1. Introduction

Single-molecule (SM) microscopy has become a ubiquitous tool in cellular imaging. With ultra-sensitive detection methods, it is now possible to count, locate and track biological molecules in their cellular environment [1]. Thereby, the composition, structure and spatial dynamics of molecular assemblies can be assessed with a spatial resolution of few nanometers and a temporal resolution down to the millisecond. This opens up a window into a complex molecular organization that could not be accessed by the conventional microscopy techniques of classical biology and biochemistry.

Common to all SM experiments is the analysis of the fluorescence signal emitted by individual molecules. Since fluorophores behave as point sources, their image corresponds to the point-spread function (PSF) of the optical system. In an aberration-free system, the PSF is described by an Airy function and has a lateral extension σ_l on the order of λ/2NA (λ being the emission wavelength and NA the objective numerical aperture) [2]. Importantly, all organic dyes and fluorescent proteins (FPs), the most commonly used markers in cell biology, are subject to photodestruction processes. In practice this means that, on average, a limited number of photons can be detected per molecule (on the order of up to 1000 photons for FPs and several thousands for dyes [3, 4]). In other words, each fluorescent molecule has a certain “photon budget” that should be optimally used to transfer as much molecular information as possible to the macroscopic world (in general a CCD camera).

In most tracking and super-resolution imaging experiments, one is primarily interested in obtaining a precise localization of single molecules. To do so, the PSF is analyzed by means of Gaussian fitting, center of mass estimation, or wavelet filtering to determine its center position [5–9]. Importantly, the localization precision varies as ~σ_l/√N where N is the number of photons detected in the fluorescence spot [10]. The localization precision can therefore be well below the diffraction limit 0.6 λ/NA, a property that is central to all SM based super-resolution microscopy techniques, as in (fluorescence) photoactivation localization microscopy ((F)PALM) [11, 12] and stochastic optical reconstruction microscopy (STORM) [13] (see reference [14] for a recent review of SM based super-resolution microscopy). Note however that for a given N, reaching the best localization possible is contingent on our a priori knowledge of the PSF. Therefore these techniques are very sensitive to optical aberrations that would degrade the PSF shape.

An additional and recurrent problem in single molecule imaging concerns the localization of the molecules along the optical axis (z-axis). While it is relatively easy to locate a molecule in the image plane (with a typical resolution of 10 to 50 nm), it is difficult to do so along the optical axis with comparable resolution. Depending on its z position with respect to the image plane, the point source will be imaged on the detector as a more or less defocused spot. Unfortunately, analysis of this defocused signal cannot precisely determine the source position along the z-axis due to: (i) the symmetry of the PSF deformation above and below the focal plane, (ii) the small change in the PSF shape for molecules within the focal depth of the objective (typically ~0.5 μm).

In recent years, several methods have been developed to achieve sub-diffraction axial localization. The first demonstration was based on the use of a cylindrical lens in the optical emission path to break the symmetry of the optical signal on both sides of the focal plane and by taking advantage of the astigmatism of the corresponding PSF [15]. With this approach, three-dimensional super-resolution STORM imaging was demonstrated, with an axial resolution of 50 to 100 nm, 2 to 5 times that achieved in the perpendicular plane [16]. This successful method, however, suffers from several practical limitations mainly related to the lack of control of the optical distortion induced by the lens. As a matter of fact, the effect of the lens is not limited to induced astigmatism but it also introduces significant optical aberrations, which can affect the pointing precision. A second method consists of imaging the signal of a single molecule in two axially separated planes. With this simultaneous bi-plane detection, the z position of a molecule between the two planes can be determined with a precision similar to that achieved with the astigmatic approach [17, 18]; the advantage of this technique is that the lateral localization precision is not coupled with the axial position of the fluorophore. Other less commonly used methods are based on PSF engineering with a double-helix shape [19] or interferometric measurements of the single molecule emission [20]. The latter technique achieves the best z-location precision (sub-20 nm 3D resolution with FPs), but at the cost of considerable experimental complexity (4pi measurement system and triple interferometric detection of emitted photons). Moreover, it remains limited to fixed samples. In all the methods above, the depth over which one can determine the position of the molecules is limited to about 1 μm. Moreover, since it is fully determined by the optomechanical components of the set-up, this depth cannot be tuned in a rapid and flexible manner. This is a clear drawback for an optimized use of the “photon budget”, as determined by the different types of fluorophores and biological applications

