the expected ${\displaystyle (x_i,y_i)}$ values were compared with the experimental ${\displaystyle (x_i,y_i)}$ values. The standard deviation of the differences, which was considered to be the alignment precision in both ${\displaystyle x}$ and ${\displaystyle y}$, was determined to be ~11 nm for each coordinate.

Samples were illuminated continuously with both 405 nm (activation beam) and 561 nm (measurement beam). Since the number of inactive mEos3 molecules decreased during data acquisition, the intensity of the activation laser was increased from 30 W/cm² to 35 W/cm² over forty 1000-frame movies. As a measure of the rapid photocycling in our experiments, ~80% of the mEos3 molecules in an imaging field were activated and photobleached in ~60 s at an imaging speed of 100 Hz.

The localization precision in both the x and y dimensions can be estimated by (Mortensen et al., 2010):

where s is the standard deviation of the Gaussian fit (~140 nm), a is the effective pixel size (240 nm), b is the background noise (~3 photons/pixel), and N is the total number of photons collected in the spot. Equation 2 yielded an average static localization precision of σ_x,y = ~17 nm for single mEos3 molecules in PALM experiments (N ≈ 350 photons).

Brightfield images of the NE were taken at the beginning of each movie (1000 frames, 10 s duration), and the NE position was determined essentially as described earlier (Yang and Musser, 2006b). The pixel intensities within a row across the NE were fit with a 1D Gaussian function. Peak positions from rows covering the useful area were fit with a cubic function, which was considered to trace the NE.

Clusters of three or more fluorescent spots with a maximal distance from their centroid of 200 nm were considered to arise from a single NPC. Inter-cluster distances were >400 nm. A normal to the NE that passed through a cluster centroid was defined as the transport axis of an NPC. Individual NPC transport axes were aligned (translated and rotated) to overlay fluorescent clusters.

Origin 7 (OriginLab, Northampton, MA) was used to convert the x- and y-coordinates of aligned and overlaid fluorescent spots into a 20 × 20 matrix with a bin size of 20 nm. Contours were plotted in intervals of 12.5% of the maximum bin.

Two major sources of error contribute to the broad distributions of PALM localizations along the transport axis (σ > 80 nm: Figure 7). The error in the NE position (σ_NE) is likely substantial due to NE spatial fluctuations, a non-smooth path of the NE, and heterogeneity near the NE, which affects the accuracy of the bright-field NE localization algorithm. Since previous work has demonstrated resolution of activities occurring on opposite sides of the NPC with particle distribution widths of <50 nm (which includes localization error) (Sun et al., 2013; Sun et al., 2008), we estimate an upper limit of ~45 nm for σ_NE. The second major source of error is the particle localization error (σ_x,y = ~17 nm; see Localization Precision section). While other sources of error exist (Musser and Grünwald, 2016), σ_NE and σ_x,y dominate and yet are insufficient to explain the broad particle distributions along the transport axis. The most likely additional major contributors to the broad PALM distribution maps are multiple anchoring sites for the labeled proteins and motion of the FG-polypeptides themselves. Higher resolution data is required for more precise conclusions on FG-polypeptide translational mobility and more refined maps of FG-polypeptide distributions and anchoring sites.

For p-PALM, the emission light was separated with a 50% polarizing beam splitter cube (PBS201, ThorLabs, Newton, NJ) mounted in an Optosplit III beamsplitter (Cairn Research, Kent, UK). The two polarization components were imaged simultaneously on the two halves of the EMCCD camera. A system-dependent factor, ${\displaystyle g}$, corrects for differences in photon collection efficiency by the p- and s-detection channels, and must be empirically determined to calculate the polarization according to:

where ${\displaystyle I_p}$ and ${\displaystyle I_s}$ are the intensities measured in the two detection channels. To estimate ${\displaystyle g}$, we measured the intensities of mEos3 molecules in 92% glycerol, which rapidly rotate on the data collection timescale (Figure 3—figure supplement 1), and therefore, ${\displaystyle <p{>}_{\text{cir}}}$ is assumed to be 0. Under these conditions ${\displaystyle g=<I_s/I_p>=0.92}$.

