Investigating the effects of temporal correlation on accurate molecular brightness determination

Abstract

Number and brightness, and photon counting histogram analyses, are powerful techniques for measuring stoichiometry in living cells. However, the accuracy of the results can be compromised by data temporal correlation, which increases with shorter acquisition times and slower molecular dynamics. Through simulations, we demonstrate that failing to account for data temporal correlation systematically underestimates the true molecular brightness. To overcome this limitation, we introduce the mixed assays to evaluate brightness analysis method, which decorrelates the data, allowing for accurate calculation of molecular brightness and achieving better precision with fewer data points.

Significance

This study provides insights into the effects of temporal data correlation on molecular brightness calculations, a key parameter in fluorescence imaging and molecular dynamics. By demonstrating how temporal correlations can distort brightness estimates and significantly increase data volume and computational demands, we underscore the necessity of data decorrelation strategies to enhance accuracy and efficiency. These findings have broad implications for quantitative imaging, where precise brightness measurements are critical for assessing molecular interactions, concentrations, and dynamics within complex biological systems. This work advances the precision of molecular brightness estimation and supports the development of optimized data processing protocols in fluorescence microscopy and related analytical fields.

Introduction

Molecular brightness, defined as the number of photons emitted per molecule per second, is a key parameter of fluorescent molecules that precisely determines the oligomeric state of molecular complexes. The oligomeric state is associated with various biological processes, including the life cycle of viral particles in living cells (1) and the mechanisms underlying neurodegenerative diseases (2). Subcellular localization and cellular function can influence a molecule’s oligomeric state (3,4,5,6). Monitoring these changes provides valuable insights into complex biological processes at the molecular level within live cells.

Over recent decades, various fluorescence-based methods have been developed to quantify molecular brightness, each leveraging different aspects of fluorescence signal behavior. Techniques that analyze fluorescence fluctuations have gained prominence for their quantitative biophysical applications.

Chen and co-workers first introduced the determination of molecular brightness through photon count distributions (PCH analysis) derived from fluorescence intensity measurements in an open observation volume (7,8). While single-point PCH analysis is widely used for its straightforward implementation, it requires large data sets (around 1,000,000 data points) and lacks a comprehensive view across the entire cell, necessitating extensive data collection for pixel-level imaging. The number and brightness method was developed to overcome these challenges, allowing for molecular brightness mapping by analyzing intensity distributions across image pixels (9,10). In raster scan acquisitions using number and brightness analysis, the intensity of each pixel in an image is uncorrelated with that of the subsequent image due to the time delay between successive frames. In camera-based imaging, pixel intensities are captured simultaneously within a single image, requiring careful adjustment of acquisition times. This often involves extending acquisition times to minimize temporal correlation between images. Lately, the mean segmented Q value was introduced to overcome effects in molecular brightness value calculation due to molecular photobleaching or insufficient overlap between cell size and observation that may affect fluorescence intensity fluctuations (11).

Although all these methods can extract molecular brightness values, they cannot track molecular dynamics. Spatiotemporal correlation spectroscopies have enabled dynamic studies of molecules. The pair correlation function, developed by Digman and co-workers, captures protein dynamics between cellular compartments and measures diffusion barriers (12,13,14,15,16), but does not provide information on molecular brightness. More recently, Hinde et al. introduced pCOMB, a technique that combines moment analysis with spatiotemporal correlation, allowing the study of molecular transport and brightness (17).

While previous studies typically provide recommendations for acquisition parameters, the effects of temporal correlation between data points on molecular brightness calculations have not been thoroughly examined. In this study, we use numerical simulations to systematically investigate the impact of temporal data correlation on brightness calculations. We employ moment analysis to evaluate how molecular brightness results change with varying numbers of correlated data points. Our findings demonstrate that temporal correlation distorts the calculated brightness unless the total number of data points significantly exceeds the correlated subset, underscoring the importance of using uncorrelated data for accurate brightness measurements. We also examine the influence of the diffusion coefficient on data correlation and molecular brightness accuracy.

We propose three strategies to accurately determine molecular brightness while minimizing the effects of temporal correlation. The first method involves selecting a large subset of uncorrelated data points, which, while effective, requires substantial data and computational resources. The second method uses shuffled data points, eliminating temporal correlation, which reduces data requirements and computational costs while maintaining accuracy in brightness values. Finally, by grouping data points with sufficient temporal spacing, temporal correlation within each group is effectively removed, while correlation between groups is preserved. This approach allows for precise molecular brightness measurements within each group while maintaining the temporal information necessary to capture dynamics through pair correlation molecular brightness analysis.