Here, we demonstrate the use of adaptive optics (AO) techniques for 3D single-molecule imaging in cell biology. Originally developed for astronomical observations in the 70s and 80s, AO is a technology in which the wavefront distortion is actively measured and corrected with a deformable mirror [21]. In astronomy, it can effectively compensate for fluctuations in the index caused by atmospheric turbulence. The use of AO in biological and medical microscopy is much more recent and less established. Most studies have focused on correcting the excitation beam profiles to improve image quality in linear and nonlinear scanning microscopy (confocal imaging, two-photon, second and third harmonic imaging) [22–27]. In wide-field microscopy, a pioneering effort has shown how AO could be fruitfully employed for aberration correction and motionless focusing [28, 29]. However, the benefits of AO in the specific context of single molecule imaging and super-resolution microscopy have not yet been explored.

In our experiments, AO techniques serve two important and complimentary purposes. First, AO enables the correction of the aberrations induced by the optical system and the sample itself. Hence it allows restoring the ideal PSF shape required for optimal analysis. Second, the AO mirror can be used to reshape the PSF in a controlled and reversible manner. By doing so, one can induce well-calibrated and z-dependent PSF shapes from which the position along the z-axis is extracted with high resolution. Below, we investigate in more details the case of astigmatic deformations, that leads to a z-localization precision down to 15-40 nm and over a focal depth of ~800 nm. The use of AO for 3D localization is supported by super-resolution images of actin filaments in fixed cells and tracking of quantum dot-labeled transmembrane proteins in live cells.

2. Experimental set-up

Our experimental set-up was based on a standard epifluorescence microscope (Nikon Ti-E) equipped with an oil-immersion objective (100X, 1.49 NA). Epifluorescence excitation was provided by either a metal-halide lamp or a laser line (405 or 561 nm). In the emission pathway, we placed an adaptive optics system (MicAO^TM, Imagine Optic) between the microscope and the EMCCD camera (Andor Ixon3 DU897). The tube lens (TL) of the microscope (200 mm of focal length) and the lens L1 imaged the pupil of the objective on a deformable mirror (DM) (mirao 52-e, Imagine Optic) (Fig. 1(A) ). Although the physical diameter of the pupil of the objective was 5.96 mm, its effective diameter was only 5.32 mm when imaging into water-based samples due to the effective NA ~1.33. In order to image the effective pupil of the objective onto the DM (15 mm in diameter), we used a lens L1 with a focal length of 500 mm. This lens, in combination with the TL, produced a magnification of 2.5X. A second lens L2, with the same focal length as L1, formed an image of the sample with the corrected wavefront (WF) in a secondary image plane. To implement the wavefront sensor (WFS) light path, a flip mirror (FM) was added in the optical setup to deviate the light coming from the DM to a Shack Hartman WFS (Haso First, Imagine Optic). A conjugation lens (CL) allowed the pupil conjugation from the DM to the WFS. For a direct comparison between the adaptive optics system and the conventional astigmatic imaging, we used in the second side port of the microscope a cylindrical lens (system N-STORM, Nikon) followed by an EMCCD camera.

 

Fig. 1 A) Experimental set-up. B) Measurement of the wavefront on the WFS: with the factory calibration of the DM (B1), after correction using an artifical star (B2), after the genetic algorithm (B3). The color scale corresponds to the distortion of the wavefront (in μm). C-E) Cross-sections of the PSF of a bead immobilized on a coverslip: PSF without the deformable mirror (C), after correction with deformable mirror (D). In each case, the spots are an image of the bead on the camera at positions 300 nm, 0 and – 300 nm (top to bottom), corresponding to the dotted lines. E) Measurement of the wavefront with a cylindrical lens in the optical pathway (E1) and after substracing the Zernike modes corresponding to an astigmatic distortion (E2). Experimental wavefront when using astigmatic deformation with amplitude A = 0.2 μm (E4). F) Cross-section of the PSF when using an astigmatic deformation, see also supplementary Media 1.

Download Full Size | PDF

For the mirao 52-e DM, there exists a linear relationship between each actuator and the WF modification. The voltages applied in the DM can be defined as a vector V. Similarly, the obtained wavefront can be described in a discrete way and represented as a vector P where each element corresponds to the phase in a certain square region. The relationship between these two vectors is determined by a matrix IM such that IM·V = P. In order to find IM experimentally, we performed unitary tests of V and measured the corresponding P for each deformation. For example the first line of IM corresponds to the phase obtained when the first actuator of the DM is used. Thus, by doing a pseudo-inversion of the matrix IM, one can determine the desired voltages in order to obtain a given phase profile P: V = IM⁻¹·P .

