Measuring true localization accuracy in super resolution microscopy with DNA-origami nanostructures

Most resolution criteria (Rayleigh [1], Sparrow [2], Houston [3]) used today are a variation of a common scientific approach: they try to define a minimum distance d for which the sum of the diffraction patterns of two point objects are imaged like two individual objects instead of just one. This type of arbitrary resolution criterion additionally disregards noise of any kind [4]. Noise due to photon statistics is inherent in fluorescence imaging and inevitably adds some level of uncertainty; additionally detector noise or sensitivity as well as fluorescent background from the sample are examples of noise sources that can further spoil the separation of two point emitters, or worse feign resolution where there should be none [5].

Authors reporting on super resolution microscopy (SRM) methods based on stochastic single molecule emitter switching have traditionally used a totally different resolution measure: the precision with which the position of a single emitter can be estimated. Prominent representatives of stochastic switching nanoscopy are stochastic optical reconstruction microscopy (STORM) [6], and photoactivated localization microscopy [7], and similar additional variants [8]. These methods all work in principle by shutting down most emitters in a random fashion, such that the remaining emitters are essentially single entities that can be individually localized. By repeating this process with a different subset of emitters, a full super resolution image can be synthetically assembled after mathematically fitting all detected events (i.e. extracting individual coordinates and precision values).

In general terms, nothing speaks against using position precision of single emitters as resolution measure for all type of microscopy techniques, including nanoscopy methods based on targeted switching, such as stimulated emission depletion microscopy (STED [9]), reversible saturable optical fluorescent transitions [10], ground state depletion [11] and structured illumination microscopy [12]. Targeted switching nanoscopy in essence does not shut off emitters in a random fashion, but instead separates emitters (single or few) within certain well-defined regions that are moved across the sample, in order to render a super resolved image.

To allow for trustful resolution criteria to be estimated by localization precision one however needs a well-defined phantom. In other words, for all microscopy techniques, the problem in claiming an absolute resolving power lies in testing how well a recorded image truly resembles the specimen. Localization precision merely measures the deviation of the estimated position when the same emitter pattern is imaged many times with different realizations of noise. This means that the figure of merit for resolution (i.e. a precision estimate) is just a measure of repeatability. Localization precision is not necessarily the same as localization accuracy, which denotes the deviation from the true position of the emitters in the sample [13, 14]. In particular stochastic switching methods are known to stray systematically from the true value due to effects of molecular dipole orientation, for example [15–17]. They are by their parallel nature also more susceptible to sample drift than microscopy based on targeted methods. Additional sources for bias are e,g. detector noise and background signal [14]. As a result, repeated measurements on the same fixed dipole (or an estimation of the deviation using statistical methods on a single measurement) may indicate high precision but in fact hide poor accuracy.

Unfortunately, accuracy is experimentally hard to determine from a recorded image, as it requires an a priori knowledge of the sample. In order to get an accurate fix on true positions, one would need a highly repeatable sample (i.e. a phantom) with a well-known structure and several reference points that can be exactly pinpointed. The estimated imaged position of the emitters can then be related to the true position established by the reference points and the quality of the estimation (i.e. the localization accuracy) can be worked out. Presently, biological test samples in use are for instance actin filaments [18] or microtubule [19], since there is at least some degree of a priori knowledge about those structures. Yet, these cytoskeletal structures are far from being ideal test samples in a sense that absolute position accuracy is hard to infer from the structures as fluorescence labeling and biological variability can introduce structural variances.

In contrast, DNA-origami offers a way to manufacture structures on the required length scale with high confidence and repeatability. Actually, the use of the system has already been demonstrated as a molecular ruler for fluorescence microscopy [20–22]. There, the authors have both looked at the localization precision of single spots as well as the distance between double-emitters. However, variation of the distance between multiple nanorods has not been, to our knowledge, used to extract the true localization accuracy or for a comparison of nanoscopic measurements. Also, for the first time we report on localization accuracy for STED.

Here, we show how to make use of DNA-origami, and in particular its repeatability, in order to report localization accuracy instead of just precision measures. By imaging a simple DNA-origami system with fluorophores attached to it, an estimate of true resolution in terms of localization accuracy is generated. This allows better comparison of nanoscopy methods as well as the generation of more unbiased claims of resolving powers. By varying the fluorophores and the specifics of the sample preparation (embedding medium, etc), this approach can be used to characterize the faithfulness for virtually any type of fluorescent microscopy technique.

