Fast Three-Dimensional Single-Particle Tracking in Natural Brain Tissue

Abstract

Observation of molecular dynamics is often biased by the optical very heterogeneous environment of cells and complex tissue. Here, we have designed an algorithm that facilitates molecular dynamic analyses within brain slices. We adjust fast astigmatism-based three-dimensional single-particle tracking techniques to depth-dependent optical aberrations induced by the refractive index mismatch so that they are applicable to complex samples. In contrast to existing techniques, our online calibration method determines the aberration directly from the acquired two-dimensional image stream by exploiting the inherent particle movement and the redundancy introduced by the astigmatism. The method improves the positioning by reducing the systematic errors introduced by the aberrations, and allows correct derivation of the cellular morphology and molecular diffusion parameters in three dimensions independently of the imaging depth. No additional experimental effort for the user is required. Our method will be useful for many imaging configurations, which allow imaging in deep cellular structures.

Introduction

Fitting point-spread function (PSF) models to images of single molecules allows computation of molecule positions with nanometer localization accuracy (LA), well below the diffraction limit (1). Such techniques have been extended to three dimensions by exploiting a relationship between features of a particles’ PSF and its axial distance to the focal plane. However, extraction of the axial position has proven difficult because the axial symmetry of the PSF prevents distinguishing whether a particle is above or below the focal plane, and the shape of the PSF contains only limited information about the axial position of the particle within the depth of field of the microscope.

Changing the light path to induce an intentional astigmatism in the PSF breaks the symmetry of the PSF and provides additional axial information. The simplest method is the introduction of a cylindrical lens into the light path (2–4). Slightly better performance is achieved by using adaptive optics (5). Similar LA can be achieved by imaging the particles at two axially separated focal planes so that the axial position is unambiguously defined (6,7). Another method, which features a nearly constant axial LA over the depth of field of the microscope, creates a double-helix PSF and extracts the axial position from the angle between the two lobes (8).

All these techniques assume that a single three-dimensional PSF (and consequently a constant relationship between the PSF and the relative axial position of the particle) is valid throughout the sample. However, when imaging several tens of micrometers deep in the tissue, the mismatch in the refractive indices (RI) between the objectives’ embedding medium and the biological specimen leads to depth-dependent spherical aberrations of the PSF (9). Therefore, the computed positions of contemporary techniques would exhibit a systematic error (SE). The position is simply incorrect, although the LA might be high.

Three approaches try to tackle this issue. 1) Strategies that minimize the influence of the refractive index mismatch (RIM) use appropriate objectives to match the RI of the respective sample (10), or compensate aberrations by adding a correction lens (11) or adaptive optics (12). 2) Methods that perform prior depth-dependent calibration measurements accept the aberrations and adjust the axial relationship according to the imaging depth (4,13). 3) Computational techniques predict the aberrated three-dimensional PSF for a certain depth and fit it to the acquired data (14,15). These methods are generally labor-intensive, owing to the specialized experimental implementation and parallel manual calibration for each acquisition. For dynamic analyses in living tissue, time-consuming calibration procedures are not favorable, and often in conflict with the experimenter’s demands.

Therefore, we focused on a fast online calibration method for astigmatism-based three-dimensional single-particle tracking (SPT) techniques. Our method determines the varying relationship between the width of the two-dimensional PSF and the relative axial position of a particle directly from the acquired two-dimensional image stream. This is accomplished with only minor preconditions and without individual calibration procedures. Thus, our method intends to make three-dimensional SPT in living tissue (for instance, brain slices) readily available.

The basic idea of this method has been published by us in Sokoll et al. (16), but here we considerably extend the initial results and incorporate it into a complete three-dimensional SPT workflow. We evaluate our method on a spinning-disk, confocal microscopy imaging configuration that we selected as a compromise for temporal resolution, axial sectioning capability in packed tissue, and broad availability.

