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Universal inverse modeling of point spread functions for SMLM localization and microscope characterization

Abstract

The point spread function (PSF) of a microscope describes the image of a point emitter. Knowing the accurate PSF model is essential for various imaging tasks, including single-molecule localization, aberration correction and deconvolution. Here we present universal inverse modeling of point spread functions (uiPSF), a toolbox to infer accurate PSF models from microscopy data, using either image stacks of fluorescent beads or directly images of blinking fluorophores, the raw data in single-molecule localization microscopy (SMLM). Our modular framework is applicable to a variety of microscope modalities and the PSF model incorporates system- or sample-specific characteristics, for example, the bead size, field- and depth- dependent aberrations, and transformations among channels. We demonstrate its application in single or multiple channels or large field-of-view SMLM systems, 4Pi-SMLM, and lattice light-sheet microscopes using either bead data or single-molecule blinking data.

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Fig. 1: Concept and modalities of uiPSF.
Fig. 2: Application of uiPSF to multichannel ratiometric dual-color SMLM and 4Pi-SMLM.
Fig. 3: In situ PSF modeling from single blinking fluorophores.
Fig. 4: Ratiometric three-color imaging of the SC using an in situ PSF model.
Fig. 5: FD-PSF modeling.

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Data availability

The data that support the findings of this study are publicly available at Zenodo. Data for Fig. 2 are available at https://doi.org/10.5281/zenodo.10823308 (ref. 78). Data for Fig. 3 are available at https://doi.org/10.5281/zenodo.10823244 (ref. 79). Data for Fig. 4 are from ref. 60. Bead data and the first 800 frames of cell data for Fig. 5 are available at https://doi.org/10.5281/zenodo.10824517 (ref. 80). Example data for uiPSF are available at https://doi.org/10.5281/zenodo.10602740 (ref. 81). Source data are provided with this paper. Other datasets are available from the corresponding author upon reasonable request.

Code availability

uiPSF software is available at https://github.com/ries-lab/uiPSF, which includes source codes, example Jupyter notebooks and tutorials for localization analysis using the PSF models generated from uiPSF. The uiPSF software package is also available in Supplementary Software 1. The microscope acquisition software for single-objective systems was developed with Micro-Manager v2.0 (https://micro-manager.org/) and EMU v1.1 (https://github.com/jdeschamps/htSMLM/). The control software for the 4Pi-SMLM microscope is available at https://github.com/kiwibogo/Microscope-Control/tree/master/. The custom localization data analysis and rendering software is available at https://github.com/jries/SMAP/. The localization software for large FOV data is available at https://github.com/Li-Lab-SUSTech/FD-DeepLoc/.

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Acknowledgements

We thank Y. E. Katrukha and L. C. Kapitein from Utrecht University, the Netherlands; B. Hajj and L. Regnier from Institut Curie, France; K. J. A. Martens from Bonn University, Germany; A. Tschanz from the European Molecular Biology Laboratory, Germany; S. Khan and S. Pani from the University of New Mexico; Y. Li from Purdue University; J. A. Camarena from Arizona State University for testing the software. We thank X. Liu from the University of Hong Kong for valuable suggestions on inverse modeling. We thank A. M. E. Pacheco for preparing the Nup96-mMaple sample. This work was supported by the European Research Council (grant no. CoG724489 to J.R., no. 802567 to Y.S.) and the European Molecular Biology Laboratory. This work was conducted with support from the University of New Mexico Office of the Vice President for Research Program for Enhancing Research Capacity. This research was supported by grants from NVIDIA and utilized an NVIDIA A6000 GPU. S.L. was supported by EMBL ARISE fellowship no. 945405. S.L. and K.A.L. were supported by National Institutes of Health grant 1R01GM140284. Y.L. was supported by National Natural Science Foundation of China (62375116); Key Technology Research and Development Program of Shandong (2021CXGC010212); Shenzhen Medical Research Fund (B2302038); Shenzhen Science and Technology Innovation Commission (grant no. JCYJ20220818100416036 and KQTD20200820113012029); Guangdong Provincial Key Laboratory of Advanced Biomaterials (2022B1212010003); a startup grant from Southern University of Science and Technology. L.-R.M. was supported by the German Federal Ministry of Education and Research (BMBF; project SIMALESAM, FKZ 01|S21055A-B). R.H. was supported by the German research foundation (Deutsche Forschungsgemeinschaft, project TR 1278, TP C04, 316213987) and the Leibniz science campus Infecto-Optics, project HotAim 2.0. C.K. was supported by the German research foundation (Deutsche Forschungsgemeinschaft, project SFB TR166 ‘Receptorlight’, TP B05). This project has received funding from the Israel Science Foundation (grant no. 450/18 to Y.S). Y.S. was supported by the Zuckerman Foundation, and by the Donald D. Harrington fellowship.