Our goal is to estimate the localization accuracy of 2D fluorophore positions using a simple well-characterized nanoscale test specimen for two of the most popular SRM methods, STORM and STED. The simplest such sample consist of two point-like emitters separated by a robustly resolvable, highly conserved and uniform distance. For creating such a sample we chose to use DNA-origami to fabricate 140 nm long nanorods, whose design (figures 1, S1 is available online at stacks.iop.org/NJP/19/025013/mmedia), apart from being one of the simplest approaches of defining two positions with a smallest amount of material used, provided a stiff scaffold for the attached fluorophores, reducing the contribution of the sample to the variability of the measured localization accuracy [22]. To provide a sample suitable for both SRM techniques in terms of signal strength and resolvability we designed the nanorods to have twelve–twelve designated single-stranded DNA binding sites spaced approximately 100 nm apart along the rod, where we selectively attached AlexaFluor488- or CAGE 552-modified complementary oligonucleotides, giving us nanorods with two points of emitters separated by approximately 100 nm (figures 1(A), (B)). Although DNA origami's applicability to produce a robust platform for such applications have already been established [20–22], we characterized the nanorods in order to have an estimate of the extent of the sample's contribution to the variance of localization accuracy by measuring the two potential, main sources of variance originating from the sample: the variance of the distance between binding sites caused by distortion of the structure and by the sub-optimal incorporation of anchoring oligos (figure S3). We estimated the contribution of the rod's distortion to the variance of distance by measuring the longitudinal length distribution of the DNA-origami structures on negative-stained transmission electron microscopy (TEM) micrographs (figures 1(C), S5), as it has shown to be a faithful representation of the designed and solution phase dimensions of DNA nanostructures following the same design paradigm as our nanorods by works of other groups [23–27] and by our comparative measurements using Cryo-EM (figure S2). As expected, the rods' measured average length of 141.596 nm closely matched the designed length of 140. 365 nm with the relatively small variance of 16. 638 nm² (figure S3(A)). We measured the variability originating from the incomplete incorporation of fluorophores to be even smaller, as with the measured 96.057% occupancy we estimated the distance variance to be 0.0521 nm² (figure S3(B)). Because of the high homogeneity and precision DNA-origami provides the magnitude of the variances originating from the sample, as discussed before, are much lower than one observes in biological samples. This along the fact that the effect of the measured variances would contribute uniformly to the measurements with both methods, and that they were well below the resolution limit of the probed SRM techniques convinced us that our DNA-based structures provide a suitable sample to be used as a reference to compare localization accuracy.

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For each nanorod, we estimated the position of the individual emitters at both ends, using a maximum-likehood estimation for STED or centroid estimation on localized clusters for STORM data. For N nanorods this gives us a set of positions $({{\boldsymbol{x}}}_{j},{{\boldsymbol{y}}}_{j}),\,j=1\ldots N.$ These are samples of normal distributed random variables ${{\boldsymbol{X}}}_{j}\sim {\mathscr{N}}({{\boldsymbol{x}}}_{j}^{* },{\rm{\Sigma }})$ and ${{\boldsymbol{Y}}}_{j}\sim {\mathscr{N}}({{\boldsymbol{y}}}_{j}^{* },{\rm{\Sigma }})$ with a covariance matrix ${\rm{\Sigma }},$ which we assume to be diagonal with equal elements $\sigma$ along the diagonal. The ${{\boldsymbol{x}}}_{j}^{* }$ and ${{\boldsymbol{y}}}_{j}^{* }$ are the unknown true positions of each emitter. Now, one would be tempted to estimate the position accuracy using the variance (or standard deviation) of the localization of a single emitter, as it is usually done [28]. However, this is a biased estimator since the position of an emitter in the image may be shifted with respect to its true position in object space [15–17]. We instead choose to use the distance measure ${\boldsymbol{D}}=\parallel {\boldsymbol{X}}-{\boldsymbol{Y}}\parallel$ between two imaged emitters, which has the following advantages: first, we can average over a large number of equally spaced samples from the same population (the position of two emitters is never the same, but the distance is highly conserved) and second, we may use one emitter of each nanorod as a reference point to the other emitter. Simply put, the first emitter defines a coordinate origin for measuring the position of the second emitter and vice versa.

