Localization accuracy of gold nanoparticles in single particle orientation and rotational tracking

The Single Particle Orientation and Rotational Tracking (SPORT) technique, which utilizes anisotropic plasmonic gold nanorods and differential interference contrast (DIC) microscopy, has shown potential as an effective alternative to fluorescence-based techniques to decipher rotational motions on the cellular and molecular levels. However, localizing gold nanorods from their DIC images with high accuracy and precision is more challenging than the procedures applied in fluorescence or scattering microscopy techniques due to the asymmetric DIC point spread function with bright and dark parts superimposed over a grey background. In this paper, localization accuracy and inherited uncertainties from unique DIC image patterns are elucidated with the assistance of computer simulation. These discussions provide guidance for researchers to properly evaluate their data and avoid making claims beyond the technical limits. The understanding of the intrinsic localization errors and the principle of DIC microscopy leads us to propose a new localization strategy that utilizes the experimentally-measured shear distance of the DIC microscope to improve the localization accuracy. © 2017 Optical Society of America OCIS codes: (180.0180) Microscopy; (180.3170) Interference microscopy. References and links 1. R. Swaminathan, C. P. Hoang, and A. S. Verkman, “Photobleaching recovery and anisotropy decay of green fluorescent protein GFP-S65T in solution and cells: Cytoplasmic viscosity probed by green fluorescent protein translational and rotational diffusion,” Biophys. J. 72(4), 1900–1907 (1997). 2. T. Ha, T. A. Laurence, D. S. Chemla, and S. Weiss, “Polarization spectroscopy of single fluorescent molecules,” J. Phys. Chem. B 103(33), 6839–6850 (1999). 3. Z. Salamon and G. 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Localization of single molecule or nanoparticle probes with high accuracy and precision is an essential requirement in single particle tracking experiments.The concept and principle of fluorescence imaging with one-nanometer accuracy (FIONA) [19], which allows the localization of single fluorescent molecules with nanometer precision via curve fitting of the approximately Gaussian shaped point spread functions (PSF), can be applied in rotational tracking.However, localizing anisotropic imaging probes is more challenging than localizing isotropic imaging probes because the 3D orientation of anisotropic probes may give rise to low signal intensities and/or asymmetric intensity distributions, which could result in significant localization errors [20][21][22].Efforts have also been devoted to the search of optimal PSFs as the fitting model for accurate localization of emitting dipoles.For example, a threeaxis dipole PSF was found to best approximate the experimental PSF of a nanorod in Surface Enhanced Raman Scattering (SERS) microscopy [23].
A greater challenge in localization is met in the case of DIC microscopy, which has an asymmetric PSF with bright and dark parts superimposed over a grey background.This non-Gaussian PSF cannot be fitted with a simple mathematical function.Therefore, a correlation mapping algorithm has been implemented to localize isotropic particles in 2D [24] and 3D [25].However, the correlation mapping algorithm does not work properly when anisotropic plasmonic nanoparticles are imaged in SPORT experiments, because the model changes constantly as a nanorod rotates to different orientations.To overcome this challenge, we have developed a dual-channel imaging system to localize gold nanorods with high accuracy in the bright-field channel at the gold nanorod's transverse localized surface plasmon resonance (LSPR) wavelength and track the rotational motions in the DIC channel at the gold nanorod's longitudinal LSPR wavelength [26].
In the present study, we attempt to further address the challenges of superlocalization in DIC microscopy with the assistance of computer simulation.The intrinsic localization errors caused by the DIC imaging principle are quantified from both theoretical and experimental images of gold nanorods.Based on this quantitative understanding, a new localization strategy is proposed to improve the localization accuracy of gold nanorods in SPORT experiments.

Localization uncertainty for inclined gold nanorods
As a simplified case within the framework of classical electrodynamics, anisotropic gold nanorods are deemed as ideal electric dipoles scattering radiation.The polar angle ψ and azimuthal angle θ of an inclined dipole (Fig. 1(a)) are defined based on the two polarization directions determined by the configuration of the polarizers and Nomarski prisms in the DIC microscope.Gold nanorods are imaged at their longitudinal LSPR wavelength and exhibit nearly all bright, nearly all dark, or half bright -half dark image patterns depending on their orientations.A set of samples DIC images of a gold nanorod with a fixed polar angle of 90° (lying flat on the surface) and varying azimuthal angles are shown in Fig. 1