Except for the data collected in 92% glycerol (see Figure 3—figure supplement 1), only molecules that lasted for three or more frames were used for polarization analysis. Polarization values were only calculated for the second frame of each trajectory. This approach was designed to only include intensities that reflected photon emission over the entire frame (i.e., ensuring that photoactivation or photobleaching did not occur during data collection) and to weight each trajectory/molecule equally. Both of these constraints are essential to accurately estimate rotational mobility. Note that pooling p-PALM images would increase the integration time and the number of photons collected, both of which influence the measurement (Figure 2—figure supplements 2A,5). Pixel-dependent background intensities were determined by averaging a 1000-frame movie collected at the end of the experiment after complete photobleaching of mEos3.

For a single fluorophore molecule, the photons collected in a single image correspond to hundreds of excitation and emission cycles, during and between each of which the probe might rotate. To decipher how polarization measurements are affected by the various conditions that could occur in p-PALM experiments, we estimated the photons collected (intensities measured) in the p- and s-polarization channels using rotational random walk simulations. Rotational random walk trajectories were simulated using Microsoft Excel with the RiskAMP Monte Carlo Simulation Engine (https://www.riskamp.com).

A freely diffusing spherical particle has identical rotational diffusion constants for the molecule's three principle rotational axes. Such a particle reasonably approximates most globular proteins (Loman et al., 2010). However, for an mEos3 molecule tethered to an FG-polypeptide, restrictions on rotational mobility due to the tether point can be expected. mEos3 has a β-barrel structure (Zhang et al., 2012). It was tethered to FG-polypeptides via its N- or C-terminus (or both), each of which are at the bottom of this β-barrel. For simplicity, we assumed that the tether point was on the rotational z-axis, and that this was coincident with the β-barrel axis. Restricted movement of the tether point thus results in D_z > D_x ≈ D_y (Appendix 1—figure 1). We assumed that the excitation and emission transition dipole moments of mEos3 are parallel, which is the case for GFP and many fluorophores (Ha et al., 1999; Inoué et al., 2002), and therefore, we usually more simply refer to these as the transition dipole. The transition dipole was assumed to be in the yz-plane of the molecule, and, since the transition dipole of GFP is ~60° from the β-barrel axis (Inoué et al., 2002), we assumed this to be approximately true for mEos3 as well (Zhang et al., 2012). Appendix 1—figure 2 demonstrates the effects of varying the angle (γ) between the transition dipole and the rotational z-axis.

The initial output of the rotational random walk simulations was the number of photons collected in each of the two polarization channels. The approach is briefly summarized here – details follow in subsequent sections. The dipole's initial orientation was chosen randomly. For each time step, the dipole was rotated around its three principle rotational axes via an angular step randomly chosen from a normal distribution defined by its rotational diffusion constant. Three decisions were then used to determine if a photon was collected, and if so, to which detection channel it went: (1) the excitation of the probe was stochastically determined based on its excitation probability, given the illumination ellipticity and the orientation of the dipole; (2) the probability of photon collection was determined based on the dipole’s orientation and the solid angle subtended by the objective NA; and (3) if a photon was collected, it was partitioned into either the p- or s-channel depending on probabilities determined by the dipole's orientation. Most rotational random walk steps did not result in the collection of a photon. For circular excitation, the number of steps (N_s) for most simulations was 1400, leading to an average of ~352 photons collected under high D_r conditions (Figure 2—figure supplement 5), which matches well with the average of ~300–400 total photons collected in the two polarization channels in p-PALM experiments (Supplementary file 1). Under these simulation conditions, an average of 341 total photons are expected based on the average excitation efficiency (2/3) and the photon collection efficiency of the objective (~36.5%). For linear excitation, N_s = 2800 was used to compensate for the lower average excitation efficiency (1/3). Thresholding (see later) is responsible for the higher than expected average total photons collected, particularly under low D_r conditions (Figure 2—figure supplement 5).

