Point-Spread Function Deformations Unlock 3D Localization Microscopy on Spherical Nanoparticles

Abstract

Nanoparticles (NPs) have proven their applicability in biosensing, drug delivery, and photothermal therapy, but their performance depends critically on the distribution and number of functional groups on their surface. When studying surface functionalization using super-resolution microscopy, the NP modifies the fluorophore's point-spread function (PSF). This leads to systematic mislocalizations in conventional analyses employing Gaussian PSFs. Here, we address this shortcoming by deriving the analytical PSF model for a fluorophore near a spherical NP. Its calculation is four orders of magnitude faster than numerical approaches and thus feasible for direct use in localization algorithms. We fit this model to individual 2D images from DNA-PAINT experiments on DNA-coated gold NPs and demonstrate extraction of the 3D positions of functional groups with <5 nm precision, revealing inhomogeneous surface coverage. Our method is exact, fast, accessible, and poised to become the standard in super-resolution imaging of NPs for biosensing and drug delivery applications.

Keywords: DNA-paint; nanoparticles; plasmonics; point-spread function; single-molecule localization microscopy; surface functionalization.

Figure 1

Commonly employed methods for single-molecule localization microscopy on nanoparticles for mapping of functional groups are hampered by PSF distortion. (a) Schematic of the process of localizing single emitters on a nanoparticle (NP) with localization microscopy. The surface of a functionalized NP is covered with chemical groups, here depicted as DNA strands. In single-molecule localization microscopy, light-emitting fluorescent dyes (red dots) bind transiently to functional groups of interest. Since only a small subset of available functional groups is labeled at each moment in time, each dye creates an isolated spot of light on the camera. The dye is localized (black cross) by fitting a point-spread function (PSF) model to the spot, to estimate the true position of the emitter (red cross) on the NP. Since the functional groups are transiently bound by a fluorophore, this process can be repeated sequentially for many frames in order to reconstruct a map of all functional groups on the NP. Due to PSF distortions, however, there is a discrepancy, often substantial, between the true and fitted emitter position, referred to as a mislocalization. (b) Theoretical and experimental PSFs for a single freely rotating fluorophore in the presence and absence of an NP, imaged on top of a glass coverslip. A single fluorophore (left column) creates a rotationally symmetric PSF, which may be modeled well by a 2D Gaussian PSF. When the fluorophore is located next to an NP (100 nm gold spherical NP, columns 2–4), the PSF does not resemble a 2D Gaussian but is distorted in a manner that depends on the position of the fluorophore relative to the NP. Scale bar applies to all PSFs in (b). Schematics are not to scale.

Figure 2

Numerical validation of our analytical PSF model. (a) Schematics of the analytical and numerical approaches for calculating the PSF for an emitter (red dot) in close proximity to an NP (gold sphere) in water on top of a glass coverslip (blue rectangle). In the analytical model, we directly and analytically derive the electromagnetic fields everywhere in the system. In the numerical approach, the fields are computed in a finite element manner on the discretized system. (b) Visual depiction of the steps of our analytical PSF model. Briefly, the dipole (red) and scattered (orange) fields are decomposed into plane waves (green), which are refracted at the water–glass interface, projected onto the far-field hemisphere (gray), collected by the objective (blue ellipse), and focused onto the camera. The far-field hemisphere is parametrized by a polar angle θ, which is further used in (d). (c) Difference between analytically and numerically calculated PSFs for a horizontally oriented dipole emitter on top of a 200 nm gold spherical NP, as a function of mesh sizes in the numerical calculation. The numerical PSF converges toward the analytical PSF for decreasing mesh size (left green y-axis). However, the computational time (right orange y-axis) drastically increases for the numerical approach (solid line) for smaller mesh sizes. In comparison, the computational time for the analytical PSF (dashed line) is four orders of magnitude faster and not depending on discretization. The “max PSF difference” is defined as the relative intensity difference of the most different pixel between the numerical and analytical PSFs. (d) Comparison of the electric far-fields and the PSFs of the analytical and numerical approaches for a fixed dipole emitter positioned with various orientations close to different NPs. The left column shows schematics of the NP-dipole (red arrows) configurations. The second column shows the comparison of the absolute, real, and imaginary components of the x-component of the electric far-field calculated using the analytical (respectively yellow, green, and orange lines) and numerical (black dashed lines) methods (y- and z-components shown in Figure S7). The third and fourth columns show the normalized numerical and analytical PSFs. Scale bar applies to all PSFs. Parameter values used in the two approaches can be found in Tables S1 and S2.

Figure 3

Workflow for localizing single emitters on the surface of a gold nanoparticle using our analytical PSF model. (a) DNA-PAINT experiments are performed on 100 nm gold spherical NPs. Single-stranded DNA docking strands (green, not to scale) are bound to the NP surface with a thiol bond, which mimics functional groups on the NPs. To these, freely diffusing single-stranded complementary DNA imager strands (red), conjugated to ATTO655 (red star), transiently bind, creating fluorescent bursts on the camera. (b) Example time trace of total detected number of photons (per 100 ms) from a single NP shows multiple binding events. (c) Zoom of the orange-outlined event in (b), of which the frames of the binding event itself (green shaded area) are averaged to obtain the PSF. (d) Each experimental PSF is fitted with our analytical model resulting in a fitted theoretical PSF. The high similarity between the experimental PSFs and the fitted theoretical PSFs shows that the analytical model perfectly describes the experimental data. The fit has four free parameters: the position of the emitter relative to the NP (expressed in spherical coordinates ϕ and θ), the number of photons, and the background intensity. By PSF fitting, the position of each emitter on the 100 nm spherical NP surface is obtained, expressed in spherical coordinates. These positions are projected onto the spherical NP to map the binding sites in 3D. Scale bar applies to all PSFs.

Figure 4

Inhomogeneous surface coverage of functional groups on gold nanoparticles. (a) All localizations (green dots) of 828 PSFs from the same single NP. (b) Area-conserved 2D mapping of the NP surface to visualize the distribution of all localizations. (c) Localization density map resulting from rendering each localization as a normalized 2D asymmetric Gaussian, where the two Gaussian axes represent the localization uncertainty in the polar and azimuthal directions, respectively. (d) Distribution of localization uncertainties of all 828 localizations, with an uncertainty of 4 ± 1 nm (mean ± std). (e) Localization maps of four different NPs, containing respectively 477, 533, 882, and 561 localizations, highlighting the particle-to-particle variation in the distribution of accessible docking strands over the NP surface.

Figure 5

Localizing single emitters on dielectric nanoparticle. (a) DNA-PAINT experiments are performed on 1 μm polystyrene spherical NPs. Single-stranded DNA docking strands (green, not to scale) are bound to the neutravidin-functionalized NP surface via biotin–neutravidin interactions. To these, freely diffusing single-stranded complementary DNA imager strands (red), conjugated to ATTO655 (red star), transiently bind. (b) Exemplary experimental PSFs (left) with their fitted theoretical equivalents (right), highlighting the unique PSF shapes not observed for the gold nanoparticles. Scale bar applies to all PSFs. (c) All localizations (green dots) of 719 PSFs from the same single NP. (d) Area-conserved 2D mapping of the NP surface to visualize the distribution of all localizations. Due to limited depth of focus, the PSFs on the lower section of the NP surface (blue shaded area) are too dim to be fitted. (e) Distribution of localization uncertainties of all 719 localizations, with an uncertainty of 15 ± 10 nm (mean ± std) on the surface (Methods).