Introduction

Super-resolution techniques includes scanning methods, such as Stimulated Emission Depletion (STED) microscopy1,2, and single-molecule localization microscopy (SMLM) techniques like STORM3,4, and PALM5,6 have become essential tools in the life sciences for visualizing cellular and tissue structures with high spatial resolution. Rather than imaging the target molecules themselves, such as specific cellular proteins, the fluorophores, as reporters, attached to the molecules of interest are detected7,8,9,10. As a result, an ideal labeling strategy should precisely position the fluorophores as close as possible to their targets to minimize displacement between the target molecule and the reporter, ensuring greater localization accuracy.

Immunofluorescence, one of the most widely used labeling methods, has been indispensable in biological research for over 80 years due to its high sensitivity and specificity11,12,13,14,15. In this approach, a known antibody labeled with a fluorophore binds specifically to the antigen on the target protein. A key parameter-the distance between the fluorophore and the target protein, known as the linkage error-can compromise localization accuracy in SMLM imaging16,17,18. Understanding and minimizing this linkage error is, therefore, critical to advancing the accuracy of super-resolution imaging.

As the precision of single-molecule localization microscopy (SMLM) improves, antibody size has become crucial for affecting the linkage errors and further enhancing localization accuracy in immunofluorescence techniques. Immunofluorescence methods have evolved from indirect to direct labeling to reduce the antibody size. Conventional indirect immunolabeling, which uses primary and secondary antibodies, introduces significant linkage errors due to the large size of the antibody complex (10–15 nm, 150 kDa), which can displace fluorophores from target proteins by more than 20 nm16,17,19. Even direct conjugation of fluorophores to primary antibodies still yields linkage errors of up to 13 nm20. In contrast, nanobodies (single-___domain antibodies, 15 kDa, 4 nm) offer a promising solution21,22,23,24,25. They allow for direct labeling of native protein targets, combining the molecular specificity of genetic tagging with the high photon yield of organic dyes, while significantly reducing linkage errors. Thus, nanobodies have the least impact on the accuracy of immunofluorescence results.

Linkage errors in antibody labeling are influenced not only by antibody size but also by structural flexibility, including variations in geometric and rotational angles of antibody fragments and the random distribution of fluorophores on antibodies (Fig. 1). Notably, these geometric and rotational angles can differ widely between antibody types and even within the same type26. Directly measuring structural flexibility and quantifying its effect on linkage errors remains challenging in experimental settings; consequently, simulations have been used to explore this effect. Previous simulations have shown that geometric angles, such as the rotation of the Fab fragment relative to the Fc region and the restricted binding orientation of the primary antibody’s Fab fragment, contribute significantly to linkage error magnitude20,26. However, the combined effects of molecular size and structural flexibility on linkage errors have yet to be fully understood through either experimental or simulated approaches.

Fig. 1: Schematic of three different labeling strategies.
figure 1

a Three types of antibodies labeled on a microtubule. The details of antibody structures include the nanobody (i.e., S reporter) (b), the primary Y antibody (i.e., M reporter) (c), and the secondary antibody bound to the Y antibody (L reporter) (d). b1–d1 3D representations of the structures used in the experiment. b2–d2 Diagrams of cross-sections illustrating microtubule labeling of different antibodies, with structural flexibility represented by the positions of fluorophores (red dot or red cycle in this figure), the geometric angles \(\phi\), and the rotational angles angle \(\theta\). b3–d3 Diagrams of cross-sections illustrating microtubule labeling of different antibodies without structural flexibility. “4 nm” and “12.5 nm” represent the molecular sizes for the nanobody and the Y antibody, respectively, and d is the diameter of the microtubule measured by the labeling strategy without any structural flexibility.

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Previous studies have assessed linkage errors in antibody labeling by calculating mean and variance through multiple experimental trials, where the presence of variance suggests that linkage errors may follow a distinct distribution19,20. This distribution has typically been represented using mean and variance based on the assumption that linkage errors conform to a normal distribution27. However, no studies to date have examined the actual distribution patterns for each labeling strategy. Additionally, experiments alone cannot elucidate the mechanisms underlying these distributions—specifically, how label size and structural flexibility might jointly influence them. Given the dimensions of antibodies (over 2 nm), traditional fluorescence nanoscopy, with a resolution greater than 10 nm, cannot accurately measure individual linkage errors, even for indirect immunofluorescence (~20 nm), and is therefore inadequate for determining the full probability distribution of linkage errors.

MINFLUX (Minimal Photon Fluxes) is a cutting-edge fluorescence nanoscopy technique28,29,30,31. This technology enables researchers to examine structural details and molecular interactions with remarkable precision. While individual localization precision in MINFLUX falls within the range of 1–3 nm, this does not directly define the microscope’s overall resolution. True resolution is influenced by multiple factors, including localization accuracy (such as deviation from the actual molecular position), potential drift, and sample-induced effects. The FRC analysis or similar validation techniques in our Supplementary Note 1 can help validate the effective imaging resolution. The ultra-high resolution achieved by MINFLUX enables precise analysis of linkage error distributions, which is critical for identifying artifacts or inaccuracies in labeling protocols32,33. In this study, leveraging the superior localization precision of MINFLUX, we aim to reveal the probability distribution of linkage errors across three labeling strategies: nanobody labeling, primary labeling, and secondary labeling. The structure of this paper is as follows: In the experimental section, we use MINFLUX imaging to measure the radii of microtubules—well-characterized models with known radii (12.5 nm) and structures—to determine linkage errors for each labeling strategy by calculating the difference between measured and actual known radii18,19. Then, we examine the distributions of linkage errors for these three strategies, comparing them with an idealized normal distribution. In the simulation section, we employ existing simulation software to validate our experimental results and subsequently develop a regression model to infer how label size, rotational angles, and geometric angles jointly contribute to the observed distribution of linkage errors.