In general, the variance of the distance is ${\rm{Var}}({\boldsymbol{D}})={\rm{Var}}({\boldsymbol{X}})+{\rm{Var}}({\boldsymbol{Y}})-{\rm{Cov}}({\boldsymbol{X}},{\boldsymbol{Y}}),$ but for distances larger than the resolution, we assume that the two localizations are independently resolved and ${\rm{Cov}}({\boldsymbol{X}},{\boldsymbol{Y}})=0.$ Furthermore, we expect the variances for both labeled positions to be the same on average and thus approximate ${\rm{Var}}({\boldsymbol{X}})={\rm{Var}}({\boldsymbol{D}})/2.$ With this, we can estimate localizability by estimating the variance of the distance and dividing by two. Importantly, since ${\boldsymbol{D}}$ captures any apparent position shifts due to e.g. dipole orientation, we expect ${\rm{Var}}({\boldsymbol{X}})={\rm{Var}}({\boldsymbol{D}})/2.$ to be an unbiased estimator reporting on the variance of localization with respect to the true position in object space, i.e. localization accuracy.

In order to confirm this, we conducted Monte Carlo simulations by artificially generating a large number of images of two-point emitters at specified distances d, estimating their positions from the images [29] and then estimating ${\sigma }_{{\rm{d}}}={\rm{Var}}({\boldsymbol{D}})/2$ from the distances ${{\boldsymbol{d}}}_{j}=\parallel {{\boldsymbol{x}}}_{j}-{{\boldsymbol{y}}}_{j}\parallel .$ This was compared to the biased position variance ${\sigma }_{{\rm{s}}}$ from single emitters obtained by simply taking the localization precision of each emitter position [8]. We also compared this to the usual position estimator [28] given by ${s}^{2}/P,$ i.e. the Cramér–Rao lower bound (CRLB) for estimation of a single emitter under Poisson noise. Here, $s$ is the standard deviation of the width of the Gaussian point spread function (PSF) and P is the average number of fluorescent photons collected from one emitter.

Figure 2 shows the simulation results for the estimated localizability for DNA nanorods imaged with a Gaussian PSF with standard deviation s, i.e. FWHM $f=2\sqrt{2\,\mathrm{ln}\,2}s\,.$ The expected number of detected photons from each emitter was set to 100. At this point we have not yet introduced any position bias, i.e. the images of single emitters are not shifted with respect to the true position [15–17]. Also, we have not restricted our analysis to a specific microscopy method yet. For $d\gg 2s,$ ${\sigma }_{{\rm{d}}}$ is only slightly above the CRLB, confirming that ${\sigma }_{{\rm{d}}}$ can be faithfully taken to estimate the position precision. ${\sigma }_{{\rm{s}}}$ is close to the theoretical bound, too, showing that the accuracy estimated from a single emitter is correct for the case of unbiased positions with large separation. As d approaches 2s, ${\sigma }_{{\rm{d}}}$ and ${\sigma }_{{\rm{s}}}$ are expected to increase as we are not working in the single emitter regime anymore and the position estimations start to mutually affect each other. However, for ${\sigma }_{{\rm{d}}},$ this bias increase is less steep and less influential for the localization (see figure 2(A)). Conveniently, this prevents us from having to use very long nanorods to test for example confocal or widefield microscopes, where large nanoscale DNA structures might lead to an increased spread in emitter separations. For nanorods longer than 200 nm the variance originating from the sample is becoming a more significant factor as at these lengths curling and bending of the structure are harder to avoid, as we have also observed (see figure S4), leading to an associated significantly higher variance of true emitter distance [22].

