Single Particle Tracking: From Theory to Biophysical Applications

Review pubs.acs.org/CR

Single Particle Tracking: From Theory to Biophysical Applications Hao Shen,† Lawrence J. Tauzin,† Rashad Baiyasi,‡ Wenxiao Wang,‡ Nicholas Moringo,† Bo Shuang,† and Christy F. Landes*,†,‡,§ †

Department of Chemistry and ‡Department of Electrical and Computer Engineering, §Smalley-Curl Institute, Rice University, Houston, Texas 77251, United States ABSTRACT: After three decades of developments, single particle tracking (SPT) has become a powerful tool to interrogate dynamics in a range of materials including live cells and novel catalytic supports because of its ability to reveal dynamics in the structure− function relationships underlying the heterogeneous nature of such systems. In this review, we summarize the algorithms behind, and practical applications of, SPT. We first cover the theoretical background including particle identification, localization, and trajectory reconstruction. General instrumentation and recent developments to achieve two- and three-dimensional subdiffraction localization and SPT are discussed. We then highlight some applications of SPT to study various biological and synthetic materials systems. Finally, we provide our perspective regarding several directions for future advancements in the theory and application of SPT.

CONTENTS 1. Introduction 2. Theoretical Foundations for Superlocalization and SPT 2.1. Optical Diffraction Limit 2.2. Fourier Optics Provide a Tool To Predict Point Spread Functions 2.3. Fisher Information and the Cramér−Rao Lower Bound 2.4. Localization Methods 2.5. Simulating Experimental Data 2.6. SPT Methods 2.7. Mean Square Displacement (MSD) Analysis 2.8. Ergodic Hypothesis in SPT 3. Revealing Dynamic Processes Using 2D SPT 3.1. Two-Dimensional Interfacial Dynamics 3.2. Imaging 2D Structures with SPT 3.3. Enhanced Spatial Resolution as a Stepping Stone to Improved 2D SPT 3.4. Correlation Analysis 3.5. Differential Interference Contrast (DIC) Microscopy 4. Revealing Dynamic Processes Using 3D SPT 4.1. Multifocal Plane Microscopy (MPM) 4.2. Astigmatic 3D Detection Using a Cylindrical Lens 4.3. Engineered 3D Point Spread Functions Using Phase Masks for 3D Imaging and SPT 4.4. High Temporal Resolution in Single Particle Tracking 5. Active Feedback Tracking 5.1. ABEL Trap 5.2. Confocal Based Active Tracking © 2017 American Chemical Society

5.3. Nonconfocal Based Active Tracking 6. Conclusion and Perspective Author Information Corresponding Author ORCID Notes Biographies Acknowledgments References

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1. INTRODUCTION Single-molecule microscopy1−9 and super-resolution microscopy10−14 have revolutionized our ability to study fundamental biology15−17 and general surface sciences.18−24 Single-molecule techniques have enabled us to understand details about, for example, the movement of molecular motors inside cells,25,26 and mechanisms of biomolecular separation efficiency at chromatographic interfaces.27−33 Such advancements in the fundamental understanding of these areas could dramatically influence our daily lives in the future. Single particle tracking (SPT), as illustrated in Figure 1, has played a central role in many of the advances in single-molecule microscopy.34−37 In a typical SPT measurement, the molecule of interest is imaged through an optical microscope by a photon detector and its motion is recorded in a continuous fashion to generate a trajectory. Like other single-molecule phenomena, while the behavior of a single trajectory might be stochastic, the statistical behavior from many trajectories may reveal additional

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Special Issue: Super-Resolution and Single-Molecule Imaging Received: December 14, 2016 Published: May 18, 2017 7331

DOI: 10.1021/acs.chemrev.6b00815 Chem. Rev. 2017, 117, 7331−7376

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fluorescence correlation spectroscopy super-resolution optical fluctuation imaging (fcsSOFI; see section 3.4 for details).50 Aside from the case in which the analyte of interest is itself emissive, in SPT a fluorescent tag is attached to the analyte molecule and it is the emission of this tag that is tracked. Indeed, the development of SPT techniques is closely tied to the continually growing library of fluorescent labels available for experimental use.51,52 Ideally, the tagging species needs to be small enough to not interfere with the mobility and functionality of the analyte; on the other hand, it must be bright enough, i.e. exhibit high quantum yield, and have a robust photobleaching lifetime to be measured over reasonable periods of time. Unfortunately, few fluorescent tags can fully meet these two requirements, as there is a trade-off between size and robust photophysical properties. While Robert Brown’s use of an optical microscope to observe the motion of pollen suspended in water led to our earliest understanding of Brownian motion,53 the modern era’s advancements in SPT began with the ability to video track motion. For instance, De Brabander et al. microinjected 40 nm AuNPs stabilized with polyethylene glycol and bovine serum albumin (BSA) into the cytoplasm of PTK-2 cells, where the motion of the AuNPs was manually tracked via video-enhanced contrast microscopy.54 The AuNPs, invisible to the naked eye, strongly scattered light and were thus clearly seen as black dots after background subtraction and contrast enhancement. Polystyrene beads,55 latex beads,56 and silica particles57 were other early analytes of choice during the inception of SPT. For example, Gratton et al. tracked the 3D motion of 500 nm polystyrene spheres using a two-photon light microscope.58 In 1993, one important milestone in SPT was developed by Betzig and Chichester, in which a single fluorescent molecule was detected at room temperature.59 Using single fluorescent molecules as tags for analytes was soon adapted to SPT. Due to the small physical sizes of organic dyes (1−2 nm), they have been attached to a variety of molecules including proteins,60 antisense DNA strands,61 mRNA,62 polymer chains,63 and antibodies.64 However, because some organic dye molecules are highly toxic to living cells, the successful synthesis of fluorescent proteins in the mid-1990s was another important advance.65,66 The significance of fluorescent proteins is that they can be genetically encoded, making live cell imaging possible. Since then, there has been an explosion of SPT studies of various systems in living cells, including gene expression,67,68 protein DNA interaction,69,70 and dynamic processes on cell membranes.39,71 An alternative choice for tagging is quantum dots (QDs),72−75 which have high quantum yields and do not suffer from photobleaching. QDs used for fluorescent tags are generally larger than either organic dyes or fluorescent proteins, but are smaller than AuNPs or polymeric spheres: a typical QD tag has an overall diameter of 10−30 nm including an inorganic core and necessary stabilization ligands. It is important to note that, despite many advances in single fluorescent probes, the current library of SPT tags leaves considerable room for improvement through optimization of sizes, quantum yields, and photobleaching lifetimes. Single fluorescent molecules are often the tagging species of choice; thus the development of highly sensitive detectors has been crucial for advances in SPT. The core components of state-of-the-art detectors are semiconductor materials that can convert photons into electrical signals based on the creation of electron−hole pairs in low light environments.76,77 Depending on the excitation geometry and sample conditions, two types of

Figure 1. Schematic representation of SPT. Particles are localized by finding the central locations of their point spread functions (A), localizations belonging to the same particle at different times are connected using algorithms such as nearest neighbor (B), and their chronological series of linked localizations form a time series trajectory (C). Dynamic information is extracted based on a statistical analysis of the trajectories (D), through, for example, mean square displacement (MSD) analysis.

chemical or physical properties combining diffusion, adsorption−desorption dynamics, folding−unfolding dynamics, trapping, or other processes.35,36,38,39 There are other techniques that can provide dynamic information about two-dimensional (2D) and three-dimensional (3D) motion, namely fluorescence recovery after photobleaching (FRAP)40−42 and fluorescence correlation spectroscopy (FCS),43,44 both of which have been extensively reviewed previously. Briefly, in FRAP, an intense laser beam is focused onto a small region to photobleach fluorescently labeled molecules of interest. After the laser illumination stops, the fluorescence intensity is recorded as fluorescent species in the surroundings diffuse back into the bleached region through Brownian motion. The rate of diffusion can then be readily calculated from the time-varying fluorescence signal.45,46 FCS is an autocorrelation based technique that measures the fluorescence intensity fluctuation as a function of time due to the diffusion of fluorescently labeled molecules. Autocorrelation analysis of the temporal fluctuations in fluorescence intensity provides information such as the diffusion coefficient, the analyte concentration, and the hydrodynamic radius, via the Stokes−Einstein equation.47−49 Although FRAP and FCS offer dynamics information based on real-time observations, they can be surpassed by SPT in two ways. First, FRAP and FCS only provide average information, and function-dependent heterogeneity is lost for analytes such as nanoparticles or biomolecules. Second, although FRAP and FCS provide millisecond temporal resolution, their spatial resolutions are still diffraction limited (see section 2.1 for details), which limits their application for nanoscale processes. However, it is also worth noting that some recent studies have combined super-resolution approaches with FCS, such as 7332


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Figure 2. Scheme of biophysical time scales relevant to single-molecule studies. “Cy” stands for cytoplasm and “Mb” stands for membrane. Reprinted with permission from ref 77. Copyright 2007 Taylor & Francis.

detectors are most commonly used: point detectors and widefield detectors.78,79 Point detectors such as photomultiplier tubes (PMT) and avalanche photodiodes (APD) have little to negligible readout noise and dark counts, and can operate at picosecond temporal resolution; therefore they are commonly used in confocal based experiments.77,80 However, rasterscanning is needed for imaging purposes; thus the applications of point detectors are limited in large data output scenarios. High sensitivity wide-field detectors are ideal for singlemolecule imaging. The most commonly seen wide-field detectors are cameras based on charge-coupled devices (CCD). To achieve single-molecule sensitivity, an image intensifier is placed in front of the CCD chip. These so-called “intensified CCD (ICCD) cameras” have quantum efficiencies ranging from 20 to 50%.77 Another popular choice is the electron multiplying CCD (EMCCD) camera, which utilizes a shift register and an output amplifier to achieve large data output and high sensitivity simultaneously. The quantum efficiency of an EMCCD camera can be >90%. The drawback of EMCCD cameras is that the stochastic electron multiplication mechanism also amplifies the shot noise in the signal, resulting in lower photodetection efficiency. Another option for wide-field detection is the scientific complementary metal oxide semiconductor (sCMOS) camera. Unlike CCD based cameras that utilize the entire pixel area for photon collection, sCMOS cameras have supporting electronic devices integrated into each pixel, meaning that only part of the entire pixel is photon sensitive. Also, sCMOS cameras usually have intense dark noise and readout noise, so single photon detection cannot be currently achieved on a sCMOS camera.81 The quantum efficiency of sCMOS cameras is ∼70%, lower than that of EMCCD cameras. However, simulation results in previous work suggest that sCMOS cameras can outperform EMCCD cameras in spot localization, because the sCMOS does not amplify shot noise.81 The temporal resolution of singlemolecule experiments is usually dictated by the frame rate of the detector. CCD based cameras have native frame rates of ∼100 fps, while sCMOS cameras can achieve several hundred frames per second. Although many advances have been made in wide-field detectors, the temporal resolution of these cameras cannot cover the entire time scale for single-molecule studies (Figure 2). To overcome these limitations, novel photon detectors have been designed in the past few years, such as single-photon avalanche photodiode detector (SPAD) arrays to simultaneously achieve large data throughput and fast frame rates up to several thousand frames per second.77,79,81,82 Recent reviews of the latest detector technologies discuss their applications, advantages, and disadvantages in detail.79,81

Moreover, not only are engineers making scientific cameras more sensitive and with faster frame rates, scientists are also improving the excitation and detection geometries to achieve an effectively faster temporal resolution. We will discuss some of the recent progress in pushing the temporal resolution of single-molecule techniques in section 4.4. Although advancements in SPT techniques have evolved in the past three decades with the development of tagging species, improved photon detectors, and more complicated excitationdetection pathways, the goal of SPT has remained unchanged: to understand the motion of single particles or molecules in various environments and to extract mechanistic descriptions of their dynamics. With modern advances, SPT can achieve tens of microseconds temporal resolution with down to 1 nm spatial resolution.83 In comparison to imaging macroscale objects, imaging a nanoscale tagged analyte under an optical microscope presents unique challenges. Because of the diffraction of light, one cannot resolve any structures smaller than ∼250 nm with light from the visible spectrum using traditional microscopy methods. In the past decade, several methods have been developed to overcome this diffraction limit. Although they are all termed as “super-resolution localization” or “sub-diffraction imaging”, they fall into two different categories. The first type, represented by techniques such as stimulated emission depletion (STED) microscopy,84,85 ground state depletion (GSD) microscopy,11,86 and reversible saturable optical linear fluorescence transitions (RESOLFT),87,88 are deterministic methods where the emission signals from point-like objects are directly modulated. In STED, for example, a red-shifted donut-shaped depletion laser pulse forces excited fluorophores into the ground state; therefore, spontaneous fluorescence emission cannot occur in the periphery of the excitation spot. Implementation of this depletion laser beam allows the residual fluorescent region to be much smaller as compared to the original diffraction limited size, and ∼30 nm lateral resolutions are achieved. The other category, known as stochastic functional techniques and represented by stochastic optical reconstruction microscopy (STORM)12 and photoactivated localization microscopy (PALM),10,52,89 use mathematical analyses to push the resolution limit. The diffraction-limited image of a single fluorophore or any point-like emitter can be analyzed to determine the emitter position if a sufficient number of photons are collected. 2D STORM and PALM usually provide lateral resolution between 20 and 40 nm. These 2D super-resolution techniques were later modified for 3D imaging as well.90−95 Welsher and Yang summarize the spatial and temporal resolutions for the commonly used superresolution techniques (Figure 3) in their recent publication.83 7333


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geometries for SPT, and the representative methods to achieve 2D superlocalization. We also present SPT studies at inorganic and synthetic polymeric surfaces in this section to demonstrate this technique being applicable to a wide range of systems. Sections 4 and 5 further extend SPT to 3D superlocalization. In section 4, we cover 3D techniques including multifocal plane microscopy, astigmatism based microscopy, and phase modulation-based techniques. Finally, in section 5 we introduce active feedback tracking and highlight the technique with highest spatiotemporal resolution to date.

We will discuss some of these methods as they relate to 2D and 3D SPT in later sections.

2. THEORETICAL FOUNDATIONS FOR SUPERLOCALIZATION AND SPT 2.1. Optical Diffraction Limit

The diffraction of light limits the resolution of a conventional microscope such that a point-like emitter is imaged instead as a blur known as the point spread function (PSF). For an optical microscope, the diffraction limit is around 250 nm, defined by λ d = 2NA , where λ is the wavelength of light and NA is the numerical aperture. It is the overlapping of PSFs from emitters spaced closer together than the diffraction limit that defines the resolution of traditional fluorescence microscopy. One of the primary goals of modern SPT is to localize individual emitters with subdiffraction limited resolution. While techniques exist to localize emitters with overlapping PSFs,99,100 most SPT experiments are based on imaging and subsequently localizing distinctly separated PSFs. Localization techniques exist that do not require a PSF model, but for fitting-based techniques prior knowledge of the PSF is necessary. The PSF is the impulse response of the optical system, and can be experimentally determined by imaging a single point source of light. Determining the PSF theoretically requires the derivation of equations to describe the shape of the PSF. High precision models such as the Richards− Wolf model101 and the Gibson−Lanni102 model are rigorous methods to calculate highly accurate PSFs, but they involve evaluating integrals. The Airy disk PSF (eq 1, Figure 4A) describes the intensity at point (x,y) for the paraxial wide-field fluorescence microscopy PSF, where J1(x) is the first-order 2π Bessel function of the first kind and kem = λ for emission

Figure 3. Spatiotemporal resolutions of various imaging methods. The left bound of each rectangle stands for the best spatial resolution for that particular method, while the right bound stands for the largest applicable scale. Reprinted with permission from ref 83. Copyright 2015 Royal Society of Chemistry.

There are a few previous reviews on the subject of SPT.35,36,39,71,96−98 Most of these reviews focus on the optical implementation to achieve single particle detection, and applications in biophysics. In this review, we aim at providing the readers a broad view of SPT, from the theoretical foundations to the recent developments in optical instrumentations and analytical methodologies, and finally discuss the application of SPT in a variety of different systems including living cells, as well as organic and inorganic surfaces. We start our discussion by presenting an overview of the theoretical analysis and modeling for superlocalization and the various ways to achieve SPT in section 2. This section also introduces some of the commonly used data analyses in SPT, such as extracting mean square displacement (MSD) and calculating the corresponding diffusion coefficients. The important ergodic assumption and some nonergodic examples are also discussed in this section. In section 3, we focus on 2D SPT techniques and their applications. We introduce a range of excitation

em

wavelength λem and is a valid representation in many cases.103−105

Figure 4. Comparison of the Airy disk (A) and Gaussian (B) point spread function models. (C) Comparison of cross section of a pixelated image, along with best fit Gaussian and Airy models. (D) Gaussian and Airy curves as in (C), but with a logarithmic scale. While the Gaussian approximation is valid near the peak, it fails to capture the ring structure of the Airy disk. (C) and (D) reprinted with permission from ref 106. Copyright 2014 Nature Publishing Group. 7334


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⎡ ⎤2 J1(kem NA x 2 + y 2 ) ⎥ ⎢ = 2 ⎢ k NA x 2 + y 2 ⎥ ⎣ ⎦ em

example, an isotropic point emitter at the front focal plane is described by a delta function. Its Fourier transform is constant valued at all points in space,109 equivalent to an infinite plane wave. When simulating experimental situations, the aperture functions as a low-pass filter, yielding the circle function shown as a discrete representation in Figure 5A. The Fourier transform

(1)

In general, the equations that most precisely describe the PSF can be prohibitively complex, but in most fitting-based single-molecule localization algorithms the approximate Gaussian PSF model (eq 2, Figure 4B) suffices. Small and Stahlheber describe the reasoning behind the use of the Gaussian approximation in their review “Fluorophore localization algorithms for super-resolution microscopy”.106 Fitting the Airy PSF shown in Figure 4A can be computationally demanding, and the improved speed of the Gaussian PSF (shown in eq 2 and Figure 4B), with Gaussian standard deviation σxy and amplitude A, is appropriate in most experimental situations,104,106,107 particularly for freely rotating emitters.108 ⎛ x2 + y2 ⎞ ⎟ PSFGaussian = A exp⎜⎜ − 2σxy2 ⎟⎠ ⎝

(2)

Parts C and D in Figure 4 compare the fits for the Gaussian and Airy PSF models for a cross section of pixelated data.106 While the Airy and Gaussian PSFs appear very close, the logscale plot in Figure 4D shows that the Gaussian PSF fails to capture the effect of the ring structure of the Airy PSF. Since most of the intensity of the PSF for an isotropic emitter is contained in the bright central lobe, the Gaussian PSF is an appropriate approximation for the 2D PSF. In fact, for high NA, a Gaussian model with background is a better approximation than the Airy disk PSF.108 2.2. Fourier Optics Provide a Tool To Predict Point Spread Functions

Figure 5. (A) Discrete representation of a plane wave truncated by the aperture. (B) Fourier transform of (A), resulting in a diffractionlimited jinc function, rather than a delta function. The magnitude squared of the jinc function is the Airy disk PSF.