Results

Diagram of the difference between antibody labeling strategies

In this study, we used microtubules as the labeling target, a well-characterized model structure in immunofluorescence research. Microtubules exhibit a straight appearance and a circular, tubular cross-section with a diameter of approximately 25 nm. To examine labeling accuracy, we tested three distinct strategies for tagging microtubule target proteins: direct labeling with a nanobody (S reporter), direct labeling with a primary “Y” antibody (M reporter), and indirect labeling using a secondary antibody bound to the primary Y antibody (L reporter), as shown in Fig. 1. Each labeling approach involves molecular components of different sizes and structures. The nanobody measures approximately 4 nm, the Y antibody is about 10–15 nm, and the combined primary and secondary antibodies reach roughly 20–30 nm. Figure 1a illustrates a microtubule labeled with each of these three reporters, highlighting differences in molecular size and structure. The S reporter is the smallest, while the L reporter is the largest. These varying labeling sizes allow us to evaluate how label size and structure impact the linkage errors in immunofluorescence.

Figure 1b1–d1 presents structural details for each antibody type. Nanobodies (S reporter) consist solely of the variable region of the heavy chain (VHH) from camelid antibodies, making them unique in their compact size of approximately 15 kDa (Fig. 1b1). Nanobodies have three complementarity-determining regions (CDRs)—CDR1, CDR2, and CDR3—responsible for antigen binding. CDR3 is often longer and more flexible than in conventional antibodies, allowing nanobodies to recognize a broader range of antigens, including hidden or unique epitopes. In contrast, traditional IgG antibodies (M reporter) have a “Y” shaped structure with distinct functional regions (Fig. 1c1). The Fab region can bend and rotate, adapting to various antigen surfaces, while the hinge region between the Fab and Fc segments provides additional rotational flexibility. Indirect immunolabeling (L reporter) further enhances structural flexibility by introducing two hinge regions—one in the primary “Y” antibody and one in the secondary antibody (Fig. 1d1). Consequently, the structural flexibility increases progressively from the S reporter to the M reporter, reaching its peak in the L reporter due to the added hinge regions.

In addition to the Fab’s rotation and swinging motions, structural flexibility includes the rotation and swinging of other antibody fragments and the stochastic labeling of fluorophores to antibody amino acid residues (Fig. 1b2–d2). These rotational and swinging motions are defined by the rotation angle θ and the swing angle ϕ, respectively. Both the rotation and the swing have one degree of freedom for the S reporter(i.e., θ1 and ϕ1), three degrees of freedom for the M reporter, and five degrees of freedom for the L reporter. Specially, the rotation angle and the swinging angle of the labeled Fab for the Y antibody are θ1 and ϕ1, and for the secondary antibody, are θ3 and ϕ3. Therefore, the degrees of freedom of these angles increase progressively from the S reporter to the M reporter and are greatest in the L reporter. These angles affect the labeling uncertainties. Moreover, the stochastic labeling of fluorophores to the amino acid residues of the antibody (red circle in b3–d3) and the angular range of protein epitopes on the microtubule surface ϕ0 also contribute to the labeling uncertainties.

Here, we provide a mathematical definition for the linkage error. The linkage error ei is defined as the difference between the radius ri measured by MINFLUX and the actual radius Ri of the microtubule for each segment, i.e., ei = ri -Ri, i = 1, 2,…, n, where n is the number of segments. Then, the average linkage error for a special labeling strategy is \(e={\sum }_{i=1}^{n}{e}_{i}/n\). Due to the differences in the molecular sizes of the antibodies and the labeling uncertainties originating from their structural flexibility, the values of linkage errors \({e}_{i}\) vary not only between labeling strategies, but also exhibit a distribution within each strategy.

When antibodies are modeled as rigid structure, the linkage errors (ei) ideally correspond to the sizes of the antibody reporters. These values are set to 4 nm, 12.5 nm, and 25 nm for the S, M, and L reporters, respectively (Fig. 1b3–d3). Given that biological assemblies consist of multiple molecules, labeling errors frequently manifest in opposing directions, doubling the effect of the tag offset on the measurement. Therefore, the diameters of the structures of interest, specifically microtubules, measured by each labeling strategy are d = 33 nm, d = 50 nm, and d = 75 nm, respectively. These values correspond to the sum of the actual known microtubule diameter (D = 25 nm) and twice the reporter size. In practice, due to structural flexibility and experimental variability, the measured microtubule diameters differ across labeling strategies, even when reporters of the same size are used. In the following sections, we examine the microtubule diameters obtained with each labeling strategy using MINFLUX and calculate the corresponding linkage errors.

Non-normal distribution of the diameters for the microtubules measured by each labeling strategy

To evaluate differences in linkage errors caused by different antibody labeling strategies in experiment, we used MINFLUX microscopy for qualitative and quantitative analysis. Microtubules, with a known diameter (D = 25 nm), were chosen as an in situ standard. Fluorescence microscopic imaging was performed on fixed BSC-1 cells stably expressing tubulin, labeled using three distinct antibody strategies (Fig. 2): (1) direct immunofluorescence with AF647-tagged anti-ALFA nanobodies (S reporter, Fig. 2a), (2) direct immunofluorescence with AF647-tagged anti-tubulin antibodies (M reporter, Fig. 2b), and (3) indirect immunofluorescence with anti-tubulin primary antibodies and AF647-tagged secondary antibodies (L reporter, Fig. 2c). These methods allowed direct visualization of hollow microtubule structures, with notable differences in appearance based on the labeling strategy. Under MINFLUX imaging, microtubules labeled with direct immunofluorescence appeared thinner than those labeled by indirect immunofluorescence (Fig. 2a4–c4), revealing clear clusters of individual antibodies. Nanobody labeling results in lower localization density despite the smaller size and enhanced binding efficiency of nanobodies, which offer certain advantages. However, these benefits may be offset by steric hindrance or accessibility limitations, which can hinder effective binding in certain cellular contexts. The localization accuracy of MINFLUX fluorescence imaging, as illustrated in Supplementary Fig. 1, achieves a precision of approximately 3 nm. In contrast, no significant differences between the three labeling markers were observed under confocal or STED microscopy (Supplementary Fig. 2). These findings confirm that MINFLUX microscopy can effectively distinguishes the diameter values d (or corresponding linkage errors e) associated with each labeling approach, offering insights into the accuracy of different immunofluorescence methods. To confirm that MINFLUX imaging captures the microtubule’s maximal cross-section, we aligned the imaging focal plane with the area of highest intensity in the confocal images (Supplementary Fig. 3). This approach ensures that the observed hollow diameter of the microtubule represents its largest dimension.