Materials and Methods

Sample preparation

We prepared organotypic slice cultures as described in detail in Biermann et al. (17). Briefly, 350-μm-thick hippocampal slices from postnatal day 4–6 Bl6 wild-type mice were prepared and cultured. We used a hand-held Helios gene gun (Bio-Rad, Hercules, CA) to transfect the slices on the day of dissection. The plasmid DNA (pCMV-GFP-GPI) was coupled to gold microcarriers and shot into the tissue. For imaging, slices between DIV5 and DIV21 were used and the secondary antibody Qdot655 was connected to the primary monoclonal antibody anti-GFP mouse IgG to detect the molecular motion of GPI-GFP. For incubation, we cut individual slices from the membrane and added a dilution of $1:1000$ antibody-coupled quantum dots (QDs) and extracellular solution to the slice. After 5–10 min of incubation, the slices were transferred to the imaging configuration.

Microscopic setup

We attached the slices to the coverslip using a net that was clamped at our custom-built perfusion chamber. This allowed us to observe QD-labeled molecules in brain slices at 33–35°C using a confocal setup, consisting of an upright microscope (model No. BX51WI; Olympus, Melville, NY), a spinning disk (model No. CSU X1; Yokogawa, Newman, GA), a back-illuminated thinned electron-multiplying charge-coupled device camera (iXon^EM + 897, 512 × 512 pixels of size 16 × 16 μm²; Andor Technology, Belfast, Northern Ireland), and an oil immersion objective (UAPON 100XO TIRF, 1.49 NA; Olympus). QDs were excited by a tunable laser (Genesis, 488-nm wavelength, maximum power 1000 mW; Coherent, Santa Clara, CA), typically adjusted to 400 mW. Time-lapse images were acquired with a temporal resolution of 30 Hz with up to 1000 images per stream, using the software IQ (Andor Technology). The required astigmatism of the PSF was established by introducing a plano-convex cylindrical lens (model No. LJ1516RM, f = 1000 nm; Thorlabs, Newton, NJ) at distance d_cl = 4.5 cm directly in front of the camera chip. This distance represents a compromise between a reasonable astigmatism (so that well-distinguishable correlation functions are obtained) and the fact of the astigmatism additionally blurring the PSF (which lowers the effective signal/noise ratio (SNR)).

Axial correlation function

Our online calibration method adjusts astigmatism-based three-dimensional SPT techniques to the aberrations induced by the RIM so that they are applicable deep in living samples. At first, the influence of the RIM will be quantified and an axial model that describes the induced aberrations is established.

To determine the influence of the RIM on the axial correlation function that represents the correlation between the size of the PSF and the relative axial position, we first analyzed 30-nm-diameter fluorescent beads with a 580–605-nm emission wavelength at different imaging depths, using an oil immersion objective (UAPON 100XOTIRF objective, 100× magnification, 1.4 NA; Olympus). The beads were dissolved in Mowiol (mixture ratio: 0.1–1 μL beads solution to 2 mL Mowiol) and drops of this solution were put on coverslips for curing.

The axial correlation curves for beads at depths of 0–30 μm are presented in Fig. 1 a. They are obtained by axial scanning of ∼20 beads at each depth and plotting of the average full width at half-maximum (FWHM) from elliptical Gaussian fits to the two-dimensional PSF at each relative axial scan position. Each bead was scanned once with an axial step size of 20 nm over an axial range of 2 μm and a temporal resolution of 10 Hz. For negative relative z positions the focal plane is above the center of the particle, and for positive positions, below.

Figure 1.

(a) Measured axial correlation curves of immobilized beads at different depths in Mowiol. (b) Fits of the proposed correlation function (in red) to calibration measurements at selected depths. The individual axial fitting range is indicated (dashed vertical lines) at the bottom of each graph. To see this figure in color, go online.

The well-established parabolic correlation function holds true only for beads located directly under the coverslip, where the RIM is effectively negligible. With increasing depth the three-dimensional PSF becomes more and more distorted, resulting in increasingly skewed parabolic correlation curves. Also, the effective resolution at the center of the particle decreases as a function of the depth.

We propose to model the aberrations of the axial correlation function by extension of the parabola in Schütz et al. (18) to

$$\mathit{\sigma }\left(z\right)={\mathit{\sigma }}_0\times \sqrt{1+{\left(\frac{z}{d}\right)}^2}+m_r\times z.$$ (1)

Here, z is the relative axial position, d is the focus depth, and σ₀ is the FWHM in focus. The extra parameter m_r controls the skewness of the correlation function. We established this model empirically and it can be considered as adding a straight line with depth-dependent slope to the symmetric correlation function at zero depth.

