## Quantifying the Localization Precision of Single Fluorescent Emitters

### Tips & Tricks for Single-Molecule Localization Microscopy

- Fig. 1: Experimental procedure to calculate the localization precision exp. (a) M = 100 simulated images of the same single emitter. (b) The localizations obtained from the images in (a) and localization precision exp calculated from the variance of the localizations.
- Fig. 2: Theoretical calculation of the localization precision. (a) On the left, a simulated emitter image with a high photon number N = 500, a suitable pixel size a = 100 nm according to the Nyquist criterion, and no photon background b = 0. On the right, the localization determined by MLE and the localization precision determined from the CRLB. (b-d) The influence of (b) a photon background b = 20, (c) a low photon number N = 50, and (d) a large pixel size a = 200 nm.

**Knowing the precision with which single fluorescent emitters are localized is essential for a correct interpretation of single-molecule localization microscopy (SMLM) data. Here, we give a brief overview of the most popular strategies to calculate the localization precision in SMLM.**

**Introduction**

Localizing individual fluorescent emitters, typically with a precision in the order of ten nanometer, is the fundamental principle of SMLM. Exact knowledge of the localization precision is therefore important, not only because it directly affects the spatial resolution achieved by SMLM, but also because it is required for a correct analysis and visualization of SMLM data. However, calculating the localization precision is not a trivial task, as it depends on a variety of experimental parameters, such as the number of detected photons from the emitter, the fluorescence background, the camera pixel size, and the used localization algorithm [1]. Here, we briefly summarize the most popular approaches to quantify the localization precision, and point out a number of pitfalls that are sometimes overlooked.

**Experimental Procedure**

Individual fluorescent emitters in SMLM are switched or activated to the fluorescent “on-state” for a limited amount of time. If this time exceeds the camera exposure time, or if the on-state is occupied more than once (e.g. due to switching or blinking), the same emitter can be independently localized in multiple camera frames. As illustrated in figure 1, the localization precision *σ*_{exp} can then be calculated using the unbiased estimator of the variance

where *M* is the number of frames, x_{i} the localization in frame *i*, and *x* is the average localization. A similar expression is valid for the y-direction. However, the application of this experimental procedure is limited, considering the requirements on the duration of the on-state or the switching/blinking behavior. Moreover, unambiguous grouping of localizations originating from the same emitter is only possible for sufficiently low emitter densities, and is prohibited by sample motion.

Approaches based on a nearest neighbor distance analysis are applicable for high emitter densities [2], but these only yield an average value for *σ*_{exp}.

**Theoretical Model**

Considering the limitations of the experimental procedure, it is common practice to theoretically calculate the localization precision instead. For this purpose, the observed image of the emitter is often modeled as a 2D Gauss with a uniform photon background, integrated over a camera pixel grid. The noise is usually modeled as Poisson distributed shot noise. Under these assumptions, the localization precision *σ*_{LS} using the popular least-squares (LS) algorithm is given by [3]

where *Ν* is the number of detected emitter photons, *s* the standard deviation of the 2D Gauss, *a* the pixel size, and *b* the photon background. The localization precision can thus be arbitrarily improved if more photons can be collected. An alternative approach is to theoretically calculate the localization precision using the Cramér-Rao lower bound (CRLB), i.e. the lowest variance any unbiased estimator can attain [4]. With the same assumptions as for σ_{LS}, the CRLB yields [3]

A closed-form expression for *σ*_{CRLB} has been reported as well [5]. Considering the meaning of the CRLB, this expression describes the best possible localization precision, independent of the localization algorithm. Maximum likelihood estimation (MLE) can achieve the CRLB and should therefore be preferred over LS [3], although the difference in performance is small for a high background [5]. As illustrated in figure 2, the theoretical approach has the important advantage over the experimental procedure that the localization precision can be calculated from a single image of the emitter. However, these theoretical models do not account for effects arising from drift correction or image channel alignment. Furthermore, they are only reliable as long as the image and noise model are good descriptions of reality.

**Photon Number**

The theoretical calculation of the localization precision requires converting the camera pixel values into photon numbers. This conversion can be done using the photon transfer curve (i.e. the variance of the pixel values plotted as a function of their mean) [6]. If only shot noise is assumed, the curve is linear, and the photon conversion factor can be determined from the slope. However, this is not a good approximation when using an electron-multiplying charge coupled device (EMCCD) camera. Such cameras produce excess noise, which has the same effect as doubling the variance of shot noise [5]. This means that the photon conversion factor is overestimated by a factor of two. The above expressions for *σ*_{LS} and *σ*_{CRLB} therefore need to be multiplied with a factor of √2. Excess noise is not present in scientific complementary metal-oxide-semiconductor (sCMOS) cameras. However, calculation of the photon conversion factor is complicated by the presence of fixed pattern noise in such cameras [7].