Fig. 2: The distribution of diameters measured from MINFLUX imaging of labeled microtubules for each reporter.
figure 2

a–c Confocal and MINFLUX images obtained using different labeling strategies in fixed BSC-1 cells stably expressing tubulin. The first two lines are confocal imaging and the last two lines are MINFLIUX imaging. a Direct immunofluorescence labeling with anti-ALFA nanobodies, i.e., S reporter. b Direct immunofluorescence labeling with rabbit anti-tubulin antibodies, i.e., M reporter. c Indirect immunofluorescence labeling with mouse anti-tubulin antibodies and secondary mouse antibodies, i.e., L reporter. d–f Experiments and simulations of the probability distribution for the diameter d around the center of microtubule. The top row is the experiment, the bottom row is the simulation. The five circles represent the minimum value, first quartile, average value, second quartile, and the maximum value, respectively. The minimum and maximum values are defined as the average of the top 10 smallest and the top 10 largest values, respectively. d Direct immunofluorescence labeling with anti-ALFA nanobodies, i.e., S reporter. e Direct immunofluorescence labeling with rabbit anti-tubulin antibodies, i.e., M reporter. f Indirect immunofluorescence labeling with mouse anti-tubulin antibodies and secondary mouse antibodies, i.e., L reporter. g Lateral intensity distribution of individual microtubule segments calculated from MINFLUX images, with peak-to-peak distances (reflecting the diameter value, d) calculated. h Measurement of peak-to-peak distances from at least 1000 microtubule segments across three kinds of reporters. The MINFLUX images have pixel sizes of 4 nm (a3–c3) and 1 nm (a4–c4, close-up), and the localization accuracy is visualized by a Gaussian distribution with a standard deviation of 2 nm (Fig. S1). Normal dist. S: defined as the artificial normal distribution of the same mean and the same standard deviation as the experimental data for S reporter; Normal dist. M: defined as the artificial normal distribution with the same mean and standard deviation as the experimental data for M reporter; Normal dist. L: defined as the artificial normal distribution with the same mean and standard deviation as the experimental data for L reporter.

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Based on the images in Fig. 2a–c, the diameter d and its probability distribution were quantitatively measured. Technically, the diameter d is measured by quantifying the lateral distribution of microtubule filaments, reflected by the peak-to-peak distance17,18. One example shows that the peak-to-peak distances were evidently different, i.e., d = 29.5, 53.6 and 69.5 nm for these three reporters, respectively (Fig. 2g). Based on the peak-to-peak distances, we obtain ~1000 values of d (n1 = 1003 for S reporter, n2 = 1038 for M reporter, and n3 = 1015 for L reporter) measured from ~1000 microtubule segments for each reporter. The mean and the standard deviation (SD) for these at least 1000 values of each reporter were (30.0 ± 3.9 nm, 49.4 ± 7.3 nm and 65.6 ± 9.9 nm). The probability distribution of these ~1000 d values around the microtubule is plotted for each reporter (Fig. 2d1–f1). It is observed that the thickness of the ring, from the smallest to the largest value of d, differs between different reporters. The ring is thinnest for the S reporter and thickest for the L reporter. Therefore, the range of the d values is shortest the S reporter and largest for the L reporter. For each reporter, five key parameters for each distribution are taken into account, including the minimum value, the first quartile, the average value, the second quartile and the maximum value. The depth of the red represents the probability of the value, and surprisingly, the deepest red ___location (about 27.5 nm for the S reporter, and 43.6 nm for the M reporter) is not around the circle of the average value (d = 30.0 nm for the S reporter and d = 49.4 nm for the M reporter), but closer to the first quartile (d = 26.8 nm for the S reporter and d = 43.8 nm for the M reporter). Interestingly, there are two deepest red locations for the L reporter, which are about 60.5 nm and 71.9 nm. These two values are not around the circle of the average value (d = 65.6 nm), but closer to the first quartile (d = 57.9 nm) and the second quartile (d = 72.9 nm), respectively. Accordingly, this is quite different from the normal distribution, where the probability of the values is largest around the average value.

The probability distribution of these values around the microtubule center is plotted for each reporter through numerical simulations (Fig. 2d2–f2). In accordance with Fig. 2d1–f1, we observe that the thickness of the ring, from the smallest value of d to the largest value of d, differs between the reporters. The ring is thinnest for the S reporter and thickest for the L reporter (d2–f2). Therefore, the range of the d values is shortest and largest for the S reporter and for the L reporter, respectively. For each reporter, five key parameters of the distribution are taken into account, including the minimum value, the first quartile, the average value, the second quartile and the maximum value. The results shown in (d2–f2) are consistent with those in Fig. 2d1–f1.

Furthermore, we compare the probability distribution of the d values with the artificial normal distribution (Fig. 2h). The solid lines represent the distribution curves based on experimental data for the three types of reporters, while the dashed lines represent the corresponding normal distribution curves calculated using the same mean values (i.e., 29.9 nm, 49.4 nm and 65.8 nm) and standard deviations (i.e., 3.9 nm, 7.3 nm and 9.9 nm) for each reporter of the experimental data. The experimental distributions for each reporter deviate significantly from a normal distribution, as confirmed by the Jarque-Bera test (p = 0.001). Particularly, the distribution is skewed to the left and flatter than the normal distribution (skewness = 0.058, 0.311 and 0.134, and kurtosis = 2.17, 2.33 and 2.23, for the three reporters, respectively, compared with skewness = 0.0 and kurtosis = 3.0 for normal distribution). Note that there are two peaks for the L reporter. Thus, the distribution of the diameter d for the experimental data differs from the normal distribution.

