Localization Precision in Stepwise Photobleaching Experiments

Abstract

The precise determination of the position of fluorescent labels is essential for the quantitative study of biomolecular structures by various localization microscopy techniques. Localization by stepwise photobleaching is especially suited for measuring nanometer-scale distances between two labels; however, the precision of this method has remained elusive. Here, we show that shot noise from other emitters and error propagation compromise the localization precision in stepwise photobleaching. Incorporation of point spread function-shaped shot noise into the variance term in the Fisher matrix yielded fundamental Cràmer-Rao lower bounds (CRLBs) that were in general anisotropic and depended on emitter intensity and position. We performed simulations to benchmark the extent to which different analysis procedures reached these ideal CRLBs. The accumulation of noise from several images accounted for the worse localization precision in image subtraction. Propagation of fitting errors compromised the CRLBs in sequential fitting using fixed parameters. Global fitting of all images was also governed by error propagation, but made optimal use of the available information. The precision of individual distance measurements depended critically on the exact bleaching kinetics and was correctly quantified by the CRLBs. The methods presented here provide a consistent framework for quantitatively analyzing stepwise photobleaching experiments and shed light on the localization precision in some other bleaching- or blinking-assisted techniques.

Introduction

Superresolution microscopy based on the localization of single molecules has enabled the retrieval of structural information with biomolecular specificity and a resolution of few nanometers. The method of choice for obtaining high-precision distance measurements in sparse samples is localization by stepwise photobleaching (1,2). Data analysis typically proceeds in two steps. First, the time point of step-like bleaching events is determined from intensity-time traces, either manually or with the help of step-finding routines. Second, this knowledge is exploited for the fitting of fluorophores with a model for the microscope’s point spread function (PSF) to determine their positions and distance. This principle essentially circumvents the fundamental resolution limit for the parallel localization of two (or multiple) fluorophores in a single image (3). As an impressive example, the distance between two Cy5 fluorophores was measured with a precision of less than 2 nm in an optimized stepwise photobleaching experiment (4).

However, the precision of individual stepwise photobleaching measurements has remained largely elusive. One problem is that various localization procedures have been used in practice, including center-of-mass calculation (4), independent fitting of subtracted images from subsequent plateaus (5–7), sequential fitting after subtraction of previous fits (2), and global fitting of all plateaus (1). How these different procedures affect the localization precision has not been addressed systematically.

Another problem is more fundamental: the stochastic nature of fluorescence emission gives rise to shot noise, an unavoidable scattering of photon counts around the expected average value. As with any kind of noise, this noise enters the uncertainty of fitting parameters. Quantitative estimates for the localization error known as Crámer-Rao lower bounds (CRLBs) have been derived for the fitting of isolated fluorophores plus a homogeneous background (8,9) or for multiple overlapping fluorophores in single images (10,11). Despite common practice, these theoretical descriptions are not directly applicable to stepwise photobleaching. For example, the removal of a signal by subtraction leaves its shot noise behind (12). In the case of a fluorophore, this noise has a particular amplitude, shape, and position and is quite different from a homogeneous background. Moreover, it is not clear how constraints that are obtained from fitting of fluorophores in other images refine the CRLBs of multi-PSF fitting.

Here, we derive fundamental lower bounds for the localization precision in the presence of shot noise from other emitters and generalize these bounds to the practical situations of interest. We use Monte Carlo simulations to verify the correctness of the theoretical descriptions and benchmark different image analysis strategies in terms of their localization precision. Finally, we discuss implications for stepwise photobleaching and other high-density imaging techniques.

Materials and Methods

CRLB calculations

Numerical calculations of CRLBs were performed using Mathematica 8.0 (Wolfram Research, Champaign, IL). For the image models and noise models, pixelated Gaussians were used (Eq. S1 in the Supporting Material). For the inversion of matrices that were close to singular, we used Mathematica’s in-built function PseudoInverse. Analytical derivations were also performed with Mathematica using smooth Gaussian PSF models. Details and example code can be found in the Supporting Material accompanying this article online.

Monte Carlo simulations and fitting

Monte Carlo simulations and fitting were done in MATLAB (The MathWorks, Natick, MA). Sample images were generated by random sampling from a Poisson distribution using pixelated Gaussians as the image model.