Fourier optics techniques provide a method by which researchers can understand the propagation of light as it passes through an optical system, important for purposes such as simulating the optical response of a specific system and engineering PSFs for 3D or supertemporal resolution microscopy (see section 4.4). Fourier optics is based on the paraxial limit of the angular spectrum method, and uses a spatial Fourier transform to represent an optical wave as a superposition of plane waves with different wave vectors.109,110 The continuous spatial Fourier transform in 2D, F( f x,f y), of a function f(x,y) is given by

of the truncated plane wave is the function shown in Figure 5B, called by names such as besinc, jinc, or sombrero function. The squared magnitude of this function represents the PSFthe classic Airy diskdisplaying a finite width due to the diffraction of light about the aperture. When observing an input located at a distance d in front of a thin lens, the optical wave at the back focal plane can be written as

∞

F(fx , f y ) =

∫ ∫−∞ f (x , y)e−2π i(f x+f y) dx dy x

y

(3)

g (x , y ) =

where x and y are the traditional spatial Cartesian coordinates, x′ and y′ are coordinates of spatial frequency, and i = −1 , while the analogous discrete transformation is given by ∞

F(x′, y′) =

(5)

where f is the focal length, and ⎛x y⎞ F ⎜ , ⎟ = -{f (x′, y′)} = ⎝ λf λf ⎠

∞

∑ ∑ f (x , y)e−2π i(x ′ x + y ′ y) −∞ −∞

⎛ (f − d)(x 2 + y 2 ) ⎞ ⎛ x y ⎞ A ⎟F ⎜ , ⎟ exp⎜iπ iλf λf 2 ⎠ ⎝ λf λf ⎠ ⎝

(4)

⎛ xx′ + yy′ ⎞ exp⎜ −2π i ⎟ dx′ dy′ λf ⎠ ⎝

for the discrete valued function f(x,y). Conceptually, a 2D spatial Fourier transform switches between spatial coordinates and spatial frequency for an optical wave.111 With the angular spectrum representation, an optical system is reduced to a series of phase shifts, convolutions, and Fourier transforms.109 Interestingly, a thin lens performs an analog 2D spatial Fourier transform on incident light. Specifically, the optical wave at the image plane (back focal plane) is the Fourier transform of the optical wave at the front focal plane.109 As an

∞

∫ ∫−∞ f (x′, y′) (6)

is the Fourier transform of the optical wave at the input plane. When d = f such that the input is at the front focal plane, as is the case for a 4f system used for engineering 3D PSFs as described in section 4.3, the relationship between g(x,y) and f(x′,y′) is an exact Fourier transform.109 When the input source is not located in the focal plane, the quadratic factor in eq 6 7335


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Figure 6. Simulated diffraction limited PSF recorded in images with different magnifications. (A) Optical plane wave with uniform phase pattern, truncated by the lens aperture, approximated with a 200 × 200 pixel image. Using a discrete 2D Fourier transform and specifying different N, where N is the number of pixels across one side of the final image, will generate images with different pixel sizes (B−E). Shared color map represents normalized intensity.

remainder of the work, the localization precision of an estimator will refer to the standard deviation, such that the fundamental limit is the square root of the CRLB. The CRLB is calculated by inverting the Fisher information matrix, var(θ̂) ≥ I−1(θ). As shown by Ober et al., the localization precision varies as the inverse square root of the number of detected photons. 105 When calculating the fundamental limit on precision for localization with an Airy

results in more complicated defocused PSFs that can be used to engineer PSFs as discussed in section 4.3.110 Discrete Fourier transforms such as the fast Fourier transform (FFT) can be used to generate the pixelated images that will be formed in a diffraction-limited system, useful for testing localization methods with ground-truth simulated data. Relating the formulas for discrete and continuous Fourier mλ transformation reveals the relationship p2 = 2m 1NA , where p2 is 2

the size of square pixels capturing the image g(x,y) and m1 (m2) is the number of pixels in the image f(x′,y′) (g(x,y)). Figure 6 shows an aperture-restricted plane wave (A) and the resultant FFT with different numbers of pixels in the transform (B−E).

PSF, Ober et al. obtain δx = δy =

λem , 2π NA Nphoton

where λem is the

wavelength of emitted light, NA is the numerical aperture, and Nphoton is the number of detected photons. This fundamental limit cannot generally be reached in experiment, as it is based on knowledge of photon arrival positions.105 Fisher information can also be used to determine the optimum magnification for single particle experiments by taking into account the effects of pixelation. Assuming the number of photons collected by each pixel of the detector follows a Poisson distribution with Poisson noise, the Fisher information matrix is given by

2.3. Fisher Information and the Cramér−Rao Lower Bound

Before discussing the estimators that are used in the localization step of SPT algorithms, it is important to understand the concepts of estimator accuracy and precision. When estimating a parameter based on collected data, accuracy and precision describe the bias and standard deviation of the estimator, respectively. The bias of an estimator is b(θ̂) = E[θ̂] − θ, the difference between the expected value of the estimator and the actual value of the parameter being estimated. In practice, the expected value of the estimator is approximated by averaging the estimated values obtained from a large number of trials. An unbiased estimator is preferable to a biased estimator. The standard deviation of an estimator is the square root of the variance, and describes how close multiple estimations of the same parameter will be to each other. A high precision estimator will have low standard deviation, which means that the results will be closely clustered together. The lower limit on the variance of an estimator is given by the Cramér−Rao lower bound (CRLB).105,112,113 Determining the variance and bias of an estimator requires simulated data with known ground truth. Most commonly used estimators are unbiased, but the variance of an estimator can differ greatly, and has been studied in a number of works.104,107,114 For this reason, most development of estimators focuses on minimizing the variance. However, even given an isolated, well-behaved point spread function, there is a fundamental limit on the precision of any localization estimate, the CRLB. In the

Fisherij =

⎡⎛

⎞

∑ E⎢⎜⎝ ∂ ln(Poiss(Ik + λ))⎟⎠ k

⎢⎣ ∂i

⎛∂ ⎞⎤ ⎜ ln(Poiss(Ik + λ))⎟⎥ ⎝ ∂j ⎠⎥⎦

for i , j = x , y (7)

where Poiss(Ik + λ) is the Poisson random variable, Ik is the expected number of emitted photons collected by pixel k, and λ is the average number of photons related to Poisson noise in each pixel. Ik is calculated by integrating the PSF over pixel k, which is dependent on magnification and pixel size. When using a symmetric Gaussian PSF, Fisherxy = Fisheryx = 0, and the CRLB can be determined by inverting the diagonal elements separately. Therefore, to determine the localization accuracy of the x-coordinate estimate of the center of the PSF 7336


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Figure 7. Illustration of particle localization by least-squares fitting to a Gaussian PSF. (A) Simulated image of an isotropic emitter located at the position of the red cross. (B) Associated histogram of photon counts. (C, D) Results of a weighted least-squares estimate of a Gaussian PSF model to the data in (A), with estimated localization located at the blue dot. 2 ⎡⎛ n ⎞⎞ ⎤ ∂ ⎛ (Ik + λ) ⎢ Fisherxx = ∑ E ⎜ ln⎜ exp( −(Ik + λ))⎟⎟ ⎥ ⎢⎣⎝ ∂x ⎝ ⎠⎠ ⎥⎦ n! k

⎛ ∂Ik ⎞2 1 = ∑⎜ ⎟ ⎝ ⎠ ∂ + λ) x ( I k k

computationally slow compared to nonfitting methods. However, they can provide higher precision and often give more information about the emission event, such as the width of the PSF.119 The most straightforward localization method is to calculate the intensity weighted centroid of the selected region (center of mass). Although calculating the centroid is a very fast, singleiteration method, the results are sensitive to noise, the relative position of the PSF in the selected region, and even the relative position in the pixel.114,119−121 However, due to the advantage in speed, centroid estimation is still useful to estimate particle positions in active tracking to provide instantaneous feedback.122 The most commonly used fitting methods are least-squares fitting (LS) and maximum likelihood estimation (MLE). A number of methods exist for solving these optimization problems, requiring the experimenter to carefully develop the objective function.104 Though only an approximate representation of the PSF, the Gaussian model is often used in these algorithms.106,107,123 Least-squares fitting with a Gaussian PSF model is a standard method to find the center of a PSF in single-molecule fields.107,114,124 The least-squares fitting method is an iterative fitting-based localization method that requires little knowledge of camera noise and is less sensitive to point spread function model misspecification, though it lacks precision in low photon count situations.104,106 An illustration of a weighted leastsquares localization algorithm is shown in Figure 7. The main principle is to search for the parameters that minimize the weighted square error between the fit and the data. It can be written as

(8)

and std(x) ≥

CRB =

1 Fisherxx

(9)

The theoretical limit on localization precision should be determined for a given experimental setup in order to test the performance of the estimator. The parameters of the experiment should be set such that the localization precision for the estimator approaches the theoretical limit to ensure precise localizations. Abraham et al. also created software packages to estimate the theoretical limitation of localization accuracy under certain experimental conditions based on Fisher information.104 2.4. Localization Methods

The choice of localization method for a given experiment involves a trade-off between computational cost, desired precision, and knowledge of noise or PSF statistics prior to fitting. Without a valid localization method, only rudimentary or correlation based analysis methods will work. In general, SPT requires a low density of emitters and high purification of the solution.115 One can track single particles with precision beyond the diffraction limit because of the prior knowledge that each observed PSF is the image of a single emitter. Under this assumption, one can resolve the position of the particle more than 10 times better than the diffraction limit by finding the center of the isolated PSF.116,117 Many different methods have been developed to find the center of the isolated PSF, and can be broadly categorized by whether or not they fit a PSF model.106,118 Usually, PSF fitting algorithms are iterative and

K

arg min ∑ θ

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k=1

[xk − fk (θ)]2 σk 2

(10) DOI: 10.1021/acs.chemrev.6b00815 Chem. Rev. 2017, 117, 7331−7376

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Figure 8. Working principle of the radial symmetry method. (A) Simulated recorded PSF with shot noise. The true center of the PSF is labeled by the red “×”. This image is generated from simulated Airy disk PSF with much smaller pixel size without noise (B). (C) Gradient of the intensity at each pixel corner is calculated using intensities of four neighbor pixels based on the image in (A). The gradients are represented by orange arrows. Yellow lines are extensions of the arrows to illustrate how to find the center of the PSF using all the gradients. Circles are the pixel centers. (D) The center calculated by radial symmetry method (orange circle) matches with the true center (red “×”). The center is calculated by finding the point of minimal overall distance to all the yellow lines, as shown in (C). Reprinted with permission from ref 119. Copyright 2012 Nature Publishing Group.

where εxy is the noise. All the unknown parameters in the Gaussian function are contained in a1−5. However, this approximation ignores noise and assumes the background is removed. When the noise level is large noise cannot be simply ignored, and when the local particle density is high the background is difficult to estimate and remove. Therefore, this fitting method can replace Gaussian fitting only when the noise level is small and local particle density is low.127 The most time-consuming part of PSF fitting is the iterative process. A variety of nonfitting methods have been developed to decrease computational time without decreasing resolution.119,128 The previously mentioned centroid method has been adapted to a number of fast algorithms;121,129 the fluoroBancroft algorithm uses the principle of triangulation130 and the Fourier domain localization algorithm uses fast Fourier transforms.131 The radial symmetry method is one of the more promising methods.119 The radial symmetry method calculates the center of the PSF through one-time linear algebra calculation.119 As illustrated in Figure 8, first the gradient of the intensity at each pixel corner is calculated, which represents a direction in the image. This direction and the corner position (xk,yk) give a line:

for measured pixel intensity values xk, k = 1, ..., K, expected variance in pixel k σk2, and prediction of the intensity of pixel k f k(θ), based on the PSF model with θ as the parameters to be estimated. The parameters that are required depend on the PSF model being usedfor the Gaussian PSF model the relevant parameters are the Gaussian width, usually defined either in terms of the standard deviation σ, or in terms of full-width at half-maximum (fwhm = 2 2 ln 2 σ ) of the Gaussian profile; the mean, or center, position in Cartesian coordinates, (x0,y0); the amplitude (peak intensity of the Gaussian); and the background noise parameters. The initial guess of these parameters will dramatically influence the performance of the fitting. Usually, the initial guess of the Gaussian width for one instrument should always be the same, which is related to the fwhm of the diffraction limited PSF: an initial guess of (x0,y0) will take the location of the pixel with peak intensity, the peak intensity minus the background as the initial amplitude, and use the minimum intensity as the initial background. Least-squares fitting is accurate and robust, but is often the most timeconsuming part of an SPT algorithm.119,121,125,126 Maximum likelihood estimation (MLE) is an iterative fittingbased localization method that reaches the CRLB (if any estimator can)104,105,126 and requires detailed statistical knowledge of the PSF and the noise.106 Rather than fitting the collected intensity data to the PSF model, MLE calculates the likeliest set of parameters to generate the observed data. Except for a few cases where the MLE can be calculated analytically,105 the MLE estimator is found by calculating the likelihood for a set of parameters and then iteratively updating the parameters to improve the likelihood until the algorithm converges. The MLE is especially noteworthy due to the fact that it reaches the CRLB if it is possible for any estimator to reach it. For this reason, MLE tends to be more precise than least-squares estimation,126 though it has been shown that, at high noise levels, both methods approach the CRLB.104,119 Anthony and Granick noted that taking the log of a Gaussian function to generate a polynomial function allows 2D Gaussian to be simplified as polynomial fitting, which is much faster:127

y = yk + mk (x − xk)

where mk is the slope of this line. Notice this slope can be infinite for vertical lines. In the program, the author replaces the infinite with a large number. The distance between the center of the PSF (xc,yc) to this line is dk =

2

≈ a1x + a 2x + a3y + a4y + a5

(y − yk ) − mk (x − xk) 1 + mk 2

(13)

This center should minimize the weighted overall distance ∑k dk 2wk , which is the point where the derivatives of this objective function over xc and yc are zeros. This condition gives us two independent equations with two unknowns (xc and yc):119 xc ∑ k

⎛ ⎞ ⎡ ⎛ (y − y0 )2 ⎞⎤ (x − x0)2 ⎟⎥ + εxy⎟ ln(Ixy) = ln⎜A exp⎢ −⎜⎜ + ⎜ ⎟ ⎢⎣ ⎝ 2σx 2 2σy 2 ⎟⎠⎥⎦ ⎝ ⎠ 2

(12)

−mk 2wk 2

mk + 1

+ yc

∑ k

mk wk 2

mk + 1

=

∑ k

mk wk(yk − mk xk) mk 2 + 1 (14)

xc ∑ k

(11)

−mk wk 2

mk + 1

+ yc

∑ k

wk 2

mk + 1

=

∑ k

wk(yk − mk xk) mk 2 + 1 (15)

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The weight wk is proportional to the magnitude of the gradient and the inverse of the distance to the center at the corner position (xk,yk). Knowing wk assumes that we already know the center. Fortunately, the radial symmetry method is not very sensitive to the weight. Using the estimated center based on the centroid of the magnitude of the gradient is good enough for the calculation. The radial symmetry method has been shown to be as accurate and robust as Gaussian fitting, but 100 times faster.119 2.5. Simulating Experimental Data

While the precision of a localization method can be estimated from repeated experimental trials, testing with simulated data is an important step in determining the localization bias and precision. Testing with experimental data requires accurate and precise prior knowledge of the position and orientation of single particles,106 which is not often feasible. Simulated images are a benchmark used for most localization108,126,131 and dynamics-tracking algorithms,125,132−136 though some tests neglect this step.116,130 Photon counts in each pixel are usually simulated by a Poisson random variable with a Poisson parameter set by the PSF, which can be generated using Fourier optics as discussed in section 2.2 or obtained through integration of a theoretical PSF model over the pixel region. This description of photon arrivals as a Poisson random variable is known as “shot noise”. Shot noise is due to the uneven distribution of photons,106 where each photon arrives at a position drawn from the PSF (normalized as a probability density function), and will be present even under ideal circumstances with no scattered photons or detector noise. Due to the discrete nature of pixels, the map of Poisson parameters (equivalent to the pixelated PSF) encodes subpixel information as asymmetry. Since localization procedures can favor certain subpixel positions,120 the PSF used as the Poisson parameter should be generated with random center positions across the area of a pixel. Background noise is modeled in different ways depending on the number of collected photons and the properties of the camera. The background noise varies based mainly on the camera and temperature of the sensor array.120 If the noise distribution of the camera is determined experimentally, noise intensity values for each pixel can be chosen directly from the distribution.120 In most circumstances, noise is modeled as Gaussian or Poisson. Approximations often use additive Gaussian white noise,137 though for more precise simulations Poisson noise should be used to model scattered photons, dark current on the CCD chip, and autofluorescence, while the Gaussian model is more appropriate for noise arising from the CCD camera readout.105 An example of simulated image generation is shown in Figure 9.

