# Systematic Assessment of Burst Impurity in Confocal-Based Single-Molecule Fluorescence Detection Using Brownian Motion Simulations

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Molecular Position Dispersion

^{2}/s, using the numerical PSF model. The results for the same simulations using the Gaussian PSF model are summarized in Figure S1.

^{2}/s), as well as for the simulation results in lower concentrations (31 and 15.5 pM) at a constant diffusion coefficient value (90 μm

^{2}/s; see Figure 3), and in lower diffusion coefficient values (22.5 and 5.625 μm

^{2}/s) at a constant concentration (62 pM; see Figure 4), using the numerical PSF model. The results using the Gaussian PSF model are also shown in Figures S3 and S4.

#### 2.2. Pure and Impure Single-Molecule Bursts

^{2}/s; see Figure 5), and as a function of different diffusion coefficients, at a constant concentration (62 pM; see Figure S6), using the numerical PSF model. These results using the Gaussian PSF model are shown in Figures S7 and S8.

^{2}/s), and as a function of different diffusion coefficients, at a constant concentration (62 pM), both when using the numerical and the Gaussian PSF models.

^{2}/s, using the numerical PSF model).

#### 2.3. Alternative Quantification of Burst Impurity using Burst Photon Timestamp Autocorrelation

_{bursts}(N > 1) for photons in bursts, using Equations 3 and 4 (Materials and Methods), respectively. The best fit values of <N> and of P (N > 1) are reported in Table S2.

^{2}/s. Figure S14 also shows the best fitted results to the model of fluorescence autocorrelation of molecules freely diffusing in 3D in a confocal-based setup. As a first step, and as an additional validation, simulations using different diffusion coefficients, and a constant concentration, at a constant volume, should yield the same value of <N>, within error ranges. Figure S15 shows that indeed the values of <N> retrieved from fitting, were the same in within the error ranges for different conditions in simulations having different diffusion coefficients, but the same concentration, per given burst analysis parameter values and per PSF model. <N> was consistently lower when using the numerical PSF model, compared to when using the Gaussian PSF model. That is expected, because the width of the numerical PSF model is smaller than the Gaussian PSF model, and hence less molecules cross it, in average (see Figure 1).

#### 2.4. Quantification of Molecule Diffusion Times from Burst Photon Timestamp Autocorrelation and from Burst Width Analysis

#### 2.5. Improving the Accuracy of Mean FRET Efficiency Estimation

#### 2.5.1. Simulation of smFRET Measurement of a Mixture of Two Species with Two FRET Efficiencies

^{2}/s. We repeated this simulation four times, taking into account either numerical or Gaussian PSF models, and with simulations times of either 60 or 180 s. Then we allocated donor and acceptor photon timestamps for ten out of fifteen molecules, according to E = 0.75, and the leftover five molecules, according to E = 0.50. Then, we analyzed the results for bursts using a sliding window of m = 10 consecutive photons, with different values of the instantaneous photon rate threshold, F = 6, 11, and 21. We collected the FRET efficiencies of all bursts into FRET histograms and fitted them with a sum-of-two-Gaussians’ function. The results of this procedure are shown in Figure 8, for the 60 s simulation using the numerical PSF model, and the best fit values of all of these simulations are reported in Table S3. Additionally, the fitting results are shown, after fixing the populations’ fraction to a fixed value of f = 0.6666, which is the value that was simulated (Figure S20).