For the laboratory frame, we assumed a spherical coordinate system where φ describes the angle from the z-axis (optical axis), and θ describes rotation around the z-axis from the x-axis. Random walk steps consisted of three angular sub-steps, one each around the molecule's three principal rotation axes, which were randomly oriented at the beginning of each simulation. The molecule's coordinate system was continuously updated and output to the laboratory frame using vector cross product multiplication to determine perpendicular unit vectors. Rotations were calculated by quaternion multiplication as follows. A rotation of the unit dipole vector a = a_xi + a_yj + a_zk by an angle ϕ around the unit rotation axis vector u = u_xi + u_yj+ u_zk was calculated as:

where a' is the rotated vector and the quaternion (q) and its conjugate (q⁻¹) were given by,

Angular sub-step sizes were randomly selected from a normal distribution centered around the starting position with a variance of ϕ² = 2Dτ (with D = D_x, D_y, or D_z, as required for the rotational axis), where τ is the duration of each angular step, or ϕ² = 2Dτ_m, where τ_m is time between excitation and emission of the molecule. The τ_m at each step was randomly chosen from an exponential decay defined by the fluorescence lifetime (τ_F) of mEos3, which was assumed to be 3.5 ns (Adam et al., 2011). Each rotational random walk cycle included an excitation step, rotation during the fluorescence lifetime (three independent sub-steps defined by D_x, D_y, and D_z), and rotation after the photon was emitted (three independent sub-steps). Since t is the image integration time, τ = t/N_s.

Excitation probability is proportional to the square of the magnitude of the electric field along the direction of the transition dipole (Forkey et al., 2000). Therefore, since we used a narrow-field excitation approach (i.e. with a very low NA for excitation), the excitation probability at each time step was set to (sin²φ)(cos²θ + (1/ε)sin²θ), where the ellipticity is defined by ε = E_x/E_y, and where E_x and E_y correspond to the average magnitudes of the electric fields along the x- and y-laboratory axes. Typical values used in the simulations were ε = 1 (circularly polarized light) and ε = 100 (linearly polarized light).

For a fixed unit dipole with laboratory frame components (x, y, z), the emission intensities collected in the p- and s-channels are:

where

are constants determined from Axelrod's expressions (Axelrod, 1979) normalized such that i_p + i_s = i_tot for θ_obj = 180° (all light collected) (Ha et al., 1999). Assuming an angular semiaperture of θ_obj = 74.1° estimated from an immersion oil index of refraction of n = 1.518 and NA = n sin(θ_obj)=1.46, these constants were calculated as K₁ = 0.38, K₂ = 0.012, and K₃ = 0.15 (but see Appendix 1—figure 4). For values of θ_obj < 180°, i_p + i_s + i_e = i_tot where i_e is the intensity of photons that escaped detection. Thus, the probability that an emitted photon escapes detection is:

and the probability that the photon is collected is 1 – P_e. If a photon is collected, the probability that the photon is partitioned into either the p- or s-channel is:

Note that i_tot ends up being simply a scaling factor that disappears upon calculation of these probabilities. Note also that the high NA implies that photons can be collected in either the p- or s-channels from molecules oriented with a z-axis component, which reduces the number of measured polarization values near the ±1 limits (Figure 2).

Equation 12 was additionally verified using simulations in which it was determined if the propagation direction of the emitted photon allowed it to be captured based on θ_obj. The photon’s propagation direction was randomly chosen assuming that the propagation direction is proportional to sin²φ_k (Forkey et al., 2000), where φ_k is the angle between the transition dipole and the propagation direction. Identical results were obtained.

In order that the simulation results more accurately reflected experimental intensity distributions and noisy single molecule data, emission intensities based on photon counts were broadened and background noise was added. Whereas histograms of total photon intensities from the simulations were Gaussian under high rotational mobility conditions, experimental intensity histograms were approximately three-fold wider and were better described by log-normal distributions (e.g. Figure 6E), consistent with previous observations (Mutch et al., 2007). The broader experimental intensity distributions likely arise from a variety of factors, including the amplification noise of the EMCCD camera (Chao et al., 2013), differential focusing, irregularities within different light paths (permeabilized cells provide complex scattering and refractive index changes), as well as dirt and aberrations/variations in the optics and pixel quantum efficiencies (Mutch et al., 2007). Conformational differences of the probe that affect photon output are also possible. The simulated intensity distributions were therefore broadened by random selection of intensities from log-normal distributions with scale parameter μ* = the total photons collected at each time step and shape parameter σ* = 0.24. The value of the shape parameter was guided by fits to experimental intensity distributions and agreement of the final simulation results to experiment. The value of σ* has essentially no effect on <p> and Var(p) (Figure 2—figure supplement 4). Log-normal distributions have an advantage over normal distributions for the purpose described here, particularly for lower intensities, since negative values cannot occur. Background noise was added to the p- and s-intensities recalculated from the new total intensities, partitioned according to the original photon numbers. Background noise (in photons) was normally distributed with σ_p = 15 and σ_s = 15 for circular excitation, and σ_p = 22 and σ_s = 15 for linear excitation, which are the average noise levels in p-PALM experiments over the 5 × 5 pixel regions of interest used to obtain fluorescence intensities. This approach yielded intensity distributions, ps-photon scatterplots, and polarization histograms that resembled the experimental data (Figure 6).