Nonnormal probability distribution of the linkage errors for each labeling strategy and simulation results validating the experimental results

The linkage error ei for each segment is the difference between the radius ri (ri = di/2) of the microtubules measured by MINFLUX images and the actual radius Ri of the microtubules, where i = 1, 2, 3, …, n. The values of the radius ri are obtained in Fig. 2, and here we introduce the value of the actual radius Ri. The actual radius Ri for the microtubules is regarded to be around 12.5 nm in previous studies34. Accordingly, we consider three conditions for the distribution of Ri, including all the values of Ri being equal to 12.5 nm, the values satisfying a normal distribution, and the values satisfying a uniform distribution (Fig. 3 and Supplementary Fig. 4).

Fig. 3: Distribution of linkage errors (e) and corresponding simulation results for three reporters.
figure 3

Linkage error distributions for experimental data (n₁ = 1003 for S reporter, n₂ = 1038 for M reporter, n₃ = 1015 for L reporter) and simulations (n = 30,000 per reporter). a Experimental distribution of e (calculated as ei = ri − Ri, where Ri = 12.5 nm), with shape matching the diameter distributions in Fig. 2h. b Simulated baseline distribution of e under the same constant R (12.5 nm), compared to an artificial normal distribution.

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Based on the experimental data, in the first condition, when all the values of Ri are equal to 12.5 nm, the mean and standard deviation for the linkage error e are 2.5 ± 1.9 nm, 12.2 ± 3.6 nm and 20.3 ± 4.9 nm for the three reporters, respectively. The distribution of e is regarded as the baseline distribution and is compared with the artificial normal distribution of the same mean and standard deviation in (a). It is evident that the distribution of e differs from the corresponding artificial normal distribution. Particularly, it is skewed to the left and is flatter than the normal distribution (skewness = 0.058, 0.311 and 0.134, and kurtosis = 2.17, 2.33 and 2.23, for the three reporters, respectively, compared with skewness = 0.0 and kurtosis = 3.0 for the normal distribution). Note that there are two peaks for the distribution of the L reporter. Thus, the distribution of the linkage error e for experimental data differs from the normal distribution.

In Fig. 3b, we reproduce the experimental results by numerical simulations (for simulation details, please see the “Methods”), and then find the key factors influencing the distribution of linkage errors. The values of linkage errors e are not significantly different between the numerical simulations and the experimental data for the L reporter (p = 0.67, t-test), M reporter (p = 0.81, t-test) and S reporter (p = 0.68, t-test), respectively. Thus, the simulations are valid for further analysis. In accordance with Fig. 2, it is evident that the distribution of e is different from the corresponding artificial normal distribution in (b). The distribution is also evidently different from a normal distribution; specifically, the distribution is skewed to the left for the simulation data, and the shape of the distribution is flatter than the normal distribution (skewness = 0.04, 0.328 and 0.102, and kurtosis = 2.2, 2.35 and 2.2, for the three reporters and, respectively, compared with skewness = 0.0 and kurtosis = 3.0 for normal distribution). Thus, the distribution of the linkage error e for simulation data also differs from the normal distribution.

To further analyze the distribution of the linkage error, we examine whether the distribution of e is altered from the baseline distribution if the values of the actual radius Ri differ between the microtubule segments within the same labeling strategy (Fig. S4). In Fig. S4a, c, it is observed that the distribution is not visibly different from the baseline distribution, respectively, if the values of Ri satisfy a normal distribution or a uniform distribution, respectively. Therefore, the moderate difference between the values of actual radius R does not influence the nonnormal probability distribution of the linkage error e.

Moreover, the distribution of the linkage errors e from experimental data is compared with the artificial normal distribution in terms of probability within a special range of e values (Table 1). It is visible that the probabilities within the ranges (Mean − SD, Mean + SD), (Mean − 2 SD, Mean + 2 SD) and (Mean − 3 SD, Mean + 3 SD) are different from those of the normal distribution for each reporter in Table 2. Particularly, in the range of (Mean − SD, Mean + SD), the probabilities for all three reporters are smaller than those of the normal distribution. In other words, the probability distribution is flatter than the normal distribution. This is consistent with the values of kurtosis <3.0 (i.e., 2.17, 2.33 and 2.23) for each reporter, where kurtosis = 3.0 corresponds to a normal distribution.

Table 1 Probability for the special range of e values
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Table 2 Influence of the parameters on the distribution of linkage errors
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Comparison of the probability distributions of linkage errors across three labeling strategies

These three distributions of linkage errors from the MINFLUX images are compared, revealing difference in their shapes due to the following results. First, the shapes are tested to be different between these three reporters through Kolmogorov–Smirnov test (p = 0.00). Second, the values of skewness and kurtosis vary among the reporters (skewness = 0.058, 0.311 and 0.134, and kurtosis = 2.17, 2.33 and 2.23, for the three reporters). In other words, the distribution for the M reporter is most pronounced left skewness, while the distribution for the S reporter is flattest. Third, in Table 2, it is visible that the probabilities are different between these three reporters in the range of (Mean − SD, Mean + SD), (Mean − 2 SD, Mean + 2 SD) and (Mean − 3 SD, Mean + 3 SD), respectively. For example, in the range of (Mean − SD, Mean + SD), the L reporter has the smallest probability, while the M reporter has the largest.

It is observed that these three distributions are not completely separated, and there are intersecting parts between them (Figs. 3 and S4). Typically, the probability of an ei value for the S reporter being larger than the minimum value (average of top 10 smallest values) of ei for the M reporter is 8.67%. Similarly, the probability of ei value for the M reporter being larger than the minimum value of ei for the L reporter is 67.44%. Conversely, the probability of an ei value for the L reporter being smaller than the maximum value (average of top 10 largest values) of ei for the S reporter is 0%, while the probability of an ei value for the L reporter being larger than the maximum value of ei for the M reporter is 39.21%. Note that there are no intersecting parts between the distributions of the S reporter and the L reporter. Because these three distributions are not completely separated, we suggest that it is not enough to determine whether the values of the linkage errors are larger or smaller between reporters based solely on the mean value.

