Brain Extracellular Space as a Diffusion Barrier

Abstract

The extracellular space (ECS) consists of the narrow channels between brain cells together with their geometrical configuration and contents. Despite being only 20-60 nm in width, the ECS typically occupies 20% of the brain volume. Numerous experiments over the last 50 years have established that molecules moving through the ECS obey the laws of diffusion but with an effective diffusion coefficient reduced by a factor of about 2.6 compared to free diffusion. This review considers the origins of the diffusion barrier arising from the ECS and its properties. The paper presents a brief overview of software for implementing two point-source paradigms for measurements of localized diffusion properties: the real-time iontophoresis or pressure method for small ions and the integrative optical imaging method for macromolecules. Selected results are presented. This is followed by a discussion of the application of the MCell Monte Carlo simulation program to determining the importance of geometrical constraints, especially dead-space microdomains, and the possible role of interaction with the extracellular matrix. It is concluded that we can predict the impediment to diffusion of many molecules of practical importance and also use studies of the diffusion of selected molecular probes to reveal the barrier properties of the ECS.

Figure 1

ECS geometry. Panel A shows electron micrograph of a region of rat cortex with several nerve fibers, together with other neuronal and glia extensions, surrounding a dendritic profile containing mitochondria. The ECS has been outlined in red but the true width is likely underestimated because of shrinkage during the fixation and processing of the tissue. Note that some fibers form bundles; this may produce anisotropic diffusion in some brain regions. Scale bar approximately 1 μm. Micrograph courtesy of Dr. C. B. Jaeger. Panel B shows simplified schematic of a small region of ECS between a group of neurons (green) and glia (red). The ECS may harbor dead-space microdomains in the form of local expansions, or voids (V), or invaginations (I) of cellular elements or glial wrapping around cells.

Figure 2

Setup for diffusion measurements using the real-time iontophoretic or pressure (RTI/RTP) methods and integrative optical imaging (IOI) method. A brain slice or dilute agarose gel is placed in a chamber that is perfused with oxygenated physiological saline solution on the stage of a compound microscope. For RTI/RTP, an extracellular probe ion, most commonly TMA⁺, is released from a glass micropipette and detected with an ion-selective microelectrode (TMA⁺-ISM), positioned about 100 μm away. The resulting diffusion curves (concentration versus time) are amplified, digitized and stored on a computer enabling an appropriate diffusion equation to be fitted to the data. In dilute agarose gel, the free diffusion coefficient, D, and transport number, n_t, are measured. In the brain slice, the effective diffusion coefficient, D^*, volume fraction, α and loss factor, k′, are measured. For IOI, a fluorescent molecule, here dextran (3000 M_r, i.e. 3 kDa) labeled with a fluorescent dye, is briefly pressure injected and a time-series of images captured using a CCD-equipped camera. An appropriate diffusion equation is fitted to the intensity profiles measured along selected image axes enabling D or D^*, to be extracted in agarose gel or brain slice, respectively. Also note that the methods are not confined to brain slices but also may be used in vivo (modified from ref. 10).

Figure 3

Graphical User Interface (GUI) for the Wanda software. This program is responsible for instrument control and data acquisition under the RTI/RTP paradigm. See text for further details.

Figure 4

Graphical User Interface (GUI) for the Devida software. This program is responsible for microscope and associated instrument control and image acquisition under the IOI paradigm. See text for further details.

