Evidence from mathematical modeling that carbonic anhydrase II and IV enhance CO2 fluxes across Xenopus oocyte plasma membranes

Abstract

Exposing an oocyte to CO2/HCO3 (-) causes intracellular pH (pHi) to decline and extracellular-surface pH (pHS) to rise to a peak and decay. The two companion papers showed that oocytes injected with cytosolic carbonic anhydrase II (CA II) or expressing surface CA IV exhibit increased maximal rate of pHi change (dpHi/dt)max, increased maximal pHS changes (ΔpHS), and decreased time constants for pHi decline and pHS decay. Here we investigate these results using refinements of an earlier mathematical model of CO2 influx into a spherical cell. Refinements include 1) reduced cytosolic water content, 2) reduced cytosolic diffusion constants, 3) refined CA II activity, 4) layer of intracellular vesicles, 5) reduced membrane CO2 permeability, 6) microvilli, 7) refined CA IV activity, 8) a vitelline membrane, and 9) a new simulation protocol for delivering and removing the bulk extracellular CO2/HCO3 (-) solution. We show how these features affect the simulated pHi and pHS transients and use the refined model with the experimental data for 1.5% CO2/10 mM HCO3 (-) (pHo = 7.5) to find parameter values that approximate ΔpHS, the time to peak pHS, the time delay to the start of the pHi change, (dpHi/dt)max, and the change in steady-state pHi. We validate the revised model against data collected as we vary levels of CO2/HCO3 (-) or of extracellular HEPES buffer. The model confirms the hypothesis that CA II and CA IV enhance transmembrane CO2 fluxes by maximizing CO2 gradients across the plasma membrane, and it predicts that the pH effects of simultaneously implementing intracellular and extracellular-surface CA are supra-additive.

Keywords: buffers; competing equilibria; intracellular pH; reaction-diffusion; surface pH; tortuosity factors.

Fig. 1.

Main features of the revised mathematical model. Following the approach of Somersalo et al. (22), we assume that the oocyte is a perfectly symmetric sphere of radius R, surrounded by a thin layer of extracellular unconvected fluid (EUF) of thickness d. The EUF is in turn surrounded by the bulk extracellular fluid (BECF), which is an infinite reservoir for all solutes. Within the BECF, convection could occur (though it is not included in the present model), but no reaction or diffusion. Solutes can diffuse between the BECF and EUF. In both the intracellular fluid (ICF) and EUF, reaction and diffusion occur, but no convection. The buffer reactions occurring in the ICF and EUF (illustrated in the inset in the bottom left corner) are: the CO₂/HCO₃⁻ buffer reactions (including the slow conversion of CO₂ into H₂CO₃ and vice versa) and a single non-CO₂/HCO₃⁻ buffer (HA/A⁻). New features of the model include (starting from the inside of the cell and moving outward): 1) reduced water content of cytosol, 2) reduced cytosolic diffusion constants, 3) refined CA II activity, 4) layer of intracellular vesicles (magnification in top left corner), 5) reduced membrane CO₂ permeability, 6) microvilli (magnification in top right corner), 7) refined CA IV activity, 8) vitelline membrane (magnification on lower edge), and 9) a new simulation protocol for the delivery and removal of the bulk extracellular CO₂/HCO₃⁻ solution. The inset in the bottom right corner explains the spatial discretization for the numerical solution near the plasma membrane (PM). EM, extracellular surface of the PM; IM, intracellular surface of the PM.

Fig. 2.

Summary of the key quantities of interest for describing intracellular-pH (pH_i) and surface pH (pH_S) transients in our physiological experiments during the addition of CO₂/HCO₃⁻. This figure schematically illustrates the transients of pH_i (in red) and pH_S (in green) and some of the experimentally measurable key quantities as we expose an oocyte to a CO₂/HCO₃⁻ solution. For pH_i, key quantities are time delay (t_d), the maximal rate of pH_i descent [(dpH_i/dt)_max], time constant for pH_i decay (τ_pHi), and the difference between initial and final pH_i (ΔpH_i). For pH_S, key quantities are time to peak (t_p), maximal height of the pH_S trajectory (ΔpH_S), and the time constant for pH_S decay (τ_pHS).

Fig. 3.

Effect of the new experimental protocol on the shape of the pH_S trajectory and on the time to peak. A: pH_S trajectory predicted by the model when using the protocol of Somersalo et al. (22). In using this original protocol, we assume that, at the outset of the simulation, the CO₂/HCO₃⁻ is already at the plasma membrane. B: pH_S trajectory predicted by the model when using the revised protocol. Here we assume that the oocyte is initially superfused with the CO₂/HCO₃⁻-free ND96 solution until we switch the bulk solution to one containing CO₂/HCO₃⁻ and allow the CO₂ and HCO₃⁻ to diffuse to the cell surface.