A threshold was implemented to distinguish single molecule signals from background noise in experimental measurements. For rapidly rotating particles, distinguishing single molecule signals from background was fairly straightforward. However, slowly rotating particles have strongly preferred transition dipole orientations, and therefore, intensity histograms were broader for such molecules, as expected, since excitation probabilities and emission intensities depend on dipole orientation. In particular, dipoles with a strong z-component yielded fewer emission photons primarily due to poor excitation efficiency, and therefore, molecules with such preferred orientations were not reliably detected in our experiments. Our criteria for spot selection was ≥100 photons in either the p- or s-detection channel.

To ensure more accurate modeling of the experimental data, a 100 photon threshold was also implemented for the simulation results. When using linear excitation, this threshold eliminates ~50% of the p values < 0 when D_r is low (Regime V; compare Figure 2B and Figure 2—figure supplement 1B). Consequently, under these conditions, a 100 photon threshold increases <p>_lin from ~0 to ~0.24 (Figure 2—figure supplement 4A). This is a major change, and indicates an extreme sensitivity to imaging and analysis parameters. In contrast, Var(p)_lin and Var(p)_cir values are less sensitive to this threshold for all values of D_r (Figure 2—figure supplement 4B C). Notably, however, Var(p)_cir has a wider dynamic range than Var(p)_lin at low D_r values, indicating greater sensitivity to changes in rotational mobility. For circular excitation and small D_r values, the threshold eliminates ~20% of the simulation values, which predominantly increases the low D_r asymptote in Var(p)_cir plots. Under these conditions, the thresholding selects against particles oriented with a strong z-component, for which x- and y-intensities are similar and low, leading to the selective elimination of p values near zero (see Figure 2D) and an increase in Var(p)_cir. Background noise also contributes to increasing this asymptotic value, although noise predominantly influences the high D_r asymptote (Figure 2—figure supplement 4C). In contrast, the 100 photon threshold does not affect the high D_r asymptote since at high D_r values all measurements are above the threshold. Notably, thresholding has little influence in most of the region where Var(p)_cir is most sensitive to D_r (Figure 2—figure supplement 4C). These simulations indicate that measuring <p>_lin is best for rotational diffusion regimes I-III and measuring Var(p)_cir is best for regimes III-V.

Single molecule and bulk fluorescence data yield fundamentally different information under conditions of low rotational mobility. This is directly observed in Figure 2A, where the single molecule and ensemble polarization results are compared for linear excitation. The emission intensities collected in the p- and s-polarization channels, I_p and I_s, are mixtures of the intensities emitted along the x-, y-, and z-axes of the laboratory frame, I_x, I_y, and I_z, respectively:

where K₁, K₂, and K₃ are defined in Equations 9-11. For x-polarized linear excitation, I_y = I_z, and therefore:

Thus, the corrected polarization (p_c) and anisotropy (r_c) values calculated for individual molecules in p-PALM images are given by:

Using these expressions, the corrected values for the mean polarization (<p>_lin,c) and the mean anisotropy (<r>_lin,c) measured in p-PALM experiments are given by:

and are plotted in Figure 2A and Figure 2—figure supplement 3, respectively. The bulk polarization (p_bulk) and anisotropy (r_bulk) were estimated by summing the intensities from 10,000 different molecules with random initial orientations:

These expressions yield low D_r limiting values of 0.5 and 0.4, respectively (Figure 2A and Figure 2—figure supplement 3), as expected (Lakowicz, 2006). The bulk and p-PALM results differ for two reasons (discussed more fully in the text): (1) the microscope objective mixes polarizations (Equations 15 and 16); and (2) the values calculated from single molecule data weights each molecule identically (i.e. the polarization/anisotropy of each molecule counts the same no matter how many photons are emitted), whereas the values corresponding to bulk conditions weights the contribution of each molecule to the measured polarization/anisotropy (a single measurement) depending on the number of photons emitted. The bulk polarization and anisotropy under circular excitation are always 0.