**Axial Localization Precision**

Several implementations of SMLM allow to localize single emitters in 3D. In this case, not only the lateral, but also the axial localization precision should be calculated. The experimental procedure mentioned above can be trivially extended for this purpose. However, the limitations for the lateral case are equally valid in 3D. Theoretical models of the axial localization precision using the CRLB have therefore been proposed for several 3D localization schemes [1]. In case of the commonly used astigmatic, double-helix, and biplane methods, convenient closed-form expressions for *σ*_{CRLB} have been published [5]. The axial localization precision for these SMLM implementations is typically worse than the lateral one, while the lateral localization precision itself is lower compared to the conventional 2D case.

**Precision versus Accuracy**

The localization precision does not necessarily describe the full localization uncertainty. Even in the imaginary case of an infinite precision, the localization can still be inaccurate if the localization algorithm is biased [1]. A common source of bias is the (implicit) assumption of isotropic photon emission. Many emitters commonly used in SMLM (e.g. organic dyes or fluorescent proteins) behave as electric dipoles and emit photons accordingly. Depending on the dipole orientation, the anisotropic photon emission can result in emitter images with an intensity peak shifted from the emitter position. This shift can lead to a localization bias up to several tens of nanometers when using a 2D Gauss as model for the emitter image [8]. More sophisticated models that account for the dipole behavior are therefore required to capture these inaccuracies. Other solutions are the use of dedicated polarization optics [8], or a labeling strategy that allows the dipole to rotate freely. Besides dipole effects, there are other causes for localization inaccuracies, such as emitter motion during camera exposure and label displacement [1].

**References**

[1] Hendrik Deschout, Francesca Cella Zanacchi, Michael Mlodzianoski, Alberto Diaspro, Joerg Bewersdorf, Samuel T Hess & Kevin Braeckmans: Precisely and accurately localizing single emitters in fluorescence microscopy, Nature Methods, 11(3) 253-266 (2014) doi: 10.1038/nmeth.2843

[2] U. Endesfelder, S. Malkusch, F. Fricke, M. Heilemann: A simple method to estimate the average localization precision of a single-molecule localization microscopy experiment. Histochemistry and Cell Biology, 141(6) 629-638 (2014) doi: 10.1007/s00418-014-1192-3

[3] K.I. Mortensen, L.S. Churchman, J.A. Spudich, H. Flyvbjerg: Optimized localization analysis for single-molecule tracking and super-resolution microscopy. Nature Methods, 7(5) 377-381 (2010) doi: 10.1038/nmeth.1447

[4] R.J. Ober, S. Ram, and E.S. Ward: Localization accuracy in single-molecule microscopy. Biophysical Journal, 86(2): 1185-1200 (2004) doi: 10.1016/S0006-3495(04)74193-4

[5] B. Rieger, and S. Stallinga: The Lateral and Axial Localization Uncertainty in Super-Resolution Light Microscopy. Chemphyschem. 15(4) 664-670 (2014) doi: 10.1002/cphc.201300711

[6] J.R. Janesick: Photon Transfer: DN to λ. 2007, Washington: SPIE Press.

[7] Fang Huang, Tobias M P Hartwich, Felix E Rivera-Molina, Yu Lin, Whitney C Duim, Jane J Long, Pradeep D Uchil, Jordan R Myers, Michelle A Baird, Walther Mothes, Michael W Davidson, Derek Toomre & Joerg Bewersdorf: Video-rate nanoscopy using sCMOS camera-specific single-molecule localization algorithms. Nature Methods, 10 (7) 653–658 (2013) doi:10.1038/nmeth.2488

[8] Mikael P. Backlund, Amir Arbabi, Petar N. Petrov, Ehsan Arbabi, Saumya Saurabh, Andrei Faraon & W. E. Moerner: Removing orientation-induced localization biases in single-molecule microscopy using a broadband metasurface mask. Nature Photonics, 10(7) 459–462 (2016) doi: 10.1038/nphoton.2016.93

**Contact
Dr. Hendrik Deschout**

Centre for Cellular Imaging

University of Gothenburg

Gothenburg, Sweden