(b).
To understand how a dipole's orientation affects its localization in DIC microscopy, we first take into account the errors that already exist in the localization based on the scattering PSF from a dipole tilted relative to the horizontal plane.The errors in localization are associated with both the azimuthal and polar angles of the dipole.The simulated ellipticalshaped amplitude scattering PSFs of inclined dipoles with various polar and azimuthal angles (images not shown) were generated with a Python script modified from a published program [21], and they were used as inputs for the DIC simulation program [9,27].The deviation of the coordinates of the brightest spot from the center coordinates of the dipole brings about the intrinsic localization errors [28].Lower signal intensities associated with smaller polar angles also lead to localization errors [29].
Conventional single particle localization methods rely on curve fitting of fluorescence or scattering images.2D Gaussian functions are often used as an approximate model to fit the emission/scattering PSF.However, the DIC PSF cannot be fitted by Gaussian functions due to the shadowcast half-bright and half-dark images resulted from interference.Moreover, a lateral shear between the bright part and the dark part of a DIC image further complicates the localization.In our previous SPORT studies, localization was done by assuming a roundshaped DIC image and simply looking for the image center.The simplified method was adequate for the previous works where superlocalization was not necessary or the DIC image patterns were kept relatively constant within short time periods [8,17].
It should be noted that the present study focuses on the cases with the polar angles ranging from 45° to 90°, which are more commonly observed in tracking nanorods on cell membranes or functionalized surfaces.Outside of this polar angle range, the nanorods are tilted to an even larger degree relative to the horizontal plane and give rise to the donut-shaped scattering PSFs [20], which complicates the DIC image patterns [30] and requires further development of computer simulation.

Lateral shear of DIC microscope
Quantitative characterization of the DIC image patterns, which is essential to improve the accuracy and precision in localizing gold nanorods in SPORT, requires the knowledge of DIC optical components.In principle, DIC microscope is one particular kind of shearing interference optical system [31].Illumination light is separated into two mutually orthogonally polarized plane light and laterally sheared by the birefringent (Nomarski or Wollaston) prisms because of the different refractive indices of the ordinary ray and extraordinary ray.Lateral shear has significant effects on contrast, resolution, sensitivity, optical sectioning depth, and localization accuracy and precision [31].The shear distance is determined primarily by the geometry of the prisms, the wavelength of the incident light, and the focal distance of the optics [32,33].Therefore, it is required to experimentally measure the shear distance for a given microscope configuration.
Several methods have been developed to measure the shear distance [26,[31][32][33][34][35].In the present work, the lateral shear distances of two commercial Nomarski prisms (Nikon standard/high-contrast prisms (Part # 100XI) and high-resolution prisms (Part # 100XI-R)) were measured by using the modified bright-field method described in our previous work [26].This procedure is illustrated in Fig. 2. Table 1 lists the measured shear distances for various combinations of the type of Nomarski prism, microscope objective's numerical aperture (NA), and illumination wavelength (540 or 700 nm).At least 10 measurements were carried out for each configuration.As predicted by the theory, longer wavelength and larger NA correspond to shorter shear distance and smaller shear angle.For example, the measured shear distance with the standard Nomarski prisms at 700 nm is ~7.5% smaller than that at 540 nm.

Experimental and simulated DIC images
The simulated DIC images of gold nanospheres and nanorods with different shear distances are shown together with their corresponding experimental DIC images (Fig. 3).As expected, the images obtained with the high-resolution Nomarski prisms (100XI-R), which results in a smaller shear distance, are stretched to a less extent along the shear direction and the bright part and the dark part are positioned closer when compared to the images obtained using the standard Nomarski prisms (100XI).
Localization errors in DIC microscopy should be assessed based on the corresponding shear distance measured for a specific configuration.The discussion in this section will be focused on the results acquired with the following parameters: the high-resolution Nomarski prisms (100XI-R), 100x oil-immersion NA 1.40 objective, and illumination wavelength of 700 nm.We first investigate the experimental and simulated DIC images of a gold nanorod with a fixed polar angle of 90° (flat on the surface) and varying azimuthal angles (Fig. 1).The cylindrical shape of a gold nanorod with a two-fold symmetry suggests a challenge in differentiating the supplementary azimuthal angles of θ and 180°-θ.Indeed, the experimental results published previously showed nearly identical bright and dark DIC image intensities and patterns at any pair of supplementary azimuthal angles.However, the simulated results presented here suggest the DIC image patterns at the supplementary azimuthal angles (except for 0° and 180°) are not identical.For example, the dark lobe in the simulated DIC image at 45° is nearly vertical, where the dark lobe in the simulated image at 135° is nearly horizontal (Fig. 1(d)).These differences stem from the oval-shaped scattering PSFs, which are used as the input for the DIC simulation program.These differences are also visible in the experimental DIC images (Fig. 1(b)), suggesting an additional element of uncertainty in localization.Furthermore, the experimental DIC intensity traces (Fig. 1(c)) show a broader valley at ~90° compared to the simulated traces (Fig. 1(e)).These differences can be attributed to the noise in the real images, the approximation of the theoretical equations used in simulation, the limit in imaging resolution, and the actual morphology of the gold nanorods that have not been taken into account in the theoretical simulations, for example, the rod's end-cap geometry affects the longitudinal surface plasmon band in real cases.