Maximum likelihood estimation (MLE) was performed using the same image model (with or without fixed coefficients) by minimizing the negative log-likelihood for Poisson-distributed data. Because MLE with Poisson statistics requires that the expected number of photons and its variance match, image subtraction (A − B) leads to incompatible data (and sometimes even negative photon counts). The model for the extra noise $\Delta \mu ={\mu }^{A+B}-{\mu }^{A-B}$ theoretically needs to be added to both the data and the fitting function in the likelihood estimator $L\sim p\left(x_k+\Delta {\mu }_k|{\mu }_k^{A+B}\right)$ (derived analogously to Snyder et al. (13); see also the Supplemental Material in Huang et al. (14)). However, for the independent fitting of individual difference images we replace the missing signal by a constant offset $L\sim p\left(x_k+c|{\mu }_k^{A-B}+c\right)$ that is treated as a free fitting parameter and perform MLE under the boundary condition $x_k+c\ge 0$. Least-square fitting does not require any special adaption.

Throughout this work, we assume that pixel values are given in photons. For experimental data, camera counts must be (approximately) converted into photons before fitting, as previously described by others (9,15). We analyzed 1000 region-of-interest images (21 × 21 pixels for Figs. 1, b and c, 2, c and d, 3, and 4, c and d; 13 × 13 pixels for all others) per condition and excluded nonconvergent fits from further analysis. Example code can be found in the Supporting Material. The standard deviation of best estimates was taken as a measure of parameter precision.

Figure 1.

Fundamental localization precision in the presence of shot noise from a second fluorophore. Fluorophore 1 with intensity N₁ is localized in the sum image using the known parameters of fluorophore 2 as fixed parameters. (a) Localization precision as a function of the intensity of the second fluorophore at the same position. (b and c) Localization precision (b) parallel or (c) perpendicular to the positional shift δ_x2 of the second emitter. Shown are the localization precision from simulations (open circles), CRLBs (solid lines), and the analytical approximation according to Eq. 3 (dashed lines). In b and c, N₁ = 10,000 photons. The size of the Gaussian PSFs was s₁ = s₂ = 1.5 pixels. To see this figure in color, go online.

Figure 2.

Error propagation in sequential fitting. (a) Localization precision versus the intensity of the second fluorophore. Shown are simulations (open circles), the ideal CRLBs (solid lines), and the error-propagated CRLBs according to Eq. 4 (dashed lines). n_A = n_B. (b) Localization precision as a function of the relative number of images of the second (n_B) or both (n_A) fluorophores. (c and d) Decay of propagated errors in the direction (c) parallel or (d) perpendicular to the direction of separation between fluorophores. Dashed lines: theory accounting for the spatial overlap (see Supporting Material). N₁ = N₂ = 10,000 photons, s₁ = s₂ = 1.5 pixels. To see this figure in color, go online.

Figure 3.

Localization precision in global fitting as a function of separation distance between fluorophores. (a and b) Localization precision for x₂ and y₂ of the second fluorophore. (c and d) Localization precision for x₁ and y₁ of the first fluorophore. Shown are simulations (open circles), the CRLBs for global fitting (solid lines), and the error-propagated CRLBs for sequential fitting (dashed lines). N₁ = N₂ = 10000 photons, s₁ = s₂ = 1.5 pixels. To see this figure in color, go online.

Figure 4.

Localization precision of difference images using MLE with a variable homogenous offset. (a and b) Localization precision (a) versus the intensity of the second emitter or (b) as a function of the number of images n_B and n_A. (c and d) Localization precision for (c) x₁ and (d) y₁ versus the separation between fluorophores. Shown are simulations (open circles) and the CRLBs for fitting of difference images (solid lines). N₁ = 10,000 photons, s₁ = s₂ = 1.5 pixels. To see this figure in color, go online.