Under single molecule conditions, the variance in the measured polarization values, Var(p), provides a measure of the width of polarization histograms (see Figure 2). The variance was calculated as Var(p) = <p²> – (<p>)², where <p²> is the average square of the measured polarization values. Theoretically, using circular excitation, Var(p) = <p²> since 

 = 0 for all D_r values. Consequently, since in most cases we obtained <p>_cir ≈ 0 (see Supplementary file 1), we assumed that Var(p)_cir was equivalent to the experimentally determined <p²>_cir. As defined here, Var(p) does not exist under bulk conditions since the information from all individual molecules is integrated by the measurement into a single value. This fundamentally explains why circular polarized excitation provides rotational mobility information under slow rotation conditions in a single molecule p-PALM experiment, whereas no information is obtained in a corresponding bulk experiment.

A general analytical solution for the Var(p)_cir dependence on the rotational diffusion constants is a complex problem. A few special cases are described in the Appendix. Guided by these results, it became clear that Pade-like approximations could be used for curve fitting, which enables extraction of rotational diffusion constants from measurements of Var(p)_cir. Here, we summarize our findings from a range of simulations where D_x, D_y, D_z, t, γ, background noise and threshold values were varied.

The three principle rotational axes, x, y, and z, of a particle are characterized by the rotational diffusion constants D_x, D_y, and D_z, respectively, with an average rotational diffusion constant given by D_r = (D_x + D_y + D_z)/3. For a freely diffusing sphere, D_x = D_y = D_z = D_r. For a sphere whose rotational motion is restricted via a tether point on its rotational z-axis, D_x = D_y, the average perpendicular rotational diffusion constant is D_⊥ = D_xy = (D_x + D_y)/2, and D_r = (2D_⊥ + D_z)/3. D_xz and D_yz are similarly defined. For a particle that is tethered via multiple constraints, is highly geometrically asymmetrical, or has its rotational diffusion differentially constrained by other means, D_x ≠ D_y ≠ D_z.

For a particle behaving similar to a freely diffusing sphere (i.e., D_x ≈ D_y ≈ D_z), we find that (e.g., see Appendix 1—figure 3):

where α, β, and a_1-3 are fit parameters that depend on γ, noise, the average number of photons collected, and the threshold. The value of α is determined by the high mobility limit (D_rt → ∞; lower asymptote), and is the same regardless of the relationship between the rotational diffusion constants. The value of β (with a small contribution from α) determines the low mobility limit (D_rt → 0; upper asymptote). In most cases, a₁ ≈ 1 works well. Using the quadratic equation, D_r is obtained from the positive root of Equation 25:

which allows calculation of D_r from a Var(p)_cir measurement.

In more complex situations (e.g., D_x ≈ D_y << D_z or D_x << D_y << D_z), we find that Var(p)_cir is well-approximated by:

The β₂ value determines an intermediate plateau (e.g., see Appendix 1—figure 3). Parameters from simpler situations can be used as a guide to fit data from more complex situations and often work well, but the parameter values are not strictly the same. We emphasize that Equations 25 and 27 are empirical solutions used to obtain smooth fits of simulation data. As far as we know, the constants a_1-5 do not have a physical interpretation.

The simulation results were used to estimate the number of experimental data points required to obtain a reasonable estimate of Var(p) and the error in the measurement. Datasets of 500–2000 measurements were randomly divided into 4 equivalently sized datasets, and means and standard deviations (SDs) were calculated. As shown in Figure 2—figure supplement 6, an experimental Var(p)_cir value obtained from ~2000 (4 × 500) measurements differs from the ‘true’ value by an average of ~2–3%. The ‘true’ value was included in the range defined by the mean ±SD approximately 85% of the time, and thus this range defines the 85% confidence interval. Errors (SDs) for all experimental <p> and Var(p) values were calculated in this manner (from 4 equivalently sized datasets), and are summarized in Supplementary file 1.