#### 2.5.2. smFRET Experimental Results of a Mixture of Two Species with Two FRET Efficiencies

## 3. Discussion

_{D}, and the mean number of molecules in the EDV, at any given moment, <N>. However, while FCS is useful when measuring higher concentrations (nM and above), in which the noisy signal has a clear mean, and the information is found in the temporal fluctuations about the mean signal, in concentrations used in confocal-based SMFD (<100 pM) there is not one mean signal, but rather two characteristic Poisson processes, with two different mean rates (the BG process, when no molecule crosses the EDV, and the signal process, when a molecule crosses it). Accordingly, analysis of fluorescence autocorrelation curves of photon timestamps only from bursts was a more promising route, however, their analysis required assuming the EDV has a constant Gaussian shape. The EDVs, following burst analysis, as are shown from molecular position histograms in this work, change shape and width. Therefore, estimates retrieved from best fit values of <N>, such as P (N > 1), the probability of having more than a single molecule in the EDV (see Equations (3) and (4)), were biased and less credible as estimates of burst impurity derived directly from knowing the molecular identity and position of each photon timestamp. Indeed, the trends we reported as a function of the actual burst impurities were sometimes different than the ones using P (N > 1). In addition, the best fit values of the diffusion time had large error ranges, rendering them less useful when trying to quantify relations to the diffusion time through the EDV. Burst widths, however, served as an accurate quantity for that purpose.

_{D}and <N>, and the mean burst width) are the same within the error ranges, for all burst analysis parameter values tested in this work. It means that the values acquired in the 60 s simulations are representative, and that longer simulation times may increase the accuracy of the retrieved values. In that context, it is worth mentioning that further increasing the simulation time was impossible due to the large file sizes produced for the molecules’ 3D trajectories (~50 Gb per 60 s, using the simulation conditions described in Materials and Methods).

## 4. Materials and Methods

- Numerically-calculated PSF. It was derived from a model of a realistic PSF of a typical 60x water immersion objective with a numerical aperture of 1.2, with a sample mounted on top a 150 μm coverglass and with sample excitation at a wavelength of 532 nm (see Figure 1). We modeled the excitation PSF using a vectorial electromagnetic simulation, PSFLab [15]. This model includes effects of refractive index mismatch as well as mismatch between objective lens correction and coverglass thickness. The model we used is based on the excitation PSF taken to the power of two, to calculate the multiplication of the excitation PSF with the detection PSF [25,26], that is assumed to be similar to the excitation PSF. In general, the multiplication of the excitation and detection PSFs should be convolved with the pinhole profile [25,26]. The convolution with the pinhole function was not considered as the simulations mimic measurements with overfilling of the objective lens back aperture, rather than underfilling. A convolution with the pinhole profile would be required in cases of underfilling the back aperture of the objective lens.
- A Gaussian-shaped PSF with standard deviations of 180 nm in the x and y directions and 880 nm in the z direction.

^{2}/s), with ten of them having a mean FRET efficiency of 0.75, and five with a mean FRET efficiency of 0.5. The two groups of molecules were simulated in the same rectangular box and had the same diffusion coefficient and the same molecular brightness (photon rate of 200 KHz at the maximum intensity of the PSF). The simulated BG rates of the donor and acceptor fluorescence detection channels were 1.5 and 0.8 KHz, respectively. Additionally, assessing the accuracy of the retrieves mean FRET efficiency values from smFRET of a mixture of two subpopulations was also carried out on experimental data, acquired in previous works [21]. These were microsecond alternating laser excitation (μsALEX) smFRET measurements of two FRET constructs with Cy3B and ATTO 647N as the donor and acceptor dyes, respectively, labeling a pair of bases in a dsDNA with a sequence of the lacCONS promoter. The FRET histograms of bursts from these measurements were achieved on all bursts that had at least 20 acceptor photons, after directly exciting it with a 635 nm laser. The photons arising after donor excitation were analyzed as the simulation results were analyzed.

_{D})]

^{2}[1+(1/κ

^{2})(τ/τ

_{D})]}

^{−1}

_{D}is the mean diffusion time through the EDV, and κ is the z-to-x ratio of the EDV (taken with a value of 7, in the fits of the autocorrelation functions to this model). The best fit <N> was also used for the calculation of the probability for having more than one molecule crossing the EDV, according to Equation 3 in the case of autocorrelation of all photons, and according to Equation 4 in the case of autocorrelation of only the photons inside bursts.

