Using DNA origami nanorulers as traceable distance measurement standards and nanoscopic benchmark structures

In recent years, DNA origami nanorulers for superresolution (SR) fluorescence microscopy have been developed from fundamental proof-of-principle experiments to commercially available test structures. The self-assembled nanostructures allow placing a defined number of fluorescent dye molecules in defined geometries in the nanometer range. Besides the unprecedented control over matter on the nanoscale, robust DNA origami nanorulers are reproducibly obtained in high yields. The distances between their fluorescent marks can be easily analysed yielding intermark distance histograms from many identical structures. Thus, DNA origami nanorulers have become excellent reference and training structures for superresolution microscopy. In this work, we go one step further and develop a calibration process for the measured distances between the fluorescent marks on DNA origami nanorulers. The superresolution technique DNA-PAINT is used to achieve nanometrological traceability of nanoruler distances following the guide to the expression of uncertainty in measurement (GUM). We further show two examples how these nanorulers are used to evaluate the performance of TIRF microscopes that are capable of single-molecule localization microscopy (SMLM).


Calibration of the stage micrometre
To calibrate DNA origami nanorulers first the EMCCD camera pixel size in the object space of the applied microscope system was traceably calibrated, because the camera pixels represent the length scale used for distance measurements in superresolution (SR) microscopy. A pixel size calibration is commonly done using a traceably calibrated stage micrometre.
The calibration of the stage micrometre was performed by the optical linewidth calibration tool of PTB. This system is based on a modified commercial microscope (Zeiss AXIOTRON) and validated both for linewidths and pitch calibrations. The sample is Koehler-illuminated in epi-configuration at a central wavelength of 365 nm (spectral bandwidths 20 nm) with an illumination numerical aperture of NAC = 0.9 and imaged by a microscope objective with a lateral magnification of m = 150 and a numerical aperture of NAO = 0.9. To realise a scanning microscope, a slit aperture is placed in the image plane in front of a photomultiplier. This aperture again is imaged de-magnified into the object plane by the microscope objective, where it has a size of 3700 nm × 90 nm. Linewidth and pitch measurement are based on the measurement of the transmitted light intensity using a photomultiplier, while the sample is moved highly precisely and interferometrically controlled in the focal plane using a onedimensional high-precision nanopositioning stage. The edge positions are deduced from the measured signal profile using in case of pitch measurements a 50 % threshold criterion. The pitch of individual grating lines can be obtained as the difference of the left edge or right edge positions of the corresponding lines or as the mean value of both values.

The stage micrometre
For the pixel-size calibration we used a stage micrometre (Präzisionsoptik Gera, 2 mm in 200 parts ( Figure S1a, b). This calibration sample is a cost-efficient and common tool for every microscopy lab. However, being handled in ordinary laboratory environment (and not in clean room facilities) typical pollution is identified within the area of the calibration structure. Furthermore, the lines show distinct structure errors at some places. These circumstances must be taken into account for the uncertainty evaluation of the calibration process.

Calibration Interval
The distances between the adjacent left edges and between adjacent right edges of the first 13 lines (line 0.0 mm to 0.12 mm) were calibrated. The middle of the scan diaphragm (length = 3.7 µm in object coordinates) is placed approximately 15 µm underneath the upper edge of the short pitch structures for measuring ( Figure S1c). This measuring track is chosen such that no large structure errors occur within the track. Five measurement series each with 10 repetitions were carried out. Therefore, each value is the average of more than 50 single measurements. The measured distance values varied within 2 nm to 5 nm (standard deviation) depending on the line measured.