Improving localization accuracy with measured shear distance
Based on the understanding of orientation-dependent DIC image patterns, we have developed an improved localization strategy, which takes the lateral shear between the bright part and dark part of the DIC images into considerations.
The initial step of the new strategy is to examine the DIC image patterns.When both the bright and dark parts of a DIC image have sufficiently high signal to noise ratio (S/N) (azimuthal angle θ near 45° or 135°), the weighing algorithm is applied to find the centroid coordinates of both the bright and dark parts and then take their midpoint as the centroid coordinates of the gold nanorod (see Experimental Section for details).On the other hand, when a DIC image is dominated by either a bright or dark part (azimuthal angle θ near 0°, 90° or 180°), the centroid coordinates of this dominant part can be easily found with the weighing algorithm; however, the S/N of the other half of the image becomes too low to warrant sufficient localization accuracy and precision.In such cases, we should discard the localization information from the weak part of the image and take the following more accurate measurement: shifting the centroid coordinates of the dominant part by half of the shear distance in both x and y.Specifically, for the DIC microscope used in our experiments, the final centroid coordinates are calculated as the weighed bright center (for a DIC image dominated by the bright part) minus half of the shear distance or the weighed dark center (for a DIC image dominated by the dark part) plus half of the shear distance.The improved localization accuracy offered by the new localization strategy is demonstrated in Fig. 4(a) for gold nanorods lying flat on the surface (polar angle ψ = 90°).Even though both the bright and dark centroid positions determined by the weighing method are clearly influenced by the azimuthal angles, the new strategy provide rather consistent accuracy for the entire angle range (0-180°).The collective standard deviations of both the x and y centroid coordinates for all azimuthal angles are ~7 nm.
For inclined gold nanorods (polar angle ψ < 90°), as the polar angle gets smaller, the decreasing DIC image contrasts and increasing complexity in image patterns result in larger localization errors.The localization results gold nanorods with polar angles of 75°, 60°, and 45° (images not shown) are displayed in Fig. 5.At the polar angle of 45°, variations in the determined centroid coordinates (Fig. 5(c)) are much more significant than the other two polar angles because of the much larger uncertainties in the weighing algorithm to find the centroid coordinates of the bright and dark parts.

Localization accuracy and precision in real DIC images
The new localization strategy has also been tested experimentally by measuring the distance between two neighboring gold nanorods during a 360° in-plane rotation (Fig. 6).The DIC image patterns of these spin-casted gold nanorods suggest that they are nearly perfectly flat on the surface (ψ ~90°), and the angle between their long axes is estimated to be ~50° from their DIC intensities.From the 72 measurements during the 360° rotation (in 5° steps), the standard deviation of the distance calculated between the two nanorods is 53.6 nm.Using particle 1 as the reference with the coordinates set at (0, 0), the positions of particle 2 are shown in Fig. 6(c).Rather consistent localization accuracy is demonstrated in nearly the entire 360°.The deviation of the distance of particle 2 relative to particle 1 from the average distance is the largest when the link between the two particles is oriented at around 25°, 95°, 200°, and 280°.A close examination of the DIC images (Fig. 6(a)) shows that at these angles the two particles display opposite image patterns: a dominantly bright image and a dominantly dark image.Because the largest localization errors are associated with these image patterns, the errors in the measured distances are also the largest.Future efforts will be required to deal with even more complicated DIC images from highly inclined gold nanorods (polar angle ψ < 45°).Furthermore, sacrificed image contrast as a trade-off for faster temporal resolution can significantly impact the ability of the SPORT technique to resolve rapid rotational motions in complex environments.Just like in all fluorescence or scattering-based SPT experiments, the desires of higher sensitivity, faster imaging rate, better spatial and angular resolution, and robust automation will continue to drive the technical development and enable novel single-molecule/nanoparticle based studies.