Materials and methods

Numerical simulations were conducted using the software SimFCS developed by the Laboratory of Fluorescence Dynamics (www.lfd.uci.edu). Data analysis was performed using Python code designed specifically for this study and available on GitHub. (https://github.com/EstradaLab-BiophotonicsGroup/MATE-BA.git).

Simulations

Two sets of simulations were conducted to evaluate the accuracy of the mixed assays to evaluate brightness analysis (MATE-BA) in determining molecular brightness across two scenarios: single-point and line scan acquisitions. In both cases, 3000 particles were set to freely diffuse on a 3D grid of 128 pixels using a pixel size of 50 nm/px and a 3D Gaussian-shaped observation volume with fixed lateral and axial widths W0 = 0.3 μm and Wz = 1.8 μm, respectively. The observation volume shape and sizes remained constant throughout all simulations.

To study the effect of temporal data correlation on molecular brightness determination in a single-point experiment, we simulated intensity data acquisition with an acquisition time of 1 μs. The molecular brightness $ϵ$ was set to 500,000 counts per molecule per second (cpms). The diffusion coefficient was then varied from D = 0.5 μm²/s to D = 400 μm²/s. A total of 128,000,000 data points were simulated for a diffusion coefficient of D = 0.5 μm²/s. For higher diffusion coefficients (D = 1–400 μm²/s), each simulation had a total of 12,800,000 data points. The molecular brightness value and acquisition time remain unchanged.

For molecular brightness calculations using line scans, we simulated 200,000 lines, each with 64 pixels and an acquisition time, $t_p$, of 1 μs per pixel. To emulate different oligomeric states, we conducted four individual simulations with varying molecular brightness: monomers (500,000 cpms), dimers (1,000,000 cpms), tetramers (2,000,000 cpms), and octamers (4,000,000 cpms). We did not account for diffusion coefficient changes due to oligomer size, using a fixed diffusion coefficient of D = 1 μm²/s for all simulations. To perform line scans temporal autocorrelation of molecular brightness values, only one oligomeric state was considered. A total of 4000 molecules with a molecular brightness $ϵ$ of 300,000 cpms was simulated. The acquisition time was set at 5 μs per pixel. The rest of the parameters remained unchanged.

Calculation of molecular brightness from n data points by moment analysis

Previous studies (9) have demonstrated that molecular brightness can be determined through the analysis of the first (mean intensity) and second (variance) moments of the intensity distribution. Considering a set of n data points, intensity mean and variance are defined as:

$$⟨I⟩=\frac{{\sum }_{i=1}^n{I}_i}{n}$$ (1)

$${\sigma }^2=\frac{{\sum }_{i=1}^n{\left(I_i-⟨I⟩\right)}^2}{n-1}$$ (2)

where $I_i$ is the i-th intensity value. The molecular brightness and the quantities from moment analysis are related by:

$$ϵ\equiv \frac{{\sigma }^2}{⟨I⟩}-1$$ (3)

Given a total amount of data points, N, selecting a subset of $n$ data point values allows for the generation of N/n data sets. Each data set calculates a molecular brightness value, resulting in N/n values of molecular brightness.

Molecular brightness values from consecutive time-correlated and consecutive time-uncorrelated data points

We calculated molecular brightness values using a custom Python routine, analyzing two types of data subsets: one consisting of n consecutive data points, and another with decorrelated sets of n shuffled intensity data points. For a diffusion coefficient of D = 0.5 μm²/s, n ranged from 100 to 100,000,000. For higher diffusion coefficients, n ranged from 100 to 12,000,000.

MATE-BA method

We present the MATE-BA method, which enables the calculation of molecular brightness values while avoiding temporal correlation between data points used for their determination. By temporally decorrelating data points, MATE-BA can accurately compute the molecular brightness value from a set of temporally consecutive intensity values. To achieve this, MATE-BA orderly mixed intensity values acquired in a point or line scan experiment by interspersing data at regular time intervals distances.