Pixel size calibration using the traceable stage micrometre
As the distances between the lines are known the calibration of the pixel size in the object space of the microscope system under test can be performed as sketched in Figure S2a. For this purpose, we placed the stage micrometre on the microscope stage with the lines of the scale aligned parallel to the pixel columns of the EMCCD-image and imaged it with transmitted white light. Then we analysed it within the calibrated interval using customized software based on LabVIEW 2011 to determine the pixel size and estimated the uncertainty following the GUM 1 . As shown in Figure S2b the distance between two lines of the stage micrometre is determined by measuring the difference between the edge positions 1 and 2 of them.
Due to diffraction and residual optical aberrations the edges are not infinitely sharp but show an intensity gradient ( Figure S2c). Thus, an edge localization criterion is needed. For this purpose, we used the intensity contrast of the lines and the space between them. The mean intensities 〈 〉 as well as 〈 〉 were determined for both regions from pixel-columns sufficiently far away from the edge profiles (> 5 pixel) to ensure a negligible impact edge profiles. The intensity at the edge was then defined as the arithmetic average of these two mean values and the linearly interpolated pixel positions corresponding to this threshold were determined as illustrated in Figure  S2d.
In this way, the distances and the corresponding uncertainties were determined within a measuring window matching the size and position of the slit used in stage micrometre calibration (~4 µm / 40 pixels height, center 15 µm above the bottom of the lines). All values were averaged within this window. Every line-to-line distance was measured on 100 different images. The corresponding mean value 〈 〉 was used to calculate the pixel size using the following model equation: The calibrated stage micrometre is placed on the objective of the microscope illuminated by transmitted white light. The EMCCD camera serves as detector. The edges are defined via their intensity. b) The distance is measured between corresponding edges. c) The edges show an approximately linear intensity gradient. d) To localize the edges the medium intensity on and between the lines is determined. The threshold (grey) was chosen to be the medium intensity between them. The pixel surpassing that threshold and the both surrounding pixels were identified (red) and fitted with a linear function. According to the fit, the position of the edge was then calculated as the corresponding coordinate to the threshold intensity.
The reference distance and its uncertainty is known from the stage micrometre calibration. In principle, the measured mean distance needs to be corrected for a number of effects caused by the non-perfect orthogonality of the lines to the pixelrows ( ), the roughness of the edges ( ), impurities on the sample and optics ( ), imperfect positioning of the measurement window ( ), the used fluorescence filters ( ) as well as by optical distortions or aberrations resulting in a position dependent lateral magnification ( ). These aberrations can e.g. be the result of a non-perfect correction of field curvature as listed in Figure 1e. Apart from the effect of a non-perfect orthogonality, the specific numerical values of these corrections are not known exactly for a single measurement. It has been assumed that these corrections have the expectation value zero. Hence, for a specific measurement, the numerical value is unchanged. However, there are uncertainties assigned to these corrections which thus increase the uncertainty of the mean value 〈 〉.
For setup no.1 the pixel size P (here: P = 101.05 nm/pixel) and its uncertainty was determined based on a single line pair of the stage micrometre. The uncertainty budget is shown in Table S1. The expanded uncertainty using the coverage factor k = 2 is U(P) = 1 nm/pixel.

Monte Carlo simulations of uncertainty contributions.
When it comes to distance measurements not only uncertainties due to the setup, but also uncertainties coming from the sample must be considered. These uncertainties can have a photophysical origin coming from the dye molecules or a structural origin coming from the DNA origami. Quantifying the single contributions is demanding because it means controlling all properties while probing a single one. Thus, an alternative approach is necessary to rationalize these influences.
To study the contributions of different measurement parameters on the distance measurement uncertainty we performed Monte Carlo Simulations, starting from the generation of signals, following the whole analysis path and terminating with the statistical analysis of multiple distance measurements on simulated nanoruler signals.
A simplified scheme of the simulation algorithm is illustrated in Figure S3. The main advantage of the simulation is the ability to control every single parameter without changing others. We probed the influence of the labelling efficiency ( Figure  S4a) as structural influence. Furthermore, we probed the photon number ( Figure S4b) and the localization per spot which can be interpreted as the influence of photobleaching or the acquisition time ( Figure S4c) as photophysical influences. We also probed the influence of the blinking kinetics (namely the ratio of on-and off-time), see Figure S4d. The exact simulation parameters are listed in Table S3.