Sample slide preparation
The 25 nm × 73 nm gold nanorod and 80 nm gold nanosphere colloidal solutions were purchased from Nanopartz (Loveland, CO).To prepare the sample solutions, 0.1 mL of colloidal gold underwent centrifugation for 10 min at 5500 rpm, followed by removal of the liquid layer and resuspension in 0.1 mL of 18.2 MΩ Milli-Q water.The solution was diluted to a concentration of 1 × 10 10 nanoparticles/mL 10 µL of the diluted solution was spin-casted onto a pre-cleaned microscope slide and covered by a glass coverslip.To prevent evaporation, the coverslip was sealed by clear nail polish.The sample slide was then placed on the microscope stage for imaging.

Instrumentation
An upright Nikon Eclipse 80i microscope and a Nikon Ti-E inverted microscope were used in the imaging experiments.These microscopes were equipped with a Nikon 1.40 numerical aperture (NA) oil-immersion condenser.Two objectives, a Nikon 100 × 1.40 NA Plan Apo VC oil-immersion objective and a 100 × 1.49 NA Apo TIRF oil-immersion objective, were used for comparison.A 540 nm or 700-nm band-pass filter (Semrock, Rochester, NY) with a bandwidth of 20 nm was inserted at the illumination side.A rotary stage was installed on the microscope for taking images of gold nanorods positioned in different orientations.At each angle, a vertical scan was done to help find the focal plane.Two scientific CMOS cameras were used to capture the DIC images: a Hamamatsu C11440-22CU, ORCA-Flash 4.0 V2 with a 2048 × 2048 pixel array and a pixel size of 6.5 μm × 6.5 μm and a Tucsen Dhyana 95 with a 2048 × 2048 pixel array and a pixel size of 11 μm × 11 μm.These cameras performed similarly in our experiments.Finally, the images were analyzed using MATLAB and NIH ImageJ [36].

Computer simulation of DIC PSFs
The scattering PSFs of dipoles in different orientations were generated by a Python script modified from a published program [21].These PSFs were then used as inputs for the DIC PSF simulation program, which was written in MATLAB and is available upon request.
The core of the DIC PSF simulation is the principle of interferometry for DIC microscopy.For the Nomarski-type DIC microscope used in our experiments, the specimen is illuminated with a plane-polarized beam split into two orthogonally polarized and mutually coherent components by the first Nomarski prism.The two components are recombined into a single beam by the second Nomarski prism and interfere with each other after the analyzer to form an image.The DIC microscope converts the phase variations caused by the specimen to intensity variations.The image is formed from the two waves that are phase-shifted to different extents and laterally shifted by a shear 2Δx along the x-axis and 2Δy along the yaxis.The 2D DIC PSF is thus derived from the interference of the PSF of the two phase-shifted waves with a bias retardation 2Δθ (expressed in radians) and the lateral shear 2Δx and 2Δy [27]: where p(x, y) is the scattering amplitude PSF for transmitted-light optics under coherent illumination, and R is the amplitude ratio, defined as the amplitude of one wave field divided by the sum of the amplitudes of the two wave fields.Δx and Δy are equal in all commercial DIC microscopes, including the Nikon microscopes used in our experiments.Equation ( 1) was implemented in a MATLAB program, which was used to generate the simulated DIC images in this paper.

Shear distance measurement
The two intermediate images of 80-nm nanospheres were taken separately by setting the polarization direction of the second polarizer to + 45° or −45° with respect to the polarization direction of the first polarizer.These polarizer settings block one of the two laterally separated beams.Then the two images were fitted with 2D Gaussian functions in MATLAB to obtain the centroid coordinates of each image.It should be noted that the intermediate images are essentially in the bright field mode; therefore, the typical scattering PSFs or Gaussian functions as a close approximation can be used to fit the images to find the centroid coordinates.

Gold nanorod localization -weighing algorithm
The DIC image pattern of a single gold nanorod consists of a bright part and a dark part over a grey background.As the first step of localization, a square is defined to cover the entire DIC image pattern.Within the square, the bright (or dark) part of the image can be identified by comparing the intensity of each pixel to the bright (or dark) intensity threshold, which is defined as the highest (or lowest) intensity value of the local background.Then the centroid coordinates of the bright part and the dark part of the DIC image are computed separately by weighing all of the pixels that have been identified as either bright or dark: Note that the pixels fall in the bright (or dark) part are used only in the calculation of the bright (or dark) centroid coordinates.Finally, the centroid coordinates of the simulated or experimental DIC images are obtained at the midpoint of the bright and dark centroids.

Funding
The work was supported by the National Institution of Health (R01GM115763).The computer simulation part was supported by the National Science Foundation (CBET 1604612).