To test the MATE-BA method, we first computed molecular brightness values using intensity data points spaced by a time interval distance J (1 < J < Ptot/100), with each brightness value computed from n = 100 data points. We generated a molecular brightness distribution comprising j = Ptot/(J × n) elements for each value of J. As J increased, j decreased, eventually falling below 100 elements per distribution. We applied the following approach to ensure that each distribution contained at least 100 $ϵ$ values: a pivot point was randomly chosen between the first and J-th values. Starting from this pivot, 100 elements spaced by J were selected to compute molecular brightness. This process was repeated until all distributions contained 100 independent, nonduplicated $ϵ$ values.

To evaluate the ability of the MATE-BA method to compute molecular brightness values from line scans accurately, we analyzed simulated data scan lines. For each pixel in the line, we compute the molecular brightness using a set of n = 100 lines, interspersed to eliminate the correlation between them. To achieve this, we first define the interval ${\delta }_{line}$, between successive lines, to create the data package used for computing the molecular brightness value.

$${\delta }_{line}=\frac{K}{n}$$ (4)

where K is the total number of simulated (or acquired) lines. We then constructed each set of lines by selecting the first line of the package and its subsequent n lines, spaced by the defined interval ${\delta }_{line}$. The molecular brightness at each pixel is calculated as in Eq. 3.

The first line of the next package is selected immediately after the first line of the previous package. This process is repeated to create packages of n consecutive lines spaced by the defined ${\delta }_{line}$ and to compute molecular brightness values. To ensure that the lines within each package are uncorrelated, the spacing between them must exceed the number of temporally correlated intensity lines. Since MATE-BA requires that the package line is constructed using only a small number of interspersed lines to achieve the true $ϵ$ value, this condition is always satisfied.

At the end of the process, we obtain ${\delta }_{line}$ values of molecular brightness from K intensity lines, with each $ϵ$ value computed exclusively from uncorrelated intensity lines. Furthermore, because lines occupying the same position across different packages remain temporally correlated, the spatiotemporal correlation of molecular brightness can be analyzed after applying MATE-BA.

Similarly to the approach independently proposed by Kohler et al., which applies segmented consecutive data points to autocorrelation function calculations (18), MATE-BA extends its application to brightness analysis.

Results

Effects of temporal correlation on molecular brightness calculation

To investigate whether the obtained molecular brightness, $ϵ$, is affected by the temporal correlation between the data used for the calculation, we computed its value by considering sets of n consecutive data points as described in calculation of molecular brightness from n data points by moment analysis. By considering n = 100 (Fig. 1 A), 10,000 (Fig. 1 B), and 1,000,000 (Fig. 1 C) intensity values to calculate each entry in the distribution, we built molecular brightness distributions using both correlated data (consecutive intensity values, hatched bars) and decorrelated data (shuffled consecutive intensity values, solid bars).

Figure 1.

Molecular brightness distributions. Each entry of the molecular brightness distributions was computed using n consecutive correlated data points (hatched bars) and n consecutive uncorrelated data points (solid bars) with n = 100, 10,000, and 1,000,000 (A, B, and C, respectively). Solid vertical lines represent the median of each distribution, and dashed vertical lines represent the $ϵ$ value used during simulations. In (B) and (C), solid dark and dashed lines overlap. The number of entries in each distribution is determined by the ratio of the total number of simulated data points, N, to n.

First, we determined whether the distributions of recovered molecular brightness are distinguishable. A Mann-Whitney test with a significance level of 0.05% was performed on molecular brightness distributions containing the same number of data entries. Using n = 100 data points to compute each molecular brightness value, a p value <2.2e−16 (machine epsilon) was obtained, indicating statistical distinguishability between the distributions (Fig. 1 A), a result that was also observed when n = 10,000 data points were used (Fig. 1 B).

However, when molecular brightness distributions were constructed using n = 1,000,000 data points to compute each $ϵ$ value, a p value of 1.0 was obtained (Fig. 1 C), indicating no statistically significant differences between distributions derived from decorrelated (solid bars) and correlated (hatched bars) data.

To assess the potential impact of the number of entries, N/n, on the statistical test results, we repeated the analysis using data sets with varying entries (Fig. S1). The results showed no significant differences, indicating that, within the analyzed range, the number of entries in the distributions does not affect the outcome.

