Aquaporin-4-dependent K(+) and water transport modeled in brain extracellular space following neuroexcitation

Abstract

Potassium (K(+)) ions released into brain extracellular space (ECS) during neuroexcitation are efficiently taken up by astrocytes. Deletion of astrocyte water channel aquaporin-4 (AQP4) in mice alters neuroexcitation by reducing ECS [K(+)] accumulation and slowing K(+) reuptake. These effects could involve AQP4-dependent: (a) K(+) permeability, (b) resting ECS volume, (c) ECS contraction during K(+) reuptake, and (d) diffusion-limited water/K(+) transport coupling. To investigate the role of these mechanisms, we compared experimental data to predictions of a model of K(+) and water uptake into astrocytes after neuronal release of K(+) into the ECS. The model computed the kinetics of ECS [K(+)] and volume, with input parameters including initial ECS volume, astrocyte K(+) conductance and water permeability, and diffusion in astrocyte cytoplasm. Numerical methods were developed to compute transport and diffusion for a nonstationary astrocyte-ECS interface. The modeling showed that mechanisms b-d, together, can predict experimentally observed impairment in K(+) reuptake from the ECS in AQP4 deficiency, as well as altered K(+) accumulation in the ECS after neuroexcitation, provided that astrocyte water permeability is sufficiently reduced in AQP4 deficiency and that solute diffusion in astrocyte cytoplasm is sufficiently low. The modeling thus provides a potential explanation for AQP4-dependent K(+)/water coupling in the ECS without requiring AQP4-dependent astrocyte K(+) permeability. Our model links the physical and ion/water transport properties of brain cells with the dynamics of neuroexcitation, and supports the conclusion that reduced AQP4-dependent water transport is responsible for defective neuroexcitation in AQP4 deficiency.

Figure 1.

Model of K⁺ and water transport in brain ECS. (A) Diagram showing neurons and astrocytes in brain surrounded by an ECS. (B) Schematic of mathematical model. Neuronal, ECS, and astrocytic compartments are shown, along with key transport mechanisms including neuronal K⁺ release (J_Kⁿ), K⁺ uptake by astrocytes (J_K^a), and osmotic water transport into astrocytes (J_V^a).

Figure 2.

Computational approach. See the Model computations section for description.

Figure 3.

Predictions of the well-mixed model. (A) Schematic of well-mixed model showing neuronal K⁺ release into ECS, and astrocytic uptake of K⁺ and water causing ECS shrinkage. (B) Time course of ECS potassium concentration ([K⁺]_e) and volume (d_e), astrocyte membrane potential (ψ), and astrocyte K⁺ and water flux (J_K^a, J_V^a) after neuronal excitation causing [K⁺]_e increase from 5 to 10 mM. Parameters: P_f = 0.04 cm/s, J_Kⁿ_o = 10⁻⁸ mol/cm²/s, Δt_n = 0.1 s, d_a = 10 µm, d_e = 2 µm, with the indicated P_K. (C) Computations as in B for P_K = 1.2 × 10⁻⁵ cm/s with the indicated P_f. (D) Computations as in B for P_K = 1.2 × 10⁻⁵ cm/s, with the indicated d_e. J_Kⁿ_o was adjusted to increase [K⁺]_e from 5 to 10 mM. (E) Effect of magnitude of neuroexcitation. Computations as in B, with P_K = 1.5 × 10⁻⁵ cm/s and J_Kⁿ_o = 5 × 10⁻⁸ mol/cm²/s, with the indicated P_f. (F) The effect of duration of neuroexcitation. Computations as in B, with P_K = 1.3 × 10⁻⁵ cm/s, J_Kⁿ_o = 8 × 10⁻¹⁰ mol/cm²/s, and Δt_n = 10 s, with the indicated P_f.

Figure 4.

