^{1}, David L. Kolin

^{1}, Katrin G. Heinze

^{2}, Benedict Hebert

^{3}and Paul W. Wiseman

^{1,3,a)}

### Abstract

Fluctuation-based fluorescence correlation techniques are widely used to study dynamics of fluorophore labeled biomolecules in cells. Semiconductor quantum dots(QDs) have been developed as bright and photostable fluorescent probes for various biological applications. However, the fluorescence intermittency of QDs, commonly referred to as “blinking”, is believed to complicate quantitative correlation spectroscopymeasurements of transport properties, as it is an additional source of fluctuations that contribute on a wide range of time scales. The QD blinking fluctuations obey power-law distributions so there is no single characteristic fluctuation time for this phenomenon. Consequently, it is highly challenging to separate fluorescence blinking fluctuations from those due to transport dynamics. Here, we quantify the bias introduced by QD blinking in transport measurements made using fluctuation methods. Using computer simulated image time series of diffusing point emitters with set “on” and “off” time emission characteristics, we show that blinking results in a systematic overestimation of the diffusion coefficients measured with correlation analysis when a simple diffusion model is used to fit the time correlation decays. The relative error depends on the inherent blinking power-law statistics, the sampling rate relative to the characteristicdiffusion time and blinking times, and the total number of images in the time series. This systematic error can be significant; moreover, it can often go unnoticed in common transport model fits of experimental data. We propose an alternative fitting model that incorporates blinking and improves the accuracy of the recovered diffusion coefficients. We also show how to completely eliminate the bias by applying -space image correlation spectroscopy, which completely separates the diffusion and blinking dynamics, and allows the simultaneous recovery of accurate diffusion coefficients and QD blinking probability distribution function exponents.

We would like to thank Dr. D. Ronis (McGill University) for helpful discussions and suggestions. D.L.K. acknowledges an NSERC CGS fellowship and B.H. acknowledges an NSERC PGS fellowship. This project was financially supported by grant support to PWW from NSERC and CIHR.

I. INTRODUCTION

II. THEORETICAL BACKGROUND

A. Temporal fluorescence correlation analysis

B. kICS

III. SIMULATION ALGORITHMS AND PARAMETERS

IV. RESULTS AND DISCUSSION

A. ACF decay curves of diffusing QDs: TICS versus FCS

B. QD blinking effects on fluorescence correlation measurements of the diffusion coefficient

1. Power-law blinking and the diffusion coefficient

2. Experimental collection parameters

C. Blinking-Diffusion an anomalous model

V. kICS

A. Diffusion coefficients

B. Blinking parameters from kICS analysis

VI. CONCLUSIONS

### Key Topics

- Diffusion
- 87.0
- Quantum dots
- 67.0
- Time series analysis
- 20.0
- Correlation functions
- 15.0
- Fluorescence
- 15.0

## Figures

A schematic illustrating the principle of FCS and its imaging variants, scanning FCS and TICS. In FCS, temporal ACFs are calculated from a time series of fluorescence fluctuations recorded from a fixed illuminated focal point in space, whereas in the imaging variants, ACFs are calculated in time from image pixels defining a sampling area in space.

A schematic illustrating the principle of FCS and its imaging variants, scanning FCS and TICS. In FCS, temporal ACFs are calculated from a time series of fluorescence fluctuations recorded from a fixed illuminated focal point in space, whereas in the imaging variants, ACFs are calculated in time from image pixels defining a sampling area in space.

Combined blinking and 2D diffusion simulation. (a) Images selected at different times from a time series with an integration time of /frame and diffusion coefficient of ; I , II , III , and IV . Image size is set to at a density of using and an PSF radius of . (b) Spatial position in time and (c) emission time trace of the highlighted point source in (a). (d) On time PDF for the simulated image time series is an inverse power law with a set “on” and “off” time exponent of 1.5.

Combined blinking and 2D diffusion simulation. (a) Images selected at different times from a time series with an integration time of /frame and diffusion coefficient of ; I , II , III , and IV . Image size is set to at a density of using and an PSF radius of . (b) Spatial position in time and (c) emission time trace of the highlighted point source in (a). (d) On time PDF for the simulated image time series is an inverse power law with a set “on” and “off” time exponent of 1.5.

Normalized temporal ACF data (●) and best fits (–) to the 2D diffusion model [Eq. (5)] calculated from image time series simulations of diffusion and combined diffusion and blinking. The images in the time series simulations were with 500 frames at a density of using and an PSF radius of . The diffusion coefficient was set to with an integration time of /frame. For the combined diffusion and blinking simulations, the “off” time PDF exponent was set to 1.5 and “on” time probability distribution exponents of 1.5 and 1.8 were selected.

Normalized temporal ACF data (●) and best fits (–) to the 2D diffusion model [Eq. (5)] calculated from image time series simulations of diffusion and combined diffusion and blinking. The images in the time series simulations were with 500 frames at a density of using and an PSF radius of . The diffusion coefficient was set to with an integration time of /frame. For the combined diffusion and blinking simulations, the “off” time PDF exponent was set to 1.5 and “on” time probability distribution exponents of 1.5 and 1.8 were selected.

