1. Introduction

Visualization, tracking and analyzing molecules/vesicles and their dynamics in cells are critical in understanding the complex biological system. The single-particle tracking (SPT) method has been successfully implemented in many biological researches for visualizing the dynamic processes inside cells [1–6]. Using centroid localization, single molecules can be localized with an precision as high as one nanometer, so the traces of target molecules can be identified with ultrahigh precision which is crucial for understanding the dynamic processes involved [7–11]. Since the molecules move in all three dimensions, SPT with three-dimensional (3D) localization ability is preferable. Many 3D localization approaches have been suggested, such as multifocal plane [12, 13], astigmatic imaging [14, 15], biplane detection [16], and double-helix point spread function [17–19]. Since live biological cells are about 10 μm thick, many applications require tracking multiple single molecules moving inside the cell is very useful in analyzing dynamic process which might occur throughout the cell, such as the study of intracellular -intercellular transport of biomolecules, binding of molecules to receptors on the cell surface, entering the cells by a receptor mediated endocytosis and then being transported in the cell whatever related to actin filaments and microtubules. Consequently, optical imaging methods with depth of field (DOF) as large as ten microns are required, which is hard to achieve. Focus stacking using MUM [20] is capable of imaging different focal planes and tracking over 10um depth in the cell. However, this system requires several cameras simultaneously, and the setup is too expensive and complex.

There is one other promising solution called distorted grating (DG) [21, 22] and DH-PSF [23] combination microscopy DDCM [24]. With DDCM, particles within a layer thicker than 10μm can be imaged and localized with 3D nano-resolution in only one shot. DDCM combines the extended DOF of the DG with the 3D nano-localization of DH-PSF. Experimentally, DDCM is implemented by introducing a bifunctional phase mask (PM) (φ) into traditional fluorescent imaging system. The PM is designed as the superposition of two phase patterns for DG and DH-PSF (φ_DG and φ_DH), respectively. The DG part is designed to simultaneously image three layers spanning a certain distance (h_DG) matching the effective localization depth of the DH-PSF part, h_DH. As a result, wherever the particle is in the cell, it can be imaged in one of the three sections of the detector, corresponding to the three diffraction orders of the DG part. The resultant image of the molecule consists of two lobes. The former also supplies rough estimation of the position of the particle, while its precise position is determined by the latter, i.e., the angle of the two lobes in the exact image section.

In this paper, we introduce a DDCM system and demonstrate it for 3D multi-molecule tracking. The key innovation is the use of a custom designed, lithographically fabricated composite PM combing the functions of DG and DH-PSF for the efficient extension of DOF. This DDCM represents a major step towards realizing 3D multi-particle tracking in thick samples. We present details of the mask fabrication and system set up process in section 2 and the experimental characterization in section 3. Section 4 describes the application to 3D multi-particle tracking.

2. Method

Based on previous the theoretical analysis and numerical simulation in [24], the bifunctional PM was fabricated on a fused quartz substrate with Ion Beam Etching (IBE) (Fig. 1). The PM has 256 phase steps and 336 × 336 pixels with a pixel size of 15μm × 15μm. The clear diameter is about 5 mm. The effective localization depth of the DH-PSF part, h_DH, is designed to be 4.94 μm, and the distance between the image plane for the ± 1st order and the 0th order, h_DG, is designed to be 4.94μm. So the total effective DOF is 14.82 μm theoretically. The measured transmission of the fabricated mask is 81%, and the actual diffraction efficiency ratio of the three diffraction orders (−1st, 0th, 1st) is 0.79:1:0.85. The PM was designed to diffract photons to the three diffraction orders equally. The discrepancy between the designed and measured value is likely due to the fabrication precision.

 

Fig. 1 Design of composite the PM. (a), (b), and (c) The image shows the simulation result of composite PM [24]. (d)The actual designed PM using micro-fabrication photoetching method.

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Next, we built a complete DDCM using this bifunctional PM on a commercial microscope (IX81, Olympus), shown in Fig. 2. 100-nm fluorescent beads (TetraSpeck Microspheres, molecular probe Co.) deposited on a glass slide surface were imaged. The fluorescent beads were excited by a 640 nm laser. The peak emission wavelength of the fluorophore beads is 670 nm. The signal from fluorescent beads was collected by an oil immersion objective (NA 1.4, 100 × , Olympus). After a tube lens (f_tube = 180 mm), the light was modulated with a 4f system (f = 200 mm). The PM was mounted on the Fourier plane of the 4f system. The signal was detected by an EMCCD (iXon 885, Andor) with a pixel size of 8 μm*8 μm.