Fig. 1 b illustrates for selected depths that the proposed correlation function is capable of approximating the induced aberrations independently of the imaging depth. The fit residuals are only in the range of ±5 nm.

To allow for unambiguous determination of the axial position, we inserted a plano-convex cylindrical lens into the light path of the spinning-disk, confocal microscopy setup. This slightly shifts the focal plane for one lateral direction along the axial axis so that the two-dimensional PSF becomes elliptical. Furthermore, the major axis is switching halfway between the two focal planes as is illustrated on the left in Fig. 2 a. The two correlation curves then exhibit an axial shift Δ_f (Fig. 2 a), which allows to unambiguously define the axial position. Note that the real center of the particle is at the vertex of the correlation curve unaffected by the lens, and therefore their vertex represents the zero position.

Figure 2.

(a) Shifted individual correlation functions acquired in confocal mode. The values Δ_f and Δ_σ are illustrated. The additional offset Δ_σ is an inherent artifact that appears for confocal systems (see Section S1 in the Supporting Material). The true particle center is at the vertex of the y curve. (Left) The corresponding images. (b) FWHM and focal shift as a function of the imaging depth. To see this figure in color, go online.

There is also an offset in the FWHM value of the curve vertices, here denoted as Δ_σ. We identified Δ_σ as being an inherent artifact of using a cylindrical lens with confocal setups (see Section S1 in the Supporting Material).

Therefore, next to the focal shift, the FWHM shift also has to be known for confocal setups. The course of both shifts as a function of the imaging depth is depicted in Fig. 2 b. It is apparent that they are virtually constant and independent from the imaging depth, yielding Δ_σ = 60.25 ± 5.71 nm and Δ_f = 278.40 ± 45.37 nm, respectively, as the average values. A notable deviation is apparent for the measurements at zero depth, which could be an artifact of the astigmatism produced by the cylindrical lens. However, because we are interested in imaging molecular dynamics within the samples, this effect can be neglected. The constant FWHM shift can simply be subtracted from the measurements and does not yield a loss of generality of our approach.

Axial online calibration method

To compute the correct axial position, the parameterization of the correlation function at this imaging depth has to be known. We now present our online calibration procedure that determines the axial correlation function directly from the two-dimensional image stream.

The key ideas of our method are as follows: because the correlation function is defined by three parameters, it can be reconstructed by knowing three coordinates of the curve; for the purpose of obtaining these coordinates, the inherent molecular particle motion can be used and, on that account, the redundancy introduced by the astigmatism can be exploited. Our method basically consists of three steps:

• 1)identification of vertex frames,
• 2)computation of the three required coordinates, and
• 3)estimation of the curve parameters.

Vertex frame identification

To estimate the FWHM of the vertex of the correlation curve, the inherent molecular motion is exploited instead of avoided. It is assumed that during the acquisition of a single two-dimensional image stream, some particles pass the two artificial focal planes of the imaging configuration several times.

Fig. 3 a presents just the axial component of a particle. Owing to three-dimensional diffusion, obstacles, or the curvature of surfaces, the purely axial speed is unlikely to be constant. However, for simplicity we assume that the particle basically has some directed motion along the axial axis. Then, the respective focal planes in the x and y directions are crossed at different frames and the vertex point of the curves corresponds to the moment when a particle is in either focal plane. To identify those images, where a particle is in either focal plane and consequently represent a vertex position, the change of the FWHM values during particle movement is analyzed.

Figure 3.

Basic principles of the algorithm. (a) Trajectory of a particle that moves along the axial axis and the corresponding FWHM measurements. (Red marks) Vertex identification model. (b) Derivation of the three required coordinates (in red) by exploitation of the redundancy introduced by the cylindrical lens. To see this figure in color, go online.