A probe’s rotational mobility is determined by its three principle rotational diffusion constants D_x, D_y, and D_z, which are related to the average rotational diffusion constant by D_r = (D_x + D_y + D_z)/3. The relationship between the values of these rotational diffusion constants can have a significant effect on the <p> and Var(p) obtained for a given D_r value. A critical parameter is the angle (γ) between the transition dipole and the major (dominant) rotational axis. Values of γ far from the magic angle of 54.7° (Axelrod, 1989) have a substantial effect on p-PALM measurements under highly anisotropic conditions (Appendix 1—figures 1–3).

Appendix 1—figure 1

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Appendix 1—figure 2

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Appendix 1—figure 3

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In most of the reported experiments, the mEos3 probe was tethered to FG-polypeptides via its N- or C-terminus, both of which are at the bottom of its β-barrel structure (Zhang et al., 2012). Consequently, we assumed that the tethered probe’s rotational z-axis was well-approximated by its β-barrel axis. The transition dipole of GFP subtends an ~60° angle from the β-barrel axis (Inoué et al., 2002), and we assumed that this is approximately true for mEos3 (Zhang et al., 2012). With these constraints, if the principal rotational diffusion constants were non-identical, we assumed that z-axis rotation of the probe at the tip of an FG-polypeptide would be the most facile (Appendix 1—figure 1A). Rotational random walk simulations for various D_z/D_xy ratios assuming D_z > D_xy (where D_⊥ $\equiv$ D_xy = D_x =  D_y) indicated that for γ = ~60° (i.e, near the magic angle), the D_z/D_xy ratio has little detectable influence on the relationship between <p>_lin or Var(p)_cir and D_r (Appendix 1—figure 1B,C). These simulations thus suggest that the mEos3 probe on tip-labeled FG-polypeptides will yield p-PALM data similar to that expected for a free particle.

For single molecule fluorescence experiments, a high NA objective is typically desired to maximize photon collection efficiency. However, the ranges of <p> and Var(p) obtained in p-PALM experiments depends on the NA, and both are broader at lower NA values (Appendix 1—figure 4). A major factor that leads to a reduction in the nominal NA is the spherical aberration that results from a refractive index mismatch, which occurs when using an oil immersion lens for an aqueous sample (Appendix 1—figure 5) (Mondal and Disaspro, 2014). An additional, albeit minor, factor reducing the NA can be a high scattering sample, which preferentially results in the loss of photons emitted at high angles relative to the transport axis due to the longer path in the scattering medium (Theer and Denk, 2006). Both of these factors contribute to reducing the NA in the reported p-PALM experiments.

Appendix 1—figure 4

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Appendix 1—figure 5

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The influence of refractive index mismatch/spherical aberration on p-PALM measurements became apparent in control experiments on freely diffusing mEos3 in glycerol solution. Under our experimental conditions, translational motion was too fast to obtain p-PALM data on freely diffusing mEos3 in normal buffer. However, in 92% glycerol, the translational motion of freely diffusing mEos3 was significantly reduced and p-PALM experiments yielded <p>_lin = 0.47–0.50 (Supplementary file 1; Figure 3—figure supplement 1). Notably, the estimated D_r of ~3 × 10⁴ rad²/s (Stokes-Einstein-Debye relation; [Loman et al., 2010]) in this high viscosity solution (η ≈ 380 cP; [Lide, 1998]) is in rotational mobility Regime III, indicating that θ_eff < 45° (NA_eff < 0.94). Thus, the NA_eff was lower when using a 92% glycerol solution (n = 1.46; θ_eff < 45°) than an aqueous sample (n = 1.33, θ_eff ≈ 50–60°). This was surprising since spherical aberration should be worse for an aqueous sample than a glycerol sample when using an oil immersion objective due to a larger refractive index mismatch (Appendix 1—figure 5). Additional aberrations may have been introduced by the dual-polarization imaging system (see Materials and methods).

The range of <p>_lin for mEos3 within the NPC was ~0.41–0.46 for the various conditions tested (Supplementary file 1). This range is higher than that in Region III of Figure 2A (~0.39). Thresholding, background noise, and ellipticity were not able to explain these differences (Figure 2—figure supplements 4,7), but a reduced NA can. Our linear excitation data (Supplementary file 1) are consistent with an NA_eff ≈ 1.02–1.15 in water (θ_eff ≈ 50–60°) (Appendix 1—figure 4), which is substantially below the nominal value of the NA = 1.46 (θ_obj = 74.1°). The reduced NA is likely a consequence of spherical aberration with a minor contribution from scattering in the sample.