Figure 9. Example of image simulation. (A) Pixelated Airy disk PSF used as the parameters for the Poisson random variable to generate the shot noise image in (B). (B−D) Noise elements for simulated image. (B) Shot noise represents the statistical nature of the arrival of discrete photons. (C) Background Poisson noise uses a constant Poisson parameter, while (D) Gaussian noise is a zero-mean, constant standard deviation random variable for each pixel. By randomly generating the three noise distributions, multiple simulated images can be generated for the same pixelated PSF. (E) Combined image with the three noise sources to generate a single simulated image.

stand established methods, and to combine modules with different algorithms for a customized SPT solution. The SPT competition published in 2014133,138 considered 14 different algorithms covering 48 different situations in an attempt to provide an objective comparison. While other comparisons of SPT methods had been previously published,114,139−142 these studies were limited in scope,133 and the need for a broader comparison was recognized.37 The SPT competition, as discussed by Chenouard et al., provided training data with ground truth for four dynamics scenarios (Brownian motion, directed motion, and random switching with either random or fixed directed motion components), three different particle density levels, and four different SNR levels. Figure 10 shows examples of the four different dynamic scenarios included in the competition. Ideally, having different research groups implementing their own algorithm on a common data set with standard evaluation metrics provides a more objective comparison than previous studies.133 Even though not all the algorithms were applied to all the situations, most of the algorithms were designed for broad applications. Importantly, it was concluded that no single algorithm performed best for all situations simulated, which means it is better to narrow down the target application when choosing or designing an SPT algorithm.138 Moreover, real experimental situations are usually more complicated than those presented in the competition. For example, the competition did not consider uneven background, which is usually unavoidable due to astigmatism and tracking in complicated environments. However, this competition provides a library for future

2.6. SPT Methods

SPT algorithm development is a relatively mature field but still has room to grow.35 SPT algorithms can usually be classified into several independent modules: signal-to-noise ratio (SNR) enhancement, particle identification and localization, particle tracking, and post dynamic/kinetic analysis.125 Some published methods are specially focused on one module, such as the fitting methods discussed in section 2.4, and the denoising methods used in more general imaging processing.137 This classification into modules also matches the recent SPT competition, as shown in Figure 10.133,138 Of course, this is only a rough grouping. For example, particle density levels will influence both particle identification and particle tracking. But a simplified classification will help researchers to better under7339


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Figure 10. Simulated SPT data for different experimental situations for the four transport dynamics used in the 2014 SPT competition. For each scenario, a typical snapshot image (i−iv) and analyzed trajectories over multiple images (v−viii) are shown. Colors are randomly assigned to different trajectories. Scenario 1: Brownian diffusion imaged by wide-field microscopy (i, v). Scenario 2: Directed movement (ii, vi). Scenario 3: Randomly switching between Brownian diffusion and directed movement with random directions imaged by confocal microscopy (iii, vii). Scenario 4: Randomly switching between Brownian diffusion and directed movement with restricted directions imaged by 3D scanning confocal microscopy (iv, viii). Not shown here are the four SNR levels and three density levels that were also used for simulating data. Reprinted with permission from ref 133. Copyright 2014 Nature Publishing Group.

linking133). Key concepts for SPT by filtering are the state vector xt, which describes the particle, and the posterior distribution of the state vector p(xt|y1:t), which represents how likely the particle is to be in a certain state given a series of measurements yt.142 The measurements are obtained by topdown and bottom-up methods: top-down measurements are taken from the predicted position x̂t and surrounding points from the elliptical validation region defined with the innovation covariance matrix, while bottom-up measurements are based on deterministic localizations from the image, assigned to the appropriate filter through a global nearest-neighbor scheme.133,143 In this application, localization was carried out by using the spot enhancing filter to enhance Gaussian-like particles142,144 and intensity-weighted centroid or Gaussian fitting,133 though other deterministic localization techniques could be used. Each tracked particle has its own filter, so determination of particle position automatically links positions together into trajectories, rather than separating localization and tracking into separate modules.143 The filter computes the particle position from multiple measurements along with their associated weights, given by the probability that each measurement generates a Gaussian spot when compared to measured pixel intensities.143 In order to track particles in close proximity, a reweighting term is applied by penalizing positions with high support for neighboring objects.133,143 SGC, submitted to the SPT competition by I. F. Sbalzarini, Y. Gong, and J. Cardinale, utilizes weighted centroid localization with combinatorial optimization for path linking over several frames for computationally fast tracking and robust performance across a wide SNR range.133 One key difference between SGC and GR (described above) is that SGC requires no prior knowledge about the motion of the particle, only requiring assumptions about the size of the particle image, about the limit on particle speed, and that particle disappearances occur on short time scales.133,145 Another difference is that it is fully a deterministic SPT method, and includes particle localization and tracking as independent modules. Image restoration is performed by convolution of

researchers to search for the method they need for a particular situation. This is extremely important for both 2D and 3D tracking, considering all the proposed tracking setups are different. The results of the SPT competition show that “the quest for better particle tracking methods remains,” and provides a baseline for researchers to use when pursuing them.133 Due to the modular nature of SPT, different portions of an SPT algorithm can often be swapped out,142 so algorithms can be generated by adapting methods that have already been developed. While a complete summary of all the algorithms in use today is beyond the scope of this work, a few representative algorithms are detailed below. The first two algorithms described are from the SPT competition: the method submitted by W. J. Godinez and K. Rohr (GR) and the method submitted by I. F. Sbalzarini, Y. Gong, and J. Cardinale (SGC). GR was considered the most accurate overall when counting the number of times it landed in the top three for different scenarios, though this metric favors methods that submitted results for all scenarios. SGC was the second most accurate by this ranking, as well as being the fastest algorithm based on computation time. It is difficult to compare even two methods across all metrics, but in general, at the lowest SNR level GR better determined the particle velocities, while SGC better determined the particle localizations. This may be due to the iterative localization technique utilized by SGC.133 Notably, GR has superior localization accuracy at low SNR for the case of diffusion with directed motion, which may be due to its use of a theoretical dynamic model in execution. GR, submitted to the SPT competition by W. J. Godinez and K. Rohr, is a probabilistic SPF algorithm that utilizes Kalman or interacting multiple model (IMM) filters with multiple measurements.133,143 Probabilistic approaches, which generally include a filtering step,142 provide a means to handle the uncertainty of the measurement and movement of particles.143 They make up an important class of SPT algorithms, and operate in a fundamentally different way from more traditional deterministic algorithms (of the 14 algorithms in the SPT competition, only two of them used filtering for particle 7340


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Figure 11. (A) Representative m0, m2 plane where potential particles designated as true particles are shown as crosses and rejected particles are shown as open circles. (B) Confocal image from virus tracking experiment with reversed image intensities. The crosses represent the location of particles accepted during the discrimination step; larger or darker spots are internalized virus particles that are excluded from tracking. Inset provides detail in the indicated region. Reprinted with permission from ref 145. Copyright 2005 Elsevier.

Figure 12. Detection strategy for high-density SPT. (A) Hypothesis test to determine if the selected region contains any PSF. (B) Exhaustive particle detection process. The first iteration detects and fits the most intense PSF. Subtracting the first fit allows the weaker PSF to be detected in the next iteration. (C) Detection strategy applied to an experimental image. The left column shows images after last deflation in each iteration. The middle column shows identified peaks in each iteration. The right column shows the fitting images used for deflation in each iteration. Reprinted with permission from ref 147. Copyright 2008 Nature Publishing Group.

particles will be in close proximity.133 Local maxima are used to initially estimate particle localizations and are refined through iterative weighted centroid calculation. Nonparticle discrimination is implemented through an algorithm based on the zero

frames with a kernel combining background removal and noise removal. The algorithm operates under the assumption that particles will overlap over the course of their trajectory, which could limit the algorithm’s usefulness in scenarios where 7341


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and second intensity moments about each potential particle.145,146 The zero and second intensity moments are given by, respectively m0 =

Atf (x

∑

MSD(n δt ) =

N−1−n

∑ + i , y + j)

i2 + j2 ≤ w2

1 m0

∑

(18)

where r(t) is the position of the particle at time t, n δt is the time lag MSD is calculated at, and δt is the smallest time interval resolved.148,149 MSD is calculated for each trajectory to estimate the population MSD as a function of time. The time interval δt between each position measurement defines the discrete steps of the MSD. Calculation of MSD for small time lags includes more trajectories that contribute to the average, increasing precision.132 In practice, the maximum value of n δt N for which MSD is calculated is bounded, such as n < 10 ,149 to avoid the imprecise estimates for larger time lags. The increasing overlap of trajectories for increasing time lag also complicates estimation of MSD due to the dependence of consecutive trajectories,132 though other definitions of MSD which only include nonoverlapping trajectories exist to address this.148 Diffusion is an inherently random process, and as such any calculation of MSD and diffusion coefficient will have a statistical variance. The diffusion coefficient and other applicable parameters can be determined by fitting the MSD vs time plot to an appropriate diffusion model, which will be detailed in section 2.8. For an overview of some common diffusion models and their functional form, see ref 150. Brownian diffusion, directed diffusion, and anomalous subdiffusion are described briefly below. For Brownian diffusion, the diffusion coefficient is constant and can be determined from the slope of a linear fit to MSD vs time using the Einstein relationship, MSD(t) = 4Dt, for diffusion in 2D. Brownian diffusion is a valid approximation when diffusion is unhindered and is occurring in a sufficiently large space. Specifically, the Brownian diffusion model is appropriate when the total measurement time is much less than L2/4D for the characteristic length of the space L. When diffusion is confined to a smaller region, the limit of MSD as t → ∞ for 2D diffusion is proportional to the area of confinement.148,150 It is important to be aware of the pitfalls when visually inspecting plots of MSD vs time for a small number of trajectories. Due to the inherent randomness in a random walk, a particle undergoing Brownian diffusion can easily appear to undergo confined diffusion or directed diffusion over certain periods.148,150 Detection methods with a proper null hypothesis of ideal diffusion can be employed to determine if an alternate type of motion is present.151 Directed diffusion, or diffusion with flow, alters the transition probability such that the MSD will be quadratic with respect to time, given by

(i 2 + j 2 )Atf (x + i , y + j)

i2 + j2 ≤ w2

[r((j + n) δt ) − r(j δt )]2

j=1

(16)

and m2 =

1 N−1−n

(17)

where i and j are pixel indices, x and y are the indices of the candidate pixel, and Atf is the filtered frame at time t.145 When viewing the m0, m2 plot of all the potential particles, clustering will occur based on their intensity distributions. Each potential particle is assigned a 2D Gaussian profile in the m0, m2 plane, and each point is then assigned a score which is the value at its location for the summation of all of the rest of the Gaussian profiles.145 Assuming that the majority are true particles, the true particles are identified by scoring over a user-defined threshold, indicating that they are the most tightly clustered. Figure 11 A shows an example of clustering in the m0, m2 plane. Trajectory linking is carried out through a combinatorial optimization problem, though other methods could be substituted. For each frame there is an association matrix that defines linking between particles. The association matrix is restricted such that each particle is assigned to exactly one particle or to a dummy particle in the next frame. By calculating particle linking across multiple frames, the algorithm can deal with temporary disappearance of particles. Initial particle linking is determined by nearest neighbors, and a greedy hillclimbing algorithm is used to find the configuration of the association matrix that minimizes a cost function based on location, total intensity, and the second intensity moment. A technique employed by this method is assigning an infinite cost to particles that are separated by a set cutoff distancethis reduces computation time by eliminating the need for extraneous calculations.145 One of the SPT algorithms not included in the SPT competition is Sergé’s multiple-target tracing algorithm.147 Sergé et al. applied hypothesis tests in particle detection and particle tracking to improve the sensitivity of the algorithm, especially for high-density tracking situations with probes having a large distribution of brightness, as shown in Figure 12. Their algorithm first identifies all the local intensity maximums and then applies a statistical test to each selected region to determine if there are PSFs in the region (Figure 12A). Once they determine there are PSFs in the region, they fit the PSF and subtract the fit from the image. After that, they repeat the same process (hypothesis test, PSF fitting, PSF subtraction) on the leftover image until no PSF can be found in the image (Figure 12B,C). This is a great strategy to analyze high-density SPT images; however, overfitting problems need to be considered for uneven background or misspecified PSFs.

MSD(t ) = 4Dt + V 2t 2

(19)

where V is the constant velocity flow rate for directed diffusion in 2D. Plotting MSD as a function of time will result in a quadratic curve, with the effect of the flow rate dominating for larger values of t, while the effect of diffusion dominates for small values of t. If the motion is identified as directed diffusion, curve-fitting the plot of MSD vs time can be used to estimate D and V.152,153 Anomalous subdiffusion can occur due to interactions with obstacles that alter diffusion, which can be observed as an MSD

2.7. Mean Square Displacement (MSD) Analysis

The diffusion coefficient D is one of the fundamental properties used to describe particle motion in SPT. A number of methods have been developed to determine the diffusion coefficient, with MSD analysis as one of the fundamental tools. MSD can be calculated a number of ways, but a common definition is 7342


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behavior of many others, therefore large statistics is the key for such observations.

plot with a negative curvature. The time-dependent diffusion coefficient for anomalous subdiffusion approaches zero for long times, though certain obstacle structures can result in the diffusion dynamics shifting to hindered diffusion at large time scales.150 This is described by Qian et al.148 as a higher, localized diffusion coefficient that dominates for short time scales, with the long-term diffusion being determined by the effective diffusion coefficient. The localized diffusion coefficient is observed for distances less than the characteristic separation of the mobile or immobile obstacles, since the diffusing particles are effectively “in between” obstacles.

3. REVEALING DYNAMIC PROCESSES USING 2D SPT 3.1. Two-Dimensional Interfacial Dynamics

The most basic way to extract dynamics information from a sample is to directly observe the phenomenon in question. Typically, in fluorescence measurements, this is done using wide-field microscopy. In its most basic implementation, transport is observed as a 2D projection on the sample plane. For this reason, observation of planar samples is common. Because of the limited photon budget when using singlemolecule emitters, it is common to use total internal reflection fluorescence (TIRF) instead of standard wide-field illumination as TIRF limits the excitation volume to a very narrow depth (69° are usually achievable). The incidence angle is higher than the critical angle of the sample−solution interface (typically the interface of glass and water) and thus results in total internal reflection and the formation of an exponentially decaying evanescent wave at the interface.159 Emitted fluorescence is captured by the same objective with filters removing undesired background such as scattered laser light. Fluorophores are imaged by capturing sequential images using a high sensitivity camera. Modern cameras can achieve frame rates in excess of 100 frames per second for 1024 × 1024 pixels. Higher frame rates are achievable if the readout area of the chip is reduced,77,81,160 or alternatively using CMOS based SPAD arrays.77,79,161 TIRF may also be achieved through the use of a prism. To achieve prism-TIRF, a prism is coupled to the side of the sample opposite the objective and the excitation light is coupled into the prism at the desired angle. As in objectiveTIR, an exponentially decaying evanescent wave is generated at the sample interface with fluorescence emission collected by the objective. Prism-TIRF requires a more complicated sample geometry, but does not require the use of an expensive high NA objective.162 After acquisition, image stacks are processed to increase the signal-to-noise ratio of each image, identify fluorescent event locations, and link events across multiple frames. Additional analyses for SPT and super-resolution image processing may also be conducted as described in the following sections. Confocal imaging is also an effective method of imaging single fluorophores with minimal background interference. Confocal imaging involves the raster scanning of a laser focal volume across the sample and detecting the fluorescence signal using a single point detector such as an APD or PMT. Unfortunately, the slow nature of raster scanning makes confocal imaging useful only for systems devoid of dynamics on time scales faster than ∼1 s.163 As discussed in section 2.6, there exist a wide variety of SPT methods and there is no one method that can be singled out as the best due to the wide variety of potential use cases, but objective comparisons of these methods exist and suggest that certain methods work best in the low signal-to-noise regime of single-molecule measurements.114,139−142,164,165 It was found

2.8. Ergodic Hypothesis in SPT

For the discussion of trajectory lengths and true representations of dynamic populations and subpopulations, one must consider the ergodic hypothesis and its applicability to SPT. The ergodic hypothesis states that it is reasonable to produce ensemble MSD values of a system from SPT time averages, given the single particle can be tracked over a long period of time.154 However, the ergodic hypothesis is based on the assumption that the observation time is infinite. In SPT experiments, the observation time must exceed the characteristic time of the diffusion to fulfill the ergodic hypothesis, which is often challenging to achieve given the photobleaching of emitters and the finite recording time of detectors. Consequently, there are a few experimental observations in which the particle diffusion shows nonergodic behavior in intracellular particle transport.155,156 The first nonergodic behavior was reported by Bouchaud et al.157 and then Weigel et al.155 Here we highlight the later one as an example for ergodicity breaking. The observed anomalous diffusion in intracellular transport is usually explained by the nonergodicity of the system. Weigel et al. reported that nonergodic processes exist in the plasma membrane. By tracking GFP-Kv2,1 (labeled by QDs) expressed in embryonic kidney (HEK) cells, they observed anomalous diffusion behavior (Figure 13A), which is indicated by double exponential curve fitting. By comparing the time averages and ensemble averages of the single particle trajectories, it is claimed that the Kv2,1 transport dynamics exhibit nonergodic behavior (Figure 13B). Weigel et al. showed that the breaking of ergodicity can be experimentally noticed only when the behavior of a single particle was compared to the average

