Effect of extracellular volume on the energy stored in transmembrane concentration gradients

The amount of energy that can be retrieved from a concentration gradient across a membrane separating two compartments depends on the relative size of the compartments. Having a larger low-concentration compartment is in general beneficial. It is shown here analytically that the retrieved energy further increases when the high-concentration compartment shrinks during the mixing process, and a general formula is derived for the energy when the ratio of transported solvent to solute varies. These calculations are then applied to the interstitial compartment of the brain, which is rich in ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{-}$ ions and poor in ${\mathrm{K}}^{+}$. The reported shrinkage of this compartment, and swelling of the neurons, during oxygen deprivation is shown to enhance the energy recovered from NaCl entering the neurons. The slight loss of energy on the part of ${\mathrm{K}}^{+}$ can be compensated for by the uptake of ${\mathrm{K}}^{+}$ ions by glial cells. In conclusion, the present study proposes that the reported fluctuations in the size of the interstitial compartment of the brain (expansion during sleep and contraction during oxygen deprivation) optimize the amount of energy that neurons can store in, and retrieve from, their ionic concentration gradients.

Figure 1

Diagram of paths plotting volume $V_1$ of the high-concentration compartment against solute concentration $K_1$, both in arbitrary units, starting from an initial state (1, 1). Different paths represent different values of parameter $a$ in Eq. (3), hence different ratios of solvent to solute transport. The paths and their construction are described in greater detail in Sec. 3d.

Figure 2

Energy density in joules per liter retrieved from the physiological concentration gradient of ${\mathrm{Na}}^{+}$ (Table 1). Work done was calculated for four different mixing strategies during which either the volume of interstitial space (curve labeled $V_1\text{const}$), or the ${\mathrm{Na}}^{+}$ concentration therein (curves labeled $K_1\text{const}$), was held constant. In the first case ($V_1\text{const}$) mixing was complete. In the latter cases ($K_1\text{const}$), the interstitial compartment shrank during mixing from its initial volume fraction $\alpha$ either to zero ($\omega$ = 0), or to half its initial volume $\alpha$ ($\omega$ = $0.5\alpha$), or to a final volume that was less than $\alpha$ by an absolute value of 0.2 [$\omega =\text{max}(0,\alpha -0.2$)].

Figure 3

Work done by mixing strategies with various ratios of solvent to solute transport. The quantity $a^{-1}$ plotted in (a) and (c) represents the concentration at which solute either left (positive values) or entered (negative) interstitial space. Panels (a) and (b) illustrate mixing of the physiological ${\mathrm{Na}}^{+}$ gradient of Table 1, for (c) and (d) the ${\mathrm{K}}^{+}$ gradient was used. Since these gradients have opposite directions, a contraction of interstitial space corresponds to a contraction or expansion, respectively, of the high-concentration compartment of ${\mathrm{Na}}^{+}$ or ${\mathrm{K}}^{+}$. During mixing, the volume of interstitial space shrank by 50% ($w$ = 0.5) from its initial value $\alpha$ for all curves, except for the curve labeled $w$ = 1.0 in (d), which represents mixing at constant volume. After mixing, the concentration $K_1^{\circ }$ in interstitial space was that of complete mixing $K_e$ [Eq. (8)], except for the curve “$K_1\text{const}$” where $K_1^{\circ }=K_1^{*}$ and for the curve “partial mixing” where $K_1^{\circ }=0.8K_1^{*}+0.2K_e$. Mixing in the “two-step” case was accomplished in two stages: first mixing at constant $K_1$ followed by mixing at constant volume.

Figure 4

Effect of the buffering of solute particles on the work done, at constant volume $\alpha$, by a complete mixing of the ${\mathrm{K}}^{+}$ gradient of Table 1. The buffer was located in the interstitial compartment. Curves of increasing amplitude represent increasing buffering capacity, corresponding to values of $\gamma$ in Eq. (27) equal to 0, 1, 2, 4, and 8, or to fractions of 0, 50, 66, 80, or 89% of the ${\mathrm{K}}^{+}$ ions being buffered.

Figure 5

Total work done by the four ion species of Table 1 when they mix, at constant volume, from their physiological concentrations to those at Donnan equilibrium (solid black curve labeled Donnan), and total work that would be done if they were able to mix completely (dotted black curve labeled complete). Colored curves plot, for each component ion species, the work of mixing to Donnan equilibrium.

Figure 6

Energy retrieved by mixing of the three major ion species during ischemia (Table 2). The left column, (a) and (b), represents the plain model in the absence of buffering (Sec. 3d). In the right column, (c) and (d), ${\mathrm{Cl}}^{-}$ and ${\mathrm{K}}^{+}$ ions were buffered in the interstitial compartment, using parameters ${\gamma }_{\text{Cl}}$ = 1.1 and ${\gamma }_{\text{K}}$ = 1.2 in Eq. (27) (Sec. 3e and Appendix pp5). The bottom graphs, (b) and (d), plot the fractional change $w$ of the volume of the interstitial compartment during the mixing of each ion species separately. The vertical black bar in (c) indicates the total energy density (276 J/l) at $\alpha =0.195$, which is the size of the interstitial compartment at which mixing of each of the three ion species required an identical volume change in (d).