Results

Ideal localization precision in the presence of shot noise from other fluorophores

Derivation

Fundamental lower bounds for the variance of parameter estimates are given by the diagonal elements of the inverse of the Fisher information matrix $\langle {\theta }_i^2\rangle \ge {\left({\mathbf{I}}^{-1}\right)}_{ii}$. Let ${\mu }_k\left(x,y\right)$ denote the mean value of the image model function $\mathbf{\mu }$ with parameters $\mathbf{\theta }=\left\{{\theta }_1,{\theta }_2,\mathrm{...},{\theta }_n\right\}$ in the $k$-th pixel at position $\left(x,y\right)$ in the image plane. Assuming that the data in each pixel are distributed with mean value ${\mu }_k$ and variance ${\mathrm{var}}_k$, the entries of the Fisher information matrix $I_{i,j}$ are given by Eq. 1 (for a detailed derivation, see Supporting Material) (8). Here, the sum runs over all pixels in the region of interest.

$${\text{I}}_{i,j}=\sum _k\frac{1}{{\mathrm{var}}_k\left(x,y\right)}\frac{\partial {\mu }_k\left(x,y\right)}{\partial {\theta }_i}\frac{\partial {\mu }_k\left(x,y\right)}{\partial {\theta }_j}$$ (1)

The variance term in the denominator normalizes the information content contained in the derivatives of the signal: the larger the noise in a pixel, the smaller is its (weighted) contribution to the goodness of the fit. The shot noise associated with the fluorescence signal follows Poisson statistics, and the variance is thus equal to the expected signal ${\mathrm{var}}_k={\mu }_k$. Additional noise sources that are not part of the image model can be incorporated into this variance term as well. If the noise sources are independent, the total variance in each pixel can be written as their sum (8,16). This approach has been used to account for Gaussian electronic readout noise in CCD (8) or sCMOS cameras (14). Here, we use the same approach to study the effect of isolated shot noise from other emitters on the localization precision.

As a PSF model, we consider the case of a freely rotating fluorophore in focus whose PSF is commonly approximated by a two-dimensional Gaussian (17,18) of the form ${\mu }_l\left(x,y\right)=N_l/2\pi {\tilde{s}}_l^2\cdot \text{exp}\left(-{\left(x-x_l\right)}^2/2{\tilde{s}}_l^2\right)\text{exp}\left(-{\left(y-y_l\right)}^2/2{\tilde{s}}_l^2\right)$. Here, $N_l$ is the total number of photons, $\left(x_l,y_l\right)$ is the fluorophore position, and ${\tilde{s}}_l$ is the effective width of the PSF. The latter accounts for the finite pixel size $a\times a$ and is related to the true Gaussian width $s$ by the relation ${\tilde{s}}^2=s^2+a^2/12$ (15,16,19,20). Analytical CRLBs for parameter errors are calculated by inserting this PSF model into Eq. 1 and replacing the sum by an integral over the detection plane (Fisher matrix entries and CRLBs for all model parameters are given in the Supporting Material). In the following, the localization precision $\sigma$ is referred to as the square root of the mean-squared errors ${\sigma }_x^2=\langle {\left(\Delta x\right)}^2\rangle$ in the x direction and ${\sigma }_y^2=\langle {\left(\Delta y\right)}^2\rangle$ in the y direction.

Dependence on intensity

For an isolated emitter, we substitute $\mu =\mathrm{var}={\mu }_1$ and obtain the well-known result ${\sigma }_{x1}^2={\sigma }_{y1}^2={\tilde{s}}^2/N_1$. For the combined photon noise from the emitter itself and a second emitter, we substitute $\mu ={\mu }_1$ and $\mathrm{var}={\mu }_1+{\mu }_2$. For the special case in which both emitters are at the same position, an exact solution for the localization precision can be derived and is given by Eq. 2:

$${\sigma }_{x1}^2={\sigma }_{y1}^2=\frac{{\tilde{s}}^2}{N_1}\cdot \left(1+\frac{N_2}{N_1}\right)$$ (2)

When the second fluorophore is dim compared with the first ($N_2$ « $N_1$), the localization precision of the first fluorophore is given by its intrinsic shot noise. In the other limit, the localization precision worsens according to the square root of the intensity $N_2$ of the second emitter. Turnover between the two cases occurs when both emitters have approximately the same intensity. Then, the localization precision is a factor of $\sqrt{2}$ worse compared with the isolated case.