To evaluate the accuracy of the molecular brightness values obtained using n data points for each entry in the distribution, we compared the medians of the distributions with the simulated molecular brightness value ${ϵ}_{sim}$. As shown in Fig. 1, the molecular brightness calculated using n = 100 consecutive correlated intensity points exhibited a larger difference between the distribution median and the simulated value ${ϵ}_{sim}$ in comparison with calculations using n = 1,000,000 data points. As n increases, the distribution median approaches the simulated molecular brightness value. In contrast, when brightness is calculated from uncorrelated data, the difference between the distribution median and the simulated true molecular brightness value remains consistent, regardless of the number of data points used. In conclusion, as expected, using fewer uncorrelated data points in the data set for molecular brightness calculations yields a computed value that more closely approximates the molecular brightness used in the simulations.

To further assess the impact of the number of data points, n, used to determine the molecular brightness value $ϵ$, we calculate the mean intensity, the variance, and their ratio, varying n from 10² to 10⁸. Fig. 2 shows the medians of: the mean intensity (A), variance (B), and molecular brightness (C) distributions, computed using sets of n consecutive data points given a total of N/n entries. Solid lines represent the median of each distribution, while the variability is depicted by the median absolute deviation (MAD).

Figure 2.

Median values of: mean intensity (A), variance (B), and molecular brightness (C) obtained distributions. Each entry was computed using sets of consecutive data points (100 < $n$ < 1 × 10⁸). The median value of each distribution (solid lines), and corresponding MAD in hatched (correlated data) and solid fill (uncorrelated data) are shown in cpms. Arrows indicate the number of data points $n$ required for the distribution’s median to differ by 5% from the simulated $ϵ$ value: $n\approx$ 2200 for uncorrelated data and $n\approx$ 1,000,000 for correlated data.

Our results show that, although the difference between the medians of distributions from correlated and uncorrelated data grows as the number of data points (n) used to compute each variable decreases, this difference occurs within distinct ranges. For uncorrelated data, $ϵ$ median values deviate by less than 10% from ${ϵ}_{sim}$, regardless of n. In contrast, for correlated data, the $ϵ$ median values gradually increase until n reaches about 15% of the total correlated data points (n = 1500). Beyond this threshold, as n continues to grow, the $ϵ$ medians of the correlated data converge with those of the uncorrelated data. Additionally, the MAD error remains consistently lower for distributions derived from uncorrelated data (Fig. 2).

The analysis of the medians of the mean intensity and variance distributions for correlated data (hatched fill, Fig. 2, A and B, respectively) reveals that these values are nearly identical when n is small (e.g., n = 100). Consequently, the median of the $ϵ$ distribution calculated from correlated data (hatched fill, Fig. 2 C) approaches zero, consistent with the theoretical relationship described in Eq. 3. As the number of correlated data points used for parameter calculation increases, the medians of the mean intensity and variance distributions exhibit a concomitant rise. Notably, the variance increases at a greater rate than the mean intensity, resulting in a progressive increase in the median of the $ϵ$ distribution. This trend continues until the median of the correlated data $ϵ$ distribution converges with that of the uncorrelated data. In contrast, for decorrelated (shuffled) data, the medians of the mean intensity and variance distributions remain stable and invariant to changes in n, maintaining consistency across all conditions.

To assess the potential impact of the number of entries, N/n, on the distribution results, we repeated the analysis using distributions containing 100 entries each (Fig. S2). The results show no differences, indicating that the number of entries within the distributions does not affect the outcome.

Finally, we determined the number of data points, n, required for the medians of distributions derived from correlated and decorrelated data to exhibit identical MAD values and a 5% deviation from the simulated ${ϵ}_{sim}$ (indicated by arrows in Fig. 2 C). For the correlated data, this condition is satisfied when each entry of the molecular brightness distribution is calculated using n $\approx$ 1,000,000 data points. On the contrary, for uncorrelated data, only n $\approx$ 2200 data points are needed to achieve the same criteria.

In conclusion, accurately determining molecular brightness requires substantially fewer data points when using uncorrelated data. In contrast, correlated data demand nearly 500 times more data points to achieve a molecular brightness value comparable to the simulated value.

Impact of the diffusion coefficient on the data requirements for molecular brightness calculation

So far, we have shown that molecular brightness calculation is affected by the temporal correlation of the data. As pointed out in the introduction, a set of consecutive data points will be temporally correlated if the acquired intensity originates from the same molecule. The extent of this correlation depends on the acquisition parameters and, in the case of diffusion, the molecular diffusion coefficient, D (Note S1). To investigate the impact of D on the recovered molecular brightness values, we analyzed the distributions of molecular brightness obtained from simulated data with varying diffusion coefficients (Fig. 3). Each entry in the distributions was calculated using n consecutive data points. The simulated diffusion coefficients spanned a wide range to represent diverse molecular dynamics (19,20) (Fig. S3).