The corresponding vertex identification model is defined as follows: a vertex frame is any image where the measured FWHM in one lateral direction is smaller than those of the two neighboring images. This is indicated by red mark 1 in Fig. 3 a. The FWHM values in the other lateral direction have to be either constantly increasing or decreasing in this neighborhood; see red mark 2. Finally, red mark 3 refers to the requirement that the FWHM value of v must be smaller than that of the opposite lateral direction. Only if all three rules apply, can an image be expected to represent a particle in focus. Because the model includes just a few frames, this holds true irrespective of whether a particle has directed motion as in our simplified scheme, or if it just crosses a single focal plane and returns. However, it is also apparent in the top graph of Fig. 3 a that, owing to the temporal sampling, it is unlikely that a particle is always acquired exactly in focus. Because the true axial distance to the center of the particle can be assumed to be stochastic, a robust estimate for the FWHM value of the two vertices can then be computed by the median FWHM of all identified vertex frames f_vx and f_vy. To improve the vertex frame identification, it should be computed on smoothed images (see Section S2 in the Supporting Material).

Computation of the three required coordinates

The redundancy introduced by the astigmatism is exploited to derive estimates for two more coordinates, located left (l) and right (r) of the vertex. Because the curves are identical and just axially separated by Δ_f, the parameters l and r are chosen to be the points at axial distance ±Δ_f from v. Then all three coordinates can be estimated from the identified vertex frames.

From Fig. 3 b, it is apparent that in this setting ly corresponds to the vertex vx, and rx to the vertex vy. Because the widths in both lateral directions are always known, the FWHM value of ly can be computed from the median of the FWHM values of the vertex frames f_vx in the lateral x direction. Similarly, the FWHM value of rx can be computed from the median of the FWHM values of the vertex frames f_vy in the lateral y direction. The FWHM value of ly is the same as the FWHM value of lx at distance −Δ_f from vx. Respectively, the FWHM value of rx is the same as the FWHM value of ry at distance +Δ_f from vy. Consequently, they can be jointly used to construct each individual axial correlation function.

Because the absolute axial position of a particle is not of interest, the axial coordinate of v is simply zero and the three general coordinates are finally given by

$$\left(\begin{array}{c}v\\ l\\ r\end{array}\right)=\left(\begin{array}{cc}0& {\tilde{\mathit{\sigma }}}_{xy}\left(f_v\right)\\ -{\Delta }_f& {\tilde{\mathit{\sigma }}}_y\left(f_{vx}\right)\\ +{\Delta }_f& {\tilde{\mathit{\sigma }}}_x\left(f_{vy}\right)\end{array}\right).$$ (2)

Here, $\tilde{\mathit{\sigma }}$ denotes the median operation for any number of σ-values. Note that, for simplicity, only the three general coordinates v, l, and r are stated. The extra notation of whether they belong to the x or y curve can be omitted because vx_x is simply vy_x − Δ_f, and so on.

Estimation of the parameterization

The parameters m_r, σ₀, and d of this axial correlation function can be computed by setting up a nonlinear equation system (see Section S3 in the Supporting Material). However, to avoid the evaluation of ill-conditioned expressions, we use a numeric root-finding procedure. We employed least-squares-based nonlinear curve fitting, which has the advantage that complex solutions can be prevented and instead a near-real solution is always obtained.

To maintain the true vertex of the fitted curve at v, an additional shift, z_s, is required (compare to Section S3 in the Supporting Material). To enforce this additional side condition, although there is now one coordinate less than required parameters, the key idea is the substitution of z_s by the remaining parameters. For this, z_s is defined to be the axial shift of the true vertex position that originates from the original parabola being skewed by means of the term m_r⋅z. The value z_s is illustrated in Fig. S2 b in the Supporting Material and can be computed by

$$z_s=\pm \sqrt{\frac{m_r^2{d}^4}{{\mathit{\sigma }}_0^2-m_r^2{d}^2}}.$$ (3)

This simply corresponds to the root of the first derivation, because the original vertex is by definition at v_x = 0 (see Eq. 2). Because the RI of the specimen is typically much smaller than that of the immersion medium, the RIM leads to axial correlation functions with positive shift z_s. However, owing to the manufacturing variance of the coverslip thickness, the curve may also be skewed in the opposite direction. This occurs most likely at low imaging depths. Errors in the coordinate estimation may also result in that phenomenon. Therefore, the appropriate solution of Eq. 3 is selected according to whether l_y or r_y is larger.