The non-normal distribution of linkage errors among reporters, the overlap between different reporters, and the differences in their distribution shapes can be attributed to structural flexibility. This is because the structural flexibility of antibodies may influence their binding to fluorescent dyes or markers, and there may also be certain systematic and chance errors during the experimental process, leading to variations in labeling positions and differences in signals27,35. Additionally, the secondary and tertiary structures of antibodies may impact their behavior under different labeling methods, further contributing to variations in linkage errors36. Thus far, we cannot examine how structural flexibility affects the value of linkage errors in the experiment. As a result, the key factors influencing the values of linkage errors cannot be found through this experiment. In the simulation section, we will reveal the key factors and their joint effects on the distribution of the linkage errors for each reporter.

Key factors influencing the distribution of linkage errors revealed by the simulations

Based on the reproduced data, the key factors (parameters) influencing the distribution of linkage errors e are listed in Table 2. These parameters can be divided into three categories: the molecular sizes L, the geometric angles \(\phi\), and the rotational angles angle \(\theta\). The degrees of freedom are 15, 9 and 3 for L reporter, M reporter, and S reporter, respectively. It is clear that all the molecular sizes and geometric angles play a significant role in the distribution of linkage errors e (p ≤ 0.05, T-test), while all the rotational angles do not significantly affect the distribution for the S reporter and the M reporter (p > 0.05, T-test). Similarly, four out of five molecular sizes, three out of five geometric angles, and one out of five geometric angles significantly affect the distribution for the L reporter (p ≤ 0.05, T-test). It is visible that the coefficients for these significant angles are negative. Therefore, structural flexibility reduces the values of the linkage errors. The top two regression coefficients for the angles are \({\phi }_{1}\) and \({\phi }_{2}\) for the M reporter, and \({\phi }_{2}\) and \({\phi }_{3}\) for the L reporter. These findings are consistent with those in ref. 20. Among the lengths of the antibody fragments, interestingly, the coefficients of both L3 for the M reporter and L5 for the L reporter are negative. Therefore, molecular sizes and geometric angles are key factors influencing the distribution of linkage errors. Additional contributing factors are thoroughly examined in the dedicated section titled “Factors influencing the geometric model” within the Supplementary Note 5.

Discussion

The linkage error significantly limits the resolution achievable in SMLM imaging when using immunofluorescence techniques. Notably, the magnitude of linkage errors varies across experimental trials, even for identical protein targets and labeling strategies. The application of Monte Carlo simulations in conjunction with STORM imaging to indirectly infer linkage error was pioneered by Früh et al.20. Their work not only introduced this approach but also established a robust theoretical framework by elucidating the non-normality of these errors. Due to the resolution limitations of STORM (~10 nm), their study required intricate mathematical derivations to address the inherent uncertainty in distinguishing between linkage errors and localization inaccuracies. In this study, we enhance the existing methodology by optimizing the parameters and extending the model within the Monte Carlo simulation framework, based on the program code of Früh et al. This refinement enables a more comprehensive analysis of linkage errors, improving the robustness and accuracy of the simulation outcomes. By leveraging the precision of individual localizations of MINFLUX, we directly measure linkage errors at the single-molecule level. This approach greatly eliminates the ambiguities inherent in traditional super-resolution techniques, providing unprecedented precision in the analysis of molecular connections.

Using MINFLUX imaging, we have now been able to investigate and characterize the probability distributions of linkage errors across three distinct antibody labeling approaches. These approaches include direct immunofluorescence labeling with a nanobody (S reporter), direct immunofluorescence labeling using a primary “Y” antibody (M reporter), and indirect immunofluorescence labeling where a secondary antibody binds to the primary Y antibody (L reporter). To experimentally measure linkage errors, we selected microtubules—a well-defined structural model—as the labeling target. The linkage error was calculated as the difference between the radius of microtubules observed in MINFLUX images and their known actual radius (~12.5 nm). Then, we obtained the probability distribution of ~1000 values of linkage errors for each labeling strategy, and accordingly, performed statistical analysis on these three distributions.

Our experimental results reveal that the distributions of linkage errors for the three labeling strategies deviate markedly from a normal distribution, i.e., they are left-skewed and exhibit a flatter shape. Additionally, the distributions vary distinctly between labeling strategies. Specifically, the M reporter shows the most pronounced left skew, while the S reporter’s distribution is the flattest. Interestingly, the L reporter’s distribution displays a bimodal pattern with two distinct peaks. Although these distributions differ in shape, they are not entirely separated, i.e., an overlapped range of linkage errors was observed between strategies. This study provides new insights into the variability and characteristics of linkage errors for each labeling strategy.

These findings offer guidance for the statistical analysis of linkage error values. The nonnormal distribution of linkage errors for each labeling strategy indicates that mean and standard deviation alone are insufficient to describe the data distribution accurately. Instead, additional parameters, such as the minimum, first quartile, peak, median, and maximum values, as well as the skewness and kurtosis, are necessary to more fully characterize each distribution. Moreover, the differences between the distributions, along with their areas of overlap, make it inadequate to simply compare the mean linkage errors among reporters to assess which reporter has a larger or smaller linkage error. A more comprehensive analysis is essential to capture the distinct distributions and relationships of linkage errors across different labeling strategies.

Our experimental findings were supported by the simulations, which showed a left-skewed and a flatter distribution for each labeling strategy, and the L reporter’s distribution displays a bimodal pattern with two distinct peaks. Further regression analysis revealed that both the molecular size of the reporter and the geometric angles of the reporter influence the probability distribution of linkage errors. The simulation results contribute to understanding the mechanism underlying the linkage errors.

The distribution of linkage error values revealed in this article can be used to help map out the molecules of interest in SMLM. Because of the existence of labeling uncertainty, the ___location of reporter fluorophores can be corrected by the values of the linkage errors calculated in this article by two methods. In the first method, the coordinates of the target locations can be adjusted by subtracting the mean value of the linkage errors. This method may lead to certain deviations from the actual structure because the values of the linkage errors do not follow the normal distribution, and the standard deviation is not small. In the second method, the coordinates of the locations can be corrected by the distribution of the linkage errors. This method may avoid this certain deviation, provided that the order of the locations can match the order of the linkage-error values. Therefore, our findings may contribute to improving the accuracy of biological structure imaging.

