We present a structured illumination microscopy based point localization estimator (SIMPLE) that achieves a 2-fold increase in single molecule localization precision compared to conventional centroid estimation methods. SIMPLE advances the recently introduced MINFLUX concept by using precisely phase-shifted sinusoidal wave patterns as nanometric rulers for simultaneous particle localization based on photon count variation over a 20 μm field of view. We validate SIMPLE in silico and experimentally on a TIRF-SIM setup using a digital micro-mirror device (DMD) as a spatial light modulator.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Super-resolution (SR) imaging has become an enabling technology to access sub-diffraction information on the nanoscale structure and dynamics of molecular building blocks . Different SR concepts have been introduced: Single molecule localization microscopy (SMLM) methods such as PALM  and STORM  process information in the optical detection path to estimate single particle positions of sparse emitters by centroid fitting to the diffraction-limited photon distribution on a pixelized camera. Each individual photon contributes to a measure of the actual particle position with a standard deviation s given by the point spread function (PSF) of the imaging system. The localization precision of SMLM scales with s ∝ λ/NA, with λ being the fluorescence emission wavelength and NA the numerical aperture of the detection objective. An increasing number of photons N results in a higher localization precision given by the standard error of the mean , with conventional SR localization methods reaching Δx = 5 – 20 nm [4, 5]. In contrast, excitation-based SR concepts such as PSF-shaping  or structured illumination microscopy (SIM) [7–10] have been implemented. SIM enables sub-diffraction imaging by exciting samples features with periodic patterns to obtain high frequency information via the Moiré effect . Typically, 3 × 3 phase-shifted and rotated sinusoidal interference-patterns are used and image stacks are mathematically processed to reach a 2-fold resolution improvement . MINFLUX [12,13], on the other hand, was recently introduced as a radically new concept for particle localization at minimum photon budget with up to 8-fold improvement in localization precision compared to SMLM. In MINFLUX, a single molecule emitter is probed with the central part of a doughnut-shaped illumination pattern over a restricted area of diameter L ≃ 50 nm. The particle position is estimated via a triangulation principle from photon count variations at different scan positions with a localization precision  MINFLUX is, however, limited to a restricted sub-diffraction field of observation and low temporal sampling due to sequential scanning of single molecule emitter positions in addition to technical challenges of accurate doughnut generation and positioning. Here, we present a structured illumination microscopy based point localization estimator (SIMPLE), that allows for the simultaneous localization of isolated emitters via detection of photon count modulations on a camera-based system. We show that phase-shifted sinusoidal excitation patterns can be used as nanometric rulers for deriving the actual positions of multiple single molecule emitters in a micron-sized field of view (FOV), thereby overcoming current spatial and temporal sampling restrictions of MINFLUX.
2. Basic concept and principle of SIMPLE
The principle of the SIMPLE method is exemplified in Fig. 1. A sinusoidal wave pattern illuminates the sample and modulated photon numbers are collected from isolated emitters depending on their relative position within the excitation pattern. Phase shifting of the illumination pattern can be performed with nanometric precision (<1 nm) to probe the actual particle position through photon number variations ni for different phase shifts Φi.
The measured photon counts ni can be fitted to the known sinusoidal illumination pattern to derive the actual particle position based on the assumption of a quasi linear response of the fluorescence emission. As x − y positions can be determined independently, using two orthogonal illumination patterns is sufficient for homogeneous 2D localization of a particle emitter. This principle differs both from conventional SIM and standard SMLM approaches which rely on complex image reconstruction via Fourier methods  or centroid fitting of a Gaussian function to the diffraction-limited photon distribution , respectively. The localization precision of SIMPLE can be evaluated using error propagation analysis and estimation of the Cramer-Rao Lower Bound (CRLB) as derived previously [12, 16, 17]. In brief, a simplified estimation is based on considering photon shot noise () that will introduce localization errors (see Fig. 1(b)). Using k = 3 equidistant phase shifts, we can solve three equations for the detected photon counts ni using a sine function I = A/2 (1 + cos (2π (x + ϕ))) + b with amplitude A, phase ϕ and background b:Fig. 3(c)). Smaller phase shifts can further enhance the localization precision at the expense of unequal localization gain in the illumination pattern.