Figure 13. Statistical analysis of Kv2.1 channel trajectories. (A) The cumulative distribution function of square displacement of a clustered channel with tlag = 0.1 s. The curve is fit with both single and biexponential distributions, and the biexponential model gives smaller residuals. (B) Distribution of MSD values calculated from individual trajectories and ensembles in clustered channels for tlag = 0.1 s. Adapted with permission from ref 155. Copyright 2011 National Academy of Sciences. 7343


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is not at all representative of classical Brownian diffusion (see section 2.7 for details) and requires a new model to explain.166 Several models of subdiffusion have been developed to explain nanometer scale dynamics. Other research groups have also reported findings similar to those of the Granick group, and various attempts have been made to determine appropriate models to explain the data.167−180 Generally speaking, anomalous diffusion is a nonspecific phenomenon and describes the case when diffusion deviates from expected Fickian behavior. There are of course many causes for this deviation. Activated diffusion is one such case.157 Diffusion in a lipid bilayer system is often explained as lipid transport resulting in fluctuating local heterogeneities resulting in diffusivities that change over time.175 Various other crowding effects can also result in non-Fickian diffusion.173,179,180 For characterizing interfacial diffusion, fractional Brownian motion and the continuous time random walk are often used.181 Because of the generality of anomalous diffusion, it is important that the model used to explain the results has physical justifications. Understanding interfacial transport mechanisms is extremely important for optimizing the function of membranes and chromatographic stationary phase materials. Recently Schwartz and co-workers have used SPT to determine that the primary mechanism for biological transport at stationary phase materials is best classified as desorption mediated diffusion.63,182−187 This desorption mediated diffusion can be modeled as a continuous time random walk, and is yet another potential candidate when anomalous diffusion is observed.166,184 This type of diffusion consists of repeated surface associations punctuated by excursions of the diffusor into the bulk medium. This is schematically represented in Figure 15A. Using single-molecule tracking, the repeated adsorption events from the same emitter can be monitored and appear as hops across the surface as bulk excursions tend to occur faster than the frame rate of the camera being used to observe. When molecules adsorb to the interface they remain for a period of time, known as the waiting time. Distributions of the waiting times for a variety of emitters are shown in Figure 15B. For single-molecule dyes, proteins, and polymer chains, the waiting time distribution for adsorption to a hydrophobic silane interface was found to roughly obey a power law distribution with an exponent of approximately 2.5.184 These results have been replicated for a variety of interfaces and diffusor types. Using SPT, desorption mediated transport

that, generally, methods involving the preprocessing of images prior to localization performed better.114,125,139−142,164 Selection of a proper algorithm along with careful experiment design can allow for new insights using these relatively common methods. Even in the simplest case of 2D transport, the diffusion of a molecule at a solid−liquid or liquid−liquid interface, SPT measurements have recently provided a new understanding of the underlying mechanisms at play. It has been known that interfacial processes are often non-Fickian (their MSD does not increase linearly with time), but it has also recently been demonstrated using SPT that even when a process appears Brownian, it can also exhibit anomalous diffusion.166 Single-molecule tracking analyses from the Granick group have revealed deviations from Fickian diffusion behavior in surprisingly simple systems.166 Fluorescent colloidal beads were measured diffusing along linear phospholipid bilayer tubes and within F-actin filament networks. The data for beads diffusing on linear tubes is shown in Figure 14. The MSD as a function

Figure 14. Demonstration of non-Fickian diffusion with linear MSD. (A) MSD curves (with a slope of 1) for fluorescent particles diffusing on lipid bilayer tubes without (top line) and with (bottom line) 40% cholesterol. Curves are plotted on a log−log scale. (B) Logarithmic plots of the displacement probability plotted against linear displacement normalized by particle diameter for several representative values of time step: 60 ms (squares), 0.6 s (circles), 3 s (crosses), and 5.8 s (triangles). Adapted with permission from ref 166. Copyright 2009 National Academy of Sciences.

of time is shown in Figure 14A. The linear appearance would suggest that the transport process being observed is Fickian. However, a deeper investigation reveals that this is simply not the case. When the normalized displacement distributions, determined from hundreds of trajectories, are examined for different observation times (Figure 14B), a shift is observed from an exponential decay (linear in the plot) to Gaussian. This

Figure 15. (A) Desorption mediated diffusion schematic showing intermittent surface association events. (B) Waiting time distributions showing the waiting time between displacements of >0.2 μm. The data sets were translated vertically to allow easier interpretation (the PEG, Atto6G, and BODIPY data were shifted by a factor of 101, 10−2, and 10−3, respectively). The dashed line represents a curve with a power law exponent of −2.5. Adapted with permission from ref 184. Copyright 2013 American Physical Society. 7344


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with the films as a function of pH likely involves a combination of electrostatics and sterics. Hydrophobic interactions are also expected to play a role.188 These studies of tunable interfacial diffusion were later extended to study the effects of functionalization on desorption mediated diffusion.171 It was found that ligand density determined the density and thus the availability of binding sites for adsorption.

modes have been observed for small molecule organic dyes,171,184,188 polypeptides/proteins,20,189,190 and long chain polymers.183,186 While the initial measurements focused on silanized hydrophobic model surfaces, desorption mediated dynamics have been observed over polymer matrices,170,183 at the oil−water interface,20,190 and functionalized substrates for separations.171 There is also evidence that for certain types of diffusors, namely long chain polymers, in-plane diffusion occurs during the adsorption period.183 Desorption mediated transport is also a key factor in more complicated tunable polymer interfaces. When exposed to a 10 bilayer film composed of alternating layers of poly(allylamine hydrochloride) (PAH) and poly(acrylic acid) (PAA), small molecule ionic probes were found to exhibit desorption mediated dynamics.188 Furthermore, tuning the solution pH post assembly allowed charge dependent tuning of the intermittent surface interactions. The single frame displacement histograms and analyses for this study are summarized in Figure 16. For the anionic probe, Alexa 555, interaction with the

3.2. Imaging 2D Structures with SPT

SPT is particularly useful in the study of diffusion in highly confined environments and when combined with anisotropy calculations can yield subdiffraction limited structural information. Such tracking experiments typically yield one-dimensional trajectories. Since the mid-2000s there have been several studies of confined diffusion in polymer matrixes, liquid crystals, and mesoporous silica.191−195 Diffusors are also expected to be rotationally restricted in such structures.195−197 Pramanik et al. recently reported the first quantitative analysis of lateral and orientational confinement of single molecules in mesoporous silica.191 In order to quantify the dipole emission angle of diffusing N,N′-bis(octyloxypropyl)perylene-3,4,9,10tetracarboxylic diimide (C11OPDI) molecules within the pores, circularly polarized excitation light was used to excite all molecules. Detected fluorescence was split into orthogonal polarizations and imaged on separate halves of a CCD camera. This type of detection scheme is known as single-molecule emission dichroism (SMED).198 The two orthogonal images detected for C11OPDI diffusing in silica mesoporous films with two different templates are shown in Figure 17. The onedimensional pores can be clearly observed as well as the dependence of dipole angle on the pore direction. Accounting for the depolarization of high NA optics (a2) and the average dipole orientation (ϕ) of each molecule, equations relating the emission in each channel to the maximum allowed wobble angle (θmax) within the pore were developed (eqs 20 and 21). IV ∝ cos2 ϕ(1 − cos3 θmax ) +

1 2 (a + sin 2 ϕ) 2

(2 − 2 cos θmax − cos θmax sin 2 θmax ) IH ∝ sin 2 ϕ(1 − cos3 θmax ) + Figure 16. Single frame displacement histograms for Alexa 555 and Rhodamine 6G at three different pH conditions. (A, B) Displacement distributions for Alexa 555 and Rhodamine 6G in HCl (pH 3.5). (C, D) Displacement distributions for Alexa 555 and Rhodamine 6G in MB water (pH 5.7). (E, F) Displacement distributions for Alexa 555 and Rhodamine 6G in Tris buffer (pH 8.7) Black and gray symbols represent experimental data and Markov chain Monte Carlo approximations, respectively. Adapted from ref 188. Copyright 2014 American Chemical Society.

(20)

1 2 (a + cos2 ϕ) 2

(2 − 2 cos θmax − cos θmax sin 2 θmax )

(21)

From the average wobble angle, an estimate of the pore diameter can be produced. For the pores studied in the experiment, an ensemble wobble angle of 19 ± 3° corresponded to an average pore diameter of 1.3 ± 0.2 nm, much smaller than the expected pore size of around 4 nm. This is attributed to hydrophobic interactions restricting the dye molecule to the center of the pore. A combination of particle tracking and stochastic frequency analysis has been used to analyze transport in biological systems. Recent experiments in live cells have demonstrated the applicability of SPT tracking in these complex and confined environments, providing new information on DNA replication and repair. In order to determine how the protein MutS, known to be involved in DNA mismatch repair, searches for errors, the Biteen group constructed strains of Bacillus subtilis that natively expressed MutS conjugated with the fluorescent label PAmCherry1 schematically represented in Figure 18A.199 Previous experiments targeting these structures were at the ensemble level and thus were unable to extract information on

substrate decreased as pH was increased and the long distance transport mode indicative of hopping was suppressed (Figure 16A,C,E). When a cationic probe (Rhodamine 6G) was used, roughly the opposite trend was detected (Figure 16B,D,F). Ionic transport on multilayer films was found to be governed by complex interaction mechanisms involving electrostatic, hydrophobic, and steric forces. Electrostatic interactions were used to explain the interaction dynamics of anionic Alexa 555. As the outermost film layer became more positively charged at lower pH values, the anionic probes were more likely to interact with the surface resulting in the observed desorption mediated diffusive behavior. In contrast, the interaction of cationic R6G 7345


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location and dynamics of MutS were determined. It was found that MutS binding seems to only occur at the replisome, a phenomenon that is distinctly different compared to the process in eukaryotic cells.199 These and other examples demonstrate that, even in the crowded complicated environment of a bacterium, SPT can reveal useful information with careful experimental design.201,202 3.3. Enhanced Spatial Resolution as a Stepping Stone to Improved 2D SPT

Although SPT involves the quantization of dynamic properties including transport and hopping, the field has benefited from multiple exciting advances in single-molecule fluorescence microscopy to improve the spatial resolution of static analytes. In 2006, several super-resolution imaging methods were independently developed paving the way for subdiffraction limited characterization of dynamic processes.10,12,203 In contrast to STED,84,85 discussed in section 1, these methods are largely similar in that they all rely on the repeat stochastic localization of emitters to build up an image with fine detail that is not detectable using conventional light microscopy. The techniques differ in the types of probes used and the method of repeat localization. Point accumulation for imaging in nanoscale tomography (PAINT) relies on the diffusion of fluorophores that will stochastically bind to the structure of interest. Repeat binding events are monitored allowing the creation of a superresolved image (Figure 19A,B).203 PAINT requires isolated binding events and is the most closely related to dynamics measurements that will be discussed later in this section. PALM52 (Figure 19C−F) and STORM (Figure 19G) are very similar in methodology.10,12 The original publications differed only in systems to which the technique was applied, specifically the type of probes used. Both use an activation laser to selectively excite a small number of fluorophores in a heavily labeled region. The stochastically excited fluorophores are resolvable individually allowing the subsequent reconstruction of super-resolution images containing detail at a resolution of ∼20 nm.10,12 Reconstruction is accomplished by localizing detected PSFs with high precision as discussed in section 2.4. The aggregated blinking or binding events from many (typically hundreds to thousands) of sequential frames are plotted to

Figure 17. Polarization resolved diffusion in 1D structures. s and p polarization resolved wide-field images (summed over 100 frames) for C7OPDI- and C1PDI-doped CTAB-template mesoporous silica films. The double-ended arrows designate the detected polarizations. Reprinted (adapted) from ref 191. Copyright 2013 American Chemical Society.

the location and dynamics of MutS with any specificity.200 By using a combination of PALM and SPT (Figure 18B,C) the

Figure 18. Location and dynamics of MutS in live B. subtilis. (A) Labeling scheme for MutS−PAmCherry. RBS, ribosome binding site. (B) Representative frames showing the photoactivation of a single copy of MutS−PAmCherry in a cell. Lines above the images correspond to the initial activation pulse then imaging laser. (C, lower left) Photoactivated localization microscopy52 reconstruction assembled from localized MutS− PAmCherry molecules and (C, right) single-molecule trajectories of MutS−PAmCherry in (C, upper left) a liveB. subtilis cell. The red arrow indicates a region of MutS accumulation. White dashed lines indicate the computer-detected cell boundary (scale bars: 1 μm). Adapted with permission from ref 199. Copyright 2015 National Academy of Sciences. 7346


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Figure 19. continued Reprinted with permission from refs 10, 12, and 203. Copyright 2006 The American Association for the Advancement of Science (ref 10), 2006 Nature Publishing Group (ref 12), and 2006 National Academy of Sciences (ref 203).

create the reconstructed image. These were all originally imaging techniques, and extensive reviews can be found elsewhere.13,14,17,204,205 The importance of super-resolution imaging methods such as PAINT to SPT applications is that their subdiffraction localization algorithms can be adapted to determine interfacial dynamics with high temporal and spatial resolutions. Typically, the PAINT variant mbPAINT (motion-blur points accumulation for imaging in nanoscale topography) is used for this purpose and is based on the diffusion and adsorption properties of fluorophores to simultaneously image and extract kinetics at a nanoscale level. In such methods, small fluorescent diffusing probe molecules can be selectively observed only when adsorbed to the target structure at the interface, and are made unobservable by motion blur when freely diffusing (D ∼ hundreds of μm2/s) in bulk solution compared to the detector temporal resolution (tens of hertz). Several research groups have utilized mbPAINT to extract the kinetics of DNA hybridization.206−211 DNA hybridization experiments are an excellent test bed for mbPAINT because probe DNA sequences embedded at the surface will capture only the corresponding target DNA sequence in solution with high selectivity. A typical mbPAINT experiment with DNA hybridization is schematically represented in Figure 20A. One of the first applications of mbPAINT was to directly measure the first order kinetics of DNA origami hybridization.208 The use of mbPAINT allowed for the monitoring of individual hybridization events. When matching target ssDNA bound to its corresponding immobilized probe ssDNA on the surface, it created a bright event. The dark time could also be monitored (by counting remaining bright spots over time represented in Figure 20B, inset), allowingd the determination of the association rate kon (Figure 20C). As would be expected for a first order process, the reciprocal of the dark time increases linearly with concentration. A linear fit yielded a rate constant kon of 2.3 × 106 M−1 s−1, which was comparable to previously reported ensemble values. After verifying that excitation laser power did not influence the determination of koff, it was confirmed that koff did not depend on concentration, again confirming a first order rate process.208 In addition to the kinetics determination, the use of mbPAINT allowed for the simultaneous acquisition of subdiffraction limited images of the DNA origami structures. Improving the methodology of single-molecule DNA hybridization assays is a current area of active research. Two major obstacles to accurate DNA hybridization kinetics determination are nonspecific adsorption to the substrate and the influence of labels on the stability of the hybridized DNA structures. Nonspecific adsorption can be minimized by using very high surface coverages of probe DNA as demonstrated by Peterson and co-workers.209 Ensuring that the concentration of surface probe sites was at least 100 times more concentrated than the optically resolvable limit reduced nonspecific binding to negligible levels.209 Interestingly, the desorption time histogram from these experiments (Figure 20B) was best fit with a double exponential decay suggesting an additional

Figure 19. Summary of super-resolution imaging techniques. (A, B) Demonstration of PAINT. Image of a supported lipid bilayer on glass, probed by Nile red. (A) Standard fluorescence image. (B) Highresolution synthetic image obtained by locating 2778 single Nile red probes collected in 4095 frames.203 (C−F) Demonstration of PALM. Comparative summed-molecule TIRF (C) and PALM (D) images of the same region within a cryo-prepared thin section from a COS-7 cell expressing the lysosomal transmembrane protein CD63 tagged with the PA-FP Kaede. The larger boxed region in (D), when viewed at higher magnification (E), reveals smaller structures not resolvable in typical TIRF. In a region where the section is nearly orthogonal to the lysosomal membrane, the most highly localized molecules fall on a line of width of ∼10 nm (inset). In an obliquely cut region [(F), from the smaller boxed region in (D)], the distribution of CD63 within the membrane plane can be discerned.10 (G) demonstration of STORM. STORM images of RecA-coated circular plasmid DNA. Indirect immunofluorescence images with switch-labeled secondary antibody taken by a total internal reflection microscope (top); the reconstructed STORM images of the same filaments (bottom). Scale bars, 300 nm.12 7347


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Figure 20. mbPAINT can be used to determine DNA hybridization kinetics. (A) Schematic showing the method of DNA hybridization detection. (B) Sample single-molecule residence time histogram (black circles) representing ∼3500 10-mer target ssDNA visits on a probe surface fit to a double-exponential decay function. (inset) Images showing single-molecule locations. These images represent a time series and show the disappearance of tracked molecules along with new binding events. (C) The reciprocal of the interevent lifetime τd increases linearly with the concentration of the imager strand. The linear fit yields an association rate (the rate of new binding events) kon of 2.3 × 106 M−1 s−1 at a salt concentration of 600 mM NaCl. The dissociation rate (the rate at which the DNA strands unbind) koff is independent of the imager strand concentration, as expected for a first-order reaction. Reprinted from refs 206, 208, and 209. Copyright 2013 (ref 206), 2010 (ref 208), and 2016 (ref 209) American Chemical Society.