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Authors

Contributions

J.R., Y.L., R.H. and Y.S. conceived the project. J.R., Y.L. and K.A.L. provided supervision. S.L., J.H., J.C., C.K. and L.-R.M. wrote the software. S.L. and J.C. analyzed the data. S.L., J.C., J.R., B.F. and D.X. acquired the data. S.L., J.C., Y.L. and J.R. wrote the paper with input from all authors.

Corresponding authors

Correspondence to Yiming Li or Jonas Ries.

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Nature Methods thanks Ryan McGorty and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editor: Rita Strack, in collaboration with the Nature Methods team. Peer reviewer reports are available.

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Extended data

Extended Data Fig. 1 In situ PSF model reduces the local deformation in z measured by the ring distance of Nup96.

(a) A subregion of the reconstructed Nup96 using the in situ PSF model. (b, c) XZ view of selected region in (a) from bead and in situ PSF models. (d) and (e) are the ring distance of Nup96 as a function of z-position at different depth reconstructed by using beads PSF model and in situ PSF model respectively. The points of various colors represent different imaging depths of Nup96. The straight lines represent the linear regression of the ring distance versus z-position (of the NPC center) for each imaging depth, with k representing the corresponding slope. An oil immersion objective lens (NA = 1.43) was used for imaging. (f) and (g) have the same meaning as (d) and (e), respectively, but imaged by a silicone oil immersion objective lens (NA = 1.35). Standard imaging buffer (n = 1.35) were used. Scale bars, 1 µm (a), 500 nm (b).

Source data

Extended Data Fig. 2 Effect of pixelation using the vectorial PSF modelling method.

(a) Test on simulated data. Data were simulated from vectorial PSF model, with a binning of 5 in both x and y dimensions, and without extra blurring to the PSF model. (b) Test on experimental data. The estimated blur factor is slightly smaller when adding a binning of 5 in the forward model. However, the estimated PSF model and the localization bias are nearly identical with or without additional binning in the forward model. In contrast, under the same blur factor but without binning (bin=1), the axial bias increases dramatically and the estimated PSF model differs substantially. The z/x/y bias vs. z are defined as in SI Note 2 Eqs. 2.37 and 2.38. Black lines and shaded areas in localization plots indicate mean and standard deviation of the x, y and z biases over >20 beads. Scale bar, 1 µm. A.U., arbitrary unit.

Source data

Extended Data Fig. 3 Estimation of voxel-based PSF model of a lattice light-sheet system from bead data.

The data were collected by imaging beads in agarose gel at sample stage positions from −50 µm to 50 µm, with a step size of 50 nm. Translation of the sample stage will translate the beads both in x and z dimensions in the coordinate of the detection objective. For this data, translating the sample stage by 50 nm, will translate the beads by 40.8 nm in x and 28.9 nm in z relative to the detection objective. (a) Localization in x of the deskewed data used for inverse modelling. (b) The raw data stack of each bead was deskewed using the above translation relationship between x and z dimensions. Only shifts of integer pixels were applied to maintain the photon statistics of the raw data. The deskewed bead stacks were used for inverse modelling. The sub-pixel shifts were incorporated in the forward model so that the estimated PSF model has no skew as in (d). (c) localization of the data used for inverse modelling, sub-pixel shifts were applied to the localization result in (a) to remove the skew effect. The z/x/y bias vs. z are defined as in SI Note 2 Eqs. 2.37 and 2.38. Black lines and shaded areas in localization plots indicate mean and standard deviation of the x, y and z biases over >20 beads. (d) Estimated PSF model. (e) Comparison of an example bead stack and its corresponding forward model. Scale bars, 1 µm (b,d,e).

Source data

Extended Data Fig. 4 Incorporating refractive index mismatch aberrations in the estimated PSF model.

Bead data at different imaging depth were collected by imaging beads in agarose gel at stage positions from −1 µm to 5 µm, with a step size of 20 nm. Bead data at the coverslip were collected at stage positions from −1.5 µm to 1.5 µm, with a step size of 10 nm. (a) Localization of beads in agarose gel using the PSF model estimated from beads at the coverslip. (b) Localization of beads in agarose gel using the PSF model estimated from beads at the coverslip and modified by adding an index mismatch aberration at the estimated imaging depth of each bead. Therefore, each bead stack was localized by its own modified PSF model. Depth is defined as the estimated emitter’s z position to the coverslip. The z/x/y bias vs. z are defined as in SI Note 2 Eqs. 2.37 and 2.38. (c) Comparison of the PSF model at the coverslip, bead data at different imaging depth and their corresponding PSF models. With increasing depth, the pupil size slightly decreases as the effective numerical aperture (NA) decreases from 1.43 to 1.33 (the refractive index of water), and the pupil phase shows larger spherical aberration kind of patterns. Scale bars, 1 µm (c).