Figure 3 shows the comparison between SIMPLE and SMLM in the presence of a background noise of b = 0, 2, 4 and 6 counts/pixel. The theoretical limit of classical SMLM localization precision as described by Mortensen et al.  is shown in black, highlighting the localization precision improvement of SIMPLE over SMLM. Three equidistant phase shifts are sufficient to determine the emitter’s position. A higher number of phase shifts Φi slightly lowers the localization precision due to a spread of total photons across multiple images (), particularly relevant for low photon counts as obtained for dim fluorophores or fast imaging conditions. A typical camera read-out noise of 1 photon/pixel sets a lower bound to σ that will be increased by additional noise sources such as sample background, auto-fluorescence and out-of-focus fluorescence. Equidistant phase shifting yields a 1.8 to 3.1-fold improvement in localization precision depending on the emitter position relative to the wave pattern and is largely independent of the number of detected photons. A reduced modulation depth m < 1 of the excitation pattern will further lower the localization precision. The residual illumination generates an offset of 1 − m unmodulated photon counts which can be considered as an increase in the effective background noise of the sample (see Table 3.1).
3.1. In silico validation of the SIMPLE concept
To validate the SIMPLE concept under realistic parameter settings, we developed an extensive simulation pipeline to generate synthetic imaging data that include relevant sources of experimental uncertainties. A fluorescent emitter is placed at a random location in the field of view. The initial phases in x and y direction, relative to the illumination pattern, are also randomized. Intensity-modulated emission counts are obtained for a series of k phases in each x and y direction, assuming a linear response of the fluorophore to the illumination. The appropriate photon spread is modeled by a Gaussian PSF on a pixelated camera chip, including photon shot noise with a Poissonian distribution. In addition, background noise is added to each pixel to mimic additional noise terms from the optical detection system or sample such as camera read-out noise, dark-current or background fluorescence. The simulations further enable to include particle characteristics such as emitter size, blinking or photo-bleaching and optical characteristics of the excitation pattern as the sinusoidal wavelength and modulation depth m. Realistic units are set by the average number of emitted photons per second and the illumination time. Synthetic data are read into our analysis pipeline to obtain the number of photons on selected pixels for an estimation of the particle position. First a pre-estimation of the emitter’s location on the pixel grid is performed. This step is achieved on all associated image triplets of the FOV using a threshold for the standard deviation of photon counts during the phase shifts. Selected candidates are used for cropping the surrounding pixels with specific masks to read the number of photons for each selected emitter during phase modulation. Next, the intensity profile is fitted with a sinusoid, providing an accurate estimation of the position. Each emitter, for which the corresponding relative position on the illumination pattern has been determined through fitting, can be mapped back onto the pixelated grid of the camera. The absolute distance between the emitters is calculated as shown in Fig. 2, counting the number of wave periods and separating them accordingly. Complementarily, event detection can be obtained from summing phase shifted images to achieve a homogeneous illumination condition to reconstruct a wide-field image and using centroid-based event detection schemes as available by the ThunderSTORM plugin of Fiji  which is used for comparison to SIMPLE.
The optimal pixel selection can be set in dependence on the relation PSF size/pixel size and the background noise level as shown in Fig. 4. Masks that are too small will reduce the available photon information while masks that are too big will enhance the fluctuations in the case of high background noise.
3.2. Experimental setup
We experimentally validated the SIMPLE concept on a novel custom-built TIRF-SIM setup (Fig. 5). The optical design is based on the use of a spatial light modulator (SLM) that allows for ultra-fast pattern generation with high modulation depth and ultra-precise phase shifting of orthogonal patterns. In general, the concept of SIMPLE relies on the exact knowledge of the patterns’ phase shifts, which we accomplish by using a digital micro-mirror device (DMD) as SLM. The DMD (DLP V-9501 VIS, VIALUX, pixel pitch 10.8 μm) can operate at ∼17 kHz switching rate, allowing for fast imaging in the millisecond range. The de-magnification of the DMD plane to the sample plane amounts to a factor of 300, resulting in a sinusoidal interference pattern on the sample with a period of 170 nm (see Fig. 5(b–g)). This corresponds to an excitation numerical aperture (NA) of 1.44 of the first orders. The DMD also serves as master clock for the trigger protocol.