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Next, we establish that the mutual distance precision ${\sigma }_{{\rm{d}}}$ indeed reports on position bias. To this end, we assume that position bias is normally distributed and we repeat our simulations with the position of each emitter image shifted by a random vector sampled from a normal distribution with variance ${\sigma }_{{\rm{bias}}},$ which might in a real sample be caused by drift or fixed transition dipoles, for example. Since the variance of the sum of two independent normally distributed variables is the sum of the individual variances, the correct position accuracy is ${s}^{2}/P+{\sigma }_{{\rm{bias}}}.$ The comparisons are shown in figure 2(B). Here we used d = 2.5s to stay just in the single emitter regime. As expected, the traditional estimator ${\sigma }_{{\rm{s}}}$ only incorporates deviations due to noise and thus fails to capture the true position accuracy. In contrast, our estimator based on distance ${\sigma }_{{\rm{d}}}$ correctly reports position accuracy, even under conditions with high readout noise (see figure 2(C)). Please note that in figures 2(B), (C), ${\sigma }_{{\rm{s}}}$ is taken in the usual way for methods using stochastic single molecule switching, i.e. sequentially. Thus, ${\sigma }_{{\rm{s}}}$ is not influenced by the other emitter, in contrast to figure 2(A).

Next, we applied our method to two common nanoscope methods, namely STED and STORM. To this end, we used DNA-origami nanorods with two groups of point-emitters attached, seperated by approximately 100 nm (figure 1). Figures 3(A) and (B) shows representative fluorescence nanoscopy images and figures S6–7 shows raw overview images of the imaged nanorods for each microscopy method, which allows a visual inspection of the nanoscale structures. From the images, the localization accuracy for ${\sigma }_{{\rm{d}}}$ is then deduced. The claimed and expected resolution for the single molecule stochastical STORM image would normally be taken from its generated precision histogram (see figure 3(D)), which has a mode of 8 nm or 64 nm² localization precision. This is an exceptional value that is a result of the high number of photons recordable from the caged dye. Yet, this figure does not testify on the correlation between recorded image and specimen ex ante and on top, the precision distribution is skewed with a long tail towards poorer localization precision. Only the distance measurements reveal with confidence that the localization accuracy is indeed in this range, namely 70.2 nm² (figure 3(D)). The localization accuracy for STED (${\sigma }_{{\rm{d}}}=\,62.5$ nm²) is found to be slightly better and moreover accurately complies with what one would expect using the FWHM $(f=54.6\,{\rm{nm}})$ and the number of photons (P = 8.4) estimated from the STED-images together with the equation $\tfrac{{s}^{2}}{P}=64.0$ nm². The underlying reason for localization precision and localization accuracy being similar is most likely that STED is known to show little position bias caused by fluorophore orientation [15] and, additionally, since read-out is spatially and temporally correlated, STED is locally not very susceptible to sample drift. For STORM, the similarity between the values shows that we have successfully reduced drift and that we have dipoles that are rotationally diffusing. We note that while the distance estimated with STED (104.1 nm) comes very close to the designed distance (105.5 nm), STORM with an estimation of 98.5 nm seems to slightly underestimate it. We attribute this to the well known, minimizable but not entirely avoidable, phenomena that is inherent to localization-based imaging techniques, namely the occurrence of simultaneous multi-molecular emission events. In the case of our nanorods, this can mistakenly result in single localizations with a position closer to the middle of the rod, leading to a shrinking of the overall measured distance [30].

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In this work, we show that true localization accuracy in fluorescence microscopy can be determined from measurement of the distance variation of a sample containing pairs of emitters at an accurate and repeatable separation. In particular, super resolution methods based on stochastic switching are sensitive to position bias, predominantly caused by changes due to drift and fluorophore orientation [14, 30]. We carefully note that while our data shows little such effects, we can safely conclude this only from localization accuracy measurements via the emitter distance, as the very same effects precisely do not manifest themselves in the localization precision (figures 2(B), (C)).

The fact that STED shows approximately the same accuracy comes not as a surprise when considering that 2D-STED on a single emitter results in roughly four times less signal in total for a FWHM decrease by a factor of two. These 'low-resolution' photons are carved away in the outer parts by the STED-donut as the area of effective fluorescence shrinks quadratically with decreasing width. (Note that we assume peak height to remain the same.) These effects counter-act in the equation $\tfrac{{s}^{2}}{P},$ resulting in a localization accuracy for a single emitter that is approximately equal for STORM and STED.