We next verified the theoretical prediction of Eq. 2 by simulations. To satisfy the conditions $\mu ={\mu }_1$ and $\mathrm{var}={\mu }_1+{\mu }_2$, we chose $\mu ={\mu }_1+{\mu }_2$ as the model function and used the true values of the second fluorophore as fixed parameters in the fitting procedure. The finite pixel size was accounted for by a binned Gaussian PSF model that had a constant value throughout the square pixels (9). This pixelated Gaussian PSF was also used to numerically calculate CRLBs using Eq. 1. Fig. 1 a shows the localization precision obtained from simulations along with the analytical and numerical CRLBs. The excellent agreement between the simulations and the CRLB proved that the MLE fitting achieved the theoretical minimum uncertainty. The exact match between the analytical formula and the CRLB verified that the analytical derivation and the correction for the finite pixel size were correct.

Dependence on distance

Whereas a close-by emitter affects the localization precision, a sufficiently distant emitter should not. To investigate this transition, we shifted the position of the second emitter by $\left({\delta }_{x2},{\delta }_{y2}\right)$ relative to the first emitter. The combined variance term in the Fisher matrix prohibited the derivation of an analytical solution for the CRLB. Only for the limiting case of $N_2$ » $N_1$ may we set $\mathrm{var}={\mu }_2$ and obtain a closed-form expression for the localization precision (see Supporting Material). To zeroth order, the effect of combined noise can then be approximated by a linear combination of the localization errors for the limiting cases (Eq. 3) (16). Here, the subscript $\xi \in \left\{x,y\right\}$ is used to shorten the notation for the x or y direction, respectively.

$${\sigma }_{\xi 1}^2\approx \frac{{\tilde{s}}^2}{N_1}\cdot \left(1+\frac{N_2}{N_1}\left(1+{\delta }_{\xi 2}^2/{\tilde{s}}^2\right){\mathrm{e}}^{-\left({\delta }_{x2}^2+{\delta }_{y2}^2\right)/{\tilde{s}}^2}\right)$$ (3)

The localization precision is expected to decay exponentially for larger separations and is anisotropic parallel and perpendicular to the shift direction. Equation 3 agreed with the numerical CRLB and with simulations in the limiting cases of full overlap and full separation of the two fluorophores (Fig. 1, b and c). The transition in between, however, showed a more complex behavior. Perpendicular to the offset, the localization precision monotonically improved with increasing separation, with a typical transition distance of little more than the PSF half-width (Fig. 1 c). Parallel to the axis through both emitters, the localization precision showed a worsening at intermediate separations before it decayed (Fig. 1 b).

Neither the decay length nor the intermediate worsening was correctly predicted by Eq. 3. The simple linear combination thus yielded an incorrect, overoptimistic estimate of the localization precision. It was previously shown that an empirical correction (20) could substantially improve the accuracy of the linear superposition that was derived for the combination of intrinsic shot noise with a constant background (16). Consequently, an empirical correction to Eq. 3 is discussed in the Supporting Material (Figs. S8–S10). However, we note that the numerical CRLB calculations are more accurate and thus should be preferably used to assess the localization precision.

Limits for different image analysis procedures

The ideal CRLBs we have discussed so far were derived and achieved under the assumption that we had perfect knowledge about the second fluorophore. In practice, however, the information from independent images is limited and the achievable localization precision will be worse. In the following, we investigate CRLBs for three common image analysis procedures that make different use of the available information. For all cases, we assume that we have $n_A$ images (A) that contain two fluorophores, and $n_B$ images (B) that contain only the second fluorophore. In stepwise photobleaching, these correspond to the plateaus before and after the second-to-last bleaching step. We seek the CRLB for the localization precision of the first fluorophore. The parameters (position and intensity) of both fluorophores are supposed to remain constant throughout all images.

Sequential fitting

Sequential fitting proceeds from the last to the first plateau. The parameters of the second emitter are first determined by a single-PSF fit from images B, and these parameters are then used as fixed parameters in the double-PSF fit for the first emitter in the common images A.