_{Burst}(N>1) = [1 – Pois(0) – Pois(1)]/[1 – Pois(0)]

- The fraction (percentage) of photonic impurity—all photons of bursts not originating from the main burst molecules, divided by all the burst photons in the simulation.
- The fraction of impure bursts—the fraction of bursts that included photons from more than a single molecule.
- The mean burst impurity—the mean over all bursts of the fraction of burst photons not originating from the main molecule in the burst.
- The molecular position dispersion in x, y, and z—standard deviation of all molecular positions in each burst photon, calculated for each dimension.
- The mean burst width—the mean of all burst widths/durations, where the burst duration is calculated as the time interval between the first and last photon timestamps in a burst.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Models of the point-spread function (PSF). The PSF was modeled either as Gaussian-shaped or numerically calculated by using the model of an excitation PSF calculated with PSFLab [15] for a typical 60x water immersion objective with a numerical aperture of 1.2, with a sample mounted on top a 150 μm coverglass and with sample excitation at a wavelength of 532 nm, where the PSF that was used was the calculated excitation PSF, taken to the power of two. For more information, see Materials and Methods.

**Figure 2.**The positions of diffusing molecules when they emitted photons that were detected and selected by the burst analysis, either with minimal burst analysis parameter values (m = 5 and F = 6; left panels) or with stringent burst analysis parameter values (m = 10, F = 6 and burst size threshold, sz = 40; right panels). In the top, central, and bottom panels we show the 2D projections at the yz, xz, and xy planes, respectively. Each dot in the scatter plots is an emitted photon. These results are for the simulation of molecules in a concentration of 62 pM, where the diffusion coefficient of the molecules was 90 μm

^{2}/s. The colors of the points correspond to the burst number out of the overall number of bursts. In each panel, the 1D projections are also shown as histograms. The black, brown, and yellow contour lines align the position of the numerical PSF model (see shapes of PSF models in Figure 1).

**Figure 3.**The molecular position dispersion as a function of burst search criteria and experimental conditions (different concentrations). Shown are the standard deviation of molecular positions in the z (left) and x (right) coordinates (the values in the y coordinate are the same as the ones in the x coordinate, in within the error ranges), when they emitted photons that were detected and selected by the burst analysis. The error values were calculated as the uncertainty of the standard deviation. All values are reported in Table S2. The assessment of the molecular position dispersion here is shown as a function of different concentrations for molecules diffusing with a constant diffusion coefficient of 90 μm

^{2}/s, and in simulations using the numerical PSF model.

**Figure 4.**The molecular position dispersion as a function of burst search criteria and experimental conditions (different diffusion coefficients). Shown are the standard deviation of molecular positions in the z (left) and x (right) coordinates (the values in the y coordinate are the same as the ones in the x coordinate, in within the error ranges), when they emitted photons that were detected and selected by the burst analysis. The error values were calculated as the uncertainty of the standard deviation. All values are reported in Table S2. The assessment of the molecular position dispersion here is shown as a function of molecules diffusing with different diffusion coefficients, at a constant concentration (62 pM), and in simulations using the numerical PSF model.

**Figure 5.**The occurrence and level of impure bursts as a function of burst search criteria and concentrations. Different burst analysis parameter values for different concentrations of molecules. The relative occurrence of impure bursts (left) was calculated as the fraction of bursts with an impurity level larger than 0 (error ranges calculated as the 95% confidence intervals), as the fraction of non-single-molecule bursts, and hence as the fraction of impure bursts. The level of impurity (right) was calculated as either the mean of all burst impurity levels (black; error ranges calculated as the standard error) or as the fraction of impure photons from all bursts relative to all burst photons (red; no error ranges, as the calculation was performed over all photons). The assessment is shown as a function of different concentrations for molecules diffusing with a constant diffusion coefficient of 90 μm

^{2}/s, and in simulations using the numerical PSF model.