Fig. 4.

Effect of intracellular vesicles on the pH_i transient during the addition of CO₂/HCO₃⁻. A: pH_i trajectories, at a depth of ∼50 μm. The six simulations correspond to six increasing values of the tortuosity factor λ. B: time delay as a function of λ. The six colored dots correspond to the six simulations in A. C: maximal rate of pH_i change (negative direction) as a function of λ. The six colored dots correspond to the six simulations reported in A.

Fig. 5.

Effect of vitelline membrane on pH_i and pH_S transients during the addition of CO₂/HCO₃⁻. A: pH_S trajectories. The six simulations correspond to six increasing values of the tortuosity factor γ. B: maximal height of the pH_S spike (positive direction) as a function of γ. The six colored dots correspond to the six simulations in A. C: pH_i trajectories, at a depth of ∼50 μm, corresponding to the simulations in A. D: maximal rate of pH_i change (negative direction) as a function of γ. The six colored dots correspond to the six simulations in A.

Fig. 6.

Simulations of pH_i and pH_S transients for the addition and removal 1.5% CO₂/10 mM HCO₃⁻ in 5 mM HEPES. A: “control” oocyte“. B: ”CA II“ oocyte. C: ”CA IV“ oocyte. The pH_i transients, calculated at a depth of ∼50 μm into the oocyte, are represented by the red lower trajectories, and pH_S, by the green upper trajectories. At the indicated time, we switched the extracellular solution from CO₂/HCO₃⁻-free ”ND96“ to 1.5% CO₂/HCO₃⁻ (see Eq. 9). The dashed black lines through the initial portions of the pH_i transients for CO₂ application and removal represent the maximal rates of pH_i change (negative or positive direction), as predicted by the model. The upward and downward arrows near the pH_S records represent maximal changes in pH_S (positive or negative direction).

Fig. 7.

Comparison of the physiological data with predictions of the mathematical model for the addition and removal of extracellular 1.5% CO₂/10 mM HCO₃⁻. A: maximal rates of pH_i change (negative or positive direction) produced by the addition or removal of extracellular CO₂/HCO₃⁻. B: maximal changes in pH_S (positive or negative direction) caused by the addition or removal of extracellular CO₂/HCO₃⁻. The physiological data come from Fig. 2 of the first paper in this series (14) and Fig. 3 of second paper (15). Note that here we report the means ± standard deviation (and not SE, as in the previous two papers). The data for model predictions come from Fig. 6 of the present paper.

Fig. 8.

Dependence of (dpH_i/dt)_max and ΔpH_S on A_i and A_S. A: dependence of (dpH_i/dt)_max on A_i, for several fixed values of A_S (i.e., each set of symbols is a surface CA isopleth). B: dependence of ΔpH_S on A_i, for several fixed values of A_S (i.e., surface CA isopleths). C: dependence of (dpH_i/dt)_max on A_S, for several fixed values of A_i (i.e., intracellular CA isopleths). D: dependence of ΔpH_S on A_S, for several fixed values of A_i (i.e., intracellular CA isopleths). In these simulations, we assume that a hypothetical oocyte with the intracellular composition (i.e., initial pH_i and ΔpH_i) of a ”CA IV“ oocyte is exposed to a bulk solution containing 1.5% CO₂/10 mM HCO₃⁻. The physiological data are plotted on the ordinate as mean ± standard deviation; the position of these points along the abscissa is arbitrary.

Fig. 9.

Simulations of pH_i and pH_S transients for the addition of graded levels of extracellular CO₂/HCO₃⁻ in 5 mM HEPES. A: ”control“ oocyte”. B: “CA II” oocyte. C: “CA IV” oocyte. The pH_i transients, calculated at a depth of ∼50 μm, are represented by the red lower trajectories, and pH_S, by the green upper trajectories. At the indicated time, we switched the extracellular solution from CO₂/HCO₃⁻-free “ND96” to 1) 1.5% CO₂/10 mM HCO₃⁻ (left), or 2) 5% CO₂/33 mM HCO₃⁻ (center), or 3) 10% CO₂/66 mM HCO₃⁻ (right). The dashed black lines through the initial portions of the pH_i transients for CO₂ application represent the maximal rates of pH_i change (negative direction) predicted by the model. The upward arrow near the pH_S records represent maximal changes in pH_S (positive direction).

Fig. 10.