Source data

Extended Data Fig. 5 Validation of in situ PSF estimation using bead data in agarose gel.

The images of these beads were collected at stage positions from −1 µm to 25 µm. For each bead, the per-frame z position was used as an independent variable to simulate the in situ single molecule data. (a) The PSF estimated from the beads on the cover glass was used to localize the bead data at different depths. (b) The in situ PSF estimated by uiPSF was used to localize the bead data at different depths. Compared to the PSF model estimated from the beads on the cover glass, the in situ PSF model fitted z-positions exhibit a good linear relationship with the objective stage positions.

Extended Data Fig. 6 Comparison of Zernike-based PSF modelling using uiPSF and INSPR.

Comparison between the input amplitude and the estimated amplitude from the in situ PSF modelling using uiPSF (a) and INSPR (b) for 21 Zernike modes (fringe indices) applied by a deformable mirror (DM). Single-molecule data were collected at each of the 21 Zernike modes and were analyzed by uiPSF and INSPR. Panel b reproduced from ref. 41, Springer Nature America. As shown in (a) and (b), the Zernike aberrations returned by uiPSF have lower residual aberrations, compared to INSPR. Comparison of uiPSF (c) with INSPR (d) for fitted z-position versus objective z-position. We imaged the single fluorophores (AF647) immobilized on a coverslip at different objective z-positions. The fluorophores’ z-position fitted by uiPSF is linear relative to the objective z-position (c), while there is a discontinuity at −300 nm for INSPR (d), indicating a mismatch between the INSPR PSF model and the single-molecule patterns.

Source data

Extended Data Fig. 7 Comparison of pixelated and Zernike pupil based in situ PSF estimation of Tetrapod blinking patterns generated from a phase plate.

(a) Comparison of estimated PSF and pupil from bead data and single molecule blinking patterns. The phase patten generated from a phase plate is more complex than the ones from a deformable mirror, both the pixelated and Zernike pupil based modelling methods can retrieve the pupil function from single molecule blinking data. However, the Zernike-based method requires accurate initial values of the Zernike coefficients, otherwise, it fails to retrieve the accurate pupil. (b, c) Estimated Zernike coefficients from Zernike-based in situ PSF modelling with (b) or without (c) accurate initial values of the Zernike coefficients. (d) Example raw camera frame used for in situ PSF modelling. Scale bar, 1 µm (a), 10 µm (d). ADU, analog to digital unit.

Extended Data Fig. 8 Validation of in situ PSF modelling for imaging a thick sample.

Imaging of Nup96-SNAP-AF647 in U2OS cells was performed using an oil immersion objective lens (NA = 1.5) and imaging buffer (n = 1.406, Methods) on a single-channel SMLM system equipped with a deformable mirror (DM). (a) A sandwiched sample of U2OS cells was prepared for an imaging depth of 25 µm. (b) Top view of the reconstructed Nup96 using the in situ PSF model. (c) Enlarged XY view of the selected region in (b). (d) XZ view of the selected region in (c) reconstructed from bead and in situ PSF models, respectively. Scale bars, 2 µm (b), 1 µm (c), and 0.5 µm (d).

Extended Data Fig. 9 Application of uiPSF for aberration correction using a deformable mirror.

Imaging of Nup96-SNAP-AF647 in U2OS cells was performed using an oil immersion objective lens (NA = 1.5) and imaging buffer (n = 1.406, Methods) on a single-channel SMLM system equipped with a deformable mirror (DM). (a) Top view of the reconstructed Nup96 using the in situ PSF model with and without uiPSF guided aberration correction by AO. (b) Zernike aberrations calculated from single molecule data using uiPSF with and without AO correction. (c, d) XZ view of the selected region in (a) for samples with and without AO correction. After uiPSF guided aberration correction, the reconstructed image quality improved notably as demonstrated by the resolved double ring structure. Scale bars, 1 µm (a), 500 nm (c-d).

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Supplementary Figs. 1–21, Supplementary Notes 1–7 and Supplementary Tables 1–3.

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Supplementary Software 1

uiPSF software package, including source codes, example Jupyter notebooks and tutorials for localization analysis using the PSF models generated from uiPSF.

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Liu, S., Chen, J., Hellgoth, J. et al. Universal inverse modeling of point spread functions for SMLM localization and microscope characterization. Nat Methods 21, 1082–1093 (2024). https://doi.org/10.1038/s41592-024-02282-x

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