During the fitting procedure, z_s is not considered as a free parameter, but instead computed by use of Eq. 3 and these parameter estimates. Consequently, the true vertex of the skewed parabola is always enforced to be at v_x = 0. As a result of this substitution, the actual underdetermined problem can be solved using just three coordinates because the fitting procedure indirectly maintains the additional side condition. This procedure yields adequate real approximations to the wanted correlation function, as is exemplarily shown in Fig. S2 a.

Results and Discussion

We evaluate our three-dimensional SPT workflow on synthetic, semisynthetic, and real data. To achieve this, we implemented a complete SPT workflow: briefly, our particle detection procedure is based on the spot-enhancement filter of Sage et al. (19) in combination with a watershed algorithm for separation of neighboring particles. For shape estimation of multiple neighboring particles, we propose the application of an expectation maximization algorithm (20). Our tracking procedure draws on a local linking strategy based on the Hungarian algorithm (21), and considers blinking, splitting, and merging events.

Performance evaluation on synthetic data

Because the aberrations induced by the RIM and the cylindrical lens leave the symmetric distribution along the lateral directions intact, the center coordinates are independent of the individual FWHMs. Consequently, we analyze the axial LA in isolation from the lateral LA. With that said, the lateral position of simulated particles is kept constant, the shape of a particle is only varied according to its relative axial position, and the simulated trajectory has just an axial component. The details of SPT data simulation are described in Section S4 in the Supporting Material.

The following evaluation configurations always comprise 50 realizations for each data point. To allow for comparable results at different depths, the simulated trajectories are identical across depths.

Quantifying the systematic error and the localization accuracy

In contrast, the LA relates to the standard deviation of estimated positions, but the total LA is again the arithmetic mean of all LAs within the axial range. The SE is the difference of the estimated value from the correct axial position that exists independently of the actual SNR of the images. It is solely defined by the deviation of the calibration curves from the true underlying axial correlation functions of the data. However, the estimation of the correlation function depends on the SNR and therefore, the SE that is obtained by the online calibration method depends on it.

Fig. 4 a presents the SE as a function of the relative axial position and the SNR. For a certain axial position, it is computed as the arithmetic mean of the absolute SEs of all realizations. Because the principle course of the curves is similar for all depths, this is depicted for 10 μm depth. Clearly, the SE deteriorates at both sides from the center between the two focal planes. This is reasonable because the absolute deviation of the estimated and the true curves typically increases with distance to the center, for any deviation in the determined coordinates. Furthermore, the SE decreases for higher SNRs, indicating that the approximation of the correlation functions improves with the quality of the images.

Figure 4.

(a) The SE as a function of the relative axial position and the SNR. (b) The total SE as a function of the imaging depth and the SNR. (c) The axial LA as a function of the relative axial position and the SNR. (d) The total axial LA for the online and the prior (dashed line) calibration method as functions of the imaging depth and the SNR. To see this figure in color, go online.

To allow for representation of the SE as a function of the imaging depth and the SNR, we define the total SE, computed as the arithmetic mean of all SEs within an axial range of ±300 nm from the center between the two focal planes. This is presented in Fig. 4 b. The total SE exhibits an apparent tendency for deterioration toward higher imaging depths. This is most obvious for the first 10 μm, where the axial correlation function changes most in consequence of the RIM.

This deterioration is likely to result from the fact that the axial correlation curves become generally flatter with increasing depth. Therefore, the FWHM differences between frames decrease for the same axial increment of a particle. However, the SE depends more notably on the SNR and is almost eliminated if noise is not present, which proves the working principle idea of our online calibration to be true.

Fig. 4 b presents the SE if only a calibration at zero depth is used, which, in the following, is denoted as “prior calibration”. It results in worse SE than the prior calibration at zero depth. However, for any other depth, the SE is always substantially (≥50%) reduced by our online calibration method.

The axial LA cannot be directly computed. Instead, we rendered 30 individual images for each realization and axial position. The axial LA then relates to the standard deviation (SD) of the corresponding estimated positions.