Figure 3.

Median values of molecular brightness distributions as a function of n for 3 diffusion coefficient values. (A) D = 0.5 μm²/s, (B) D = 10 μm²/s, and (C) D = 100 μm²/s. The median value of each distribution (solid lines), and corresponding MAD in hatched (correlated data) and solid fill (uncorrelated data) are shown in counts per second per molecule. Arrows indicate the number of data points n required to compute each distribution’s entry for its median value (solid line) deviates by 5% from the simulated ${ϵ}_{sim}$ value (dashed line). The right arrows correspond to correlated data: n$\approx$ 18,000,000 (A), n$\approx$ 220,000 (B), and n$\approx$ 60,000 (C), while the left arrows correspond to uncorrelated data: n$\approx$ 15,000 (A), n$\approx$ 3000 (B), and n$\approx$ 7200 (C).

The median value of distributions derived from decorrelated data remains consistent with the simulated molecular brightness value, irrespective of the molecule’s diffusion coefficient. In contrast, for distributions derived from correlated data, the median values exhibit variation with D.

Building on the findings in the section effects of temporal correlation on molecular brightness calculation, we identified the required number of data points, n, to achieve identical MAD and a 5% deviation from the simulated ${ϵ}_{sim}$. For uncorrelated data, approximately 5000 data points are sufficient, regardless of the diffusion coefficient used in simulations (left arrows, Fig. 3). In contrast, the required n for correlated data varies with the diffusion coefficient. For slow dynamics (D = 0.5 μm²/s), approximately 18,000,000 points are needed (right arrows, Fig. 3 A). As the diffusion coefficient increases, the required n decreases to 220,000 data points for D = 10 μm²/s and 60,000 data points for D = 100 μm²/s (right arrows, Fig. 3, B and C). These results demonstrate that, using uncorrelated data, as proposed in this study, requires at least an order of magnitude fewer data points to achieve comparable accuracy with that obtained with correlated data.

Impact of the time interval distance on the molecular brightness calculation

Having established that temporal correlation among data points distorts the recovered $ϵ$ value, we explored an alternative approach to decorrelate the data. As described in MATE-BA method, molecular brightness values were computed by selecting nonmixed subsets of data points spaced by a time interval, J, sufficiently large to eliminate temporal correlations (Fig. 4 A). Fig. 4 B shows the results for various spacing intervals (1 < J < 100,000). The MAD of the distributions remains similar across all J values because the number of data points used to calculate the molecular brightness is held constant. As shown in Figs. 1, 2, and 3, increasing the number of data points improves precision. When data points are not spaced (J = 1), the results align with those shown in Fig. 1 A. For J < 100, the distribution’s median remains largely unchanged compared with the J = 1 case. However, as J increases, the distribution’s median approaches the simulated molecular brightness value (dashed line).

Figure 4.

Impact of the time interval distance on the molecular brightness calculation. (A) Schematic representation of the n data points selection separated by a time interval distance J for the $ϵ$ calculation. (B) Molecular brightness distributions for varying time interval distances, J. The distribution median value (solid line) and corresponding MAD (hatched fill) are shown. Arrows indicate the minimum time interval distance J required for the median of the distribution to differ by no more than 1% from the simulated value (dashed line).

Furthermore, when J exceeds the number of correlated data points (J > 22,500 in this example), the difference between the median of the molecular brightness distribution and ${ϵ}_{sim}$ becomes approximately 1% (arrows, Fig. 4 B). This indicates that, by selecting appropriately time-spaced subsets to ensure the data points are uncorrelated, an accurate molecular brightness value can be obtained with only n = 100 data points. This approach eliminates the need for random mixing and reduces computational processing times.