With our fluorescently labeled DNA-origami structures, localization accuracy can be unambiguously measured. These nanostructures can be incorporated in specimens as fiducial markers and performance probes. If needed, the accuracy achievable with a particular labeling strategy can be measured by replicating the same situation on the nanorods, i.e. by attaching conjugated antibodies or fluorescent proteins to the DNA structure [31]. For gauging 3D resolution, nanopillars [32] could be used in much the same way.

Finally, we would like to point out that localization accuracy should not be confused with resolution. For stochastic switching methods, the degree of correlation between specimen and image not only depends on localization accuracy; for example, on/off times of the fluorophores have to be carefully balanced in order to guarantee sparsity of emitters in each frame as much as possible. With targeted switching methods, most specimen of interest will contain dense emitter clusters that inherently cannot be resolved to the single emitter limit if they are closer than the width of the PSF. Nonetheless, because localization accuracy reports not only on PSF-size, but also on the photon levels at which it can be achieved, it gives a better idea than PSF-size alone. In order to obtain a full picture of the achievable resolution, nanorod samples with different emitter spacing could be tested in order to check for the emitter separation that just results in an increase in localization accuracy as shown in figure 2(A). In any case, localization accuracy is eminently suitable to analyze day-to-day performance of microscopes using the same technique, and to get an accurate fix on true localization accuracy.

Simulations

All Monte Carlo simulations were performed in MATLAB. We model the image of an emitter at position $({x}_{0},{y}_{0})$ by assuming a Gaussian PSF of the form

$$\begin{eqnarray*}&&h(x,y)=\,\displaystyle \frac{P}{2\,\pi \,{s}^{2}}{{\rm{e}}}^{-\displaystyle \frac{{(x-{x}_{0})}^{2}+{(y-{y}_{0})}^{2}}{2{s}^{2}}}.\end{eqnarray*}$$

For realistically simulating a dual emitter nanorod, we sum two PSFs at random positions with the constraint that their distance is either fixed or modified by a random bias for each position. We subsequently add Poisson noise and Gaussian noise, if applicable.

Light microscopy

Position estimation

The procedure for estimating the positions of emitters (for simulated as well as experimental data) depends on the type of microscopy. For STED and our simulations, we take raw images $R(k)$ and perform a maximum-likelihood-estimation by minimizing [33]

$$\begin{eqnarray*}&&{\chi }^{2}=2\displaystyle \sum _{k}^{K}{\upsilon }_{\theta }(k)-R(k)\,-2\displaystyle \sum _{k,R(k)\ne 0}^{K}R(k)\,\mathrm{ln}\,\displaystyle \frac{{\upsilon }_{\theta }(k)}{R(k)},\end{eqnarray*}$$

where ${\upsilon }_{\theta }(k)$ is the model function ${\upsilon }_{\theta }(k)={h}_{1}+{h}_{2}.$ The sum runs over all pixels $k=1\ldots K$ of the image. $\theta =\{{x}_{0}^{1},{y}_{0}^{1},{s}^{1},{P}^{1},{x}_{0}^{2},{y}_{0}^{2},{s}^{2},{P}^{2}\}$ are the parameters to be determined. For minimization, we use MATLAB's fminsearch function. STORM data was first processed using the ZEN software (Zeiss) with default settings. Then, we selected emitter pairs in MATLAB and either performed k-means clustering to segment the localizations into two groups for each emitter and used the centroid of each cluster as the position estimation for the corresponding emitter, or performed least-squares fitting on the molecular density.

Materials used for DNA-origami nanorod fabrication

Design of the DNA-origami nanorod

Folding and quality assessment of the DNA-origami nanorods

Electron microscopy imaging of the DNA origami nanorods

Cryo-electron microscopy imaging of the DNA origami nanorods

Purification of the DNA-origami nanorods after folding

Labeling and purification of DNA-origami nanorods

Immobilization of DNA-origami nanorods on coverslips

Measurement of site-occupancy of DNA-origami nanorods

Measurement of length-distribution of DNA-origami nanorods

This work was supported by the European Commission under the Seventh Framework Programme (FP7), as part of the Marie Curie Initial Training Network, EScoDNA (GA no. 317110), the Märta and Gunnar V Philipson foundation, the Erling Persson family foundation and by the Swedish Research Council (VR 2015-04198) and 2013-6041.

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.