Fig. 4 c depicts the axial LA as a function of the relative axial position and the SNR. This is done again for the exemplary depth of 10 μm. It is apparent that besides the improvement as a function of the SNR, the axial LA improves if a particle is located above the focal plane. This can again be explained by the course of the calibration curves. Their course is flatter if the particle resides below the focal plane. Therefore, already small deviations in the estimated FWHM result in substantial deviations in the axial position.

The total axial LA, also computed as the arithmetic mean of all LAs within the specified axial range, is presented in Fig. 4 d. Similar to the SE, it improves for higher SNRs and deteriorates as a function of the imaging depth. This is again owed to the generally flatter curves at larger depths. Fig. 4 d also presents the course of the axial LA for the prior calibration. At zero depth the axial LA is similar to that obtained with the online calibration method for any SNR. This was expected because the online calibration method aims at improving the SE. The axial LA should actually not be affected. However, surprisingly, there is almost no deterioration toward larger imaging depths for the axial LA of the prior calibration. This observation can be explained by looking at the measured axial trajectories.

Visual analysis of the trajectories

Exemplary measured and true axial trajectories at selected depths are plotted in Fig. 5 for SNR = 10.

Figure 5.

(a and b) Measured trajectories determined by the online (a) and the prior (b) calibration method at selected depths. (Red lines) True simulated trajectories. To see this figure in color, go online.

The measurement approximates the true trajectory quite well at zero depth for both methods. The course of the particle movement is followed correctly and the absolute positions are met. However, the principal course is preserved for larger depths only for the online calibration method. Systematic deviations from the trajectory are more pronounced further away from the axial center between the two focal planes. As expected, large systematic deviations are observed using prior calibration. The measured trajectories hardly represent the true course of the trajectory. Instead, the particles are erroneously observed to move in a quite narrow axial range. This happens because, owing to the influence of the RIM, large axial step sizes are only represented by small changes in the FWHM values. By computation of the axial position using the steep calibration curves from zero depth, the axial increment is notably underestimated.

The axial LA deteriorates notably with depth for the online calibration. Furthermore, it gets worse if the particle is measured below the center of the focal planes. For prior calibration, the axial LA is erroneously measured to remain almost constant as a function of the depth. This is because the estimation errors are downscaled using the steep calibration curves. It must be concluded that the achieved axial LAs, provided in the literature, are likely to be underestimated if the RIM was neglected.

Impact on the calculated diffusion parameters

Fig. 6 plots the change of D_l for the online and the prior calibration method as functions of the depth for the respective values of the SNR. For varying depths, the SNR was fixed to 10, while for varying SNRs a depth of 10 μm was chosen. The first eight mean square displacement (MSD) points were used for D_l computation.

Figure 6.

(a and b) Observed courses of D_l as functions of the imaging depth and the SNR. To see this figure in color, go online.

Although the online calibration method cannot prevent that the measured absolute MSD values can deviate from the true values (see Section S5 in the Supporting Material), it still allows us to determine the diffusion coefficients with not more than ∼18% deviation, almost independently of the depth and the SNR. In contrast, using prior calibration results in deviations from the true values of up to ∼70% for this configuration. The strong dependency from the imaging depth is clearly observable.

Application to real data

In real experiments, coverslip tilt, coverslip thickness variation, or spherical aberration due to errors in the optical alignment, may introduce artifacts in the image creation process. Therefore, we conducted identical analyses on semisynthetic data that already included these effects (see Section S6 in the Supporting Material). We obtained similar results, but with larger absolute errors. As a result of these findings and previous calibration results with varying coverslip thickness and incorrect adjustment of the correction collar, we can conclude that additional aberrations mainly affect the actual parameterization, although the proposed model remains a valid approximation.

For evaluation of our method in real brain slices with mobile molecules, the ground truth is unavailable. Only the plausibility of the output can be tested. Therefore, we monitored the mobility of GPI-GFP using QDs. Two-dimensional image streams were acquired at depths between 1 and 20 μm at an interval of 1 μm. Only trajectories of length ≥20 were considered for analysis.