**Figure 6.**The correlation of burst impurity with molecular position dispersion. Different burst analysis parameter values for different concentrations of molecules. The mean burst impurity levels (error ranges calculated as the standard error) were compared against the molecular position dispersion in the z coordinate (error ranges calculated as the uncertainty of the standard deviation), as a function of different burst analysis parameter values (from left to right: varying m values, varying F values, varying burst size threshold values, and varying burst width threshold values), for different simulation conditions (from top to bottom: Different concentrations at a constant diffusion coefficient value of 90 μm

^{2}/s in simulations using the Gaussian PSF model, different concentrations at a constant diffusion coefficient value of 90 μm

^{2}/s in simulations using the numerical PSF model, different diffusion coefficients at a constant concentration of 62 pM in simulations using the Gaussian PSF model, and, different diffusion coefficients at a constant concentration of 62 pM in simulations using the numerical PSF model).

**Figure 7.**The molecular positions of pure and impure bursts photons, as a function of varying instantaneous photon rate threshold values, F (numerical PSF model)—shape and amplitude. We the histograms of molecular positions in the z coordinate of impure photons (red), burst photons of impure bursts (yellow), of pure bursts (green), and of all bursts (black), both un-normalized (left) to assess the weight of burst impurity, and normalized (right) to assess the histogram shapes. These results refer to the simulations in concentration of 62 pM and diffusion coefficient of 90 μm

^{2}/s, using the numerical PSF model.

**Figure 8.**Simulations of single-molecule Förster resonance energy transfer (smFRET) experiments with two FRET efficiency subpopulations—increasing the value of the photon rate threshold, F, improves the accuracy of the retrieved mean FRET efficiency. From top to bottom, each panel shows the resulting FRET histogram (blue), the best fit sum-of-two-Gaussians (red), the best-fit mean FRET efficiencies (orange and cyan vertical lines; dimmer lines show the error ranges), and the simulation ground-truth mean FRET efficiency values (dashed red and green vertical lines). These results are for the 60 s simulation of molecules a concentration of 62 pM, where the diffusion coefficient of the molecules was 90 μm

^{2}/s, using the numerical PSF model, and the molecules were split to 10 with E = 0.75 and 5 with E = 0.5. The number of bursts in each histogram is also reported in each panel. The best fit values and the fitting error values are also reported in Table S3.

**Figure 9.**smFRET measurements of a mixture of two FRET constructs with different mean FRET efficiencies—increasing the value of the photon rate threshold, F, improves the accuracy of the retrieved mean FRET efficiency. From top to bottom, each panel shows the resulting FRET histogram (blue), the results of the best fit models (black), the best-fit mean FRET efficiencies (dashed red vertical lines; orange vertical lines show the error ranges), and the expected mean FRET efficiencies of each FRET sub-population (black vertical lines; calculated from the best fit results of fits of single FRET population to single Gaussian functions in the left and center panels). These results are for the measurements of FRET lacCONS promoter construct labeled with Cy3B at the nontemplate strand and ATTO 647N at the template strand at registers +4 and -8, respectively (left), +2 and -15, respectively (center) and a measurement of their mixture (right). The best fit values are also reported in Table S4.

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**MDPI and ACS Style**

Hagai, D.; Lerner, E. Systematic Assessment of Burst Impurity in Confocal-Based Single-Molecule Fluorescence Detection Using Brownian Motion Simulations. *Molecules* **2019**, *24*, 2557.
https://doi.org/10.3390/molecules24142557

**AMA Style**

Hagai D, Lerner E. Systematic Assessment of Burst Impurity in Confocal-Based Single-Molecule Fluorescence Detection Using Brownian Motion Simulations. *Molecules*. 2019; 24(14):2557.
https://doi.org/10.3390/molecules24142557

**Chicago/Turabian Style**

Hagai, Dolev, and Eitan Lerner. 2019. "Systematic Assessment of Burst Impurity in Confocal-Based Single-Molecule Fluorescence Detection Using Brownian Motion Simulations" *Molecules* 24, no. 14: 2557.
https://doi.org/10.3390/molecules24142557