Methods

Materials

For cell culture and sample preparation, MEM culture medium, fetal bovine serum, and non-essential amino acids were obtained from Gibco, sodium borohydride was purchased from Sigma, and paraformaldehyde and glutaraldehyde were obtained from Electron Microscopy Sciences.

For immunofluorescence labeling, the following primary antibodies were used: Rabbit anti-α-tubulin antibody labeled with Alexa Fluor 647 (Abcam, ab190573) and biotin-conjugated mouse anti-α-tubulin antibody (Abcam, ab74696). The secondary antibody and its source were: goat anti-mouse IgG secondary antibody labeled with Alexa Fluor 647 (Thermo Fisher, A32728). The nanobody and its source were: the anti-ALFA nanobody conjugated to Alexa Fluor 647 (NanoTag, N1502).

For imaging, we used a single molecule imaging buffer (Standard Imaging Biotechnology Ltd), which enhances light scintillation, mitigates damage to target proteins and fluorescent dyes caused by reactive oxygen species generated during fluorescence imaging, and extends imaging time.

Cell culture

The BSC-1 cells (ATCC, CCL-26, BeNa Culture Collection, China) were cultured in MEM (11095-080, Gibco) medium at 37 °C and 5% CO2, supplemented with 10% Fetal Bovine Serum (FBS, 10099-141, Gibco) and 1% Non-Essential Amino Acids (NEAA, 11140-050, Gibco) of MEM (100X). Cells were inoculated in 12-well plates placed with 14 mm L-PLL coated cover glass and cultured for 48 h before fixation.

Sample preparation

For fixation, the cells were wetted once with PBS and extracted with a preheated extraction solution (0.1 M PIPES, 1 mM EGTA, 1 mM MgCl2, 0.2% Triton X-100) for 60 s at room temperature. The extract was gently aspirated with the tip of a gun and added with 4% paraformaldehyde (157-8, Electron Microscopy Sciences) and 0.1% glutaraldehyde (16220, Electron Microscopy Sciences), both preheated to 37 °C, and fixed at room temperature for 10 min. The cells were rinsed three times with PBS, followed by an additional brief wash in PBS and immersion in a 10 mM sodium borohydride (71320, Sigma) solution in PBS for 5 min, and the samples were rinsed with PBS until there were no air bubbles. After the washing was completed, the cells were incubated with the sealing solution (50% BSA (10% BSA in PBS), 5% donkey serum, and 10% Triton X-100 (5% Triton X-100 in PBS) in PBS) for 1 h at room temperature and rinsed for 30 min. Staining primary antibody: The cells were incubated overnight at 4 °C with the primary antibody solution (dilution 1:200), diluted in cell containment buffer (1% BSA in PBS), and the samples were washed six times with PBS (1X) for 5 min each time. Staining second antibody: the cells were incubated with the secondary antibody solution (dilution concentration 1:1000), diluted with cell-confinement buffer (1% BSA in PBS), for 45 min at room temperature, and the samples were washed with PBS (1X) six times for 5 min each time. The gold spheres were incubated for 20 min and then washed twice with PBS to remove away any floating gold spheres, minimizing their effect on subsequent imaging. The biological samples were stored at 4 °C prior to imaging.

STED microscope imaging

In this study, STED image acquisition was performed using an Abberior STED microscope (Abberior Instruments), where the prepared biological samples were placed on the microscope. The 488 nm confocal scanning of the microscope was utilized to identify the cells and find the focus. We selected a suitable area in the confocal scan image and adjusted the laser intensity and pinhole size in the software. All STED imaging was performed using a 0.78 AU pinhole. In our experiments, the pixel size was of 10 nm × 10 nm, and the pixel dwell time was 10 µs. For super-resolution images with the size of 300 pixels × 300 pixels, the acquisition time was 0.9 s. Super-resolution optical imaging was performed using a power of 640-nm excitation laser beam of 0.06 mW and a power of 775-nm doughnut-shaped inhibition laser beam up to 0.24 mW.

MINFLUX microscope imaging

MINFLUX microscopy is a novel microscopy technique based on the localization of individual fluorescent molecules. Unlike camera-based Single Molecule Localization Microscopy (SMLM,) techniques such as PALM/STORM, MINFLUX drives a beam that scans around the preset position of the fluorophore, localizing it based on the number of photons detected. Specifically, a laser with zero central intensity is used as excitation, forming a horizontal donut shape in 2D MINFLUX and a spatially hollow sphere in 3D. Based on the predicted zero-intensity position, the known shape of the excitation beam, and the number of photons emitted by the fluorescent molecule at different positions of the excitation beam, the fluorescent molecule can be precisely localized after several iterations. During each iteration, the donut-shaped excitation beam is centered on the most recently obtained fluorophore localization, and the resolution can be further improved by gradually shrinking the diameter of the donut’s movement. Throughout the iterative process, the position information of the fluorescent molecules detected in each detection is accumulated into the amount of information of the next detection.

Prepared samples were fixed in a sample holder, and data were collected using an Abberior MINFLUX microscope (Abberior Instruments) equipped with a 1.4 NA, 100\(\times\) oil objective and an avalanche photodiode. Cells were identified and focused using 488 nm confocal scanning with Imspector software (v.16.3.15636-m2205-win64-MINFLUX, Abberior Instruments). All data collection was carried out using the default pinhole diameter of 0.78 AU. Data acquisition for the MINFLUX microscope was conducted according to a predefined set of parameters detailed in a text file (Imaging_2D.json). The critical parameters for the 2D iteration sequence are as shown in Table 3. For MINFLUX imaging, the samples were stabilized by incubating them with 10% gold nanorods (A12-40-980-CTAB-DIH-1-25, NanoStone Inc) in PBS for 15 min, followed by multiple PBS washes. The stabilization system of the microscope was activated before starting MINFLUX measurements, with stabilization accuracy typically below 1 nm. The region of interest was selected in the confocal scan image, and the laser power and pinhole size were adjusted in the software. Finally, start the MINFLUX measurement on the region of interest. Once the cell was in focus, the microscope’s stabilization system was activated to ensure stabilization accuracy remained below 1 nm. The region of interest was specified in the confocal scan image, and the excitation wavelength was set to 642 nm with an intensity of 3%. The recording channel is tuned to 650–685 nm. Subsequently, the appropriate sequence was selected for data acquisition.