 

Fig. 2 Schematic of the DDCM. The laser is expanded and collimated after an excitation filter (EF), and then sent to excite the fluorescent beads with an objective (Obj) combined with a tube lens(TL1) whose focal length is f = 180 mm. Fluorescence signal is collected by the same objective and split with laser beam by a dichromatic mirror(DM). A 4f relay system consisting of two achromatic lenses (F1 and F2, f = 200 mm) and a composite PM mounted at the Fourier plane is inserted before the detector (iXon 885, Andor).

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3. Calibration

Considered that for each of the three imaging sections corresponding to the three diffraction orders, the change rates between the angle of the two lobes and the relative defocusing distance may vary in different ways, calibration is necessary before data acquisition. For each imaging section, one calibration curve should be acquired to connect the exact axial position and the angle. In order to eliminate the Brownian motion of the fluorescent beads, the beads were mixed with 0.5% low melting temperature agarose, and then spread on the glass slide. The sample was then displaced in 100 nm steps along z axis with a piezoelectric stage (P-561.3cd, PI). The motion of the stage and the exposure of the camera were synchronized to make sure the stage is stable during each exposure. The total movement of the sample was 15μm which was close to the effective DOF of the DDCM system. One single bead in the sample was imaged and analyzed. When the bead move from z = −7.5 μm to z = 7.5 μm, its two lobe images appeared in the above, middle, and below imaging sections one after another successively. The relationship between the actual axial positions and the angles of the two lobes measured from the images for the three orders are shown in Fig. 3. The angle changed from −84.72° to 84.08°, −84.89° to 86.96°, and −88.63° to 82.39°, when the bead moved from −7.26 μm to −2.06 μm, from −2.37 μm to 2.23 μm, and from 2.1 μm to 6.69 μm, respectively. All the following data analysis and image reconstruction were based on these calibration curves.

 

Fig. 3 Calibration of the DDCM system by imaging a fluorescent bead. The movement of bead is controlled by a nano piezoelectric stage. The bead was moved in 100 nm steps along z axis and imaged. Three representative images of the bead are shown in (a), (b), and (c), that are three images of the bead in the −1st, 0th and + 1st diffraction order section which corresponded to −6 um, −1 um and 6 um along with the z axial position. (d) Calibration curves for the 0th and the ± 1st diffraction orders.

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To quantify the 3D localization precision of the DDCM system, 100 images of the bead at each axial position were acquired and analyzed. Localization precision was denoted by the standard deviation of coordinates of the 100 localized positions. The exposure time was set as 30 ms to make sure the photon number for each lobe in the 0th order is about 5000. Consequently, for the photon numbers in the + 1st and the −1st orders were 4100 and 3500 respectively. The average x, y and z localization precision for the each order can be calculated from the corresponding localization precisions at all positions for the corresponding order which is shown in Fig. 4, i. e. σ_-1st (x, y, z) = (6.5 nm, 9.2 nm, 23.4 nm), σ_0th(x, y, z) = (3.7 nm, 2.8 nm, 10.3 nm), and σ_+1st(x, y, z) = (5.8 nm, 6.9 nm, 18.4 nm), respectively. Overall, the localization precision for the 0th order image is best, and that for the −1st order is a bit better than that for the + 1st order. The former is reasonable since the diffraction efficiency of the 0th order is higher than the other two. The latter can also be explained, because collective efficiency of the −1st order is higher than that of the + 1st order because when the bead is imaged in −1st order section, it is closer to the objective than the similar situation for the + 1st order.

 

Fig. 4 Estimation of the localization precision in x, y (upper) and z (lower) for the −1st, 0th and the + 1st diffraction orders.

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Based on the above results, the localization precisions for different detected photons are analyzed. The localization precisions for single beads as a function of collected photons from 500 to 6000 photons for each diffraction order are shown in Fig. 5. For fluorescent labels with higher emission rate, such as semiconducting polymer dots (Pdots) [25, 26], the performance of DDCM will be much better.

 

Fig. 5 Localization precision as function of the number of collected photons for the −1st(a), the 0th(b) and the + 1st(c) diffraction order.