Figure 21. Ligand clustering effects on protein adsorption kinetics and separations. (A, B) Cartoons of protein interacting with (A) engineered clusters and (B) individual ligands immobilized on a porous support. (C, D) Super-resolution images showing adsorption is only detectable at ligand clusters. (E) Relating rate constants through a theoretical model predicts that engineered clusters would induce more efficient protein separation. Adapted with permission from ref 27. Copyright 2014 National Academy of Sciences.

ization. For example, by combining adsorption and desorption kinetics with a statistical mechanical model of chromatography, these experiments suggest that much of the broadening that occurs in typical protein ion-exchange separations occurs due to stochastic ligand clustering (Figure 21E).27

adsorption pathway and was attributed not to bleaching, which was ruled out in control experiments, but rather to desorption of the double stranded DNA duplex following binding. This suggests that consideration must be given to the chosen immobilization scheme. The stabilizing effects of labeling DNA with fluorescent dyes have recently been demonstrated.212 Competitive single molecule binding assays have been demonstrated as a solution to this potential issue.207 Stochastic based super-resolution imaging methods have been adapted to understand the mechanisms driving the adsorption of proteins onto chromatographic substrates (Figure 21).27 We studied the dynamics of adsorption of a model protein, α-lactalbumin, to clustered charge vs individual argininamide ligands (Figure 21A, B) on agarose substrates.27,29 Super-resolution images are produced by localizing multiple adsorbed dye-labeled proteins. A low protein concentration ensures that only a few fluorophores are adsorbed at a time within a single frame. The previously described 2D superlocalization methods can thus achieve simultaneous ∼30 nm precision as well as dynamic information about desorption and adsorption statistics.27,206 Potentially important conclusions can be drawn from super-resolved spatial and kinetic character-

3.4. Correlation Analysis

It is important to contrast SPT with other methods for determining transport from single-molecule image series. Correlation methods can be applied to existing movies taken on conventional wide-field microscopes with little to no modification of the instrument or data collection protocols. Modern correlation-based methods for determining transport dynamics at the single-molecule level are rooted in fluorescence correlation spectroscopy (FCS), originally an ensemble analysis technique developed by Elson and Magde.49,213 Image correlation spectroscopy (ICS) was developed for confocal scanning microscopes and operated in the spatial domain.214 Later it was extended into the temporal domain and was used to characterize dynamics in live cells.215,216 While ICS allows for the measurement of dynamics over larger areas than conventional FCS, the temporal resolution is low with accessible time scales of tens of seconds to minutes. The 7348


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Figure 22. (A) Imaging of DiI on C18-modifed surface (30 ms acquisition time, 128 × 128 pixels). The expanded inset shows the various imaging FCS observation areas, and the 3D intensity plot shows the molecular PSF of an emitter over the 8 × 8 pixel area. Molecular spots had an average signal-to-noise (S/N) ratio of ∼4. (B) Normalized autocorrelation functions for varying probe region sizes fit to eq 20. An example residual plot is from the 8 × 8 pixel autocorrelation included. (C) Plot of 1/τ versus 1/ω2 showing the expected linear dependence of the diffusion time extracted from the fits in (B) to the width of the observation area. Adapted from ref 219. Copyright 2014 American Chemical Society.

Gaussian focal volume as in traditional FCS, but as a convolution of the microscope PSF and the imaging area.221 The linear relationship between the imaging volume and diffusion time is shown in Figure 22C. In 2009 Dertinger et al. developed super-resolution optical fluctuation imaging (SOFI), which generates super-resolved spatial information from the autocorrelation analysis of sequential wide-field images. SOFI uses wide-field image series and high order correlations to improve spatial resolution of images by up to a factor of 5. Autocorrelation is performed on the signal transient from each pixel in the image stack over time. The resolution enhancement scales with the square root of the correlation order. The analysis of images can be performed iteratively allowing a balance of computation time and image quality to be determined.222 We have combined SOFI and an imaging FCS based method to map super-resolution spatial information combined with diffusion dynamics in porous structures with fluorescence correlation spectroscopy SOFI (fcsSOFI).50 fcsSOFI allows for super-resolution spatial mapping of diffusion on a surface or within confined pores by using the diffusion of molecules near the surface or within pores as the source of fluctuations, which are subsequently analyzed using correlation analysis. The process is illustrated in Figure 23. Diffusion was simulated in side-by-side pores separated by a distance less than the diffraction limit (300 nm, Figure 23F). The diffusion coefficient was different in each pore with Dleft = 1 × 105 nm2/s and Dright = 1 × 104 nm2/s. Five hundred frames of diffusion were simulated and sample images are shown in Figure 23A. Intensity traces for sample pixels in each pore are shown in Figure 23B,C. A second order autocorrelation is performed on the intensity transient from each pixel. The correlation curves for the intensity traces in Figure 23, parts B and C, are shown in

Gratton group extended ICS by accounting for the fast raster scanning motion of the focal volume in a typical commercial confocal imaging system allowing access to much faster time scales, on the order of the maximum scan speed.217,218 In recent years, ICS has been modified to capture faster dynamics by combining ICS concepts with fluorescence correlation spectroscopy analysis methods.219−221 Called imaging FCS, this technique is similar in concept to ICS but uses frames of data acquired consecutively on CCD or CMOS chips. Because raster scanning to generate an image is not required, faster time scales are accessible than would be achievable with confocal imaging. Typically, EMCCD cameras can image with integration times on the order of tens of milliseconds. By decreasing the readout area of the CCD chip used for the experiment, faster time scales (∼1 ms) are accessible. The theoretical framework for this method was adapted from ICS and FCS for wide-field imaging. For a typical experiment, a subregion of pixels varying in size from 8 × 8 to 2 × 2 pixels (Figure 22A) was imaged for ∼40 000 frames. The signal in this region was averaged over all pixels for each frame to generate a fluctuating intensity trace over time. Following autocorrelation of this trace, the resulting curve was fit with a Brownian diffusion equation adapted from FCS. Generalized as G (τ ) = A *

1 +B 1 + τ /τ1/2

(22)

where τ1/2 is the diffusion time. From this, the diffusion coefficient can be determined. Sample autocorrelation curves and fits for the hydrophobic dye DiI diffusing on a silanized glass interface are shown in Figure 22B. The extracted diffusion time corresponds to the imaging area that is modeled, not as a 7349


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sections are plotted in Figure 23J. The resolution enhancement is clearly apparent. The extracted diffusion coefficients from each pixel were mapped to a color and plotted to produce the diffusion map in Figure 23K. The images were then fused producing the super-resolution diffusion map in Figure 23L.50 Second order correlation was used to produce the fcsSOFI images shown in Figure 23, but further resolution enhancements can theoretically be achieved by using higher order correlations at the expense of computational resources. For the second order correlation, the expected resolution enhancement is ≲√2. A blind deconvolution would increase the resolution enhancement further to ≲2. For higher order correlations, the resolution enhancement is expected to scale with the correlation order n and fall within the range σxy/√n < σn < σxy, where σ is the width of the Gaussian PSF of a diffraction limited emitter. Higher correlation orders come with increased computational cost, which is expected to scale with the correlation order squared.50 3.5. Differential Interference Contrast (DIC) Microscopy

DIC microscopy is often used in conjunction with other 2D fluorescence imaging and tracking techniques, or can perform tracking without the use of other techniques. Unlike fluorescence based methods, DIC microscopy probes the effective optical path length of the sample.223 DIC microscopy creates an image by using two orthogonally polarized, spatially offset light rays. As the rays pass through the sample, the relative phase will drift as they encounter different optical environments. Upon recombination, this phase difference is used as an image contrast and allows DIC microscopy to resolve objects that would be transparent in other forms of optical microscopy.223 DIC is often used to colocalize cells and probes when fluorescence tracking is used, such that the position of the probe is known relative to the rest of the cell.224−226 It is especially convenient because, as a transmission technique, it can be implemented on the same microscope assembly as an epifluorescence tracking experiment. DIC microscopy is highly sensitive and as such was used in some of the earliest single particle tracking experiments. It was used to study the motion of kinesin coated latex beads along microtubules as early as 1988.227 Nanometer spatial resolution and microsecond temporal resolution are possible with DIC tracking, but its usefulness is limited by the fact that it requires enough material to change the effective optical path length of the sample; therefore, single molecule resolution has not been achieved.227,228 Even relatively recent iterations of this technique use relatively large (>100 nm in diameter) beads.228 Additionally, the optics involved tend to complicate data analysis;229 therefore, tracking using DIC microscopy remains a challenge.

Figure 23. fcsSOFI analysis of 1D pores. (A) Two emitters are simulated diffusing in two different pores shown by the colored lines. (B, C) Example intensity traces from a single pixel in each pore. (D, E) Autocorrelation of each intensity trace is fit to extract D. (F) Ground truth pore locations. (G) An intensity-averaged image, showing pore locations are within diffraction limit. (H) Sample trajectories from conventional SPT. (I) fcsSOFI image maps the value of G2(τ) from (D,E) as the source of the contrast. (J) Resolution comparison between diffraction limit, SPT image, and fcsSOFI vs ground truth shown in (F−I). (K) Dcalc for each pixel. (L) Final fcsSOFI image generated by fusing (I) and (K) Scale bars = 300 nm. Reprinted from ref 50. Copyright 2015 American Chemical Society.

4. REVEALING DYNAMIC PROCESSES USING 3D SPT Recent developments offer exciting possibilities to understand more complex structure−function dynamics in 3D, but also present new challenges. For example, fast data collection rates and the ability to transfer, save, and process large amounts of data are important. Modifications of wide-field methods are attractive because they intrinsically capture events in a large area (∼10 μm) in one focal plane, and thus are inherently highthroughput. Moreover, as discussed earlier, TIRF microscopy significantly suppresses background signal and noise from defocused events not in the focal plane. TIRF microscopy has been a powerful tool to study events near the plasma

Figure 23, parts D and E, respectively. Each autocorrelation curve is fit with the standard model for Brownian diffusion modified for wide-field analysis. The value of the first time lag in G2(r,τ) for each pixel is plotted to produce the fcsSOFI image (Figure 23I). A comparison of the fcsSOFI image with conventional particle tracking (Figure 23H) and the average intensity image (Figure 23G) are also given, and sample cross 7350


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Figure 24. Design of bifocal plane microscopy. (A) Layout of bifocal plane microscopy. Usually, two excitation laser beams with different frequencies are used. Two beams are focused in different planes. In the original design, the high frequency beam (488 nm laser) is focused at the glass−water interface in TIRF mode and the corresponding signal is recorded by camera 1, and the low frequency beam (543 nm laser) is focused at a focal plane above the glass−water interface in wide-field epifluorescence mode and the corresponding signal is recorded by camera 2. Target probes are double labeled to be detected in both excitations. (B) Simplified light path to explain the working principle of bifocal plane microscopy. Two lenses represent the objective (left) and the tube lens (right). The focal plane in light color is the standard focal plane, and the corresponding image plane represents camera 1 in (A). The focal plane in dark color is the adjustable focal plane, and the corresponding image plane represents camera 2 in (A). Adapted with permission from ref 236. Copyright 2004 IEEE.

membrane230,231 and observe interfacial transport dynamics of biological macromolecules.170,171,231 However, combining the two techniques cannot provide depth information about single emitters in 3D especially in cellular matrixes. Wide-field microscopy combined with electronically controlled piezo stages and fast detectors provides a simple solution: scanning over different z depths and recording the corresponding images, as discussed in section 5.58,232,233 However, the movement of the piezo stage can effectively reduce the SNR of the recorded images due to hysteresis.234 In contrast, lock-in methods that rely on piezo stages to actively track a single emitter are low-throughput and can easily lose a fast moving emitter due to temporal limitation in the feedback systems.235 In the following sections, we discuss a range of solutions that are aimed at the ultimate goal of 3D SPT: high-throughput particle localization in 3D efficiently and accurately.

nm laser was focused above the glass−water interface in a widefield excitation geometry with adjustable focal distance. The novel microscope design of simultaneous detection in both TIRF mode and epifluorescence mode provides more sensitivity in the setup. Accordingly, the target probe is labeled with two fluorescent molecules. Mixed fluorescent signals are split equally into the respective detection paths by a beam splitter (Figure 24A). Signal excited by the 488 nm laser is recorded by camera 1, while signal excited by the 543 nm laser is recorded by camera 2 on an adjustable translational stage in order to ensure precise calibration of the system. Emission filters are used in each light path to eliminate the laser reflection and signal cross talk. It should be noted that this experimental design is very similar to two-color single molecule Förster resonance energy transfer (FRET) setups;237 therefore using a long pass beam splitter would save more photons when separating the signals excited by two lasers. During the early stages of biplane microscopy, the photon budget was less of a concern prior to the single-molecule explosion after 2006.10,12 Biplane microscopy provides better resolution in all three dimensions compared to conventional wide-field microscopy.236,238 The theoretical spatial resolution can be determined by Cramér−Rao lower bound (CRLB) simulations, which involve calculations using the Fisher information matrix (see section 2.3 for details).238 For a typical separation of two planes (∼500 nm) and considering the magnification at the camera and noise, biplane microscopy shows a better and more stable resolution (∼20 nm in x and y, and ∼100 nm in z) in 3D when the fluorescent probe is near or in-between the two focal planes, as compared with standard Gaussian PSFs.238 This can provide an ∼1 μm depth detection range for 3D SPT. However, for these spatial resolutions to be achieved, precise

4.1. Multifocal Plane Microscopy (MPM)

MPM, developed by Ober and co-workers,236 is one method of 3D SPT that has been widely adopted, and records simultaneous images at different focal planes in sample space without sacrificing the temporal and spatial resolution of the resulting trajectories. The initial design of MPM recorded events in two different focal planes simultaneously to extract a particle’s axial location, and was coined “biplane 3D microscopy” (Figure 24A).236 In a typical microscopy system, each focal distance has a corresponding image distance (Figure 24B). In biplane 3D microscopy, two cameras are arranged at different imaging distances allowing signal to be recorded at two different focal planes, as depicted in Figure 24B. In the initial design, two excitation lasers were used with different wavelengths (488 and 543 nm). The 488 nm laser was utilized to image glass−water interfaces in TIRF mode, while the 543 7351


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Figure 25. Using bifocal plane microscope to observe transport process from sorting endosomes to exocytic sites. The lower focal plane is focused on the glass−water interface to detect the membrane. The higher focal plane is focused 0.6 μm higher than the membrane plane. The MHC class I related receptor, neonatal Fc receptor (FcRn), is shown in green. The detected tubule is highlighted in orange. Cartoons in the third row explain the corresponding cellular events. By detecting in two focal planes simultaneously, tubule extending (0.51−2.38 s, leftward arrows), partial fusion with the plasma membrane (2.38−2.89 s, upward arrows), and retracting and detaching (3.23−8.33 s, leftward arrows) are observed. Adapted with permission from ref 239. Copyright 2007 National Academy of Sciences.

laser, in comparison to the original bifocal plane microscopy. Compared to the double plane microscopy discussed earlier (Figure 24), only one emission filter is needed to block the laser scattering and reflection. After the emission filter, the signal is split into four paths and collected by four cameras. The four cameras are placed behind tube lenses with increased distance to capture signal from different detection planes. In this setup, an objective with small NA is utilized, effectively decreasing the photon flux transmitted through the detection path and lowering the SNR at camera 4. As a result of the aforementioned advancements in MPM, the study of complex intercellular processes within large detection volumes has been achieved by producing 3D single particle trajectories.242−244 An example of these results can be seen in Figure 26B−D, where single QD-labeled transferrin molecules dynamically transport between adjacent cells. The transferrin was initially observed in one cell (highlighted by the red arrow in Figure 26B), and then undergoes exocytosis, quickly moving to the target plasma membrane of another cell. This process is then followed by endocytosis into the recipient cell (Figure 26C,D). From these observations it is noted that transferrin has a very short travel time between cells and short resident time before being internalized by the recipient cell. The 3D detection capability reveals the vertical distance traveled by the transferrin while uncovering specific interfacial regions of the plasma membrane at which endocytosis occurs. MPM is a powerful tool for tracking 3D events in cellular environments. Recent developments in MPM instrumentation have focused on detection path modifications to enhance 3D detection capabilities, leaving the excitation path free to be adapted with other microscopy techniques. This has given rise to methods such as biplane FPALM super-resolution microscopy.245 The main drawback of MPM is a great loss in the photon flux at a detector when the emission is split to a large number of detectors, which is a problem particularly when using organic fluorescent molecules with low quantum yields. The next two techniques discussed in sections 4.2 and 4.3 improve upon the photon budget in the detection path and achieve 3D SPT in a unique and novel manner.

calibration of the system is necessary. This is also true for other techniques that will be discussed in later sections. Although biplane microscopy is powerful in its ability to study 3D processes, it does involve strict experimental criteria that must be met in order to explore complex biological environments in 3D. Prabhat et al. elegantly utilized biplane microscopy to study 3D cellular processes, specifically the highly debated dynamic process of intracellular transport carried out by sorting endosomes.239 The dynamic intracellular transport inside the plasma membrane is a popular target for wide-field microscopy, which was probed in Prabhat’s experimental setup using a TIR geometry given its advantage of high SNR at a given interface (Figure 25). The second focal plane in this work was within the intracellular volume (0.6 μm from the membrane, Figure 25) to track the dynamic process of endosome fusion and exocytosis carried out by FcRn receptors. As shown in Figure 25, the details observed in the intermediate plane can reveal the location of the endosome as it undergoes transport to the cellular membrane. However, if this was the only plane that was imaged, important mechanistic 3D information would be lost. More specifically, the details of exocytic fusion, exocytosis, and detachment, including the time scale over which these dynamic events occur, would have been lost without simultaneous imaging of the membrane plane. This highlights the tremendous insight and advancements biplane microscopy have made for probing important and complex intracellular trafficking events. It must be noted in the outlined experiments that the lateral position of the emitter in both image planes should be spatially correlated. In principle, MPM is not limited to only two planes. Ober and co-workers have developed three- and four-plane microscopies to enhance the 3D detection capabilities.239−241 However, in their four-plane microscopy, only two lasers and two fluorescent labels were used: one for the TIRF mode detection at the glass−water interface and the other for three focal planes that were focused in the bulk sample environment. The crossover among different light paths (corresponding to signal from different focal planes) would highly diminish the respective SNRs without the proper separation of wavelengths.239 The same group also simplified the instrumentation by using only one laser and one fluorescent label,241 as shown in Figure 26A. When using four detectors, the excitation path effectively becomes a typical wide-field excitation path with one