Table 3 Key parameters of the MINFLUX 2D iteration sequence
Full size table

Data analysis

Data export

The experimental data were acquired on an Abberior MINFLUX/STED microscope (Abberior Instruments) using Imspector Software (Abberior Instruments). Each MINFLUX/STED data were exported using Imspector Software (Abberior Instruments). The exported file contains a set of recorded parameters for all valid localization and also includes invalid localization attempts that were discarded.

Analysis of microtubule transversal profiles

In order to analyze and count the linkage errors caused by antibody labeling of different molecular sizes, a software was chosen that detects fiber-like structures and can automatically determine the transversal profile along these structures in reconstructed MINFLUX and STED images18. In detail, the MINFLUX and STED images were first convolved with a Gaussian blur to compensate for noise discontinuities or holes. The thresholding algorithm35 converts the grayscale image into a binary-valued image. Using Lees algorithm36, the expanded lines were reduced to one pixel width. Then we found the coordinates of all pixels that were still connected to the line, and then we calculated their continuity. Discontinuous points were used as breakpoints, and all coordinates were connected to a new line. Lines that were smaller than the minimum required length were discarded. Note that the shape and gradient of the line depend on the smoothing parameter. Perpendicular to the derivative, a line profile was extracted from the original image at each coordinate point. The averaged profiles for each spline were fitted with the following functions (Eqs. (1)–(5)):

$${{\mathrm{Gaussian}}}:y=h{e}^{\frac{{-(x-c)}^{2}}{{2w}^{2}}}+b,$$
(1)

where h is the intensity, c is the center, b is the offset, and w is the variance of the distribution. The best choice for individual profiles is as follows:

$${{\mathrm{Bi}}}-{{\mathrm{Gaussian}}}:y={h}_{1}{e}^{\frac{{-(x-{c}_{1})}^{2}}{{2w}_{1}^{2}}}+{h}_{2}{e}^{\frac{{-(x-{c}_{2})}^{2}}{{2w}_{2}^{2}}}+b$$
(2)

The best choice for profiles containing a dip is as follows:

$${{\mathrm{Tri}}}-{{\mathrm{Gaussian}}}:={h}_{1}{e}^{\frac{{-(x-{c}_{1})}^{2}}{{2w}_{1}^{2}}}+{h}_{2}{e}^{\frac{{-(x-{c}_{2})}^{2}}{{2w}_{2}^{2}}}+{h}_{3}{e}^{\frac{{-(x-{c}_{3})}^{2}}{{2w}_{3}^{2}}}+b,$$
(3)

where y describes the theoretical intensity profile of microtubules.

The best choice for profiles exhibiting a dip and high background signal is as follows:

$${{\mathrm{Cylinder}}}:y=\left\{\begin{array}{cc}h\left(\sqrt{{r}_{2}^{2}-{(x-c)}^{2}}-\sqrt{{r}_{1}^{2}-{(x-c)}^{2}}\right), & {{\mathrm{if}}}{||x||} < {r}_{1}\hfill \\ h\left(\sqrt{{r}_{2}^{2}-{\left(x-c\right)}^{2}}\right),\hfill & {{\mathrm{if}}}{||x||}\ge {r}_{1},{||x||} < {r}_{2}\\ 0,\hfill & {{\mathrm{else}}}\hfill \end{array}\right.,$$
(4)

where \({r}_{1}\) and \({r}_{2}\) denote the inner and outer cylinder radius. Due to the nonlinearity of the cylinder function, the quality of the fit depends strongly on the initial estimates of the parameters.

Multi-Cylinder is defined as:

$$y = \,{{\mathrm{cyl}}}\left({i}_{1},c,25{e}_{x}/2-2a,25{e}_{x}/2-a\right)+{{\mathrm{cyl}}}\left({i}_{2},c,42.5{e}_{x}/2,42.5{e}_{x}/2+a\right)\\ +{{\mathrm{cyl}}}\left({i}_{3},c,25{e}_{x}/2+a,25{e}_{x}/2+2a\right)+b,$$
(5)

which includes the theoretical dimensions of microtubules, leaving fewer degrees of freedom. Note that the fitted intensity (h) provides a good estimate of the relative labeling density.

Analysis of MINFLUX localization precision

An emitting molecule is usually localized directly under the microscope several times in succession by repeating the last two MINFLUX iterations. These successive localizations are assigned to the same event via the same trace ID (exported parameter TID). To estimate the localization precision of a measurement, we calculated the standard deviation for each molecule (with at least ten localizations having the same exported parameter TID) as \({\sigma }_{r}=\sqrt{\left({\sigma }_{x}^{2}+{\sigma }_{y}^{2}\right)/2}\) with the standard deviations of the x and y coordinates as determined by the microscope. The median \({\sigma }_{r}\) represents the stated localization precision. The TIDs from the MINFLUX data were imported into a custom analysis script written in MATLAB (R2017a) to calculate the positioning accuracy in an automated manner.