Most rotational walk simulations reported in this paper assume θ_obj = 74.1°, which corresponds to a non-scattering sample medium and aberration-free imaging with a sample index of refraction matching the immersion oil. These simulations are valid for determining the role of various parameters. However, since simulations with θ_eff = 50–60° more closely model the experimental conditions on permeabilized cells, they were used for more precise interpretation of the results (see Mixed populations section in main text). Importantly, our major conclusions are not affected by a moderate uncertainty in the NA_eff.

The sensitivity of the p-PALM approach to binding interactions became apparent when mEos2 (McKinney et al., 2009) was used as a probe. We observed significantly reduced rotational mobility for the probes on mEos2-Nup98 relative to mEos3-Nup98 (Appendix 1—figure 6), consistent with the finding that mEos2 forms dimers (Hoi et al., 2010; McKinney et al., 2009). Dimerization of mEos2 was likely promoted by the high local concentration of such probes within the FG-network. Since the mEos3 probe is a true monomer (Zhang et al., 2012), the reduced rotational mobilities reported here under certain conditions requires an alternate interpretation. As we have argued in this paper, molecular crowding is a reasonable explanation for the low rotational mobilities observed for probes in the FG-network of NPCs under some conditions. Nonetheless, the mEos2 results indicate that p-PALM can be used to directly monitor the binding of fluorescent molecule to a binding partner.

Appendix 1—figure 6

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Since the p-PALM parameters <p> and Var(p) are averages over the entire population within the dataset, the rotational mobility behavior of individual sub-populations in more complex samples may not be accurately reflected in these values. In the main text, one example of a mixed population sample (containing molecules with at least two distinct rotational mobilities) was discussed (Figure 6). Here, we further discuss this example and identify additional datasets containing mixed populations.

The <p> and Var(p) values from rotational random walk simulations for a variety of conditions were obtained by running the simulations many times for individual molecules and averaging the results. In order to augment and verify the simulations, and guide the choice of the functional form of fits to the simulation data (Equations 25 and 27), we performed analytical calculations for a few special situations under circularly polarized illumination. For simplicity, we have dropped the ‘cir’ subscripts on <p> and Var(p) in this appendix.

Spherical coordinates were defined as:

Using Equation 7 (Materials and methods) and including the excitation probability, the intensity collected from m dipole orientations under high photon flux is:

where ε is the ellipticity, as defined earlier (Materials and methods). For all of the derivations in this appendix, we have assumed circular excitation (ε = 1) and that the dipole does not rotate between excitation and emission (D_r < ~10⁶ rad²/s). Consequently,

and similarly,

Substituting the expressions above into the definition of polarization in Equation 3 (Materials and methods) and assuming that g = 1, the polarization p measured for one molecule in elapsed time t for m dipole orientations (analogous to the rotational walk steps) is given by:

where A = (K₁ – K₂) /(K₁ + K₂) and B = 2K₃ /(K₁ + K₂). As discussed earlier, K₁, K₂, and K₃ depend on the NA of the microscope objective (Equations 9–11). For θ_obj = 74.1°, A = 0.939 and B = 0.765.

The upper limit of Var(p) is attained when the particles are rotationally immobile, which corresponds to either short collection times or arrested rotational diffusion. This is formally defined as the limit where D_rt → 0. Under these conditions, θ and φ are constant for a given individual particle, and thus, Equation 31 reduces to:

where the φ-dependence has been subsumed into the function H, which depends on A, B, and φ. Assuming an isotropic distribution of dipole orientations, the second moment of the polarization distribution, <p²>, for a very large collection of such particles is obtained by averaging p over all angles. Recalling that <p> = 0 and Var(p) = <p²> for circular excitation (see Materials and methods, Figure 2, and Equations 46 and 65), we get:

The integration over θ equates to π, which leads to:

As A and B are constant for a given experiment, numerically integrating using values for θ_obj = 74.1° yields:

Thus, for D_rt → 0 (immobile) and α = 0 (infinite photons), the value of β in Equation 25 is β₀ = 1/0.2529 = 3.954.