4.2. Astigmatic 3D Detection Using a Cylindrical Lens

More recent 3D imaging techniques such as astigmatic imaging allow for higher photon counts through PSF engineering.91,92,246−248 These methods only detect one focal plane of 7352


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imaging non-Gaussian PSFs allow the z position information to be extracted from the shape of the PSF. Astigmatic 3D detection techniques use a weak cylindrical lens to slightly shift the focal plane in one lateral dimension thereby encoding the particle’s z position in the shape of the recorded PSF. As shown in Figure 27, a cylindrical lens with a focal distance much larger than the tube lens is mounted in the detection path.91 The cylindrical lens causes shifts of the focal distance along the x and y directions. Usually, the focal distance of the cylindrical lens is about 1 m, while the focal distance of an imaging lens is about 10 cm.91 To make our discussion easier, we follow Figure 27 and assume the focal distance in the x direction has been changed by the cylindrical lens. In some studies, the focal plane remains unchanged with the addition of a cylindrical lens.246 Therefore, the corresponding Gaussian PSFs have different widths in two directions at the focal plane. However, other studies also suggest that the focal plane is the image plane where the corresponding Gaussian PSF is symmetric in shape, meaning that the focal plane is above the current image plane shown in Figure 27.91 The difference between these two cases is subtle, corresponding to a difference of a few millimeters in the camera’s location. In the development of astigmatic 3D detection, multiple methods to perform 3D SPT have been proposed. In 1994, Kao and Verkman were the first to develop astigmatic 3D detection to track rhodamine-labeled beads. They calculated the center of mass to determine the x and y positions of emitters, and the widths of the PSF in the x and y directions to determine the z position.246 Later in 2007, Holtzer et al. applied the same technique to track the 3D diffusion of quantum dots in a cellular environment.247 At that point, Gaussian fitting had been demonstrated to be one of the most accurate and precise methods to calculate the lateral positions of a standard 2D PSF.119,127 In Holtzer’s work, 2D Gaussian fitting was used to extract the x and y positions and the fwhm in both x and y dimensions (wx and wy). wx and wy are compared with the fwhm’s of standard calibrated PSFs to determine the z position.247 As mentioned in section 2.4, even though 2D Gaussian PSF fitting achieves relatively high precision, it often requires intensive computational time in comparison to determining an emitter’s center of mass (section 2.4).119,127 Huang et al. developed another approach based on calibration data to determine the z position.91 Typical calibrated wx,calib and wy,calib at different z positions are shown in Figure 27B. The z position of a new emitter can be found by minimizing the following:

Figure 26. Four focal plane MPM scheme and a 3D tracking application. (A) Principle of MPM. The fluorescence signal is split into four different light paths (represented by different colors) using three 50:50 beam splitters. Different light paths correspond to different detection planes. Each detection plane matches with one detector (CAM1 to CAM4) at a calibrated focal distance from the tube lens. (B) Example tracking images of quantum dot labeled transferrin traveling between adjacent cells (scale bar = 5 μm). Four images are recorded by four detectors at the same time, and overlapped with the segmented plasma membrane profiles (in green). The red arrow in the second image highlights the quantum dot labeled transferrin receptor that transports between cells during the measurement. The trajectory of the transferrin receptor is depicted in 2D (C) and in 3D (D), with time dimension being color-coded. Adapted with permission from ref 241. Copyright 2012 Elsevier.

(wx ,new 0.5 − wx ,calib 0.5)2 + (wy ,new 0.5 − wy ,calib 0.5)2

The PSF will be rejected if the minimized distance is larger than the predefined threshold.91 Astigmatic 3D detection allows researchers the freedom to combine it with the excitation geometries of other techniques. A cylindrical lens can be used to produce non-Gaussian PSFs, such as the diamond-like PSF used by the Wang group to probe the 3D motion of fluorescent polystyrene beads in cylindrical pores.249 As the PSF is not Gaussian, more complicated correlation based analysis algorithms are implemented to extract the 3D localization information from the corresponding images. Huang et al. have coupled the detection techniques of astigmatic 3D detection with the excitation setups used in STORM for 3D super-resolution imaging.91 The complicated excitation setups utilized in STORM are readily coupled with

the sample space, but through the use of a cylindrical lens encode the depth information of the emitter within the shape variation of the PSF. As a result of the inherent nature of this technique, photon loss can be greatly minimized. Traditional Gaussian PSFs are centrosymmetric, thus not allowing the axial location of the emitter to be encoded in the PSF. In astigmatic 7353


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Figure 27. Principle of using a weak cylindrical lens to encode an emitter’s 3D position. (A) Detection path of the 3D microscope using a cylindrical lens. A weak cylindrical lens is added in the typical detection path so that the focal points in both the x (green light path) and y (blue light path) directions are shifted. PSFs corresponding to different z positions are shown in the right panel. Two-dimensional Gaussian function with different widths in x and y directions can be used to calculate the 3D center position of the emitter. (B) Gaussian widths in the x and y directions (wx and wy) as a function of z. Details about measuring the calibration data and the fitting are described in ref 91. Reprinted with permission from ref 91. Copyright 2008 American Association for the Advancement of Science.

an astigmatic based 3D detection setup.91 Lien et al. has used a cylindrical lens for 3D detection with temporal focusing of multiphoton excitation microscopy, which provides deeper 3D imaging in thick tissues.250 Spille et al. combined the astigmatic 3D detection path with light sheet excitation and a feedback controlled three-axis translation stage, which can provide long time 3D tracking capability deep inside living tissues.122,251 Here we will highlight the work by Spille et al. (Figure 28)122,251

around the nucleus in Chironomus tentans salivary gland cells covering 100 μm in depth (Figure 28), which cannot be accomplished with traditional 3D microscopy.122 However, the particle tracking relies on feedback and as such can only follow one or a few particles for an extended period of time. The feedback voltage is based on the 3D position of one selected particle. The z position of the particle is roughly determined by the ratio between wx and wy for a fast estimation enabling realtime image processing taking only 50 μs per frame. The 3D position of the target particle can be determined in high resolution later by post data processing. Because of the careful instrument design, they were able to track one targeted particle for 800 frames over 12.8 s for an axial distance of 60 μm.122 Later, they focused on Balbiani ring (BR) mRNA and rRNA diffusion near the nuclear envelope to investigate the kinetic properties of the two types of RNA molecules (Figure 29).251 For certain time steps and positions the mRNA molecules exhibited reduced mobility, which was believed to be a direct observation of the mRNA sampling different nuclear pores along the nuclear membrane. Diffusion types, MSD, and dwell time distributions can be extracted from the long-lived SPT trajectories. Overall, in comparison to MPM, astigmatic 3D detection is versatile and can be used in many excitation designs, has no additional photon loss due to signal collection, and uses one detection path and one detector. However, overlapping of PSFs from different depths can introduce challenges in the data analysis (“decoding” process). Because only a weak cylindrical lens is added in the detection path, the detection range is similar to the diffraction limit of the standard PSF in the axial direction. Later we will see that the detection range can be further improved through manipulating the phase information on the signal. Cylindrical lenses break the axial symmetry of the standard PSF generated by spherical lenses,91 and this resulting asymmetry encodes the axial information. We will see in section 4.3 that a cylindrical lens is just one type of depth encoding technique for 3D detection. Yet, broad applicability and convenient data analysis make the use of cylindrical lenses a useful and cheap alternative to other more complicated and expensive 3D detection methods.

Figure 28. Application of 3D microscopy combining light sheet microscopy and astigmatic imaging. (A) Profile of a C. tentans salivary gland cell nucleus (in green) and trajectories of fluorescent beads, with different color labeling different trajectories. (B) One representative trajectory that is tracked more than 4500 frames. Adapted with permission from ref 122. Copyright 2012 Optical Society of America.

Light sheet microscopy combined with an astigmatic 3D detection path can simultaneously achieve a large detection range and nanometer scale spatial resolution. Light sheet microscopy utilizes a thin light sheet to excite only one layer of specific depth such that the background signal from fluorescent emitters in the bulk solution is suppressed. Light sheet excitation provides approximately uniform illumination in the sample for a large horizontal area. The x−y−z translation stage provides coarse 3D tracking capability as deep as 100 μm, and the cylindrical lens further provides ∼10 nm resolution along with fast feedback in the z direction. This method was demonstrated to track particles up to 200 μm inside live tissues and provides a useful tool to study the intranuclear dynamics.122,251 Spille et al. tracked 20 nm fluorescent beads 7354


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Figure 29. mRNA diffusion paths near the nuclear envelope produced by 3D SPT with a cylindrical lens. (a) Three-dimensional representation of an mRNA diffusion trajectory. (b) Overlay of the nuclear envelope (green), maximum intensity projection of mRNA signal (red), and trajectory data (white) over a time course of 15.4 s. Scale bar, 2 μm. Adapted with permission from ref 251. Copyright 2014 Oxford University Press.

Figure 30. Microscope detection path with 4f system and a phase mask for 3D detection. A 4f system (in dashed box) is added in a typical microscope detection path. Lens 1 and lens 2 are identical with comparable focal distance as the focal lens (FL). A transparent phase mask is mounted in the middle of lens 1 and lens 2 (the Fourier plane). The phase mask for the double-helix PSF is shown at right. Different colors represent different phase delays caused by the thickness of the material. The rotated PSFs corresponding to different z positions are shown in the right panel. Adapted with permission from ref 261. Copyright 2016 Nature Publishing Group.

4.3. Engineered 3D Point Spread Functions Using Phase Masks for 3D Imaging and SPT

lenses placed a distance equal to 2f, with f being the focal distance, as highlighted in Figure 30. The plane in the middle of the two lenses is called the Fourier plane, because wave functions of the corresponding signals at this plane are Fourier transforms of the signal in the focal plane of the sample. Emitters at different depths exhibit different phase distributions at the Fourier plane. If a phase mask is added with a specific phase pattern designed to break axial symmetry, then the corresponding asymmetric PSFs can be obtained in the final image of a system of interest. These asymmetric PSFs better indicate the axial positions of emitters. The use of phase masks that produce a double-helix (DH) PSF, and related improvements, is one of the most successful examples of 3D SPT.257−260 DH PSFs use the relative orientation between the two lobes to indicate the axial position of fluorescent emitters (Figure 30). The idea of a rotating PSF originated from a study using

By manipulating phase via Fourier optics, as discussed in section 2.2, we can acquire more freedom and options in encoding depth information in the detection path compared to the previously covered 3D imaging methods. According to the optics wave equations, the depth information is embedded in the phase of the wave. Depth encoding using Fourier optics couples phase information via intensity interference. Earlier developments in Fourier optics led to the invention of holograms,252 phase contrast microscopy,253 and DIC microscopy.254,255 In Fourier optics, the function of a lens is mathematically equivalent to taking a Fourier transform of signal from the focal plane (see section 2.2 for more details). Therefore, the signal can be manipulated in the phase domain after a lens. The ideal application of this model is the 4f system.256 A 4f system is usually composed of two identical 7355


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Figure 31. Three-dimensional tracking of single mRNA−protein complex in a yeast cell. (A) Overlapping of the particle tracking trajectory in (B) and (C) (in red box) on top of the bright field image of the cell (in gray scale). (B) Two-dimensional projection of the 3D particle tracking trajectory. Relative time is shown in different colors as indicated by the color bar. (C) Three-dimensional plot of the same trajectory. The movement and location of the mRNA−protein complex in the z dimension becomes clear in 3D. (D) Example raw images showing 3D DH PSFs overlapped with the yeast cell. Different z positions are indicated by the orientation of the two lobes. The relative time of each image is 0, 0.233, 5.78, and 6.99 s. Scale bar is 2 μm. Adapted with permission from ref 267. Copyright 2010 National Academy of Sciences.

Figure 32. Tetrapod masks and corresponding 3D PSFs for large range 3D tracking. (A, E) Phase mask design for 6 μm (A) and for 20 μm (E) detection range. (B, F) Corresponding simulated PSFs at different z positions. (C, G) Corresponding experimental measured PSFs at different z positions. Both scale bar and detection range are larger for the phase mask (E). (D, H) Theoretical resolution calculated based on the CRLB. Adapted from ref 271. Copyright 2015 American Chemical Society.

finite energy to generate propagation-invariant wave fields.262 In earlier work, Piestun et al. found that certain types of basic cylindrically symmetric mode (Gauss−Laguerre) combinations generate waves with a rotating pattern in the final image plane.262 To generate this exact rotating PSF, both the amplitude and phase of the signal are required. However, manipulating the amplitude distribution in the Fourier domain would result in a loss of photons of more than 90%,258 which is not suitable for single-molecule studies. Later an optimized design was developed that only required the manipulation of the phase pattern of the signal to generate a rotating PSF, which is called the DH PSF.258 Soon after, this phase mask was used in 3D super-resolution imaging263 and SPT.257,264,265 Generating DH PSFs is achieved by adding a 4f system in the detection path and modifying the phase pattern in the Fourier plane. Initially, reflective spatial light modulators (SLM) were

used to provide the desired phase pattern. However, SLMs only reflect linearly polarized light, potentially resulting in a 50% photon loss.266 Later transparent phase masks, fabricated by gray-level lithography, provided another option of inducing a DH PSF with higher photon efficiency.266 Unfortunately, the phase mask fabricated via lithography is restricted to a specific optimum wavelength, and the phase pattern cannot be reprogrammed once fabricated. Considering their complementary advantages and disadvantages, both types of phase masks are used nowadays. Figure 30 shows a typical light path used in a transparent phase mask setup. Double-helix PSFs allow 3D tracking with a 2 μm range in the z direction with a 50 nm resolution without any assistance from a piezo stage.264,267,268 The use of DH PSFs yields better high throughput tracking capabilities in 3D compared to MPM and astigmatic 3D detection, giving larger depth detection 7356


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Figure 33. Enhancement of DH PSF microscopy with light sheet excitation. (A) Isolated DH PSF obtained from light sheet fluorescence microscopy and (B) epi-illumination microscopy. (C) Intensity profile of the cross section for the two white lines in (A) and (B), indicating the SNR improvement. (D) Three-dimensional trajectory of single fluorescent beads. (E) Linear fitting of the 3D MSD of single fluorescent beads. Adapted with permission from ref 275. Copyright 2016 IEEE.

ranges as well as higher localization precision.269 Thompson et al. have used DH PSFs to study transport dynamics of mRNA in live cells (Figure 31A),267 which is the first step to understanding the important role of mRNA dynamics in gene expression and regulation. These details could not be discovered without the advancements in DH PSF microscopy. As shown in Figure 31B,C, directed motion in the z direction could be easily confused as local Brownian motion if this system was only monitored in 2D (shown in the trajectory between 6 and 7 s). This work has classified different modes of transport, such as Brownian motion, confined diffusion, and directed motion for each single mRNA trajectory, and quantitatively characterized different types of motion. It has provided a much clearer picture of how mRNA functions inside cells by further revealing the specific interactions during confined diffusion and the mechanism of directed motion. After the application of the phase engineering for DH PSFs, other phase mask designs have been proposed to further improve the 3D detection capability.94,270−274 One of the goals is to establish a general approach for variations of phase masks to be designed in the future and to understand the connection between the phase pattern and the performance of the 3D PSF.258,272−274 Recently, Shechtman et al. proposed a new tetrapod phase mask using shape extension as the indicator of the z position (Figure 32).271,272 The behavior of the tetrapod PSFs are very similar to the PSFs generated using a cylindrical lens. However, the intensity profiles of the tetrapod PSFs are much more complicated. These PSFs are designed for detection over a large axial range. When including high frequency components in the phase pattern, the detection range is increased, as is the size of the PSFs in 2D.271 There is however a trade-off between peak intensity and detection range, which

could be problematic if the probes of interest are single organic fluorescent molecules that exhibit a low quantum yield. For the purpose of a large detection range, a 3D detection path combined with an additional feedback tracking system may offer an alternative solution.122,251 The peak intensity of most current engineered PSFs used to encode 3D localization is much lower than that of traditional Gaussian PSFs, leading to low SNR which correspondingly results in low localization and tracking precision for 3D SPT measurements.275 To improve the SNR, the phase masks are designed to maximize the peak intensity of their corresponding PSFs as long as they can satisfy their basic functions for the 3D detection.258 When the phase mask efficiency is optimized, subtle experimental design could help to further boost the SNR. Light sheet microscopy combined with a feedback circuit can effectively suppress the background noise from the bulk solution and has been applied to enhance the SNR in astigmatic 3D images, as discussed in section 5. Yu et al. proposed to combine light sheet microscopy excitation method with the DH PSF 3D microscope to enhance the SNR of the system. Figure 33A,B compares the DH PSF with and without light sheet excitation. The 1D cross section of the DH PSF indicates an SNR increase by a factor of 2.4, which therefore improves the precision of the particle localization and tracking results. In the application of 3D tracking experiments with this setup, a feedback circuit is operated to ensure that the target particle is excited at all times by the light sheet within an observation interval. Finally, they demonstrated the ability of tracking single particles in 3D using fluorescent beads, with one representative trajectory shown in Figure 33D with fits used to extract the 3D diffusion coefficients (Figure 33E).275 Even though light sheet excitation can only be used to track a single 7357


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Figure 34. (A) Multifocus optical elements are appended to a wide-field fluorescence microscope after the primary image plane (at the camera port). Two relay lenses ( f1 = 150 and f 2 = 200 mm) create a Fourier plane and a second image plane. The multifocus grating (MFG) is placed in the Fourier plane and followed by the chromatic correction grating (CCG) and prism. A dichroic mirror (purple) splits the color channels onto separate cameras. (B) Schematic of the MFG. The basic unit of MFG is optimized to distribute light evenly into the central 3 × 3 diffractive orders. The CCG reverses the dispersion of MFG, and the followed prism directs the images to the camera. (C) The instant focal stack recorded on the camera is computationally assembled into a 3D volume. (D) Movement in 3D of the two centromere clusters (black and gray). Bottom right, separation between the centromere clusters over time. Rapid movement (phase I) is followed by slow movement (phase II). (inset) Average speed during phases I and II (n = 5 cells). Adapted with permission from ref 276. Copyright 2013 Nature Publishing Group.

particle at a time, it provides a potential solution when the target fluorescent signal is restricted and of low quantum yield. Phase modulation in the Fourier domain also allows a novel variant of MPM.276 Abrahamsson et al. proposed a novel phase pattern to project images from different focal planes onto different regions of a camera chip. The experimental setup resembles the DH PSF 3D microscope, as the signal at the primary image is transferred into the Fourier domain with the first image lens (Figure 34A). The multifocus grating (MFG) (Figure 34B) is located exactly at the Fourier plane, and it encodes the signal in two aspects. First, the basic pattern MFG evenly splits the signal from a single fluorescent emitter into nine branches, which correspond to nine regions on a camera chip, as indicated in Figure 34C. The MFG also provides a phase aberration such that different regions on the camera chip correspond to different focal depths in the object space. The chromatic aberration is corrected by a chromatic correction grating (CCG) and then finally imaged on the camera by a second image lens. The advantage of achieving MPM using phase modulation is that this technique allows for a large depth detection range (∼4 μm) with fast signal acquisition speeds compared with other MPM techniques. Abrahamsson et al. applied this technique to track Saccharomyces cerevisiae expressing Cse4-GFP to study yeast centromere separations, and they observed biphasic behavior during cell division. The fast separation speed (phase I) reaches 20 nm/s over ∼2 min initially, followed by a slow separation speed (phase II) 3 nm/s over ∼15 min, as indicated in Figure 34D. This observation is made possible by the large field of view and fast detection speed provided by this 3D SPT method.