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4. Tracking experiment

Finally, the tracking ability of the system with extended DOF is demonstrated by multi-particle-parallel-tracking. The preparation of sample is different from that for the above experiments. Beads were mixed with 75% glycerol aqueous solution to increase its mobility (Fig. 6). During the experiment, a series of images (200 frames) of the beads were acquired with an exposure time of 80 ms. Positions of the beads were estimated with the DDCM algorithm. Three 3D tracing trajectories of the three beads are shown in Fig. 6(b). The beads are located at different positions at different instants, and their separation along z can be over 12 μm. Even so, they can also be localized and tracked successfully without any z-scanning parts.

 

Fig. 6 Three- dimensional tracking of three beads simultaneously. (a) One image from the movie (media 1) of the dynamic imaging of beads. (b) Trajectories viewed in 3D (top) and projected on x-z plane (bottom left) and x-y plane (bottom right), respectively.

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To further test the feasibility of our method to directed motion dominated by diffusive processes, we studied the diffusion of 100 nm fluorescent beads in glycerol-water mixtures representing media of different, but well characterized viscosities. Beads in solution were tracked with 10ms exposures time. Experiments were performed for water-glycerol mixtures of 100:0, 80:20, 60:40, 40:60, 20:80, 0:100. The mean square displacement (MSD) data in r versus lag time averaged over the trajectories of fluorescent beads is shown in Fig. 7. A fit to the MSD for r is shown, and the trend is clearly linear. The diffusion coefficient D extracted from the slope of the fit was found to be 3.26 μm²/s for pure water. Using the viscosity of the pure water solution at 195 K of 9.548 × 10⁻⁴ Pa s and the Stokes-Einstein relation for a spherical particle, we obtained a hydrodynamic radius of 69 nm for the individual fluorescent beads. The large value for the radius is likely due to the statistical error.

 

Fig. 7 MSDs for 100nm beads in different water-glycerol mixtures with a linear fit to the slope of the data. The MSD shown in green line were derived from the data in Fig. 6.

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Imaging in live cells introduces many potential complications into a super-resolution tracking experiment. In order to assess the feasibility of this approach on particle tracking in live cells, we studied preliminarily the endocytic process of the macrophages using the DDCM. The Raw-264.7 cells were grown to confluence in phenol-red DMEM containing 10% FCS into incubator. Before imaging, the medium was changed to the imaging buffer solution containing no serum. The cells were washed three times by 1% PBS solution, then added 2 ml the imaging buffer with 4 nM TransFluoSpheres, 0.04 µm (488/560), and the imaging was started immediately. During the imaging process, the cell sample dish was maintained at 37 °C by using an objective warmer. Total imaging time was less than 30 min. We have tracked the 3D dynamics of fluorescent beads across different depths in the live-cell monolayer. Figure 8(a) shows a typical DH-PSF image of the beads at a time point. In Fig. 8(c) the entire trajectory is shown. On the basis of this experiment, it seems reasonable that the DDCM could be used effectively for live cell tracking experiments. We will present more detailed cell application results in future studies.

 

Fig. 8 Tracking fluorescent beads in a live cell in 3D. (a) White light image of the Raw-264.7 cell. (b) Fluorescence image of the emitter (media 2). (c) Three-dimensional trajectory of the bead from the red smaller boxed region in (b), showing a variety of diffusive and linear transport characteristics. (d) The trajectory from the red smaller boxed region in (c) when viewed at higher magnification.

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5. Conclusion

In conclusion, we demonstrated a DDCM system for thick samples with extended DOF over 14 μm. With a specially designed PM, the point spread function is transformed into the double helix point which contributes to the 3D nano-localization capabilities of the DDCM system. Multiple object planes can be imaged simultaneously at different areas on a single image plane, which contributes to the extended DOF. Experiments demonstrated that with DDCM, multi-particle moving spanning more than ten microns in z axis can be parallel-tracked successfully. Taken into account of practice factors, the speed would be limited by the localization precision required. Higher localization precision means more photons need to be collected and more exposure time for a certain fluorophore. Actually, to get the corresponding localization precision in an extended DOF, the exposure time for DDCM is about three times that of normal 3D tracking approach DH-PSF. Nevertheless, such kind of limitation can be compensated for by using ultra-bright fluorescent probes, such as semiconducting Pdots. DDCM will become a potential powerful tool for dynamic tracking molecule in living cells.