Table S3
Simulation conditions for the data shown in Figure S4. A "x" indicates that the simulation has been done for multiple values of the corresponding property (see graphs As mentioned in the manuscript the most important aspect is the question if the uncertainties are random or systematic. Random uncertainties are not problematic since their overall influence is experimentally observed and can be minimized by large statistics which is straightforward with DNA origami nanorulers. Systematic errors are much more critical. They must be corrected or suppressed by proper experimental conditions if possible. It is expected that all influences but the blinking kinetics produce random uncertainties because they blur the signals without shifting their centre. The simulations confirm these expectations with the following limitations: The photon number can cause systematic distance uncertainties when it is so low, that the distance of the nanoruler is near the resolution border. For the simulated ruler length of 30 nm one should have at least ~3000 photons per camera frame. At our measurements conditions, we typically detected 18000 ± 3000 (mean±SD) photons per camera frame. Since this is far more intensity than needed to resolve a 30 nm distance the photon number is only contributing to the random uncertainty.
In case of the labelling efficiency it is assumed that every dye on the nanoruler has an equal missing possibility when the labelling efficiency is smaller than 100%. Unequal labelling efficiencies for the different strands could lead to systematic uncertainties.
It is generally known from localization microscopy that double localization can occur at improper blinking kinetics, giving a false localization between two emitters. This can lead to a systematic shortening of the measured distance. This is demonstrated by the simulation results in Figure S4d. Consequently, a low duty cycle is mandatory for achieving accurate distances. This influence can be neglected when ton/toff is tuned to be much smaller than 0.05. This is especially possible for DNA-PAINT because the method allows adjusting the blinking kinetics arbitrarily by choosing proper imager concentrations and binding sequences 2 . In our experiments, the on-off-ratio was about 0.02, i.e. in a regime where no significant shortening of the distances occurs. For our experiments, we however estimated an uncertainty ( ) representing a nonresolvable influence that could be caused by this effect.
The question arises whether these influences are the only ones acting on the measurement uncertainty or if there are other influencesmost likely geometrical inhomogeneities increasing the uncertainty. To answer this question, we performed additional simulations with mixed influences. The values of the input parameters including their standard deviations were chosen to be in typical ranges known from experiments. They are listed in Table S4. Potential geometrical inhomogeneities were simulated by adding Gaussian distributed noise with defined standard deviations (SD) to the distance between the marks. The results for different SDs can be seen in Figure  S5. While experimental SDs of the distance are in the order of ~4-5 nm simulations without distance-noise have an overall SD of ~1 nm. Output SDs comparable to the experimental results are only reached when also the SD of the input distance is chosen to be 4-5 nm.
Thus, it is likely that the main random uncertainty contribution in the experiments is geometrical inhomogeneity of the sample. The reason can be small defects of single structures as well as slight differences in their 3D orientation (note that we are just measuring the 2D-projection in this work). However, the reproducibility of the mean distances show that a statistical analysis of the structures is still reasonable for calibration purposes.

Table S4
Simulation conditions for the data shown in Figure S5. The "x" indicates that the simulation was done for multiple values of the corresponding property (see graph).

The influence of false-positive signals
An uncertainty contribution that is difficult to quantify comes from the data analysis method. We have used software tools that are able to identify and analyse distances of DNA origami nanorulers automatically in a fast and usually satisfying way (e.g. with the GATTAnalysis software or the software package CAEOBS 3 ). However, corresponding algorithms cannot be perfect as is illustrated in Figure S6. So, there is always a possibility of analysis artefacts leading to uncertainties that have to be taken into account when accurate measurements are needed. Filtering the results by hand can circumvent these artefacts, but can lead to uncertainties due to subjective picking of structures. We estimate the uncertainty contribution to be 1 nm. Here, this order of magnitude is estimated by the difference of both types of analysis (i.e. automatic and manual), which is (0.7≈1 nm). Furthermore, our analysis is based on picking signals within a superresolved image that are likely to be the result of a nanoruler because of a spot-to-spot distance within the expected range. However, there is also the unlikely possibility of e.g. two dimerized, broken nanorulers with one mark each in the expected distance range. This ends up with a non-zero possibility of picking such false-positive structures. Overall, we estimated the corresponding uncertainty contribution ( ) by the difference between the mean distance obtained by automatic and manual analysis of an exemplary SR-image: To avoid an underestimation of this influence we presume an uncertainty of 1.5 nm here, including potential uncertainty contributions form false-positive structures and subjective picking of structures. Since we have no information about the corresponding probability distribution, we estimate this value to be the interval of a rectangular distribution. This leads to a final uncertainty contribution of 1.5/√3 ≈ 0.9 nm.