Recovery of the 3D position based on complicated PSFs is a difficult computational problem. For the first two 3D techniques discussed above, a centrosymmetric 2D Gaussian is still a reasonable approximation of the generated PSFs. However, for DH PSFs or tetrapod PSFs, it is difficult to approximate the PSFs using explicit mathematical expressions. Therefore, double-helix PSFs were approximated as two Gaussian PSFs with a certain separation distance.258,277 The relative orientation between the two Gaussian PSFs was used to represent the orientation of the double-helix PSFs. However, the two lobes of the double-helix PSFs are not actually symmetric as is the 2D Gaussian function,277,278 and the distance between the two lobes changes at different orientations. Moerner and co-workers have developed a corresponding correlation method.260 They calculate the correlation between the calibration PSFs at different z positions with the experimentally measured PSFs, with the highest correlation indicating the corresponding z position. It is important to note that the correlation method is timeconsuming and computationally expensive. Another family of proposed methods is 3D deconvolution using sparse sampling techniques.279 Deconvolution methods are especially useful in the analysis of images with overlapped PSFs,279,280 but are computationally intense and vulnerable to overfitting when the SNR is low. As engineered phase masks become popular and diversified, reliable recovery algorithms will remain an area of interest in the near future. The introduction of the 4f system combined with a phase mask opened a new era of 3D microscopy and SPT. This method has no special requirements in the excitation path other 7358


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Figure 35. (A) Position trajectory of transcription protein molecule in a living cell shown by dashed line. (inset) Locations of two residence time events (τ1 and τ2) and corresponding trajectory pathways. (Scale bars are 500 and 80 nm, respectively.) (B) Displacement (r) per time lapse is the frame to frame distance the molecule traveled between two consecutive images for trajectory shown in (A). Shaded gray regions correspond to inset in (A) and corresponding distance traveled. Two microscopic residence times thresholded by r0 = 220 nm (horizontal red dashed line) to determine binding and unbinding events on a chromosome. (C) Proposed kinetic pathways of specific binding (SB), nonspecific binding (NB), and freely diffusing (FD) transcription factors generated from single molecule stroboscopic kinetic results. Reprinted with permission from ref 297. Copyright 2015 Nature Publishing Group.

to cells or tissue should also be considered.287−289 In a typical well-engineered experiment these processes take place on time scales longer than the camera integration time and thus set upper limits to the experimental time scales.284,288 Therefore, it is crucial to capture emitter signals as fast as possible to reduce these effects. Events that occur within one camera frame are assumed to have occurred at identical points in time, meaning trajectories cannot be resolved with a temporal resolution higher than the camera exposure time. Low temporal resolution thus limits the application of super-resolution microscopy to processes occurring at or slower than the acquisition time of a single frame.290,291 Fast, accurate scientific cameras161 have been in development since 1969, yet high cost and background noise make it difficult to entirely satisfy the requirements for SPT experiments. In order to improve the temporal resolution of SPT without new cameras exhibiting higher frame rates, many techniques have been proposed and implemented. One technique developed to overcome these limitations was shown by Xie and co-workers utilizing stroboscopic excitation.225 Stroboscopic techniques utilize excitation laser pulses synchronized with camera exposure times in order to increase the temporal detection of probes in a wide array of environments.70,292−296 Recently Chen and co-workers utilized stroboscopic SPT in order to investigate the complex kinetic process of gene transcription regulation in E. coli cells (Figure 35).297 This work allowed for a temporal resolution of 4 ms to be achieved, in addition to a 20 nm spatial localization of transcription regulators in vitro. As shown in Figure 35A, transcription factors of interest were tracked performing transcription processes within many areas of the cellular environment. Single trajectory analyses were implemented (Figure 35B) to determine residence times and corresponding termination of binding events on single chromosomes. Further

than the 4f configuration but is more complicated than simply using a cylindrical lens, especially for the data analysis. Due to the complexity of these PSFs, traditional fitting methods cannot analyze them, and calibration has an even greater influence on the resolution performance. Although theoretical calculations show DH PSFs provide better spatial resolution than bifocal and cylindrical lens techniques,269 reaching this theoretical limit will be difficult. Advancements in the fields of computational vision, such as machine learning, could be a promising direction to further reduce the computational burden in SPT. In addition to the previously mentioned methods, a less common technique that is worth mentioning is the use of a wedge prism to achieve 3D tracking.281 This prism splits the light from an emitter into two focal spots with a defined offset in the x direction on the detector. Movement of the emitter in the z direction creates an additional offset of one of the focal spots in the y direction on the camera. Thus, the y offset between the centers of the two spots is linearly related to the depth of the emitter. This technique was used to determine that Eg5 and two-headed kinesin molecules produce small amounts of torque as they travel along microtubules.281 A method using mirrors instead of a wedge prism to produce a similar effect has also been developed.282 These techniques are generally not useful for high throughput tracking because half of the image space is sacrificed when splitting the image, but they are useful in certain scenarios where sparse tracking of a limited area can be employed. 4.4. High Temporal Resolution in Single Particle Tracking

Super-resolution microscopy techniques enhance the spatial resolution in SPT, while the temporal resolution remains restricted, mainly by the camera frame rate.283 Temporal resolution is also influenced by dye photophysics like blinking and bleaching.284−286 In biological experiments, photodamage 7359


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5. ACTIVE FEEDBACK TRACKING Active feedback tracking methods utilize real-time particle position to generate 3D trajectories over several micrometers within a system of interest. Although active tracking is inherently low throughput, single particles can be tracked over large distances in all three dimensions while maintaining nanometer spatial resolution and submillisecond temporal resolution. Moreover, feedback based tracking allows the particle of interest to explore a large variety of local environments within a system; thus the user is not limited to a single region of interest as with the previously covered nonfeedback based systems. Limitations do exist in the range a particle can be tracked contingent on the experimental method chosen along with the photophysical effects of fluorescent labels making it challenging to monitor single particles over long distances and times. Despite these challenges, advancements in active tracking can achieve high spatial and temporal resolution trajectories to help uncover hidden mechanistic details of single particle dynamics in real time. 3D SPT was first investigated by Peters et al. to uncover the dynamics of interfacial ligand interactions with transmembrane proteins.301

analyses of concentration dependent results allowed for unprecedented kinetic details in the transcription processes to be uncovered, showing heavy concentration effects on the transcription processes at varying degrees of chromosomal condensation. This use of stroboscopic SPT provided kinetic cues into the complex dynamics occurring in transcription processes in vitro and allowed for a kinetic model of transcription to be determined (Figure 35C). This work highlights the kinetic details that can be uncovered using stroboscopic techniques; however, the native frame rate of the camera still limits the achievable temporal resolution of the system, which can be overcome with other techniques. Recently we developed Super Temporal-Resolved Microscopy (STReM),283 in which the temporal resolution can be enhanced 20 times greater than the native camera frame rate. STReM encodes the temporal information in the PSF by rotating a DH phase mask in the Fourier domain, such that the orientation of the DH PSF lobes encodes subframe details, rather than depth, information.283 Figure 36A shows the 4f

5.1. ABEL Trap

The Anti-Brownian ELectrophoretic Trap (ABEL), established by Cohen et al.,302 is a feedback based method that allows for a particle to be physically trapped within a microfluidic system based on a feedback generated potential to offset intrinsic Brownian motion.303 This applied potential induces electrophoretic forces to iteratively counteract intrinsic Brownian drift via region-specific electrodes assembled within the microfluidic device (Figure 37).304 The applied potential electric field directly interacts with the trapped particle via electroosmotic forces to restore the particle to the center of the microfluidic cell iteratively after each acquisition period. The location of the particle was previously determined via camera based (CCD) fluorescence microscopy; however, to measure smaller biological particles exhibiting high diffusion coefficients in solution, high-speed orbiting lasers (discussed in section 5.2) have been implemented to improve the temporal resolution of ABEL.304−306 Reconstruction of a particle’s trajectory can be statistically estimated from feedback potentials applied throughout the trapping period along with photon locations by utilizing Kalman filter feedback, ADF, and MLE based algorithms as previously discussed in section 2.6 (Figure 37).304−307 Trapping nanoscale particles via electrophoretic forces is advantageous in comparison to optical trapping methods, given electrophoretic forces are stronger and are not dependent on second order field components (i.e., polarization). Although induced trapping forces are strong, ABEL is a noninvasive technique utilized in many applications including the exploration of DNA dynamics and the ability to perform single-protein Förster resonance energy transfer studies.308,309 It must be noted that this method is not truly tracking a particle; however, particles of interest can freely diffuse in solution and be monitored for longer periods while avoiding commonly used tethering chemistries, which can alter the inherent behavior of a particle interacting in a given environment.310 Additionally, ABEL allows for fluorescent properties to be observed over long time periods with high spatial and temporal resolution. This method is unique in its ability to precisely control the position of a single particle in a feedback system, thus allowing the particle of interest to explore its natural dynamics in solution.

Figure 36. Application of STReM to SPT with high temporal resolution. (A). A DH PSF is generated by installing a DH phase mask in the Fourier domain. (B). The DH PSF rotates simultaneously with the DH phase mask. (C, D). Simulated and experimental 2D trajectories observed in traditional microscopy. The ground truth trajectory is overlaid in (C). The temporal resolution in (D) is the camera exposure time, 100 ms. (E, F). Simulated and experimental 2D trajectories enhanced by STReM. The colored trajectories denote the recovered emitter movement with high temporal resolution. Figure adapted from ref 283. Copyright 2016 American Chemical Society.

scheme as discussed in section 4.3, in which a DH PSF is formed by adding a DH phase mask in the Fourier plane.258,262 Although the DH phase mask was initially designed to extract information about the depth of an emitter,262 in STReM, the experiment is constrained to 2D by exciting the samples in TIR mode, and the DH phase mask is rotated 180° within one camera exposure time so that temporal information about the emitter is uniquely encoded in the orientation of the PSF (Figure 36B). We demonstrated both in simulation and in experiments that STReM is able to enhance the temporal resolution in single particle tracking. In the detection of an analyte in fast 2D motion, only the spatial information can be resolved in high resolution without STReM, while the temporal resolution remains the same as the camera exposure time of 100 ms (Figure 36C,D). A complicated trajectory of a fastmoving emitter is captured with STReM (Figure 36E,F), with color denoting the temporal coordinates. The trajectories are recovered by solving an L1 norm constrained optimization problem with the alternating direction method of multipliers (ADMM) algorithm.261,298−300 The temporal resolution is improved 20-fold compared with the native camera frame rate, which makes it a powerful tool for future SPT applications. 7360


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Figure 37. (left) Fundamental setup of the feedback tracking in the ABEL trap with four electrodes (one in each lateral direction). The red laser spot outlines an orbiting confocal laser exciting a tracked particle; subsequent photons are detected on a single photon counter. The feedback system then relies on Kalman filtering methods to improve lateral localization of the particle (relies on intrinsic mobility and diffusion of trapped particle). As a result of this improved localization, restoring electrokinetic forces are applied (indicated by green arrows) to a particle within the microfluidic chamber in order to return the particle to the center of the trap. (right) Three-dimensional rendering of the microfluidic device in which the depth of the chamber is roughly 600 nm in area outlined by the red square.311 Adapted from ref 304. Copyright 2011 American Chemical Society.

Figure 38. Schematic of tetrahedral confocal based 3D tracking optical configuration. Tetrahedron detection volume in sample space is created through coupling of two fiber optic cable pairs before signal is detected on single photon detectors. Roughly 8% of wide-field emission light is detected on the EMCCD to prior to the fiber optic cables in order to extract contextual information on emitter location. Real-time feedback lock in detection is achieved by iteratively positioning the sample such that the emitter is in the center of the detection volume through the use of a threedimensional piezo scanner.232,315 Adapted from ref 232 and with permission from ref 315. Copyright 2010 American Chemical Society (ref 232) and 2015 Society of Photo Optical Instrumentation Engineers (ref 315).

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Figure 39. (A) Produced single particle 3D trajectory of a quantum dot labeled receptor on a mast cell (color gradient indicative of time further shown in parts B and D). (B) Photon counts for trajectory in (A) over time. (C) Representative contextual image of emitter location from EMCCD. (D) Axial localization of emitter showing a range of roughly 4 μm. (E) MSD (blue) of receptor over time fit to single exponential (red), exhibiting compartmentalization. (F) Photon correlation measurements indicative of antibunching for trajectory in (A) lasting roughly 14 s. (G) Fluorescent lifetime measurements (∼16 ns) with respect to laser excitation (red) and exponential fit (black). (H) Autocorrelation of photon detection rate dictated by intrinsic blinking of quantum dot label utilized. Adapted from ref 232. Copyright 2010 American Chemical Society.

after illumination acting as pinholes or by focusing four lasers into the sample volume of a confocal microscope.232,312−315 Trajectories of actively tracked particles are constructed in real time from the feedback-driven position of four SPADs, which can actively track an emitter over ∼20 μm in lateral and axial directions (Figure 38). Due to the noninvasive nature of this method, Werner and co-workers have investigated the complex process of protein transport in ligand induced endocytosis in live cells while extracting time-resolved information on the tracked proteins (Figure 39).232 Moreover, photon pair correlation analyses are representative of photon antibunching, indicating only one emitter was in the confocal volume throughout tracking periods (Figure 39).232 This method has also been able to actively track diffusivity variations in genetically fluorescent protein oligomers and experimentally achieve tracking of fast-moving proteins at speeds exceeding most intracellular protein transport processes.313 Most often quantum dots are used to label the particle of interest in order to provide a large photon flux; however, the intrinsic nature of blinking with these labeling methods can make tracking challenging. The challenges of photoblinking and large cellular based autofluorescence were recently overcome by DeVore et al.315 in live cells through the incorporation of time-gated data acquisition in the tetrahedral based active tracking setup shown in Figure 38. This was achieved by using novel nonblinking quantum dots along with a time-gated detection feedback system coupled with a pulsed laser excitation (Figure 40). These advancements allow for 3D SPT to be carried out in various cell lines where large photon fluxes originating from cellular autofluorescence are present. Moreover, these techniques allow for further investigation into the complex mechanistic details in particle transport processes in live cells (Figure 40).315

Figure 40. (A, B) Image of QD labeled receptor on rat mast cells without time gate detection (A) and (B) with time gate detection (4 ns). Comparison of two images shows significant decrease in background emission from intracellular structures. Each image has a linear scale between minimum and maximum pixel counts applied. Adapted with permission from ref 315. Copyright 2015 Society of Photo Optical Instrumentation Engineers.