Monte Carlo simulations

Compared to the conventional empirical approach, the Monte Carlo simulation method may more closely mimics the genuine scenario and accordingly, it can model the actual physical process. Therefore, in this study, the geometric conformation of the microtubule-IgG and antibody complex was simulated using a Monte Carlo approach. This simulation extends the framework of the program code developed by Früh et al.20 and was implemented in MATLAB 2019a (Mathworks). This approach further demonstrates the high accuracy of the spatial localization of MINFLUX and the distribution of the linkage errors. Supplementary Tables 1–3 display the length characteristics of the Fab and Fc fragments of three distinct antibodies tagged with fluorescent Alexa Fluor 647 (AF647). The evaluation of the cryo-electron microscopy structure of PDB 5SYF was used to establish the microtubule epitope sites20. Direct immunofluorescence labeling of anti-ALFA nanobody and rabbit anti-tubulin antibody, and indirect immunofluorescence labeling of mouse anti-tubulin antibody plus a secondary mouse antibody, were randomly sprinkled as Gaussian distributions on fragments of the three different antibodies, with 30,000 molecules mimicked for each labeling. Within the angular range, the geometric and rotational angles for both direct and indirect immunofluorescent labeling were consistently chosen (refer to Supplementary Tables 1–3). Furthermore, the schematic diagram (see Fig. 1) refers to the ___location of the fluorescent marker dispersion, the geometry of the three separate antibodies, and the differences between the three antibodies labeled on the microtubules. The diameter value d in a Monte Carlo simulation was characterized by figuring out the Euclidean distance between the fluorescent marker’s ___location and the microtubule center. The diameter value d is then transformed into linkage error using the formula\({e}_{i}=({d}_{i}-{D}_{i})/2\). The skewness and kurtosis values were quite comparable, and the simulation findings did not differ substantially from the MINFLUX actual results by t-test (\(p > > 0.05\)) (Fig. 3).

Compared to previous methods20, the current model has the following significant advancements:

  1. (1)

    New model additions

    The simulations have been extended to incorporate geometric parametric modeling of nanobodies, an aspect not considered in previous models.

  2. (2)

    Expanded geometric (φ) and rotational (θ) angles

  1. (a)

    Enhanced fab segment dynamics in primary antibodies

    Earlier models constrained the geometric angle (φ₁) and rotation angle (θ₁) of the Fab segment to ±22.5° under the assumption of a uniform binding direction. The geometric angle (φ₁) and rotation angle (θ₁) have been expanded to ±30° (Table S2), integrating cryo-electron microscopy34 and single-molecule FRET data4. This refinement more accurately captures the flexibility of Fab segments during antigen binding (Fig. 1c2), enabling more dynamic oscillations and improving the simulation of antibody-epitope interactions.

  2. (b)

    Independent parameterization of secondary antibody degrees of freedom

Earlier models did not independently define the orientation of the secondary antibody’s Fab segment, approximating it based on the hinge movement of the primary antibody’s Fc region (φ₂ = 15°–127.6°). Independent dynamic parameters have been established for secondary antibodies, specifying the rotation angle (θ₃) and geometric angle (φ₃) in indirect labeling (L reporter). The rotation angle (θ₃) varies freely from 0° to 360°, while φ₃ is constrained between 67.5° and 112.5° (Table S3), reflecting the omnidirectional binding probability of the secondary antibody’s Fab segment to the Fc epitope.

  1. (1)

    Increased degrees of freedom and improved multilayer structure

  1. (a)

    Refinement of the primary antibody Fc segment

    Previous models primarily accounted for the hinge length (L₂ = 3.217–6.072 nm) and geometric angle (φ₂ = 15°–127.6°) of the Fc segment, without incorporating conformational variations. The dynamic hinge length parameter (L₂ = 2.520–7.239 nm, Table S3) has been refined to include both stretching and compression phases of the hinge region.

  2. (b)

    Parameterization of secondary antibody complexes at multiple levels

Earlier models included five degrees of freedom (L₁, φ₁, θ₁ for the primary antibody and L₂, φ₂ for the Fc segment). This has been expanded to 15 degrees of freedom (Table S3) for indirect labeling (L reporter), incorporating parameters for the Fab segment (L₃, φ₃, θ₃) and Fc segment (L₄, L₅, φ₄, φ₅, θ₄, θ₅).

  1. (1)

    Experimental calibration and non-uniform angular distributions

    Earlier approaches assumed a uniform angular distribution, disregarding conformational preferences under physiological conditions. A non-uniform distribution for key angles (φ₄) has been introduced based on single-molecule FRET4 and cryo-electron microscopy34 (Table S3, footnote e). Instead of uniform sampling, φ₄ follows a Gaussian distribution (110° ± 30°), more accurately reflecting antibody conformation under physiological conditions.

  2. (2)

    Multifactor regression analysis

Unlike previous models, the current approach integrates a linear regression model (Table 2) to quantify the independent effects of molecular size (Fab/Fc length), geometric angles (φ), and rotational angles (θ) on linkage error distributions.

Multifactor linear regression: in this study, we used multifactor linear regression analysis to assess the effects of multiple structural parameters of antibodies on linkage errors to identify and quantify the role of these parameters in linkage errors. This method provides a systematic and quantitative basis for analyzing the factors affecting linkage error, which helps to optimize these key parameters in subsequent experiments. First, a dataset including 15 structural parameters such as Fab segment length, geometric angles and torsion angles, as well as linkage error values was extracted and organized from the simulated data. Then, in MATLAB 2019a (Mathworks), a data table was constructed for subsequent regression analysis. Using the fitlm function in MATLAB, we built a multifactor linear regression model with linkage errors as the dependent variable and the structural parameters of the three different antibodies as independent variables. The generalized mathematical form of the model is given below:

$${{\mathrm{Linkage}}}\,{{\mathrm{Errors}}}={\beta }_{0}+{\beta }_{1}\times {L}_{1}+{\beta }_{2}\times {L}_{2}+\cdots +{\beta }_{15}\times {\theta }_{5},$$
(6)

where \({\beta }_{0}\) is the intercept, and\({\beta }_{1}\) through \({\beta }_{15}\) are the regression coefficients for the respective variables. Each coefficient represents the average effect of that independent variable on the linkage errors, controlled for by the other variables.

The multifactor linear regression analysis generated estimated coefficients, p values, etc., for each parameter. These statistics were used to determine whether the effect of each variable on linkage errors was significant. If the p value is less than 0.05, the variable is considered to have a statistically significant effect on linkage errors.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.