The improved fit-model
Usually, distances in SR images are measured by fitting the cross section of two signals with the sum of two Gaussian functions in order to determine the difference between their centre. This procedure is however not efficient for DNA origami nanorulers: the need to project the structure on? one axis of the coordinate system takes a lot of time for large statistics and causes a systematic projection uncertainty in addition.
To circumvent this problem, we measured the distance between the two marks on a DNA origami by isolating their individual superresolved signals and histogramming the 2D-distances between all corresponding localizations like recently introduced 3 . This results in a bimodal distribution containing the distribution within the spots and between the spots that can be approximated by the function.
The parameter 0 represents the distance between the two marks. For its uncertainty determination two contributions have to be considered: the first one ( ) describes the uncertainty due to the fitting of the parameter (output of the LabView-routine "Nonlinear Curve Fit Intervals.vi"). This uncertainty contribution is stochastic and can be decreased by averaging when the mean distance of a nanoruler ensemble is determined. On the other hand, the fit model is an approximation which results in a systematic uncertainty ( ). Simulating two Gaussian distributed localization spots with a distance of a = 30 nm and different standard deviations (SD), we estimated that this uncertainty should not be larger than 0.5 nm as shown in Figure S7. However, note that the approximation only holds for s<<a. This means that this fit model is not suitable for the measurement of distances near the resolution limit. Because we have values of SD of about 5 nm in our measurements, it is a proper approximation in our case. Additionally compared to the fit model we have recently used 3,4 this model is improved by the factor √ . The old model is still a suitable approximation but with a systematic uncertainty of ∆a≈s²/a. Under usual imaging condition this results in (∆a ≈ 0.5-1 nm) but can get worse at lower resolutions.

Figure S 7
Difference ∆a between the measured and the simulated distance vs. the standard deviation s of the simulated Gaussian distributions. In the experimental regime ∆a oscillates around 0 nm indicating that there is no significant systematic uncertainty within this interval.

Summary of different uncertainty contributions and their consideration in the measurement uncertainty budget.
As depicted in Figure 1e the development of the nanoruler distance uncertainty determination starts with an Ishikawa-diagram to visualize potential influences on the measurement. To make this development more transparent Table S5 provides a summary of these influences, their effect on the measurement and how they are considered with respect to the measurement uncertainty budget. Considered in the uncertainty budget of the nanoruler calibration as part of the overall random uncertainties.

Non perfectstaples
Staples with false sequence can influence the distance of single nanorulers.
The probability is equal for every staple so the uncertainty should be random.
Considered in the uncertainty budget of the nanoruler calibration as part of the overall random uncertainties.

Labeling efficiency
Missing single labels can influence the distance of single nanorulers. The probability is equal for every label so the uncertainty should be random.
Considered in the uncertainty budget of the nanoruler calibration as part of the overall random uncertainties.

Imperfect geometry of nanorulers Design simplifications
Simplifying assumptions during the nanoruler design lead to systematic differences between the nominal distance and the measured distance.
Not considered as an uncertainty: aim of the calibration is the proper determination of the actual distance.

3D-distortions
Distortions can occur from the DNA origami design, e.g. if a simple rectangular structure is not twist corrected.
Part of the reasons for the differences between nominal and measured distance: not considered as an uncertainty Width of marks Causes a slight broadening of the spots that leads to additional random uncertainties.
Considered in the uncertainty budget of the nanoruler calibration as part of the overall random uncertainties.

Sample drift
Sample drift can be caused by temperature gradients or other external influences and causes broadening of the spots. It has to be corrected or suppressed.
Drift has been suppressed to a nondetectable level for the calibration measurements. The uncertainty of this suppression has however been considered in the budget of the nanoruler calibration representing the result of environmental influences. Immobilization strategy Part of the chemical environment of the nanoruler and can have a systematic influence on the distance due to electrostatic forces.
Not relevant: calibration procedure holds for a single slide of nanorulers with a single immobilization strategy: Influences are part of the difference between nominal and measured distance. Buffer components Considered in the uncertainty budget of the nanoruler calibration as part of the overall random uncertainties. Especially salt concentrations can have an influence due to the interaction of cations with the DNA backbone.
Not relevant: calibration procedure holds for a single slide of nanorulers with a single immobilization strategy: Influences are part of the difference between nominal and measured distance.

Exemplary drift correction for comparison with Figure 4
As mentioned above, drift can be corrected by tracking the signal of fiducial markers over time. This is most commonly done in single-molecule localization microscopy (SMLM) while drift suppression like shown in Figure 4 of the main text is rather unusual. To give a comparison for the same sample we show some data here where drift is not suppressed but corrected.
We use a commercial product, namely GATTAquant HiRes beacons made for DNA-PAINT and the corresponding analysis software (GATTAnalysis by GATTAquant GmbH). As illustrated in Figure S8. localization positions of the fiducial markers are plotted over time. Then a polynomic of desired order or a spline (sliding average) is fitted to the data points in order to calculate a proper correction function. If the nanoruler spots do not show a detectable asymmetry after drift correction, possible resulting uncertainties resulting from drift can be neglected.