The detailed optical path of the setup comprises the excitation laser with 500 mW power at 473 nm wavelength (gem473, LaserQuantum), followed by a 10× telescope (AC080-010-A-ML and AC254-100-A-ML, Thorlabs) to expand and collimate the laser beam. The diffracted light from the DMD is collected by a lens (AC508-500-A-ML, Thorlabs) and passes through a quarter wave plate (AQWP05M-600, Thorlabs), generating circular polarization. Two neutral density filters (NDC-50C-2, Thorlabs) provide fine control over the intensities of the individual diffracted beams. A custom aluminum mask is placed in the Fourier plane of the DMD and serves as spatial filter to block unwanted diffraction orders originating from the binary stripe pattern. Only the required first orders pass through and create the sinusoidal intensity pattern on the sample. A segmented polarizer (colorPOL VIS087 BC3 CW01, CODIXX) then generates azimuthal polarization, which is necessary to achieve maximum modulation depth of the sinusoidal pattern at the sample. The combination of the fast-switching DMD with the segmented polarizer implies only mechanically fixed and stable components with no moving parts. This circumvents the low time resolution in classical SIM configurations and furthermore guarantees an accurate determination of the phase shift by solely changing the displayed pattern on the DMD. The beams are relayed by two lenses (#49-367, Edmund Optics, AC254-300-A-ML, Thorlabs) and then reflected by two identical, but rotated dichroic beam splitters (ZT405/473/561rpc, AHF) to eliminate detrimental polarization effects for s- and p-polarized reflected light. The diffraction spots are projected close to the edge of the back focal plane of the objective (UAPON 100XOTIRF, Olympus), which is mounted onto a z-piezo stage (N-725.2A PIFOC, Physik Instrumente). The beams interfere at the focal plane at an effective NA of 1.44 and generate the desired sinusoidal illumination pattern on the sample. Sample positioning can be performed by a piezo stage (P-545.3C8H, Physik Instrumente). The emitted fluorescence is collected by the same objective lens, passing through the dichroic mirror, the tube lens and a combination of emission filters (ZET405/473/561NF, AHF; FF01-503/LP-25, Semrock; HQ515/30m, Chroma) to reject unwanted excitation light and achieve lower residual background. A sCMOS camera (Zyla-4.2P-CL10, Andor) detects the fluorescence signals in the mode “rolling shutter global clear external triggering (non-overlap mode)”. The size of the FOV on the sample is defined by a circular mask in the displayed DMD pattern and amounts to approximately 27 μm.
3.3. Pattern period calibration
For pattern period calibration a dense distribution of single Alexa Fluor488 molecules was repeatedly illuminated with phase shifted sinusoidal patterns. Data were analysed automatically by an adapted code of fairSIM  written in Java, extracting the shift vector length of the parameter estimation that corresponds to the pattern period of the excitation pattern. As shown in Fig. 5(b) we revealed a sinusoidal illumination pattern period of ℓ = (169.0 ± 0.3) nm.
4. Results and discussion
As a proof of concept, we applied SIMPLE to the localization of single immobilized Alexa Fluor488 dyes. We recorded image sequences with k = 3 equidistant phases of the sinusoidal illumination pattern and subsequent localizations of the same molecule were used to obtain the localization precision Δx. Figure 6 shows a representative image series of one single Alexa488 dye.
To validate the parallel detection of multiple single molecule emitters in a large field of view we placed single Alexa488 dyes on a glass slide and derived their positions via the SIMPLE method and standard SLML point localization. Sparse single molecule emitters were modulated with three phase shifted sinusoidal illumination patterns multiple times to reveal consecutive positions of all molecules in the FOV. This allows us to generate proper statistics for comparing localization precision enhancement of SIMPLE compared to SMLM in a parallel detection scheme. Figure 7 shows a reconstruction of a summed stack for three phases. The corresponding emission intensities to the three phases over the time series for two indicated emitters and the SIMPLE fitting routine are depicted in Fig. 7(b).
The obtained localization precision of SIMPLE in Fig. 7 outperforms conventional centroid fitting using ThunderSTORM  applied to the sum of three phase images being equivalent to homogeneous illumination. The achieved localization precision for a single molecule is Δx = 4.3 nm (Δy = 4.5 nm) for SIMPLE in contrast to Δx = 8.8 nm (Δy = 10.9 nm) for SMLM, and 〈xall〉 = 4.7 nm (SIMPLE) versus 9.4 nm (SMLM) for 40 particles averaged across the FOV (Fig. 7(c)). We further demonstrate the performance of our localization approach by precisely repositioning single emitters in (20.0 ± 0.4) nm steps on a piezo driven stage, for mapping particle localizations on the principle of a nanometric ruler (Fig. 8).