Visual analysis

Fig. 7 presents exemplary measured three-dimensional trajectories. The lateral extent of an axon is clearly recognizable in Fig. 7, a and d. As expected from three-dimensional measurements (22), the distribution of particles is highest at the axon’s lateral border. By encoding the axial position in the color, Fig. 7 d also reveals that the particles in the center of the image run on top of the axon, whereas those on the left are located below the axon. This conclusion is justified by the observation that, in the center of the image, the trajectories in the inner area of the axon have a higher axial position than those at the lateral border. The situation is vice versa on the left. In addition, the entire axon seems to follow the axial course of a bridge.

Figure 7.

Exemplary measured three-dimensional trajectories in brain slices. (a and d) Two-dimensional views of an axon. (a) Color distinguishes individual trajectories; in (d) it encodes the axial position. (b and e) Three-dimensional views of two intersecting axons. The meaning of the colors is the same as in (a and d). (c and f) Three-dimensional and front view of a part of an axon. To make the morphological structure more visually apparent, the trajectories are presented with position markers only. The color encodes the axial position. To see this figure in color, go online.

The second example (Fig. 7, b and e) reveals that the algorithm allows us to distinguish the axial course of two intersecting axons. Apparently, the horizontal axon proceeds ∼400 nm above the perpendicular axon.

Even the tubular structure of an axon can be determined. It is clearly observable in Fig. 7, c and f. It also becomes apparent that the axial LA considerably decreases if a particle moves below the center of the focal planes. For comparison we also simulated trajectories at an axon and obtained identical results (see Section S7 in the Supporting Material).

Molecular dynamics analysis

The courses of D_l for the individual directions are provided as functions of the imaging depth in Fig. 8 a. The absolute diffusion values are well compatible with the range observed for lateral diffusion (17). The axial diffusion is similar to the lateral diffusion and a deterioration toward larger imaging depths cannot be observed for the first 10–15 μm. If prior calibration is applied, the axial diffusion is consistently underestimated and deteriorates as a function of the depth. The apparent breakdown of the performance of our method for depths ≥15 μm is likely to be attributed to the SNR, which drops below 6 for such depths.

Figure 8.

(a) Individual and (b) combined diffusion results as functions of the depth. The presented data includes 18,098 trajectories in total, with at least ∼300 trajectories per imaging depth. It was obtained from two slice cultures, five brain slices, and 125 acquisitions. To see this figure in color, go online.

Fig. 8 b provides D_l for the combined diffusion as functions of the imaging depth. Apparently, two- and three-dimensional diffusion is similar for the online calibration method, but using prior calibration the three-dimensional diffusion is significantly underestimated. These results indicate that our method might not be sensitive enough to resolve differences between two- and three-dimensional diffusion. Here, it could be reasonable to extract the rough morphology first and then update the length of the trajectories with respect to this information.

Conclusions

We studied the performance of our online calibration extensively on synthetic, semisynthetic, and real brain-slice data. We show that the correct principal course of the underlying trajectories can be detected at any depth. This is a result of the capability to at least halve the SE at any depth and SNR. In contrast, the axial LA was found to deteriorate more notably over depth and SNR, which in consequence yielded deviations from the absolute positions and MSD values. Nevertheless, and more importantly, it could be shown that correct diffusion coefficients can be obtained almost independently of the imaging depth and SNR.

These results indicate that our online calibration method is indeed able to automatically adjust for the aberrations introduced by the RIM, which makes fast nanoscale three-dimensional SPT experiments deep in living samples readily available. This is now possible with virtually no additional experimental efforts or compromises for the user. Only the focal shift has to be adjusted for the setup. We could also clarify how critical it is to neglect the influence of the RIM: the true axial motion and, as a consequence, the axial LA, are significantly underestimated.

The most critical factor is the relatively poor SNR, which is for the given setup especially limiting at imaging depths ≥15 μm. This can be improved at the cost of temporal resolution or by using more advanced imaging configurations. For example, combination of our method with light sheet fluorescence microscopy is one of the most promising options, but two-photon microscopy is also a compatible configuration. We expect that standard WF imaging in primary cultures will profit from the described algorithm as well.

Author Contributions

S.S., K.T., and M.H. designed research; S.S. performed research; S.S., Y.P., and M.H. contributed analytic tools; S.S., B.B., and M.H. contributed experimental tools; S.S. analyzed data; and S.S. and M.H. wrote the article.

Editor: Antoine van Oijen.