Molecular brightness from interspersed lines: The MATE-BA method

Having established that selecting adequately time-spaced data points is effective for recovering reliable $ϵ$ values, we extended this approach to raster scanning acquisition experiments. We used subsets of interspersed, uncorrelated data points as indicated in MATE-BA method to derive a molecular brightness value with a precision within 5% with respect to its corresponding simulated value, using only 100 data points. Additionally, a spatiotemporal correlation analysis of the molecular brightness values can be performed due to the temporal correlation between consecutive subsets of data points. By simulating line scan acquisition as described in simulations, we identified that 351 lines exhibited temporal correlation in this example. Fig. 5 shows molecular brightness distributions obtained from consecutive (hatched fill) and interspersed (solid fill) lines. Simulations were performed for four oligomeric states: 1-mers (red), 2-mers (green), 4-mers (violet), and 8-mers (yellow). Molecular brightness distributions built from correlated data were obtained using n = 100 consecutive lines (hatched fill, Fig. 5 A) and n = 50,000 consecutive lines (hatched fill, Fig. 5 B). Molecular brightness distributions from uncorrelated data (solid fill, Fig. 5 C) were calculated by performing MATE-BA analysis using n = 100 interspersed lines. We obtained the median (solid vertical line) for each distribution and compared it with the simulated molecular brightness (dashed vertical line).

Figure 5.

Molecular brightness distributions obtained from correlated lines (hatched fill) and uncorrelated lines (solid fill) for a set of four different oligomeric states: 1-mers (reds), 2-mers (greens), 4-mers (violets), and 8-mers (yellows) (color online). Dashed vertical lines represent the simulated molecular brightness value. Solid vertical lines are the median value of the molecular brightness distribution. (A) Molecular brightness distributions using n = 100 correlated consecutive lines, (B) using n = 50,000 correlated consecutive lines, (C) using n = 100 uncorrelated (interspersed) lines (${\delta }_{line}$ = 2000).

When n = 100 (Fig. 5 A), consecutive (correlated) lines were used to calculate the value of the molecular brightness and the median values of the distributions approach $ϵ$ = 0 cpms. In all simulated cases, since the number of consecutive lines used to compute each entry is below the number of temporally correlated lines (234 lines), there is a significant discrepancy between the medians of the distributions (hatched fill, Fig. 5 A) and the simulated molecular brightness value, ${ϵ}_{sim}$.

To minimize this discrepancy, it was necessary to increase at least 500 times the number of lines used in the molecular brightness analysis, ensuring that the correlated lines comprised less than 1% of the total (Fig. 5 B). As a result, the median of the molecular brightness distribution for each oligomeric state showed only about a 5% difference from its corresponding simulated $ϵ$ value. This result was consistent across all four oligomeric state distributions constructed from n = 50,000 consecutive lines. Finally, when we applied the MATE-BA method (Fig. 5 C), the medians of the molecular brightness distributions constructed from interspersed lines for each oligomeric state exhibited the same deviation from the simulated values observed in the previous case. This result was obtained by analyzing n = 100 interspersed lines, specifically chosen to eliminate temporal correlations. It is important to highlight that the number of correlated lines is inherently related to the diffusion coefficient. As D increases, the number of correlated lines decreases, which supports the validity and robustness of analyzing consecutive lines in this context. However, to investigate the temporal correlation of molecular brightness in the slow dynamics regime (0.01 μm²/s < D < 1 μm²/s), the MATE-BA approach is required, as it accounts for the increased number of correlated lines characteristic of these conditions.

The MATE-BA method enables the calculation of $ϵ$ values using only a few data points. This results in a notable reduction in the amount of acquired data needed and computational costs during the analysis. Also, because the MATE-BA method avoids temporal correlation between data points used for molecular brightness calculation, it is possible to calculate the molecular brightness value for a wide range of diffusion coefficients, extending the analysis to very slow dynamics.

Spatiotemporal correlations extracted from a series of molecular brightness values determined by MATE-BA

In MATE-BA, the ability to extract spatiotemporal correlations from molecular brightness measurements requires that the temporal shift used for brightness computation be significantly shorter than the characteristic correlation time of the dynamic process under investigation. This condition ensures that the relevant temporal fluctuations are retained in the resulting time series, enabling reliable quantification of the system’s dynamic behavior.

Fig. 6, A and B shows the autocorrelation function of the molecular brightness series ($G_{ϵ}$ ($\delta r$ = 0)) computed by pCOMB (correlated consecutive lines) (A), and MATE-BA (uncorrelated interspersed lines) (B). In both cases, positive correlation curves with similar characteristic times (${\tau }_c^A$ = 16 ms and ${\tau }_c^B$ = 14 ms) corresponding to the expected value for the simulated diffusion coefficient (D = 1 μm²/s) are obtained. This result demonstrates that MATE-BA retains information about temporal structure in the data. Additionally, Fig. 6 C compares the distributions of true molecular brightness obtained using pCOMB (correlated consecutive lines) and MATE-BA (uncorrelated interspersed lines) analysis.