5.2. Confocal Based Active Tracking

Tetrahedral based active tracking utilizes a stage scanning confocal microscope, coupled with single-photon avalanche photodiodes configured in a feedback loop such that particle location is highly sensitive in the axial and lateral dimensions while producing ∼5 ms temporal resolution.232,312−315 Utilizing confocal based detection is advantageous in these applications given the small illumination volume allows for a high signal-tonoise ratio to be achieved in complex environments while minimizing phototoxicity with low illumination powers (mW) and providing time-resolved fluorescence measurements of tracked particles. The use of a tetrahedral detection volume for 3D SPT was first used by Berg et al.316 and has recently been advanced in its capabilities with four coupled fiber optic cables 7362


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Figure 43. Fundamentals of signal detection using an optical cantilever. As an emitter is displaced in sample space, the signal difference between the two detectors is used to determine emitter location (inset upper left). The plot shows the normalized signal difference between the two APDs as a function of particle displacement in sample space. The signal differences were simulated for optical cantilevers of varying lengths (distance from back aperture of objective to detector). Optical cantilevers become more sensitive to smaller displacements as the length is increased. The relationship shown here was used to optimize the optical response for SPT with a cantilever length of a few meters. Adopted with permission from ref 83. Copyright 2015 Royal Society of Chemistry. Figure 41. Orbital lock-in tracking setup. (A) Gray area representative of orbiting confocal laser beam orbiting and exciting an emitter being tracked. Emission signal modulation is monitored in two planes to determine the lateral location and axial location from the difference in signals at the two image planes. The location of subsequent laser orbits is determined by a feedback loop driven by a piezo mirror in real time. (B) Setup of excitation and detection paths for orbital tracking showing simultaneous wide-field imaging on a CCD and multiple excitation wavelengths (488, 561, and 633 nm). The two focal planes monitored in (A) are detected two APDs as shown in (B). Adopted with permission from ref 38. Copyright 2011 Royal Society of Chemistry.

the spatiotemporal resolutions of 3D tracking is orbital tracking.38 This method was inspired by the work of Enderlein,317 which originally was designed for tracking fluorescent single particles in 2D. The evolution of this concept into the 3D tracking domain evolved from the work of Levi et al.,318 initially based on scanning FCS319 and was then applied to study single protein aggregation dynamics in solutions.320 This lock-in technique is based on a laser scanning system where a confocal laser beam orbits a single fluorescent particle during tracking periods. A feedback loop then repositions the laser beam via piezo controlled mirrors, or galvanometric based mirrors, to re-center the emitter after each time step of the system’s response. The emitter’s lateral location is determined in real time using fast FFT calculations on fluorescent intensity

An additional active 3D tracking method utilized to investigate many biological systems of interest while improving

Figure 42. (A) Three-dimensional trajectory of artificial virus (shown in blue) in live eGFP-tagged (green) human hepatome cells. Gray projections are 2D projections of the blue trajectory to show relative distance traveled in all three dimensions. Included are two wide-field z-slice images from sample space further shown in (B). Arrows in (B) are indicative of lateral microtubule network motion with blue trace still representing original trajectory from (A). Adopted with permission from ref 322. Copyright 2009 John Wiley and Sons. 7363


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produced a spatial resolution of 20 nm and a time resolution of 32 ms that could follow an emitter up to 10 μm in all three dimensions. Another method shown by Katayama et al.322 uses a onephoton excitation, while simultaneously monitoring two focal planes above and below the particle through confocal pinholes; the resulting signal modulation from these two channels is used to determine the axial location of an emitter (Figure 41). A piezo based objective holder is used in this method to center the emitter in the detection volume between each time step of the trajectory; it is the confocal pinholes that are adjusted iteratively in order to extract axial locations. As a result of the objective not continuously oscillating during measurements, higher NA water immersion objectives were used to simultaneously acquire wide-field images of the current focal plane where the emitter is located (Figure 42). The ability to extract a trajectory and the contextual widefield images of a system can give tremendous insight into the dynamics within local confinements of biological systems. Dupont et al.323 utilized the orbital tracking method outlined by Katayama et al.322 in order to investigate the anisotropic nature of cellular diffusion of nanoparticles and how that information is not recovered in traditional 2D measurements of the same cell types. Moreover, Strano324 and co-workers tracked single wall carbon nanotubes (SWCNTs) to explore local cellular viscosities and active transport velocities within various cellular regions in HeLa cells. The use of SWCNTs is advantageous due to their large Stokes shift and the inherent emission in the near-infrared region reducing autofluorescence background signals and allowing deep tissue detection of fluorescent signal.325 In addition, recent advances have been made in the capabilities of orbital tracking to further improve the temporal resolution and axial tracking range of 3D orbital tracking. Lanzanò and Gratton were able to achieve SPT in 3D by designing a piezo feedback controlled system from a commercially available confocal microscope.326 They demonstrated the viability of this setup to achieve SPT with the transport of vesicles within epithelial cells. Furthermore, Gratton and co-workers were able to improve the temporal resolution of orbital tracking to 8 ms and increase the tracking range of the system by 2.5 times.327 This was achieved utilizing a commercially available electrically tunable lens (ETL) to replace the use of piezo based stages, effectively eliminating the movement of the objective in the previously mentioned setups. The ETL allows for the divergence of the laser to be quickly changed at the back aperture of the objective by varying the focal length of the lens with an applied current, thus promptly controlling the image plane of the system. Moreover, the tunable range of the ETL is 500 μm in comparison to a piezo stage actuator with a range of only 200 μm, thus allowing much longer axial tracking regimes to be probed in a system. As a result of the objective movement being eliminated, a high NA oil immersion objective could be used, allowing for more photons to be captured while not affecting the photobleaching rates of emitters. This novel setup extracts 3D trajectories of single fast-moving genomic loci inside the nucleus of live cells.327 This work highlights the low cost conversion of current scanning microscopes to achieve unprecedented axial tracking ranges and temporal resolution for orbital tracking 3D SPT. In order to push the frontier of the spatiotemporal regimes (Figure 3) of 3D active tracking, 3D multiresolution microscopy (3D-MM) was established by Yang and

Figure 44. (A) Overhead view of PS-QD-peptide approaching protrusion of live cell. Inset is zoomed in on the area where particle interacts with cell surface. (B, C) High-resolution views of trajectory in (A) from two different vantage points with color gradients still indicative of time. (D) Trajectory represented in (A) with color gradient representing diffusion over the evolution of the trajectory. (E) Two-photon LSM image of cell expressing GFP-tubulin with overlaid high resolution trajectory from (A) with scale bar representing 5 μm. (F) Three-dimensional trajectory of PS-QD-peptide exploring cell surface uncovering small semicircular protrusions along the cell membrane. (G, H) Diffusional states analysis and corresponding diffusional gradient plotted within the trajectory represented in (F). Adopted with permission from ref 328. Copyright 2014 Nature Publishing Group.

modulations as the emitter deviates from the center of the detection volume as previously shown by Kis-Petikova and Gratton321 (Figure 41A). The orbit of the confocal beam rotates at a diameter equal to the width of the emitter’s PSF. Orbital tracking is uniquely advantageous given the FFT calculations used to produce the location of a tracked particle are not affected by homogeneous background noise (e.g., autofluorescence or shot noise) in a given environment. Lateral coordinates of the tracked particle can be determined through various experimental techniques. Levi et al.58 utilized twophoton illumination coupled with a piezo nanopositioner to move the objective in order to probe two focal planes, separated by half the width of the PSF. The fluorescent intensities at the two focal planes are then compared to determine the axial location of the emitter. This method 7364


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Figure 45. (A) Three-dimensional tracking setup using two-photon multiplexing in which four laser beam pulses arrive in sample space every 3.3 ns iteratively. Lock-in tracking is achieved with scanning mirrors and a piezo objective stage (axial focus). (B) Photon counts of emitter located in the center of the detection volume showing separation of signal consistent with time delay of laser pulses from beam multiplexer. (C) Detection volume geometry from various vantage points. (D) Experimental image results of 100 nm bead with four beam excitation using half-wave plate (HWP), polarizing beam splitter (PBS), dichroic mirror (DM), and beam dump (BD) (scale bar is 1 μm). Adopted with permission from ref 331. Copyright 2015 Nature Publishing Group.

Welsher.83,328 The demand to increase the spatial and temporal resolutions of these tracking methods is advantageous to investigate highly dynamic processes occurring on much faster time scales than previously covered methods. The inception of 3D-MM stemmed from the work of Cang and Yang,329,330 which created a confocal based setup, forming an intensity gradient in the axial direction of their detection path to extract the z location of a tracked particle of interest. This method did suffer from a low spatial resolution of 210 nm; however, it achieved a temporal resolution of 1 ms. This method was modified330 to encode the lateral location of an emitter by splitting the signal with a prism to two SPADs. The difference in the signal at the detectors was used to determine the lateral locations, while the axial location was still determined using an intensity gradient as mentioned.329 This method was most recently improved with 3D-MM using an elegant combination of optical cantilevers in the detection path to achieve a 3D target locking feedback system in combination with simultaneous two photon-laser scanning microscopy (2P-LSM) detection to image large contextual features in the region of the emitter.83,328 The use of optical cantilevers in the 3D lockin tracking portion of the setup allows for the displacement in lateral directions to be magnified roughly 3000× such that the system achieves high sensitivity in particle position in sample space while achieving a large sensitivity in optical response from diverging photons on the APD in the detection path (Figure 43). 3D-MM has achieved a temporal resolution of 10 μs and a spatial resolution of 10 nm in both the lateral and axial dimensions. 3D-MM was used to track polystyrene-QD labeled Tat peptide (HIV-derived) interacting on the surface of NIH3T3 cells (Figure 44).83,328 This study uncovered unprecedented 3D details in the anisotropic diffusion of the labeled peptide near protrusions on the cell membrane. This example further highlights the powerful information that can be learned from the use of novel active 3D tracking systems used to acquire dynamic details from uncharted spatiotemporal domains while providing insight into the relationships of

structures to biological interactions in highly dynamic environments. 5.3. Nonconfocal Based Active Tracking

A recently developed method, coined “TSUNAMI” (Tracking Single Particles Using Nonlinear and Multiplexed Illumination) by Dunn331 and co-workers, utilizes two-photon (2P) microscopy in order to achieve lock-in 3D SPT at large penetration depths (200 μm) in highly scattering environments. The excitation beam, focused through a high numerical aperture objective, is pulsed (13 ns duration) and separated into four beams to form a tetrahedral shaped detection volume in sample space (Figure 45A−C). Each separated beam is delayed by 3.3 ns respectively, shown in Figure 45B, providing evidence of only one emitter present in the detection volume during tracking periods. The inception of this method is based on nonlinear microscopy, in which the implementation of passive pulse splitters has been shown to reduce the intrinsic photobleaching rates in 2P microscopy in addition to delivering a controlled number of equal energy subpulses to the system.332 This is coupled with a beam multiplexer setup used by Cheng et al.333 allowing for multiplexed 2P illumination to be achieved with periods of demultiplexing to extract the axial location of the particle. The range to which a particle can be actively tracked using this method is limited to the range of piezo scanners (Figure 45A). The unique capabilities of TSUNAMI lie in the ability to perform 3D SPT with a single detector while simultaneously achieving 2P multicolor microscopy; in addition, time-resolved fluorescent measurements can also be conducted with this technique. TSUNAMI was used to track single growth factor receptors tagged with 10 nm fluorescent beads as they were actively transported across the membrane of tumor spheroids at depths of up to 100 μm (Figure 46). TSUNAMI can achieve a spatial resolution of 22 nm laterally and 90 nm in the z-direction with temporal resolutions ranging from 1 ms to 50 μs dependent on the photon budget emitted (i.e., labeling method) from the tracked particle. Lastly, the axial resolution information extracted from TSUNAMI was 7365


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Figure 46. Deep 3D SPT in cancer spheroids. (A) Three-dimensional contextual image of spheroid captured utilizing 2P scanning microscopy with red stain for the plasma membrane and blue stain for the nuclei. The green image plane 50 μm into the spheroid is indicative of the plane in which tracking will be carried out with white circle highlighting area of subsequent trajectory. (B) Cross-section representation of location of trajectory in black circle which is zoomed in within the inset showing time dependence of the trajectory. (C) Zoomed-in image of trajectory in (B) originating from within the cell. (D) Trajectory plotted with no contextual background from same perspective as shown in (C). (E) Instantaneous velocity over the evolution of trajectory in (C) and (D). Adopted with permission from ref 331. Copyright 2015 Nature Publishing Group.

improved with the application of an MLE based algorithm.334 From the Monte Carlo simulations executed using this MLE on TSUNAMI trajectories the axial resolution was increased 1.7fold. Advancements in the field of 3D SPT have been indispensable in probing complex biological systems in situ, further elucidating diffraction-limited mechanistic details of biologically important molecules carrying out their functions. The aforementioned techniques have allowed for dynamic processes to be tracked at the single-molecule level in 3D, which has uncovered rich contextual structures and anisotropic diffusional details within single-molecule trajectories. In our review of 3D SPT we covered two main categories of techniques: intrinsically high throughput tracking techniques followed by active tracking methods, which are inherently low throughput. Although both types of 3D SPT have their advantages and drawbacks, improving the spatial and temporal resolutions of 3D SPT is imperative to expanding the systems where SPT is feasible.

systems. The evolution and development of SPT in the past three decades is the result of continuous efforts from chemists, biochemists, physicists, and engineers to further advance its utility and applicability. As a result, SPT has become a rapidly growing and interdisciplinary field where no single review can cover all the relevant topics. We aimed to provide both enough depth through a brief description of underlying theories and enough breadth by highlighting a range of methodologies utilized in emitter localization and trajectory reconstruction, recent instrument developments to achieve 2D and 3D subdiffraction tracking, as well as some of their applications in cell biology and other materials. As for future directions, it is almost certain that SPT as a field has not reached full maturity. In particular, more efforts are needed in the search for brighter and more photostable tagging species that induce minimal perturbations to the analyte or host matrix. As we have discussed in previous sections, QDs seem to be the best choice to date given their high quantum yield and relatively small physical sizes. However, it is well-known that most QDs suffer from photoblinking,335 which makes trajectory reconstruction in later stages of data analyses more difficult. Moreover, QDs also require capping ligands for studies conducted within cellular environments, thus setting limitations

6. CONCLUSION AND PERSPECTIVE We hope we have shown that SPT is a powerful tool for interrogating complex dynamic processes in a wide variety of 7366


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on their applications. Breakthroughs in the development of better fluorescent reporters would greatly benefit the entire field of optical microscopy,336−338 including SPT studies. Alternate excitation/detection schemes that do not rely on linear fluorescence microscopy also present opportunities for improvement.339,340 We also predict the incorporation of modern signal processing methods in SPT. It will be important to address the “big data” issues plaguing the newest 3D SPT applications due to the challenges introduced by transferring, saving, reconstructing, and analyzing large volumes of data that are intrinsically sparse in information. At the same time, as was found for the earlier SPT challenge,138 we do not foresee that there is a single one-size-fits-all solution. We introduced a few representative algorithms in section 2, and hopefully with the aid of this review, readers can find or develop the most suitable algorithm based on their unique sample conditions. In the future, the integration with compressive imaging methods341−345 and machine learning algorithms125,346,347 might allow SPT researchers to reach the ultimate goal of real-time analysis. Finally, we suggest that the types of analytes studied using SPT will continue to expand. In section 3, we focused on the most common applications of SPT including studying diffusion in biologic systems, and on inorganic and polymeric surfaces. Additionally, we reviewed various diffusion models and the important ergodicity assumptions and their roles in this section. To date ergodicity breaking has been reported mostly in biophysical systems; whether it is a universal behavior that also exists in other systems remains a fundamental question to address in future studies. Recent examples of new applications include a study of the free-radical polymerization mechanism and a review of how nanoscale structure governs mass transport in a range of materials.348,349 Achieving a mechanistic understanding of a broad range of nanoscale chemical, physical, and biological processes will be possible in part by continuing advances in SPT.

work focused on understanding the mechanisms of interfacial transport at tunable polymer interfaces at the single molecule level. He received his B.S. in chemistry from Louisiana State University, where he performed research for Dr. Bin Chen and Dr. Jayne Garno. Rashad Baiyasi was born in Michigan, where he received his B.S. in physics from Saginaw Valley State University. He is currently a graduate student at Rice University in Houston, TX. His work focuses on improving computational analysis techniques for super-resolution imaging and single particle tracking. He is particularly interested in the shifting and splitting of fluorescent emission in the vicinity of plasmonic nanoparticles. Wenxiao Wang is a graduate student in the Landes research group in the Department of Electrical and Computer Engineering at Rice University. His current research is focused on applying computational method to improve super-resolution microscopy, including both the spatial resolution and temporal resolution. Nicholas Moringo obtained his B.S. in chemistry from the University of California, Irvine, in 2014. He joined the Landes group at Rice University in 2015, where his work has focused on single protein− polymer interactions for modeling chromatography based protein separation systems. As a recent Ph.D. candidate he was awarded the NSF GRFP Fellowship in 2017 and plans to further pursue his interest of protein separation systems at the single molecule level. Bo Shuang graduated from Rice University in 2016 with a Ph.D. in physical chemistry under the supervision of Prof. Christy F. Landes. His research focused on strategies of data analysis for 2D and 3D single molecule microscopy. He is working for Dow Chemical Co. in Freeport, TX, focusing on chemical process optimization. Christy F. Landes is an associate professor in the Departments of Chemistry and Electrical and Computer Engineering at Rice University in Houston, TX. Her research group develops nextgeneration spectroscopic tools to understand dynamics at soft nanoscale interfaces.

Corresponding Author

ACKNOWLEDGMENTS The authors acknowledge the Welch Foundation (Grant C1787) and the National Science Foundation (Grant CHE1151647) for support of this work.

*E-mail: cfl[email protected].

REFERENCES

AUTHOR INFORMATION

ORCID

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Hao Shen: 0000-0002-2798-5861 Nicholas Moringo: 0000-0002-0044-255X Christy F. Landes: 0000-0003-4163-6497 Notes

The authors declare no competing financial interest. Biographies Hao Shen is a postdoctoral research associate working with Prof. Christy F. Landes in the Department of Chemistry at Rice University. He received his Ph.D. from Cornell University in 2014 under the supervision of Prof. Peng Chen. During his Ph.D., he studied heterogeneous catalysis and electrochemical catalysis at the single molecule level. His current research is mainly focused on exploring protein−synthetic polymer interactions using single molecule spectroscopy. Lawrence Tauzin is a postdoctoral research fellow working for Dr. Stephan Link at Rice University studying the mechanisms of noble metal photoluminescence. He received his Ph.D. from Rice University in 2016 under the supervision of Dr. Christy F. Landes. His Ph.D. 7367


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