Fig. 1 .
Fig. 1.(a) Schematic illustration of a fixed dipole with polar angle ψ and azimuthal angle θ.One of the polarization direction (x-axis) may be referred to as the dark optical axis because a gold nanorod would generate a nearly all dark image when its long axis is aligned with this polarization direction; similarly, the other orthogonal polarization direction (y-axis) may be called the bright optical axis in accordance to the presentation of nearly all bright DIC image.These definitions of optical axes are used consistently in all the experimental and simulated images of gold nanorods in this study.(b) Experimental DIC images of a gold nanorod with the azimuthal angles from 0° to 180° in 15° steps.Note that 0° and 180° are equivalent due to the symmetry.(c) The corresponding bright-part intensity (blue) and dark-part intensity (red) traces for the experimental DIC images in b.(d) Computer simulated DIC images of a gold nanorod with a polar angle of 90° and azimuthal angles from 0° to 180° in 15° steps.(e) The corresponding bright-part intensity (blue) and dark-part intensity (red) traces for the simulated DIC images in d.

Fig. 2 .
Fig. 2. Two intermediate bright-field images of an 80-nm gold nanosphere obtained when the angle between the first and second polarizer is (a) 45° and (b) −45°.(c) The merged image.The two crosses are the centers of the two intermediate bright-field images (a) and (b) by fitting with 2D Gaussian functions.The distance between the two centers is the shear distance.The shear direction is from northwest to southeast, which is indicated by the relative positions of the two centers.

Fig. 3 .
Fig. 3. Experimental and simulated DIC images of (a) a 40 nm × 118 nm gold nanorod (90° polar angle and 45° azimuthal angle) and (b) an 80-nm gold nanosphere under 540 or 700 nm incident light with either Nikon standard/high-contrast (100XI) or high-resolution (100XI-R) prisms.It is worth noting that the measured shear distance is not affected by the size and shape of the nanoparticle when other configurations are the same.

Fig. 4 .
Fig. 4. (a) Localization of the simulated gold nanorod images with a constant polar angle of 90° and various azimuthal angles.Red and blue: the x and y coordinates of the weighed centers of the bright part.Black and green: the x and y coordinates of the weighed centers of the dark part.Cyan and magenta: the x and y coordinates of the midpoint between the weighed centers of the bright part and dark part.Grey and yellow: the x and y centroid coordinates of the gold nanorod when either bright or dark part dominates.Yellow squares are hardly visible in the figure because they are closely overlapped with grey squares.(b) Standard deviations of the localized x (black) and y (red) coordinates of gold nanorods at four polar angles: 45°, 60°, 75°and 90°.Each data point is calculated from the simulated images with various azimuthal angles from 0 to 180°, in 5° step size.The localization is determined from the weighed centers of the bright part and the dark part of the images.

Fig. 5 .
Fig. 5. Localization of the simulated gold nanorod images with constant polar angles at 75° (a), 60° (b) and 45° (c) and with various azimuthal angles.Red and blue: the x and y coordinates of the weighed center of the bright part.Black and green: the x and y coordinates of the weighed center of the dark part.Cyan and magenta: the x and y coordinates of the midpoint between the weighed centers of the bright part and dark part.Grey and yellow: the x and y centroid coordinates of the gold nanorod when either the bright or dark part dominates; they are calculated as the weighed bright center minus half of the shear distance or the weighed dark center plus half of the shear distance.

Fig. 6 .
Fig. 6. (a).The DIC images of two neighboring gold nanorods (labeled with 1 and 2) during 360° in-plane rotation.The scale bar represents 2 μm.(b) The bright (blue) and dark (red) DIC intensity traces of particle 1 (solid square) and 2 (hollow square) during the 360° in-plane rotation.(c) Localization of particle 2 relative to the position of particle 1.The radius of the green circle indicates the average distance between these two particles.
In summary, we have studied the localization accuracy of gold nanorods in DIC microscopy imaging.The inherited localization errors from the asymmetric scattering PSF of a tilted dipole and the unique DIC patterns of gold nanorods with various polar angles and azimuthal angles have been discussed based on computer simulated DIC images.The localization accuracy in real examples of DIC microscopy imaging has also been demonstrated by measuring the distance between two neighboring gold nanorods at various orientations.These discussions provide guidance for researchers to properly evaluate their data and avoid making claims outside the technical limits.The dependence of localization accuracy on the instrument-specific lateral shear adds an additional layer of complexity to SPORT in DIC microscopy, compared to the fluorescence or dark field microscopy-based methods.A refined localization strategy has been proposed to use the experimentally measured shear distance to compensate for the localization errors.