Comparison of the physiological data with predictions of the mathematical model for the addition of graded levels of extracellular CO₂/HCO₃⁻. A: maximal rates of pH_i change (negative direction) produced by the addition of extracellular CO₂/HCO₃⁻. B: maximal changes in pH_S (positive direction) caused by the addition of extracellular CO₂/HCO₃⁻. The physiological data are taken from Fig. 10 of the first paper in this series (14) and Fig. 10 of the second paper (15). Note that here we report the means ± standard deviation (and not SE, as in the previous two papers). The data for model predictions come from Fig. 9 of the present paper.

Fig. 11.

Simulations of pH_i and pH_S transients for the addition of extracellular 1.5% CO₂/10 mM HCO₃⁻, with graded increases in levels of extracellular HEPES. A: “control” oocyte. B: “CA IV” oocyte. The pH_i transients, calculated at a depth of ∼50 μm, are represented by the red lower trajectories, and pH_S, by the green upper trajectories. At the indicated time, we switched the extracellular solution from CO₂/HCO₃⁻-free “ND96” to: 1) CO₂/HCO₃⁻ + 1 mM HEPES, or 2) CO₂/HCO₃⁻ + 5 mM HEPES, or 3) CO₂/HCO₃⁻ + 25 mM HEPES. The dashed black lines through the initial portions of the pH_i transients for CO₂ application represent the maximal rates of pH_i change (negative direction) predicted by the model. The upward arrows near the pH_S records represent maximal changes in pH_S (positive direction).

Fig. 12.

Simulations of pH_i and pH_S transients for the addition of extracellular 10% CO₂/66 mM HCO₃⁻, with graded increases in levels of extracellular HEPES. A: “control” oocyte. B: “CA IV” oocyte. This figure differs from Fig. 11 in that, here, the CO₂/HCO₃⁻ levels are ∼6.6-fold higher. The pH_i transients, calculated at a depth of ∼50 μm, are represented by the red lower trajectories, and pH_S, by the green upper trajectories. At the indicated time, we switched the extracellular solution from CO₂/HCO₃⁻-free “ND96” to 1) CO₂/HCO₃⁻ + 1 mM HEPES, or 2) CO₂/HCO₃⁻ + 5 mM HEPES, or 3) CO₂/HCO₃⁻ + 25 mM HEPES. The dashed black lines through the initial portions of the pH_i transients for CO₂ application represent the maximal rates of pH_i change (negative direction) predicted by the model. The upward arrows near the pH_S records represent maximal changes in pH_S (positive direction).

Fig. 13.

Supra-additivity of simultaneous implementation of intracellular (i.e., cytosolic) and extracellular-surface CA activity. A: dependence of (dpH_i/dt)_max on A_i, for two fixed values of A_S (i.e., surface CA isopleths), a replot of selected data from Fig. 8A. B: dependence of ΔpH_S on A_i, for two fixed values of A_S (i.e., surface CA isopleths), a replot of selected data from Fig. 8B. In all simulations for this figure, we assume that a hypothetical oocyte with the intracellular composition (i.e., initial pH_i and ΔpH_i) of a “CA IV” oocyte is exposed to a bulk solution containing 1.5% CO₂/10 mM HCO₃⁻. The points labeled “∼Ctrl” represent our hypothetical oocyte but with the CA properties of a “control” oocyte (i.e., A_i = 5, on the 150 A_S isopleth). The points labeled “∼CAII” represent our hypothetical oocyte but with the CA properties of a “CA II” oocyte (i.e., A_i = 1,000, on the 150 A_S isopleth). These points (lavender vectors) approximate the physiologically observed (dpH_i/dt)_max and ΔpH_S values for oocytes injected with CA II. The points labeled “CA_S” represent a hypothetical oocyte but in which we raise A_S to 10,000 but leave A_i = 5. These points (red vectors on left side of panels) represent (dpH_i/dt)_max and ΔpH_S values that are far smaller than those physiologically observed with oocytes expressing CA IV. The points labeled “CA_i” represent a hypothetical oocyte but in which we raise A_i to 40 but leave A_S = 150. These points (blue vectors) represent (dpH_i/dt)_max and ΔpH_S values that are far smaller than those physiologically observed with oocytes expressing CA IV. The sum of the red and blue vectors produce the brown vectors, which indicate the effects of simply adding CA_i and CA_o. Note that the asterisks (at the head of the brown vectors) represent (dpH_i/dt)_max and ΔpH_S values that are still much smaller than those physiologically observed with oocytes expressing CA IV. Finally, the points labeled “CAIV” represent the combination of simultaneously raising A_S from 150 to 10,000 and raising A_i from 5 to 40 in our simulations (green vectors). These points do indeed approximate the physiologically observed (dpH_i/dt)_max and ΔpH_S values of oocytes expressing CA IV.