The double-PSF fit with fixed, exact parameters for the second fluorophore had been shown to reach the fundamental CRLB (see Fig. 1). However, the parameters of the second emitter now contain errors. The signal in an image containing several fluorophores is a weighted sum of their PSFs, with the fluorophores’ intensities serving as weights. The uncertainty in the position of the second fluorophore ${\sigma }_2^2$ thus adds to the localization error of the first fluorophore according to the laws of error propagation. For fully correlated signals, the effective localization precision is given by Eq. 4. Here, ${\sigma }_{1,CRLB}^2$ is the ideal lower bound.

$${\sigma }_1^2={\sigma }_{1,CRLB}^2+{\left(\frac{N_2}{N_1}\right)}^2\cdot {\sigma }_2^2$$ (4)

Fig. 2 a shows the localization precision as a function of the intensity of the second emitter at the same position, and Fig. 2 b shows its dependence on the number of images that determine the precision of the first fit. The data from simulations approached the ideal CRLB only when the second emitter was dim or many images were used to determine its parameters. The data were well described by Eq. 4, which assumes a complete correlation between the two errors and accounts for a weighting by the relative intensities of the fluorophores.

The simple relation (Eq. 4) does not hold true when fluorophores are separated in space. This is easily seen from the limiting case of nonoverlapping PSFs, where the error of the second fluorophore is meaningless for the localization of the first. Actually, the localization precision in the direction of emitter separation showed a biphasic decay over the length scale of one or two times the PSF half-width (Fig. 2 c). The error in the perpendicular direction decayed monophasically over a shorter range (Fig. 2 d).

We hypothesized that the distance dependence of the error correlation was governed by the sensitivity of the fit (represented by the diagonal Fisher matrix entries for x₁ and y₁) to systematic intensity differences between the fitted and true images of the second fluorophore. Fisher matrix entries varied for each pixel over the region of interest, which is commonly observed in multi-PSF fitting (14), as did the intensity differences, and we speculated that their integrated overlap entered the error propagation (see Fig. S1 and Supporting Material). This theory captured the main characteristics of the correlation in the simulations, including biphasic and monophasic trends (Fig. 2, c and d, dashed lines). Although individual fits were correlated in a more complicated way (see Fig. S2), the localization precision in sequential fitting was well explained by the ideal CRLBs plus error propagation.

Global fitting

For global fitting, each image was modeled by an appropriate number of PSF models whose parameters were shared between images that contained the same fluorophore. All parameters were then estimated simultaneously in a single optimization process.

To calculate CRLBs for global fitting, we followed the derivation outlined by Liu and co-workers (21) for determining the 3D localization precision from two images at different focal planes. Let the independent data sets (= images from different plateaus) and corresponding image models be denoted by the superscript $m\in \left\{A,B,\dots \right\}$. The summation for calculating the Fisher matrix entry $I_{i,j}$ then runs over all image models $m$ and all pixels $k$ (Eq. 5). Here, the parameter set $\mathbf{\theta }$ contains all unique parameters of all PSF models.

$${\text{I}}_{i,j}=\sum _m\sum _k\frac{1}{{\mathrm{var}}_k^m\left(x,y\right)}\frac{\partial {\mu }_k^m\left(x,y\right)}{\partial {\theta }_i}\frac{\partial {\mu }_k^m\left(x,y\right)}{\partial {\theta }_j}$$ (5)

Simulations showed that global fitting achieved the same precision as the sequential fixed parameter fit when the two fluorophores were at the same position (Fig. S3). Also, the rigorous CRLB for global fitting and the error-propagated CRLB for sequential fitting fully agreed in this case. This equivalence of fitting procedures is explained by the fact that there is no independent information about the second fluorophore in the common image. Global fitting in this case effectively proceeds analogously to sequential fitting.

However, global fitting provided a significant advantage over sequential fitting with increasing emitter separation (Fig. 3). The localization precision for the second fluorophore substantially improved in an anisotropic fashion (Fig. 3, a and b). This improvement was expected from the increasingly independent information in the common image, but was not fully explained by the additional information from an independent double-PSF fit (Fig. S4, a and b). The localization precision of the first fluorophore was only slightly improved at intermediate separations (Fig. 3, c and d). This improvement was fully explained by the better precision of the second fluorophore and error propagation (Fig. S4, c and d). We conclude that error propagation again governed the localization precision of the first fluorophore in the common image, whereas the combined information improved the localization of the second fluorophore.

Image subtraction

Subtracting images after a bleaching step from images before a bleaching step leaves behind the signal from the bleached fluorophore, which then is fitted to the PSF model. One advantage of image subtraction is its simplicity: the fit has few parameters, and each bleaching step is usually analyzed independently.