Plot of the percent relative error in the recovered diffusion coefficient from a 2D diffusion model [Eq. (5)] fit to ACFs calculated for simulation image time series of combined blinking and diffusion with varying “on” time PDF exponents (a) for a range of diffusion coefficients typically encountered in transport studies of biological molecules in cells and cell membranes. To capture the full range of diffusion coefficients, we select three sampling rates that span three orders of magnitude. The percent relative error in the recovered diffusion coefficient is independent of the set and shows a strong dependence on the sampling rate, which is defined as the number of frames per characteristic diffusion time (b). In (b), was set to with an integration time that varied between 0.03 and . For all combined diffusion and blinking simulations, the “off” time PDF exponent was set to 1.5 and the “on” time PDF exponent was varied from 1.5 to 2.2. For each set of conditions, results are the average of 20 simulations. Error bars are relative standard deviation compared to the set diffusion coefficient.

Plot of the percent relative error in the recovered diffusion coefficient from a 2D diffusion model [Eq. (5)] fit to ACFs calculated for simulation image time series of combined blinking and diffusion with varying “on” time PDF exponents (a) for a range of diffusion coefficients typically encountered in transport studies of biological molecules in cells and cell membranes. To capture the full range of diffusion coefficients, we select three sampling rates that span three orders of magnitude. The percent relative error in the recovered diffusion coefficient is independent of the set and shows a strong dependence on the sampling rate, which is defined as the number of frames per characteristic diffusion time (b). In (b), was set to with an integration time that varied between 0.03 and . For all combined diffusion and blinking simulations, the “off” time PDF exponent was set to 1.5 and the “on” time PDF exponent was varied from 1.5 to 2.2. For each set of conditions, results are the average of 20 simulations. Error bars are relative standard deviation compared to the set diffusion coefficient.

(a) A comparison of the percent relative error in the recovered diffusion coefficient from a pure diffusion [Eq. (5)] and anomalous diffusion model [Eq. (6)] fit for image time series simulations at selected temporal sampling rates and with varying “on” time PDF exponent. The was set to . For each set of conditions, results are the average of 20 simulations. The inset plot shows the recovered values for the corresponding blinking diffusion model fits in (a). (b) A plot of the percent change in the relative error on the diffusion coefficients recovered from a fit to the anomalous diffusion model, as calculated in comparison to the diffusion coefficients recovered form a purely diffusive model. The relative error is defined as . Error bars are calculated following uncertainty propagation rules (Ref. 44).

(a) A comparison of the percent relative error in the recovered diffusion coefficient from a pure diffusion [Eq. (5)] and anomalous diffusion model [Eq. (6)] fit for image time series simulations at selected temporal sampling rates and with varying “on” time PDF exponent. The was set to . For each set of conditions, results are the average of 20 simulations. The inset plot shows the recovered values for the corresponding blinking diffusion model fits in (a). (b) A plot of the percent change in the relative error on the diffusion coefficients recovered from a fit to the anomalous diffusion model, as calculated in comparison to the diffusion coefficients recovered form a purely diffusive model. The relative error is defined as . Error bars are calculated following uncertainty propagation rules (Ref. 44).

kICS analysis of two simulations which combine blinking and diffusion with different values. (a) Normalized, circularly averaged, log-transformed kICS correlation functions for for one simulation with and another with . In both cases, was set at , the integration time was , and the images in the time series simulations were with 500 frames. The blinking properties are manifested as different -intercepts for the plots from each simulation. (b) A plot of the slopes from (a) at different values of as a function of with linear regression fit that yields from the regression slope for both simulations. The set was the same, which is measured as equivalent regression slopes (within error) for each simulation, as also seen for the single plots in (a).

kICS analysis of two simulations which combine blinking and diffusion with different values. (a) Normalized, circularly averaged, log-transformed kICS correlation functions for for one simulation with and another with . In both cases, was set at , the integration time was , and the images in the time series simulations were with 500 frames. The blinking properties are manifested as different -intercepts for the plots from each simulation. (b) A plot of the slopes from (a) at different values of as a function of with linear regression fit that yields from the regression slope for both simulations. The set was the same, which is measured as equivalent regression slopes (within error) for each simulation, as also seen for the single plots in (a).

A plot of the percent absolute relative error in the recovered diffusion coefficient as a function of temporal sampling for both kICS analysis and TICS analysis using a 2D diffusion model fit [Eq. (5)] to the temporal ACF for image time series simulations of combined blinking and diffusion with set and 2.2 and a set .

A plot of the percent absolute relative error in the recovered diffusion coefficient as a function of temporal sampling for both kICS analysis and TICS analysis using a 2D diffusion model fit [Eq. (5)] to the temporal ACF for image time series simulations of combined blinking and diffusion with set and 2.2 and a set .

(a) Change in QD ensemble blinking behavior in time for various set “on” time PDF exponents, as illustrated in the plot of as a function of the lag parameter . was calculated from the fit intercept of the log transformed kICS correlation function for and integration time of . (b) Recovered on time PDF exponents from the fit of to (○) and (◻) averaged over 20 simulations. Error bars are standard deviations.

(a) Change in QD ensemble blinking behavior in time for various set “on” time PDF exponents, as illustrated in the plot of as a function of the lag parameter . was calculated from the fit intercept of the log transformed kICS correlation function for and integration time of . (b) Recovered on time PDF exponents from the fit of to (○) and (◻) averaged over 20 simulations. Error bars are standard deviations.

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