Figure 6.

Spatiotemporal correlation from B values. (A) Autocorrelation of molecular brightness values computed from temporally correlated lines and (B) interspersed, temporally uncorrelated lines show a positive correlation. Each panel displays both the pixel-by-pixel autocorrelation curves and their average. The extracted correlation time (${\tau }_c^A$ = 16 ms and ${\tau }_c^B$ = 14 ms) matches the expected value for a diffusion coefficient of D = 1 μm²/s. (C) Molecular brightness distributions. Each histogram entry was obtained from n = 100 consecutive correlated (hashed distribution) and interspersed uncorrelated (solid distribution) intensity values. The dashed black line indicates the simulated molecular brightness, while solid lines represent the median of each distribution. The obtained distribution from interspersed, temporally uncorrelated lines shows an unbiased median molecular brightness value with respect to the value used in the simulations.

The improved accuracy and precision obtained using MATE-BA demonstrates that it not only retains dynamic information but also enhances the fidelity of molecular brightness estimation.

Discussion

As shown throughout this work, temporal correlation among data points can significantly bias molecular brightness determination. We simulated temporally correlated data using the Brownian model for particles diffusing in a 3D grid to identify conditions that lead to miscalculations of $ϵ$ values and propose accurate alternatives. When data subsets used for $ϵ$ determination were temporally correlated, the distributions of the recovered molecular brightness value were significantly skewed, resulting in values beneath the simulated ones (hatched fill in Figs. 1, A, B, 2 C, 3, and 5 A). For this reason, we investigated three distinct strategies to decorrelate the data used to compute $ϵ$.

• 1)Using a subset of n consecutive (correlated) data points, sufficiently large to lose correlation among distant points.
• 2)Shuffling the original data to create a subset of n consecutive but decorrelated data points.
• 3)Extracting subsets of interspersed (uncorrelated) data points.

The first strategy requires long data sets, which involve prolonged exposure of cells to irradiation and significant computational resources to analyze the large volume of data. The second method requires fewer data points and computational costs, making it useful where temporal correlation between molecular brightness values is not required. The third strategy also relies on a limited number of data points. However, unlike the previous approaches, it yields molecular brightness values that are temporally correlated. This occurs because, while the interspersed intensity values used for each individual molecular brightness calculation are uncorrelated, the overall data set remains correlated with adjacent data sets due to slight temporal shifts. The ability of MATE-BA to extract spatiotemporal correlations from molecular brightness values depends on selecting a temporal shift that is short relative to the characteristic timescale of the dynamic process under study. This approach becomes particularly useful when spatiotemporal correlation analysis on molecular brightness values is desired, enhancing the capabilities of the pCOMB technique by improving the accuracy and reliability of brightness estimation under various sampling conditions.

Together, these results show that, regardless of the decorrelated strategy used, the recovered $ϵ$ value converged to the simulated molecular brightness value with a smaller n, reducing computational time and improving accuracy (solid fill, Figs. 1, 2 C, 3, and 5). These findings underscore the importance of managing temporal data correlation for accurate molecular brightness calculations, while maintaining the temporal information necessary to capture dynamics through pair correlation molecular brightness analysis (Fig. 6).

Acknowledgments

We express our sincere gratitude to our colleagues Francesco Palomba, Lorenzo Scipioni, and Michelle Digman from the University of California (UCI) for their insightful discussions and feedback throughout this study and to Dr. Elizabeth Hinde for providing experimental data for initial testing purposes. We also acknowledge the financial support from Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT) and Universidad de Buenos Aires through grants PICT 2020-02718, PIDAE 2020-3980, PIDAE 2025-5201, and UBACyT 20020220100088BA. Our thanks go to the members of Estrada’s group for their constructive feedback and continuous encouragement. I.S. thanks the School of Sciences of the University of Buenos Aires for the travel award and FUNDACEN for their support during his visit to UCI. I.S. is also grateful for a PhD fellowship from the National Scientific and Technical Research Council (CONICET). L.C.E. is a member of CONICET.

Author contributions

I.S. and L.C.E. contributed to the development of the MATE-BA method. I.S. conducted the simulations, wrote custom data analysis routines, and prepared the figures. Both I.S. and L.C.E. co-wrote the manuscript. L.C.E. supervised the project.

Declaration of interests

The authors declare that they have no commercial or financial relationships that could be interpreted as potential conflicts of interest.

Editor: Sudipta Maiti.