Imaging barriers to diffusion by pair correlation functions

Abstract

Molecular diffusion and transport are fundamental processes in physical, chemical, biochemical, and biological systems. However, current approaches to measure molecular transport in cells and tissues based on perturbation methods such as fluorescence recovery after photobleaching are invasive, fluctuation correlation methods are local, and single-particle tracking requires the observation of isolated particles for relatively long periods of time. We propose to detect molecular transport by measuring the time cross-correlation of fluctuations at a pair of locations in the sample. When the points are farther apart than two times the size of the point spread function, the maximum of the correlation is proportional to the average time a molecule takes to move from a specific location to another. We demonstrate the method by simulations, using beads in solution, and by measuring the diffusion of molecules in cellular membranes. The spatial pair cross-correlation method detects barriers to diffusion and heterogeneity of diffusion because the time of the correlation maximum is delayed in the presence of diffusion barriers. This noninvasive, sensitive technique follows the same molecule over a large area, thereby producing a map of molecular flow. It does not require isolated molecules, and thus many molecules can be labeled at the same time and within the point spread function.

Figure 1

Schematic of the spatial pair-correlation method. The fluorescence intensity is rapidly sampled (compared with the motion of the particles) at several points in a grid (labeled 1, 2, 3, 4). As particles migrate, they appear at different points of the grid. Only the same particle will produce an average cross-correlation with a given time delay at two different points in the grid. For example, a fluctuation in intensity at position 1 will statistically correlate with a fluctuation of the intensity at position 2 if the same particle is moving (with some delay) to position 2. Fluctuations at position 3 in the grid (the other side of the barrier in blue) will never correlate with the fluctuations at position 1 or 2. If we map the amount of correlation between pairs of points (1,2 and 2,3), we see a discontinuity in the correlation between 2 and 3 but not between 1,2 and 3,4. If instead of an impenetrable barrier we have obstacles, as shown in the right panel, we could observe the same particle on the other side of the obstacle but with a delayed correlation. If we cross-correlate the intensity fluctuations at each point of the grid, we produce a map of molecular flow with a resolution given by the size of the PSF shown in light blue, which is ∼250 nm in the plane of the grid.

Figure 2

PCFs. (A) Particle observed at time t = 0 at the origin can be found at a distance r with a probability (shown schematically by the shaded parabolic shape) proportional to the fluorescence intensity at a given distance. (B) The fluorescence intensity is calculated at different distances from 0 to 2 μm in steps of 0.2 μm from the origin along a vertical line with respect to the plot in part A. For this calculation, the waist of the PSF was 0.3 μm and the diffusion coefficient D = 1.0 μm²/s. (C) Intensity carpet for simulation of 500 particles diffusing on a plane with a diffusion coefficient of 0.1 μm²/s shown in the color-coded image. The warmer colors correspond to higher intensities. (D) PCFs at different pixel distances from 1 to 6. The pCF at a distance of 6 pixels (0.9 μm) falls below zero. (E) pCF(10) calculated at a distance of 10 pixels for different values of the diffusion coefficient. The maximum of the pCF(10) function moves at longer times as the diffusion coefficient decreases. The amplitude of the correlation remains approximately constant. When two molecular species with different diffusion constants are present simultaneously, there are two maxima of the pCF due to the different time delays of the two species in reaching a given distance. At short correlation times, the pCF function can be negative, indicating spatial antibunching.

Figure 3

Simulation of particles diffusing in restricted zones of a membrane. The particles cannot cross the boundaries of the confinement zones shown in the upper part of the figure. Barriers to flow appear as “dark” vertical lines in the pCF carpet (no probability of finding a particle at a distance of 3 pixels along the orbit, pCF(3)). In this simulation the transient confinement zone size is 3.2 μm and the distance between zones is 12.8 μm. Transient confinement zones as small as 200 nm result in “visible” barriers to diffusion in the pCF representation.

Figure 4

Beads in solution. The panels at left show the intensity carpet, autocorrelation (ACF, log time presentation) and the pair-correlation carpet calculated at a distance of 5 pixels along the orbit (pCF(5)). Along the orbit, 1 pixel corresponds to 0.15 μm. The plot shows the pCF calculated at a distance of 1–6 pixels.

Figure 5

Detection of barriers to diffusion. (A) DiO in MEF cells. A barrier to diffusion is associated with a macroscopic “unknown” object in the membrane. The pCF(8) shows a discontinuity in the correlation at a given position along the orbit. The detail of the pCF at column 107 (out of 256 columns) along the orbit shows the molecules' delay in reaching a distance of 8 pixels (900 nm). (B) GAP-EGFP diffusing in two different cells. Obvious obstacles to diffusion are observed at the junction between cells. In this case, molecules never cross the cellular membrane barrier.

Figure 6

GAP-EGFP. Diffusing proteins are clearly visible in the ACF curve. The transport of molecules is uniform and isotropic at the basal membrane, although there are two or more diffusing components, as shown by the pair-correlation plot calculated at different distances of 8 and 16 pixels. The pCFs in the graph are averaged along the entire orbit. The pair-correlation operation was applied from 0 to 30 s and from 30 s to 60 s. In the first time segment, the correlation carpet is not uniform along the orbit (indicated by the red arrow), hinting at the existence of barriers to diffusion during the first time segment, but this obstacle to diffusion is not observed at later times (segment 30–60 s). The detail in the figure shows a projection of pCF(8) on the spatial axis. At specific locations the pCF seems to go to zero, indicating a barrier for diffusion, but in a different time segment of the same measurement the barrier appears to have dissipated.