Figure 5

Visualization, using DReAMM software, of ensembles of cubes and associated ECS employed in MCell simulations. Panel A shows a perspective view from the top of just the 4,096 voids in an entire 32×32×32 cubes and void geometry (equivalent to 32,768 cubes). As described in the text, every eighth cube has been replaced by a void (here visualized as extracellular matrix, colored green, see also Fig. 7). To aid visualization, the remaining cubes are not shown but they fill most of the gaps between the voids. Many simulations used 64×64×64 cubes with voids (equivalent to 262,144 cubes). Panel B shows an enlarged view of a stack of just 4×4×4 cubes with all cubes present showing the location of the voids. The stack is taken from the larger ensemble with ECS conforming to α = 0.2, which is a typical value found in experiments. A total of 10,000 molecules (shown in red) are shown at at t = 0.1 ms after release diffusing from a point-source in the center through the spaces between the cubes and also within the voids. Using flat projection, a view of the entire ensemble from the top is obtained showing that diffusing molecules have explored the vicinity of several cubes by this time. Using the DReAMM program the cube surfaces are displayed in transparent gray in order to visualize the molecules that are moving in the lower layers of the geometry. Note that the exaggerated size of the red molecules is used as an aid to visualization and is not a feature of the simulation (in MCell, molecules are regarded as point structures). Panel C shows the same diffusion paradigm as detailed for Panel B in a geometry with uniformly distributed cubes and no voids. Panel D shows a larger region of the geometry, with 8×8×8 cubes with voids, seen from a perspective that emphasizes the 3D structure. As in panel A, to aid visualization, each void is filled with extracellular matrix (shown in green). There are 1300 matrix molecules in each void, giving a concentration of 10 μM and they forms the basis for reaction simulations depicted in Fig. 7.

Figure 6

Tortuosity of ECS versus volume fraction with and without voids. The solid line represents the plot of Eq (13) with a value β = 2.91, obtained by a non-linear curve fit to the MCell simulation data (filled circles) for a geometry with cubes and voids. Each cube was had side 2a = 0.6 μm and 64×64×64 cubes were used. The spacing between cubes was adjusted so that the total volume fraction corresponded to the depicted values. Note that the minimum possible volume fraction was α = 0.128 and this occurred when the spaces between cubes approached zero and then α was determined only by the missing cube. The dashed line corresponds to the case where no void is present, i.e. Eq (12) with the geometry made up of a uniform distribution of cubes.

Figure 7

Visualization of molecules interacting with matrix in voids. Green molecules represent the matrix, the diffusing molecules are red the molecular complex between the matrix and diffusing molecules is represented in blue. A total of 10,000 molecules were released from a point source (red dot in the middle of panel A) at the center of a geometry comprising 32×32×32 cubes with every eighth cube removed to form voids. The matrix was concentrated the in voids at a value of 10 μM and volume fraction was 0.28 (excluding the voids) and 0.37 (including the voids). Forward and backward rate constants for binding and unbinding of diffusing molecules to the matrix were set (somewhat arbitrarily) to k_f = 5.6×10⁸ M⁻¹.s⁻¹ and k_b = 2.2×10³ s⁻¹; when combined with the matrix concentration averaged over the whole ECS this gave an R-value of 0.6 (see Eq 20). The free diffusion coefficient was D = 7.4×10⁻⁶ cm² s⁻¹, which is the value for Ca²⁺ at 23 °C. Panels show a small region of the ensemble of cubes, looking down through the semi transparent surfaces. They show molecules diffusing from the source point at t = 0 (Panel A) and then exploring their microenvironment at successive times, t = 0.1 ms (Panel B), t = 0.5 ms (Panel C) and t = 1 ms (Panel D). As time passes an increasing amount of complex (blue) is formed. Visualization with DReAMM.

Figure 8

Multiplicative properties of tortuosities. Curve labeled ‘λ_g’ represents tortuosity computed, using Eq (11), from an MCell simulation in a medium with cubes and voids (i.e. one cube in eight removed) as a function of time after release of 5000 Ca²⁺ molecules from origin. Curve labeled ‘λ_m’ represent a similar tortuosity calculation in a medium containing only matrix at a concentration of 2.34 μM (i.e. no geometry present). Curve labeled ‘λ’ represents tortuosity calculated in a medium with cubes and voids but with a quantity of matrix in the voids such that the matrix concentration averaged over the whole ECS is also 2.34 μM. Finally, ‘λ_c’ represents the product of the λ_g and λ_m curves. It is seen that λ and λ_c are close in value. Parameters: k_f, k_b and D as for Fig. 7; α = 0.2.