When images are subtracted, signals that are present in both images are removed, but their noise adds up in the same way as for image addition. We thus derived CRLBs for fitting of the processed image by replacing the variance term in Eq. 1 by the sum of all fluorescent sources in the original images. For simple addition/subtraction of two raw images, the noise from the second fluorophore was doubled. The CRLB for the localization precision had the same form as the fundamental CRLB, but with a prefactor of 2 for the intensity of the second fluorophore (replace $N_2\to 2N_2$). More generally, when a different number of images $n_A,n_B$ were available, their linear combination resulted in a prefactor of $1+n_A/n_B$.

We investigated the extent to which least-squares minimization (LSq) or MLE with some adaptions to difference images (see Materials and Methods) reached this CRLB. Fig. 4 shows the results for MLE with a variable offset. The fits were well described by the CRLB for difference images when the fluorophores were at the same position (Fig. 4, a and b); however, they resulted in a substantially worse localization precision at intermediate distances (Fig. 4, c and d). Accounting for the missing signal of the subtracted fluorophore can improve the fit performance (Fig. S5) but requires the parameters from the fit of the second fluorophore; thus, it loses the sole advantage of image subtraction and comes at the cost of a worse CRLB in comparison with sequential fitting. The most commonly used LSq fitting of difference images approached a localization precision of ∼4/3 times the CRLB (Fig. S6). This prefactor was also previously found in the context of fitting an isolated fluorophore (19). In the presence of a homogeneous background, all three fitting versions showed a more similar behavior, but they did not always reach the CRLB (Fig. S7).

In summary, image subtraction added additional noise to the images that negatively affected the CRLBs. Moreover, the missing signal led to inconsistencies that prevented the fits from reaching even these inferior CRLBs.

Application to distance measurements

Thus far, we have shown how the localization precision of two fluorophores is affected by the fluorophore intensities, background, relative position, the number of images, and the fitting procedure. The calculation of CRLBs from fitted parameters can be used to assess the precision of individual measurements. One important application is the determination of a distance $d$ between two labels. The error ${\sigma }_d$ of the measured distance results from the individual localization errors as ${\sigma }_d^2={\sigma }_1^2+{\sigma }_2^2$ and thus will also depend on the aforementioned parameters.

Fig. 5 shows results for global fitting of two fluorophores with the same intensity. The expected error of the distance according to the CRLBs of the individual localizations decreased with an increasing number of images in each plateau in a nontrivial way (Fig. 5 a). For small distances, the precision depended more strongly on the length of the second plateau. This finding is explained by error propagation in the localization of the first fluorophore in the common images. For larger distances, the precision was dominated by the length of the first plateau, which is a direct result of reduced error propagation and increasingly independent information about the second fluorophore in the common images (see also Fig. 3).

Figure 5.

Precision of distance measurements in stepwise photobleaching using global fitting. (a) Theoretical dependence on the number of images per plateau. The error of distance measurements was calculated from the CRLB for the positions of emitters 1 and 2 in global fitting. Contour plots depict the precision of the distance (color coded) for different distances (d = 0, 0.75, 1.5, and 3 pixels). n_A and n_B, number of images in the first and second plateaus, respectively. (b) Results from simulations with exponential bleaching kinetics. The scatterplot shows the estimated precisions versus the distance from individual fits. Shaded: theoretical confidence intervals according to the Rice distribution. Simulation parameters: N₁ = N₂ = 1000 photons/frame, bg = 30 photons/pixel, d = 0.14 pixels, s₁ = s₂ = 1.5 pixels, τ_1/2 = 40 frames. (c) The distance distributions (histograms) for different precisions fully agreed with the expected Rice distribution (red lines; Eq. 4 in Churchman et al. (22)). To see this figure in color, go online.

In experiments, the lengths of plateaus follow an exponential distribution and thus have different probabilities. We simulated this behavior by Monte Carlo simulations with two emitters at a fixed distance $d=0.14$ (see Supporting Material). We used global MLE fitting to estimate the parameters of the two fluorophores, and then calculated the CRLBs using these fitted parameters. Fig. 5 b shows a scatterplot of the calculated precision of the distance versus the fitted distance. Individual fits showed a substantial variation in both fitted values and precision. The distribution of measured distances broadened with increasing error (Fig. 5 c). This behavior was quantitatively described by the Rice distribution (22,23) using the correct distance and the respective standard deviations, without any free fitting parameters. The individual fits also fell well within the confidence intervals for this expected distribution (Fig. 5 b). We conclude that the precision of individual distance measurements was correctly calculated from the CRLBs and that the analysis procedure reached these bounds.