3. Wavefront correction

We used the AO system to compensate for the optical aberrations of the microscope (Fig. 1(B)) with two complementary methods: first a WFS-based method and a then sensorless approach based on the evaluation of the image. In the former case, the light collected from a point-like emitter (i.e. a fluorescent bead) was used as an artificial star [30], and its wavefront measured with the WFS. With this information, the shape of the DM was directly calculated in order to compensate the aberrations of the system and obtain a flat measure of the wavefront of the artificial star (Fig. 1(B2)). The latter method performs an iterative algorithm based on a merit factor, which evaluates the quality of each image obtained with a given shape of the DM. We used a genetic algorithm [31] in order to converge toward an optimal DM deformation. From a general standpoint, this algorithm takes the numerical description (the genes) of the individuals of a population and evaluates the fitness of each individual by using a merit function. After that, only the two best individuals are selected to generate the next population. This is done by using the genetic operators (crossover and mutation) in a certain number of genes. The evaluation and the procedure are repeated until convergence of the algorithm. In our case, the individuals are the different shapes of the DM. Each individual can be described by the coefficients (the genes) of the decomposition of the shape in the base of the Zernike polynomials. In our experiments, the merit function was computed on the obtained image after applying a shape to the DM. We used fiduciary markers (bright fluorescent beads) present in the sample for XY drift correction to define the merit function as the maximum fluorescence intensity collected from such markers. We considered that the system was corrected when the merit function reached a plateau over 10 consecutive iterations. In case of high numerical apertures, the spatial frequency needed to describe the spherical aberration present in samples with index mismatch is very important especially in the borders of the pupil of the objective [23]. It is also worth noticing that the crosstalk between the Zernike modes [32] can make the convergence of the genetic algorithm difficult due to its non-linear dependence. We thus ran the genetic algorithm in two consecutive steps. First we limited the correction to the first order of the non-radially symmetric Zernike modes (astigmatism, coma and trefoil). Next, in order to correct the spherical aberrations produced by the small index mismatch between the immersion media and the sample, we launched a second correction using up to the seventh order of the spherical aberration. This procedure took typically from a few tens of seconds to a few minutes. We could therefore apply a new correction for each field of view, although different experiments on the same specimen with a single correction gave us similar results. With this strategy we greatly reduced the exploration of the large number of potential solutions tested by the genetic algorithm and obtained remarkably uniform wavefronts (Fig. 1(B3)).

4. Reshaping the PSF with a deformable mirror

To evaluate the performance of our optical microscope and the deformable mirror, we first recorded the PSF for 40-nm beads deposited on a glass surface and covered with water to mimic the case of single molecules in cultured cells. Cross-sections of volumetric renderings are used to visualize the light intensity distribution in the PSF. These renderings are formed by alpha-blending the images of 25 nm-step z-series for a stationary 40 nm bead on a coverslip. Without correction with the deformable mirror, we noted a clear uneven distribution of light intensity in the PSF (Fig. 1(C)), indicative of spherical aberrations. These aberrations, which can be attributed to imperfections of the optical system as well as to the positioning of the fluorescence source at a glass/water interface, can be largely eliminated by correcting the PSF with the deformable mirror (Fig. 1(D)). Importantly, the transmission loss due to the Micao system was less that 10%, thus marginally affecting the x-y localization precision (LP) (supplementary Fig. 5 ).

 

Fig. 5 Plot of x- and y-localization precision at focal plane for microscope configuration (A) without use of deformable mirror, (B) with use of deformable mirror, and (C) with DM-induced astigmatism (amplitude 0.20 μm).

Download Full Size | PDF

Next, we used the mirror to impose an additional controlled distortion of the PSF. An interesting case is that of an astigmatic deformation in which the x and y focal planes do not coincide. As shown in past reports, this provides a way to determine the z-position of a point emitter with sub-diffraction precision [15, 16]. Traditionally, astigmatic imaging has been implemented by positioning a cylindrical lens in the emission pathway. We tested this method in our set-up and measured the wavefront of the light that traveled through the cylindrical lens (Fig. 1(E1)). We found that the quality of the astigmatic deformation produced with cylindrical lenses was not very high. The many additional aberrations introduced by the cylindrical lens can be clearly revealed by subtracting the Zernike modes giving rise to the astigmatic deformation of the measured wavefront (Fig. 1(E2)). A much more favorable situation was obtained when using the DM (Fig. 1(E3), Fig. 1(F), and supplementary Media 1). To create an astigmatic distortion, the wavefront was modified by a term:

5. 3D localization using astigmatic imaging

Astigmatic PSFs were analyzed using a 2D asymmetric Gaussian fit $a+b\exp \left(-{\left(x-x_0\right)}^2/2w_x^2-{\left(y-y_0\right)}^2/2w_y^2\right)$, based on a modified version of the recently published Multi-Target Tracking (MTT) algorithm [33]. The PSF widths w_x and w_y were measured in a 6 nm-step z-series and correlated to the z-position, relative to the focal plane. We performed these measurements with astigmatism induced with the DM and the cylindrical lens (Fig. 2(A) and 2(B), respectively). Note the asymmetry of the astigmatic deformation at both sides of the focal plane in the case of the cylindrical lens (Fig. 2(B)), in contrast with the good symmetry of the measure with AO (Fig. 2(A)). In turn, these measurements served as a calibration curve for the determination of the z position of individual molecules. For that purpose, we fitted the difference of x- and y-widths Δw = w_x - w_y using a 3rd-degree polynomial curve (Fig. 2(C)). Bead calibration curves were formed by averaging the raw data points of at least 5 high-SNR beads, for which the z-positions had been first aligned.

 

Fig. 2 A-B) Widths w_x and w_y of bead astigmatic PSF with respect to distance from the focal plane for AO-induced astigmatism with an amplitude A = 0.20 μm (panel A) and a cylindrical lens (panel B). Data are computed with a Gaussian fit of the PSF in a 6-nm step z-series. (C) Calibration curve of the difference (Δw = w_x - w_y) in different astigmatic conditions.

Download Full Size | PDF

Figure 2(C) shows that the slope of the calibration curve increases with the degree of astigmatism of the PSF (determined by the amplitude A). This implies that changes in PSF shape relative to the distance to the focal plane are more pronounced at the higher astigmatism levels, which is potentially beneficial for z-localization. However, other considerations limit the degree of astigmatism that can be practically used. First, the PSF profile can only be approximated by a Gaussian curve when it is within ~0.5 μm from the focal plane. At a larger distance, the profile is more complex and its width cannot be simply evaluated by Gaussian fitting. Second, a large degree of astigmatism will lead to a greatly asymmetric PSF. As a result of this elongated PSF and of the spatial spread of the emitted photons, the precision in the localization of the PSF center (x₀ or y₀) and the determination of the PSF widths is reduced. In our experiments, we found that an amplitude A = 0.2 μm offered a favorable trade-off for the 3D localization of individual molecules.

The inverse-square relationship between photon number and z-localization precision (z-LP) is illustrated in Fig. 3(A) (corresponding to the known trends in x,y-LP). This plot is constructed from the analysis of beads of different intensities imaged at the focal plane for the amplitude of the distortion equal to 0.10, 0.15, and 0.20 μm, as well as for the cylindrical lens. An important observation in Fig. 3(A) is the improved z-LP with increased PSF astigmatism. These results concord with those of Fig. 3(B), in which the z-LP dependency with respect to the distance to the focal plane is investigated. In this case, raw data from beads in the range of 3000 to 4000 photons per frame are fitted with a quadratic curve. Over the range of roughly −300 nm to 300 nm, the 0.20 μm amplitude provides a better z-LP.

 

Fig. 3 A) Plot of z-localization precision at the focal plane with respect to number of photons. B) Quadratic curve fits of the z-localization precision with respect to the distance from the focal plane. C) “Stair graph” of a single bead displaced along z in consecutive steps of 50 nm with a piezo stage. Between two steps, 100 frames were acquired. The data (circle) correspond to the estimated localization in each frame and the plain line to the expected position.

Download Full Size | PDF

The quality of depth discrimination is conveyed with the “stair plot” of Fig. 3C. Here, 100 images of a bead immobilized to a coverslip are captured consecutively at different z-positions, each separated by piezo-controlled 50 nm steps. Observed is an excellent correlation of experimental measurement with the expected displacement; as the z-LP degrades toward the −300 nm and 300 nm extremes, the mean value of the data points corresponds closely with the 50 nm steps.

