Transient anomalous subdiffusion: effects of specific and nonspecific probe binding with actin gels

Abstract

When signaling molecules diffuse through the cytosol, they encounter a wide variety of obstacles that hinder their mobility in space and time. Some of those factors include, but are not limited to, interactions with mobile and immobile targets or obstacles. Besides finding a crowded environment inside the cell, macromolecules assemble into molecular complexes that drive specific biological functions adding additional complexity to their diffusion. Thus, simple models of diffusion often fail to explain mobility through the cell interior, and new approaches are needed. Here we used fluorescent correlation spectroscopy to measure diffusion of three molecules of similar size with different surface properties diffusing in actin gels. The fluorescent probes were (a) quantum dots, (b) yellow-green fluorescent spheres, and (c) the beta isoform of Ca(2+) calmodulin-dependent protein kinase II tagged with green fluorescent protein. We compared various models for fitting the autocorrelation function (ACF) including single component, two-component, and anomalous diffusion. The two-component and anomalous diffusion models were superior and were largely indistinguishable based on a goodness of fit criteria. To better resolve differences between these two models, we modified the ACF to observe temporal variations in diffusion. We found in both simulated and experimental data a transient anomalous subdiffusion between two freely diffusing regimes produced by binding interactions of the diffusive tracers with actin gels.

FIGURE 1

Data generated from various ACF models. A) Single component, two component and anomalous diffusion autocorrelation data generated from Eq. 10, 12, and 18, respectively. B) The numerical inversion of Eq. 24 on the data generated from a one component ACF shows a flat temporal spectrum of the apparent diffusion (squares) that returns an apparent D of 15 μm²/s. The data generated from the anomalous diffusion ACF shows a power law dependence (triangles). The fitted values with a power law model are α= 0.6 and Γ = 6.33 μm²/s^α. C) The result of numerical inversion for D_app(t) for the case of data generated with the two-component model of the ACF Eq 12 is shown. This result shows a transition between two normal modes of diffusion. The transition region from x ms to x ms was fit with an anomalous diffusion model which returned an α = 0.72.

FIGURE 2

FCS data generated with SimFCS. A) Shows the simulated data for either one diffusive component (black curve) and for the case of A↔B at three different conditions i) R = 0 (red curve), ii) R = 667 s⁻¹ (green curve), and iii) R = 0.0667 s⁻¹ (blue curve). Refer to the Methodology section for additional details concerning the simulations. Evaluating the ACF, the only significant difference is in the case of (ii) where the rate of transition of A↔B is on the same time scale as the transition time of the molecule through the focal volume. However, when D_app(τ) is extracted there are clear differences in the spectra as shown in panel (B). The single component data (black trace) shows a flat line across the temporal spectrum as anticipated. With two component data (red trace) there is a period of normal diffusion at short timescales (<1 ms) which then shows a downward decay from 1-100 ms. When kinetics are added, where the rate of transition between two diffusing states is slow (0.0667 s⁻¹; blue trace), we again see a region of normal diffusion that shows a downward trend up to ~100 ms. When the transition rate is fast (667 s⁻¹; green trace), the curve is flat, as in the one component case (black curve) and the amplitude has decreased to reflect the average diffusion of the two components.

FIGURE 3

Cartoon representations of Quantum Dots, Yellow Green spheres and eGFP-βCaMKII and their respective surface properties. Quantum Dots from Invitrogen (Qtracker® 565) are coated with polyethylene glycol with a MW of 5000 Da. Fluorescent spheres are sulfate coated giving them an effective negative charge at neutral pH. A surface rendered model of eGFP-βCaMKII, (enhanced green fluorescent protein is shown in green, βCaMKII is shown in purple); below the structure, the linear sequence of amino acids, residues 354 to 392, of βCaMKII responsible for actin binding is shown . The individual molecules are shown scaled by the radius of gyration extracted from FCS (see Table 1).

FIGURE 4

Fluorescence correlation analysis of Quantum Dots (Qdots), YG spheres and eGFP-βCaMKII diffusing in buffer. A) Raw data and fits using a single component model and their residuals. B) Same data as that shown in panel A fitted with the anomalous diffusion model. Residuals show a slightly better fit than the single component model. C) The average diffusion at τ_Da coefficient of three independent experiments is plotted for each sample. The error bars represent the standard deviations. D) α exponent from the anomalous model fit for the same samples in buffer. A straight line at α=1 (normal diffusion) is shown as a reference.

FIGURE 5

Fluorescence correlation spectroscopy of Quantum Dots in buffer, G-actin and F-actin cross linked with filamin. A) Raw data and fitting using the anomalous diffusion model. Residuals for each fit are shown below the plot. B) The average diffusion at τ_Da are presented and the error bars represent the standard deviation from three independent experiments. Panel C) shows the average and standard deviation of the anomalous exponent for the same samples.

FIGURE 6

Fluorescence correlation spectroscopy of YG-spheres in buffer, G-actin and F-actin/filamin. A) Raw data and fitting with the anomalous diffusion model. Residuals for each fit are shown below the plot. B) The average diffusion at τ_Da are presented and the error bars represent the standard deviation from three independent experiments. Panel C) shows the average and standard deviation of the anomalous exponent for the same samples.

FIGURE 7

Diffusion as a function of time of Qdots and YG spheres in buffer and in F-actin with filamin. A) Diffusion profile for Qdots in buffer shows a constant value for most of the time domain. For the case of Qdots in F-actin/filamin (red curve) there is a slight downward trend in the data. The fit of the same data to the anomalous model returned an α value of 0.91. This value was used to fit a line to the curve with extending end points until a deviation (> 0.05%) was detected and revealed that the spectral domain that fell within this criteria was from 1 to 100 ms (magenta line). B) Diffusion spectra of YG spheres under the same conditions as in panel A. In buffer, the profile is again flat for almost the entire time-domain. In F-actin/filamin the regime at which power law fit maintained an anomalous exponent of 0.82 ranges from 1-100 (magenta line).

FIGURE 8

Fluorescence correlation spectroscopy of eGFP-βCaMKII in buffer, G-actin and F-actin cross linked with filamin. A) Raw data and fitting using the anomalous diffusion model. Residuals for each fit are shown below the plot. B) The average apparent diffusion values in the same conditions are presented as bar plots. Day-to-day variations are shown as the error bars representing the standard deviation. Panel C) shows the average and standard deviation of the anomalous exponent for the same samples.

FIGURE 9

Diffusion as a function of time of eGFP-βCaMKII in buffer, G-actin and in F-actin/filamin. The diffusion profile in buffer shows a constant value for most of the time domain but this does not hold true for diffusion in G-actin or F-actin/filamin. In G-actin the regime at which the power law fit maintained an anomalous exponent of 0.64 ranges from 1-100 ms (red line). The fit of βCaMKII in F-actin/filamin at which the power law fit maintained an anomalous exponent of 0.89 was over the time domain of 0.1 to 10 ms (purple line)