Discussion

Consequences for stepwise photobleaching

All tested analysis procedures reached similar localization precision with a second fluorophore at short distances (<1 pixel; Figs. 2–4). The similar performance implied that the removal of a noisy fluorophore signal by image subtraction (which effectively adds noise) had the same effect as using its limited information for fitting. This equivalence can be verified analytically from Eqs. 2 and 4 (see Supporting Material). However, the analysis procedures differed substantially under practical conditions and when the localization precision of both fluorophores was considered. Compared with fitting of original images, the CRLB for image subtraction was worsened by the summation of the homogeneous background and was not always reached by the fitting routines. Global fitting provided a substantial advantage over sequential fitting for the localization of the first fluorophore at larger separations. From a practical perspective, the calculation of error-propagated lower bounds for sequential fitting is only approximate and becomes tedious for more than two fluorophores. Global fitting was simply optimal for all fluorophores under all conditions. The respective CRLBs for all parameters can be directly calculated from the fit result via the comprehensive Fisher matrix (Eq. 5). Global fitting thus provides the most consistent framework for quantifying stepwise photobleaching experiments.

The CRLBs for two emitters given in this study can be easily generalized to an arbitrary number of emitters. However, the localization precision quickly worsens with an increasing number of fluorophores due to the accumulation of propagated errors. In addition, more fluorophores result in larger shot noise and shorter plateaus, which complicates the identification of early bleaching steps in the intensity-time trace. Taken together, these drawbacks restrict the use of stepwise photobleaching to a handful of fluorophores.

Relation to other localization microscopy techniques

Several superresolution methods isolate the fluorescence from single emitters by image subtraction, taking advantage of dye blinking or bleaching (24–26). In principle, the CRLBs presented here for difference images are relevant to quantify the localization precision in these cases. However, calculation of the CRLBs requires knowledge about the number, intensity, and position of the noise sources. This contrasts with the main advantage of image subtraction, i.e., that the localization of an emitter in a difference image does not require any knowledge about the subtracted signal sources. Thus, a rigorous calculation of CRLBs is not feasible for most image subtraction techniques.

Nevertheless, we may estimate the localization precision of a blinking/bleaching emitter in a difference image based on the density of the fluorophores. According to Eq. 2, the presence of noise from $n$ subtracted, equally bright, and substantially overlapping emitters affects the localization of the remaining emitter as ${\sigma }^2={\sigma }_0^2\left(1+2n\right)$, where ${\sigma }_0$ denotes the precision of a single emitter without background. As an example, when 13 dyes are present within an area of ∼ $100\times 100$ nm², the localization precision from a difference image is ∼5 times worse than that achieved by classical stochastic optical reconstruction microscopy (see also Supporting Material), which is a typical value that has also been found experimentally (25). Because the acquisition times for image subtraction techniques are shorter than those required for stochastic optical reconstruction microscopy, the choice of methods becomes a trade-off between speed and precision, similar to the case of multi-PSF fitting.

An alternative method called Bayesian analysis of blinking and bleaching (3B) maximizes the probability that the observed images originated from N blinking fluorophores at certain positions (27). Hence, it combines the two tasks that typically are performed sequentially in analyses of stepwise photobleaching experiments; however, it does not provide estimates for the parameter precision. Although calculating CRLBs by following a procedure similar to that used for global MLE fitting of multiple images seems attractive, it is not feasible because 3B does not provide a definite estimation of the blinking history, but rather integrates over different possible realizations. Hence, the results from 3B analysis are not amenable to the estimation methods presented here.

Conclusions

The descriptors of localization precision in stepwise photobleaching experiments presented here complement results from other techniques (recently reviewed in Deschout et al. (28)). The use of correct CRLBs helps to improve the reliability of the results and the precision of stepwise photobleaching, i.e., of distance measurements. We further propose that filtering of data according to their precision could be used to meet a required resolution (4). Finally, the precision information allows one to study the heterogeneity of distances or patterns within a population, e.g., to investigate different conformations of a biomolecule.