6. 3D single molecule super-resolution imaging and tracking

We applied our three-dimensional single molecule localization scheme using adaptive optics to two different systems relevant to cellular imaging: 3D photoactivated localization microscopy (PALM) of the actin cytoskeleton in fixed cells and 3D tracking of quantum dots (QD) bound to diffusing transmembrane proteins in living cells (Fig. 4 ). In the former experiments, HeLa cells were transfected with an actin-binding-peptide (LifeAct) fused to the photoconvertible fluorescent protein tdEos (ABP-tdEos) [34]. We first performed a direct comparison between the use of the cylindrical lens and the AO-based approach with a level of astigmatism deformation matching that of the lens (Fig. 4(A)). The pre-converted form of tdEos (488 nm excitation, 515 nm emission) allowed us to acquire diffraction-limited images of actin bundles (grey scale image in Fig. 4(A)). We next imaged in 3D PALM the converted form of the fluorophore (561 nm excitation, 607 nm emission) using either the cylindrical lens or the AO-based system. Super-resolution data were acquired in similar conditions (imaging duration, photoactivation and imaging intensity…). Given the density of the ABP probes in the cells, we could acquire the two PALM images consecutively without exhausting the pool of photoactivatable proteins (31638 individual detections for the cylindrical lens and 27493 SM detections for the AO acquisition). Figure 4(A) shows the superior quality of the 3D images when using the AO system. A second example is presented in Fig. 4(B) in which one can clearly distinguish the positions of the crossing actin bundles relative to one another (see also supplementary Media 2). In this image, the z-localization precision was ~40 nm, as estimated by the mean SNR of the detected molecules.

 

Fig. 4 B) Conventional image and three-dimensional PALM images of actin bundles in fibroblasts transfected with ABP-tdEos, with cylindrical lens and AO. The color scale represents molecular density. Scale bar, 2 μm; bounding box, 8.6μm x 13.7μm x 0.6 μm. B) 3D PALM images of actin bundles with AO and controlled astigmatism of A = 0.2 μm. Scale bar, 1 μm; bounding box, 10μm x 5.4μm x 1.3μm. Inset, diffraction limited 2D image of the same region. See also supplementary Media 2. C) Three-dimensional trajectory of quantum dot bound to a transmembrane protein diffusing in the plasma membrane of a cultured HeLa cell. Scale bar, 1 μm; bounding box, 10μm x 5.4μm x 1.3μm. In B and C, the color bar corresponds to the z position, in nm. See also supplementary Media 3 and Media 4.

Download Full Size | PDF

The second biological example is that of membrane proteins labeled with quantum dots (QD) and diffusing on a HeLa cell membrane [35, 36]. The protein corresponded to a transmembrane domain of the PDGF receptor fused to an extracellular epitope (AP tag) that can be biotinylated in live cells. They were labeled with a diluted solution (1 nM for 10 minutes) of streptavidin-coated QDs (QD605, Invitrogen) [37]. By analyzing the shape of QD fluorescence spots in a sequence of images (acquisition time 50 ms), we could reconstruct individual trajectories in 3D with a z-precision of 15 nm (supplementary Media 2 and Media 3). As shown in the illustrative example of Fig. 4(C), trajectories reveal the diffusion of the proteins on a 2D surface within a 3D volume.

7. Conclusions

Here we demonstrate the potential of adaptive optics for single molecule imaging in biological samples. In SM microscopy, improving the detection of the photons emitted by each single fluorophore is key to maximize the amount of information that one can obtain from an experiment. The use of an AO system allowed us to reduce the aberrations of the system, effectively recovering photons otherwise lost due to the WF imperfections. Furthermore, the possibility to accurately modify the wavefront of the light emitted by single molecules enables a controlled and dynamic shaping of the PSF. With tunable astigmatic imaging, we could access axial position of individual molecules with sub-diffraction axial resolution. Note that PSF shapes other than that induced by an astigmatic deformation can be explored with this system. As shown by applications in super-resolution microscopy and single particle tracking in live cells, AO systems can be readily used to address questions on the 3D organization and dynamics of molecular complexes in fixed and live cells. In a longer term, a key benefit of AO will be its application to imaging in thick samples, where the degradation of the PSF can be detrimental to the detection of individual low photon yield fluorophores (such as FPs). Thus, we anticipate that the ability to optimally use the budget of photons of each molecule combined with the possibility to finely tune the PSF shape will be decisive factors to further extend single molecule imaging techniques from cultured cells to tissues, slices or full organisms.

Supplementary information

Supplementary Fig. 5