Single-molecule orientation localization microscopy for resolving structural heterogeneities between amyloid fibrils

Abstract

Simultaneous measurements of single-molecule positions and orientations provide critical insight into a variety of biological and chemical processes. Various engineered point spread functions (PSFs) have been introduced for measuring the orientation and rotational diffusion of dipole-like emitters, but the widely used Cramér-Rao bound (CRB) only evaluates performance for one specific orientation at a time. Here, we report a performance metric, termed variance upper bound (VUB), that yields a global maximum CRB for all possible molecular orientations, thereby enabling the measurement performance of any PSF to be computed efficiently (~1000× faster than calculating average CRB). Our VUB reveals that the simple polarized standard PSF provides robust and precise orientation measurements if emitters are near a refractive index interface. Using this PSF, we measure the orientations and positions of Nile red (NR) molecules transiently bound to amyloid aggregates. Our super-resolved images reveal the main binding mode of NR on amyloid fiber surfaces, as well as structural heterogeneities along amyloid fibrillar networks, that cannot be resolved by single-molecule localization alone.

Fig. 1.

Orientation measurement precision using various point spread functions (PSFs) for 380 signal photons and 2 background photons per pixel. (a) Average (over orientation space) trace R_Σ of the CRB R, versus the trace Γ_Σ of our variance upper bound (VUB) for various PSFs. Blue: unpolarized standard (std), red: polarized standard (polar), yellow: bifocal microscope (bif), purple: bisected (bi), green: double helix (DH), aqua: tri-spot (tri), and maroon: quadrated (qua) PSF. Dark line: Γ_Σ = mean of R_Σ. Inset: error of Γ_Σ relative to average R_Σ. Bars indicate minimum/maximum error relative to the mean (circle) over all of space. (b) Fisher information (FI) lower bound [Γ⁻¹]_jj of each orientational second moment m_j. (c) Trace Γ_Σ of VUB for various refractive indices. Dotted line: refractive index of water. Simulated orthogonally polarized images of (d) a fixed molecule with orientation (θ = 90°, ϕ = 45°) and an isotropic emitter using (e) the polarized PSF and (f) the tri-spot PSF. Insets: normalized noiseless PSF images. Scale bars: 400 nm.

Fig. 2.

Transient amyloid binding (TAB) SMLM and SMOLM images acquired using Nile red (NR). (a) SMLM image of a network of Aβ42 fibrils. Color bar: localizations per bin (20 × 20 nm²). Inset: diffraction-limited image. (b) TAB SMOLM image, color-coded according to the mean azimuthal (ϕ) orientation of NR molecules measured within each bin. Inset: Main binding mode of NR to β-sheets, i.e., dipole moments aligned mostly parallel to the long axis of a fibril (its backbone). (c-g) All individual orientation measurements localized along fibril backbones within the white boxes in (b). The lines are oriented and color-coded according to the direction of the estimated ϕ angle. Red dashed lines depict fibril backbones estimated from the SMLM image. Scale bars: (a,b) 1 μm, (f,g) 100 nm. Orientation-localization data are available in Dataset 1 [30].

Fig. 3.

Structural heterogeneity of Aβ42 fibrils revealed by TAB SMOLM imaging. (a) SMLM image of fibril bundles. Color bar: localizations per bin (20 × 20 nm²). Inset: fibril cross-sections at the locations denoted by green and purple lines with measured full-width at half-maximum (FWHM) thicknesses. (b) TAB SMOLM image corresponding to (a), color-coded according to the mean (ϕ) and standard deviation (σ_ϕ) of the azimuthal orientation measured within each bin. Inset: zoomed (i) thin and (ii) thick fibril regions isolated from background structures. (c) Histograms of measured azimuthal orientations relative to the fibril backbone within the regions denoted in (b), showing standard deviations (σ_ϕ) of (i) 18° and (ii) 37°. (d) Measured wobbling areas (Ω) corresponding to the localizations in (c), yielding median wobbling areas (Ω_med, cyan) of (i) 0.07 sr and (ii) 1.89 sr. (e-h) TAB SMLM and SMOLM images of another fibril field of view, as in (a-d). Although fibril regions (i) and (ii) in the inset of (f) show little difference in apparent width, the measured orientation distributions contain significant differences. The standard deviations of azimuthal angles σ_ϕ are (i) 20° vs (ii) 30°, and the median wobbling areas (Ω_med, cyan) are (i) 0.07 sr vs (ii) 1.12 sr. Scale bars: 1 μm. Orientation-localization data are available in Dataset 1 [30].