Anomalous extracellular diffusion in rat cerebellum

Abstract

Extracellular space (ECS) is a major channel transporting biologically active molecules and drugs in the brain. Diffusion-mediated transport of these substances is hindered by the ECS structure but the microscopic basis of this hindrance is not fully understood. One hypothesis proposes that the hindrance originates in large part from the presence of dead-space (DS) microdomains that can transiently retain diffusing molecules. Because previous theoretical and modeling work reported an initial period of anomalous diffusion in similar environments, we expected that brain regions densely populated by DS microdomains would exhibit anomalous extracellular diffusion. Specifically, we targeted granular layers (GL) of rat and turtle cerebella that are populated with large and geometrically complex glomeruli. The integrative optical imaging (IOI) method was employed to evaluate diffusion of fluorophore-labeled dextran (MW 3000) in GL, and the IOI data analysis was adapted to quantify the anomalous diffusion exponent dw from the IOI records. Diffusion was significantly anomalous in rat GL, where dw reached 4.8. In the geometrically simpler turtle GL, dw was elevated but not robustly anomalous (dw = 2.6). The experimental work was complemented by numerical Monte Carlo simulations of anomalous ECS diffusion in several three-dimensional tissue models containing glomeruli-like structures. It demonstrated that both the duration of transiently anomalous diffusion and the anomalous exponent depend on the size of model glomeruli and the degree of their wrapping. In conclusion, we have found anomalous extracellular diffusion in the GL of rat cerebellum. This finding lends support to the DS microdomain hypothesis. Transiently anomalous diffusion also has a profound effect on the spatiotemporal distribution of molecules released into the ECS, especially at diffusion distances on the order of a few cell diameters, speeding up short-range diffusion-mediated signals in less permeable structures.

Figure 1

Schematic of the rat cerebellum granular layer. Granular layer, the deepest of the cerebellar cortical layers, contains numerous glomeruli wrapped by thin astrocytic processes. Glomeruli are giant complex synapses where dendrites of granular cells receive both an excitatory glutamatergic input via mossy fiber terminals, and an inhibitory GABAergic input from local Golgi cells. Granular cells send information via parallel fibers to the molecular layer for further processing. The axons of Purkinje cells represent the main output from cerebellar cortex. (GL, granular layer; ML, molecular layer; Gl, glomerulus; Mf, mossy fiber; Gr, granular cell; Go, Golgi cell; Pf, parallel fiber; Pu, Purkinje cell.) To see this figure in color, go online.

Figure 2

Diffusion of dex3 in agarose gel and in rat GL. (A) Four images of diffusing marker molecules selected from a typical time-series recorded in the agarose gel. The white scale bar is 200 μm in length and the pseudocolor bar (valid also for C) represents fluorescence intensities. (B) Intensity profiles (black) extracted along the dashed line superimposed on the first image in (A), together with the fitted Gaussian curves (orange). (C) Four images of diffusing marker molecules selected from a typical time-series recorded in the rat GL. The white scale bar length is 200 μm. (D) Intensity profiles (black) extracted along the dashed line superimposed on the first image in (C), together with the fitted Gaussian curves (red). (E) The mean-square horizontal displacement as a function of time for the agarose gel (orange circles, experiment; black line, linear fit) and for the rat GL (red circles, experiment). Note the pronounced nonlinearity after dex3 release in the rat GL.

Figure 3

Diffusion of dex3 in the rat and turtle GL at high magnification. (A) Six images of diffusing marker molecules selected from a typical time-series recorded in the rat GL. The white scale bar is 50 μm long and the pseudocolor bar (valid also for C) represents fluorescence intensities. (B) Intensity profiles (black) extracted along the dashed line superimposed on the first image in (A), together with the fitted Gaussian curves (red). (C) Six images of diffusing marker molecules selected from a typical time-series recorded in the turtle GL. The white scale bar is 50 μm long. (D) Intensity profiles (black) extracted along the dashed line superimposed on the first image in (C), together with the fitted Gaussian curves (green). (E) The mean-square horizontal displacement as a function of time for the rat GL (red circles, experiment) and for the turtle GL (green circles, experiment; black line, linear fit). Note the more pronounced nonlinearity in rat GL.

Figure 4

Quantification of anomalous diffusion (see Eq. 5). (A) Representative logarithmic plot of mean-square horizontal displacement as a function of time for the agarose gel (10× objective), rat GL (40× objective), and turtle GL (40× objective). Lines obtained by linear fitting represent anomalous exponents 2.05 (agarose gel), 5.24 (rat GL), and 2.57 (turtle GL). (B) A summary of all experiments obtained in the agarose gel, in the rat GL and in the turtle GL. Normal diffusion corresponds to anomalous exponent d_w = 2 (dashed line). Note the high d_w in a rat GL but also the slightly elevated d_w in a turtle GL. The corresponding average values and other statistical results are summarized in Table 1. To see this figure in color, go online.

Figure 5

Numerical simulations of normal and anomalous extracellular diffusion with MCell program. (A) An example section of a three-dimensional model composed of large parcels (see B). (Left image) One shell was made transparent to expose the eight cubes within. (Right image) Most elements were removed to reveal the diffusing extracellular particles (red dots). (B) Cross sections of five different elements used to build the simulation environments. Adjacent elements touch at their boundaries (dotted lines). (C) Free, anomalous, and hindered diffusion regimes (see text for details). Without wrapping, no anomaly is apparent (gray and yellow circles). Smaller wrapped elements result in the same anomalous exponent as larger elements but with earlier onset and shorter duration of the anomaly (blue versus red circles). Changing the dead-space geometry affects both the timing and the anomalous exponent (red versus green circles). (D) Anomalous exponents d_w and durations Δt extracted from the data displayed in (C). Color coding is the same as in (C).

Figure 6

Numerical simulations of the dead-space compartmentalization. (A) A single parcel element with dead space between the inner cube and its surrounding shell. Deleting subsets of the dividing walls suspended between the cube and the shell provided seven different compartmentalizations with 1, 2, 3, 4, 6, 8, and 12 subcompartments (pockets), each with a single entrance (see text for details). As an example, one pocket of the 12-pocket arrangement is outlined (dashed line) along with its entry (dotted line). (B) Free, anomalous, and hindered diffusion regimes. Example data sets obtained in geometries containing 1, 6, and 12 pockets are shown. (C) Anomalous exponents d_w and durations Δt extracted from simulated diffusion in all seven geometries. The anomalous diffusion phase becomes progressively shorter when the total dead space is divided into more compartments but the anomalous exponent is largely unaffected.

Figure 7

An effect of anomalous diffusion on diffusion-mediated signals. Over a short diffusion distance of 20 μm, more complex geometry leads paradoxically to faster signaling. (A) Parameters obtained from experiments in the rat and turtle GL (Turtle, D^∗ = 46; Rat, D^∗ = 10, anomalous D = 253, d_w = 4.82; distance unit 1 μm; time unit 1 s) were used to calculate the concentration time courses. The ECS concentration wave arrives faster and lasts for a shorter duration in the geometrically more complex rat GL. (B) If strictly normal diffusion was assumed in both structures, the geometrically more complex rat GL would have lower D^∗ and therefore experience slower diffusion, reversing the timing shown in (A). The estimate of D^∗ in the rat GL is necessarily inaccurate but that does not alter the order of concentration peaks. To see this figure in color, go online.