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.
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.
(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).
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.
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