Predictions of the diffusion-limited model for high astrocyte water permeability. P_f = 0.04 cm/s for computations in this figure. (A) Schematic showing nonlinear and newly added mesh elements in astrocyte cytoplasm used for numerical solution of the diffusion equation. See text for explanation. (B) Time course of [K⁺]_e, d_e, ψ, J_K^a, and J_V^a after neuroexcitation for the indicated cytoplasmic diffusion coefficients, D_a. Parameters: P_K = 1.2 × 10⁻⁵ cm/s, J_Kⁿ_o = 10⁻⁸ mol/cm²/s, Δt_n = 0.1 s, d_a = 10 µm, and d_e = 2 µm. (C and D) Astrocyte spatial distribution of [K⁺]_a and [non-K⁺]_a at the indicated times, with parameters as in B, for D_a = 10⁻⁸ cm²/s (C) and t = 5 s (D). (E) Effect of duration of neuroexcitation. Computations as in B, with P_K = 1.3 × 10⁻⁵ cm/s, J_Kⁿ_o = 8 × 10⁻¹⁰ mol/cm²/s, and Δt_n = 10 s.

Figure 5.

Water permeability effects in the diffusion-limited model. (A) Time course of [K⁺]_e and d_e for the indicated P_f and D_a. Parameters: P_K = 1.2 × 10⁻⁵ cm/s, J_Kⁿ_o = 10⁻⁸ mol/cm²/s, Δt_n = 0.1 s, d_a = 10 µm, and d_e = 2 µm. (B) Effect of magnitude of neuroexcitation. Computations as in B, but with J_Kⁿ_o = 5 × 10⁻⁸ mol/cm²/s to increase [K⁺]_e from 5 to 30 mM. (C) Effect of duration of neuroexcitation. Computations as in C, with P_K = 1.3 × 10⁻⁵ cm/s, J_Kⁿ_o = 8 × 10⁻¹⁰ mol/cm²/s, and Δt_n = 10 s. (D) Spatial distributions of [K⁺]_a and osmolarity (Φ_a) in astrocyte cytoplasm for the indicated P_f and D_a, with parameters as in C.

Figure 6.

Modeling of experimental observations of altered K⁺ dynamics in AQP4 deficiency. (A) Single neuron firing (Δt_n = 0.1 ms). Time course of [K⁺]_e modeled for wild type (WT) mice (P_f = 0.04 cm/s) and AQP4 knockout mice (AQP4^−/−, P_f = 0.001 cm/s, d_e = 2 or 2.4 µm) for well-mixed model (WMM, left) and diffusion-limited model (DLM, D_a = 10⁻⁹ cm²/s, center). Parameters: P_K^a = 1.2 × 10⁻⁵ cm/s, J_Kⁿ_o = 3.5 × 10⁻⁷ mol/cm²/s, and Δt_n = 0.1 ms. (right) Relative increase [K⁺]_e after neuroexcitation in WT vs. AQP4 knockout mice (Δ[K⁺]_e^−/−/Δ[K⁺]_e^+/+). (B) Repetitive pulsed neuronal excitation. (left and center) [K⁺]_e in response to 20 Hz stimulation (Δt_n = 0.1 ms firing) for 10 s. Parameters: P_K^a = 1.2 × 10⁻⁵ cm/s and J_Kⁿ_o = 2.1 × 10⁻⁷ mol/cm²/s. (right) [K⁺]_e at 20 s (broken line) after neuroexcitation and relative half-times for return of [K⁺]_e to baseline (t_1/2^−/−/t_1/2^+/+). (inset) P_f dependence of t_1/2 in the DLM (at fixed d_e of 2 µm), shown on a normalized y scale (denoted <t_1/2>). A similar P_f dependence was seen for [K⁺]_e (not depicted). (C) Prolonged neuronal firing. (left and center) [K⁺]_e responses to 20 Hz stimulation for 30 s. Parameters: P_K^a = 1.2 × 10⁻⁵ cm/s and J_Kⁿ_o = 3.45 × 10⁻⁷ mol/cm²/s. (right) Summary of t_1/2^−/−/t_1/2^+/+. (inset) P_f dependence of <t_1/2> in the DLM (at fixed d_e of 2 µm), shown on a normalized y scale.