In conclusion we have shown that sinusoidal wave patterns can be used as nanometric rulers for localizing the position of single molecule emitters with improved precision by a factor of 2 compared to standard SMLM methods. Similar to MINFLUX, particle localization is extracted from the photon count variation in an inhomogeneous illumination field. This enables to enhance localization precision with a minimal photon budget by the contribution of positional information in low illumination regions. Using a periodic wave function as an illumination pattern, we further demonstrate here that the concept of MINFLUX can massively parallelized from diffraction limited areas to a micron-sized field of view of >10 μm. Using a DMD as spatial light modulator we show that fast and exact nanometric phase shifting of sinusoidal illumination patterns can be employed to achieve a gain in localization precision of a factor of 2 compared to classical centroid fitting methods used in STORM/PALM and single particle tracking (SPT). A similar approach was recently presented in an article by Cnossen et al.  using piezo-mounted gratings to reveal enhanced localization precision, which was termed SIMFLUX. SIMPLE, MINFLUX and SIMFLUX are based on measuring photon number variations ni in a shifted periodic excitation pattern, thereby setting limits to single molecule movements during sample illumination and the excitation shifting period. Effects of motion blur and emitter movement during illumination are also well known in SPT experiments and affect the determination of diffusion modes or structural confinement sizes [22–24]. However, a reduction of photon numbers in SIMPLE, MINFLUX and SIMFLUX allows for improved temporal resolution and higher dynamic sampling rates at comparable localization precision to standard SMLM methods . In the case of fixed samples, methods such as PALM/STORM [2,3] and DNA-Paint  achieve nanometric structural resolution based on adjusting optimal sample labeling density, number of active emitters and obtaining a sufficient set of emitter localizations in consecutive images. For this purpose, specific labeling strategies and buffer systems were developed to set fluorophore switching rates and molecular binding kinetics for obtaining maximal structural resolution . In SMLM imaging, frame rates are adjusted to gather the maximum photon signal during the on-state of fluorophore blinking/binding that determines localization precision. In comparison, SIMPLE requires multiple image acquisitions (three images for each direction) for localizing a single emitter. This demands, for optimal imaging speed, to capture photon counts of the emitter during the on-state before bleaching or blinking occurs. DMDs providing very high temporal control of illumination patterns, the limitation in image acquisition times is mainly due to the camera read-out speed, ensuring a high total detection rate of fluorescence spots (also referred to as “miss/hit probability” in the sample). Furthermore, SMLM requires four times the number of photons compared to SIMPLE to achieve a similar localization precision, therefore reducing the acquisition time of SIMPLE four-fold. Note that the emitter’s density is a key factor that determines the false positive/negative rates of emitter localizations in SMLM methods  and accordingly provides an inherent quality criteria for SIMPLE to detect emitters within a diffraction-limited distance by changes in the modulated intensity response that can be extracted to avoid mis-localizations.
Altogether, SIMPLE improves localization precision at reduced photon numbers compared to standard SMLM approaches , the method will thus enable to minimize detrimental phototoxicity and photobleaching for live cell applications and single particle tracking that will be validated in future studies.
“la Caixa” Foundation (ID 100010434, LCF/BQ/IN18/11660032); Marie Skłodowska-Curie Horizon 2020 (713673, 754558, 642157); Spanish Ministry Centro de Excelencia Severo Ochoa (SEV-2015-0522); MINECO’s Plan Nacional (BFU2017-86296-P); Generalitat de Catalunya (CERCA); Fundació Privada Cellex.
L.R. built the theoretical framework, developed simulation and data analysis tools, and contributed to the setup design and implementation; J.Z. implemented the setup, performed experiments and contributed to the initial study design; C.K. prepared samples and contributed to supplementary data analysis; F-C.W. provided data; T. H. advised on the manuscript and provided data; V.R and S.W. designed the study, performed simulations and data analysis, supervised the work and wrote the manuscript with the contribution of all authors.
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